Design of Ultra Wideband Antenna Matching Networks: Via Simplified Real Frequency Technique

Design of Ultra Wideband Antenna Matching Networks Binboga Siddik Yarman Design of Ultra Wideband Antenna Matching N...

Author: Binboga Siddik Yarman

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Design of Ultra Wideband Antenna Matching Networks

Binboga Siddik Yarman

Design of Ultra Wideband Antenna Matching Networks Via Simplified Real Frequency Technique

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Dr. Binboga Siddik Yarman College of Engineering Department of Electrical-Electronics Engineering Istanbul University 34320, Avcilar, Istanbul Turkey [email protected]

ISBN: 978-1-4020-8417-1

e-ISBN: 978-1-4020-8418-8

Library of Congress Control Number: 2008926115 2008 Springer Science+Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

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Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

This book is dedicated to my wife Sema for her endless patience and love

Preface

Antennas, antenna matching networks (or equalizers), antenna switches and phase shifters of antenna arrays are the most critical components of ultra wideband communications systems. As a whole, they constitute, what we call is the “antenna system”. It is obvious that these critical components are placed in the front-ends of the communication systems. In other words, transmitters and receivers start with antenna systems. If the antenna system is wideband, then the wireless set up may have a strong chance to be wideband. Otherwise, no matter how good is the rest of the communication system, the system’s bandwidth is bounded by the antenna gears. Commercially, one can find variety of antenna design handbooks and/or Computer Aided Design (CAD) tools to construct antennas. On the other hand, as far as design of wideband practical matching networks is concerned, there is not much available for the design engineers on the market place. The topic is hot but difficult to comprehend. Over the last 30 years, the design methods known as the real frequency techniques (RFT), which was initiated by Prof. H.J. Carlin of Cornell University, provided excellent solutions to construct power transfer networks for many applications. Furthermore, the Simplified Real Frequency Technique (SRFT) has been verified as the most suitable one to design matching networks and microwave amplifiers for antennas. Therefore, this book is solely devoted to design ultra wideband practical antenna matching networks employing SRFT. We tried to make the topic as simple as possible for the designers by providing “ready to use” software (S/W) tools developed on (SRFT). Once the measured data is plugged into SRFT design tool, the complete lossless matching network is obtained at the output. The book starts with the historical review of the real frequency techniques (Chapter 1), electromagnetic field theory and antenna performance related definitions (Chapter 2). Then, major issues of antennas employed in modern cellular communication systems are covered in Chapters 3–6. Thus, the first 6 chapters of the book are prepared to orient the new comers exposed to wireless communication engineering. We recommend theses chapters to be covered in junior and senior classes in Communication Engineering Departments. Chapter 7 fully covers the scattering parameters from the design perspective. In this chapter, emphasis is given to lossless two-ports which is essential to construct matching networks for antennas. vii

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In Chapter 8, basic concepts and analytic theory of broadband matching is introduced. Several practical examples are presented. In Chapter 9, Simplified Real Frequency Technique (SRFT) to design ultra wideband Matching Networks is introduced. In this chapter, the reader is guided with several examples to develop her or his own design tool employing SRFT on the MatLab environment. Eventually, a home made design tool is provided by us to facilitate the work of the reader. This home made tool is developed on MatLab utilizing Levenberg-Marquard non-linear optimization algorithm. Chapter 10 is devoted to real life problems to construct antenna matching networks. The first problem is to design an ultra-wideband matching networks for a short monopole antenna which may be utilized for military and commercial purposes. In the second problem, we design a multi-band matching network for a PIFA antenna employed for the cellular bands of GSM 900/1800 and CDMA 1900. As it is well known, nonlinear optimization requires excellent initials. Therefore, in Chapter 11, we introduce initialization techniques for SRFT design algorithm. Once the SRFT design is completed, it may be necessary to retrofit the original design to production technology. In this regards, commercially available S/W tools such as ADS and AWR can be utilized. Therefore, in Chapter 12 and 13 basic ingredients of ADS and AWR are covered and examples are presented. From the graduate education and research point of view, Chapters 7–11 can be used for masters and Ph.D programs. Furthermore, we strongly believe that the book will be useful for research managers and design engineers employed by commercial wireless communication companies as well as government and military agencies. Vanikoy, Istanbul, Turkey

Binboga Siddik Yarman

Acknowledgments

I should emphasize with great pleasure that it would not be possible to put this book without mental guidance of Prof. H.J. Carlin of Cornell University throughout my professional life. He deserves a big applause for initiating the real frequency technique which facilitates design and implementation process of any kind of matching network including antenna systems. Sincere gratitude are extended to Prof. Y. Tokad of National Research Center (or in short TUBITAK) of Turkey, who passed-away in 2001 with rigor, Prof. D.C. Youla of New York Polytechnique Institute, Prof. P.P. Civalleri of Torino University, Italy, Prof. A. Fettweis of Ruhr University, Germany, Prof. E. Tutuncuoglu of Istanbul University and Prof. N. Fujii of Tokyo Institute of Technology. They all provided with me their immense support furnished with outstanding research ideas and environments. I would like to thank to my former Ph.D students Dr. Metin Sengul of Kadir Has University and Dr. Ali Kilinc of Elma Corp. who partially help me with SRFT S/W development especially with the synthesis of driving point functions and also with the figures of the book. It must be acknowledged that recent SRFT practical antenna matching networks were designed and physically built within the European Union Project NEWCOM (Department 3 of Network of Excellence for Wireless Communication Contract No IST NoE 507325.) under guidance of group leaders Prof. M. Hein of Ilmenau Technical University of Germany, Prof. A. Rydberg of Uppsala University, Sweden. Vanikoy, Istanbul, Turkey

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Contents

1 Real Frequency Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binboga Siddik Yarman

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2 Antenna Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhaskar Gupta

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3 Antennas for Mobile Wireless Communication . . . . . . . . . . . . . . . . . . . . . 39 Peter Lindberg 4 Challenges in Mobile Phone Antenna Development . . . . . . . . . . . . . . . . . 45 Peter Lindberg 5 Design Techniques for Internal Terminal Antennas . . . . . . . . . . . . . . . . . 67 Peter Lindberg 6 Terminal Antenna Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Peter Lindberg 7 Description of Lossless Two Ports in Terms of Scattering Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Binboga Siddik Yarman 8 Analytic Approaches to Antenna Matching Problems . . . . . . . . . . . . . . . 139 Binboga Siddik Yarman 9 Simplified Real Frequency Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Binboga Siddik Yarman 10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Binboga Siddik Yarman 11 Initialization of Simplified Real Frequency Technique . . . . . . . . . . . . . . 257 Binboga Siddik Yarman xi

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12 Analysis and Optimization of Matching Networks-I . . . . . . . . . . . . . . . . 281 Metin Sengul 13 Analysis and Optimization of Matching Networks-II . . . . . . . . . . . . . . . 293 Metin Sengul Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Contributors

Binboga Siddik Yarman College of Engineering, Department of Electrical-Electronics Engineering, Istanbul University, 34320, Avcilar, Istanbul, Turkey, [email protected] Bhaskar Gupta Jadavpur University, India, [email protected] Peter Lindberg Laird Technologies AB, Mobile Antenna Systems, Research Department, Isafjordsgatan 5, 164 22 Kista, Sweden, [email protected] Metin Sengul Kadir Has University, Engineering Faculty Electronics Engineering Department, 34083, Cibali-Fatih, ˙Istanbul, Turkey, [email protected]

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Chapter 1

Real Frequency Techniques A Historical Review Binboga Siddik Yarman

Historical Review No matter how good the circuit production technology is, we, as the design engineers, are in a position, to squeeze the best electrical performance out of it. As far as the design of communication systems are concerned, the most critical issue is the error free transmission and reception of electrical signals over the band of operation. In order to squeeze the limits of the production technology, we must be able to provide best of signal transmission over a wide frequency band. Signal transmission starts with a properly designed antenna and its matching network. This way of understanding design problems guides us to construct antenna matching networks (or equalizers) with high gain and wide frequency bandwidth as much as possible. Thus, we are facing to gain-bandwidth limits of antennas both at receiver and transmitter ends of the communication systems. This problem is called broadband matching and occupies a significant part of the existing literature. The analytic theory behind of it, is elegant, difficult to comprehend and hard to access beyond simple problems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Furthermore, depending on the design problem, using analytic theory, we may not be able to reach to a solution. If the solution exists it may result in sub-optimal electrical performance with complicated circuit structures to be manufactured [12, 13, 14, 15]. Therefore, almost all design engineers prefer to work with computer-aided ad-hoc or brute force design techniques to construct antenna matching networks (or equalizers). In these techniques, regardless what the gain-bandwidth limits of the devices to be matched, first a circuit topology is selected for the matching network to be designed then, element values of the selected circuit is determined to optimize the electrical performance of the matched system. Beyond simple cases, utilization of ad-hoc design techniques

B.S. Yarman College of Engineering, Department of Electrical-Electronics Engineering, Istanbul University, 34320, Avcilar, Istanbul, Turkey e-mail: [email protected]

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

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penalizes circuit performance despite the use of outstanding manufacturing technology. This means waste of technology, waste of time and eventually waste of money. If we have a chance to do it better to save power and bandwidth which in turn saves production cost and brings more profit why not do it? In 1977, Prof. H.J. Carlin of Cornell University brought a new vision to the problem of broadband matching by introducing a new matching network design method called “Real Frequency Technique” [12]. This technique, simply by passes the analytic theory, and provides a reasonable estimate for the gain bandwidth limits of the devices to be matched. It also reveals the lossless matching circuit automatically with almost optimum gain-bandwidth performance. The initial version of the real frequency technique so called the real frequency-line segment technique was good to solve only single matching problems as well as to design single stage microwave amplifiers. At that time, I had an outstanding opportunity and pleasure to work with Prof. Carlin to generalize his real frequency-line segment technique to handle double matching problems during my Ph.D work [15, 16, 17]. After a short period of time, we came up with an idea to handle any kind of matching and microwave amplifier design problem via a scattering approach which is called “Simplified Real Frequency Technique-SRFT” [18] It turnout to be that SRFT is very practical and straight forward to implement; and, it is naturally suited to design microwave circuits [19]. In 1980s, during my professional work at RCA David Sarnoff Research Center, Princeton, NJ, USA, we designed several microwave amplifiers for satellite transponders, antenna arrays with matching networks employing SRFT [20]. In 1990s and 2000s, the real frequency techniques had found wide application in the field of microwave engineering [22, 23, 24, 25, 26, 27, 28] and their usage had been extended to construct matching networks and amplifiers in two-kinds of elements; namely with lumped and distributed elements [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. In these years, SRFT had also been employed to model measured data obtained from active and passive devices [41, 42, 43, 44, 45]. Recently, utilizing SRFT, we have completed several designs of multi-band antenna matching and switch networks for cellular communication systems [26, 27, 28]. An alternative real frequency method which is called “a parametric approach” was proposed by A. Fettweis of Ruhr University [46, 47, 48, 49] and it is generalized and elaborated as in [49, 50, 51]. For the sake of completeness, it should be mentioned that the “real frequency design vision” is expanded to construct digital phase shifters which are the major building blocks of ultra wideband smart antenna arrays for cellular communication systems [52, 53, 54, 55, 56, 57, 58, 59, 60]. After all above, it is well understood that, proper design of front and back-ends, more specifically, antennas, antenna matching networks and phase shifters are essential components to construct ultra wideband communication systems. Therefore, this book is devoted to design especially, ultra wideband antenna matching networks using SRFT. Over all these years, we have been continuously asked by many researchers, designers and students to provide the necessary Real Frequency-S/W Tools to design

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antenna matching networks. In order to partially satisfy the expectations of our readers, this book is organized in such a way that designers are encouraged to develop their own SRFT-design programs. In order to facilitate the work of the designers/engineers and students, we included many practical design examples with related computer algorithms and MatLab programs in this book. If the readers carefully follow the design instructions presented in this book, they will able to develop their own SRFT design packages. Even though this book is devoted to SRFT designs, just to provide a broad understanding, let us summarize all the real frequency techniques to design matching networks.

Real Frequency Techniques The real frequency techniques can be grouped under the following headings:

r r r r

Line Segment Technique (LST) Direct Computational Technique (DCT) Parametric Approach Simplified Real Frequency Technique (SRFT) Now, let us summarize the hard lines of the above techniques.

(a) Line Segment Technique (LST): In this method, the lossless matching network or equalizer is described in terms of its driving point input immitance FQ ( jω) = A Q (ω) + j B Q (ω) over the entire angular real frequency ω axis. It is assumed that FQ is either minimum reactance or minimum suseptance. In this case, if the real part A Q (ω) is represented as a linear combination of straight N lines with unknown break points Rk such that A Q (ω) = ak (ω)Rk then, the k=1

imaginary part B Q (ω) is also expressed by means of the linear combination of N same break points Rk as B Q (ω) = bk (ω)Rk . We should mentioned that if k=1

FQ ( jω) is minimum function (i.e. minimum reactance or minimum suseptance) then its imaginary part B Q (ω) is generated directly from the real part A Q (ω) by means of Hilbert transformation relation. Thus, bk (ω) = H [ak (ω)] where H [.] designates the Hilbert transformation operator. In LST, the unknown break points Rk are determined via optimization of the transducer power gain of the matched system. Eventually, numerical data obtained for the driving immitance FQ must be modeled as a realizable positive real function so that it is synthesized as a lossless two port in resistive termination yielding the desired matching network with its element values. In short, we can say that LST provides an excellent insight to the matching problem under consideration yielding a good

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estimate about the gain-bandwidth limit of the complex load to be matched to a resistive generator. Details are omitted here. Interested reader is referred to references [12, 13, 14, 15] (b) Direct Computational Technique (DCT): In this method, lossless matching network is also expressed in terms of its positive real minimum immin ( p) tance function FQ ( p) = d QQ ( p) in complex variable p = σ + jω It is well known that FQ ( p) = Ev( p) + odd( p) can be uniquely determined from its 2 even part A Q ( p) = Ev( p) = ND(( pp2 )) ≥ 0; ∀r eal( p) ≥ 0. For many practical cases, N ( p 2 ) is selected as N ( p 2 ) = p 2k and the denominator polynomial is given in full coefficient form as D( p 2 ) = d Q ( p).d Q (− p) = D0 + D1 P 2 + . . . + Dn p 2n . In this case, the unknowns of the matching problem are the coefficients {D0 , D1 , . . ., Dn }and they are determined via optimization of the transducer power gain of the matched system. Details are omitted here. . . . Interested readers are referred to references [16, 17]. In short we can say that DCT provides excellent solution for general matching problems where a complex generator is matched to a complex load. In the past, we have used DCT to design various kinds of HF-Antenna matching networks with success (c) Parametric Approach : As in LST and DCT, in parametric approach, lossless equalizer is described by means of its minimum driving point immetance n Ak function FQ ( p) which is given in its parametric form as FQ ( p) = p+ pk k=1

where Ak are the residues evaluated at the Left Half Plane (LHP) poles pk = αk + jβk ; αk > 0. Here, the unknowns of the matching problem are chosen as the poles pk and they are determined via optimization of the transducer power gain of the matched system. In the optimization process of the transducer power gain, it can be shown that the gradient of the gain function can explicitly be expressed in terms of the poles pk . This fact can be viewed as a significant advantage of the optimization algorithm. Details are omitted here. Interested readers are referred to references [46, 47, 48, 49, 50, 51].

Simplified Real Frequency Technique (SRFT): As oppose to other techniques, in SRFT Lossless equalizer is described in terms of its real normalized scattering parameters in a very handy way. Therefore, it is suitable to design antenna matching networks, as well as microwave amplifiers [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. When it is combined with modeling techniques [41, 42, 43, 44, 45], it yields outstanding solutions to any kind of matching problems. Details are covered in the rest of the book. It is hoped that this book will be useful to all students and designers who are exposed to construct wireless communication systems and seeks the best out of their designs as far as the electrical performance of the designed circuits, usage of the state of the art technology and production costs are concerned.

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References Analytic Theory of Broadband Matching: 1. R. M. Fano, Theoretical limitations on the broadband matching of arbitrary impedances, J. Franklin Inst., vol. 249, pp. 57–83, 1950. 2. E. S. Kuh, and J. D. Patterson, Design theory of optimum negative-resistance amplifiers, Proc. IRE, vol. 49, no. 6, pp. 1043–1050, 1961. 3. D. C. Youla, A new theory of broadband matching, IEEE Trans. Circuit Theory, vol. 11, pp. 30–50, March 1964. 4. W. K. Chien, A theory of broadband matching of a frequency dependent generator and load, J. Franklin Inst., vol. 298, pp.181–221, Sept. 1974. 5. W. H. Ku, M. E. Mokari-Bolhassan, W. C. Petersen, A. F. Podell, and B. R. Kendall, Microwave octave-band GaAs-FET amplifiers, Proc. IEEE MTT-S Int. Microwave Symp., Paolo Alto, pp. 69–72, 1975. 6. W. K. Chen, Explicit formulas for the synthesis of optimum broadband impedance matching networks, IEEE Trans. Circuits and Systems, vol. 24, pp.157–169, April 1977. 7. H. J. Carlin and B. S. Yarman, The double matching problem: Analytic and real frequency solutions, IEEE Trans. Circuits Syst., vol. 30, pp. 15–28, Jan. 1983. 8. D. C. Youla, H. J. Carlin, and B. S. Yarman, Double broadband matching and the problem of reciprocal reactance 2n-port cascade decomposition, Int. J. Circuit Theory and Appl., vol.12, pp. 269–281, Feb. 1984. 9. W. K. Chen, T. Chaisrakeo, Explicit formulas for the synthesis of optimum bandpass butterworth and chebyshev impedance-matching networks, IEEE Trans. Circuits Syst., vol. CAS-27, no. 10, October 1980 10. Y. S. Zhu and W. K. Chen, Unified theory of compatibility impedances, IEEE Trans. Circuits Syst., vol. 35, no. 6, pp. 667–674, June 1988. 11. C. Satyanaryana and W. K. Chen, Theory of broadband matching and the problem of compatible impedances, J. Franklin Inst., vol. 309, pp. 267–280, 1980.

Real Frequency Line Segment Technique (LST) 12. H. J. Carlin, A new approach to gain-bandwidth problems, IEEE Trans. Circuits Syst., vol. 23, pp. 170–175, April 1977. 13. H. J. Carlin and J. J. Komiak, A new method of broadband equalization applied to microwave amplifiers, IEEE Trans. Microw. Theory Tech., vol. 27, pp. 93–99, Feb.1979. 14. H. J. Carlin and P. Amstutz, On optimum broadband matching, IEEE Trans. Circuits Syst., vol. 28, pp. 401–405, May 1981. 15. B. S. Yarman, Real frequency broadband matching using linear programming, RCA Review, vol. 43, pp. 626–654, Dec. 1982.

Direct Computational Technique (DCT) 16. B. S. Yarman, “Broadband Matching a Complex Generator to a Complex Load”, PhD thesis, Cornell University, 1982. 17. H. J. Carlin and B. S. Yarman, The double matching problem: Analytic and real frequency solutions, IEEE Trans. Circuits Syst., vol. 30, pp. 15–28, Jan. 1983. 18. H. J. Carlin and P. P. Civalleri, On flat gain with frequency-dependent terminations, IEEE Trans. Circuits Syst., vol. 32, pp. 827–839, Aug. 1985.

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Simplified Real Frequency Technique (SRFT) 19. B. S. Yarman, A simplified real frequency technique for broadband matching complex generator to complex loads, RCA Review, vol. 43, pp. 529–541, Sept. 1982. 20. B. S. Yarman and H. J. Carlin, A simplified real frequency technique applied to broadband multi-stage microwave amplifiers, IEEE Trans. Microw. Theory Tech., vol. 30, pp. 2216–2222, Dec. 1982. 21. B. S. Yarman, A dynamic CAD technique for designing broad-band microwave amplifiers, RCA Rewiev, 1983. 22. B. S. Yarman, Modern approaches to broadband matching problems, Proc. IEE, vol.132, pp. 87–92, April 1985. 23. P. Jarry and A. Perennec, Optimization of gain and vswr in multistage microwave amplifier using real frequency method, European Conference on Circuit Theory and Design, vol. 23, pp. 203–208, Sept. 1987. 24. L. Zhu, B. Wu, and C. Cheng, Real frequency technique applied to synthesis of broad-band matching networks with arbitrary nonuniform losses for MMIC’s, IEEE Trans. Microw. Theory Tech., vol. 36, pp. 1614–1620, Dec. 1988. 25. M. Sengul and B. S. Yarman, “Real Frequency Technique without Optimization”, ELECO 2005 4th International Conference on Electrical and Electronics Engineering, 07–11 December 2005, Bursa-Turkey. 26. P. Lindberg, M. S¸eng¨ul, E. C ¸ imen, B. S. Yarman, A. Rydberg, and A. Aksen, (2006), A Single Matching Network Design for a Dual Band Pifa Antenna via Simplified Real Frequency Technique, The first European Conference on Antennas and Propagation (EuCAP 2006), 6–10 November 2006 Nice, France. 27. B. S. Yarman et al., A Single Matching Network Design for a Double Band PIFA Antenna via Simplified Real Frequency Technique, Asia Pacific Microwave Conference, December 13–15, 2006, Yokohama, Japan, ISBN-90 2339-11-0, IEEE Catalog No: 06TH8923, pp.THOF-45, 1–4. 28. B. S. Yarman et al., Design of Broadband Matching Networks, ECT January 24–27, 2007, Okinawa, Japan. (Invited Talk), pp. ECT-07, pp. 35–40.

SRFT with Mixed Lumped and Distributed Elements 29. B. S. Yarman and A. Aksen, An integrated design tool to construct lossless matching networks with mixed lumped and distributed elements, IEEE Trans. on CAS, vol. 39, pp. 713–723, March 1992. 30. A. Aksen, Design of Lossless Two-Ports with Lumped and Distributed Elements for Broadband Matching, PhD thesis, Ruhr University at Bochum, 1994. 31. A. Sertbas¸, A. Aksen, and B. S. Yarman, “Construction of some classes of two-variable lossless ladder networks with simple lumped elements and uniform transmission lines”, IEEE APCCAS’98, Asia Pacific Conference on Circuits and Systems, November 24–27, 1998, Chiangmai, Thailand. 32. A. Sertbas¸, A. Aksen, and B. S. Yarman, “Explicit Formulas for A Special Class of TwoVariable Resonant ladder Networks with Simple Lumped Elements and Commensurate Stubs”, ELECO’99, Bursa, 1–5 December 1999, Electronics Vol. 1, pp. 132–135 33. B. S. Yarman, A. Sertbas¸, and A. Aksen, “Construction of analog RF circuits with lumped and distributed components for high speed/high frequency mobile communication MMICs.” European Conference on Circuit Theory and Design ECCTD’99, 29 August-2 September 1999, Stresa-Italy 34. A. Aksen and B. S. Yarman, “Cascade Synthesis of Two Variable Lossless Two-Port Networks of Mixed, Lumped Elements and Transmission Lines: A Semi-Analytic Procedure”, NDS-98,

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36.

37.

38.

39.

40.

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The First International Workshop on Multidimensional Systems, 12–14 July, 1998, Technical University of Zıelona Gora, Poland. A. Aksen, A. Sertbas¸, and B. S.Yarman,. “Explicit Two-Variable Description of a Class of Band-Pass Lossless Two-Ports with Mixed, Lumped Elements and Transmission Lines”, NDS-98, The First International Workshop on Multidimensional Systems, 12–14 July, 1998, Technical University of Zıelona Gora, Poland. A. Aksen and B. S.Yarman, A Real Frequency Approach to describe lossless two-ports formed with mixed lumped and distributed elements, International Journal of Electronics and Com¨ vol. 6, pp. 389–396, November 2001. munications (AEU), B. S. Yarman and E. G. Cimen, “Design of a broadband microwave amplifier constructed with mixed lumped and distributed elements for mobile communication”, ICCSC ’02. 1st International Conference on Circuits and Systems for Communications, 2002. Proceedings, 26–28 June 2002 St. Petersburg, pp. 334–337. B. S. Yarman, A. Aksen, and E. G. Cimen, “Design and simulation of miniaturized communication systems em symmetrical lossless two-ports constructed with two kinds of eleme” ISCAS ’03 Proceedings of the IEEE Internation Symposium on Circuits and Systems, 2003, vol. 2, May 25–28, 2003, Thailand, pp. 336–339 A. Aksen and B. S. Yarman, “A parametric approach to describe distributed two-ports with lump discontinuities for the design of broadband MMIC’S” ISCAS ’03 Proceedings of the IEEE International Symposium on Circuits and Systems, 2003, vol. 1, May 25–28, 2003, Thailand. A. Sertbas¸ and B. S. Yarman, A Computer-Aided Design Technique for Lossless Matching Networks with Mixed, Lumped and Distributed Elements, International Journal of Electronics and Communications (AEU), vol. 58, pp. 424–428, 2004.

Modeling via Real Frequency Techniques 41. B. S.Yarman, A. Aksen, and A. Kilinc¸, “An Immitance Based Tool for Modelling Passive ¨ 55 One-Port Devices by Means of Darlington Equivalents”, Int. J. Electron. Commun. (AEU) (2001) No.6, pp.443–451 42. B. S. Yarman, A. Kilinc¸, and A. Aksen, “Immitance data modeling via linear interpolation techniques”, ISCAS 2002, IEEE Int. Symp. on Circuits and Systems May 2002, Scottsdale, Arizona ABD. 43. A. Kilinc¸, H. Pinarbas¸i, M. S¸eng¨ul, and B. S. Yarman, A Broadband Microwave Amplifier Design By Means of Immitance Based Data Modelling Tool, IEEE Africon 02, 6th Africon Conference in Africa, 2–4 October 2002, George, South Africa, vol. 2, pp. 535–540. 44. B. S.Yarman, A. Kılınc¸, and A. Aksen, Immitance Data Modelling via Linear Interpolation Techniques: A Classical Circuit Theory Approach, Int. J. Circuit Theory and Appl., vol. 32, pp. 537–563, 2004. 45. B. S Yarman, M. Sengul, and A. Kilinc, “Design of Practical Matching Networks with Lumped Elements Via Modeling”, IEEE Trans. CAS-I, vol. 54, No.8, August 2007, pp.1829–1837

Parametric Approach 46. A. Fettweis, Parametric representation of brune functions, Int. J. Circuit Theory and Appl., vol. 7, pp. 113–119, 1979. 47. J. Pandel and A. Fettweis, Broadband matching using parametric representations, IEEE Int. Symp. on Circuits and Systems, vol. 41, pp. 143–149, 1985. 48. J. Pandel and A. Fettweis, Numerical solution to broadband matching based on parametric ¨ representations, Archiv Elektr. Ubertragung, vol. 41, pp. 202–209, 1987.

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49. B. S. Yarman and A. Fettweis, “Computer Aided Double Matching via Parametric Represantation of Brune Functions”, IEEE Trans. on CAS vol. 37, No:2, February, 1990, pp.212–222. 50. E. G. Cimen, A. Aksen, and S. B.Yarman, “A Numerical Real Frequency Broadband Matching Technique Based on Parametric Representation of Scattering Parameters”, Melecon’98 9th Mediterranean Electrotechnical Conference, May 18–20, 1998, Tel Aviv, ˙Israil 51. A. Aksen, E. G. C¸imen, and B. S.Yarman, “A Numerical Real Frequency Broadband Matching Technique Based on Parametric Representatkion of Scattering Parameters”, IEEE APCCAS’98, Asia Pasific Conference on Circuits and Systems, November 24–27, 1998, Chiangmai, Thailand.

Digital Phase Shifters for Antenna Arrays with RFT vision 52. B. S. Yarman, et al., Low Loss Millimeter-Wave Digital Phase Shifters Suitable for Monolithic Implementation, Proceedings of the IEEE ICAS, Montreal, Canada, pp.1235–1238, 1984. 53. B. S. Yarman, Design of Digital Phase Shifters Suitable for Monolithic Implemetation, Bulletin of the Technical University of ˙Istanbul, vol. 38, pp. 183–205, 1985. 54. B. S. Yarman, “New circuit configuration for designing 0–180 digital phase shifters”, IEE Proceeding, vol. 134, pt. H, No:3, June 1987. ◦ ◦ 55. B. S. Yarman, Novel circuit configurations to design loss balanced 0 –360 digital phase ¨ Vo. 45, pp.96–104, 1991. shifters AEU, ¨ and B. S. Yarman, New Circuit Topologies to Construct Wide Phase Range/Wide 56. M. Un, Frequency Band Digital Phase Shifters, ECCTD 95, August 27–30, 1995, ˙Istanbul. 57. B. S Yarman, “Low Pass T-Section Digital Phase Shifter Apparatus”, US. Patent number: 4.630.010, December 16, 1986. 58. B. S. Yarman, “Low Pass PI-Section Digital Phase Shifter Apparatus”, Patent Number: 4.614.921, September 30, 1986. 59. B. S. Yarman, “PI -Section Digital Phase Shifter Apparatus”, US. Patent Number: 4.604.593, August 5, 1986 60. B. S. Yarman, “T-Section Digital Phase Shifter Apparatus”, US. Patent Number: 4.603.310, July 29, 1986.

Further Reading 1. H. W. Bode, Network Analysis and Feedback Amplifier Design, Princeton, NJ: Van Nostrand, 1945. 2. V. Belevitch, Classical Network Theory, San Francisco: Holden Day, 1968. 3. H. J. Carlin and P. P. Civalleri, Electronic Engineering Systems Series, J. K. Fidler (ed), Wideband Circuit Design, Boca Raton: CRC Press LLC, 1998. 4. W. K. Chen, Broadband Matching, Theory and Implementations, 2nd ed., Singapore: World Scientific, 1988. 5. B. S. Yarman, “Broadband Networks”, Wiley Encyclopedia of Electrical and Electronics Engineering John G. Webster, Editor, Vol 2, pp. 589–605, 1999, John Wiley&Sons corp.

Chapter 2

Antenna Fundamentals Bhaskar Gupta

Electromagnetic Waves-Radiation and Propagation Visible lights, X rays, gamma rays, radio waves used for communication-all are different forms of electromagnetic waves. These waves are associated with both time varying electric and magnetic fields that follow the characteristics of waves, viz., representing physical phenomena getting repeated at a different location at a later time instant. Actually the different manifestations of such waves depend only on the frequency involved. A complete picture of various waveforms corresponding to different spectral regions is presented below in Fig. 2.1. A stationary distribution of charges gives rise to an electrostatic field. When such charges are in motion with uniform velocity, corresponding to a steady current, they give rise to a magnetostatic field. This is an expected result since Oerstead and Ampere established experimentally that a flowing current sets up a magnetic field whose intensity depends only on the current enclosed by it. However, when charges are in accelerated (or decelerated) motion, they give rise to electromagnetic fields composed of electric and magnetic fields existing simultaneously. This was the cardinal discovery by J.C. Maxwell, the Scottish mathematician during the middle of nineteenth century. Indeed, this path-breaking discovery changed the future course of world order. But the obvious question is, why does it happen and what led Maxwell to this formulation? It seems really interesting to note that almost all properties of electric and magnetic fields and even their nature of interdependence were known well before Maxwell. Still electromagnetic waves could not be postulated. To understand why, let us summarize the knowledge existing thereunto in this regard in the following equations:

B. Gupta Jadavpur University, India e-mail: [email protected]

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

9

10

Equation No. (2.1)

(2.2)

B. Gupta

Integral Form − →− → H . dl = I

Differential Form

Name of Law

Significance

− → − → ∇× H = J

Ampere’s Law

− → ⭸B − → ∇× E =− Faraday’s Law ⭸t

Relates magnetic field to its source as current

⭸Φ − →− → E . dl = − ⭸t

(2.3)

− →− → D . ds = Q

− → ∇. D = ρ

(2.4)

− →− → B . ds = 0

− → ∇. B = 0

Fig. 2.1 Electromagnetic spectrum

Gauss’ Law

Relates time varying magnetic field to associated i.e. induced electric field Relates electric flux to its source as electric charge

Gauss’ Law for Since magnetic flux magnetic field lines are always closed with no source or sink, as there is no isolated magnetic single pole, r.h.s. of these equations (Eq. 2.4) are zero.

2 Antenna Fundamentals

11

For any current, we must have the equation of continuity satisfied as ⭸Q − → − → J . ds = − (integral form) ⭸t (2.5) ⭸ρ − → i.e. ∇. J = − (differential form) ⭸t ⭸ρ ⭸Q = 0 i.e. = 0, the equation of continuity reduces For steady current, as ⭸t ⭸t − → − → − → to ∇. J = 0. Now, we get from Eq. (2.1), that ∇. J = ∇. (∇ × H ) = 0 This condition is valid only for steady currents and not for time varying cur− → − → − → rent. Here E and H are the electric and magnetic field intensities, whereas D − → − → and B are the electric and magnetic flux densities respectively. I and J stand for electric current and corresponding density (per unit area) respectively and Q and ρ represent electric charge and corresponding density (per unit volume) respec− →− → − →− → tively. ψ = D . ds and Φ = B . ds are the total electric and magnetic flux respectively. This apparent inconsistency of Ampere’s Law was actually removed by the genius of Maxwell. Before him, current used to be defined as ‘the rate of flow of free charges’. But Maxwell modified the definition as ‘the rate of change of charges’. The minute difference in wording carried a huge amount of information within it. What Maxwell meant was that current may exist even without any flow of free charges simply by the redistribution of charges (even bound ones) in space. We now know that this is exactly what happens within the dielectric of a capacitor. With application of ac signal to the capacitor plates, at every instant the charge distribution on the capacitor plates (and hence the flux or displacement connecting them) goes on changing. Maxwell called this contribution to the total current as ‘Displacement Current’ and the other contribution as ‘Conduction Current’. With introduction of the notion of displacement current, we find that Eq. (2.1) gets modified as

⭸ψ − →− → (integral form) — remember Gauss’ Law: ψ = Q H . dl = I + ⭸t

− → − → − → ⭸D and ∇ × H = J + (differential form). ⭸t Hence we obtain from Eq. (2.1) − → − → − → ⭸D ∇. (∇ × H ) = ∇. ( J + ) ⭸t

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B. Gupta

− → ⭸D − → − → ∇. J = ∇. (∇ × H ) − ∇. ⭸t

or

=0−

⭸ − → (∇. D ) (assuming that path of integration around infinitesimally small ⭸t

region chosen is independent of time) ⭸ρ (according to Eq. (2.3)) =0− ⭸t =−

⭸ρ ⭸t

which is exactly tallying with Equation of Continuity for Time Varying Current (Eq. (2.5)). Thus the inconsistency of Ampere’s Law got removed. At the same time, this analysis opened up the vista to a fantastic new probability – that of an electromagnetic wave. The existence of such had been experimentally verified within few decades of its postulation by pioneering radio engineers like H. Herz, Sir J.C. Bose and others. This is a classic case of theory prognosticating experiment, contrary to the most common practice of development of theory to explain experimental observations. In fact, if the current through a conductor changes either its speed or direction or both i.e. mobile charges in the conducting element undergo change in velocity – a changing magnetic field is created consequently. This time varying magnetic field in turn sets up an electric field according to Faraday’s law of induction. This time varying electric field consequently drives a current in the dielectric region surrounding the conductor but outside it. The current in the dielectric is set up as a displacement current since no free charge carrier is available therein. However, according to Ampere’s law, this current is associated with a magnetic field similarly with time. The process gets repeated and builds up on itself. The natural consequence is a propagating wave having both electric and magnetic fields associated with it. The process may be visualized as shown in Fig. 2.2 for conceptualization without considering as the actural reality. Thus electromagnetic waves are radiated and propagated through dielectric media. The whole idea rests on the fact that current may be supported in a dielectric

Fig. 2.2 Propagation of electromagnetic waves

2 Antenna Fundamentals

13

also, in the form of displacement current. But for the concept of displacement current, the entire theoretical premises for the same would have collapsed. In fact, now the four Maxwell’s equations (to be more specified, MaxwellHeavyside equations since Sir Oliver Hearyside was actually responsible for neatly arranging Maxwell’s theory in a lucid form with formulation of the following four equations) are read as modified versions of equations (2.1–2.4). They are reproduced hereunder as

Equation No. (2.6) (2.7) (2.8) (2.9)

Integral Form

Differential Form

− ⭸ψ →− → H . dl = I + ⭸t − ⭸Φ →− → E . dl = − ⭸t − →− → D . ds = Q − →− → B . ds = 0

− → − → − → ⭸D ∇× H = J + − → ⭸t ⭸B − → ∇× E =− ⭸t − → ∇. D = ρ − → ∇. B = 0

− → − → Combining these, we readily get the so-called wave equations for E or H . The − → derivation for electric field wave equation in a source free dielectric region ( J = ρ = 0) with the assumption that path of integration is stationary is shown below. − →

⭸B − → ∇× ∇× E =∇× − ⭸t From Eq. (2.7), we get =−

⭸ − → ∇ ×μH ⭸t

− → − → substituting B = μ H ⭸ − → ∇× H ⭸t − → ⭸ ⭸D = −μ . ⭸t ⭸t − → ⭸2 E = −με 2 ⭸t

= −μ

− → − → (substituting D = ε E ) − → − → − → But ∇ × (∇ × E ) = ∇(∇. E ) − ∇ 2 E → − → − → 1 − = −∇ 2 E as ∇. E = ∇. D = 0 according to Eq. (2.8) with ρ = 0 ε

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B. Gupta

− → ⭸2 E − → ∴ ∇ 2 E = με 2 ⭸t Similarly we can derive − → ⭸2 H − → ∇ 2 H = με 2 ⭸t Here ␧ and ␮ are the permittivity and permeability of the medium respectively. The solution to any of these in an unbounded (i.e. infinite in cross-section) space is − → E = f 1 (x − v0 t) + f 2 (x + v0 t) − → and H = f 1 (x − v0 t) + f 2 (x + v0 t) where f 1 ( ) and f 2 ( ) are arbitrary functions and 1 v0 = √ . με The solution here has been carried out using rectangular co-ordinates such that ∇2 =

⭸2 ⭸2 ⭸2 + 2+ 2 2 ⭸x ⭸y ⭸z

and ⭸ ⭸ = = 0 (assuming symmetry in cross-sectional yz plane as the structure is ⭸y ⭸z infinitely large along these dimensions). Indeed, the f 1 (x − v0 t) solution is a wave moving along +x direction with velocity v0 and the f 2 (x + v0 t) solution is another wave moving in the –x direction with the same velocity. It can further be shown that in such unbounded space, the electric fields and magnetic fields are mutually orthogonal and both are orthogonal

Fig. 2.3 Electric and magnetic field patterns in a propagating wave

2 Antenna Fundamentals

15

to the direction of wave propagation. For sinusoidal waves, the propagating wave − → − → amplitude patterns of electric ( E ) and magnetic fields ( H ) are shown below in Fig. 2.3.

Antennas – What, Why and How? ‘Antennas’ are transducers i.e. devices which transform electric energy to electromagnetic energy and vice versa. This energy transformation enables an electric signal to traverse a wireless path. Thus, antennas are integral and most essential components of all wireless communication links. They are the terminal blocks used at both ends of the link. On the transmitter side, an antenna is the last block responsible for radiation and on the receiver side, it is the first block responsible for reception of signal from free space. The fundamental question is: why do they radiate? Indeed, if we can answer this question we can say for sure that they can receive electromagnetic waves also by virtue of reciprocity. To answer this question, let us consider a simple wire antenna. When a two-wire line is opened up and bent, we get our desired wire antenna in the form of a standard dipole (formed by two linear arms). The process is shown below in Fig. 2.4.

(a) Open circuited transmission line

(b) Bending of arms of the two wire line

(c) Dipole antenna

Fig. 2.4 Construction of a dipole antenna

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B. Gupta

Fig. 2.5 Current standing wave pattern on dipole antenna

The current flowing on either arm of the dipole will get reflected from the open circuited ends and set up a standing wave. The standing wave amplitudes may be plotted along the dipole length as depicted. Now, at every point along the dipole we get a time varying value of current whose maximum magnitude is limited by the standing wave pattern. As a result, a time varying magnetic field is created in space surrounding the antenna. This in turn induces a time varying electric field which sets up a displacement current in the dielectric medium and the process builds up on itself, as explained in the previous section and shown in Fig 2.5. Obviously, this leads to formation of electromagnetic waves i.e. radiation. Conversely, when an electromagnetic wave is incident on it, it sets up an antenna current by reciprocity which causes an electric voltage to appear at the antenna terminals. Essentially, the radiation mechanism from more complex antennas is also similar. Thus the antenna acts as a transition between electrical circuits and dielectric media supporting propagating electromagnetic fields. Consequently, it can be modeled both ways-either using circuit quantities like voltages and currents or using field quantities. Even if the linear element would have carried a steady current, it would have been reflected at the open-circuited ends and radiation would have occurred since a change in the direction of current and consequently its velocity would have occurred due to reflection at the open ends. This way spurious radiation may take place in many cases where it is actually unwarranted and one has to safeguard against it.

Different Types of Antennas All antennas can be broadly classified into two categories: i. linear antennas and ii. aperture antennas. As the name suggests, linear antennas are those which radiate from a linear element fed by a time varying current. Examples are a) dipole, b) monopole with ground plane, c) folded dipole, d) loop, e) helix, f) conical helix, g) vee, h) rhombic antenna etc. Their structures are depicted in Figs. 2.6(a–h).

2 Antenna Fundamentals

a) Dipole

c) Folded dipole

e) Helix antenna

g) Vee antenna

Fig. 2.6 Different linear antennas

17

b) Monopole with ground plane

d) Loop antenna

f) Conical helix

h) Rhombic antenna

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B. Gupta

Commonest among these is the dipole, which has been already discussed qualitatively. However, for efficient operation, such an antenna must have an appreciable length in terms of wavelength. From the preceding discussion, it seems apparent that a center fed half-wavelength long dipole is actually fed at the center of standing wave maximum i.e. antinode existing on it. As the current variation with time along this standing wave is responsible for radiation, a center fed half-wave dipole is the obvious choice for an efficient linear radiator. However, for frequencies even up to MF (300 KHz–3 MHz), wavelength is too large to facilitate fabrication of a halfwave dipole with considerable mechanical ease. So, only one of its two quarter-wave long arms is often placed on ground (which is an infinite conducting surface at zero potential) or a sufficiently large conducting plane simulating ground. The electromagnetic image of this arm completes the dipole. Folded dipole is a modified form of dipole to provide different impedance value and larger band width. Loop is the complementary structure of dipole whereupon the radiated electric and magnetic fields are interchanged in behavior. Helix or conical helix combine features of both the loop and straight linear current elements and hence are capable of providing circularly polarized radiation too, either in the axial direction or in a direction transverse to it. Vee antenna is a bent form of dipole. All these are open-ended resonant antennas supporting standing waves. On the other hand, rhombic antenna belongs to the class of ‘Traveling Wave Antennas’ whose terminal end is terminated in a matched load so that no reflection occurs therefrom. As the frequencies of operations are scaled up, the physical size of a linear antenna rapidly diminishes. In this context, it must be stressed the length of an antenna is always expressed normalized with respect to the wavelength. A long antenna means that it is electrically long i.e. having considerable length with respect to wavelength and a short antenna means just the reverse. Consequently it means that even a long linear antenna at frequencies in the range of microwaves (1–40 GHz) and above is physically quite small. Such an antenna obviously has got limited power handling capacity. Hence at higher frequencies, linear antennas are replaced by aperture antennas which radiate from the total area of a complete two dimensional aperture. Naturally their feed mechanisms are also different – mostly in the form of waveguides. Examples are a) rectangular (sectoral or pyramidal) horns, b) conical horns, c) waveguide slots, d) paraboloidal reflectors, e) lens antennas, f) microstrip or patch antennas, g) dielectric antennas etc. Some of these are shown in Figs. 2.7(a–g) below. Horns are the obvious choice to make a waveguide radiate efficiently into free space. Slots or slot arrays may also serve the same purpose. Microstrip antennas are small in size and compatible with Microwave Integrated Circuits (MICs) and Monolithic Microwave Integrated Circuits (MMICs). They shall be dealt in a little bit more detail in a subsequent section. Paraboloidal or other shaped reflectors and lens antennas are used to generate very sharp collimated beams which radiate energy almost solely to a particular direction only and they operate in conjunction with other feed antennas located at the foci. Dielectric antennas are used to overcome the problems of corrosion and other hazards associated with metallic conductors.

2 Antenna Fundamentals

19

sectoral horn

pyramidal horn

a) rectangular (sectoral or pyramidal) horns

c) Waveguide slots

d) Paraboloidal reflector

e) Lens antenna

f) Microstrip or patch antenna

g) Dielectric antenna

Fig. 2.7 Different aperture antennas

b) Conical horn

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B. Gupta

a) Yagi-Uda array

b) Log periodic dipole array

Fig. 2.8 Some typically used antenna arrays

Usage is there also of antenna arrays i.e. collection of individual antennas called elements arranged in either linear or planar fashion. The assembly usually consists of antennas/elements of similar type and is used to direct more radiated energy around a particular desired direction in the form of a narrow beam. But there is also usage of arrays formed by different types of elements. Two commonly known examples are the Yagi-Uda and log periodic dipole arrays, where the dipole dimensions (and spacing too for the log periodic case) are nonuniform. They are shown in Figs. 2.8(a–b) respectively. Yagi Uda antenna is used to block energy flow to the backward direction and enhance it in the forward direction, whereas log periodic arrays are used for very broad band (almost frequency independent) operation.

Antennas for Different Usage The total usable frequency spectrum can be divided in decades as follows: Extreme Low Frequency (ELF) Very Low Frequency (VLF) Low Frequency (LF) Medium Frequency (MF) High Frequency (HF) Very High Frequency (VHF) Ultra High Frequency (UHF) Super High Frequency (SHF) Extreme High Frequency (EHF) Tera hertz and optical Frequencies

<3 KHz 3 –30 KHz 30–300 KHz 300 KHz-3 MHz 3–30 MHz 30–300 MHz 300 MHz–3 GHz 3–30 GHz 30–300 GHz > 300 GHz

Different antenna systems are used for operation in different spectral ranges. Some of the important classes of antennas, application-wise, are enlisted herewith. a) Broadcast Antennas: Commercial amplitude modulated radio broadcasting uses the MF range and requires equal coverage in all directions horizontally. Vertically polarized linear antennas are the antennas of choice. Since a half-wave antenna

2 Antenna Fundamentals

21

Fig. 2.9 PIFA antenna

becomes physically too large, vertically polarized quarter wave monopoles are used on ground in the form of huge masts and are called Marconi antennas. b) VHF Antennas: VHF and UHF ranges are mostly used for television broadcasting and they are horizontally polarized Yagi arrays. For good reception, constructive interference between a direct ray (from transmitter to receiver) and a ground reflected ray is used. c) Mobile and Wireless Link Antennas: Size and mobility are the prime concerns for these applications. Commonest antennas used are monopoles, patch antennas and a modified version thereof known as Printed Inverted F Antenna (PIFA). A typical PIFA structure is shown below in Fig. 2.9. d) Satellite Links and RADARs: Since these links require extremely sharp directed beams, the antennas of choice are paraboloidal reflectors and lenses. Requirements on performance indicators are even more stringent for satellite links due to much higher range of coverage. Over and above they require circularly polarized operation. Quite often a modified form of reflector arrangement with another hyperbolic sub-reflector is used, known as Cassegrain antenna shown in Fig. 2.10.

Fig. 2.10 Cassegrain antenna

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B. Gupta

Radiation Resistance ‘The radiation resistance of an antenna is that equivalent resistance which would dissipate the same amount of power as the antenna radiates when the current in that resistance equals the input current at the antenna terminals’ [1]. It may be evaluated as the ratio of the total radiated power to the square of the effective current at the antenna feed. If the total power radiated by the antenna is Pt and I is the effective i.e. rms current fed to it, then the radiation resistance referred to the input terminals is defined as Rr = Pt /I2

Antenna Impedance and Its Equivalent Circuit Input impedance of an antenna is defined as the ratio of voltage to current at the antenna input terminals or the ratio of appropriate components of electric and magnetic fields there. Viewed as a circuit element, this is the impedance presented by the antenna to its feed network. It is generally a complex quantity expressed in phasor notations as Zin = Rin + jXin where Rin is its resistive component i.e. input resistance of the antenna and Xin is its reactive component i.e. input reactance of the antenna. The input resistance has got two components: one the radiation resistance (Rr , say) and the other a lumped resistance (RL , say) to represent all other antenna losses. Losses may be incurred due to power dissipation in the ohmic resistance of the antenna and ground (if any), discharge or corona effects, losses in imperfect dielectrics close to the antenna or on it, eddy currents induced in metallic objects in near field of antenna etc. Thus we have Rin = Rr + RL Consequently the antenna efficiency, defined as the ratio of the power radiated to the power fed at the antenna input, is given by η=

Rr Rr + R L

To draw the equivalent circuit of an antenna in transmitting mode, let us assume the feed network (including generator and feed lines) be represented by a source voltage Vs having an internal impedance Zs = Rs + jXs . Under this assumption, the Thevenin’s equivalent circuit is as shown in Fig. 2.11.

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23

Fig. 2.11 Thevenin’s equivalent circuit for transmitting antenna

Fig. 2.12 Norton’s equivalent circuit for transmitting antenna

Here, aa are the antenna input terminals and VS is the voltage developed across aa with the antenna removed i.e. feed port open circuited. The Norton’s equivalent circuit for the transmitting antenna can also be developed as shown in Fig. 2.12 Here, Is = VZ SS i.e. the short circuit current at aa and YS = GS + jBS = Z1S along with Yin = Gin + jBin = Gr + GL + jBin = Z1in . In the receiving mode, a voltage Vind is induced across the antenna input terminals aa which delivers a current Iind through a load Zrx = Rrx + jXrx connected across aa . Consequently its Thevenin’s equivalent circuit may be drawn as in Fig. 2.13.

Fig. 2.13 Thevenin’s equivalent circuit for receiving antenna

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B. Gupta

The Norton’s equivalent circuit can also be drawn in the receiving mode too, as produced hereunder in Fig. 2.14. Quite obviously, in each case maximum power is delivered under the condition of conjugate matching and herein lies the importance of designing an antenna feed network properly.

Fig. 2.14 Norton’s equivalent circuit for receiving antenna

Some More Antenna Parameters Some more important parameters need to be explained and defined quantitatively to understand the operation of antennas. Hence the foregoing section attempts to serve this purpose.

Radiation Pattern The relative distribution of radiated electric field or power as a function of direction in space is the radiation pattern of an antenna. If it is a plot of field strength as a function of direction i.e. angle – it is called a Field Strength Pattern. On the other hand, if it is a plot of power radiated – it is a Power Pattern. The patterns can be calculated or measured at any given distance in the antenna far field (where only the radiated field components are present) since the relative distribution i.e. shape of the variation contour remains unchanged although the absolute values of field strength or power reduces with increase in distance from the antenna. The complete radiation pattern, in general, is a three dimensional plot. But it can be reconstructed from two-dimensional patterns taken at different sections. For example, if the radiated power per solid angle at direction (␪, φ) is P(␪, φ) then P(␪, φ) = P(␪, 0)X P(0, φ) Of maximum interest are usually two orthogonal planes called principle planes. Most commonly the two principle planes in which radiation patterns are plotted are the E and H planes. E plane is the plane containing the direction of the radiated beam maximum and that of the far field electric vector. Similarly, H plane is defined as the plane containing the directions of beam maximum of far field magnetic vector.

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25

Near and Far Fields The far field or radiation field of an antenna covers the region which is sufficiently far away from the antenna such that there is no effect of local energy storage and only the outward propagating field components are present. The near field or induction field is the region close to the antenna where antenna reactive effects and consequently local storage of energy is not negligible. From another viewpoint, far or Fraunhofer region is the one where the wavefront can be considered essentially planer and uniform i.e. rays are parallel without any phase shift between them. If the distance is somewhat smaller such that phase difference between different points along the cross-section can no longer be neglected, it is called the Fresnel region – whereas if the point of observation is so close to the antenna that neither amplitude nor phase variation along cross-section can be neglected, it is called the near region. This is a viewpoint based on diffraction theory and according to this theory, the far field should be located at a distance of at least 2D2 /␭ from the antenna where ␭ is the wavelength and D is the antenna diagonal dimension (linear). All radiation patterns and consequent parameters must be defined for the far field only.

Radiation Intensity It is defined as the power flow per unit solid angle in a given direction as illustrated in Fig. 2.15 below. − → − → − → The Poynting vector P = E × H gives the power radiated per unit area in any direction. The radiation intensity U(␪, φ) is related to its magnitude as U(␪, φ) = r2 . P (since r2 d⍀ is the area covered by a solid angle of d⍀ at source)

Fig. 2.15 The concepts of solid angle and radiation intensity

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B. Gupta

This is independent of the distance r from radiator since P ∝ The total radiated power is thus given by U (θ, φ)d⍀ Wr =

1 r2

in the far field.

Directive Gain Directive gain gd (␪, φ) in a given direction is defined as the ratio of the actual radiation intensity in that direction to the radiation intensity in that direction which would have been obtained from an isotropic source radiating the same amount of power. An isotropic source is one which radiates equally in all directions in space e.g. a point source. However, this is a hypothetical concept and in reality, all practical antennas concentrate radiated power along particular directions only. Directive gain is simply a relative measure to indicate how well radiated power is concentrated along that particular direction. Mathematically, gd (θ, φ) =

U (θ, φ) 4πU (θ, φ) 4πU (θ, φ)

= = Wr U (θ, φ)d⍀ Wr 4π

The denominator could also have been considered as the average power radiated per unit solid angle as the total solid angle in a sphere is 4␲ steradians. Expressed in decibels, directive gain can be written as G d (θ, φ) = 10 log gd (θ, φ) dB

Directivity Directivity (D) of an antenna is defined as its maximum directive gain i.e. D = gd (θ, φ)|max =

Umax (θ, φ) Uav

=

Umax (θ, φ) Wr /4π

=

4πUmax (θ, φ) Wr

4πUmax (θ, φ) = U (θ, φ)d⍀ It can also be expressed in decibels as10 log D dB. However, in common parley, the terms ‘directive gain’ and ‘directivity’ are often interchanged.

2 Antenna Fundamentals

27

Power Gain Power gain gp (␪, φ) of an antenna is obtained on replacing the total radiated power Wr by the total input power Wt in the expression for gd (␪, φ). It means that gp (␪, φ) expresses a similar parameter to indicate the directive behavior of an antenna but with respect to the total power fed to it i.e. g p (θ, φ) =

U (θ, φ) 4πU (θ, φ) = Wt /4π Wt

In decibels, we get G p (θ, φ) = 10 log g p (θ, φ) dB The maximum value of Gp is again given by 4πUmax (θ, φ) Umax (θ, φ) = G = g p (θ, φ)max = Wt /4π Wt and is also normally expressed in decibels as 10 log G = 10 log

4πUmax (θ, φ) dB Wt

Again, commonly the term ‘power gain’ or ‘antenna gain’ without any qualifier simply refers to its maximum value. It can be readily established from our earlier discussion on antenna input impedance as Wt = Wr + WL where WL is the power lost in the antenna loss resistance RL ∴

g p (θ, φ) Wr G = = D gd (θ, φ) Wr + W L Rr = Rr + R L = η (antenna efficiency)

such that G = ηD The gain of an antenna can be measured by comparing the maximum power density (Pmax ) of the Antenna Under Test (AUT) with that of a Reference Antenna (RA), whose gain is calibrated and known. Thus, G(AU T ) =

Pmax (AU T ) × G(R A) Pmax (R A)

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Different standard antennas like standard gain horns or dipoles are available commercially. Gain can thus be expressed either with respect to such a standard gain antenna or with respect to the isotropic antenna (as has been defined throughout this section). Very commonly, gain with respect to isotropic source is expressed by the artificial unit dBi rather than dB, the subscript ‘i’ referring to ‘isotropic’.

Effective Isotropic Radiated Power (EIRP) It is the product of the input power and the maximum gain i.e. EIRP = Wt · G It thus provides a measure of the total radiated power keeping in view the directive properties of the antenna.

Beam Width and Side Lobe Level Generally, typical radiation patterns show lobular structure with lobes peaking up between nulls distributed along the angular axis. The highest among these is called the ‘Principal’ or ‘Major’ lobe and the others are called ‘Secondary’ or ‘Minor’ lobes. The ratio of the peaks of major and highest minor lobe (also known as ‘Main’ and ‘Side’ lobes respectively) is called the ‘Side Lobe Level’ (SLL). It is usually expressed in dB after appropriate normalization. Usually the highest side lobe is the one closest to the main lobe. A good antenna design must ensure a low SLL so that maximum energy is effectively concentrated in the main beam. The Beam Width is defined with respect to the main lobe. Without any qualifier, by default it means the Half Power Beam Width (HPBW) which is the angular separation between two points closest to the peak of the main beam where the radiated power is halved (or reduced by 3dB) compared to the maximum value occurring at the peak. However, sometimes we are interested in the First Null Beam Width (FNBW) which is nothing but the angular spacing between the first nulls delimiting the main beam. A typical antenna beam along with a description of these concepts is illustrated in Fig. 2.16.

Beam Solid Angle/Beam Area and Beam Efficiency Beam Area (⍀A ) of an antenna is defined as the solid angle through which all radiated power would have flowed out had the radiation intensity been maintained constant at its maximum value within ⍀A . Accordingly

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Fig. 2.16 Beam width and side lobe level

Umax (θ, φ)⍀ A = Wr =

U (θ, φ)d⍀

∴ ⍀A =

U (θ, φ)d⍀ Umax (θ, φ)

Approximately, neglecting minor lobes, ⍀ A ≈ θE θH where ␪E and ␪H are the half-power beamwidths in two principal planes (usually E and H planes). The total value of beam area can again be divided into two components: one due to the major lobe (⍀M , say) and the other due to all minor lobes (⍀m , say) such that ⍀A = ⍀M + ⍀m ‘Beam efficiency’ (η M ) denotes how well energy is concentrated in the main beam and is defined as ηM =

⍀M ⍀A

Its reciprocal is known as the ‘Stray Factor’ (ηm ) and is thus defined as ηm = Needless to say, η M + ηm = 1

⍀m ⍀A

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From definition of Directivity (D), we get its relationship to Beam Area (⍀A ) as D =

Umax (θ, φ) 4π 4π = = ⍀A U (θ, φ)d⍀/4π U (θ, φ)d⍀/Umax (θ, φ)

Converted to degrees from radians, D = 41,253 ≈ 41,253 where ␪E and ␪H are ⍀A θE θH the half power beam widths in E and H planes respectively, both expressed in degrees. Stutzman [2], however, suggests that a more appropriate expression for evaluation of D from beam width in principal planes is D≈

26, 000 with ␪E and ␪H are as defined before. θE θH

The correction is necessary since the uncorrected expression is too optimistic neglecting beam shape and the minor lobes.

Effective Aperture or Effective Area If the incident power density on a receiving antenna is P for appropriately polarized incident wave which produces a received power WR across a load properly matched to the antenna, then WR = P.Ae where Ae is the effective area or effective aperture of the antenna. The actual power received is usually less than this value because of polarization and impedance mismatches. Nonetheless, this is a measure of the area actually presented by the antenna to the incident wave rather than the physical area. It is related to the physical area as Ae = ␳.A where ␳ is called the ‘aperture efficiency’. For a uniformly illuminated aperture the value of ␳ is unity, a value which gets reduced for tapered illumination of the aperture. But tapering of illumination (either in continuous form in aperture antennas or in discrete form in antenna arrays) is often required to reduce the side lobe level to desired extents. Just as the antenna efficiency denotes how much of the supplied power is radiated, the beam efficiency denotes how much of the radiated power is concentrated in the main beam and the aperture efficiency denotes how much of the physical aperture size can really be utilized for reception (and conversely for radiation, by invoking reciprocity).

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It is related to the directivity as D=

4π Ae λ2

Therefore, the larger the effective area, the higher is the directivity and consequently the sharper is the main beam.

Effective Height If a voltage V is produced across the terminals of a receiving antenna by an appropriately polarized incident electric field E, then the effective height (h) is defined as V E i.e. V = hE h=

It means that it is the appropriate value of height to be considered for the antenna to calculate the terminal voltage from incident field. More generally, under the condition of non-matched polarization, we can define a vector effective height − → ( h ) as − →− → V = h .E The term ‘Polarization’ has been applied here in its usual sense i.e. the locus of the tip of the electric field vector in an electromagnetic wave. By polarization of an antenna, what is meant is the polarization with which it would have radiated a wave.

Reflection Coefficient, Return Loss and Voltage Standing Wave Ratio (VSWR) at Antenna Input When an antenna having input impedance Zin is fed by a transmission line of characteristic impedance Z0 , reflections occur at its input port whose extent is denoted by the reflection coefficient (⌫) evaluated as ⌫=

Z in − Z 0 Z in + Z 0

This expression is also valid for a waveguide feed provided Z0 is replaced by the wave impedance Zw , since every waveguide has got a transmission line equivalent whose characteristic impedance is equal to the wave impedance. Consequently, standing waves are produced at the feed network characterized by Voltage Standing Wave Ratio (S) defined as

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S=

Vmax Vmin

where Vmax and Vmin are the maximum and minimum amplitudes respectively of the voltage standing wave pattern on the actual or equivalent feed line, as the case may be. It is related to the reflection coefficient or reflectance as S=

1 + |⌫| 1 − |⌫|

or

|⌫| = ρ =

S−1 S+1

The Return Loss (RL) is the absolute value of the magnitude of the reflection coefficient, expressed in decibels i.e. RL = 10 log |⌫| dB Of course, when the antenna is resonant (indicated by a zero value of net reactance in the antenna circuit) and its resistance is equal to the feed line characteristic resistance, we get no reflection. This situation is characterized by ⌫=0 and

S=1 RL = 0 dB

Otherwise, a part of the power fed is reflected back to the source and we have 0 < |⌫| ≤ 1, 1 < S < ∞ and 0 dB < RL < ∞ dB.

Antenna Bandwidth Like any other frequency sensitive device, by the term ‘Bandwidth’ we mean the effective operating frequency range of the antenna. Like any other case, it can also be defined either as absolute bandwidth or on relative basis with respect to the center frequency (quite often as a percentage value). It is usually expressed in either of two different means: i) Pattern Bandwidth and ii) Impedance Bandwidth Pattern bandwidth refers to the frequency range over which the radiation patterns remain essentially invariant or usable. A typical example, numerically quantified, is the Axial Ratio (AR) bandwidth of a circularly polarized antenna. Since exact circular polarization (with AR = 0 dB) can be achieved only for a particular frequency, this bandwidth refers to the frequency range over which the axial ratio remains within some maximum acceptable limit, say 3 dB.

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Nonetheless, of more practical significance is the impedance bandwidth since it is usually found to be more restrictive in most of the cases. Excepting perhaps, the case of circularly polarized antennas (where pattern band width is often evaluated as AR bandwidth), almost all other antennas show a narrower impedance bandwidth than pattern bandwidth. Impedance bandwidth refers to a maximum tolerable mismatch between the antenna and its feed and is usually expressed in terms of a maximum permissible value of VSWR or worst case value of return loss. Both are equivalent e.g. a VSWR value of 2:1 is equivalent to a return loss value of 9.8 dB. Most commonly used definitions for impedance bandwidth of antennas is either 2:1 VSWR bandwidth or 10 dB return loss bandwidth. But other limiting values are also used depending upon the particular application. At this point it is important to note that, VSWR, return loss, and impedance bandwidth of an antenna can drastically be improved by introduction of a properly designed matching network between the antenna and its feed. Therefore, within the antenna systems, proper design of matching networks plays an essential role.

Antennas on Chip and Antennas on Packages Antennas monolithically integrated on chips and/or printed on packages (as in the case of RFID) are mostly in the form of patch or microstrip antennas. Although it has been introduced earlier in this chapter, this class of antennas deserves a little more detailed description in view of their application-oriented importance. A microstrip antenna is formed by photoetching, printing or forming by some other means of an arbitrarily shaped conducting patch on a dielectric substrate, while the opposite side of the substrate is completely covered by a comparatively large metallic plane (ground plane) simulating artificial ground. The structure is reproduced once again in Fig. 2.17. Although the patch may assume any arbitrary shape, like rectangular, circular, triangular, pentagonal, annular ring, thin planar dipole etc. – rectangular and circular patches are most commonly used. The radiation mechanism from a rectangular microstrip antenna can be easily understood from a plot of electric and magnetic field lines therein, as shown below in a cross-sectional view (Fig. 2.18): It is obvious that there is a fringing of the time-varying electric field at the edges due to linkage through air, which is responsible for radiation in open space. Hence,

Fig. 2.17 Microstrip antenna

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Fig. 2.18 Field lines along microstrip

to make the patch an efficient radiator, we must ensure sufficiently large amount of fringing. It means that low dielectric constant substrates with large thickness are preferred for such antenna applications. On the other hand, microstrip lines or components would require minimal stray radiation and hence negligible fringing. Accordingly, high dielectric constant thin substrates, to which the field lines are bound more tightly, are preferred for such circuit applications. For monolithic applications using a common substrate for both antenna and associated circuitry, the choice of substrate is often a compromise between these conflicting requirements. The two radiating edges in a rectangular microstrip antenna are modeled as slots emitting radiation in free space while the length of the non-radiating edges is chosen as half-wavelength. Thus we get two radiating slots connected by a wide microstrip line or a parallel plate dielectric filled waveguide. According to array theory, radiations from the slots are in phase and thus reinforcing each other along the broadside direction and out of phase i.e. canceling each other along the end-fire directions. This is the simplest model for analysis of microstrip antennas, called the ‘Transmission Line Model’. Another less approximate model, known as the ‘Cavity Model’ considers the antenna as a leaky cavity with electric walls on top and bottom and magnetic walls on the other four sides. The internal fields within the cavity are first computed and then equivalent electric and magnetic currents are considered to be residing on the cavity walls. The radiated field is then calculated as generated by these equivalent currents as per equivalence principle. In this technique, radiated fields are considered as leakage from the imperfect cavity. Much more rigorous analytical models based on full-wave analysis are also available to describe the antenna behavior. To this end, numerical techniques like Finite Difference Time Domain (FDTD), Finite Elements Method (FEM), Method of Moments (MOM) etc. are often exploited. Various commercial packages are also available for achieving this goal. For design purposes, the non-homogeneous antenna structure is first converted to a homogeneous structure with effective dielectric constant ␧eff in which the conductive strip is embedded. The radiating edges are modeled as a two-element array spaced ␭/2 apart. Fringing is taken into account by considering a hypothetic line extension ⌬l as proposed in [3] ⌬l = 0.412h

εe f f + 0.3 w/ h + 0.264 · εe f f − 0.258 w/ h + 0.813

Then the length of non-radiating edges i.e. ‘l’ is determined as λo l + 2⌬l = λ/2 = √ 2 εe f f

λo or l = √ − 2⌬l 2 εe f f

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where ␭ is the guide wavelength i.e. wavelength in situ and ␭o is the free space wavelength. A semi empirical dispersion relation [4] is invoked for calculating ␧eff as εe f f =

εr + 1 εr − 1 + 2 2(1 + 12h/w)1/2

Here, w is the width of the patch, h is the thickness of the substrate and εr is its dielectric constant. Metallization thickness has been assumed to be zero everywhere.

Recent Advances in Antenna Engineering Although its origin can be historically traced back to more than 100 years, Antenna Engineering remains a vibrant field or research with newer developments taking place at regular intervals, almost every other day. Such developments have actually become not only imperative, but also inevitable given the rapid progress in wireless communication technology, of which Antenna Engineering is an inseparable component. To name a few of such recent developments booming on the horizon, we can site the followings: i) ii) iii) iv) v)

Adaptive and Smart Antennas Fractal Antennas Nanotube Antennas Superconducting Antennas Antennas on Photonic Bandgap Structures etc.

We are outlining very briefly sketchy overviews of few of these hereunder viz. Smart Antennas, Fractal Antennas and Nanotube antennas. Interested readers can study recently published literature on these topics, which is now available in sufficient volume.

Adaptive or Smart Antennas By ‘Adaptive’ or ‘Smart’ antennas we do not mean any single antenna, but a cluster or array of antennas. The array can adopt itself to a changing signal environment. Hence are the names. Actually the array consists of not only the individual radiating elements, but a complex control for positioning the beam adaptively. The control network is known as the ‘Beamforming Network’ and it is at the heart of the system. For a signal incident from a particular direction, the phase of the signal at each antenna element is the so-called ‘Steering Vector’ (much like the impulse response of the array) and a complete set of steering vectors for signal incident from all possible directions, is said to constitute the ‘Antenna Manifold’. It is this Antenna Manifold that is monitored, analyzed and controlled using efficient signal processing

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algorithms. Actually, adaptive arrays have been used from the Second World War days in rudimentary form when the radar beam was positioned and steered in space by using a switching matrix network known as Butler matrix. However, by 1960s more robust and efficient signal processing techniques started being viable and research on adaptive antennas gained momentum with the advent of cellular telephony during 1990s. Nowadays, such smart antennas are viewed as part of Multiple Input Multiple Output (MIMO) systems, or, putting the concept other way round, MIMO systems are considered as extension of adaptive antennas. Interference can be reduced; coverage range, channel capacity and spectral efficiency can be increased in MIMO systems by using the adaptive algorithms. Space diversity is an additional parameter introduced by smart antennas in MIMO systems. Spatial Filtering for Interference Reduction (SFIR) and Space Division Multiple Access (SDMA) concepts, as employed with smart antennas in cellular communication, result in a smaller reuse distance for same time and frequency channels i.e. better cell repeat pattern. Overall, a properly planed and optimally designed MIMO system using smart antennas can ensure considerable improvement in Quality Of Service (QOS), For an excellent comprehensive overview of the same, the reader is referred to [5].

Fractal Antennas Traditional approaches of antenna design are based on Euclidian geometry. However, a new class of innovative antennas and arrays has been developed in recent years based on fractal geometry. The word ‘Fractal’ literally means broken or irregular fragments, originally coined to describe families of complex structures possessing same sort of inherent self-similarity or self-affinity. The ideas were developed by investigating the naturally available patterns like snowflakes, mountains, leaves, clouds etc. Fractals can be constituted through stages of iteration from an initial or fundamental shape. One such example is the Sierpinski gasket, construction of first few stages of which is shown in Fig. 2.19. Thus fractals possess no characteristic size and they contain copies of themselves in different scales. Since the antenna operating frequency is usually dependent on

Fig. 2.19 Sierpinski gasket

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its electrical size, fractal geometries can be effectively exploited to develop a new class of multi frequency band antennas and miniaturized antennas of reduced size i.e. compact antennas. Starting from 1994, various fractal antennas like monopoles, dipoles, loops, volume and patch antennas all utilizing some fractal geometry or other have been reported to show multiband performance and/or compactness in size. Fractal antenna arrays have also been reported to describe arrangement of antenna elements according to fractal geometries. Such arrays are subsets of sparse arrays which possess relatively low side lobe levels (as expected from periodic arrays but not random arrays) and are also robust (as expected from random arrays but not periodic arrays). Accordingly, fractal arrays have been developed to provide desirable characteristics like low side lobes, multiband operation and reconfigurability. An excellent review on this topic is available in [6].

Nanotube Antennas Carbon nanotubes were discovered in 1999 and ever since became a hot topic of interest to researchers. It is because they have fascinating electrical properties like behaving either as semiconductor or as metal depending on the geometrical configuration. A single wall carbon nanotube is nothing but a rolled up sheet of graphene (i.e. a monoatomic layer of graphite) having a radius of the order of nanometers and length up to the order of centimeters. Naturally, such nanotubes attracted the interest of the antenna community too, apart from workers in other branches of science. Recently, carbon nanotube dipole antennas have been studied theoretically [7] for microwave and millimeter wave applications almost extending up to terahertz range. A transmission line model has been developed to analyze its radiating properties. Interestingly, there are several effects that make a nanotube line unique as compared to ordinary macroscopic transmission lines. For example, we have to consider a kinetic inductance dominating over the normal magnetic inductance and a quantum capacitance as well, apart from the usual electrostatic capacitance. As a result, the wave velocity gets reduced to the order of Fermi velocity vF (9.71 × 105 m/s for carbon nanotubes). Thus much higher frequencies of operation can be attained which has been demonstrated in [7] solving a Hallen’s type integral equation using the current distribution along the carbon nanotube transmission line. The results show some very interesting conclusions like the fact that they have inherently very high input impedance values. This is the case because at the nano scale, dc electron transport is either ballistic or associated with tunneling. In fact, it has been suggested that rather than 50 ohms as for conventional antennas, 12.9 K⍀ should be considered as the reference impedance for nano scale antennas. This may be useful in matching the antenna with other nanoelectronic devices. Further, they exhibit multiple very sharp resonances called ‘Plasmon’ resonances at frequencies for which the propagation velocity reaches a value of about 3vF – making compact multi frequency operation possible.

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Referencess 1. Collin R.E., Antennas and Radio Wave Propagation, Artech House, NY, 1997, p. 27. 2. Stutzman W.L., “Estimating Directivity and Gain of Antennas”, IEEE AP Magazine, vol. 40, Aug. 1998, pp.7–11. 3. Hammerstad E.O., “Equations for Microstrip Circuit Design”, Proc. European Microwave Conf., 1975, pp. 268–272. 4. Gupta K.C., Garg R. and Bahl I.J., Microstrip Lines and Slot Lines, Artech House, NY, 1979, p. 88. 5. Tsoulos G., “Adaptive Antennas and MIMO Systems for Mobile Communication” in ed. by Chandran S., Springer-Verlag, 2004, pp. 3–26. 6. Werner D.H. and Ganguly S., “An Overview of Fractal Antenna Engineering Research”, IEEE Antennas and Propagation Magazine, vol. 45, no.1, Feb. 2003. 7. Hanson G.W., “Fundamental Transmitting Properties of Carbon Nanotube Antennas”, IEEE Transactions on Antennas and Propagation, vol. 53, no.11, Nov. 2005.

Further Reading 1. Jordan E.C. and Balmain K.G., Electromagnetic Waves and Radiating Systems, 2nd ed., Springer Verlag, NY, 2004. 2. Ida N., Engineering Electromagnetics, 2nd ed., Springer Verlag, NY, 2004. 3. Ramo S., Whinery J.R. and Van Duzer T., Fields and Waves in Communication Electronics, 3rd ed., John Wiley, NY, 1994. 4. Cheng D.K., Field and Wave Electromagnetics, Addison Wesley, Reading MA, 1999. 5. Balanis C.A., Antenna Theory Analysis and Design, 2nd ed., John Wiley, NY, 2001. 6. Kraus J.D., Marhefka R.J. and Khan A.S., Antennas for All Applications, 3rd ed., Tata McGraw Hill, New Delhi, 2006. 7. Elliott R.S., Antenna Theory and Design, Prentice Hall India, 1985. 8. Collin R.E., Antennas and Radio Wave Propagation, McGraw Hill, NY, 1985. 9. Stutzman W.L. and Thile G.A., Antenna Theory and Design, John Wiley, 1981. 10. Skolnik M.I., Introduction to Radar Systems, 3rd ed., Tata McGraw Hill, New Delhi, 2001. 11. Garg R., Bhartia P., Bahl I. and Ittipiboon A., Microstrip Antenna Design Handbook, Artech House, Norwood MA, 2001. 12. Chandran S. (ed.), Adaptive Antenna Arrays: Trends and Applications, Springer Verlag, Berlin, 2004.

Chapter 3

Antennas for Mobile Wireless Communication Peter Lindberg

Background The field of portable terminal antenna design has witnessed a remarkable evolution during the last decade, mainly as result of the increasing requirements set by the mobile phone industry. Less than 10 years ago mobile phones were used exclusively for voice communication, utilizing a single wireless system (e.g. GSM) and a single frequency band (e.g. 900 MHz). The terminal antenna was external; either a retractable rod, or to make it smaller, folded together into a coil/helix. In the mid 90’s, frequency bands at 1800 MHz in Europe and 1900 MHz in the US were allocated to increase the network capacity, thus creating a need for antennas supporting two separate frequency bands, so called dual-band antennas. The first phone1 with this feature appeared on the market in late 1997. 1998 saw the first internal antenna (single-band however) in a commercial phone2 , a concept that had been proposed already in 1982 [1]. These antennas were non-obtrusive and thus retained the aesthetics and increased the mechanical ruggedness of the phone. Dual-band internal solutions was first presented in the literature in 1997 [2] with commercial phones using the technique available in mid-19993 . A couple of years later, the consumer requirement of global roaming (e.g. Europeans wanting to use their mobile phones in the US and vice versa) led to the development of triple band phones, featuring triple band antennas (900/1800/1900 MHz). Around the same time, Bluetooth modules, with a separate internal antenna, and FM radio receivers, using the earpiece cord as antenna, started to become standard features in phones. In early 20034 , a new cellular network called 3G (or UMTS/WCDMA), using a separate frequency band, started world-wide deployment. In about 5 years time, wireless terminals went P. Lindberg Laird Technologies AB, Mobile Antenna Systems, Research Department, Isafiordsgatan 5, 164 22 Kista, Sweden e-mail: [email protected] 1

Motorola MicroTAC8900, using a combined helical-monopole external antenna [1] Nokia 8810, using a PIFA antenna 3 Nokia 8210, using a PIFA antenna. 4 2001 in Japan 2

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

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from single system/single band to multiple systems and multiple bands. Most of these systems were not designed to work in the same terminal (e.g. by frequency planning or standardization), which means that the terminal manufacturers (including antenna designers) have to take responsibility for the inter-operability. Today, in 2008, phones in the mid-to high price range supports five different cellular bands (GSM850/900/1800/1900+3G), Wireless LAN, Bluetooth, Digital TV (DVB-H), FM radio and GPS. In the next few years, several new wireless systems such as RF-ID, UWB, WIMAX etc. will probably also be integrated in the terminal. Meanwhile, some of the “old” systems will most likely gain new frequency bands, such as WCDMA at 2.6 GHz and WLAN at 5GHz. Additionally, there is an interest in adding diversity capability for some of the systems (requiring extra receive antennas). Several of these antennas are by themselves exceedingly difficult to implement, and the close integration of all the different radiators leads to mutual EM coupling which further complicates the design phase. As more features have been added to each new generation of wireless terminals, the size of the phones has become progressively smaller. While this trend of size reduction, as demanded by the consumers, was made possible by advancements in battery technology, LCD displays and low-power/high integration circuit technology, the antennas are not as prone to size reduction since their performance (in most respects) is related to their occupied volume by laws of physics. If the volume is reduced, some penalty in performance must be paid. Additionally, during the past few years, an awareness of the possible health effects from mobile phone radiation has led to regulations [3] of the maximum local power densities induced in the user’s tissue by the mobile phones. Methods to avoid power being radiated in the direction towards the users head is thus needed, with the added benefit of increasing the total antenna efficiency. Also, the multiple possible positions of the user placing the phone in his/her hand should be considered as it will have an impact on the real-life antenna performance. Finally, modern mobile phones comes in a variety of form factors, e.g. bar, swivel, foldable/clamshell, flip, slide etc., each with its particular impact on the antenna design. In short, the professional life of terminal antenna designers has become a lot harder during the past few years. The following sections will introduce each of the problem aspects of terminal (and terminal related) antenna design together with, where appropriate, typical solutions.

Practical Limitations of Antennas for Commercial Wireless Terminals The fundamental limitations of electrically small antennas5 in terms of bandwidth and radiation efficiency were probably first examined by Wheeler in 1947 [4]. A 5 It should be noted that most mobile phone antennas are not electrically short in a strict sense as all widely accepted definitions requires the antenna to be at least smaller than the radian sphere (introduced by Wheeler in 1947 [4] and further treated in 1959 [5]), i.e. a sphere of radius a = 1/k where k is the wave number, which means that the largest dimension of the antenna must be smaller than 2/k = ␭/␲. Since a typical phone is about 100 mm long, and as the complete phone

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year later, Chu derived an approximate lower limit for the achievable radiation Q [10], which was corrected by McLean in 1996 [11]. Harrington [12] showed the effect of antenna size on gain in 1959. In addition to these fundamental restrictions imposed by physics, a terminal antenna designer also has to respect limits that comes from practical considerations, as the antenna is far from the only sub-component necessary for making the communication system work. These practical limits, and their consequences, are briefly outlined here (with a thorough treatment of the more significant aspects in the following chapter):

System Aspects All mobile phones are built around a multi-layered PCB6 on which the subcomponents (such as transceivers, DSPs, displays, battery etc.) are mounted. At least one of the layers of this PCB is completely metallized to act as a ground plane for the system. All currently used RF modules have unbalanced I/O ports with the ground plane as a reference terminal, implying that the antennas should also be implemented in an unbalanced configuration [13]. The all-pervading 50 ⍀ system impedance is exclusively used.

Available Volume Mobile phones are constantly getting smaller. While the length and width of the terminal must be of a certain size to facilitate a wide LCD screen and a conveniently large keypad, the thickness of the phone has no such restriction. The latest trend in terminal design is therefore ultra-thin phones, leading to very small heights above ground plane available to the antenna elements. This has a huge impact on patch type of antennas (such as the popular planar inverted-F antenna, PIFA) as the achievable bandwidth and radiation efficiency is proportional to this height [14]. In addition, many modern phones share the antenna volume with the speakers resonance box. While this is an efficient use of available volume it means that the antenna and speaker must be co-designed with the effect of increasing the complexity of both.

Chassis The metallized layer of the PCB, possibly together with other metallized parts of the chassis, functions as a ground plane for the various subcomponents in the terminal. can potentially function as an antenna, only antennas working at frequencies below 960 MHz, or radiators with no coupling to the chassis, should be considered as electrically small. Other proposed definitions in common text-books have smaller boundaries than Wheeler, e.g. ␭/10 in Balanis [6], ␭/8 in Schelkunoff &Friis [7], ␭/4 in Hirasawa [8] etc. An excellent discussion on the topic is given by Best [9], who suggests a limit half of that of the radian-sphere. 6 For some form factors, such as swivel, slide and clamshell types, there are in fact two PCBs (one for each part of the phone), but as the two grounds are galvanically connected together the effect is essentially the same.

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Therefore, the antenna designer typically has no mandate to make major alterations to the chassis to suit his/her needs, rendering for example PBG-surfaces [15] and slotted grounds [16, 17] unpractical. Recently however, radiators using “ground clearance”, i.e. the radiator is positioned above a PCB section without metallization, have been increasingly more common in commercial phones with certain form factors. This is perhaps a first step towards more, for the antenna performance, customized chassis layouts.

Current Consumption/Linearity Active antennas, i.e. with electronics (diodes or transistors) integrated together with the radiating element, consumes DC current thus reducing battery life time. This has so far limited widespread deployment in commercial phones. Also, all active components are intrinsically non-linear (even more so at low bias currents) which limits the use in high power Tx systems or systems co-located with high power transmitters. A more practical problem is that the active devices in many cases (such as switches) needs control voltages, implying that the antenna cannot be designed independently from the front-end module.

Price As in any consumer product, price is one of the most important parameters for a mobile phone antenna, thus excluding elaborate production techniques from being viable. Currently, the preferred choice for most radiators is single sided flexfilms glued to a plastic carrier, both for internal and external radiators. This means that only 2D metal structures are possible to implement. Since the plastic carrier can be somewhat irregular in shape and the film can be folded around edges, there is however a certain amount of freedom for slightly more complex structures. An interesting technology called LDS (Laser Direct Structuring) has recently become popular, making dual layer metallized plastic carriers (with via holes) possible. “Exotic” materials such as ceramics and ferrites are preferably avoided for cost reasons, as are active components integrated with the antenna and/or discrete matching components.

Weight One of the key figure of merits for a modern mobile phone is the weight. Hence, the antenna must be light. This is typically accomplished by using hollow plastic carriers and avoiding the use of ceramics and ferrites.

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SAR The main limitation of PIFA antennas is the limited bandwidth for low profile phones, and “new” types of antennas, such as monopoles and slots, have been suggested as a remedy. However, most of the non-PIFA/patch type of radiators suffers from excessive specific absorption ratio (SAR) values for high power (≥ 1 W) wireless systems such as GSM850/900. One way of reducing the negative impact on SAR is to place the radiator near the bottom of the phone (as opposed to the much more common placement at the top) which is naturally farther away from the head in talk position. This unfortunately means that the DC connector and external signal interface must be relocated to the side or top of the phone, which is considered unaesthetic many consumers.

Complexity A complete mobile phone is an extraordinarily complex system from an EM point of view. The presence of the battery, display, keypad, various plastics etc. in the near field can have a huge impact on the total antenna performance [18]. For instance, speakers are often co-integrated with the antenna element for volume sharing7 and can give rise to non-radiating resonances (i.e. an extra resonance is visible in the input impedance, but the radiation efficiency is almost zero as the power is instead coupled to the speaker amplifier output resistance). In addition, the complexity of the terminal makes EM modeling of the antenna excessively difficult and therefore nearly impossible to optimize using software. To retain some consistency, the RF community has agreed on using a simple PCB, typically of size 100 × 40 mm2 using FR-4, for early design evaluation. A very significant part of the scientific literature on terminal antenna design is concerned with synthesization of elaborate radiator shapes to achieve a certain performance. Almost exclusively these radiators are optimized for and evaluated using a naked PCB. However, when implemented in a real phone, these designs will most likely have some change in characteristics. Therefore, for such papers, some means of making simple adjustments to compensate for these effects are necessary to make the proposed design practically useful. This is typically not included in the papers. It should be noted though, that while the immediate practical value is limited, these proposed designs are important for estimating what performance can be theoretically achieved with a certain type of radiator.

7 The plastic antenna carrier is made as a hollow back-cavity/resonance box for the loudspeaker, while simultaneously providing a suitable distance between the antenna and the metallic chassis.

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Referencess 1. K. Fujimoto, A. Henderson, K. Hirasawa, and J. R. James, Small Antennas. Wiley, 1987. 2. Z. Liu, P. Hall, and D. Wake, “Dual-frequency planar inverted-F antenna,” IEEE Transactions on Antennas and Propagation, vol. 45, no. 10, pp. 1451–1458, Oct. 1997. 3. Federal Communication Commission (FCC), Guidelines for Evaluating the Envirornmental Effect of Radiofrequency radiation. FCC 96-326, 1996. 4. H. A. Wheeler, “Fundamental limitations on small antennas,” Proceedings of the IRE, vol. 35, pp. 1479–1484, Dec. 1947. 5. H. A. Wheeler, “The radiansphere around a small antenna,” Proceedings of the IRE, vol. 47, pp. 1325–1331, Aug. 1959. 6. C. A. Balanis, Antenna Theory, 2nd ed. Wiley, 1997. 7. S. A. Schelkunoff and H. T. Friis, Antennas – Theory and Practice. Wiley, 1952. 8. K. Hirasawa et al., Analysis, Design and Measurement of Small and Low Profile Antennas. Artech House, 1992. 9. S. R. Best, “A discussion on the properties of electrically small self-resonant wire antennas,” IEEE Antennas and Propagation Magazine, vol. 46, no. 6, pp. 9–22, Dec. 2004. 10. L. J. Chu, “Physical limitations of omni-directional antennas,” Journal of Applied Physics, vol. 19, pp. 1163–1175, Dec. 1948. 11. J. McLean, “A re-examination of the fundamental limits on the radiation Q of electrically small antennas,” IEEE Transactions on Antennas and Propagation, vol. 44, no. 5, pp. 672–675, May 1996. 12. R. Harrington, “Effects of antenna size on gain, bandwidth, and efficiency,” Journal of Research of the National Bureau of Standards, vol. 64-D, Jan. 1960. 13. S. P. Kingsley, J. M. Ide, D. Iellici, and S. G. O’Keefe, “Radio and antenna integration for mobile platforms,” in Proc. European Conference on Wireless Technology, Sept. 2006, pp. 79–82. 14. K.-L. Wong, Planar Antennas for Wireless Communications. Wiley, 2003. 15. Z. Du, K. Gong, J. Fu, B. Gao, and Z. Feng, “A compact planar inverted-F antenna with a PBGtype ground plane for mobile communications,” IEEE Transactions on Vehicular Technology, vol. 52, no. 3, pp. 483–489, May 2003. 16. M. Ali and F. Abedin, “Designing ultra-thin planar inverted-F antennas,” in Proc. Antennas and Propagation Society International Symposium, vol. 3, June 2003, pp. 78–81. 17. R. Hossa, A. Byndas, and M. Bialkowski, “Improvement of compact terminal antenna performance by incorporating open-end slots in ground plane,” Microwave and Optical Technology Letters, vol. 14, no. 6, pp. 283–285, June 2004. 18. D.-U. Sim and S.-O. Park, “The effects of the handset case, battery, and human head on the performance of a triple-band internal antenna,” in Proc. Antennas and Propagation Society International Symposium, vol. 2, June 2004, pp. 1951–1954.

Chapter 4

Challenges in Mobile Phone Antenna Development Peter Lindberg

Background The design of terminal antennas is in many respects significantly different to the design of antennas for the vast majority of other applications. This is partly related to the smallness of the antennas, which limits the achievable performance and complicates the measurements, with the complex near-field environment of the antenna element, and with the fact that the antenna, in its final implementation, is fully integrated with the front-end.1 This chapter will discuss these particular aspects of terminal antenna development. The end objective of any commercial terminal antenna design is the fulfillment of some given specification, typically provided by the terminal manufacturer and/or the network operators (for the case of mobile phones). Additionally, there are government regulations setting constraints on e.g. SAR levels2 . The specification is typically mainly concerned with electrical properties -bandwidth, efficiency etc., and mechanical properties -size, placement, interface etc. However, from a purely commercial point of view, other qualities are (at least) equally important – low cost, low weight, ease of production and verification (as discussed in Chapter 3) that introduces practical boundaries on the design options. As in the case of most massmarket electronics, a cheap solution with sufficient performance is often preferred over a slightly more expensive solution with much better performance. In particular, this is true for sub-components (such as the antenna) where the performance is not

P. Lindberg Laird Technologies AB, Mobile Antenna Systems, Research Department, Isafjordsgatan 5, 164 22 Kista, Sweden e-mail: [email protected] 1 Terminal antennas are sometimes categorized as “physically constrained small antennas” [1] since their location, near environment and allocated volume are predetermined by other (practical) considerations without possibility for optimization to improve the performance. Hence, while the total volume represented by a typical handheld terminal is not electrically small per se, this volume is in reality not readily available to the antenna designer. 2 In many cases though, mobile phone manufacturers have tougher requirements than those set by the regulations.

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

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directly noticed by the end-user (i.e. buyer) and which can be compensated for at the system level by e.g. increasing the number of base stations3 . For the service provider on the other hand, e.g. cellular network operators, the communication capability of the terminal (and hence the antenna) is highly important to relax requirements on base station density. Therefore, it is not surprising that the toughest specifications on the mobile phone antenna are given by the (US) network operators. This chapter is concerned mainly with fundamental challenges related to the electrical properties, though keeping in mind the more practical/commercial aspects.

Characterization During the design phase, the antenna is characterized using a coaxial measurement cable (so called passive measurements) connected to the antenna mounted in a preliminary terminal prototype/mock-up. Standard measured characteristics are impedance, antenna efficiency, bandwidth and SAR. Directivity, radiation pattern and polarization, which are three of the most important properties for large antennas, are seldom of interest for terminal applications. Unlike other components such as filters and amplifiers, the antenna is highly sensitive to its nearby environment. Hence, any alteration to some plastic parts or similar during the development of the terminal will require a re-tuning of the antenna. Throughout the development phase of the terminal, minor changes are frequent, leading to many antenna design iterations. For this reason, it is highly important that the implemented antenna is designed for tunabilty and that the antenna engineer fully understands the design to facilitate rapid retuning. Perhaps this is one of the reasons why commercial antennas are almost exclusively based on fairly conservative design approaches (e.g. PIFAs). For the final verification measurements, the antenna is implemented in the complete terminal and operated by the transceiver. As the system is fully integrated, it is no longer possible to measure qualities such as bandwidth, efficiency etc. Instead, the antenna is evaluated from over-the-air (OTA) measurements (also called active measurements) [2]: TIS - Total Isotropic Sensitivity (Receive mode) TRP - Total Radiated Power (Transmit mode) SAR - Specific Absorption Rate (Transmit mode) HAC - Hearing Aid Compatibility (Transmit mode)

3 While this is true for current cellular systems such as GSM, where a huge margin in the link budget is only useful for e.g. higher tolerance to fading dips (so a few dB lost in the antenna is no big deal), it will be different in upcoming systems for data transfer (i.e. internet access). In such systems, the extra SNR (signal to noise ratio) provided by a better antenna will result in a higher data throughput as the coding and modulation format is flexible and is adapted depending on available SNR. Hence, users will be able to compare different RF solutions (in different terminals) by their achieved data transfer rate, which could potentially be a strong selling point.

4 Challenges in Mobile Phone Antenna Development

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The RF transmitting performance of the complete front-end is determined by the “Total Radiated Power” in an anechoic (and possibly also in a reverberation chamber) by measuring and integrating the power density transmitted by the terminal over all angles, with the phone set to operate in its highest power class. For terminals fitted with external coaxial connectors, it is possible to compare the measured radiated output power to the power available from the transmitter and thus deduce the antenna performance. This is however not necessarily the same performance as measured in passive mode as the transmitter may not operate equally well for all load impedances within the return loss limits (in contrast to a theoretical 50 ⍀ source). Some specifications only considers the talk-position mode, typically accepting more than 10 dB total antenna loss (for GSM850/900, and around 5–6 dB for GSM 1800/1900), other considers only free-space and yet others specifies both. The receiver performance “Total Isotropic Sensitivity” is measured by lowering the power level from a base station simulator until a specified bit error rate (BER) is reported by the terminal. Again, this might not be identical to the results from passive measurements as the total noise figure of the receiver depends on the source (i.e. antenna) impedance and not only the return loss (which is given in specification). A normal receiver sensitivity in GSM850 is −108 dBm, and a typical specification for TIS is around 10 dB higher in talk position (consistent with the transmit requirement). During measurements in an anechoic chamber, the base station first transmits a signal at a fixed power level and the reported received power by the phone is recorded at all angles and at specified frequencies. For the angle with the highest power level, a specified bit sequence is repeatedly transmitted by the base station with reduced output power for each iteration. The received signal is retransmitted by the phone to the base station and compared to the original sequence, and TIS is recorded as the power level that corresponds to a BER of 2.44% for GSM and 0.1% for WCDMA. The sensitivity at all other angles are then calculated using the measured received power levels, and the final TIS value is the average over all angles. SAR is only relevant to high-power systems such as GSM850/900, and requires a fairly elaborate measurement set-up as described later in this chapter. Since mobile phones sold to the general public in the US needs to comply with FCC (Federal Communication Commission) regulations on SAR (i.e. less than 1.6 W/kg peak value), in some cases, it is the SAR that limits the Tx power rather than the maximum output power that the power amplifier (PA) can deliver [3]. HAC measurements are, in the context of antenna development, conducted to classify a mobile phone into different categories of hearing aid compatibility in a scale from M1 to M4, where a category M4 phone is more likely to work with a given hearing aid than a category M1 phone. A M3 grading meets the standard, and M4 exceeds the standard. Note that the scale is relative; it is not guaranteed that even a M4 phone will work properly with a given hearing aid. The measurement procedure is similar to that of SAR and is typically conducted using an extended SAR measurement set-up. In principle, a probe measures the E-field and H-field 1 cm above the phone (on the speaker side) in a specified area and these values are then compared to standard values specified by FCC, which determines the compatibility

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category. The exact classification procedure though is somewhat more elaborate, for instance high peak values can be removed from the measurement if they are localized (since the user can compensate by simply shifting position slightly). As the result of these measurements, as previously mentioned, also depends on other components such as filters, switches, the power amplifier (TRP, SAR and HAC) and the low-noise amplifier (TIS), the system level performance can be quite different compared to what could be expected from the passive measurements, even though each component complies with the specification. This is easily understood by noting that the standard specification on the antenna input impedance accepts the full 17–150 ⍀ range, while the filter, switch and power amplifier (PA) designs all assumes a fixed 50 ⍀ source and load. In particular, the output power of the PA is strongly dependant on the load impedance (possibly also at harmonic frequencies) and can therefore vary greatly over the frequency band (due to impedance variations). Hence, as a final part of the antenna design phase, re-tuning based on active measurement results are sometimes needed. Needles to say, this re-tuning can be very difficult and time consuming to do. The US-based Cellular Telecommunications and Internet Association (CTIA) is currently the only organization that produces OTA performance tests. In Europe, ETSI and 3GPP have produced a range of standards, but have yet to publish OTA specifications. Also, the activities in COST 273 are likely to result in OTA specifications but there is currently no known schedule for such publication.

Crosstalk of Multiple Antennas With the continuous introduction of new cellular and complementary (WLAN, DVB-H) wireless systems, the number of antennas per handset has been increasing rapidly. The main problem of co-locating several antennas in a confined volume, besides reducing the available space for each antenna and hence bandwidth (see the next section), is the mutual coupling, or crosstalk, between all elements, inevitably leading to an increased design complexity. The design of a single terminal antenna for modern mobile terminals is often extremely challenging, in particular for wide-band and/or multi band applications, and taking mutual effects of other elements into account can be very difficult and time consuming. In practice, the process is often iterative: first design one antenna, then design the other antenna taking into account the presence of the first antenna. Then the first antenna is retuned to take the presence of the second antenna into account and so on. It should be noted that it is not clear in the design phase how all non-active (i.e. those currently not being measured) antennas should be terminated to simulate a real world situation. The cellular antenna, for example, will have different terminating impedances depending on if it is in transmit or receive mode (with either the PA or LNA switched on). From a system level point of view, crosstalk is a problem in particular for systems that are active simultaneously, such as when a user is talking through a Bluetooth headset using GSM, and is most severe for low-frequency systems that

4 Challenges in Mobile Phone Antenna Development

49

radiates mainly through the chassis mode (i.e. below say 1.5 GHz). Typical problems includes

r r r

Reduced efficiency due to power being leaked into other systems instead of being radiated. High power signals from transmitters can saturate receivers of other systems, so called “blocking”. High output noise levels from transmitters can degrade the receiver sensitivity of other systems.

Naturally, the problem is more difficult the closer the systems are in frequency, where currently the most challenging inter-operability issue is between low-band GSM and DVB-H, as seen in Fig. 4.1. Crosstalk can be reduced by using balanced radiators (such as dipoles or loops) with no RF connection to the chassis [4]. The presence of the metallized chassis will however reduce the radiation resistance of the radiators, similar to the case of having a dipole antenna too close to a reflector or ground plane, thus reducing the achievable bandwidth. Also, all currently available transceivers have single-ended I/O interfaces, meaning that an expensive and lossy balun is required to drive the balanced antenna [5]. Mutual coupling can also be reduced by careful orientation of the radiating elements [6], by cutting band-notch ␭/4 slots in the ground plane [7], or by introducing an inductive link between two antennas [8] that provides a second coupling factor to cancel the original (capacitive) coupling. GSM Antenna PA Filter

PA

Wideband noise

GSM900 Tx

Coupled power DVB-H Antenna LNA

Fig. 4.1 Crosstalk between transmitting GSM antenna and receiving DVB-H

Bandwidth The single most challenging task in terminal antenna design is the attainment of a sufficient impedance bandwidth without sacrificing efficiency. Standard requirements for mobile phone antennas today is a return loss of < −6 dB within the

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frequency band. To compensate for manufacturing tolerances, some extra margin is typically required in the design phase (e.g. < −7 dB return loss at the band edges). Currently, the most difficult systems are the low frequency bands GSM 850/900 and DVB-H, where even the terminal length (which could potentially be used as part of the antenna) is shorter than half of the operating wavelength. The bandwidth of the mobile antenna is for such systems theoretically limited by the resonance frequency and the radiation quality factor of the chassis, and in practice by the amount of achievable coupling from the radiating element to the chassis. The fundamental limit for electrically small antennas was first presented in 1948 by Chu [9] (and later corrected by McLean [10]), who derived an approximate lower limit for the achievable radiation quality factor Q r . The corrected formula (usually called the “Chu limit” or “Chu-Harrington limit”), which relates the bandwidth (through Q) to the physical size, is given as Qr =

1 1 + (ka)3 (ka)

(4.1)

where k = (2␲)/␭0 is the wave number in free-space and a is the radius of the smallest sphere in which the antenna could be fitted. Eq. 4.1 is only valid for single resonant structures exciting a single (lowest) mode (TE or TM); when both the lowest TE and TM modes are excited (circular polarization) a modified expression is given in [10]. The radiation quality factor relates the reactive electric or magnetic energy W to the radiated power Prad of a tuned antenna (i.e. the antenna’s reactance j X (if non-zero) has been resonated by a series reactance − j X using a lossless series inductance or capacitance through Q r (ω0 ) =

ω0 |W (ω0 )| Prad (ω0 )

(4.2)

where ω0 is the resonant angular frequency ␻0 = 2␲ f 0. Q r only includes power dissipation through radiation (there is no fundamental limit for lossy antennas). Conductive losses Pc and dielectric losses Pd can be added to the radiated power to obtain the total accepted power by the antenna PA , and since ␻ and W are fixed, an unloaded quality factor Q 0 can be defined as 1 1 1 1 1 Q0 = (4.3) + + =η + Qr Qc Qd (ka)3 (ka) where Q c and Q d are the conductive and dielectric quality factors, respectively, and ␩ is the radiation efficiency. Material losses are more commonly expressed using dielectric (tan δ ) and magnetic (tan δμ ) loss tangents, which relates to their respective quality factors as [11] 1 = tan δ Qd

and

1 = tan δμ Qm

(4.4)

4 Challenges in Mobile Phone Antenna Development

51

The Q-value of an antenna can be directly (and exactly) calculated from Eq. 4.2., which however involves the computation of volume integrals of the electric and magnetic fields surrounding the antenna (which is feasible in simulation tools, but not very practical for measurements). More convenient is instead to use the antenna’s feed point impedance through [12] ω ω Z = Q0 ≈ Qz = 2R 2R

R 2

+

X

|X | + ω

2 (4.5)

where Z = R + j X is the antenna’s frequency dependent (untuned) feed point impedance and R and X denotes the frequency derivatives of the feed resistance R and feed reactance X. Eq. 4.5 thus enables the direct calculation of the antenna’s Q-value in a wide frequency range. It should be noted that the quality factor rather than the bandwidth is the fundamental property of the electrically small antenna. The realized fractional impedance bandwidth BW = ( f 2 − f 1 )/ f 0 of the complete antenna on the other hand is also related to (besides Q 0 ) the acceptable mismatch level V SW R ≤ S through BW ≈

S−1 √ Q0 S

(4.6)

where a perfect match to the generator at the resonant frequency f 0 , i.e. R( f 0 ) = Z 0 is assumed and any antenna reactance is resonated by a lossless series inductance or capacitance. Eq. 4.6 can therefore also be used to calculate the antenna’s Q by noting the bandwidth BW that the antenna provides for any given S with the source impedance Z 0 chosen as R and any reactance at the examined frequency f 0 resonated by a lossless series inductance or capacitance. The bandwidth BW here is defined as the so called matched VSWR bandwidth. Another common bandwidth definition is the “conductance bandwidth”, which however is not valid around anti resonances (parallel resonances) [12]. It has been shown that Eq. 4.6 can be used to accurately predict the Q value even of multi-resonant wide-band antennas, and more importantly, that the inverse relationship between Q and matched VSWR bandwidth holds for this case, provided that S is chosen sufficiently small (say S = 1.25) so as not to allow the VSWR bandwidth, to encompass multiple resonances [13]. For a perfect match at the center frequency (i.e. R0 = Z 0 ) and settings S = 5.828 (which gives the half-power VSWR bandwidth, or equivalently the –3 dB return loss bandwidth), Eq. 4.6 simplifies to the well-known inverse relationship between (half-power) fractional bandwidth and quality factor BW =

2 Q0

(4.7)

A larger bandwidth, for a certain acceptable S ≤ V SW R within the frequency band, can be achieved by mismatching the system at the center frequency, i.e. selecting Z0 = R [14]:

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1 BW = Q0

(T S − 1)(S − T ) S

(4.8)

where f 1 is the lower frequency limit f 2 is the upper frequency limit fr is the resonant frequency S is the bandwidth criterion V SW R ≤ S T is the coupling coefficient, T = Z 0 /R0 for a series resonance and T = R0 /Z 0 for a parallel resonance – R0 is the input resistance at resonance – Z 0 is the characteristic impedance

– – – – –

Maximum bandwidth, for a certain return loss (or VSWR) criterion, is achieved by selecting the coupling coefficient T for so called “optimal overcoupling” (instead of a perfect match at the resonance frequency, called “critical coupling”). The value of Topt for optimal overcoupling is obtained by finding the zero of dBW/dT from Eq. (4.8) with the solution Topt

1 = 2

1 S+ S

(4.9)

For maximum bandwidth in typical mobile phone applications, S = 3 (equivalent to a return loss of −6 dB) results in an optimum input series resistance of R0 = 30 ⍀ in a Z )0 = 50⍀ system. Entering Topt from (4.9) into (4.8) gives the maximum achievable bandwidth for a single resonator without matching network: BWmax =

S2 − 1 2Q 0 S

(4.10)

Compared to critical coupling, i.e. R0 = Z 0 , the bandwidth improvement for −6 dB return loss can be calculated using Eqs. (4.10) and (4.8) with T = 1 (which then is identical to 4.6), giving a 15compared to using critical coupling is larger for higher values of S (or return loss). As an example, consider an series RLC resonance circuit with L = 50 nH, C = 0.5 pF and R = 50 ⍀ (for an unloaded Q-value of ␻0 L/R = 2␲ f 0 L/R = 2␲(1 ∗ 109 )(50 ∗ 10−9 )/50 = 6.3). By connecting this resonator to generators of different characteristic impedances Z 0, significantly different bandwidths (at the RL < −6 dB level for instance) are achieved even though the Q-value of the resonator is constant. The concepts of overcoupling, undercoupling, critical coupling and optimal overcoupling of this resonator are graphically illustrated in Fig. 4.2. The total input impedance of the antenna close to resonance is related to the quality factor as [15]

4 Challenges in Mobile Phone Antenna Development

53

0 VSWR<3

1112

Overcoupled (0% BW)

–2

Critically coupled 1096 1053 Undercoupled

1000

Return loss (dB)

Overcoupled 1200

Optimally overcoupled

–4

Undercoupled (10.3% BW)

–6 Optimally overcoupled (21.3% BW)

–8

–10

950

–12

912

Criticallycoupled (18.4% BW)

899 800

–14

Zresonator

800

850

(0.8 – 1.2GHz)

900

950

1000 1050 1100 1150 1200

Frequency (MHz)

Fig. 4.2 Coupling of series RLC resonator to a generator with Z0=300 ⍀ (over-coupled), Z0 = 83.3 ⍀ (optimally over coupled), Z0 = 50 ⍀ (critically coupled) and Z0 = 30 ⍀ (under coupled) characteristic impedances

Z in = R0 (1 + j Qv)

or

Z in =

R0 1 + j Qv

with

v=

f f0 − f0 f

(4.11)

for the series (left) and parallel (right) resonant cases, respectively. For the computation of Q-values using Eq. 4.5 or 4.6, some feeding mechanism has to be assumed, making them unsuitable for analyzing e.g. the Q of coupled resonators such as the chassis (which is typically excited by e.g. a high-Q PIFA element). A method that circumvents the need for port excitation is the Characteristic Mode Theory, where a discrete set of orthogonal current densities (characteristic currents or eigencurrents) are given as eigenvalue solutions of the Z-matrix eigenvalue problem for the modes on conducting bodies [16, 17]. The Z-matrix, or generalized impedance matrix, is given by direct application of the standard Method of Moments. While the modes are dependent of the body’s geometry only, the relative excitation of each mode can be controlled by careful selection of feed or radiator type and placement. Unfortunately, unlike cavity type of resonators for which eigenmode solvers are readily available in commercial software, open boundary eigenvalue problems (such as antennas) is generally not directly supported in standard field solvers. In [18] this problem was solved by utilizing a plane wave illumination of the structure (in IE3D [19]) and then post-processing the resulting surface current density to identify the chassis mode resonances. By chassis mode resonance it is meant an equiphase current distribution over the (lossless) metallic surface, which when excited (and undisturbed) behaves as [18] Js r , t = Js,i r e jωi t−ωi t/2Qrad,i

(4.12)

where i means the i-th resonant mode with angular resonance frequency ␻i and Qrad,i as before. Other presented applications of characteristic mode theory for terminal antenna analysis and design are given in [20, 21, 22]. In particular, the theory

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was used by Rahola [23] to validate the optimum placement of patch type radiating element at the short-edges (voltage maximum) of the chassis for mono block (or bar) type of phones. Rahola also demonstrates the important lower-order modes of the chassis, where the first mode is the well known dipole-type distribution (resonant at 1.3 GHz) and the second mode is a full wave mode with a current minimum at the center (at 2.95 GHz). The third and fourth modes (at 4 GHz) have currents that are mainly in the direction of the short-edges (from long-side to long-side). A similar analysis was performed in [24], which also included the radiating element. Methods of measuring the quality factor of antennas are provided in [25, 26, 27]. In the previous equations and arguments, the antenna has been assumed to be sufficiently modelled as a simple series or parallel resonance circuit. This implies in particular that the total resistance (radiation plus losses) is constant throughout the frequency band of interest, which is not always the case for real antennas. For example, the radiation resistance typically increases with frequency for small antennas. For the typical case of coupled resonators, e.g. the chassis and the radiator with the chassis resonance above the operating frequency band, a tighter bound of the achievable bandwidth was presented in [28]. For more complex cases, such as dual band operation [29], the Real Frequency Line Segment Technique of Carlin [28] or Simplified Real Frequency Technique (SRFT) [30, 31] provides an insight to matching problems with an excellent estimate of the upper level of flat transducer power gain (or return loss)over the selected frequency bands. SRFT was applied in [32] for a standard dual-band PIFA antenna to estimate the achievable bandwidth using lumped components.

Impedance Matching The maximum achievable bandwidth, for a given antenna Q value and a certain acceptable reflection coefficient ⌫, using an infinite number of tuned circuits (i.e. series LC and/or parallel LC circuits) was presented by Bode in 1945 [33]

Bmax

∞

=

π 1 ln 1 Q ⌫

(4.13)

In 1948 Fano [34] presented the achievable bandwidth for a given number n of tuned circuits, expressed as a set of coupled trigonometric equations. Lopez developed more accessible equations in 1973 [35] for the case of n tuned circuits, where for the typical case of ⌫ > 1/3 Bmax

n

=

an 1 ln 1 Q ⌫

(4.14)

4 Challenges in Mobile Phone Antenna Development

55

The coefficients an are provided from numerical solutions of the Fano equations [35], as tabulated below together with the bandwidth enlargement factor ⌬B = Bn /Bn−1 . n an ΔB(%)

1 1

2 2 100

3 2.41 20

4 2.63 9.1

5 2.76 4.6

6 2.84 3.3

7 2.90 1.8

8 2.94 1.7

... ... ...

∞ ⌽ 0.0

As can be seen, one extra resonator (the antenna counts as the first (n=1)) doubles the bandwidth while the next one provides 20% extra bandwidth. Further resonators are probably not motivated in practice. For the particular case of coupled resonators in mobile phones, the bandwidth enlargement factor was thoroughly investigated first in [36] and later in [45].

Chassis Influence on Impedance Bandwidth The characteristics of small internal (e.g. PIFA) antennas mounted on handheld terminals are very different compared to when placed on an infinite ground plane, and depends on both the antenna position on the terminal chassis and the dimensions of the chassis (the length in particular). This is due to the existence of radiating surface currents on the terminal ground plane induced by the antenna element. While the typical bandwidth of a patch type antenna on an infinite ground plane is in the 1–3% range (depending mainly on the height above ground), more than 10% is regularly achieved in standard size terminals. Although this has been known since (at least) the mid 80’s [37], the significance of the chassis was not fully appreciated and analyzed until late 90’s-early 2000’s [38, 39], which was probably due to the introduction of internal antennas in mobile phones. The standard model for studying (single band) terminal antennas, as reported by Vainikainen et al. [37] in 2000, analyzes internal antennas as two coupled resonators-the antenna element itself, which supports a high-Q quasi-TEM transmission line wave-mode in the case of patch type antennas (e.g. PIFA), and the phone chassis, which supports a low-Q thick-wire dipole type current distribution. A circuit model of the PIFA and chassis combination, based on [40] and [41], is shown in Fig. 4.3. The PIFA element, being essentially a wide monopole bent towards the Shorting pin Top plate of PIFA of PIFA 1:N1

Antenna input LSC

Chassis N2:1

LP

CP

RP

Fig. 4.3 Equivalent circuit model of PIFA-chassis combination

LC

CC

RC

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chassis where the short-circuit functions as a gamma-match [42], is modelled as a high-Q series resonant circuit. The chassis of a typically sized (100 × 40 mm2 ) handheld terminal, being slightly less than ␭/2 long at 900 MHz and coupled to the PIFA at the edge (voltage maximum), is modelled as a low-Q parallel resonant circuit. Bandwidth maximas for the coupled resonators in Fig. 4.4 are obtained when both resonance frequencies are equal. For the chassis, this frequency is mainly determined by the terminal length and it therefore exists a relationship between chassis length and antenna bandwidth. As was previously noted, a 100 mm long chassis is resonant at 1.3 GHz (without loading from dielectrics and radiating element; it reduces to about 1.1 GHz in typical mobile phones [27]) and so some method of increasing the length is needed (e.g. by introducing a center slot as in [43]). At higher frequencies, such as GSM1800/1900 or UMTS 2.1 GHz, the chassis is too long for resonance (the second chassis resonance is at around 3 GHz) and needs to be shortened (e.g. by wavetraps as proposed in [44]). In Fig. 4.4 the simulated impedance bandwidth of a 2.1 GHz PIFA antenna as a function of chassis length is presented. As can be seen, a bandwidth minima is obtained for the typical length 100 mm at frequencies around 2 GHz. It is interesting to note that several antennas have recently been proposed that displays spectacular bandwidths at around 2 GHz (like 1700–2200 MHz) [45, 46, 47, 48, 49]. A common 30

Relative bandwidth (%)

25

20

15

10

PIFA

5

Feed and ground

LGP 0 20

40

60 80 100 Ground plane length, LGP (mm)

120

140

Fig. 4.4 Simulated relative impedance bandwidth (RL < −6dB) of a critally coupled PIFA antenna as a function of chassis length

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feature of all these antennas is a chassis length of 60–70 mm, which, intentionally or not, provides a high coupling to the chassis resonance (and hence wide bandwidth, see Fig. 4.4). Also interesting is the fact that none of the papers motivates the particular choice of chassis length. The lack of standardized lengths therefore makes the comparison between the different proposed radiators difficult, or even impossible. To obtain maximum coupling to the chassis resonator, for maximum bandwidth, the placement of the radiating element (e.g. PIFA) and feed/short pins are highly important [50]. For a patch type of antenna, the coupling is mainly through the E-field (or voltage) at the edge of the chassis, meaning that the best placement is at the short edges [40].

Reconfigurable Frequency Tuning The use of electrical switches, implemented using transistors, PIN diodes or MEMS, has been suggested to reconfigure the properties of an antenna. The main benefit for terminal applications is frequency band agility, mainly in order to enable smaller (and hence more narrow-band) antennas to be used. Typically, this is realized by changing the electrical structure of the antenna (e.g. by shorting a slot [51, 52]), by selection of different matching components/networks at the input port [53, 14] or by loading the antenna by reactive tuning components [54, 55]. As an added benefit, these antennas are likely more efficent in talk-position due to their lower coupling to the chassis. However, there are many problems with this technique that has so far limited it from widespread deployment in commercial phones:

r

r r r r

On/off switches are typically implemented using PIN diodes due to their high current capability and better linearity compared to varactor diodes. At microwave frequencies, PIN diodes are basically variable resistors. High resistance is achieved with no DC current while low resistance is achieved by a high DC current. High currents (>5 mA) are necessary to reduce the insertion loss of the switch (i.e. reduce the series resistance) and to increase the linearity of the device, thus reducing battery life time. FET based switches are preferable as they do not consume DC current, here the main limitation is instead the linearity [56]. MEMS switches are naturally potential candidates for frequency band selection [57] due to the high linearity and low loss, today however such devices are not commercially viable at reasonable prices. Switches needs control voltages, implying that the antenna can not be de-signed as an ad-hoc component independent of the RF module. The switches are, relative to the cost of a passive antenna, expensive. The complexity of design, production and verification is increased. The switch package can become very hot, which can lead to e.g. melting of the carrier plastic.

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Adaptive Impedance/Frequency Tuning As an alternative to simply switching between different frequency bands, there are also more ambitious proposals where a multitude of switches, placed between the antenna port and the transceiver input/output port, could be used for adaptive impedance tuning to compensate for e.g. hand/head effects [58, 59, 60, 61, 62]. Such modules, typically employed at low frequencies, are called ATU s (Antenna Tuning Units) or ITU s (Impedance Tuning Units). In contrast to frequency band switching, this however requires a continuous monitoring of the antenna impedance, which must be done with low losses (RF and DC) and at a low cost. Alternatively, varactors can be used to control the resonance frequency of e.g. patch elements [63, 64, 65]. The problems associated with frequency switching, e.g. nonlinearities, current consumption, complexity etc., are also valid for adaptive tuning systems. Recently, several vendors have demonstrated ATUs (or roadmaps of ATUs) based on MEMS (both DC switches together with capacitor banks, and switched capacitors), BST (highly linear varactors) and even CMOS switches. The choice of mismatch detector spans from simple directional couplers that only measures the magnitude of the reflection coefficient to multi-probe solutions that detects both phase and magnitude. Commercial products utilizing these modules are expected sometime in 2009.

Talk Position Three major effects (excluding HAC), all related, are introduced when the mobile phone is placed in talk position, i.e. pressed against the users ear and cheek:

r

r

r

The increased effective dielectric constant as experienced by the antenna compared to free space reduces the antennas resonance frequency (so called “detuning”). This will, for the typical case of a narrow band antenna designed for (and characterized in) free-space, lead to increased mismatch losses. The exact position of the hand and head are both effecting the magnitude of the detuning, and as this is different for different users, it is very hard to compensate for in the design phase. Cognitive antennas able to adaptively adjust the resonance frequency (or more generally, the input impedance) for different near-field environmental effects has been suggested in the literature [64] but has so far not been commercially implemented. As the human head is highly lossy at microwave frequencies (due to the high water content), the power radiated towards the user is nearly completely absorbed by the tissue. This creates a deep null in the radiation pattern, and more importantly, reduces the total antenna efficiency. The problem is exacerbated by the low wave impedance presented to the antenna in the direction towards the head compared to the free-space side, amplifying the ratio of power delivered to the head compared to radiated into free-space [66]. Specific Absorption Rate (SAR)

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Specific Absorption Rate (SAR) Due to an increasing awareness of the potential health effects from the near-fields and radiation of mobile phones since the 1990s, limits have been introduced by governmental regulating agencies in terms of peak power densities induced in human tissue, so called SAR values [67]. In the US, the Federal Communications Commission (FCC) specifies a maximum of 1.6 W/kg taken over a volume equivalent of 1 g tissue [68]. The corresponding limit in the EU, recommended by the International Commission on Non-Ionizing Radiation Protection (ICNIRP), is 2 W/kg taken over a volume corresponding to 10 g tissue. Recently, IEEE has adopted requirements similar to that of the EU [69]. It has been reported that spatial peak SAR values are located in areas close to the antennas current/H-field maxima [70]. Therefore, for 900 MHz terminal antennas with ∼100 mm chassis length, SAR peaks are generally obtained close to the vertical center of the chassis and at the short circuit of the antenna element (typically at the top-most short edge) [50, 71]. At 1800 MHz, peak SAR is normally obtained at the short circuit of the antenna element [72]. For small chassis lengths, i.e. close to a ␭/2 resonance at 1800 MHz, a SAR maxima is also seen at the chassis vertical center [50]. As a contrast to the common belief that SAR is related to peak magnitudes of antenna currents, a recent paper explains the SAR peaks by the boundary conditions of the quasi-static E-field of the antenna at the air-tissue interface [72]. The idea is here that the high real part of the tissues permittivity ␧r attenuates the perpendicular E-field component from the antenna and so the peak SAR value can then be found in regions with significant E-field components parallel to the tissue (which is less affected by the high ␧r ). For the simple case of a dipole antenna (or terminal chassis), the parallel component of E-field is strongest at the center of the dipole, which of course also corresponds to the H-field (or current) maxima, meaning that both theories (E- vs H-field) predicts the same peak SAR region (at least for this simple case). A recent paper [∗] reporting the simulated SAR of a terminal antenna over a wide frequency band (0.6–6 GHz) shows that the SAR peak(s) closely follows the current maxima(s) of the chassis current modes, indicating that, for whatever reason (strong E- or H-fields), SAR peaks are found close to current peaks. Finally, it should be noted that SAR peaks and distributions can be significantly affected by metallic objects, such as piercings, worn by the user [73]. In general, SAR is a greater problem for GSM900 compared to GSM1800 due to the higher maximum output power (+33 dBm at 900 MHz in 1/8 time slots, +30 dBm at 1800 MHz in 1/8 time slots). For passive measurements on antenna mockups, a 250 mW CW signal is applied by a coaxial feeding cable to the antenna at 900 MHz, and 125 mW at 1800 MHz. For active measurements, the phone is set to output maximum power. SAR is measured or calculated from the r.m.s. electric field strength inside the human tissue, the conductivity ␴ and the mass density ␳ from: S AR =

σ E2 ρ

(4.15)

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Fig. 4.5 The DASY measurement set-up by schimdt & partner for SAR measurements at Laird Technologies, Kista, Sweden

Probe Robot phantom

Typically, a Dosimetric Assessment System (DASY) is used to determine the SAR distribution inside a hollow phantom of the human body, see Fig. 4.5. The phantom is filled with a liquid with similar electrical properties as tissue, and a robot controlled probe measures the field densities in the volume of interest. For active measurements, the mobile phone is mounted in a specified talk position on the outside of the hollow phantom and is set to operate at maximum output power. The SAR values are calculated using the measured E-fields using the known ␴ and ␳ of the liquid.

Gain/Efficiency In talk position of mobile phones, the gain (or more relevant – the efficiency (or average gain)) is reduced mainly due to power absorption in the lossy tissue, and to a lesser extent due to a detuning effect from the high permittivity of the tissue in the near-field of the antenna [74]. The reduction of efficiency is naturally related to the SAR value, but is different in two respects. First, SAR is concerned with peak densities while efficiency is an average effect. Hence, an antenna with high SAR values does not necessarily have a low efficiency in talk position (although it is likely). Secondly, SAR depends strongly on the output power available from the power amplifier (through the filter/switch). Here it should be noted that SAR values are absolute while the efficiency reduction is relative. The low SAR value of a certain phone could be caused by a low output power available from the PA, or due to high losses in the antenna, or in an optimum scenario due to the radiation being directed away from the head. From the SAR value alone, it is not known which of these effects is responsible. Additionally, as the output power in many wireless systems is adaptive (e.g. in GSM), the SAR value by itself is not a sufficient measure of the total power induced into the operator during real usage, and it does also not provide any information about the reduction of communication range due

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Phantom

61

Measurement antennas

AUT

Fig. 4.6 Measurement set-up for passive characterization of the efficiency in talk position of an antenna mock-up mounted on a phantom head inside a near field measurement chamber

to the reduced efficiency. For these reasons, TCO Developments have suggested the complementary (to SAR) figure of merit “Telephone Communication Power” TCP. Together with SAR, TCP fully characterizes the emission characteristics of the mobile phone. Currently, an average TCP of 0.3 W for GSM phones (with the phone operating in maximum output power mode) for each band/mode/antenna over 4 different telephone positions is mandated. It should be noted that TCP is actually identical to the previously mentioned TRP (Total Radiated Power) which is regularly specified by mobile phone manufacturers, it is however not clear if the requirements are identical. A picture of a typical measurement set-up for efficiency measurements in talk position is shown in Fig. 4.6. As a final note, there is an non-obvious connection between efficiency reduction in talk position and impedance bandwidth [50] in free-space. This is due to the fact that large bandwidths can only be obtained by strong coupling to the low-Q chassis resonator, which has a dipole type current mode distribution and hence similar efficiency reduction. In contrast, the radiating element is often shielded by the very same chassis (e.g. for patch type of radiators) and also has a larger distance to the tissue.

Form Factors Modern hand held terminals comes in a variety of shapes, or so called form factors. Concerning mobile phones, the “bar” or “mono block” used to be the dominant type, mainly competing with the “flip” type which had a cover face for keypad protection and holding the microphone. Today, other forms such as the “swivel”, “foldable/clamshell” and “slide” are fairly common together with various other, more experimental concepts. See Fig. 4.7 for the most popular versions. The swivel etc. have become popular as they allow a large keyboard (sometimes even

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(a) Bar

(b) Clamshell

(c) Slider

(d) Swivel

(e) Flip

Fig. 4.7 Various mobile phone antennas

QWERTY-style) and a large display while preserving a small in-pocket size. From an antenna designers view-point, these additional forms presents both challenges as well as opportunities. As an example of the latter, antennas for the clamshell type, which has two separate chassis sections (like a wide dipole), can be advantageously implemented by simply feeding the two chassis sections, thus realizing extremely large bandwidths. This technique has been proposed for DVB-H reception [75, 76] and dual-band cellular applications [77]. As an example of the former, most forms except the bar can be operated in two states: open and closed. Preferably, the implemented antenna should work equally well in both states.

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Chapter 5

Design Techniques for Internal Terminal Antennas Peter Lindberg

Planar Inverted-F Antenna (PIFA) The planar inverted-F antenna (PIFA) is currently the most popular internal antenna for mobile phones. In its most basic form, the PIFA consists of a rectangular patch element, a feed pin and a short circuit pin, see Fig. 5.1. The operating principle of the antenna is easily understood by first considering a standard quarter wavelength monopole antenna over ground. At 900 MHz, the required height above ground is about 8 cm, which is too much for many applications (e.g. terminals). To reduce the total height but still maintain self-resonance, the monopole can be bent down towards ground and form a so called “inverted-L” antenna, see Fig. 5.2. As a consequence of the induced anti-phase currents in the ground plane, the radiation resistance drops significantly compared to the monopole (from ∼ 37 ⍀ to a few ohms (depending on the height), see Fig. 5.3). Hence, the inverted-L antenna is more narrow-band than the monopole, and the input impedance at resonance presents a large impedance mismatch in a typical 50 ⍀ system. To increase the real part of the input impedance, the gamma-match can be implemented by moving the feed point along the length of the antenna and with the antenna short-circuited at the previous feed location, forming a so called “inverted-F” antenna1 . The introduction of the short-circuit pin adds shunt inductance – with a magnitude set by the thickness and length of the pin, and closeness to the feed point – to the series resonance circuit provided by the antenna, thereby enabling the selection of the real part of the input impedance at the resonance frequency and hence a good match to any

P. Lindberg Laird Technologies AB, Mobile Antenna Systems, Research Department, Isafjordsgatan 5, 164 22 Kista, Sweden e-mail: [email protected] 1 Another common technique of increasing the input impedance is to “fold” the monopole [1] (not to be confused with “bending” the monopole, which is typically done to conform with space restrictions). It is however not clear how this concept can be extended to dual band applications, although such designs have been presented in the literature [2, 3] (without much explanation of the physical principle behind the dual band functionality).

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

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WAn t

LAnt

Side view

Antenna element

WGP

HAn t

Feed and short pins

Ground plane x = feed o = short circuit

LGP Top view

Fig. 5.1 Schematic layout of PIFA antenna mounted on a typical terminal ground plane

system impedance2 . The bandwidth however remains low, but can be increased by increasing the size of the radiator [4]. This is typically done by making the horizontal part of the antenna very wide, thus forming the standard planar inverted-F antenna. An example of such a transition from monopole to inverted-F including equivalent circuit schematics is shown in Fig. 5.3. The circuit values provided are derived from a 900 MHz (8 cm high) monopole bent into a 10 mm high inverted-L (and F) antenna. The series inductance in Fig. 5.3(f) corresponds to the feed pin

/4

Monopole

/4

Inverted-L antenna

/4

Inverted-F antenna

Fig. 5.2 Conceptual transition from monopole to inverted-F antenna

2 An alternative explanation of the increased input impedance is the following. Consider the current distribution along an inverted-L antenna with the feed point short-circuited to ground. At the end of the L, the current is zero and hence the input impedance at this point is very high. At the short-circuit, the current has a maximum (due to the λ/4 length) and hence the input impedance here is low. Therefore, the input impedance can simply be chosen by proper location of the feed point along the length of the L-section. This is identical to the well known method of offset-feeding slot/notch antennas to select the input impedance.)

5 Design Techniques for Internal Terminal Antennas

69 1.0 0.8

1.0 Zmonopole (0.8-1.0 GHz)

Zinverted-L

1.0

(0.8-1.0 GHz)

0.9 0.9

0.9 Zinverted-F

0.8

(0.8-1.0 GHz)

0.8 (a) Monopole

(b) Inverted-L

0.7 pF

0.8 pF

45 nH Antenna input (d) Monopole

36

37 nH

(c) Inverted-F 6 nH

2.4

Antenna input

Antenna input

(e) Inverted-L

0.8 pF

37 nH 4 nH

2.4

(f) Inverted-F

Fig. 5.3 Input impedance (a–c) and equivalent circuit model (d–f) of monopole, inverted-L and inverted-F antennas. Circuit values taken from a 900 MHz (8 cm) monopole bent into a 10 mm high inverted-L (and F) antenna

length from ground to the location of the shunt inductance (or short circuit). The size of the rectangular patch is determined by the resonant frequency approximately as c fr es = √ε 4 e f f (Want + L ant )

(5.1)

where the εe f f factor (the effective dielectric constant, similar to the case of microstrip lines or patch antennas) is included to take the effect of dielectric loading (from e.g. carrier plastics) into account. To save space, the PIFA element is typically meandered or bent, which preserves the resonant length while reducing the area requirement. As described in the previous chapter, the bandwidth of PIFA antennas in terminal applications is mainly determined by the coupling to the lowQ chassis resonance (rather than the Q of the PIFA). This coupling is maximized by placing the PIFA along the short edge of the terminal and the feed and short pins as close to the corner as possible [5]. Also, the bandwidth can be increased by removing the chassis metallization below the PIFA element (so called “ground clearance”). The antenna is however now called a “monopole” or “IFA” instead of a PIFA. More importantly, the antenna now loses its two most principal advantages that are responsible for its popularity in the first place: low SAR values [6] (and the related effect of lower sensitivity, in terms of frequency detuning and efficiency reduction, to the presence of the users head [7]) and the installation above the terminals circuitry (i.e. reusing the space within the phone). Modern requirements of bandwidth together with the latest trend of thin terminals is nevertheless forcing antenna manufacturers to implement monopole type of antennas instead of PIFAs.

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Monopole/ IFA Antenna The external monopole, typically either retractable or coiled into a helix (to save space), used to be the dominant antenna type for mobile phones until it became all but replaced by internal PIFAs a few years ago. Today, as a result of the latest trend of ultra-thin mobile phones, for which PIFAs can not provide sufficient bandwidths, the monopole is experiencing a major come-back. This time, however, the monopole is hidden within the cover of the phone and thus enjoys the same mechanical and aesthetical advantages as other internal antennas such as PIFAs. It should be noted that while the monopole can achieve significantly larger bandwidths compared to PIFAs, they suffer from two major drawbacks: high SAR values (and low efficiencies in talk-position) and the need for “ground clearance”, i.e. removal of chassis metal below3 . The SAR problem is usually solved by placing the antenna at the bottom of the phone instead of the more standard top position behind the display. The bottom side of the phone is typically reserved for cable interfaces, which then have to be relocated to the side. This also means that the interface is placed where most people grip their phones, making e.g. talking in the phone while charging the battery slightly inconvenient. The need for ground clearance means that parts of the PCB area is no longer available for circuit modules. While the monopole antenna has a higher input radiation resistance than the PIFA (and hence larger bandwidth), it is typically still not sufficient for a good match to a 50 ⍀ source. The standard matching technique is to use a shunt inductance at the feed point, either using a discrete component or having it printed as part of the antenna pattern. This shunt inductance has an identical effect as the short circuit in a PIFA (or IFA), and so most monopoles should arguably be called IFA:s instead. This is of course just a matter of semantics. Typically, a terminal antenna is called a PIFA if large parts of the radiating element is located above chasses metallization, and it is called a monopole if the chassis metal is largely removed. The more general terms “on-ground” and “off-ground” antennas are becoming increasingly popular, which probably makes more sense than the PIFA/monopole distinction. An example of a monopole antenna is shown in Fig. 5.4 together with the simulated return loss. Two galvanically connected resonators (usually called “branches”) are used to provide a dual resonance frequency response (as discussed in the section “Multiband/Broadband antenna design techniques” later in this chapter), which together with a parasitic resonator gives a bandwidth at the upper frequency band which is sufficient to cover the popular cellular bands GSM1800/1900/UMTS2100 (1710– 2170 MHz) in addition to either GSM900 or GSM850. Such antennas are usually called “quad band” (“triple band”, i.e. GSM900/1800/1900 can easily be achieved with monopoles without any parasitic element) in the mobile phone industry. One of the biggest current challenges in the terminal antenna area is the design of a passive “penta band” radiator, i.e. an antenna that simultaneously (i.e. no switches) covers

3 The notion “below” might need some explanation. Typically, terminal antennas are placed on carrier plastic modules that provides a certain height above the chassis level. This is also true for monopoles, but while PIFAs are placed on top of metallized areas, a monopole needs this metallization to be removed.

5 Design Techniques for Internal Terminal Antennas 40

71

0

4

With parasitic element

21

69

900 MHz element Parasitic 1800 MHz element element

No metal

Return loss (dB)

–3 –6 –9 –12 –15 –18 –21 0.8

(a) Layout

1.0

1.2 1.4 1.6 1.8 Frequency (GHz)

2.0

2.2

(b) Return loss

Fig. 5.4 Layout of monopole/IFA antenna (a) and simulated return loss (b) withand without a parasitic resonantor

the frequency bands GSM850/900/1800/1900/UMTS2100, and which consumes a reasonable volume. A proposed radiator type that can give that kind of bandwidth is the notch antenna, as described later.

Folded Inverted Conformal Antenna (FICA) A modification of the PIFA antenna, called FICA (Folded Inverted Conformal Antenna), has recently been proposed [8, 9] where a multi-resonance frequency response is obtained by utilizing different resonant modes within the same structure. In contrast, PIFAs (and monopoles) are typically made multi-band by including multiple branches/patches (usually one for 850/900 MHz and one for 1800/1900 MHz) each excited at its lowest λ/4 mode, with all radiators sharing the allocated antenna volume. Hence, as each resonator occupies only a fraction of the total available volume, the achievable bandwidth is also (at least in principle) smaller compared to a single band antenna. The multi-band operating principle of the FICA is instead based on having three different resonant modes that each utilize (nearly) the entire antenna volume. An example of a FICA is shown in Fig. 5.5 together with a return loss plot showing the three excited resonances. The radiator has typically a U-folded (to conform with the available space) elongated structure (mostly) symmetrical around the chassis center including a slot that is short-circuited in both ends. The multi-mode excitation can be explained by the superposition principle, as illustrated in Fig. 5.6. A voltage source is connected to the feeding pin and the other pin is short circuited to ground (providing the unbalanced feeding). The short circuit is then replaced by two series voltage sources with opposite phase, while the voltage source at the feeding pin is split into two in-phase sources. From here, it is easy to see that two possible modes exists, the common mode – with in-phase currents at the feed and short pins – and the differential mode – with anti-phase currents at the feed and short pins. This decomposition into different modes is similar to the standard analysis of folded dipole antennas.

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8

Return loss (dB)

–3

29 6

100

–6 –9 –12 Common mode Differential mode

–15 –18

Slot mode

–21 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Frequency (MHz)

(a) Layout

(b) Return loss

Fig. 5.5 Layout of FICA antenna (a) and simulated return loss (b). All dimensions in mm

Antenna

Antenna

=

+ V –

+ V/2 – + V/2 –

+ V/2 – +V/2

Unbalanced feed

Antenna

=

+ V/2 –

Antenna

+ V/2 –

+

+ V/2 –

Common mode

– +V/2

Differential mode

Fig. 5.6 Decomposition into common and differential modes of antenna with unbalanced feeding

λ /2 @ 1.48 GHz

λ/4 @ 0.82 GHz

(a) Common mode

(b) Differential mode

3λ/4 @ 1.98 GHz

(c) Slot mode

Fig. 5.7 Current distributions of the 3 resonant modes in a typical FICA antenna

5 Design Techniques for Internal Terminal Antennas

73

To provide a good match to a 50 ⍀ system, each resonance must have a current maxima at the feed terminal (corresponding to a series resonance). This excludes the excitation of the lowest order (λ/2) slot resonance due to the current minima at the center (assuming a feed point fairly close to the center), providing a very high input impedance. Instead, the slot is excited at the frequency corresponding to a 2x(3λ/4) total slot length. For most applications, it is desirable to bring the differential and slot mode close together at around 1800–1900 MHz which will give a wide upper frequency band. This is however not trivial in practice as all three resonances are strongly coupled and can not be tuned separately (unlike in e.g. PIFA design). Nevertheless, the concept has been utilized in several mobile phones (mainly from Motorola) which indicates that good performance can be obtained in practice despite the high design complexity. The current distributions corresponding to the common mode, differential mode and slot mode in the antenna shown in Fig. 5.5 is shown in Fig. 5.7.

Notch Antenna The notch antenna, or quarter wave slot (or just slot), has been proposed in the scientific literature for various terminal applications. The main benefits of the slots are the wide bandwidths that can be achieved (due to the high coupling to the chassis – if correctly placed), and the suitability for low-profile terminals as the slot is located in the chassis and do therefore not need any volume. However, the slot/notch has probably never been used in a real commercial terminal, most likely because of one or two of the following serious disadvantages: excessive SAR values (even higher than monopoles) and more complicated circuit floor planning. To maximize the coupling to the chassis (in order to increase the bandwidth), the slot should preferably be placed in the current/H-field maxima of the chassis, which is found in the center (below 3 GHz). For the slot to function properly, no metal can be placed above/below the slot, effectively cutting the chassis in two equal size parts (with a ground-to-ground connection at one side). Any signal routing between these two parts must take place over/at the ground-to-ground connection to minimize the coupling to the slot [10], which is a serious restriction compared to the standard case where the full PCB is available for signal traces4 . Nevertheless, for some terminal form factors the slot could perhaps be a suitable candidate antenna. For instance, consider the phone (bar type) shown in Fig. 5.8. Here, the phone is already, from a metallization point of view, essentially divided into two parts with the keypad and display in each section. The implementation of such an antenna is exemplified in Fig. 5.9, where the slot is fed by a microstrip stub. This type of feeding has the benefit of adding additional degrees of freedom to

4 The main PCB in most terminals can have 10–15 metal layers to facilitate the necessary interconnections, so the removal of a large transversal section of PCB area therefore represents a large layout inconvenience.

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PIFA

Slot Slot

(a) Back side

(b) Front side

Fig. 5.8 Pictures of front and backside of typical bar-type mobile phone without plastic cover. Suitable placement of notch antenna indicated

the design which can be used for impedance matching. In particular, the microstrip stub (open at the end) is a particularly suitable feeding technique in this case as it provides capacitance below its (λ/4) resonant frequency and inductance above it, while the slot is inductive below and capacitive above. Hence, the slot-stub combination is equal to that of a coupled series and parallel resonant circuit, providing “double tuning” (see Chapter 4 ) that approximately doubles the bandwidth of the radiator (slot) alone. Other types of feeding includes direct connection of the two feeding terminals across the gap (where the real part of the input impedance (e.g. 50 ⍀) is determined by the distance to the short circuit), or using a T-match (i.e. feeding a strip inside the slot, which is a similar (or dual) technique to folding a dipole [11] – i.e. reducing the input impedance (dual to increasing the impedance of the dipole)). For measurement purpose, the feeding microstrip line is here connected to a coaxial connector at the symmetry line of the structure (virtual ground), thereby λg/2 @1000 MHz Ground plane (bottomlayer) 0.09 λg @ 900 MHz

0.13 λg @ 900 MHz

40

0.8 mm FR-4 PCB: εr = 4.44 tan δ = 0.02

Fig. 5.9 Layout of slot antenna and microstrip feed indicating electrical lengths of microstrip stub, notch and chassis

5 Design Techniques for Internal Terminal Antennas

75

0.8 3.0

S11 (0.8–3 GHz)

S11 (0.8–3 GHz) 0.96

1.67

3.0

0.8

(a) Slot only

(b) Chasis only

Fig. 5.10 Simulated input impedances of slot and chassis

avoiding any induced currents on the outside of the cable shield. In a real application, the microstrip would of course be connected to the transceiver module located somewhere on the PCB. The total antenna structure here consists of three parts/resonators: the notch, the stub and the chassis resonator. The notch (2 × 35 mm2 ), analyzed by making the ground plane (100 × 40 mm2 ) large, is resonant at approximately 1.7 GHz and provides inductive reactance at all other frequencies, see Fig. 5.10a. The chassis, simulated by feeding the two wide dipole halves separated by the notch with a centralized gap source and chamfering each dipole leg to avoid the effects of the slot providing extra shunt capacitance at the feed point (similar to [12]), is resonant at about 1 GHz, see Fig. 5.10b. Corresponding return loss plots are shown in Fig. 5.11. The combination of the two resonators, analyzed using a gap source at the microstrip-slot crossing, is shown in Fig. 5.12 where, as expected, the dual resonant response and the overall inductive behavior is shown. Clearly, the low radiation resistance at 0.8 GHz and 3.0 GHz of the slot is limiting both lower and upper frequency limits of the antenna. 0

Return loss (dB)

Return loss (dB)

0 –5 –10 –15 –20

–5 –10 –15 –20

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Frequency (GHz) (a) Slot only

Fig. 5.11 Simulated return loss of slot and chassis

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Frequency (GHz) (b) Chassis only

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P. Lindberg 0

0.8 3.0

–5 S11 (0.8–3 GHz)

–10 Chassis

Slot

–15 –20 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 (a) Slot + chassis

(b) Slot + chassis

Fig. 5.12 Simulated input impedance and return loss of the combination between slot and chassis

Later, a modified version suitable for wider terminals, featuring a longer slot, is presented to reduce the lower cutoff frequency. Also, it can be seen in Fig. 5.12a that series capacitance is needed at the feed point, which is provided by a microstrip stub below resonance (i.e. below 2.22 GHz, see Fig. 5.13a). This stub further provides an extra resonance. However, as all three resonators are strongly coupled and overlaps in frequency, it is difficult to attribute each resonance (e.g. in a return loss plot) to a specific resonator. The input impedance of the complete antenna, with reference plane at the microstrip-slot crossing, is shown in Fig. 5.13b. The measured antenna efficiency (including mismatch loss) of a prototype antenna, see Fig. 5.14a, is in the 70–95% range over the entire band 0.9–2.7 GHz, with the smaller values being caused by mismatch losses. Excellent agreement between simulations and measurements of return loss (Fig. 5.14b) is achieved due to the negligible influence of the

3.0

3.0

2.22

S11 (0.8–3 GHz)

0.8

S11 (0.8–3 GHz)

0.8

(a) Microstrip stub

(b) Complete antenna

Fig. 5.13 Simulated input impedance of the series microstrip stub and the complete slot antenna

5 Design Techniques for Internal Terminal Antennas

77

100

80

0

70

Return loss (dB)

Antenna efficiency (%)

90

60 50 40 30 20 10 0.8

–10

–20 Measured Simulated

–30 1

1.2 1.4 1.6 1.8

2

0.0

2.2 2.4 2.6 2.8

0.5

1.0

1.5

2.0

Frequency (GHz)

Frequency (GHz)

(a) Efficiency

(b) Return Loss

2.5

3.0

Fig. 5.14 Measured efficiency and return loss slot antenna prototype

cable placement in the virtual ground point at the chassis center. The layout of the prototype antenna is shown in Fig. 5.15. The slot can be relocated away from the chassis center to e.g. simplify signal routing. However, the coupling to the chassis diminishes with distance away from the center, and hence also the achievable bandwidth. For the case of placing the slot at one of the short edges, the antenna is actually identical to that of a monopole/IFA (and as such gives fairly low bandwidths). One of the main concerns of non-PIFA type antennas is the SAR value and antenna efficiency in talk position, in particular 2 49

49

Ground plane (bottom layer)

Microstrip (to player) 8.6

35

Notch 0.8 mm FR-4 PCB: εr = 4.44 tan δ = 0.02

1.6

RF transceiver

13.6

2 5 Measurement cable

11.2

Fig. 5.15 Layout of slot antenna. All measures in mm

40

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for low frequency cellular bands (e.g. GSM900). As was shown in [13], the SAR value increases with higher coupling to the chassis resonator for internal (PIFA type of) radiators. Hence, it is expected that radiators such as the slot previously presented that relies solely on the chassis for radiation should have excessive SAR values thus rendering it commercially unacceptable. SAR was measured on a prototype antenna using a Schmid & Partner DASY4 dosimetric assessment system, similar to the set-up described in [13]. In accordance with specifications, a human hand was not included in the measurement. To provide a representative distance to the phantom head, thin dielectric spacers of height 5 mm (simulating a typical terminal cover) were mounted on the back side of the chassis. The antenna was measured in touch position on the left side of the phantom at 890 MHz where a very low return loss was obtained. A 250 mW CW signal was applied to the antenna, corresponding to +33 dBm in 1/8 time-slots. The SAR distribution is shown graphically in Fig. 5.16 with a peak of 4.1 mW/g averaged over 1 g tissue, and a peak of 2.68 mW/g averaged over 10 g tissue. The Federal Communications Commission (FCC) limit for public exposure from cellular telephones is a SAR level of 1.6 mW/g, which is applicable in the US. In EU (and Japan, Brazil and New Zealand) the recommended SAR limit by the International Commission on Non-Ionizing Radiation Protection (ICNIRP) is 2 mW/g averaged over 10 g of tissue. This limit has also recently been adopted by IEEE [14]. While the prototype slot antennas peak value of 2.68 mW/g over 10 g is above all recommended limits, the value for an implemented phone antenna could be significantly lower due to additional losses (from the cover and other nearby objects) and also due to a less than +33 dBm output power available from the power amplifier (which is fairly common in reality). The proposed configuration, using a naked FR-4 PCB, has a lower cut-off frequency of 915 MHz, making the antenna unable to support GSM850 (824–894 MHz). While the cut-off frequency is expected to be lowered by dielectric dB

0.00

–2.64

–5.28

–7.92

–10.6

–13.2

Fig. 5.16 Measured SAR distribution of slot antenna

5 Design Techniques for Internal Terminal Antennas

79

2 49

49

Ground plane (bottom layer) Microstrip (top layer)

13.6

40 45 7.0

Slot antenna

0.8 mm FR-4 PCB: εr = 4.44 tan δ = 0.02

1.6

2 5 11.2

Fig. 5.17 Layout of modified slot antenna for wide chassis applications. All dimensions in mm

loading (from cover and other plastic parts) when implemented in a real terminal, it is anyway interesting to examine if/how the structure can be modified to enable also GSM850 coverage. As was concluded previously in this section, the notch length is mainly responsible for the bandwidth of the antenna. By increasing this length and the total width of the chassis by 5 mm, it is possible to significantly decrease the lower frequency limit. The layout of the modified antenna is shown in Fig. 5.17 and the simulated return loss is provided in Fig. 5.18. As can be seen, the bandwidth of this version is (RL < −6 dB) 810–2790 MHz. 0

Return loss (dB)

–5

–10

–15

BW –20 0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Frequency (GHz) Fig. 5.18 Simulated return loss of slot antenna with 40 mm long slot in 45 mm wide chassis

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Miniaturization Techniques Terminal antenna elements are seldom implemented in full size, instead some means of miniaturization is nearly always utilized to reduce the space/volume consumption. Since a metallized chassis is available, all terminal antennas uses this as counterpoise/ground plane making the necessary resonant length a quarter wavelength. At 1 GHz, a quarter wavelength is 7.5 cm. As typical terminals have a total thickness of about 1–2 cm, the antenna element must be bent down towards ground and having most of its length in parallel with the ground. The section parallel to ground does not contribute much to the total radiated power due to induced out-of-phase currents in the chassis forming a transmission line type of mode, so the function of this antenna is that of a short vertical monopole top loaded with an open-ended transmission line stub. The short radiating section is reflected in the low radiation resistance of such antennas, necessitating some means of rasing the total input resistance (like feed-point selection) as discussed in the previous section. It should be noted that even though the radiation resistance of such loaded antennas are low compared to full-size versions, it is still higher than that of an unloaded antenna due to the increased current amplitude in the vertical (radiating) section. Miniaturization inevitably reduces the impedance bandwidth of the antenna (and also the efficiency due to the reduction of radiation resistance) since the total volume goes down, see Chapter 4. However, as the low-Q chassis is responsible for most of the radiated power at lower frequencies, the reduction of bandwidth is not as severe as one might first expect, as will be shown by example later. There are basically four different ways of making an antenna smaller without increasing the resonant frequency, which are all basically just nuances of inductive and capacitive loading:

r r r r

Material loading – placing high permittivity (εr ) and/or permeability (␮r ) material close to the antenna. Component loading – using series chip inductors to increase the electrical length of the antenna (or capacitors in the case of small loop antennas). Here, the component is considered as a part of the antenna (for semantic reasons – otherwise the antenna is no longer “self-resonant”). Inductive loading by geometry (meandering). Capacitive loading by geometry (top hat).

Material loading is always present, intentionally or not, in the form of carrier and cover plastics. Ferrites (high ␮r material) are in practice never used due to excessive losses at frequencies above 100 MHz and for cost and weight reasons. The main techniques that are often used in practice are inductive and capacitive loading by geometry, in particular inductive loading. These concepts are illustrated by example in the following. Consider a full-size quarter-wave PIFA at 900 MHz, as seen in Fig. 5.19a. The required length needed for resonance can be reduced from 29 mm to 15 mm by folding the radiator, effectively introducing distributed inductance

5 Design Techniques for Internal Terminal Antennas 40

81 40

8

8

15 PIFA 100

29

x = feed o = shortcircuit

100

Chassis

(a) Full size

(b) Inductive loading

40

8

30

10

8

4 15

15 2 100 100

(c) Capacitive loading 1

(d) Capacitive loading 2

Fig. 5.19 Layout of PIFA antenna, full-size (a), inductively loaded (b), capacitively loading at bottom edge (c) and capacitively loading at top edge (d). All measures in mm

along the length of the antenna, obtaining e.g. the structure in Fig. 5.19b. There are of course infinitely many ways of bending the antenna together, the obtained performance of the stand-alone radiating element (i.e. not considering the effect of the coupled chassis resonance) is however not significantly dependent on the exact shape of the radiator [15, 16, 17] and there are thus no optimal solutions presented in the literature. Some recommendations on suitable geometries to optimize the current vector alignment (for maximum bandwidth and efficiency) in a given allocated volume is provided in [18], which however does not consider the case of terminal antennas (where the coupling to the chassis resonator, rather than the Q of the radiator, dominates the antenna bandwidth). It should be noted that various radiator configurations, while showing similar properties when mounted on a large ground plane (in accordance with [16, 17]), can provide fairly different characteristics in a terminal application due to the differences in coupling to the chassis. Hence, the exact shape of the radiator must be optimized for each particular terminal, which is usually done by trial and error in combination with previous experience. In general, the open ended section of the PIFA should be located at the short edge of the chassis to maximize the antenna-chassis E-field

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coupling5 , as in Fig. 5.19b. Figure 5.19c presents a typical example of capacitive loading, here in the form of the bottom edge being folded closer to ground. This is similar to the well-known technique of adding a “top hat” to short monopoles. For PIFAs, the loading can be implemented at many different places, e.g. along the four edges of the rectangle. The performance in this case is however fairly strongly dependent on the location, as illustrated by the two versions in Fig. 5.19c and d. In d, the capacitive loading takes place at the upper long-edge of the PIFA, which is the location of the E-field maxima of the chassis resonator. Hence, an increased coupling to the chassis wave-mode can be expected by this placement, which somewhat counters the bandwidth reduction due to the smaller volume of the miniaturized antenna. The results are shown in Fig. 5.20 and summarized in Table 5.1, where one additional inductively loaded antenna (“Inductive 2”) has been added (not shown in Fig. 5.19) which differs compared to “Inductive 1” (Fig. 5.19b) in that the antenna has been mirrored along the long-axis and now has the open end below the top edge of the chassis. Clearly, both inductive and capacitive loading can be utilized without significant reduction of impedance bandwidth, provided that the general rules outlined above is followed. It has however been argued that inductive loading is preferable compared to capacitive loading from bandwidth considerations [19, 20], the reasoning being as follows: the PIFA can be thought of as a quarter-wave stub with a (fairly low) radiation conductance at the open end6 . If the √ stub has a high characteristic impedance Z 0 = L/C, the radiation conductance 0

Return loss (dB)

–5

–10 Full-size Meander Captop Capbottom

–15

–20 800

820

840

860

880 900 920 Frequency (MHz)

940

960

980

1000

Fig. 5.20 Return loss of full-size and miniaturized PIFA antennas 5

The chassis has a (lowest mode) dipole type of field distribution, with E-field maximas at the short edges. For any antenna element located near the short-edge of the chassis, the antenna-chassis coupling is mainly through electric fields and is thus capacitive. To maximize this coupling, the E-field maxima of the radiating element should be located at the chassis edge. 6 This is similar to the standard model of patch antennas where radiation conductances are placed at each radiating edge.

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Table 5.1 Bandwidth reduction due to different miniaturization schemes of PIFA antennas Loading type

f0

Δf

BW

No Inductive 1 Inductive 2 Capacitive 1 Capacitive 2

902 MHz 902 MHz 900 MHz 905 MHz 902 MHz

99 MHz 78 MHz 78 MHz 70 MHz 98 MHz

11% 8.7% 9.6% 7.7% 10.9%

is transformed along the length of the stub into a “high” radiation resistance at the feed point. For example, assuming a radiation conductance of 1 ms, Z 0 = 20 ⍀ gives Rin = 0.4 ⍀, Z 0 = 50 ⍀ gives Rin = 2.5 ⍀ and Z 0 = 100 ⍀ gives Rin = 10 ⍀. The bandwidth will thus be wider for an inductively loaded antenna compared to a capacitively loaded antenna. Note that the bandwidth reduction due to a reduced antenna volume, see the previous chapter, always dominate over the bandwidth gain from meandering, the net effect is thus always a reduction of achievable bandwidth. Care must also be taken to prevent too narrow stub widths to be used (to achieve a high Z 0 ) which will increase the resistive losses. Using a similar reasoning, it can be shown that material loading using ferrites are advantageous compared to dielectrics. Unfortunately, today’s available ferrites are excessively lossy above 100 MHz making them unpractical for e.g. cellular antennas. Material loading is therefore always done using dielectrics. Probably the most common type of (intentionally) dielectrically loaded antennas are ceramic patch antennas for GPS. Here, the system bandwidth is very narrow (1575.42 MHz ±10.23 MHz) so the increased Q-value from the miniaturization is actually beneficial from a system point of view as it adds some extra filtering. Also, as the tolerance of high- material is typically really low the resonant frequency is also very stable (which is of course a requirement for small-band antennas). And finally, due to the confined near-field of the antenna inside the dielectric material, the antenna is also fairly insensitive to frequency de-tuning by nearby objects.

Multi-band/Broadband Antenna Design Techniques Besides using matching networks, which is thoroughly addressed in other sections of this book, there are several known techniques to increase the impedance bandwidth of terminal antennas, or similarly to introduce additional frequency bands for multi-band systems (such as GSM 900/1800 or WLAN 2.4/5.8 GHz). This section presents the most common and practically realizable methods used in the industry, illustrated using the standard PIFA. Conversion to other similar antenna types, such as monopoles, is fairly straight-forward.

Multiple Radiating Resonators Multiple resonances can be obtained by the simple connection of several radiators of different size/length. For instance, by galvanically connecting 2 dipoles of

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different lengths at the feed points, resonances are provided at the corresponding λ/2 frequencies of each dipole7 . The coupling between the two resonators will determine whether two disjunct resonances will appear (with a high VSWR between the resonant frequencies), or if one double-resonance (i.e. wide-band instead of dual band) is provided. High coupling can be achieved by e.g. galvanically coupling the two resonators together and/or by orienting the two resonators to have identical polarization. Loose coupling is achieved by e.g. EM coupling and/or perpendicular orientation (with the obvious disadvantage, for many applications, that the different polarization is provided at the different resonant frequencies). As an example, consider the case of two dipoles of lengths 150 mm and 120 mm with identical orientation. First the two radiators are galvanically connected (i.e. strongly coupled) at the feed points with a 2 mm gap. The resulting input impedance and return loss is shown in Fig. 5.21. As expected, two resonant frequencies are visible with a high VSWR in between. There is however a large difference in Q-value of the two resonances (Q low = 8.3 and Q high = 22.3). A third, slightly shorter, resonator would have an even higher Q-value due to the increased stored energy between the radiators, thus making the method unpractical for most applications. By decreasing the coupling between the two dipoles, a wide-band response can be obtained instead of the multi-band response given by the strong coupling. Fig. 5.22 illustrates the bandwidth of the two dipoles when they are no longer galvanically connected, i.e. the longer element is driven by the source and the shorter is coupled through the EM interaction. The separation is still 2 mm. Now, due to the weaker coupling, a loop can be seen in the impedance locus in the Smith chart, which extends the bandwidth of the antenna (as opposed to introducing a second band). The radius of the loop can be easily adjusted by either the separation between the two elements (larger separation −> smaller locus), or by changing the polarization between the two elements. 0 S11 (0.1 – 2.0 GHz)

1010 1260

Return loss (dB)

–5

–10

–15

1210 880 –20 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Frequency (GHz)

(a) Galvanic coupling

(b) Galvanic coupling

Fig. 5.21 Impedance and return loss of galvanically coupled 2-element dipole antenna

7

If both dipoles are of similar length, the frequency response will instead be that of a single (thick) dipole.

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85

0

–5

Return loss (dB)

1190

–10

–15

897 –20 S11 (0.1–2.0 GHz)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Frequency (GHz)

(a) EM coupling

(b) EM coupling

Fig. 5.22 Impedance and return loss of EM coupled 2-element dipole antenna

Completely orthogonal elements, but galvanically coupled (some means of coupling must be there), have usually higher bandwidth than EM coupled elements with similar orientation due to the minimization of stored fields between the two elements. Optimally, the loop should be centered around the center point in the Smith chart and be of a radius that corresponds to the VSWR requirements (typically less than 3). An example of such a solution is shown in Fig. 5.23. Note that this type of dual resonant solution can be obtained either through the coupling of two radiating antenna elements, or by a single-resonant antenna and a matching network8 . Another common technique is to utilize the fact that many systems have a frequency spacing of approximately a factor 2 (like GSM900/1800, WLAN 2.4/5.8) by connecting two radiators resonating at each frequency. 0 –1 –2 Return loss (dB)

VSWR < 3

–3 –4 –5 –6 –7 –8 –9 –10 0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Frequency (GHz)

(a) Impedance

(b) Return loss

Fig. 5.23 Impedance and return loss of near-optimally coupled (for S < 3 requirement) dual resonant antenna 8

Utilizing a matching network however generally leads to a more narrow band solution compared to coupled antenna resonators, due to the higher Q-values of the matching components compared to the extra radiating element. A matching network on the other hand consumes much less space and has the potential of much higher bandwidth enhancement.

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For such systems, when one radiator exhibits a low impedance series (λ/4) resonance, the other radiator has a high impedance parallel (λ/2) resonance. Hence, the RF currents will mainly flow into the (low impedance) λ/4 resonator, thus decoupling the two radiators and providing independently tunable dual band resonances. For terminal antennas, connecting two PIFA elements of different size together at the feed point was first suggested in 1997 [21], a technique that is now utilized in the cellular antenna of nearly all mobile phones. A typical dual band PIFA for 900/1800 MHz is shown in Fig. 5.24 together with the corresponding return loss plot. A small element (patch) is used for the 1800 MHz resonance and a larger, bent element is used for the 900 MHz resonance. The two resonators are typically fairly decoupled and can be adjusted individually to adjust resonant frequencies etc. The distance between short circuit and feed point mainly effects the lower resonance (due to the lower shunt reactance at 900 MHz compared to 1800 MHz) and is thus selected to provide a good impedance match there. As the electrical height of the PIFA is much bigger at 1800 MHz, the input radiation resistance is also larger meaning that a larger shunt reactance should be used (the x2 reactance automatically obtained by the x2 frequency spacing is often a suitable value). In practice, a good impedance matches at both frequency bands can always be obtained by proper selection of feed/short circuit points. The double patch type of dual band PIFA antennas are nowadays usually combined with a parasitic radiator to increase the bandwidth at the upper frequency band, typically to support GSM1800/1900 (and in some cases even UMTS 2.1 GHz) for global roaming. The technique of parasitic radiators is explained in the next section. An interesting alternative to having a different branch/patch for each frequency band is to excite different resonant modes within the same structure, which is typically done by introducing λ/2 slots in the radiating structure. This type of multiband implementation is exemplified and explained by the FICA antenna presented previously in this chapter.

0 40

8

100

900 MHz element

1800 MHz element

Return loss (dB)

–5 21

–10

–15

–20 0.6

(a) Layout

0.8

1.0

1.2 1.4 Frequency (GHz)

1.6

1.8

2.0

(b) Return loss

Fig. 5.24 Layout of dual band 900/1800 MHz PIFA antenna (a) and simulated return loss (b). All dimensions in mm

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Parasitic Resonators

Short Feed

Return loss (dB)

A parasitic resonator/radiator (or “parasite”) is recognized by it not being galvanically connected to the main antenna element (other than through perhaps a common or virtual ground). Instead, the element is EM coupled to antenna, which is done for different reasons in different applications. Perhaps the most common use of parasitic radiators are as directors and reflectors in Yagi-Uda antennas (for TV reception) to increase the directivity. For terminal applications, they are used exclusively for bandwidth enhancement9 , which is obtained due to the loose coupling to the main radiator (as explained in the section on multiple radiators). For PIFA antennas, the idea was first presented in 1997 [22] (although it had been known to work for patch antennas for a very long time, so the extension to PIFA:s was fairly obvious). The parasitic element is typically λ/4 long and grounded to the chassis at one end. The coupling to the main radiator is controlled by the physical distance between them (assuming similar polarization), which is an easily controlled parameter by the designer. An example of a parasitic radiator implementation is shown in Fig. 5.25, where the parasitic radiator has been applied to the previously discussed dual band PIFA (shown in Fig. 5.24) to obtain a triple band response. Compared to the dual band version (shown using dashed lines), the impedance bandwidth at the upper frequency band is approximately doubled due to the extra coupled resonance. Typically, parasitic radiators decreases the radiation efficiency of the antenna due to the increased field densities between the antenna and parasite and induced currents in the parasite. Also, depending on the exact implementation, the bandwidth of the lower frequency

Parasitic radiator Chassis

PIFA

m

0m

10

40 mm

(a) Layout

0 –2 –4 –6 –8 –10 –12 –14 –16 –18 –20 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 Frequency (MHz)

(b) Return loss

Fig. 5.25 Layout of triple band 900/1800/1900 MHz PIFA antenna with parasitic resonator (a) and simulated return loss (b) (with the dual band antenna displayed using dashed lines)

9 Other uses have been suggested, such as placing directors above the terminal antenna to focus the radiation pattern away from the operators head and thus reduce the SAR value (and increase efficiency). However, such techniques have probably never been implemented in commercial products.

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band can be slightly reduced (from the nearby grounded metal effectively reducing the occupied antenna volume). Parasitic radiators are exclusively used for the upper frequency band, for some reason the technique has probably never been implemented for the 850/900 MHz bands (at least not in any commercial terminal).

RF Chokes A patch element or monopole branch can electrically be cut to a shorter length by introducing a “wavetrap” or “RF choke”, i.e. a structure/component that provides a high series impedance (ideally infinite). This can be used to implement a dual frequency response by inserting the choke at a λ/4 distance from the feed. The section between the feed and choke then gives a λ/4 resonance at an upper frequency band, and the section from the feed to the open end of the patch/branch gives a λ/4 (extended by the choke) resonance at a lower frequency. This technique is fairly common in TV antennas to support both VHF and UHF frequency bands. It is also in principle applicable to terminal antennas [23] but has probably never been implemented in a commercial mobile phone, possibly as the technique does not provide any obvious advantages compared to e.g. using multi-branches. The choke is commonly implemented as a parallel LC circuit (using discrete or printed components) or as a λ/4 short-circuited transmission line stub. An example using discrete components is shown in Fig. 5.26. By using a high value L the bandwidth of the choke is higher, but the extension of the length of the longer resonator is also higher (thus reducing the bandwidth). Wavetraps have also been used to electrically reduce the length of a terminal chassis to increase the coupling to the antenna element (for bandwidth enhancement), as explained in the next section.

0

8

40 /4 @ 900

/4 @ 1800

L = 6.5 nH C = 1.3 pF 85

Return loss (dB)

15

–5

–10

–15 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Frequency (GHz)

(a) Layout

(b) Return loss

Fig. 5.26 Layout of dual band 900/1800 MHz PIFA antenna using RF choke (a) and simulated return loss (b) All dimensions in mm.

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Increased Coupling to the Chassis The terminal chassis is, compared to most practical radiators, a very low-Q resonator and the degree of radiator-chassis coupling thus determines the achievable bandwidth of a terminal antenna. This coupling is maximized if the chassis and radiator have equal resonant frequencies. While the resonant frequency of the radiator is easily selected by the antenna engineer, the chassis resonance is determined by the physical length of the chassis (which in turn is determined by the length of the terminal), and by the total dielectric loading from battery, display, plastic cover etc. At 900 MHz (and down), the chassis is typically too short and at 1800 MHz (and up) the chassis is too long. In principle, it is possible to increase the electrical length of the chassis by slot loading [5, 24] or capacitive top loading [25], with the obvious disadvantage of increasing the signal routing complexity and circuit floor planning. An example of a slot loaded chassis at 900 MHz is shown in Fig. 5.27 with the corresponding return loss. As can be seen, the achieved bandwidth can be made significantly larger. Electrically decreasing the length can however be done without excessive chassis alterations by introducing wavetraps [12] along the long edges, as shown in Fig. 5.28 for a 2100 MHz PIFA. The idea is here to introduce high impedance interfaces at a selected location to electrically shorten the chassis, typically to obtain a λ/2 chassis resonance. Since the current distribution on the chassis is mainly concentrated to the edges, it is sufficient to introduce one wavetrap “finger” at each long edge to obtain an adequate current stopping function. The length of these fingers are selected as λ/4 at the design frequency. Each finger will form an assymetrical transmission line stub together with the chassis, which will transform the short circuit at the bottom side into an circuit at the upper side. Hence, the input impedance Z in, WT seen be the currents will be large thus realizing the intended chassis shortening.

40

0 15

Return loss (GHz)

Without slot

49

35

–5 With slot –10

–15

2

–20 0.7

49

(a) Layout

0.8

0.9 1.0 Frequency (GHz)

1.1

1.2

(b) Return loss

Fig. 5.27 Layout of 900 MHz PIFA antenna with center chassis slot (a) and simulated return loss (b)

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Without wavetraps –5

Ground plane

Zin,WT /4 @ 2.1 GHz

0

Return loss (GHz)

/2 @ 2.1 GHz

Antenna element

–10 With wavetraps –15 –20 –25

x = feed o = short circuit

–30 1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

Frequency (GHz)

(a) Layout

(b) Return loss

Fig. 5.28 Layout of 2100 MHz terminal antenna with wavetraps applied (a) and simulated return loss (b)

Since the bandwidth of the wavetraps are in practice limited to about 10 percent, the increase in bandwidth is however not as large as if the chassis is actually cut at the corresponding location. The increased coupling to the chassis is clearly indicated in the return loss plots, both for the length increase and decrease, by the extra resonance (one resonance from the chassis and one from the radiator (e.g. PIFA)).

Referencess 1. S. Hayashida, T. Tanaka, H. Morishita, Y. Koyanagi, and K. Fujimoto, “Built-in folded monopole antenna for handsets,” Electronics Letters, vol. 40, pp. 1514–1516, Nov. 2004. 2. E. Lee, P. S. Hall, and P. Gardner, “Dual band folded monopole/loop antenna for terrestrial communication system,” Electronics Letters, vol. 36, pp. 1990–1991, Nov. 2000. 3. Y.-D. Lin and P.-L. Chi, “Tapered bent folded monopole for dual-band wireless local area network (WLAN) systems,” Antennas and Wireless Propagation Letters, vol. 4, pp. 355–357, 2005. 4. K. Fujimoto, A. Henderson, K. Hirasawa, and J. R. James, Small Antennas. Wiley, 1987. 5. P. Vainikainen, J. Ollikainen, O. Kivek¨as, and I. Kelander, “Resonator-based analysis of the combination of mobile handset antenna and chassis,” IEEE Transactions on Antennas and Propagation, vol. 50, no. 10, pp. 1433–1444, Oct. 2002. 6. G. Lazzi, J. Johnson, S. S. Pattnaik, and O. P. Gandhi, “Experimental study on compact, highgain, low SAR single- and dual-band patch antenna for cellular telephones,” in Proc. Antennas and Propagation Society International Symposium, vol. 1, June 1998, pp. 130–133. 7. K. H. Chan, K. M. Chow, L. C. Fung, and S. W. Leung, “SAR of internal antenna in mobilephone applications,” Microwave and Optical Technology Letters, vol. 45, no. 4, pp. 286–290, May 2005. 8. C. Di Nallo and A. Faraone, “Multiband internal antenna for mobile phones,” Electronics Letters, vol. 41, no. 9, pp. 514–515, Apr. 2005. 9. C. Di Nallo and A. Faraone, “The folded inverted conformal antenna (FICA) for multi-band cellular phones,” in Antennas and Propagation Society International Symposium, 2005 IEEE, vol. 4, July 2005, pp. 52–55.

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¨ 10. P. Lindberg, E. Ojefors, and A. Rydberg, “Wideband slot antenna for lowprofile hand-held terminal applications,” in Microwave Conference, 2006. 36th European, Manchester, UK, Sept. 2006, pp. 1698–1701. 11. C. A. Balanis, Antenna Theory, 2nd ed. Wiley, 1997. ¨ 12. P. Lindberg and E. Ojefors, “A bandwidth enhancement technique for mobile handset antennas using wavetraps,” IEEE Transactions on Antennas and Propagation, vol. 54, no. 8, pp. 2226–2233, Aug. 2006. 13. O. Kivek¨as, J. Ollikainen, T. Lehtiniemi, and P. Vainikainen, “Bandwidth, SAR, and efficiency of internal mobile phone antennas,” IEEE Transactions on Electromagnetic Compatibility, vol. 46, no. 1, pp. 71–86, Feb. 2004. 14. J. C. Lin, “Update of IEEE radio frequency exposure guidelines,” IEEE Microwave Magazine, vol. 7, no. 2, pp. 24–28, Apr. 2006. 15. S. R. Best, “A discussion on the significance of geometry in determining the resonant behavior of fractal and other non-euclidean wire antennas,” IEEE Antennas and Propagation Magazine, vol. 45, no. 3, pp. 9–28, June 2003. 16. S. R. Best, “The performance properties of electrically small resonant multiplearm folded wire antennas,” IEEE Antennas and Propagation Magazine, vol. 47, no. 4, pp. 13–27, Aug. 2005. 17. S. R. Best, “A discussion on the quality factor of impedance matched electrically small wire antennas,” IEEE Transactions on Antennas and Propagation, vol. 53, no. 1, pp. 502–508, Jan. 2005. 18. S. R. Best and J. D. Morrow, “On the significance of current vector alignment in establishing the resonant frequency of small space-filling wire antennas,” Antennas and Wireless Propagation Letters, vol. 2, no. 1, pp. 201–204, 2003. 19. K. Fujimoto and J. R. James, Mobile Antenna Systems Handbook, 2nd ed. Artech House, 2001. 20. R. C. Hansen, Electrically Small, Superdirective, and Superconducting Antennas. Wiley, 2006. 21. Z. Liu, P. Hall, and D. Wake, “Dual-frequency planar inverted-F antenna,” IEEE Transactions on Antennas and Propagation, vol. 45, no. 10, pp. 1451–1458, Oct. 1997. 22. K. Virga and Y. Rahmat-Samii, “Low-profile enhanced-bandwidth PIFA antennas for wireless communications packaging,” IEEE Transactions on Microwave Theory and Techniques, vol. 45, no. 10, pp. 1879–1888, Oct. 1997. 23. G. K. H. Lui and R. D. Murch, “Compact dual-frequency PIFA designs using LC resonators,” IEEE Transactions on Antennas and Propagation, vol. 49, no. 7, pp. 1016–1019, July 2001. 24. R. Hossa, A. Byndas, and M. Bialkowski, “Improvement of compact terminal antenna performance by incorporating open-end slots in ground plane,” Microwave and Optical Technology Letters, vol. 14, no. 6, pp. 283–285, June 2004. 25. W. L. Schroeder, C. T. Famdie, and K. Solbach, “Utilisation and tuning of the chassis modes of a handheld terminal for the design of multiband radiation characteristics,” in Proc. IEE Conference on Wideband and Multiband Antennas and Arrays, Birmingham, UK, 2005.

Chapter 6

Terminal Antenna Measurements Peter Lindberg

The main problem of measuring the performance of small antennas is the influence of the coaxial feed cable [1] that is connected to the measurement equipment (typically a Vector Network Analyzer, VNA). Not only is the far-field pattern distorted by scattering (re-radiation) of fields incident on the cable, the cable can, if not properly decoupled by baluns1 become a large contributing part of the total radiating structure [2]. As the size of the antenna system (radiator + chassis) or operating frequency goes down, the problems are increased. In addition to introducing a significant measurement error (at least if the final product does not include a cable, like a mobile phone), the repeatability is also severely reduced as the cable will be placed differently during different measurements. For this reason, even techniques that decouples the cables by introducing losses can be of practical interest (as will be discussed later). Terminal antennas are exclusively single-ended, i.e. the reference terminal is the ground plane/chassis. Therefore, terminal antennas should in principle be compatible with coaxial cables, where the coaxial shield is soldered to the chassis metal and the center conductor is attached to the feed terminal. However, this assumes that the chassis is sufficiently large to act as a proper ground plane2 . In most cases, this is

P. Lindberg Laird Technologies AB, Mobile Antenna Systems, Research Department, Isafjordsgatan 5, 164 22 Kista, Sweden e-mail: [email protected] 1

A balun, which is short for balanced to unbalanced, is anything (usually some metallic object) that prevents currents to flow on the outside of the coaxial shield. The ground return currents, which should be equal in magnitude and opposite in direction compared to the currents on the center conductor, should flow on the inside of the coaxial shield. The two sides are separated by the thickness of the shield, which is typically much bigger than the skin depth, and are thus isolated from each other. The currents on the outside of the shield is not a part of a transmission line structure and will therefore be attenuated through radiation along the length of the cable. For sufficiently small antennas, this “parasitic radiation” can actually be the dominant source of radiation and therefore provide unreliable measurement results. 2 A circular ground plane with a radius of 1–2 λ is a good and practical approximation of a perfect (infinite) ground if the antenna is placed in the center. For smaller grounds, the measured characteristics will depend on the exact radius. The reason for this is that the induced currents on the

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

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not true and the chassis instead displays standing wave patterns along its length and provides coupled resonances to the radiating element (e.g. the PIFA). Hence, the location of the measurement cable, in particular the exact place where and in what direction the cable enters/exits the chassis, will have an impact on the measured characteristics. To reduce the effects of the feed cable during measurements of terminal antennas, several suggestions have been made. Generally, the cable should be connected to an E-field minima of the antenna (if such can be located) [3]. High impedance ferrite beads are commonly employed along the coaxial measurement cable, which will partly reflect and partly absorb (depending on operating frequency) the induced power on the cable. The use of ferrites on the cable tends to result in overestimation of the impedance bandwidth and underestimation of the radiation efficiency of the antenna due to the extra losses. The increased measurement repeatability however usually motivates the use of ferrites, and the final characterization is anyway done by cable-less active measurements. As a compliment/replacement to ferrite beads, several types of baluns suitable for terminal measurements have been proposed [4], including baluns for dual band applications in [5]. At wavelengths much longer than the size of the terminal chassis, such as for FM radio, it is no longer possible to decouple the cable by such means. Instead, cable-less procedures have been suggested as a remedy. For transmitting antennas, a (battery operating) voltage controlled oscillator (VCO) with known output power is typically integrated on the terminal and the received power level is recorded with a reference antenna. For receiving antennas, the receiver power level (using e.g. diode detectors) is usually transmitted using (non-metallic) optical fibers to a measurement computer. Also, Tx/ Rx fiber-optic systems have been used for accurate impedance measurements on small antennas [6]. Such measurements are fairly time consuming compared to using standard coaxial cables, which is acceptable for verification measurements but not very suitable for the development phase were several design iterations are needed. An example of a proper cable connection to a 900/1800 MHz dual band PIFA antenna is illustrated in Fig. 6.1. As the current distribution on the chassis below 3 GHz is that of a thick dipole, the currents flow mainly along the long edges with a current maximum (and voltage/ E-field minimum) at the center of the chassis. The cable should therefore be connected perpendicular to the chassis length to minimize the induced leakage currents on the outside of the coaxial shield. The proposed cable placement is however right in the “main beam” of the antenna, thus potentially maximizing the scattering effect. For this reason, it is tempting to extend the cable away from the chassis in the direction of a radiation pattern zero, in this case in the direction of the chassis length as indicated in Fig. 6.1 (marked with “Unsuitable cable placement”). It however turns out that this is a poor choice since the length of the chassis determines its resonant frequency, which in turn determines the achievable bandwidth for unbalanced terminal antennas. And by routing the cable ground disc will flow to the edges (causing edge diffraction), then around the edges to the bottom side of the disc and continue along the coaxial measurement cable. For a large ground plane, the currents reduces to zero before reaching the edges.

6 Terminal Antenna Measurements

95

0 1800 MHz balun

Magnitude of E-field (dB)

–5 Measurement cable

Zin,balun

–10 900 MHz balun

–15

900 MHz element

–20 –25

Unsuitable cable placement

–30

1800 MHz element

–35 –60 –50 –40 –30 –20 –10 0 10 Position (mm)

20

30

40

50

60

Fig. 6.1 Simulated E-field strength 2 mm above the chassis at 900/1800 MHz as a function of chassis position. Suitable placement of measurement cable, together with baluns, is indicated

along the length of the chassis, this length is effectively increased by the length of the cable, thereby significantly affecting the chassis current distribution and all related antenna characteristics. Nevertheless, this cable routing is commonly seen in academic papers, even in top class journals. To illustrate the significance of proper cable placement, a 100 mm long cable was attached to both locations indicated in Fig. 6.1. The cable was simulated by adding a 3 mm metal strip to the chassis, and the resulting return loss plots are shown in Fig. 6.2. Clearly, by connecting the cable perpendicular to the short edge, the low frequency resonance is nearly completely

Return loss (GHz)

0

–5 Perpendicular to short edge No cable

–10 Perpendicular to long edge

–15 0.8

1.0

1.2

1.4 1.6 Frequency (GHz)

Fig. 6.2 Simulated effect of cable attachment to terminal chassis

1.8

2.0

2.2

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P. Lindberg

removed. At 1800 MHz, the effect is not as significant due to the relatively larger chassis at these frequencies [7]. For the proper cable placement, perpendicular to the long edge, only a minor frequency detuning is visible at the low frequency resonance, which is in many cases an acceptable level of accuracy. An example of a dual band balun (from [5] is also shown in Fig. 6.1. The balun consists of two series metal L-shaped “fingers” soldered to the coaxial shield away from the antenna. The principle of operation of this balun is identical to the bazooka (or sleeve) balun that has previously been successfully applied during terminal antenna measurements [8] – an unsymmetrical transmission line is formed by the coaxial shield and the metal finger, and by making the finger L = λ/4 long at the design frequency (900 or 1800 MHz) and connecting the finger to the shield at one end, the short-circuit is transformed into a high-impedance Z in ,balun , balun at the open end facing the chassis according to Z in,balun = j Z 0 tan(γ L)

(6.1)

where Z 0 is the characteristic impedance of the transmission line formed by the finger and the coaxial shield, γ = α + jβ is the propagation constant with real part α being the attenuation constant (in Neper/m) and imaginary part β being the phase constant (in rad/m). The high impedance effectively chokes any leakage currents on the shield within a limited bandwidth (∼10%) around the design frequency. At the upper frequency band, the first (low band) balun transforms the true short into a virtual short at the open end. However, the second (shorter) balun then connects in series and introduces a virtual open circuit at its open end, again suppressing any radiating currents on the outside of the coaxial shield.

Impedance Measurement During the design phase, most of the engineering time is spent on optimizing the antennas input impedance to conform with the return loss requirement (usually −6 dB). In most cases, this is done by altering the shape of the radiating element, and/or by tuning the matching network components. In this stage, the terminal antenna is typically placed on a foam like block (with εr ≈ 1, simulating free space) and connected to a VNA by a coaxial cable. Several measurement errors are of course present in this set-up: the proximity of the engineer, the influence of the cable, the table beneath the foam, the measurement instrument and other metallic objects surrounding the antenna etc. However, the instant feedback provided to the engineer by the measurement instrument outweighs this lack of accuracy. The effect of any changes made on the antenna is immediately visible on the instrument’s screen, which immensely simplifies the design process. Hundreds of small changes can be tried in a couple of hours, which in addition to being very efficient also makes the work less boring for the engineer. In contrast, computer simulations with reasonable accuracy takes at least several minutes to complete, which is ample time for an engineer to start thinking about other things

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and possibly forget what he was examining. Also, cutting and pasting copper tape in a prototype is also much faster than doing the same operation in the EM software (except for special cases, such as when all surfaces are planar). Finally, the instant measurement feedback is also useful for crude validations such as in what direction and polarization the antenna radiates (by waving a metal rod around the antenna and looking at the impedance curves moving – no movement means no radiation in that direction, lots of movement means lots of radiation in that direction, and that the antenna indeed radiates power. . .), if the cable is decoupled from the antenna (moving a finger along the cable should not affect the impedance significantly) etc. Due to these advantages of measurements compared to e.g. simulations, it is likely that the bulk of terminal antenna design will be manual also in the near future. That almost all terminal antennas are designed manually (using scalpels, copper tape and a VNA) is most easily seen by simply opening a mobile phone and looking at the geometry of the antenna. No straight lines are seen, no 90◦ angles, always strange dimensions (like line or slot widths of say 0.46 mm (instead of 0.5 mm)) etc. One could argue that the return loss of the antenna is a redundant figure of merit since the power loss due to the impedance mismatch is already included in the “antenna efficiency” metric, defined as ηant = ηrad . (1 − ⌫2 )

(6.2)

where ηrad = Prad /Pacc = Rrad /(Rrad + Rloss ) is the radiation efficiency (i.e. fraction of radiated power to the accepted power) and ⌫ is the voltage reflection coefficient defined as ⌫=

Z out − Z 0 Z out + Z 0

(6.3)

Antenna efficiency is hence the fraction of radiated power to the power available from the source (with generator impedance equal to the characteristic impedance Z0 ). As radiation patterns is not important, the antenna efficiency is indeed the most important figure of merit and if the source and load connected to the antenna were ideal, the return loss requirement could probably be completely replaced by antenna efficiency. However, the subcomponents connected to the antenna - filter, switch or amplifier, besides not having exactly 50 ⍀ interfaces themselves, are moreover designed for a specific load and source impedance (always 50 ⍀) and might not function properly when terminated with impedances too far from 50 ⍀. In particular, filters might display excessive amounts of ripple in the transfer function in case of large levels of mismatch at either port. Therefore, to ensure compatibility with other components, a certain return loss level must be ensured. As a final note, return loss is defined as the logarithm of the power reflection coefficient (i.e. voltage coefficient squared) with a negative sign, i.e. RL(dB) = −10 log(|⌫|2 )

(6.4)

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P. Lindberg

In accordance with this definition, return losses should always be positive (at least for passive components), with high return loss values associated with low reflection power loss. For instance, a return loss of 10 dB means that one tenth of the power is reflected and a return loss of 20 dB means that one hundredth of the incident power is reflected. Hence, a higher “loss” is better. As a positive decibel number, such as 10 dB, typically means that something is ten times higher than the reference, rather than a tenth of the reference, the definition of return loss is slightly counter intuitive. Some authors therefore utilizes concepts such as “reflection loss” instead, which differs to return loss in that the minus sign in Eq. 6.4 is missing. In this book, return loss is used synonymously with reflection loss, as is done regularly in both industry and academia.

Gain Measurement In addition to impedance measurements, which constitutes 99% of all measurements in the development phase, the radiation properties of the antenna must also be measured. As this is a somewhat time-consuming procedure, such measurements are preferably kept to a minimum. Only a few years ago, data sheets and scientific papers on terminal antennas contained polar plots of radiation diagrams in different cut-planes. Today, it is widely recognized that such plots are not important3 ; only the “average gain”, or “efficiency” is relevant. An antenna is considered as a good antenna if most of the power available from the transmitter is radiated (as opposed to e.g. being dissipated as heat), in what direction is not important. Of course, directivity away from the user’s head is desirable, but such a property is anyway visible from the radiation efficiency in talk position and SAR values. Recently, systems that support multiple antennas, either for diversity or MIMO (Multiple-Input-Multiple-Output), to compensate for deep fading dips have started being implemented in some terminals. Such systems typically targets high data rates and operate at high frequencies (>2 GHz). Examples include Wireless LAN at 2.4 and 5.8 GHz (IEEE 802.11g), WiMAX at 3.5 GHz etc. For these systems, the radiation patterns should be as different as possible (preferably completely orthogonal), which motivates complete pattern (in e.g. a spherical nearfield chamber) or other means of correlation measurements (e.g. reverberation chamber [9]).

3 As the position and orientation of the mobile terminal is completely unknown, a perfect isotropic pattern would be ideal (unless, of course, a directive pattern which could somehow be adaptively focused towards the base station could be implemented). On the other hand, due to the presence of the user’s head almost completely absorbing the power in that direction, directivity away from the head is preferable. In practice however, the radiation pattern is beyond the control of the antenna designer (who instead focuses on getting bandwidth and efficiency) and typically ends up fairly dipole-like due to the chassis influence.

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Referencess 1. S. Saario, D. V. Thiel, J. W. Lu, and S. G. O’Keefe, “An assessment of cable radiating effects on mobile communications antenna measurements,” in Proc. IEEE International Symposium on Antennas and Propagation, 1997, pp. 550–553. 2. C. Icheln, J. Ollikainen, and P. Vainikainen1, “Reducing the influence of feed cables on small antenna measurements,” IEE Electronics Letters, vol. 35, no. 15, pp. 1212–1215, July 1999. 3. P. J. Massey and K. R. Boyle, “Controlling the effects of feed cable in small antenna measurements,” in Proc. Antennas and Propagation Society International Symposium, vol. 2, Mar. 2003, pp. 561–564. 4. C. Icheln, J. Krogerus, and P. Vainikainen, “Use of balun chokes in small antenna radiation measurements,” IEEE Transactions on Instrumentation and Measurement, vol. 53, no. 2, pp. 498–506, Apr. 2004. ¨ 5. P. Lindberg, E. Ojefors, and A. Rydberg, “A single matching network design for a dual band PIFA antenna via simplified real frequency technique,” in Proc. 1st European Conference on Antennas and Propagation, Nov. 2006. 6. T. Fukasawa, K. Shimomra, M. Ohtsuka, and S. Makino, “Accurate and effective measurement method for small antenna using fiber-optics,” in Proc. International Union of Radio Science (URSI), Oct. 2005. 7. H. Arai, Measurement of Mobile Antenna Systems. Artech House, 2001. 8. Y. L. Chow, K. F. Tsang, and C. N. Wong, “An accurate method to measure the antenna impedance of a portable radio,” Microwave and Optical Technology Letters, vol. 23, no. 6, pp. 349–352, Dec. 1999. 9. P. Hallbj¨orner and K. Mads´en, “Terminal antenna diversity characterization using mode stirred chamber,” Electronic Letters, vol. 37, no. 5, pp. 273–274, Mar. 2001.

Chapter 7

Description of Lossless Two Ports in Terms of Scattering Parameters Binboga Siddik Yarman

Introduction In general, two ports can be described in terms of network parameters such as impedance, admittance, chain, hybrid and Scattering parameters. For many engineering applications, two-ports may be constructed to optimize the system performance under consideration. In this case, descriptive parameters of two ports must be chosen in such a way that they are easily generated on the computer to reach to the pre-set targets or goals which are linked to system performance. Impedance, admittance or hybrid parameters are some what idealized since they are measured under open or short circuit conditions. Furthermore, they may be unbounded such as open circuit impedance or short circuit admittance which is infinity. On the other hand, scattering parameters are bounded and very practical to describe linear active and passive systems since they are defined or measured under the operational conditions. As far as computer aided design or synthesis problems are concerned, they present excellent numerical stability in number crunching process. For many engineering applications, there is a demand to construct lossless twoports for various kinds of problems such as filters, power transfer networks or equalizers. Therefore, in the following first, formal definition of scattering parameters are given to describe, specifically linear active and passive two-ports. However, these definitions can easily be extended to describe n-ports. Then, some important properties of these parameters are summarized. Eventually, a technique to construct or design lossless two-ports employing the scattering description is introduced. In the literature, this method is known as the Simplified Real Frequency Technique or in short SRFT.

B.S. Yarman College of Engineering, Department of Electrical-Electronics Engineering, Istanbul University, 34320 Avcilar, Istanbul, Turkey e-mail: [email protected]

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

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Definition of Scattering Parameters Linear two-ports may be described in terms of two wave quantities called incident and reflected waves. Literally speaking, these wave quantities are defined by means of port voltages and currents. Referring to Fig. 7.1, let us define the incident waves for port 1 and port 2 as functions of time t as follows. a1 (t) =

1 v1 (t) [ √ + R0 i 1 (t)] 2 R0

1 = √ [v1 (t) + R0 i 1 (t)] 2 R0

a2 (t) =

(7.1)

1 v2 (t) [ √ + R0 i 2 (t)] 2 R0

1 = √ [v2 (t) + R0 i 2 (t)] 2 R0

(7.2)

Where v1 (t) and v2 (t) and i 1 (t) and i 2 (t) designate the port voltages and currents with selected polarities and directions as shown in Fig. 7.1. R0 is the normalization resistance which is essential to define the wave quantities. It may as well be utilized to terminate the ports under consideration. It is also known as the port normalization number. It should be noted that the above relations are preserved under the linear transformations such as Fourier or Laplace Transformation. Let the Laplace Transformation of a1 (t) and a2 (t) be a1 ( p) and a2 ( p), respectively. Then, they are given by +∞ a j ( p) = a j (t)e− pt dt;

j = 1, 2

(7.3)

−∞

where p = σ + jω is the conventional complex frequency of the Laplace Domain.

R0

I1 →

Eg

Fig. 7.1 Definition of wave quantities for a linear Two-Port [N]

a1 ←

1

b1

Zin

+ V1

S11 S12 S21

S22

Two-Port [N]

I2

+ V2

← 2

a2 ←

b2

R0

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103

Then, designating the time and the Laplace domain pairs as a j (t) ↔ a j ( p), v j (t) ↔ V j ( p) and i j (t) ↔ I j ( p), the above equations can be directly written in p. 1 V j ( p) [√ + R0 I j ( p)] 2 R0 1 = √ [V j ( p) + R0 I j ( p)]; 2 R0

a j ( p) =

j = 1, 2

(7.4)

Specifically, at port-1, the incident wave quantity can be given in terms of the excitation eg (t) = v1 (t) + R0 i 1 (t) or in its Laplace domain pair E g ( p) = V1 ( p) + R0 I1 ( p). Thus, eg (t) a1 (t) = √ 2 R0 or E g ( p) a1 ( p) = √ 2 R0

(7.5)

For many applications, actual frequency response of the systems is determined by setting p = jω. In this case, the complex port quantities ai ( jω), Vi ( jω) and Ii ( jω) are measured with properly designed equipments called network analyzers. Here, in this representation, f is being the actual operating frequency measured in Hertz (or 1/sec), ω = 2π f represents the actual angular frequency in radian/sec. For passive reciprocal two-ports which consist of lumped circuit elements such as capacitors, inductors, resistances and transformers, the voltage and current quantities are expressed by means of ratio of polynomials in the complex variable p, known as rational functions. Obviously, for passive systems, these functions must be bounded for all bounded excitations. Similarly, reflected waves quantities are defined as time and Laplace domain pairs b j (t) ↔ b j ( p) as b j (t) =

1 v j (t) [ √ − R0 i j (t)] 2 R0

or b j ( p) =

(7.6) 1 V j ( p) [√ − 2 R0

R0 I j ( p)]

j = 1, 2

At this point, it is appropriate to mention that based on the maximum power theorem, a complex exponential (or equivalently a sinusoidal) excitation eg (t) = E m e jωt (or equivalently eg (t) = E m cos(ωt) which is the real part of {E m e jωt }), with internal resistance R0 , delivers its maximum power to the input port, when it sees a driving point input impedance Z in ( p) = R0 . In this case, the maximum average power PA over a period T = 1/ f sec is given by

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B.S. Yarman

1 PA = r eal{ T

T

v1 (t)i 1∗ (t)dt}

(7.7a)

0

where v1 (t) = eg (t)/2 and i 1 (t) = eg (t)/2R0 and “∗” designates the complex conjugate of a complex quantity. Thus (7.7a) yields that PA =

E m2 4R0

(7.7b)

On the other hand, amplitude square of the incident wave is given by a1 (t)a1∗ (t) = |a1 |2 = E m2 /4R0 also measures of the maximum deliverable power of the input excitation. Therefore, we say that PA = |a1 |2 is the available power of the generator. As a matter of fact, definition of the incident wave is made in such a way that for p = jω, incident or available power of the excitation is equal to the amplitude square of the incident wave a1 ( jω) or simply, |a1 ( jω)|2 = PA . Based on the above introduction, in the Laplace Domain p = σ + jω, for any linear two-port, reflected waves b1 ( p) and b2 ( p) can be expressed in terms of the incident waves a1 ( p) and a2 ( p) such that b1 ( p) = S11 ( p)a1 ( p) + S12 ( p)a2 ( p) b2 ( p) = S21 ( p)a1 ( p) + S22 ( p)a2 ( p)

(7.8a)

Or in the matrix form,

Where Si j ( p); S( p) given by

b1 ( p) S11 ( p) S12 ( p) a1 ( p) = b2 ( p) S 21 ( p) S22 ( p) a2 ( p)

(7.8b)

i, j = 1, 2 are called the scattering parameters and the matrix S( p) =

S11 ( p) S12 ( p) S21 ( p) S22 ( p)

(7.8c)

is called the Scattering Matrix. Specifically, S11 and S22 are called input and the output reflectance, under R0 terminations and S21 and S12 are called transfer scattering parameters from port-1 to port-2 and from port-2 to port-1, respectively. It is important to note that for any linear active and passive two-ports,

r r r

for a pre-selected normalization port numbers R0 , for any excitation applied to either port-1 and port-2, and under arbitrary port terminations,

7 Description of Lossless Two Ports in Terms of Scattering Parameters

105

Equation (7.8a) holds for all measured sets of {v j (t) ↔ V j ( p) and i j (t) ↔ I j ( p); j = 1, 2}. That is to say, scattering parameters must be invariant under any excitation and termination. In other words, real normalized (or normalized with respect to R0 ohms) scattering parameters which are measured or derived over the entire frequency band (0 ≤ ω = 2π f ≤ ∞), completely describe the linear two-ports. Therefore, proper generation of the scattering parameters is essential. In the following section, we introduce a straight forward technique to generate or measure the scattering parameters for linear networks.

Reflectance Definition for Linear One Port Networks Referring to Fig. 7.2, for one ports, we can always define incident a(t) ↔ a( p) and reflected b(t) ↔ b( p) waves in terms of the one port’s input voltage vin (t) ↔ Vin ( p) and current i in ( p) ↔ Iin ( p) pair. In the Laplace Domain p, the incident wave a( p) is defined by a( p) =

1 Vin ( p) [ √ + R0 Iin ( p)] 2 R0

(7.9a)

1 = √ [Vin ( p) + R0 Iin ( p)] 2 R0 Similarly, the reflected wave b( p)is given by b( p) =

1 Vin ( p) [ √ − R0 Iin ( p)] 2 R0

(7.9b)

1 = √ [Vin ( p) − R0 Iin ( p)] 2 R0 Then, the input reflectance (or the input reflection coefficient) Sin ( p) of one-port is defined in terms of b( p) and a( p) as

R0

Iin →

a ←

+ Vin

b

Fig. 7.2 Definition of wave quantities for a one-port network

Zin

Sin

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B.S. Yarman

b( p) = Sin ( p)a( p)

(7.9c)

or b( p) a( p) Vin ( p) − R0 Iin ( p) = Vin ( p) + R0 Iin ( p)

Sin ( p) =

or =

Z in ( p) − R0 Z in ( p) + R0

or Sin ( p) =

z in ( p) − 1 z in ( p) + 1

where, voltage to current ratio Z in ( p) = Vin ( p)/Iin ( p) is the input impedance (or the actual input impedance) of one-port and z in ( p) = Z in ( p)/R0 is called normalized impedance with respect to R0 . Now, let us investigate power relations for the one-port under consideration. As discussed in the previous section, available power or the incident power of the generator is given by PA = |a( jω)|2 . Similarly, it may be appropriate to define the reflected power from one-port as PR = |b( jω)|2 . In this case, at given angular frequency ω, the net power Pin delivered to one-port must satisfy the relation Pin = PA − PR or Pin = |a|2 − |b|2 2 b 2 = |a| [1 − ] a

(7.9d)

= |a|2 [1 − |Sin |2 ] Let the Power Transfer Ratio T be defined as the ratio of the input power to the available power. Then, Pin PA

(7.10)

T = 1 − |Sin |2

(7.11)

T = or

Equation (7.11) can be expressed in terms of the normalized input impedance z in ( jω) = rin (ω) + j xin (ω).

7 Description of Lossless Two Ports in Terms of Scattering Parameters

z in − 1 2 T = 1 − z in + 1 = 1− =

107

(7.12)

2 (rin − 1)2 + xin 2 (rin + 1)2 + xin

4rin 2 (rin + 1)2 + xin

It should be noted that the above relation is valid, if the definition given for the reflected power PR = |b( jω)|2 results in the actual power Pac delivered to one-port equal to Pin = PA − PR . In other words, one has to show that the actual power delivered to one-port is Pac = Re al{Vin Iin∗ } = Pin . 1 . In fact, this is true as shown √ by the following derivations √ normalized Let vin ( jω) = Vin ( jω)/ R0 and i in ( jω) = R0 Iin ( jω) be the √ port voltage and current, respectively. Similarly, let eg ( jω) = E g ( jω)/ R0 be the normalized excitation voltage. Employing (7.9), vin ( jω) and i in ( jω) is expressed in terms of incident and reflected waves as vin ( jω) = a( jω) + b( jω)

(7.13)

i in ( jω) = a( jω) − b( jω) Thus, the actual power delivered to one-port is Pac = Re al{(a + b)(a − b)∗ } = Re al{aa ∗ − bb∗ − ab∗ + a ∗ b}

(7.14)

Notice that the quantity a ∗ b − ab∗ is purely imaginary 2 . Therefore, Pac = aa ∗ − bb∗ = |a| − |b| 2

(7.15)

2

or Pin = Pac Thus, we conclude that definition given for the reflected power PR = |b|2 is appropriate.

It should be noted that in terms of the normalized port voltage vin and current i in , actual power ∗ delivered to port 1 is invariant i.e. Pac = Re al(Vin Iin∗ ) = Re al(vin i in ) 2 If we define C = a ∗ b = A + j B, then C ∗ = ab∗ = A − j B. Thus, a ∗ b − ab∗ = 2 j B is purely imaginary. Therefore, Pac = |a|2 − |b|2 . 1

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Now, let us investigate some important properties of the input reflection coefficient Sin . Property 1. If one-port consist of passive lumped elements such as inductors, capacitors, resistors and transformers then, the input impedance z in ( p) is a positive real (PR) and rational function in the complex variable p and it is expressed as z in ( p) =

N ( p) D( p)

(7.16a)

where both numerator N ( p) and denominator D( p) polynomials are Hurwitz3 , perhaps with simple roots on the imaginary (or real frequency) axis p = jω of the complex p − plane. Thus, the input admittance yin ( p) = 1/z in ( p) is also a PR, rational function in p. In this case, the corresponding input reflectance Sin = z in − 1/z in + 1 must be a real rational function in p such that Sin ( p) =

h( p) g( p)

(7.16b)

where, h( p) = N ( p) − D( p) = h 0 + h 1 p + h 2 p2 + . . . + h n pn

(7.17a)

and g( p) = N ( p) + D( p) = g0 + g1 p + g2 p 2 + . . . + gn p n

(7.17b)

In (7.17), degree “n” of the polynomials h( p) and g( p) refers to total number of reactive elements (i.e. capacitor and inductors) in one-port under consideration. Let us go through the following example to find the input reflectance of a simple one port consist of parallel combination of a capacitor C = 1 and resistor R = 1 as shown in Fig. 7.3. Example 1. In this example, we wish to derive the input reflectance of the one-port shown in Fig. 7.3. The input impedance of the one port is given as Z in ( p) = 1/ p +1 = N ( p)/D( p) with N ( p) = 1 and D( p) = p + 1. Then, h( p) = N ( p) − D( p) = 1 − p − 1 = − p and g( p) = N ( p) + D( p) = 1 + p + 1 = 2 + p. Thus, Sin ( p) = − p/2 + p. Obviously, N ( p) = 1 and D( p) = p + 1 are Hurwitz polynomials since they include no open right half plane roots. 3

A Hurwitz polynomial must have all its roots in the closed Left Half Plane (LHP) in the complex domain p = σ + jω. Or equivalently, a Hurwitz polynomial must be free of open Right Half Plane (RHP) roots. These definitions imply that a root on the jω is allowable.

7 Description of Lossless Two Ports in Terms of Scattering Parameters Fig. 7.3 A one port consist of R//C

109

R0 Eg

C=1

Zin =

1 pC +

1 R

=

R=1

1 p+1

Property 2. Subtractions of Hurwitz polynomials results in an arbitrary polynomial with real coefficients. Therefore, the numerator polynomial h( p) is an ordinary polynomial with real coefficients. On the other hand, addition of two Hurwitz polynomials results in Strictly Hurwitz polynomial which has all its roots in the closed Left Half Plane (LHP)4 . Thus, the denominator polynomial g( p) must be strictly Hurwitz which makes the input reflection coefficient Sin ( p) regular on the jω axis as well as in the Right Half Plane (RHP) in the complex domain p = σ + jω. In other words, Sin ( p) is bounded over the entire frequency band as well as in the open RHP. It is also interesting to note that on the jω axis, positive real functions always yield non-negative real parts. Therefore, if z in ( jω) = rin (ω) + j xin (ω) is a PR function, then rin (ω) must be non-negative over the entire frequency axis which bounds the amplitude square of the input reflectance by 1 as detailed below. z in ( jω) − 1 2 |Sin (ω)| = z in ( jω) + 1 2 [rin (ω) − 1]2 + xin (ω) [r (ω) + 1]2 + x 2 in in 2

=

|h( jω)| 2 ≤ 1; |g( jω)|2

(7.18a)

∀ω

Or (7.18a) can be re-written as |h( jω|2 ≤ |g( jω)|2

(7.18b)

By analytic continuation5 , in (7.18a), the complex variable p = jω can be replaced by its full version p = σ + jω yielding Sin ( p) = h( p)/g( p) ≤ 1 for all p = σ ≥ 0 since g( p) is strictly Hurwitz (free of closed RHP zeros). Now let us verify (7.18b) for the one port shown in Fig. 7.3. 4 5

By definition, in strictly Hurwitz polynomials jω roots are not permissible. That is to say by expanding p = jω to the right half plane by setting p = σ + jω with σ ≥ 0.

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Example 2. Show that for the one port depicted in Fig. 7.3, (7.18b) holds. By Example 1, h( jω) and g( jω) are given as h( jω) = − jω and g( jω) = 2+ jω or |h( jω)|2 = ω2 and |g( jω)|2 = 4 + ω2 . Obviously, ω2 < 4 + ω2 for all ω. Thus, (7.18b) holds. Property 3. If there is no reflection from the one-port network (PR = |b( jω)|2 = 0), then, all the incident power is dissipated on the one port yielding T = Pin /PA = 1 − |Sin |2 = 1 or equivalently |Sin | = 0. In this case, z in = 1 or the actual input impedance Z in must be equal to the normalization resistance R0 . This situation describes a perfect match between the excitation with internal resistance R0 and one-port network. On the other hand, if one-port network is purely reactive (rin (ω) = 0), then |Sin ( jω)|2 = 1. In this case, one-port reflects all the incident power back to the generator. Obviously, this power will be dissipated on the internal impedance of the generator. In practice, this fact may burn the excitation source. Therefore, one has to take necessary measures to avoid this undesirable situation, perhaps by introducing an attenuation over a circulator to dissipate the reflected power. It is remarkable to note that the above properties, can be derived from (7.15) in a straight forward manner as follows. For a passive one-port network, the actual power delivered to its port must be non-negative. Therefore, Pac = |a( jω)|2 − |b( jω)|2 ≥ 0. If one-port is lossless (i.e. purely reactive yielding z in ( jω) = j xin (ω)) then, there will be no power dissipated on it. In this case, Pac = 0 and the equality holds. By analytic continuation (i.e. replacing jω by p = σ + jω; σ ≥ 0), we can state that a( p)a(− p)−b( p)b(− p) ≥ 0 for all p = σ ≥ 0. Replacing b( p) by Sin ( p)a( p), it is found that a( p)[1 − Sin ( p)]a(− p) ≥ 0; ∀ p = σ ≥ 0 or equivalently, 1 − Sin ( p) ≥ 0; ∀ p = σ ≥ 0. For lossless one-ports Sin ( p) = 1. Thus, Sin ( p) ≤ 1; ∀ p = σ ≥ 0. Because of the aforementioned properties, the rational form of Sin ( p) = h( p)/ g( p) is called the Bounded-Real (BR) function. Example 3. Referring to Fig. 7.4, let us compute the reflectance for a one port consists of a single inductor. In this case, Z in = p. On the jω axis, 2 2 jω − 1 2 = 1 + ω = 1 = |b1 ( jω)| . |Sin | = jω + 1 1 + ω2 |a1 ( jω)|2 2

R0

EG

Fig. 7.4 A one-port consist of a single inductor (a purely imaginary input impedance)

L = 1H

7 Description of Lossless Two Ports in Terms of Scattering Parameters

111

This result indicates that reflected power is equal to incident power. In other words, one-port dissipates no power if it is purely reactive as expected. Property 4. In (7.18b), amplitude square functions of |h( jω)|2 and |g( jω)|2 in real variable ω, define non-negative even polynomials H (ω2 ) and G(ω2 ) such that h( jω)h(− jω) = H (ω2 ) = H0 + H1 ω2 + . . . + Hn ω2n ≥ 0 g( jω)g(− jω) = G(ω ) = G 0 + G 1 ω + . . . + G n ω with 2

2

2n

(7.19)

≥0

H (ω2 ) ≤ G(ω2 ) At this point, we can comfortably state that one can always find a non-negative even polynomial F(ω2 ) = f ( jω) f (− jω) = F0 + F1 ω2 + . . . + Fn ω2n such that the last inequality can be transformed to an equality as follows. G(ω2 ) = H (ω2 ) + F(ω2 )

(7.20)

or F(ω2 ) = G(ω2 ) − H (ω2 ) In fact, the physical existence of F(ω2 ) can easily be verified by close examination of the Power Transfer Ratio (PTR) given by (7.11). Obviously, the Power Transfer Ratio T (ω2 ) which is defined by (7.11), describes an even, non-negative, bounded and real function over the entire frequency axis ω as follows. T = 1 − |Sin ( jω)|2 =1− =

(7.21)

H (ω2 ) G(ω2 )

G(ω2 ) − H (ω2 ) G(ω2 )

or T (ω2 ) =

F(ω2 ) ≤ 1; G(ω2 )

∀ω

Thus, F(ω2 ) appears as the numerator polynomial of T (ω2 ). Clearly, zeros F(ω2 ) are the zeros of the power transfer ratio. Here, it is interesting to note that if the passive one port consists of lumped elements and transformers then, one can immediately generates the power transfer ratio function T (ω2 ) in terms of network descriptive functions step by step as follows. In the first step, normalized input impedance z in ( p) is generated as a positive real function as z in ( p) = N ( p)/D( p) with N ( p) = a0 + a1 p + a2 p 2 + . . . .am p na and D( p) = b0 + b1 p + b2 p 2 + . . . .bn p nb such that 0 ≤ |na − nb| ≤ 1. Here, the

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real coefficients ai , b j are explicitly computed by means of the lumped element values of the one-port under consideration. In the second step, input reflection coefficient is derived as Sin ( p) = h( p)/g( p) where h( p) = N ( p) − D( p) = h 0 + h 1 p + h 2 p 2 . . . + h n p n ; h k = ak − bk ;

k = 1, 2, . . . , n and

g( p) = N ( p) + D( p) = g0 + g1 p + g2 p 2 . . . + gn p n ; gk = ak + bk ; k = 1, 2, . . . , n with n = max{na, nb}. In the third step, even functions H ( p 2 ) = h( p)h(− p) and G( p 2 ) = g( p)g(− p) are derived. At this step, polynomial H ( p 2 ) = H0 + H1 p 2 + H2 p 4 +. . .+ Hn p 2n and G( p 2 ) = G 0 + G 1 p 2 + G 2 p 4 + . . . + G n p 2n are given in terms of the coefficients of the k k polynomials h( p) and g( p) as Hk = (−1)i h i h 2k−i and G k = (−1)i gi g2k−i ; i=0

k = 0, 1, 2, . . . , n In the fourth step, F( p 2 ) of (7.21) is generated as

i=0

F( p 2 ) = F0 + F1 p + F2 p 2 + . . . + Fn p n such that Fk = G k − Hk ;

k = 0, 1, 2 . . . , n

Finally, in the fifth step, replacing p 2 by −ω2 , T (ω2 ) is generated as in (7.21). Let us run a simple exercise to generate the Power-Ratio for a one-port. Example 4. Derive the Power Transfer Ratio function for the one port depicted in Fig. 7.3 In Example 1, we have already obtained Sin ( p) = − p/2 + p = h( p)/g( p) yielding h( p) = − p and g( p) = 2 + p. Then, even polynomials H ( p 2 ) and G( p 2 ) are generated as H ( p 2 ) = h( p)h(− p) = (− p)( p) = − p 2 and g( p)g(− p) = (2 + p)(2 − p) = 4 − p 2 . Thus, F( p 2 ) = G( p 2 ) − H ( p 2 ) = 4 − p 2 − (− p 2 ) = 4 yielding T (ω2 ) = F(ω2 )/G(ω2 ) = 4/4 + ω2 . Property 5. In his remarkable PhD dissertation, it was shown by Sydney Darlington that any positive real immitance function6 can be realized as a lossless two-port in resistive termination R. However, this resistive termination can always be leveled to the normalizing resistance R0 by using a transformer which is also lossless. This property is known as the Darlington Theorem which constitutes the heart of the classical filter theory7 . 6 Immitance function is a generic name such that it either refers to an impedance or admittance function. 7 Darlington Theorem, PhD dissertation, 1939, University of New-Hampshire.

7 Description of Lossless Two Ports in Terms of Scattering Parameters

113

Now, let us exercise the Darlington theorem in the following example. Example 5. 2 p2 + 8 p + 1 is a positive real impedance function. p+4 (b) Find the zeros of the even part of z( p). (c) Synthesize z( p) as a lossless two-port in unit termination. (a) Show that z( p) =

Solution (a) In order z( p) to be a positive real function the followings must be satisfied. 1. When p = σ (real), z(σ ) must be real. In fact this is the case. 2. When σ ≥ 0, z(σ ) must be non-negative. As a matter of fact z(σ ) = 2σ 2 + 8σ + 1/σ + 1 ≥ 0 for all σ ≥ 0. 3. Now let us find the even part of z( p) which is designated by r ( p). r ( p2 ) =

4 1 1 2 p2 + 8 p + 1 2 p2 − 8 p + 1 + = [z( p) + z(− p)] = 2 2 p+4 −p + 4 16 − p 2

Close examination of r ( p 2 ) reveals that r ( p 2 ) is free of finite zeros and its zeros are at infinity of order 2. Zeros at infinity can easily be extracted by continuous fraction expansion of z( p) which is also known as long division. Thus, z( p) is rewritten as z( p) = 2 p + 1/ p + 4 which yields a series inductance L = 2 Henry, and a parallel capacitor C of 1 Farad and finally terminated in a conductance of 4 siemens. Eventually, the resistive termination can be shifted to 1 ohm by means of a transformer8 as shown in Fig. 7.5 We can extend the above properties given for the one-port reflectance Sin ( p) to cover the complete scattering parameters of lossless two-ports. However, in

L = 2H

Transformer n = 2:1

RR = 1 = 1/YR

C = 1F

Fig. 7.5 Darlington Synthesis of z( p) 8

1 zin(p) = 2p + p+4

G = YL = 4

The turn ratio n of the transformer is selected in such a way that power measured at the left port PL = VL I L must be equal to the power on the right port PR = VR .I R . On the other hand, VL /VR = (1/n)2 = 1/4 = Y R /Y L = Y R /G. Thus, the transformer ratio n = 2 results in G = 4Y R = 4 (for Y R = 1/R R = 1) as desired. In this representation, VL , VR , Y L , and Y R designate the left and the right port voltages and conductance, respectively.

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the following section, first let us consider the generation procedures of the scattering parameters for two-ports, then study the properties of the lossless twoports.

Generation of Scattering Parameters for Linear Two Ports Referring to Fig. 7.6, let {Si j ( p); i,j=1,2} be the scattering parameters of a linear passive two-port. Let E Gk be an arbitrary excitation with internal complex impedance Z Gk ( p) applied to either port-1 or port-2. Let Z Lk ( p) the any complex impedance which terminates the other end of the two-port. Since the scattering parameters are invariant under any terminations, port reflected waves b1k ( p) and b2k ( p) are always related to port incident waves a1k ( p) and a2k ( p) linearly by means of scattering parameters. b1k ( p) = S11 ( p)a1k ( p) + S12 ( p)a2k ( p) b2k ( p) = S21 ( p)a1k ( p) + S22 ( p)a2k ( p)

(7.22)

for any k (i.e. for any source and load terminations). Obviously, under arbitrary port terminations Z Gk and Z Lk , one can always measure port voltages and currents or related incident and reflected wave quantities for a selected common port normalization number R0 , satisfying the equation set given by (7.22). Then, we can always compute input and output reflectance yielding Sin = b1k /a1k = Z in − R0 /Z in + R0 and Sout = b2k /a1k = Z out − R0 /Z out + R0 , respectively. In this representation, Z in is the driving point input impedance when the two-port is terminated in Z Lk . Similarly, Z out is the output impedance when the input port is terminated in Z Gk . On the other hand, (7.22) implies that S11 and S21 can directly be measured for any excitation applied to port-1 (with an arbitrary internal impedance Z Gk ) if a2k is zero. This can easily be achieved by √ choosing Z Lk = R0 meaning that V2k = −Z Lk I2k = −R0 I2k or a2k = 1/2 R0 (V2k + R0 I2k ) = 0 . In this case, the ratio b1k /a1k measures S11 and S21 is measured as the ratio of b2k /a1k . Similarly, when port-1 is terminated in R0 forcing a1k = 0, and port-2 is derived by an arbitrary excitation, then, S22 and S12 are measured as S22 = b2k /a2k and S12 = b1k /a2k . R0 ZGk EGk

I1k

→ a1k

← b1k

+ 1

V1k

⎡ S11 ⎢S ⎣ 21

S 12 ⎤ S 22 ⎥⎦

Two-Port [N]

Sin ; Zin

Fig. 7.6 Two-Port under arbitrary terminations

I2k

+ V2k

2

← a2k → b2k

Z out ; S out

ZLK

R0

7 Description of Lossless Two Ports in Terms of Scattering Parameters

115

The above measurements can be summarized as follows. S11 =

b1k a1k

S21 =

b2k when a1k

a2k = 0 ⇒

Z Lk = R0

S22 =

b2k a2k

S12 =

b1k when a2k

a1k = 0 ⇒

Z Gk = R0

(7.23)

Once the scattering parameters are measured, input (Sin ) and output (Sout ) reflectance of the two-port can be easily be obtained in terms of these parameters. Referring to Fig. 7.7, let us first determine the output reflection coefficient Sout = b2 /a2 when port 1 in terminated in arbitrary impedance Z G . Considering (7.8b) , the ratio b2 /a2 can easily be written as Sout =

b2 a1 = S21 ( ) + S22 a2 a2

(7.24a)

On the other hand, by (7.8a), the ratio b1 /a1 is readily obtained. b1 a2 = S11 + S12 ( ) a1 a1

(7.24b)

Now, let us consider the input termination Z G as a passive one-port with incident wave aG and reflected wave bG . By definition, the reflectance SG is given by SG = bG and in terms of the impedance Z G , SG = Z G − R0 /Z G + R0 as shown in aG the previous section. However, close examination of Fig. 7.4 reveals that, at port-1, aG = b1 and bG = a1 which. Hence, (b1 /a1 ) = (aG /bG ) or (

b1 1 )= a1 SG

(7.24c)

Using (7.24c) in (7.24b) one obtains 1 a2 a1 S12 SG = S11 + S12 ( ) or = . SG a1 a2 1 − S11 SG ZL

→ bG = a1 ZG

← aG = b1

SG

1

S11 S 21

S12 S 22

Zin

Fig. 7.7 Derivation of output reflectance Sout = b2 /a2

+

← 2

a2

→ b2 Sout

E

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B.S. Yarman

Employing this result in (7.24a) one ends-up with Sout . Sout = S22 +

S12 S21 SG 1 − S11 SG

(7.24d)

Sin = S11 +

S12 S21 SL 1 − S22 SL

(7.24e)

Similarly,

Transducer Power Gain in Forward and Backward Directions At this point it is interesting to look in to the ratio specified by |S21 ( jω)|2 =

|b2 ( jω)|2 |a1 ( jω)|2

when a2 = 0.

√ By definition, b2 = 1/2 R0 (V2 −R0 I2 ) and the voltage VL across the termination R0 is VL = −R0 I0 . Then, |b2 |2 = |VL |2 /R0 which is the power PL delivered to termination R0 at port-29 . On the other hand, we know that |a1 |2 is the available power PA . Thus, |S21 |2 measures the rate of power transfer from port-1 to port-2 (forward power transfer ratio), as the ratio of power delivered to the termination R0 to the available power of the generator with internal resistance R0 . In fact, this is the formal definition of Transducer Power Gain (T P G) F under resistive terminations R0 at both ends yielding PL = |S21 ( jω)|2 (7.24f) PA where subscript F refers to power transfer in forward direction (or from port-1 to port-2). Similarly, referring to Fig. 7.7, transducer power gain (T P G) B in backward direction is given by (T P G) F =

(T P G) B = |S12 |2

(7.24g)

when a1 = 0.

Properties of the Scattering Parameters of Lossless Two-ports A lumped element lossless two-port, which consists of ideal inductors, capacitors, coupled coils and transformers, can be constructed by cascading the sections of type A, B, C, and D as shown in Fig. 7.8 Let {Si j ( p); i, j = 1, 2} designate the scattering parameters of the lumpedlossless two-port under consideration. Let R0 be the normalizing resistance for both It is interesting to note that |b2 |2 is the reflected power of port 2 which is directly dissipated on the termination R0 when a2 = 0.

9

7 Description of Lossless Two Ports in Terms of Scattering Parameters

117

Fig. 7.8 Simple lumped element building blocks which constitute lossless two-ports

input and output ports. Since the two-port is lossless, there will be no power dissipation on it. In other words, the total power delivered to its ports must be zero. Let us elaborate this statement by mathematical equations as follows. Referring to Fig. 7.9, let P1 = |a1 |2 − |b1 |2 be the net power delivered to port- 1. Similarly, let P2 = |a2 |2 − |b2 |2 be the net power delivered to port-2. Thus, the total power delivered to [N ]is given by a1 b1 − b1∗ b2∗ PT = P1 + P2 = a1∗ a2∗ a2 b2

(7.25a)

must be zero.

R0 Eg

I1 → a1 ← b1

Sin = S11 =

+ 1

V1

S11 S 21

S12 S 22

Lumped Element Lossless Two-Port [N]

h(p) g(p)

Fig. 7.9 Scattering Parameters for Lossless Two-Ports

I2 + V2

← 2

a2 →

b2

R0

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Recall that by definition b =

b1 b2

a = S 1 a2

= Sa. Then (7.25a) will be

re-written as PT = a + I − S + S

a

(7.25b)

Since the two-port [N ] is lossless, PT must be zero. Thus,

∗ 10 S S S 11 S ∗ 21 − 11 12 S21 S22 S ∗ 12 S ∗ 22 01 S11 (− jω) S21 (− jω) 10 S ( jω) S12 ( jω) =0 = − 11 S21 ( jω) S22 ( jω) S12 (− jω) S22 (− jω) 01

(7.25c)

Or in matrix form †

SS = I

(7.25d)

where the sign “†” is called the dagger and it indicates the complex conjugate of the 10 transposed matrix and I = is the identity matrix. It should be noted if we 01 take the dagger of both sides of (7.25d) it will still be equal to identity matrix that since the identity matrix I is real and symmetric. Thus, †

S S=I

(7.25e)

In the above equations, jω can be replaced by p = σ + jω such that (7.25) holds for all σ > 0 and ω. † The expression SS = I yields four equations but one of the zero equality is redundant. Therefore, the open form of (7.25d) is written as S11 ( p)S11 (− p) + S12 ( p)S12 (− p) = 1 S22 ( p)S22 (− p) + S21 (− p)S21 ( p) = 1

(7.25f)

S11 (− p)S21 ( p) + S12 (− p)S22 ( p) = 0 or simply, S11 ( p)S11 (− p) = 1 − S12 ( p)S12 (− p) S22 ( p)S22 (− p) = 1 − S21 ( p)S21 (− p) S21 ( p) S11 (− p) S22 ( p) = − S12 (− p) †

Similarly, the open form of S S = I yields

(7.26a)

7 Description of Lossless Two Ports in Terms of Scattering Parameters

119

S11 ( p)S11 (− p) = 1 − S21 ( p)S21 (− p) S22 ( p)S22 (− p) = 1 − S12 ( p)S12 (− p) S12 ( p) S11 (− p) S22 ( p) = − S21 (− p)

(7.26b)

It is interesting to note that comparison (7.26a) and (7.26b) reveals that |S11 ( jω)| = |S22 ( jω)| and

S12 ( jω) S21 ( jω) = S (− jω) S (− jω) = 1 21

(7.26c)

12

For lumped element two ports, reflection coefficients S11 ( p) and S22 ( p) can be regarded as one-port reflection coefficients Sin ( p) = S11 ( p) when port-2 is terminated in R0 and Sout ( p) = S22 ( p) when port-1 is terminated in R0 . Therefore, they must be Bounded Real (BR) rational functions. In other words, they must be both regular in the closed RHP (σ ≥ 0). If S11 ( p) = h( p)/g( p) then (7.26a) reveals that h( p)h(− p) or g( p)g(− p) F( p 2 ) g( p)g(− p) − h( p)h(− p) = S12 ( p)S12 (− p) = g( p)g(− p) G( p 2 ) S12 ( p)S12 (− p) = 1 − S11 ( p)S11 (− p) = 1 −

(7.27a)

Obviously, by (7.25), S12 ( p) is also BR rational function. Therefore, its denominator polynomial must be strictly Hurwitz g( p). In this case, let S12 ( p) be represented as S12 ( p) =

f 12 ( p) g( p)

(7.27b)

Then, f 12 ( p) will be obtained on the selected roots of F( p 2 ) = g( p)g(− p) − h( p)h(− p) by factorization. Clearly, the solution is not unique. Similarly, (7.26b) suggests that S21 ( p) =

f 21 ( p) g( p)

(7.27c)

Where f 21 ( p) is also obtained by the factorization of F( p 2 ). In this case, (7.26a) yields S22 ( p) = −

f 12 ( p) h(− p) f 21 (− p) g( p)

(7.27d)

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B.S. Yarman

Similarly, by (7.26b) S22 ( p) = −

f 21 ( p) h(− p) f 12 (− p) g( p)

(7.27e)

Hence, for lossless two-ports (7.27d) and (7.27e) demands that f 12 ( p) f 21 ( p) = f 21 (− p) f 12 (− p)

(7.27f)

Bounded Realness of the scattering parameters requires that the ratios given by (7.27f) should be free of open RHP poles. This could be achieved with possible cancellations in ηa ( p) = f 12 ( p)/ f 21 (− p) and ηb ( p) = f 21 ( p)/ f 12 (− p). Furthermore, these ratios must have unity amplitude on the jω axis since |S11 ( jω)| = |S22 ( jω)|. Henceforth, (7.27f) suggests that f 12 ( p) = μf 21 (− p). In this case, let us set f ( p) = f 12 ( p), then f 21 ( p) = μf (− p); where μ is a uni-modular constant (i.e. μ = ±1). Moreover, if the lossless two-port is reciprocal then, S12 ( p) = S21 ( p) or f 12 ( p) = f 21 ( p)

(7.27g)

which automatically satisfies (7.27f). In this case, f ( p) cannot be a full polynomial. Rather, it must be even or odd. If this is not true, lossless two-port can not be reciprocal. If f ( p) is even, i.e. f 12 ( p) = f ( p) = f (− p), then, ⌬ f 21 ( p) = μf 12 (− p) = μf (− p) = μf ( p). Thus, (7.27g) of reciprocity is satisfied when μ = +1. Therefore, even f ( p) demands μ = +1 for reciprocal networks. If f ( p) is odd, i.e. f 12 ( p) = f ( p) = − f (− p), then, by (7.27f) ⌬

f 21 ( p) = μf 12 (− p) = μf (− p) = −μf ( p). In this case, reciprocity condition of (7.27g) is satisfied if μ = −1 is chosen (i.e. f 21 ( p) = f 12 ( p) = f ( p)). Hence, we see that for reciprocal lossless two-ports, the uni-modular constant μ is specified by the ratio μ = f ( p)/ f (− p). Thus, Scattering Matrix of a lossless reciprocal two-port is given by ⎤ ⎡ h f 1 h 1⎣ f ⎦ = S= f g μf ∗ − g f −μh ∗ h∗ μf ∗

(7.28)

where “∗” designates the para-conjugate of a function x( p) (i.e. x∗ = x(− p)); In summary, for a reciprocal lossless two-port f ( p) is either even or odd polynomial ! f ( p) +1 when f is even = . and the uni modular sign μ is given by μ = −1 when f is odd f (− p)

7 Description of Lossless Two Ports in Terms of Scattering Parameters

121

The above results are summarized as follows10 . Referring to Fig. 7.9, for a lumped element-reciprocal lossless two-port, if the BR input reflection coefficient has the following form, S11 ( p) =

h( p) g( p)

(7.29a)

with real polynomials h( p) and g( p) of degree “n” then, rest of the scattering parameters are given as f ( p) g( p)

(7.29b)

f ( p) h(− p) f (− p) g( p)

(7.29c)

S12 ( p) = S21 ( p) = S22 ( p) = −

provided that f ( p) is either even or odd polynomial satisfying F( p 2 ) = f ( p) f (− p) = g( p)g(− p) − h( p)h(− p)

(7.29d)

In (7.29) all the scattering parameters must be Bounded-Real Rational functions (BR) in the complex variable p = σ + jω. From (7.29d), construction of f ( p) requires a little bit of care. Therefore, let us investigate the possible situations within the following discussions. For a given lossless two-port, one may wish to obtain the full scattering matrix perhaps, starting from the driving point input impedance z in ( p) which directly specifies the input reflection coefficient S11 ( p) = h( p)/g( p) and then by spectral factorization of F( p 2 ). In this case, first the even polynomial F( p 2 ) is generated from the given polynomials h( p) and g( p) as in (7.29d). Then, f ( p) is constructed on the selected roots of F( p 2 ), which in turn yields S21 ( p) = f ( p)/g( p). Eventually, S22 ( p) is constructed as in (7.29c). Thus, tedious derivations of (7.23) is simply omitted to form scattering matrix. However, this process requires some care as discussed in the following. In the literature, the above forms of the scattering parameters are known as the Belevitch11 canonical form. As was shown in the previous section, under the standard terminations of Fig. 7.9, |S21 |2 = |b2 |2 / |a1 |2 measures the transducer power gain in forward direction (from port-1 to port-2). In this equation, |a1 ( jω)|2 is the measure of the available power PA of port-1, and |b2 |2 specifies the power PL delivered to load R0 12 . Let us now investigate some properties of the lossless two-ports. 10

A comprehensive discussion on the subject can be found in “Wideband Circuit Design” by H.J Carlin and P.P Civalleri, CRC Press, 1997, pp. 231–235. 11 V. Belevitch, Classical Network Theory, S Francisco, Holden – Day, 1968, p.277 12 In the literature, it is common to designate Transducer Power Gain as T P G = P /P . L A

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Property 6. As was proven by Darlington, any positive real function Z in which is specified as a rational function Z in ( p) = N ( p)/D( p) or corresponding input reflection coefficient Sin ( p) = S11 ( p) = Z in ( p) − 1/Z in ( p) + 1 = h( p)/g( p) can be realized as a reciprocal lossless two-port in unit termination. In this case, proof of the Darlington’s theorem becomes straight forward, since h( p) and g( p) are specified then, the complete scattering parameters of the lossless two-port can be determined just by the factorization of the even polynomial specified by F( p 2 ) = g( p)g(− p) − h( p)h(− p). Then, f ( p) is constructed on the selected roots of F( p 2 ) = f ( p) f (− p). Obviously, this way of constructing of f ( p) is not unique. At this point, it is very important to note that if the factorization process does not yield odd or even polynomial for f ( p), we can always augment S21 ( p) = f ( p)/g( p) with unity products in the form of γr + p/γr + p with γr real and positive such that S21 ( p) = ˆf ( p)/gˆ ( p) = f ( p)(γr + p)/g( p)(γr + p) yielding odd or even ˆf ( p). The same augmentation is also extended to S11 ( p) or S22 ( p) to end up with the same denominator polynomial gˆ ( p) = g( p)(γr + p) in all the scattering parameters. √ √ Example 6. Let S11 ( p) = 1 − p/ 2 + 2 p . Generate the scattering parameters for the corresponding lossless reciprocal lumped element two-port. √ √ Solution Here, h( p) = 1 − p and g( p) = 2 + 2 p which are specified by S11 ( p). Let us now compute G( p 2 ) = g( p)g(− p) and H ( p 2 ) = h( p)h(− p). G( p 2 ) = g( p)g(− p) = 2 − 2 p 2 and H ( p 2 ) = h( p)h(− p) = 1 − p 2 . In this case, F( p 2 ) = f ( p) f (− p) = G( p 2 )− H ( p 2 ) = 2−2 p 2 −1+ p 2 = 1− p 2 or f ( p) f (− p) = (1− p)(1+ p). Since gˆ ( p) = f (− p)g( p) must be strictly Hurwitz, then f ( p) = 1 − p. At this point we have to be careful since f ( p) is neither even nor odd in the complex variable p. Rather, it is a full polynomial of degree 1. Therefore, it cannot belong to a reciprocal lossless network. However, by augmentation with the product 1 + p/1 + p, S21 ( p) can be written as ˆf ( p) 1 − p2 (1 − p) 1 + p . =√ S21 ( p) = √ = √ √ √ gˆ ( p) ( 2 + 2 p) 1 + p 2 + 2 p + 2 p2 √ 2 ˆ In√this case, √ f 2( p) = 1 − p which is an even polynomial in p, and gˆ ( p) = 2 + 2 2 p + 2 p becomes the strictly Hurwitz denominator polynomials of S12 ( p) = S21 ( p). Similarly, S11 ( p) can √ be augmented √ √ with the same product 1 + p/1 + p ˆ p)/gˆ ( p). yielding S11 ( p) = 1 − p 2 / 2 + 2 2 p + 2 p 2 = h( Finally, S22 ( p) is generated as S22 ( p) = −

ˆ p) ˆf ( p) h( ˆ p) h(− 1 − p2 =− = −√ . √ √ 2 ˆf (− p) gˆ ( p) gˆ ( p) 2 + 2 2p + 2p

7 Description of Lossless Two Ports in Terms of Scattering Parameters

123

At this point it is very important to note that augmentation results in a scattering matrix which has a common denominator gˆ ( p) for all the entries Si j ( p). Therefore, we can state that for reciprocal lossless two-ports, the scattering matrix can always be given as ⎡ ˆ h( p) ⎢ gˆ ( p) ⎢ S=⎢ ⎣ ˆf ( p) gˆ ( p)

ˆf ( p) ⎤ gˆ ( p) ⎥ ⎥ ⎥ ˆh( p) ⎦ −μ gˆ ( p)

where ˆf ( p) is either even or odd polynomial in p and the uni modular constant μ is defined as above.

Blashke Products or All Pass Functions As it is understood from the above discussions, the complex function η( p) = f ( p)/ f (− p) must define a regular function in the closed Right Half Plane p = σ ≥ 0 since the denominator polynomial gˆ ( p) = f (− p)g( p) must be strictly Hurwitz to make S22 ( p) regular in the closed RHP13 . Furthermore, on the jω axis, the amplitude function |η( jω)|2 = 1. Thus, a rational-regular (or analytic) function with unity amplitude is called a “Blashke Product” or an “All Pass” function.

Definition of Proper Polynomial f ( p)14 A real polynomial in complex variable p is called proper if it yields a regular Blashke product η( p) = f ( p)/ f (− p). At this point it is crucial to note that the numerator polynomial f ( p) of S21 ( p) = f ( p) must be proper to end up with the BR Scattering Parameters for the lossless g( p) two-ports.

Possible Zeros of a Proper Polynomial f ( p) In general, zeros of a proper polynomial f ( p) can be classified as in the following cases. A complex variable function η( p) is called analytic in a complex region if it has no singularity in . In other words it has to be differentiable in . If η( p) is rational and analytic in then it should be free of poles in because poles lying in introduces singularities for which η( p) becomes infinity. 14 In to our knowledge, this is the first introduction of the “proper polynomial definition of f ( p)”. 13

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Case A If f ( p) includes a real zero (α > 0) in the closed RHP, such that f ( p) = (α − p) ˆf ( p), then, the term (σ − p) in f ( p) appears as a factor (α + p) in f (− p) yielding gˆ ( p) = (α + p) ˆf (− p)g( p) which preserves the Scattering Hurwitz property of gˆ ( p). Therefore, real RHP zeros of f ( p) are called proper zeros. In this case, the Blashke product η( p) will include the term α − p/α + p or in general η( p) = α − p/α + p ˆf ( p)/ ˆf (− p). Obviously, in this representation, it is assumed that the term ˆf ( p)/ ˆf (− p) is regular in the closed RHP to make S22 ( p) regular (or bounded real) in the closed RHP. This can only be achieved with possible cancellations in the term ˆf ( p)/ ˆf (− p), as will be discussed in Case C. Case B If f ( p) includes a complex zero (such as p = α + jβ with α > 0) in the closed RHP, it must be accompanied with its conjugate in order to make S21 ( p) a real function. In this case, f ( p) will take the following form f ( p) = [(α + jβ) − p][(α − jβ) − p] ˆf ( p) = [(α − p)2 + β 2 ] ˆf ( p), which is a real polynomial in p. Again, assuming a regular term ˆf ( p)/ ˆf (− p) with possible cancellations, the Blashke product η( p) is written as η( p) = (α − p)2 + β 2 /(α + p)2 + β 2 ˆf ( p)/ ˆf (− p). Therefore, we say that a complex closed RHP zero of f ( p) must be matched with its conjugate pair in order to make f ( p) proper. Clearly, proper f ( p) yields a regular η( p). Case C Assume that f ( p) includes a purely real closed LHP zero pz1 = δ with δ < 0. Obviously, this zero is not proper since it introduces a closed RHP zero in f (− p) yielding un-regularη( p)15 . However, if pz1 = δ is matched with its mirror image pz2 = −δ in the closed RHP then, η( p) will include a term (δ − p)(−δ − p)/(δ + p)(−δ + p) which is equal to 1 by cancellation. Thus, we say that a proper f ( p) may include a purely real closed LHP zero if it is paired with its mirror image to make η( p) regular with cancellations. Case D Assume that f ( p) includes a complex closed LHP zero pz1 = δ + jβ with δ < 0 and β ≥ 0. As explained in the previous case, this zero can not be proper unless it is paired with its mirror image which results cancellations in η( p). Furthermore, in order to assure the reality of the polynomial f ( p), complex LHP zeros must be accompanied by their conjugate pairs yielding quadruple symmetry as shown in Fig. 7.10. Thus, we say that complex LHP zeros of a proper f ( p) must have quadruple symmetry. In this case, f ( p) should take the following form. f ( p) = [(δ − p)2 + β 2 ][(δ + p)2 + β 2 ] ˆf ( p) where δ < 0 and a ˆf ( p) describes a regular all pass function ˆf ( p)/ ˆf (− p). Case E Multiple DC zeros of f ( p) which is specified by p k in f ( p) = p k ˆf ( p), are always proper since they are cancelled in η( p). In this context, the term “un-regular” is used to indicate a singularity in η( p) at a point p = −δ > 0 in the closed RHP. 15

7 Description of Lossless Two Ports in Terms of Scattering Parameters

125

Case F Finite-non zero jω zeros of f ( p) are always proper as long as they are paired with their complex conjugates. In this case, f ( p) is expressed as f ( p) = ( p 2 + ωk2 ) ˆf ( p) where ωk designate non-zero but finite jω zeros of f ( p). All the above zeros of f ( p) is summarized in Fig. 7.10.

Transmission Zeros 16 Zeros of f ( p) coincide with the finite RHP, LHP as well as jω zeros (or real frequency zeros) of S21 ( p). Moreover, they also coincide with the zeros of the transducer power gain T P G = |S21 ( jω)|2 over the real frequency axis jω.

Fig. 7.10 Zeros of f(p)

16

Definition of the transmission zeros introduced in this book is some what slightly different then those of introduced in the existing literature by Fano, Youla, Carlin & Yarman.

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B.S. Yarman

Obviously, signal transmission stops at the zeros of T P G = |S21 ( jω)|2 . Therefore, in general, zeros of S21 ( p)S21 (− p) is called the “Transmission Zeros” of the lossless two-port. Transmission zeros of the lossless two-ports may be introduced at finite values of p or at infinity. The circuit structure (or the circuit topology) of the lossless two-port imposes the transmission zeros. Therefore, in the following properties some practical circuit topologies are introduced in connection with transmission zeros. Property 7. In the complex p-plane, general form of f ( p) may be written as follows. f ( p) = p k

nω $

( p 2 + ωr2 )

r =1

nσ r $

(σr − p)

r=

×[(αr − p)2 + βr 2 ]

nδ $ r =1

nL $

(δr2 − p 2 )

nR $ r =1

[(δr − p)2 + βr 2 ][(δr + p)2 + βr 2 ] (7.30)

r =1

where the pairs {αr > 0, βr ; r = 1, 2, .., n R }, {δr < 0, βr ; r = 1, 2, . . . , n L } designate the complex RHP and LHP zeros; the pairs {σr > 0; r= 1, 2, . . ., n σ } and {δr < 0; r = 1, 2, .., n δ } designate real RHP and LHP zeros; p k ; k ≥ 0 multiple zeros at DC of degree k; and {ωr ; r = 1, 2, . . . , n ω } designates the imaginary axis zeros of f ( p). Full form of (7.30) corresponds to a highly complicated network structure which is impractical to realize. In practice however, we generally deal with lossless ladder forms as summarized in Properties 8-10. Property 8. The simplest form of f ( p) includes no zero. In this case, f ( p) will be a constant and may be normalized at unity. Thus, the simplest form of f ( p) = 1(or constant C). This form corresponds to a low-pass L-C ladder as shown in Fig. 7.11. In this case, transmission zeros of the system will be at infinity of degree 2n; where “n” is the degree of the denominator polynomial g( p) = g0 +g1 p+. . .+gn p n which also specifies the total number of reactive elements17 in the lossless two-port under consideration.

Fig. 7.11 f ( p) = 1 yields an LC Low-pass ladder structure as a lossless two-port

17

In this case, reactive elements will be the elements of the Low-Pass Ladder (LPL) structure which is constructed as cascade connections of series inductors and shunt capacitors as depicted in Fig. 7.10.

7 Description of Lossless Two Ports in Terms of Scattering Parameters

127

Fig. 7.12 Lossless ladder with f ( p) = p k which includes zero of transmissions at DC of order 2k and at infinity of order 2(n-k)

Property 9. A little bit more complicated form of f ( p) may have only multiple zeros at DC such that f ( p) = p k . This form corresponds to a band-pass structure as shown in Fig. 7.12. In this case, integer “k” will be the count of total number of series capacitors and shunt inductors which introduce transmission zeros at DC of degree 2k. Furthermore, the difference (n − k) will be the total number of series inductors and shunt capacitors which introduce transmission zeros at infinity of degree 2(n − k). Property 10. Finite real frequency or jω zeros of f ( p) =

nω % r =1

( p 2 + ωr2 ) may either

be realized as a parallel resonance circuit in series configuration, or as a series resonance circuit in shunt configuration or as a Darlington C section with coupled coils as depicted in Fig. 7.13. Example 7. For a lossless two-port let the unit normalized bounded real input reflection coefficient is given by S11 ( p) =

(a) (b) (c) (d) (e)

− p4 + p3 + 2 p h( p) = 4 g( p) p + 3 p3 + 4 p2 + 3 p + 1

Derive f ( p) and check if it is proper. Find the transmission zeros of the lossless two-port. Comment on the possible network topology of the two-port. Construct the full scattering parameters form S11 ( p). Synthesize S11 ( p) as a Darlington lossless two-port in resistive termination which yields the transmission zeros of T (− p 2 ) = S21 ( p)S21 (− p).

Fig. 7.13 For f ( p) = p 2 + ωr2 case, realization of jωr zeros. (a) As a parallel resonance circuit is series configuration; (b) As a series resonance circuit in parallel configuration; (c) As a Darlington C-Section

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B.S. Yarman

Answer (a) The transmission zeros of the two-port described by S11 ( p) is determined from S21 ( p)S21 (− p) = F( p 2 )/G( p 2 ) where F( p 2 ) = G( p 2 )− H ( p 2 ) with G( p 2 ) = g( p)g(− p) and H ( p 2 ) = h( p)h(− p). Since h( p) = − p 4 + p 3 +2 p +0 and g( p) = p 4 +3 p 3 +4 p 2 +3 p +1 are given then, one can readily compute G( p 2 ) = g( p)g(− p) and H ( p 2 ) = h( p)h(− p) as, G( p 2 ) = p 8 − p 6 + 0. p 4 − p 2 + 1,

H ( p 2 ) = p 8 − p 6 − p 4 + p 2 + 0.

or F( p 2 ) = p 4 − 2 p 2 + 1 = ( p 2 + 1)2 . Thus, f ( p) = f (− p) = ( p 2 + 1). f ( p) = 1. f (− p) Thus, f ( p) is proper and introduces a single real frequency transmission zero at ω1 = 1( p = ± jω1 = ± j1). (b) Since the degree of the denominator g( p) is n = 4 and the degree of the proper polynomial f ( p) is n f = 2, then the lossless two-port must have transmission zeros of order 2(n − n f ) = 2(4 − 2) = 4 at infinity. (c) The real frequency zero can be realized as either a parallel resonance circuit in series configuration or a series resonance circuit in shunt configuration if and only if z in ( p) = 1 + S11 ( p)/1 − S11 ( p) or yin ( p) = 1 − S11 ( p)/1 + S11 ( p) satisfies the Fujisawa Theorem18 . If this is the case then, the term ( p 2 + 1) appears as a multiplying factor in denominator polynomials of the impedance or admittance functions as we go alone with long division of the driving point impedance function. This fact will be seen during the synthesis process of the z in ( p) = 1 + S11 ( p)/1 − S11 ( p). Otherwise, this transmission zero can be extracted using zero shifting technique yielding a Darlington C-section. (d) Since f ( p) is constructed as f ( p) = p 2 + 1 and With the existing form of f ( p), one can readily see that η( p) =

η( p) =

f ( p) =1 f (− p)

then,

18

Fujisawa’s Theorem indicates that for a given positive real impedance function z in ( p), if nω % ( p 2 + ωr2 ) and if z in (0) = c > 0 or equivalently if yin (0) = c > 0, and if z in ( p)

f ( p) =

r =1

has either pole or zero at infinity then, the zero of transmission ωr can be realized as either a parallel resonance circuit in mid-series configuration or a series resonance circuit in mid-shunt configuration by extracting the term 2L r p/L r Cr p 2 + 1 during the long division (or equivalently continuous fraction expansion) process of the driving point impedance.

7 Description of Lossless Two Ports in Terms of Scattering Parameters

S21 ( p) =

129

p2 + 1 p4 + 3 p3 + 4 p2 + 3 p + 1

and − p4 + p3 + 2 p2 p4 + 3 p3 + 4 p2 + 3 p + 1

S22 ( p) = −

(e) Let us first generate the driving point input impedance of the lossless two-port when the output port is terminated in 1 ohm. z in ( p) =

1 + S11 g( p) − h( p) 2 p3 + 2 p2 + 2 p + 1 = or z in ( p) = 4 1 − S11 g( p) + h( p) p + p3 + p2 + p + 1

which satisfies the condition of the Fujisawa’s theorem to be synthesized as a Low-Pass Ladder (LPL) with z in (0) = 1 > 0. Furthermore, it is found that z in ( p) =

2 p3 + 2 p3 + 2 p + 1 ( p 2 + 1)( p 2 + p + 1)

which includes the term ( p 2 + 1) in the denominator. This term can be immediately extracted from the input impedance yielding z in ( p) = 2 p/ p 2 + 1 + p + 1/ p 2 + p + 1. Then, we can continue with the long division process. Hence, z in ( p) =

2p + p2 + 1

1 p+

1 p+1

Final form of z in ( p) indicates that the first term 2 p/ p 2 + 1 is a parallel resonance circuit with L 1 = 2H and C1 = 1/2F in series configuration. It is followed with a shunt capacitor of C2 = 1F and, then a series inductor L 3 = 1 follows. Eventually, the lossless ladder is terminated in 1 ohm resistance as shown in Fig. 7.14.

Fig. 7.14 Synthesis of the input impedance Z in ( p) of Example 7

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B.S. Yarman

Fig. 7.15 Realization of lossless ladders

Lossless Ladders If a lumped element reciprocal lossless two-port is free of coupled coils, it exhibits a special kind of circuit topology called lossless ladder. In this configuration, simple sections of Fig. 7.10 are in tandem connection as shown in Fig. 7.15. In this figure Z i and Yi designates the series-arm impedances and the shunt-arm admittances respectively. For lossless ladders, it is straight forward to see that, the proper f ( p) is either even or odd polynomial in the complex variable p = σ + jω, having all its zeros on the jω axis, yielding η( p) = f ( p)/ f (− p) = μ = ±1. This fact can easily be seen by erasing the real and complex zeros of f ( p) from (7.30). Thus, for a lossless ladder nω $ ( p 2 + ωr2 ) (7.31) f ( p) = p k r =1

Obviously, in (7.30) f ( p) is either an even or an odd polynomial in p depending on the value of the integer “k” which is associated with the multiple zeros at DC. Furthermore, all the zeros of f ( p) must appear as poles of Z i and Yi . Let the input impedance of the lossless ladder be Z in ( p). Then, by continuous fraction expansion, 1

Z in ( p) = Z 1 +

(7.32a)

1

Y2 +

1

Z3 + Y4 +

1 ... + .... +

1 .. + R

7 Description of Lossless Two Ports in Terms of Scattering Parameters

131

Where R is the termination resistance of the lossless two port which can be removed with a transformer closed in unity resistance and

{Z i or Yi } =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ai p 1 bi p ci p di p 2 + 1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(7.32b)

At this point it would be appropriate to elaborate the synthesis of driving point functions in a different chapter.

Further Properties of the Scattering Parameters of the Lossless Two Ports For many practical manipulations, it is interesting to evaluate the determinant of a para-unitary scattering matrix ⌬S. In the following this derivation is introduced.

Derivation of ⌬S S11 S12 be a paraunitary scattering matrix. That is, S.S + = S + S = Let S = S21 S22 1 0 I = . 0 1 Determinant S is given by

⌬S = S11 .S22 − S12 .S21

(7.33)

Para-unitary condition yields that S11 .S11∗ + S12 .S12∗ = 1 S21 S11∗ S22 = − S12∗

(7.34)

Using (7.34) in (7.33) we have, ⌬S = −

S21 S21 .S11 .S11∗ − S12 .S21 = − (S11 .S11∗ + S12 .S12∗ ) S12∗ S12∗

(7.35)

Or ⌬S = −S21 /S12∗ ; For reciprocal lossless two-ports ⌬S = −S21 /S21∗ = −S21 /S21∗ The above result can be summarized in the following property.

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B.S. Yarman

Property 11. Determinant ⌬S = S11 S22 − S12 S21 of the Scattering Matrix S = S11 S12 of a reciprocal lossless two-port, which consist of lumped elements, is S21 S22 an all pass function and it is given by19 f ( p) g(− p) . f (− p) g( p)

η⌬ = ⌬S =

Obviously, on the real frequency axis has unitary amplitude. That is |⌬S( jω)| = 1.

Transfer Scattering Parameters Most of the communication systems are constructed by tandem connection of subsystems. Each of these may be described by means of scattering parameters. In order to assess the over all performance such as transducer power gain, we should be able to generate the scattering parameters of the complete structure. Thus, the Transfer Scattering Parameters are introduced to facilitate the computation of the over all performance of the communication systems as follows. Referring to Fig. 7.16, transfer scattering parameters are defined in such a way that at the input, reflected b1 and incident a1 waves are related to the output port’s incident a2 and reflected b2 such that τ11 τ12 a2 b1 = (7.36a) a1 τ 21 τ22 b2 Or in short we write

b1 a1

=T

a2 b2

(7.36b)

where T =

τ11 τ12 τ21 τ22

(7.36c)

Fig. 7.16 Definition of transfer scattering parameters

In this expression Belevitch notation is used. That is S11 = h( p)/g( p) and S12 = S21 = f ( p)/g( p)

19

7 Description of Lossless Two Ports in Terms of Scattering Parameters

133

From Eq. (7.36), relation between scattering parameters and transfer scattering parameters can easily be derived. Let us refresh our mind. Scattering parameters were defined as b1 = S11 a1 + S12 a2

(7.37a)

b1 = τ11 a2 + τ12 b2 .

(7.37b)

which is equal to

Comparison of these equations reveals that one needs to expressed a1 in terms of a2 . On the other hand, b2 = S21 a1 + S22 a2 , or a1 =

b2 − S22 a2 S22 1 =− a2 + b2 . S12 S21 S21

(7.38a)

Using Eq. (7.38a) in (7.37a) we have b1 =

S .S22 S12 − 11 S21

a2 +

S11 S21

b2

(7.38b)

thus, τ11 = S12 −

S11 .S22 S11 ; τ12 = S21 S21

(7.38c)

S22 1 ; τ22 = S21 S21

(7.38d)

and Eq. (7.38a) directly reveals that τ21 = −

It is interesting to note that, in Eq. (7.38d) τ22 is directly related to S21 which is the measure of transducer power gain. In other word, for an arbitrary two-port, TPG is given by T (ω) =

1 2 τ22 ( jω)

(7.39)

Cascaded (or Tandem) Connections of Two-Ports Referring to Fig. 7.17, let us consider cascaded connection of two-port [N1 ] and [N2 ]. Let the following sets (7.40a) and (7.40b) be the transfer scattering parameters of [N1 ] and [N2 ] respectively.

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B.S. Yarman

Fig. 7.17 Tandem connection of two-ports

b1(1)

=

a1(1)

(1) τ11

(1) τ12

(1) τ21

(1) τ22

(2) τ11

(2) τ12

(2) τ21

(2) τ22

a2(1)

=T

b2(1)

(1)

a2(1)

(7.40a)

b2(1)

and

b1(2)

=

a1(2)

a2(2)

= T (2)

b2(2)

a2(2)

b2(2)

(7.40b)

From Fig. 7.17, it is observed that connection at port-2 of [N1 ] to port-1 of [N2 ] reveals that (2)

(1) (2)

b1 a2 a2 (2) = =T (7.41) (1) (2) b2 a1 b2(2) Thus, inserting (7.41) into (7.40) we end up with (2)

(2)

(1)

a2 a2 b1 (1) (2) =T T =T (1) (2) a1 b2 b2(2)

(7.42)

Therefore, we conclude that over all transfer scattering matrix T of the network [N ] which is formed on the cascaded connection of [N1 ] and [N1 ] is given by T =

τ11 τ12 τ21 τ22

= T (1) .T (2)

(7.43a)

or (1) (2) (1) (2) τ11 = τ11 .τ11 + τ12 .τ21

(1) (2) (1) (2) τ12 = τ11 .τ12 + τ12 .τ22

(1) (2) (1) (2) τ21 = τ21 .τ11 + τ22 .τ21

(1) (2) (1) (2) τ22 = τ21 .τ12 + τ22 .τ22

At this point, it is interesting to evaluate τ22 = Since

(1) τ21

=−

(1) S22 (1) S21

,

(2) τ12

=

(2) S11 (2) S21

and

(1) τ22

τ22 =

=

1 (1) , S21

(2) τ22

1 S21

of the composite structure.

=

1 (2) S21

(1) (2) .S11 1 − S22 (1) (2) S21 .S21

(7.43b)

then,

(7.44a)

7 Description of Lossless Two Ports in Terms of Scattering Parameters

135

or equivalently, S21 =

(1) (2) .S21 S21

(7.44b)

(1) (2) 1 − S22 .S11

We should stress that (7.44b) is quite an important equation since it directly reveals the transducer power gain TC (ω) of the composite structure. S (1) .S (2) 2 21 21 TC (ω) = (7.45) 1 − S (1) .S (2) 22 11 Assume that [N1 ] and [N1 ] are lossless reciprocal two ports and they possess the scattering matrices in Belevitch canonical form such that (1) = S11

h 1 ( p) ; g1 ( p)

(1) (1) S12 = S21 =

f 1 ( p) ; g1 ( p)

(1) S22 =−

h 1 ( p) f 1 (− p) h 1 ( p) . = μ1 f 1 ( p) g1 ( p) g1 ( p) (7.46a)

and (2) S11 =

h 2 ( p) ; g2 ( p)

(2) (2) S12 = S21 =

f 2 ( p) ; g2 ( p)

(2) S22 =−

h 2 ( p) f 2 (− p) h 2 ( p) . = −μ2 f 2 ( p) g2 ( p) g2 ( p) (7.47)

then, Tc (− p 2 ) = S21 ( p).S21 (− p) =

f. f ∗ [ f 1 . f 1∗ ] . [ f 2 . f 2∗ ] = (g1 g2 + μ1 h 1∗ h 2 )(g1∗ g2∗ + μ1 h 1 h 2∗ ) g.g ∗ (7.48)

Where (a) μ1 = 1 if f 1 ( p) is even, μ1 = −1 if f 1 ( p) is odd. f ( p) (b) S21 ( p) = is the transfer scattering parameter of the composite structure g( p) from port-1 to port-2. Equations (7.45) and (7.48) are quite important for matching network designers. Therefore, it would be appropriate to make the following comments. Comments:

r r

The composite structure [N ] must include all the transmission zeros of [N1 ] and [N2 ]. If [N1 ] and [N2 ] are reciprocal-lossless two-ports having scattering matrices in the Belevitch canonical form then, [N ] must also have its scattering matrix in Belevitch form: S11 = h/g; S12 = S21 = f /g; S22 = −μh ∗ /g

136

r r

B.S. Yarman

Equation (7.48) demands that f ( p) = f 1 ( p). f 2 ( p) since both f 1 and f 2 is proper which in turn results in proper f ( p). The denominator polynomial g( p) can be constructed on the closed LHP roots of the polynomial G(− p 2 ); G(− p 2 ) = g( p).g(− p) = (g1 g2 + μ1 h 1∗ h 2 )(g1∗ g2∗ + μ1 h 1 h 2∗ )

r

Similarly, the numerator polynomial h( p) is constructed on the explicit factorization of H (− p 2 ) = h( p).h(− p) = G(− p 2 ) − F(− p 2 ) where F(− p 2 ) = f 1 f 1∗ f 2 f 2∗

TPG of the “Tandem Connection of m-Sections” Introduction of Transfer Scattering Matrix (TSM), greatly facilitates the computation of the transducer power gain of the tandem connection of m-sections.

Fig. 7.18 Cascade connection of m-section in a sequential manner; “step by step”

7 Description of Lossless Two Ports in Terms of Scattering Parameters

137

Referring to Fig. 7.18, we can generate TPG of the m-sections step by step in sequential manner. At step-1, the procedure is initialized. Here, first we take section-1 and set 2 (1) (1) (1) T1 = S21 and Sout = S22

(7.49)

At step-2, the second section is connected to section-1. Then, we set 2 (2) S21

T2 = T1 . 2 (1) (2) 1 − Sout .S11 (2) (1) Sout = Sout +

(2) (2) .S21 S12

1−

(1) (2) Sout .S11

(2) .S11

At step-k, 2 (k) S21

Tk = T(k−1) . 2 (k−1) (k) 1 − Sout .S11 (k) (k−1) Sout = Sout +

(k) (k) .S21 S12 (k−1) (k) 1 − Sout .S11

(7.50)

(k) S11

Hence, the above steps continues for k = 3, 4, . . . up to k = m The last step will bek = m. At this step, the m th section is connected to the (m − 1)th section. Thus, (m) 2 S21

Tm = T(m−1) . 2 (m−1) (m) 1 − Sout .S11 which stops the sequential procedure.

Chapter 8

Analytic Approaches to Antenna Matching Problems Basic Concepts Binboga Siddik Yarman

Introduction Design problem of antenna matching networks is one of the major concerns of the communication engineers. As the device production technology improves, data communication speed and carrier frequencies increases beyond X-Band. Thus, demand on wideband communication systems becomes inevitable. This fact forces the design engineers to produce high quality ultra-wideband and/or multi-band antennas for communication systems. A typical high frequency wireless communication system contains two major sites namely; a transmitter and a receiver as depicted in Fig. 8.1. On the transmitter site, generated signal must be properly transferred to the antenna over a preferably non-dissipative (lossless) network so that maximum signal power is pumped into the antenna over the frequency band of operation. Similarly, on the receiver site, the received signal of the antenna is transferred over a lossless matching network and dissipated at the user end within the same frequency band. The user end may be a radio or a TV set or a headphone etc. In this case, again the role of the matching network is to provide the maximum power transfer for the received signal to the end-user over the passband. This is called the classical broadband matching. In the literature, several terms are associated with the non-dissipative power transfer network such as “impedance matching network”, “equalizer”, “lossless two-port”, or “lossless network”; all being inter-changeable. The classical broadband matching theory deals with the proper design of the lossless matching networks between the prescribed terminations. It is common that the signal generation section of the transmitter can simply be modeled as an ideal signal generator in series with internal impedance Z G . This model is called the Thevenin equivalent of the transmitter site (see Fig. 8.2).

B.S. Yarman College of Engineering, Department of Electrical-Electronics Engineering, Istanbul University, 34320, A vcilar, Istanbul, Turkey e-mail: [email protected]

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

139

140

B.S. Yarman Antenna

Signal Generator

Nt

Nr Load

Transmitter Site

Receiver Site

Fig. 8.1 Concept of maximum power transfer via antenna matching networks built-in transmitters and receivers

The transmitter antenna will behave as a typical passive load termination Z L to the lossless power transfer two port [E] as shown in Fig. 8.2. Similarly, the receiver antenna can be considered as an ideal signal source with an internal impedance Z G . The user end of the receiver is also considered as a dissipa tive load Z L to the lossless two-port [E]. In the above discussion, it is evident that both transmitter and receiver sites present a similar model as far as the signal flow is concerned. In both cases, the crucial issue is the maximum power transfer from the generator Z G to the load Z L over the specified frequency band. Referring to Fig. 8.2, as far as design problem of antenna matching networks is concerned in a generic manner (for both transmitter and receiver ends), matched system performance is optimized for the maximum power transfer over the band of operation. Thus, the lossless two-port [E] is constructed accordingly. Hence, the antenna-matching problem is defined as one of “constructing a lossless reciprocal two-port or equalizer so that the power transfer from source (or generator) to load is maximized over a prescribed frequency bands”.

Fig. 8.2 Thevenin equivalent of the transmitter with internal impedance Z G and antenna as a passive load Z L

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It should be noted that until 2006, the classical literature had concerned with the design of matching networks over only a single band. However, recent advancements in communication systems require the design of multi-band matching networks. Typical examples are multi band High Frequency (HF) Multi-Band, Multi Band Cellular communication on the same platform and Multi-Input/Multi-Output (in short MIMO) communication systems with single antennas. Physical implementation of matching networks can be completed with lumped circuit elements such as capacitors, inductors or with distributed elements like transmission lines, or with mixed elements, in other words, with lumped and distributed elements. However, in this book, we will only concern with the designs problems employing only lumped circuit elements. The power transfer capability of the lossless equalizer or so called “matching” network is best measured with the transducer power gain T , which is defined as the ratio of power delivered to the load PL to the available power PA of the generator; over a wide frequency band. That is;

T (ω) =

PL (ω) PA (ω)

(8.1)

Ideally, the designer demands the transfer of the available power of the generator to the load; which in turn requires the flat transducer power gain characteristic in the band of operation at a unitary gain level with sharp rectangular roll-off as illustrated in (Fig. 8.3). But unfortunately, the physics of the problem permits the ideal power transfer only at a single frequency. In this case, the equalizer input impedance Z in is conjugately matched to the generator impedance Z G . Therefore, the design of a matching equalizer over a wide frequency band with “high” and “flat” gain characteristics present a very complicated theoretical problem. It is well known that the terminating impedances Z G and Z L impose the possible highest flat gain level over the selected frequency band B, so called the theoretical “gain bandwidth limitation” of the matched system. Before we introduce the design methodologies, let us classify the nature of the antenna matching problems based on the type of terminations Z G and Z L as follows.

T(ω) T0

Fig. 8.3 Ideal Signal transmission over a prescribed frequency band (ω1 to ω2 )

ω1

ω2

ω

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Fig. 8.4 Single Matching: Design of an antenna matching network for optimum power transfer between a resistive generator R0 and a complex load Z L = R L + j X L .

Single Matching Most of the commercially available systems are designed in such a way that the output impedance of the transmitters or the input impedance of the receivers is approximately equal to standard resistance R0 = 50⍀ (i.e. internal impedance of the generator Z G ≈ 50⍀). On the other hand, usually, measured passive antenna impedance shows a complex behavior (i.e. on the real frequency jω axis, Z L ( jω) = R L ( jω)+ X L ( jω) is a complex load impedance). In this case, the generic antenna matching network design problem becomes as the one to construct a lossless two-port [E] in between resistive generator Z G = R0 and a complex load Z L . This is called the “single matching” (see Fig. 8.4).

Double Matching As the frequency band of operation becomes higher and wider, the input impedance of a transmitter, or the output impedance of receiver can no longer be considered as resistive. In this case, for a generic antenna matching problem, the source internal impedance Z G ( jω) = RG (ω) + j X G (ω) must be complex. Hence, the design of matching network must be considered in between two complex terminations. This is called the “double matching” which refers to optimum power transfer problem from a complex generator to a complex load as shown in Fig. 8.5.

Filter or Insertion Loss Problem It should be mentioned that the classical “filter” or the “insertion loss” problem might also be considered as a very special form of the broadband matching problem

Fig. 8.5 Double Matching: Design of an antenna matching network for optimum power transfer between a complex generator Z G = RG + j X G and a complex load Z L = R L + j X L

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which deals with resistive generator and resistive load (Fig. 8.6). Actually, at low frequencies, such as Amplitude Modulation (AM) and Frequency Modulation (FM) radio communications, even antenna impedance can be considered resistive such as 60⍀, 75⍀ or 300⍀. Similarly, input and the output impedances of the antenna amplifiers may as well be resistive. In this case, the problem is to limit the power transfer over an operational band. In practice, it is said that “we should kill the gain outside the band” to allocate the out of band frequencies for other means of communications. Thus, we face the filter problem: Design of a lossless two-port between resistive generator RG and the resistive load R L for a specified frequency band. In Fig 8.6a, generator and load resistances are assumed to be different. However, employing an ideal lossless-transformer either on the load site or on the generator site, one can always end up with a classical filter design problem with standard resistive terminations R0 as depicted in Fig. 8.6b and 8.6c respectively. Therefore, in the following we will consider the classical filter problem with standard terminations R0 . Coming back to the matching problems, if the load resistance of the lossless filter is perturbed a little bit by adding a complex part to it then, the filter must act as a “single matching network” as described in Fig. 8.4. Similarly, if one perturbs the generator resistance of the filter by adding a complex part to it then, we end up with the double matching situation as shown in Fig. 8.5.

Fig. 8.6a The simplest broadband matching problem: Filter Design between resistive terminations RG and R L

Fig. 8.6b Classical Filter Problem with a transformer at the load-end √ to match R L to R0 ; n L = R L

Fig. 8.6c Classical Filter Problem with a transformer at the Generator-end√to match RG to R0 ; n G = RG

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In this respect, well-established filter design techniques may be employed for broadband matching problems where appropriate. For example, in narrow-band operations at low frequencies, the antenna impedance may be considered as a resistive termination. In this case, a lossless network must be constructed in between resistive generator and a resistive antenna. On the other hand, if appropriate, one may always consider far- end filter elements as part of the generator and load networks, which in turns converts the filter design problem to matching problems. There are two main approaches to the solution of broadband matching problems namely analytic and computer aided design (CAD) solutions. The classical procedure is through analytic Gain-Bandwith Theory. The CAD solutions are accomplished by means of the numerical optimizations which work on the transducer power gain in the passband. The analytic gain-bandwidth theory is essential to understand the nature of the matching problem, but in general, it is not accessible beyond simple problems. From the practical point of view, CAD techniques may be studied under two categories: Brute-force techniques and the real frequency solutions first introduced by Carlin. In brute-force techniques, the designer selects the circuit topology for the equalizer to be constructed, and then, determines the unknown element values of the chosen topology by means of optimization. This way of constructing matching networks heavily involves with non-linear optimization schemes and perhaps never converges in designing wideband matching networks for antennas. However, for narrow bandwidth problems, one may end up with satisfactory solutions by trail and error. On the other hand, Real Frequency Techniques (RFT) directly works on the generation of realizable network functions such as driving point impedance, admittance or bounded real input reflection coefficient to construct antenna matching networks for optimum performance. It was shown that the numerical set-up of the RFT optimization is always stable and convergent. Therefore, they are very practical, easy to use to construct ultra-wideband antenna matching networks for various kinds of applications. In this chapter, we will briefly discuss the analytic theory of broadband matching via filter design problems.

Analytic Theory of Broadband Matching Generally, the lossless matching network to be designed can be described in terms of two-port parameters (such as impedance, admittance, chain, real or complex normalized regularized scattering or transmission parameters) or by means of driving point so called Darlington immitance or bounded real (real normalized) reflection coefficient. At this point, it is appropriate to state the modified version of Darlington’s famous theorem.

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Fig. 8.7 Darlington Representation of the load impedance

Lossless Two Port

1

Z,S

Darlington Theorem Any positive real impedance (Z) or admittance (Y) functions or corresponding bounded real reflection coefficient S = (Z − 1)/(Z + 1) or S = (1 − Y )/(1 + Y ) can be synthesized as a lossless two port terminated in unit resistance. The resulting lossless two ports are called the Darlington Equivalent (Fig. 8.7). Based on the fundamental gain-bandwidth limitations introduced by Bode, the analytic approach to single matching problems was first developed by Fano (8.4) using the concept of “Darlington Equivalent” of the passive load impedance (Z L ). In Fano’s approach, the problem is handled as a “pseudo-filter” or “pseudo-insertion loss” problem, since the tandem connection of the lossless equalizer [E] and Darlington’s load equivalent [L] is considered as a whole lossless filter [F] (Fig. 8.8a). Later, Youla proposed a rigorous solution to the problem using the concept of complex normalization. Youla’s concept provided an excellent solution to handle the single matching problems; however, it should have been elaborated for double matching problems. The complete analytic solution to the double matching problem has been accomplished by the main theorem of Yarman and Carlin, which relates to the “real”, and the “complex normalized-regulized” generator and load reflection coefficients of the doubly matched system. This theorem enables the designer to fully describe the doubly matched system in terms of the “realizable”- real normalized (or unit normalized) scattering parameters after replacing generator and load with their Darlington equivalents, as in the filter design theory as shown in Fig. 8.8b. Instructional accounts of gain-bandwidth theory for both single and double matching problems have been elaborated by W.K. Chen.

RG

EG

E

RL

L

EL F11

ZL

Fig. 8.8a Fano’s definition of broadband matching problem

F22

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B.S. Yarman RG

EG

G

E

L

RL

GEL F11

ZG

ZL

F22

Fig. 8.8b Yarman-Carlin description of double matching problem via darlington’s equivalents of the generator [G] and the load [l] networks

In the following, a general overview of the analytic theory of broadband matching in conjunction with classical filter theory is reviewed and several examples are presented. In order to understand the essence of the analytic theory of broadband matching, it may be appropriate to review the filter or insertion loss problem, which perhaps constitutes the unified approach.

Filter or Insertion Loss Problem in View of Broadband Matching A typical filter or insertion loss problem is depicted in (Fig. 6). In view of broadband matching, the problem is stated as follows. “Given the resistive generator R0 and the resistive load R0 , construct the reciprocal-lossless filter two port [F]to transfer the maximum power of the generator to the load only over the pass band ω1 to ω2 ; stop it otherwise.” In this problem, it is suitable to describe the reciprocal-lossless filter two-port [F] in terms of it’s real (or equivalently unit) normalized scattering matrix F with respect to ports normalization number (R0 )1 F=

F11 F21

F12 F22

(8.2)

The system performance of the filter two-port [F] is measured with the transducer power gain T (ω) given by T (ω) = |F21 ( jω)|2

(8.3)

If the filter consists of lumped, the real normalized bounded real (BR) scattering parameters are given in the following, so called the “Belevitch” canonic form2 . In the classical literature, the scattering parameters are usually designated by the letter S ; however, in chaper, the letter F refers the scattering parameters of the filters under consideration. 2 It should be noted that in terms of input Z and output Z in out impedances, F11 = Z in − 1/Z in + 1 and F22 = Z out − 1/Z out + 1. 1

8 Analytic Approaches to Antenna Matching Problems

F11 =

h , g

F21 =

f , g

F12 = η

147

f∗ , g

F22 = −η

h∗ g

(8.4)

where η = f ∗ / f and h, f , g are the real polynomials in complex variable p = σ + jω for lumped element design. In practice, one is mainly interested in the designing reciprocal lossless two-port filters, which require equal F12 and F21 , (ie. F21 = F12 ). In this case η = f ∗ / f = ± 1, where “+” sign is applied if f is even; “−” sign is applied if f is odd It is well known that a lossless two-port must possess a bounded real para-unitary scattering matrix . That is, FT∗ F = I

(8.5)

or in open form, F11 F11∗ + F21 F21∗ = 1 or on the j ω axis

|F21 |2 = 1 − |F11 |2

(8.6a)

F12 F11∗ + F21∗ F22 = 0 or on the j ω axis

F22 = −F11∗ F12 / F21∗

(8.6b)

F22 F22∗ + F12 F12∗ = 1 or on the j ω axis

|F22 |2 = 1 − |F12 |2

(8.6c)

F11 F12∗ + F21 F22∗ = 0 or on the j ω axis

∗ / F12

(8.6d)

F11 = −F22∗ F21

where I designates a 2×2 unitary matrix, superscript “T” indicates the transpose of a matrix, and the “∗” sign indicates either para-conjugate as subscript or complex conjugate as superscript. Having used the complex frequency variable, per say “ p”, which is associated with the lumped filter design, the equation set (8.6a–8.6d) can be written in terms of the canonic polynomials h, f and g. In this regard (8.6a) becomes hh ∗ = gg∗ − f f ∗

(8.7a)

h ( p) h (− p) = g ( p) g (− p) − f ( p) f (− p)

(8.7b)

or in the open form

In terms of the canonic polynomials f and g, the transducer power gain is given by f (jω) f (−jω) = |F21 |2 T ω2 = g (jω) g (−jω)

(8.8a)

or in complex variable p, f ( p) f (− p) = F21 ( p)F21 (− p) T − p2 = g ( p) g (− p)

(8.8b)

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In essence, (8.8a) dictates the “transducer power gain” based performance measure of a lossless-reciprocal filter over the entire frequencies. When the transducer gain T is different than zero, the lossless system allows the signal transmission. However, there may be complex frequencies “ p Zi ” such that T − p 2 is zero. At these virtual points we can say that “there will be no signal transmission”. Therefore, as it is defined in the previous chapter, let us define the transmission zeros of a lumped element, reciprocal-lossless filter as follows. Definition: Zeros of T (− p 2 ) = F21 ( p)F21 (− p) is called the transmission zeros of a lumped element, reciprocal-lossless filter.

Construction of Doubly Terminated Lossless-Reciprocal Filters Based on the above theoretical overview, design of the doubly terminated lossless reciprocal filters starts with the selection of a proper transfer function T (ω2 ) which is the measure of power transfer specified by (8.8a). Then, by Belevitch, the scattering parameters are determined via proper factorization of the transfer function T in a right manner. Eventually, synthesis of the filter is completed as described in the following design algorithm.

Filter Design Algorithm Step 1: Selection of the proper Transducer Power Gain (TPG) function. Choose an appropriate transducer power gain form T ω2 = N (ω2 )/D(ω2 ) of degree “n”which includes all the desired transmission zeros of the doubly terminated system3 . At this step, any readily available form such as Butterworth, Chybeshev, Elliptic or Bessel type of function may be suitable depending on the application. Step 2: Computation of f ( p) and g( p) (a) By (8.8a), set N (ω2 ) = f ( jω). f (− jω) and replace ω2 by − p 2 and jω by 2 p.4 Thus, N (− p 2 ) = f ( p). f (− p). In this case, find the roots of N (− p ) and select the proper roots −PN k to construct f ( p) = ( p + PN k ) as described in Nk

the previous chapter. This process is called the proper factorization of N (ω2 ) 5 . (a) Obviously, N and D are even polynomials in ω and the integer n is the degree of the denominator polynomial which is specified by D(ω2 ) = D0 + D1 ω2 + . . . + Dn 2n . (b) It should be noted that TPG function T (ω2 ) is also called the transfer function. 4 Notice that when p = jω then, ( p).(− p) = ( jω)(− jω) yielding − p 2 = ω2 . 5 Referring to Chapter, f ( p) must be either even or odd polynomial to end-up with a reciprocal filter [F]. However, this may not be always possible with the given form of N (− p 2 ). For example, N (− p 2 ) = (1 − p 2 ) yields f ( p) = (1 − p) which is proper but not leads us to build a reciprocal 3

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(b) Similarly, set D(ω2 ) = g( jω)g(− jω) or D(− p 2 ) = g( p)g(− p). Then, find 2 the roots of D(− p ). Choose the closed LHP roots −PDk to form g( p) = ( p + PDk ) since g ( p) is strictly Hurwitz. This process is called the Hurwitz Dk

factorization of D(− p 2 ). Hence, the transfer scattering parameter of the lossless reciprocal filter is determined as F21 = F21 = f /g. In general, the above factorization procedure to form an arbitrary polynomial is called the spectral factorization. For synthesis purpose, full polynomial form of g( p) n gk p k must be generated as g( p) = g0 + g1 p + g2 p 2 + . . . + gn p n = k=0

Step 3: Computation of h( p) By losslessness condition of (8.7), h( p)h(− p) = g( p)g(− p) − f ( p) f (− p) or H (− p 2 ) = h( p)h(− p) = D(− p 2 ) − N (− p 2 ). Then, the arbitrary real polynomial h( p) can easily be constructed by spectral factorization of H (− p 2 ) yielding the numerator polynomial h( p) of F11 ( p) = h( p)/g( p). Step 4: Finally, the filter is constructed by means of Darlington’s synthesis procedure of driving point impedance Z = (1 + F11 ) / (1 − F11 ) as a lossless two-port in unit termination (8.12). Let us utilize the above algorithm to construct an LC low pass filter. Example 1. In this example, we will construct a 2 element low pass filter for the standard terminations R0 = 50⍀ up to 1 GHz utilizing the above algorithm. For the sake of simplicity, let us choose a Butterworth function for the transducer power gain. Construction of the Filter with Normalized Element Values: Step 1: Selection of a proper Transfer Function For the given specifications, it is proper to choose the transducer power gain function (or in short- transfer function) as T (ω2 ) = N (ω2 )/D(ω2 ) = 1/1 + ω2n . Step 2: Computation of f ( p) and g( p) (a) Obviously, the above form of TPG yields f ( p) = 1 which is the required form of a lowpass structure as introduced in the previous chapter6 . (b) One has to choose n = 2 since 2 element lowpass ladder is required. In this case, D(− p 2 ) = 1 + (− p 2 ) = 1 + p 4 . The roots are located at the points where p 4 = (−1). Note that e j(2k+1)π = −1. Therefore, four roots are located on the unit a circle7 , at filter. However, as described in the previous chapter, by an appropriate augmentation applied on the scattering parameters, one can always construct a symmetrical Scattering Matrix which results in a reciprocal lossless filter. 6 Clearly, N (ω2 ) = 1 = f ( p). f (− p). Therefore, f ( p) = 1. 7 It is noted that in complex p- plane the roots p = cos(φ ) + j sin(φ ) has amplitude | p | = k k k k cos2 (φk ) + sin2 (φk ) = 1. Therefore, they are placed on a unit circle with phases φk = (2k + 1)π/4; k = 0, 1, 2, 3, 4

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π π π + j sin (2k + 1) ; p Dk = e j(2k+1) 4 = cos (2k + 1) 4 4 √ √ 2 2 p0 = + +j 2 2 √ √ 2 2 p1 = − +j 2 2 Or

k = 0, 1, 2, 3

√ √ 2 2 −j p2 = − 2 2 √ √ 2 2 p3 = + −j 2 2

Choosing the closed LHP roots of p1 and p2 ,

g( p) =

√

√ √ √ 2 2 2 2 +j −j p+ p+ 2 2 2 2

or g( p) = p 2 +

√

2p + 1

Step 3: Computation of h( p) Clearly, H (− p 2 ) = h( p)h(− p) = D(− p 2 ) − N (− p 2 ) = 1 + (− p 2 )2 − 1 = p 4 . The above equation yields two solutions for h( p) as follows. Case A: h( p) = p 2 Case B: h( p) = − p 2 Step 4: Synthesis of F11 ( p) = h( p)/g( p) as a Darlington two-port which yields the lossless filter in unit termination. For this example, one could end-up with two different filter configuration for the selected TPG function since h( p) has dual solution in step 3. Thus for Case A, we have √ + p2 1 + F11 2 p2 + 2 p + 1 Case A: F11 = or Z in ( p) = = √ √ 1 − F11 p2 + 2 p + 1 2p + 1 Synthesis of Z in√( p) can easily by long √ be completed √ √ division which yields Z in ( p) = 2/ 2√p + 1/ 2 p + 1 = 2 p + 1/ √2 p + 1; which results in a series inductance L 1 = 2 and a shunt capacitor C2 = 2 in parallel with a termination conductance G 0 = 1 as depicted in √Fig. 8.9. For Case B F11 = − p 2 / p 2 + 2 p + 1. In this case, it would be meaningful to generate input admittance Yin for synthesis purpose.

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Fig. 8.9 Synthesis of the input impedance Z in ( p) for Example 1

√ √ √ Hence, Yin ( p) = 1 − F11 /1 + F11 = 2 p 2 + 2 p + 1/ 2 p + 1 = √2P + √ 2-porta with a shunt capacitance C1 = 2 and 1/ 2 p + 1 resulting in a lossless √ a series inductance L 2 = 2; terminated in unit resistance R0 = 1 as shown in Fig. 8.9b. Computation of the Actual Element Values So far, we computed the normalized element values of the filter. However, our ultimate goal is to design an actual lowpass filter up to f 0 = 1G H z between R0 = 50⍀ terminations. In this case, element values must be de-normalized. De-Normalization of the Element Values for the Actual Filter Let [L N k , L Ak ] and [C N k , C Ak ] be the normalized and actual inductor and capacitor pairs respectively. Then, actual elements are given by L Ak =

L Nk R0 2π f 0

(8.9a)

C Ak =

CNk 2π f 0 R0

(8.9b)

Referring to Fig. 8.9, for the filter example presented above, actual elements are found as

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For Case A: L A1 = and C A2 =

1.414 × 50 ≈ 11.252n H 2 × 3.14159 × 1 × 10−9

1.414 ≈ 4.5 p F 2 × 3.14159 × 1 × 10−9 × 50

Similarly for Case B: C A1 ≈ 4.5 p F and L A2 ≈ 11.252n H Remarks: In order to develop some feelings regarding the implementation of the analytic theory of gain bandwidth, it may be appropriate to make some comments on the practical issues and also summarize the important facts of the design as rule of tombs. Especially these comments and reviews will be useful for new comers in the field of broadband network design. a. At the first glance, when you design a circuit using rigorous mathematical theories, with a pre-set performance criteria then, you should start by selecting a mathematical form (or function) which measures the desired performance of the circuit. In filter or broadband matching network design problem, the pre-set performance measure is the transducer power gain function. Therefore, the problem starts with the proper choice of the TPG function. Eventually, this function will generate the lossless two-port which will be manufactured for the desired purpose. In selecting the performance measure function, the designer must pay attention to the following issues. b. First of all, this function must yield a realizable network. As it was established in the previous chapter, TPG function must be bounded by 1. That is, the inequality 0 ≤ T P G(ω) ≤ 1 must be satisfied over the entire real frequency axis ( f ) or equivalently over the angular frequency axis ω = 2π f . c. If the designer works with the lumped elements, the TPG function must an even rational function in the angular frequency variable ω satisfying 0 ≤ T (ω2 ) = N (ω2 )/D(ω2 ) ≤ 1. Obviously N (ω2 ) ≤ D(ω2 ); ∀ω. Furthermore, since T (ω2 ) is bounded by 1 then, denominator polynomial D(ω2 ) can never be zero (i.e. D(ω2 ) = 0; ∀ω). d. The above issues are necessary conditions to end-up with a realizable design; but they do not say any thing about the practical issues. From the practical implementation point of view, we should always avoid using couple coils or coupled elements when working with lumped elements designs. In the literature it is well established that lowpass LC structures yields excellent element sensitivity. Therefore, in building broadband networks, we always prefer to deal with series inductors and shunt capacitors. However, if the design forces, we may employ series capacitors and shunt inductors in building discrete component circuits, Microwave Monolithic Integrated Circuits (MMICs) or Silicon VLSI chips. In short, we can say that, in practice, it is always preferred to deal with ladder structures in designing passive lossless filters and broadband matching networks. e. In general, when dealing with designs using analytic theory, the structure of network topology is controlled by the numerator polynomial N (ω2 ) of TPG

8 Analytic Approaches to Antenna Matching Problems

f.

g.

h. i.

153

function. Since N (− p 2 ) = f ( p) f (− p) then, as pointed out earlier, for lowpass ladders we select f ( p) = 1 (or N (− p 2 ) = 1 or a constant K ). If the design requires bandpass ladder structure, with transmission zeros at DC of order “k” then, N (ω2 ) = ω2k is selected. In a lossless 2-port constructed on the TPG function T (ω2 ) = N (ω2 )/D(ω2 ), the total number of elements of the circuit is specified by the degree n of the denominator polynomial D(ω2 ) = D0 + D1 ω2 + . . . + Dn ω2n . In Example 1, we constructed a lowpass filter with two-elements for which TPG becomes 1 at DC (i.e. at ω = 0) and approaches to zero as ω goes to infinity with monotone-roll off. However, if TPG were chosen as T (ω2 ) = T0 /1 + ω2n then, it would be T0 at DC and the termination resistance would be R L = 1 + (1 − T0 )/1 − (1 − T0 ) = 2 − T0 /T0 which is of course, different than 1 (or equivalently R L different than 50⍀). Obviously, element values of the filter which is designed based on T (ω2 ) = T0 /1 + ω2n , depends on the value of T0 . Broadband matching problems can be regarded as special filter problems such that when the TPG function T (ω2 ) = N (ω2 )/D(ω2 ) of the filter is synthesized, it should yield desired generator and load impedances. Details of this issue will be discussed in the following section. However, let us elaborate this statement in Example-2.

Example 2. Design of a single matching equalizer for an R//C load Referring to Fig. 8.10a, let us design a reasonable matching network between a resistive generator (with internal resistance of RG = 50⍀) and a complex load which includes a shunt capacitor C A = 3.183 p F in parallel with a 50⍀ resistance (i.e.R A = 50⍀) over the frequency band of DC to f 0 = 1G H z. Construction of a Single Matching Network via Filter Design This is a simple-single matching problem over a wide-frequency band (from DC to1 GHz). As mentioned earlier, this problem can be solved by perturbing the element values of a properly designed filter. For matching problems, analytic design process starts by modeling the generator and load networks so that one can properly choose a TPG function to imbed these models in the resulting lossless network. For the present case, generator is resistive. The load network is a simple shunt capacitor C A in parallel with a resistance (in short C A //R A ). So, the load model is already given. The Darlington equivalent of the load network is a simple capacitor

Fig. 8.10a Single matching problem of Example-2

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B.S. Yarman

Fig. 8.10b Load network [L] is considered as part of the filter [F]

terminated in 50ohms resistance. In this case, referring to Fig. 8.8, the load capacitor must be part of the lossless two-port to be designed. In other words, one should be able to extract the load capacitor from the so called filter as shown in Fig. 8.10b. In order to carry out the design process analytically, it is appropriate to normalize the modeled generator and load networks with respect to the normalization resistance and frequency which is usually chosen as the upper edge of the passband. For this example, normalization resistance is R0 = 50⍀ and high-end of passband is f 0 = 1G H z. In this case, normalized value of the internal resistance of the generator is RG N = RG /R0 = 50/50 = 1. Similarly, normalized value of the capacitor is C N = ω0 R0 C A = 2 × π × 1 × 109 × 3.1813 × 10−9 ≈ 1. The normalized value of the load resistance is R N = R A /R0 = 50/50 = 1. Close examination of Example 1, leads us to the solution. For this purpose, we can employ either Case A or Case B if the design is flipped over. In this case, the end-capacitor C = 1.4142 which is considered as the parallel combination of two capacitors, namely a capacitor C1 = 0.4142 and the load capacitor C N = 1 yielding a reasonable solution for the given single matching problem as shown in Fig. 8.11. Let us now, lay-out our solution for the single matching problem under consideration in a rigorous way which constitutes the essence of the analytic theory of broadband matching. (a) First, we should select a realizable transfer function from which the load network can be extracted. Thus, from the above discussion, we see that T (ω2 ) = T0 /1 + ω2n with T0 = 1 and n = 2 is a proper choice.

Fig. 8.11 Extraction of the load capacitor from the 2-element butter worth filter

8 Analytic Approaches to Antenna Matching Problems

155

(b) Then, we should construct the scattering parameters for the lossless two-port to be designed. For this example, it is meaningful to start from the back-end by setting F22 ( p) = h( p)/g( p) so that synthesis can directly start √with the extraction of the load network. In this case, we set F22 = − p 2 / p 2 + 2 p + 1 in a similar manner to that of Case B of Example 1. Then, generate the output admittance Yout ( p) as √ 1 − F11 2 p2 + 2 p + 1 √ 1 Yout ( p) = = 2P + √ . = √ 1 + F11 2p + 1 2p + 1 √ (c) Hence, the lossless two-port starts with a shunt capacitor C = 2 = 1.4142 from which one has to extract the load capacitor C N = 1. This is trivial. After extraction, we end-up with a residual capacitor C R = 0.4142. (d) In this case, the√matching network starts with C R and continues with the series inductor L = 2. Eventually, it is terminated with the normalized generator resistance RG N = 1 as shown in Fig. 8.11. The above example is essential to understand the theory of broadband matching. Now, let us ask the following question: What would happen if the load capacitor √ were bigger than C = 2 = 1.4142? Well, in this case, one would start with a little bit more general form of a transfer function T (ω2 ) = T0 /1 + ω2n to guarantee the extraction of the load capacitor which in turn yields the value of the real constant T0 as shown in the following example. Example 3. Let the transfer function of a filter be T (ω2 ) = T0 /1 + ω4 . For various constant DC gain level T0 , examine the variation of the first element, say, capacitor C1 of the filter as a function of T0 . Solution As in Example 1 and 2 we set,

√ √ (a) f ( p). f (− p) = T0 yielding f ( p) = ∓ T0 and g( p) = p 2 + 2 p + 1. (b) To compute h( p), we set H (− p 2 ) = h( p)h(− p) = 1 + p 4 − T0 .

For the present case, zeros of H (− p 2 ) are found at p = (−1)1/4 × (1 − T0 )1/4 π or pk = e j(2k+1) 4 (1 − T0 )1/4 ; k = 0, 1, 2, 3. Let α = (1 − T0 )1/4 ≥ 0 for 0 ≤ T0 ≤ 1. π π Then, pk = α cos(2k + 1) + j sin(2k + 1) . 4 4 Or, four roots of H (− p 2 ) are given by

√

p1,2

√ √ √ 2 2 2 2 =α − ±j and p3,4 = α + ±j 2 2 2 2

If we construct h( p) on the LHP roots, we obtain,

156

B.S. Yarman

h( p) =

√ 2 √ 2 √ 2 2 p+α + α = p2 + α 2 p + α2 . 2 2

For this example, let us describe the filter from the back-end by, F11 ( p) = F22 =

h( p) ; g( p)

F12 = F21 =

√ ± T0 f ( p) = g( p) g( p)

f ( p) h(− p) h(− p) =− f (− p) g( p) g( p)

with Z out =

1 + F22 n( p) where n( p) = g( p) − h(− p); = 1 − F22 d( p)

d( p) = g( p) + h(− p) √

2 − α p + (1 − α 2 ) n( p) thus, Z out = = √ d( p) 2 p 2 + 2(1 + α) p + (1 + α 2 ) or Yout

√ 2 p 2 + 2(1 + α) p + (1 + α 2 ) 1 = C1 p + = √ 2 L2 + R ( 2 − α) p + (1 − α ) C1 = √

2 2−α

= f (T0 ) ≥ 0,

√ √ 2 + α 2 − 2 − α2 2 where L 2 = ≥ 0, √ 2−α √ √ 2 − α + α2 2 − α3 ≥ 0; R= √ 2−α

0≤α≤1

Now, let us examine the variation of C1 for the values of T0 = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1} . The result is summarized in the following table and depicted in Fig. 8.12. Close examination of Table 8.1 and Fig. 8.12 indicates that the DC gain level T0 drops as the value of the capacitor increases. Coming back to the answer of the question raised at the end of Example 2 for the single matching problem under consideration, if the capacitor C1 is bigger than 1.414 then, one lowers the DC gain value T0 down to a level, to end up with the desired load capacitor. For example, if C1 ≈ 3.4 then, it is appropriate to choose T0 ≤ 0.5 as high lighted in Table 8.1.

8 Analytic Approaches to Antenna Matching Problems

157

Variation of the Capacitor C1 5 4.5 4

Capacitor C1

3.5 3

Series1

2.5 2 1.5 1 0.5 0 1

2

3

4

5

6

7

8

9

10

DC Gain level T 0 × 10

Fig. 8.12 Capacitor variation with respect to DC Gain Level T0

It can be shown that, for an R//C load, if a lossless matching network is constructed with infinitely many elements from DC to an angular frequency ω (or equivalently angular-passband ω), the resulting ideal flat transducer power gain level T0 is given by T0 = 1 − e−2π/RCω . This means that for an R//C load, the average TPG level of an actually matched system must be always T0 ≤ 1 − e−2π/RCω . This is called the theoretical gain-bandwidth limit of an RC load. In this regard, one can compute the variation of the load capacitor with respect to ideal flat gain level employing Eq. (8.10). For nominal values of R = 1 and ω = 1, this variation is shown in Fig. 8.13. C =−

2π Rω. ln(1 − T0 )

(8.10)

Obviously, as the value of the load capacitor increases, maximum achievable flat gain level T0 exponentially reduces down to zero. For example, if the load capacitor Table 8.1 Variation of the capacitor C1 with respect to the DC gain level T0 T0

C1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

4.543288 4.269199 4.003825 3.744604 3.488471 3.231316 2.966783 2.682859 2.34777 1.414214

158

B.S. Yarman Ideal Flat Gain Level T0 versus Load Capacitor C Load Capacitor C

70 60 50 40 30

Series1

20 10 0

1

2

3

4

Series1 59.635 28.158 17.616

12.3

5

6

7

8

9

9.0647 6.8572 5.2187 3.904 2.7288

Ideal Flat Gain Level T0 × 10

Fig. 8.13 Variation of the load capacitor with respect to ideal flat gain level

is C = 2.73 the best value of the flat gain is T0 = 0.9. It should be mentioned that for lowpass matching problems, the simplest model for a given complex load includes either a single capacitor in parallel with a resistance or a single inductor in series with a resistance. In case of an inductive load the upper limit of TPG is given by T0 = 1 − e−2π R/Lω . So far, TPG of the matched system has been given in terms of the transfer scattering parameter of the filter by T (ω) = |F21 (ω)|2 = 1 − |F11 (ω)| 2 = 1 − |F22 (ω)|2

(8.11)

In this formulation, certainly, Darlington’s representation of the load network [L] is imbedded as shown in Fig. 8.8. In this case, we need to describe [L] by means of the scattering parameters.

Scattering Description of the Load network for Single Matching Problems In this section, let us assume that passive load network to be matched, is specified as a positive real impedance function in rational form as follows. Z L ( p) =

N L ( p) D L ( p)

(8.12a)

Where N L and D L are Hurwitz polynomials given by N L ( p) = a0 + a1 p + a2 p 2 + . . . + am p m D L ( p) = b0 + b1 p + b2 p + . . . + bn p |m − n| = 1 Such that 2

n

(8.12b) (8.12c) (8.12d)

8 Analytic Approaches to Antenna Matching Problems

159

Generally, any polynomial Q( p) can be expressed as the summation of its even Q e and odd Q o parts such that Q( p) = Q e ( p 2 ) + Q o ( p). In this regard, let N Le , N L o and D Le , D Lo be the even and odd parts of polynomials N L , D L respectively. Then, the load impedance is expressed as ZL =

N Le + N Lo D Le + D Lo

(8.13)

where N Le = a0 + a2 p 2 + . . . + ame p me ;

N Lo = a1 + a3 p 3 + . . . + amo p mo

D Le = b0 + b2 p 2 + . . . + bne p ne ;

D Lo = b1 + b3 p 3 + . . . + bno p no

and

m if m is even me = ; m − 1 if m is odd n if n is even ne = ; n − 1 if n is odd

m − 1 if m is even mo = m if m is odd n − 1 if n is even no = n if n is odd

Assume that the load network is modeled in Darlington sense. That is to say, it is represented as a lossless reciprocal two-port [L] in unit termination as was described in Chapter 7. Here in this section, we wish to generate the descriptive scattering parameters of [L] in terms of the load impedance Z L ( p) which is given as above. Let L = L i j ; i, j = 1, 2 be the real normalized scattering parameters of the load network in Belevitch form. Then, one can write, L 11 =

h L ( p) g L ( p)

(8.14)

f L ( p) , g L ( p) f L ( p) h L (− p) · =− f L (− p) g L ( p)

L 12 = L 21 = L 22

Referring to Fig. 8.14, we can immediately write that

L 11

NL −1 N L − DL ZL − 1 D = L = = NL ZL + 1 N L + DL +1 DL

Thus we can set h L ( p) and g L ( p) as

(8.15)

160

B.S. Yarman

Fig. 8.14 Scattering description of the load network

h L ( p) = N L ( p) − D L ( p)

(8.16)

g L ( p) = N L ( p) + D L ( p) Furthermore, by losslessness condition of [L] FL ( p 2 ) = f L ( p) f L (− p) = g L ( p)g L (− p) − h L ( p)h L (− p)

(8.17)

Certainly, factorization of Eq. (8.17) results in proper f L ( p). However, it may be appropriate to express FL ( p 2 ) in terms of the load impedance explicitly. In this manner, let us take a look at the load impedance as the summation of an even R L ( p) and an odd OL ( p) functions such that8 Z L ( p) = R L ( p) + O L ( p)

(8.18)

where 1 [Z L ( p) + Z L (− p)] 2 1 O L ( p) = [Z L ( p) − Z L (−P)] 2 R L ( p) =

(8.19)

By algebraic manipulation, it is straight forward to find that R L ( p) =

N Le .D Le − N Lo .D Lo D L ( p).D L (− p)

(8.20)

It should be noted that the even numerator polynomial N Le .D Le − N Lo .D Lo of Eq. (8.20) can properly be factorized as A L ( p)A L (− p) = N Le .D Le − N Lo .D Lo . In this case, one may suggest that It should be noted that, an even function is defined as the one which yields R( p) = R(− p) and by definition an odd function yields O( p) = −O(− p). Based on these definitions Eq. (8.17) follows.

8

8 Analytic Approaches to Antenna Matching Problems

A L ( p).A L (− p) = R L ( p) = D L ( p).D L (− p)

161

A L (− p) D L ( p)

A L ( p) D L (− p)

= F˜ L ( p). F˜ L (− p)

(8.21)

A L (− p) F˜ L ( p) = is called the Fictitious function9 D L ( p) On the other hand, one can easily show that10

where

f L ( p) f L (− p) = g L ( p)g L (− p) − h L ( p)h L (− p) = 4(N Le .D Le − N Lo .D Lo ) (8.22) Comparison of (8.19) and (8.21) reveals that zeros of A L (− p).A L ( p) coincides with those of f L ( p) f L (− p). Thus, we conclude that11 L 21 ( p) =

f L ( p) 2A L (− p) = g L ( p) g L ( p)

(8.23)

with f L ( p) = 2.A L (− p) Using (8.16) in (8.23), we obtain that L 21 =

2A L ( p)/D L ( p) 2A L (− p) = N L ( p) + D L ( p) Z L ( p) + 1

(8.24)

or L 21 = 2.

F˜ L ( p) Z L ( p) + 1

Finally, L 22 can be expressed in terms of the load impedance as, L 22 = −

L 21 L 11∗ L 21∗

(8.25a)

9 H.J. Carlin, B.S. Yarman, “Double Matching Problem: Analytic and Real Frequency Solutions”, IEEE Trans. CAS, January 1983 pp. 10 It should be noted that in terms of the load impedance quantities,

g L .g L ∗ = (N L + D L )(N L ∗ + D L ∗ ) = (N Le + N Lo + D Le + D Lo )(N Le − N Lo + D Le − D Lo ) Similarly, h L .h L ∗ = (N L − D L )(N L ∗ − D L ∗ ) = (N Le + N Lo − D Le − D Lo )(N Le − N Lo − D Le + D Lo ) Then, by straight forward manipulation, g L ( p).g L (− p) − h L ( p)h L (− p) =(N Le + D Le )2 − (N Lo + D Lo )2 − (N Le − D Le )2 + (N Lo − D Lo )2 or f L ( p). f L (− p) = g L ( p).g L (− p) − h L ( p)h L (− p) = 4(N Le .D Le − N Lo .D Lo ) = [2A L (− p)] . [2A L ( p)] 11

Obviously, zeros of A L (− p) must be proper since they are the same zeros of f L ( p).

162

B.S. Yarman

or L 22 ( p) = −

A L (−P) D L (− p) Z L (− p) − 1 . . A L ( p) D L ( p) Z L ( p) + 1

(8.25b)

Let the all pass functions be defined as η AL ( p) =

A L (− p) D L (− p) and η DL ( p) = A L ( p) D L ( p)

(8.25c)

then, L 22 ( p) = −η AL ( p).η DL ( p).

Z L (− p) − 1 Z L ( p) + 1

(8.25d)

This concludes the derivation of the scattering parameters of the load network. Now let us summarize the above results. 1. When a passive complex load Z L ( p) = N L ( p)/D L ( p) is represented as a reciprocal-lossless two-port in unit termination, then, the complete scattering parameters of resulting lossless two-port is given by Z L ( p) − 1 , Z L ( p) + 1 2 F˜ L ( p) L 21 ( p) = = L 12 ( p), Z L ( p) + 1 Z L (− p) − 1 L 22 ( p) = −η AL ( p).η DL ( p). Z L ( p) + 1 L 11 ( p) =

(8.26)

2. In the above formulation, all the finite transmission zeros of the load network coincides with the finite zeros of the even part R L ( p) of the impedance function Z L ( p) = R L ( p) + O L ( p). 3. In (8.26), a. The all pass function (or Blashke product) η AL ( p) is constructed on the finite proper zeros of the even partR L ( p). b. The all pass function η DL ( p) = D L (− p)/D L ( p) is constructed on the roots of the denominator polynomial D L ( p) of the impedance function12 Z L ( p). c. The fictitious function F˜ L ( p) = A L (− p)/D L ( p) is an analytic function in the LHP where A L ∗ = A L (− p) includes all the finite proper zeros of the even partR L ( p)13 .

12

It should be noted that D L (− p) has all its roots in the closed RHP since D L ( p) is strictly Hurwitz polynomial. 13 including finite real frequency axis i.e. jω zeros.

8 Analytic Approaches to Antenna Matching Problems

163

Loaded Equalizer and the Transducer Power Gain Referring to Fig. 8.15, let the lossless equalizer and the load de networks be ; i, j = 1, 2 and scribedby the real normalized scattering parameters = E [E] ij [L] = L i j ; i, j = 1, 2 respectively. Then, the cascade connection of [E] and [L] constitutes a passive lossless filter described by the real normalized scattering parameters [F] = {Fi j; i, j = 1, 2}. The input reflection coefficient Sin of the equalizer under complex termination is given by Eq. (7.24e) of Chapter 7. Sin = E 11 +

E 12 .E 21 SL 1 − E 22 .SL

Fig. 8.15 Single-Simple Cascade section formed with reactive elements

(8.27)

164

B.S. Yarman

Where SL is the load reflection coefficient and it is specified by the load impedance Z L such that SL = L 11 = Z L − 1/Z L + 1. As it is shown by Fig. 8.15, Sin is equal to the input reflection coefficient of the filter. In other words, F11 = Sin . As was discussed in Chapter 7, over the real frequencies, the transducer power gain of the filter is given by T (ω) = 1 − |F11 ( jω)|2 = 1 − |Sin ( jω)|2

(8.28)

At this point, it would be wise to describe the lossless equalizer in terms of its Darlington Driving Point impedance Z Q . In other words, the driving point impedance Z Q is considered as a lossless two-port in resistive termination yielding the equalizer [E]. In this case, we can immediately use the results of the previous section specified by Eq. (8.26) by flipping over the equations for the position of the complex termination Z Q . Thus, we have E 22 ( p) =

Z Q ( p) − 1 , Z Q ( p) + 1

E 21 ( p) =

2 F˜ Q ( p) = E 12 ( p), Z Q ( p) + 1

E 11 ( p) = −η AQ ( p).η D Q ( p).

(8.29)

Z Q (− p) − 1 Z Q ( p) + 1

where ZQ =

NQ = RQ + OQ DQ

(8.30)

R Q = FQ .FQ ∗ A Q∗ F˜ Q = DQ η AQ =

A Q∗ AQ

ηD Q =

D Q∗ DQ

As usual “∗” designates the para-conjugate of a function in complex variable p i.e.Z Q ∗ = Z Q (− p), F˜ Q ∗ = F˜ Q (− p), A Q ∗ = A Q (− p), D Q ∗ = D Q (− p) Open form of Eq. (8.27) reveals that Sin =

E 11 − SL .ΔE 1 − E 22 .SL

(8.31)

8 Analytic Approaches to Antenna Matching Problems

165

Where ΔE = E 11 .E 22 − E 12 .E 21 is the determinant of the para-unitary scattering E E 11 12 and it is given by15 matrix14 [E] = E 21 E 22 ⌬E = −

E 21 Z Q∗ + 1 = −η AQ .η D Q . E 21∗ ZQ + 1

(8.32)

Employing Eq. (8.33) in Eq. (8.32), and by straight forward manipulations, one can obtain a significant result for the expression of Sin = F11 . F11 = Sin = η AQ .η D Q .

Z L − Z Q∗ ZL + ZQ

(8.33)

It should be noted that over the real frequency axis jω, all pass functions yield η AQ ( jω) = η D Q ( jω) = 1. Therefore, TPG is given by ∗ 2 Z L − Z ∗Q 2 = 1 − Z Q − Z L (8.34) T (ω) = 1 − |F11 |2 = 1 − |Sin |2 = 1 − Z + Z ZL + ZQ Q L or T (ω) =

4R Q .R L (R Q + R L )2 + (X Q + X L )2

(8.35)

Similarly, the output reflection coefficient Sout of the lossless two port [F] can be derived as in above. Hence we have, L 12 .L 21 .S Q 1 − L 11 .S Q L 22 − S Q .ΔL = 1 − L 11 .S Q

F22 = Sout = L 22 + or Sout

(8.36)

where L 11 =

ZL − 1 ZQ − 1 ; SQ = ; ZL + 1 ZQ + 1

ΔL = −

L 21 Z L∗ + 1 = −η AL .η DL . L 21∗ ZL + 1

(8.37)

By straight forward manipulations, one obtains

F22 = Sout

1 0 0 1 15 See Chapter 7, Property 11. 14

That is E.E + = I =

Z Q − Z L∗ = η AL .η DL . ZQ + ZL

(8.38a)

166

B.S. Yarman

Let the load Blashke product be defined as η L ( p) = η AL ( p).η DL ( p) then16 , Z Q − Z L∗ (8.38b) F22 ( p) = η L ( p) ZQ + ZL Equation (8.38b) is highly crucial to describe the complete matched system in terms of its real normalized scattering parameters17 . Equation (8.38b) can be employed to extract Z Q ( p) from F22 ( p). For this purpose, let us take a look at the difference η L ( p) − F22 ( p). Z Q − Z L∗ 1 = η L [Z L + Z L ∗ ] η L ( p) − F22 ( p) = η L 1 − ZQ + ZL ZQ + ZL

(8.39a)

By definition, RL =

1 [Z L + Z L ∗ ] 2

Then, η L ( p) − F22 ( p) = 2η L

RL ZQ + ZL

(8.39b)

Hence, Z Q is extracted from (8.39b) as, ZQ = 2

η L .R L − ZL η L − F22

(8.39c)

The above equation set constitutes the heart of the single matching problem as follows. Obviously, F22 ( p) is the unknown back-end reflection coefficient of the special filter [F] and it should be constructed in such a way that (a) it must yield the desired shape of the transducer power gain as specified by T (ω2 ) = 1 − F22 ( jω)F22 (− jω) (b) When Z L is extracted from it, resulting driving point impedance Z Q = 2η L RL − Z L must be realizable as a Darlington 2-port which yields the deη L − F22 sired equalizer [E] for the single matching problem under consideration.

Clearly, the load Blashke product is directly generated from the given load impedance Z L ( p) = N L ( p)/D L ( p). 17 B.S. Yarman, “Broadband Matching A Complex Generator to Complex Load”, Ph.D Dissertation, Cornell University, 1982. 16

8 Analytic Approaches to Antenna Matching Problems

167

Therefore, the magic rule of thumbs to construct desirable-realizable F22 ( p) are hidden in the golden expression of γ ( p) = η L ( p) − F22 ( p).

γ ( p) = η L − F22 = 2η L

RL ZQ + ZL

(8.40)

Roughly speaking, γ ( p) vanishes at the zeros p = pzk of 2η L R L /Z Q + Z L yielding η L ( pzk ) = F22 ( pzk )

(8.41)

The above equation constitutes the major constraints of broadband matching imposed by the load network by means of the load term η L = η L A η DL = A L∗ D L∗ /A L D L = F˜ L / F˜ L∗ . Let the zeros of γ ( p) be designated by pzk = σzk + jωzk . Clearly, these zeros overlap with transmission zeros of the load network. Therefore, the complex value of η L ( pzk ) is uniquely determined from the given load impedance Z L which in turn imposes set of constraints on F22 ( p) by Eq. (8.41). Now let us drive these constraints.

r r

r

r

F22 ( p) is the R0 -normalized output reflection coefficient of the special filter from which, the load network must be extracted. In order to construct the special filter [F] for the single matching problem under consideration, we should first select a proper transfer function T (− p 2 ) to include the load network [L]. Then, the back-end reflection coefficient F22 ( p) must be constructed on the explicit factorization of F22 ( p).F22 (− p) = 1− T (− p 2 ). Now, the major question is “How can we select T (− p 2 ) to include [L]?” In this case, one should be able to device an intelligent process to generate T (− p 2 ). At the beginning of this process, over the real frequency axis T (ω2 ) must provide the desired shape of the TPG performance. For example, for a lowpass design problem, the desired shape Tn (ω2 ) may be an n th order “Butterworth” or a Chebyshev form18 . So, the choice of Tn (ω2 ) depends on the specification of the problem. Obviously, Tn (− p 2 ) results in an all pass free reflection coefficient F22n by the explicit factorization of 1 − Tn (− p 2 ). That is, F22n ( p).F22n (− p) = 1 − Tn (− p 2 )

r r

18

(8.42)

It is evident that F22 ( p).F22 (− p) = F22n ( p).F22n (− p) Eventually, one can introduce a proper all pass function into F22n to end up with a realizable F22 ( p), from which the load network [L] is extracted.

For example, order butterworth form is given by Tn (ω2 ) = 1/1 + ω2n

168

r

B.S. Yarman

Let the proper all pass function be ηx =

nx

σ j − p/σ j + p where the integer

j=1

nx can be related to the total number of reactive elements in the load network19 . Furthermore, we know by Eq. (8.39a) that F22 ( p) must include the all pass function η AL ( p). Thus, one can set the mathematical form for the realizable back-end coefficient F22 ( p) as F22 ( p) = η AL . [μ.ηx .F22n ( p)]

(8.43)

Where μ = ±1 is a unimodular constant. Where ηx must be determined to satisfy Eq. (8.42). Comparison of the above equation with Eq. (8.39a) demands that [μ.ηx .F22n ] = η DL .

Z Q − Z L∗ ZQ + ZL

In the classical paper of Youla20 Z Q ( p) − Z L (− p) Z Q − Z L∗ D L (− p) . S( p) = η DL . = ZQ + ZL D L ( p) Z Q ( p) + Z L ( p)

(8.44)

(8.45)

is called the “Complex Normalized-Regulized Reflection Coefficient” of the load. Furthermore, Youla considers η DL ( p), S( p) and 2η L D ( p)R L ( p) as power series expansions around a zero p0 of λ L ( p) = R L ( p)/Z L ( p) as21 η DL ( p) = S( p) =

∞

Br ( p − p0 )r

r =0 ∞

Sr ( p − p0 )r

(8.46a)

(8.46b)

r =0

F ( p) = 2η L ( p)R L ( p) =

∞

Fr ( p − p0 )r

(8.46c)

r =0

Employing Youla’s notation Eq. (8.45–8.46) reveal that22 S ( p) = μ.ηx ( p) F22n ( p)

19

(8.47a)

It is noted that, total number of reactive elements is also related to the transmission zeros of [L]. D. C. Youla, “A new theory of Broad-broadband Matching, IEEE Trans. Circuit Theory, Vol. March 1964, pp. 30–50. 21 Obviously, zeros of γ ( p) = 2η Z R /Z + Z = 2η Z λ/Z + Z overlaps with those L L L Q L L L Q L of λ ( p) 22 Equation (8.47b) is given as the main theorem in the classical work of Yarman & Carlin [ph.d of yarman],[carlin-yarman paper],[carlin book],[Yarman-Wiley ansiklopedia]. 20

8 Analytic Approaches to Antenna Matching Problems

169

and F22 ( p) = η AL ( p) S ( p)

(8.47b)

At this point we should note that S( p) includes the unknown parameters {σ1 , σ2 , . . . , σnx } of ηx ( p). Therefore, as function of these parameters, S( p) = f ( p, σ1 , σ2 , . . . , σnx )

(8.48)

Based on the above notation, (8.41) is simplified as η DL − S = 2

Z L λL ZQ + ZL

(8.49)

or in the series form ∞

(Br − Sr )( p − p0 ) = r

r =0

1 ZQ + ZL

∞

Fr ( p − p0 )r

(8.50)

r =0

In Eq. (8.51), coefficients Br are uniquely determined from the load network. On the other hand, coefficients Sr are specified in terms of the unknown parameters. Therefore, a handy way to determine the unknown parameters {σ1 , σ2 , . . . , σnx } of ηx is to evaluate Eq. (8.51) at the zeros p0 of λ L ( p). In this regard, Youla classifies the zeros of λ L ( p) and drives the restriction based on Eq. (8.51) as below. Before we proceed with the restriction of broadband matching let us take a look at Eq. (8.51) closely. At a zero ( p0 = σ0 + jω0 ; σ0 > 0) of order k of λ L ( p) = R L /Z L , F ( p) = 2η DL R L ( p) is also zero with the same order provided that Z L ( p0 ) is finite. In this case, one can write F ( p) = ( p − p0 )k ψ ( p) =

∞

Fr ( p − p0 )k

(8.51a)

r =0

with F0 = F ( p0 ) = 0, F1 =

yielding F ( p) =

∞ r =k

d (k−1) p d F 1 = 0, . . . , F = =0 k−1 d P p= p0 (k − 1)! dp (k−1) p= p0 (8.51b)

Fr ( p − p0 )r

Therefore, Eq. (8.5) becomes ∞ r =0

∞

(Br − Sr ) ( p − p0 ) = r

r =k

Fr ( p − p0 )r

Z Q ( p) + Z L ( p)

=2

=

λ ( p) Z L ( p) Z Q ( p) + Z L ( p)

( p − p0 )k ψ( p) Z Q ( p) + Z L ( p) (8.51c)

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B.S. Yarman

Clearly, depending on the values of Z L ( p) and Z Q ( p) + Z L ( p), at p = p0 of λ L ( p) with order k, relationship between the coefficients Br , Sr and Fr are effected. Let us investigate the coefficient relationships under the following cases. Case A Let p = p0 = σ0 + jω0 ; σ0 > 0 the open right half plane zero of λ L ( p) of order k. Due to the positive realness of impedances Z L ( p0 ) and Z Q ( p), Z Q ( p0 ) + Z L ( p0 ) must be non zero and finite. Thus, Eq. (8.51c) yields that Fr = 0;

r = 0, 1, 2, . . . , (k − 1)

(8.52a)

Br = Sr ;

r = 0, 1, 2, . . . , (k − 1)

(8.52b)

and therefore,

Case B In this case, p0 = jω0 is a real frequency axis zero of λ L ( p) = R L /Z L with order k and also it is the zero of Z L ( p). Then, this zero must appear in F ( p) = 2η L R L = 2R L with order of (k + 1) provided that Z Q + Z L = 0. For these conditions, Eq. (8.51) becomes, ∞

∞

(Br − Sr ) ( p − jω0 ) = r

r =k+1

Fr ( p − jω0 )r

Z Q ( p) + Z L ( p)

r =0

=2

=

( p − jω0 )k+1 ψ( p) Z Q ( p) + Z L ( p)

λ ( p) Z L ( p) Z Q ( p) + Z L ( p)

(8.53)

yielding, the coefficient constraints Br = Sr ;

r = 0, 1, 2, . . . , k

(8.54)

For which Z Q ( jω0 ) = 0 However, if Z Q is also zero at p0 = jω0 then, Eq. (8.54) will take the following generic form, ∞

∞

(Br − Sr ) ( p − p0 )r =

r =0

r =k+1

Fr ( p − jω0 )r

Z Q ( p) + Z L ( p)

˜ p) = ( p − jω0 )k ψ(

(8.55a)

yielding Br = Sr ; r = 0, 1, 2, . . . , (k − 1)

(8.55b)

Moreover, when both Z Q and Z L approach to zero, their real parts must vanish.

8 Analytic Approaches to Antenna Matching Problems

171

Therefore, they must act like foster functions in the neighborhood of p = jω0 . In this case, we can make the following approximations. Z L ( p) | p= jω→ jω0 ≈ j X L (ω) = X L (ω0 )( p − jω0 );

p = jω

(8.56a)

Z Q ( p) | p= jω→ jω0 ≈ j X Q (ω) = X Q (ω0 )( p − jω0 );

p = jω

(8.56b)

By Foster theorem, d XL |ω=ω0 ≥ 0 dω dXQ X Q (ω0 ) = |ω=ω0 ≥ 0 dω X L (ω0 ) =

(8.56c) (8.56d)

In this case, by equating the like wise coefficients of Eq. (8.56a) for the term ( p − jω0 )k we found Br = Sr ;

r = 0, 1, 2, . . . , (k − 1)

Bk − Sk =

X L

Fk+1 (ω0 ) + X Q (ω0 )

(8.57a) (8.57b)

or Bk − Sk 1 ≥0 = Fk+1 X L (ω0 ) + X Q (ω0 )

(8.58c)

Case C Here, in this case, we also consider a real frequency zero p = jω0 of λ L ( p) of order k for which the load impedance satisfies the inequality 0 < |Z L ( jω0 | < ∞. For the present situation there will be no cancellation occurs unless Z Q + Z L = 0. Hence, Eq. (8.51) yields Br = Sr ;

r = 0, 1, 2, . . . , (k − 1)

(8.58)

provided that Z Q ( jω0 ) + Z L ( jω0 ) = 0 However, if Z Q ( jω0 ) + Z L ( jω0 ) = 0 then, as in Case B, we can say that, in the neighborhood of p = jω0 Z Q ( jω) + Z L ( jω) ≈ j X (ω) = X (ω0 )( p − jω0 ); X (ω0 ) = d X/dω|ω=ω0 ≥ 0. Again, as in the previous case, considering the cancellation and comparing the like wise coefficients of Eq. (8.51) we find that Br = Sr ;

r = 0, 1, 2, . . . , (k − 2)

(8.59a)

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B.S. Yarman

and Bk−1 − Sk−1 =

Fk X (ω0 )

1 Bk−1 − Sk−1 ≥0 = Fk X (ω0 )

or

(8.59b)

Case D For this case, at a real frequency zero p = jω0 of λ L ( p) of order k, the load impedance goes to infinity as p approaches to jω0 . In other words, we assume that, at p = jω0 the load impedance has a pole acting like a foster function as Z L ( p)| p= jω→ jω0 = a 1 / p − jω0 → ∞. For this situation, in the neighborhood of p = jω0 we can state that Z L ( p) ≈ j X L (ω) or

Z L ( p) =

a 1 ; p − jω0

a−1 = lim ( p − jω0 )Z L ( p) p→ jω0

(8.60a)

In this case Eq. (8.51) reveals that Br = Sr ;

r = 0, 1, . . . , (k − 1)

(8.60b)

Fk−1 a 1

(8.60c)

and Bk − Sk = or Fk−1 =a Bk − Sk

1

>0

(8.60d)

The above equation sets namely, Eqs. (8.52), (8.57), (8.59) and (8.60) given for the four different classes of zeros of λ L ( p) are called gain-bandwidth restrictions imposed by the positive real impedance function Z L ( p) when it is broadband to a resistive generator over an Equalizer [E] within specified frequency band. These restrictions first initiated by Bode [x1], then expanded by Fano [x2], and generalized by Youla [x3]. Let us highlight some issues in the following remarks. Remarks

r r r

It should be noted that zeros of λ L ( p) are called the zeros of transmission by Youla. From our perspective, it may be appropriate to call them as the zeros of the load or directly zeros of lambda as it is described by Youla. However, in this book, we consider the cascaded connection of the Load Network [L]-the lossless Equalizer [E] and the Generator Network [G] as a complete matched or broadband system. The power transmission performance of this system is measured by means of the transducer power gain. Therefore, at the points p = p0 = σ0 + jω0 where the transmission of power stops is called the zeros of transmission. Clearly, transmission zeros of the matched system is imposed not

8 Analytic Approaches to Antenna Matching Problems

r

173

only by the load network but also by the equalizer and the generator networks as well. On the other hand, it is clear that, at the load end, transmission zeros overlaps with those of the zeros of Youla’s since A L .A L ∗ 2A L (− p) 2A L ( p) . =4 ∗ ZL + 1 ZL + 1 (Z L + 1) (Z L ∗ + 1) 4N L D L ∗ (8.61) =λ (Z L + 1) (Z L ∗ + 1)

L 21 ( p) .L 21 (− p) =

r

r

r

r

For the single matching problems, let us consider a point p = pd = σd + jωd where power-transmission of the matched systems stops. This point either belongs to the load network, or belongs to the equalizer, or belongs to neither of them; but belongs to the tandem connection of Equalizer [E] and the Load [L]. If this is the case, this situation is called “degeneracy”. A degeneracy occurs when the left-end of the equalizer is connected to the rightend of the load network forming a single-simple reactive cascade section with a transmission zero at p = pd = σd + jωd . Typical examples of this situation are given in Fig. 8.15. Series connection of parallel resonance circuit, parallel connections of series resonance circuit, series connection of inductors and parallel connection of capacitors are considered as single-simple reactive cascade section as shown in Fig. 8.15. In his classical paper, Youla proved that the driving point back-end impedance Z Q = 2η L R L /η L − F22 − Z L = 2η D A R L /η D A − S − Z L of [E] is a “realizable positive real function” if and only if the reflection coefficient S ( p) satisfies the gain-bandwidth constraints given by Equations (8.53), (8.58), (8.60) and (8.61)23 . On the real frequency axis, it is straight forward to shown that Z Q ( jω) = R Q (ω) + j X Q (ω) with R Q (ω) = R L ω)

1 − |S( jω)|2 |η DL ( jω) − S( jω)|2

≥ 0;

∀ω

and X Q (ω) =

r

23

2R L (ω) I m {η DL (− jω)S( jω)} − X L (ω) |η DL − S|2

We should mentioned that for many practical cases real frequency axis zeros of lambda (λ L ) could be at zero or and/or at infinity. For these cases, obviously

It should be recalled that the back-end reflection coefficient F22 of the over all system was given by F22 = η L A S

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B.S. Yarman

the above restrictions are valid. However for the zero at infinity ( p0 = ∞), the load impedance should act like a foster function with the residue a−1 = lim Z L ( p)/ p. p→∞

Now, let us summarize the gain bandwidth restrictions for single matching problems under four class in Table 8.2. Example 4. In this example, a parallel R//C load is equalized to a resistive generator over the normalized frequency band from DC to ωc = 1 employing the analytic theory of broadband matching24 . Let us apply the theory step by step in an implementation algorithm.

Algorithm: Implementation of The Analytic Theory of Broadband Matching Step 1. Drive the analytic form of the load impedance to be equalized: If the load impedance were given as a measured data then, one would need to build an analytic model for it to implement the theory. However, in the present case, it is already given in the open form as R//C impedance. Thus, Z L ( p) = R/1 + p RC. Step 2. Drive the load parameters R L , η DL , F = 2η DL R L , λ L ( p) = R L /Z L ; find the Youla’s zeros of the load network specified by λ L ( p) = R L /Z L and determine the Class of the problem: Table 8.2 Gain bandwidth restriction for single matching problems [3] Zeros of λ L of order k

Status of load the impedance Z L ( p)

Gain-bandwidth restrictions imposed by the load Z L ( p)

Class I (Case A)

p0 = σ0 + jω0 ; σ0 > 0 Zero in open RHP

Z L ( p0 ) is finite

Br = Sr ; r = 0, 1, 2, . . . , (k − 1)

Class II Case B)

Real frequency zero at p0 = jω0

Z L ( jω0 ) = 0

Br = Sr ; r = 0, 1, 2, . . . , (k − 1) Bk − Sk 1 ≥0 = Fk+1 X L (ω0 ) + X Q (ω0 )

Class III (Case C)

Real frequency zero at p0 = jω0

0 < |Z L( jω0 )| < ∞ Br = Sr ; r = 0, 1, 2, . . . , (k − 2) Bk−1 − Sk−1 1 ≥0 = Fk X (ω0 )

Br = Sr ; r = 0, 1, . . . , (k − 1) Fk−1 =a 1>0 Bk − Sk Let us apply the theory of broad band matching to the general form of an RC load of as in Example 1, 2, and 3. Class IV (Case D)

24

Real frequency zero at p0 = jω0

Z L ( jω0 ) → ∞

Here, ωc designates the cutoff frequency of the upper end of the band.

8 Analytic Approaches to Antenna Matching Problems

175

In this case, 1 p− 1 R RC . R L = [Z L ( p) + Z L (− p)] = and η DL ( p) = 1 2 1 − p2 R 2 C 2 P+ RC 2 1 F = 2η DL R L = − . and λ L ( p) = 1 − P RC 1 2 2 RC p + RC The only zero of lambda is at infinity ( jω zero with p = ∞ of order k = 1). In the mean time, at p = ∞, Z L = 0. Therefore, this is a Class II problem. Step 3. (a) Select a transfer function T ω2 which preferably includes all the zeros of the load network genuinely. (b) Then, determine the all pass free form for F22n ( p) 25 (a) Since the only zero of the load network is at infinity and the equalization will be made from DC to 1 then, it may be appropriate to choose a butter-worth form for TPG which eventually yields a low pass filter like structure. In this regard, the general form for the TPG is given by T ω2 =

T0 2n ; ω 1+ ωc

0 ≤ T0 ≤ 1

(8.62)

or T − p2 =

T0 2n ; p 2 1 + (−1) ωc

0 ≤ T0 ≤ 1

(b) In this case, F22n ( p)F22n (− p) = 1 − T (− p 2 ) = where Hn = (1 − T0 ) + (−1) 25

n

p ωc

2n

Hn ( p 2 ) G n ( p2 )

(8.63)

2n

= (1 − T0 ) 1 + (−1)

n

p (1 − T0 )1/2n ωc

A transfer function which is free of RHP zeros is called minimum-phase. Based on this definition, an all pass free function F22n is minimum phase.

176

B.S. Yarman

and G n = 1 + (−1)n

2n p ωc

Let x=

p (1 − T0 )2n ωc

(8.64)

and y=

p ωc

Then, we can factorize Hn (x 2 ) = 1 + (−1)2 x 2n and G n (y 2 ) = 1 + (−1)2 y 2n as Hn (x) = h(x)h(−x) = 1 + a1 x 2 + a2 x 4 + . . . + a(n−1) x 2(n−1) + x 2n

(8.65)

and G n (x) = g(y)g(−y) = 1 + a1 y 2 + a2 y 4 + . . . + a(n−1) y 2(n−1) + y 2n where h(x) = 1 + h˜ 1 x + h˜ 2 x 2 + . . . + h˜ (n−1) x (n−1) + x n g(y) = 1 + g˜ 1 y + g˜ 2 y + . . . + g˜ (n−1) y 2

With h˜ k = g˜ k ; shown that26

(n−1)

+y

(8.66)

n

k = 1, 2, .., (n − 1) is a strictly Hurwitz polynomial and it can be

α(n−1) = h˜ (n−1) = g˜ (n−1) = sin

1 π

(8.67)

2n

Eventually, the minimum phase reflection function F22n ( p) should be F22n ( p) =

h(x) 1 − T0 g(y)

(8.68)

yielding S( p) = ±ηx ( p)F22n ( p) It should be noted that for ωc = 1, p = y and g( p) = g(y). Step 4. Determine the gain-bandwidth restrictions imposed by the load by series expansion of η DL ( p), F( p) and S( p) about the zeros of λ L ( p) : 26

L. Weinberg, “Network design by use of modern synthesis techniques and tables”, Proc. National Electronics Conf. vol.12; October 1956.

8 Analytic Approaches to Antenna Matching Problems

177

The series expansions of the above functions about p = ∞ can be obtained as 2 2 . η DL ( p) = B0 + B1 + . . . . = 1 − + . . . .; yielding B0 = 1, B1 = − RC p 2 RC 2 1 2 F( p) = − + . . . ; yielding F0 = F1 = 0 and F2 = − . 2 2 RC p RC 2 In order to find the series expansion of S( p) let us first expand F22n ( p) about p = ∞. F22n ( p) = 1 + α(n−1)

ωc 1 − (1 − T0 )1/2n 1 1 1 1 π − . + .... = 1 − . + ... x y p p sin 2n

In the mean time, expansion of ηx ( p) =

nx r =1

p − σr / p + σr is obtained as

nx 2 σr + . . . . ηx ( p) = 1 − p r =1 Thus, we have ⎛ ⎡

⎞⎤ nx 1/2n ωc 1 − (1 − T0 ) 1 1⎜ ⎢ ⎟⎥ π S( p) = μ. ⎣1 − ⎝2 σr + ⎠⎦+. . . .. = S0 +S1 +. . . .. p p sin r =1 2n For Class II, gain-bandwidth constraints demands that B0 = S0 = 1 and B1 − S1 /F2 ≥ 0. In this⎡case, we must set the unimodular⎤constant μ = 1. This choice nx ωc 1 − (1 − T0 )1/2n ⎥ ⎢ π yields that S1 = − ⎣2 σr + ⎦. Then, the second constraint r =1 sin 2n becomes ⎡ π ⎤

nx 2 sin ⎥ 1 ⎢ 2n 1/2n 1 − (1 − T0 ) − ≤⎣ σr (8.69) ⎦ ωc RC r =1 In order Eq. (8.69) to have a meaningful solution, T0 must be less than 1. Therefore, we can not choose σr arbitrarily to end up with realizable special filter F. Condition of 0 ≤ T0 ≤ 1 demands that 1 ≥ σr RC r =1 nx

(8.70)

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B.S. Yarman

In this case, it is found that ⎧⎡ ⎫ π ⎤ ⎪2n

⎪ nx ⎨ ⎬ 2 sin ⎢ 2n ⎥ 1 − T0 ≤ 1 − ⎣1 − σr ⎦ ⎪ ⎪ ωc RC ⎩ ⎭ r =1

(8.71)

Equation (8.71) conveys useful information for broadband matching as far as implementation of the theory is concerned. (a) The purpose of the broadband matching is “to design a lossless equalizer to achieve the optimum power transfer over the prescribed frequency band from the generator to the load”. In this regard, while the gain-bandwidth restrictions are satisfied, one has to determine the unknown parameters (σr > 0) and constant gain level T0 in such a way transducer power gain is maximized as flat as possible over the pass-band. From Eq. (8.71) it is clear that insertion of the right half plane zeros σr > 0 of ηx ( p) reduces the constant gain level T0 . Therefore, one must avoid employing the all pass function ηx ( p) if possible. Furthermore, utilization of ηx ( p) complicates the circuit structure and perhaps makes the physical implementation of the matching networks impossible at the microwave frequencies and beyond. (b) Using Butterworth functions, the rectangular-idealized shape of Fig. 8.3 is obtained as the degree n approaches to infinity. Thus, the best solution for the R//C load is achieved by setting σr = 0 with n = ∞. Thus, we can state that maximum achievable flat-transducer power gain level (T0 )max is given by ⎡ (T0 )max

⎢ = 1 − lim 1 − ⎣ n→∞

2 sin

π ⎤2n

2n ⎥ ⎦ RCωc

2π = 1 − e RCωc −

(8.72)

(c) The above equation yields the ultimate gain bandwidth limit of the given R//C load. (d) Assume that the back-end of the equalizer starts with a capacitor C Q , in other words, Y Q ( p) = 1/Z Q ( p) = pC Q + Yˆ Q ( p) introducing a degeneracy together with the load capacitor C L . In this case, we see that the effective value of the back-end capacitor C = C Q + C L increases which in turn reduces the gain by Eq. (8.72). (e) Thus, it is concluded that degenerate equalization penalizes the transducer power gain of the matched system. Therefore one must avoid using degenerate solutions. Step 5. Construct the equalizer by extracting the load from the back-end reflection coefficient leading the synthesis of Z Q ( p) = 2η DL ( p)R L ( p)/η DL ( p) − S( p) − Z L ( p):

8 Analytic Approaches to Antenna Matching Problems

179

By choosing n = 1 and setting ηx = 1 (i.e choosing ␴r = 0 as recommended) let us construct S( p) as √ p + ωc 1 − T0 S( p) = F22n ( p) = F22 ( p) = p + ωc

(8.73)

and Z Q ( p) = 2

η DL ( p)R L ( p) − Z L ( p) η DL ( p) − S( p)

(8.74)

Or Z Q ( p) =

η DL ( p)Z L (− p) + S( p)Z L ( p) η DL ( p) − S( p)

Hence, Eq. (8.73) and Eq (8.74) leads to √ Rωc (1 − 1 − T0 ) Z Q ( p) = √ √ 2 − RCωc 1 − 1 − T0 p + ωc 1 + 1 − T0

(8.75)

In order to make positive real we should have 1−

1 − T0 ≤

2 ωc RC

(8.76)

To avoid degeneracy, one should use the equality resulting in ωc RC =

1−

2 √ 1 − T0

(8.77)

For example, if ωc = 1, R = 1 and √ C = 5 then,√T0 becomes T0 = 0.4 yielding the back end impedance Z Q = 1 − 1 − T0 /1 + 1 − T0 = 0.127. In this case, the equalizer will be a simple transformer with turn ratio n tr = 2.8059 to 1 as shown in Fig. 8.16

Fig. 8.16 Design of the equalizer of Example 4.

180

B.S. Yarman

Summary & Concluding Remarks Let us conclude this chapter by summarizing the implementation algorithm of the analytic theory of broadband matching and also high light some important practical issues. One can attempt to design antenna matching networks utilizing the analytic theory of broadband matching.

r r

r

r

r r r r r

In the course of the analytic design, first antenna impedance must be measured over a wide frequency range to build a reliable circuit model. Then, a reasonable circuit model has to be built in Darlington sense. That is to say, the load network must be represented as lossless two-port [L] which is terminated in unity resistance. At this step, it should be noted that transmission zeros of the load network [L] is specified by means of the transfer scattering coefficient L 21 ( p). To be specific, transmission zero of the load network are defined as the zeros of L 21 ( p)L 21 (− p). Once the load is modeled then, the next step is to choose an appropriate transfer function as the measure of the transducer power gain performance. At this step, one must make sure that the selected transfer function includes all the transmission zeros of the load network. Then, by proper factorization of the transfer function, one obtains the minimum phase back-end reflection coefficient S( p) of the equalizer. The follow-up step covers the derivation of the gain-bandwidth constraints imposed by the load network. At this stage, one may need to insert some unknown parameters as a Blashke product into S( p) without disturbing its amplitude form on the real frequency axis. Then, the unknown parameters are determined satisfying the gain-bandwidth constraints yielding realizable rational positive real back-end impedance Z Q ( p). Eventually, the back-end impedance of the equalizer is synthesized as a lossless two-port in resistive termination yielding the desired equalizer. Let us now point out some difficulties in the implementation phase. At the first place the antenna to be matched is a physical device; it does not come with its circuit model from the production line rather comes with its measured data. To build a reasonable model may be tedious and trouble some to be imbedded within the transfer function. For many practical problems, selection of the transfer function is not easy, beyond simple load models, it could even be impossible. As it was experienced from the above example, derivation of gain-bandwidth constraints may not be straight forward algebra manipulation; and perhaps, beyond simple problems, it may require the solution of non-linear equation sets. Even if one completes the above steps successfully, it may not be practical to physically implement the resulting equalizer circuit.

Finally, we can comfortably state that, based on the above difficulties, analytic theory of broadband matching has not been widely utilized in practice.

8 Analytic Approaches to Antenna Matching Problems

r r r

181

However, it is essential to understand the nature of the problem. Furthermore, one may wisely manipulate the theory to asses the ultimate performance limits of the load network to be broad banded. It has been well practiced over the years that the real-frequency-computer aided design techniques provide excellent solutions to all broadband matching problems since they utilize the essential design skills stems from the analytic theoretical of broadband matching.

Further Reading 1. H. W. Bode, Network Analysis and Feedback Amplifier Design, Princeton, NJ: Van Nostrand, 1945. 2. R. M. Fano, Theoretical limitations on the broadband matching of arbitrary impedances, J. Franklin Inst., vol. 249, pp. 57–83, 1950. 3. D. C. Youla, A new theory of broadband matching, IEEE Trans. Circuit Theory, vol. 11, pp. 30–50, March 1964. 4. H. J. Carlin and P. P. Civalleri, Electronic Engineering Systems Series, J. K. Fidler (ed), Wideband Circuit Design, Boca Raton: CRC Press LLC, 1998. 5. W. K. Chen, Broadband Matching, Theory and Implementations, 2nd ed., Singapore: World Scientific, 1988. 6. H. J. Carlin, A new approach to gain-bandwidth problems, IEEE Trans. Circuits Syst., vol. 23, pp. 170–175, April 1977. 7. H: J. Carlin and J. J. Komiak, A new method of broadband equalization applied to microwave amplifiers, IEEE Trans. Microw. Theory Tech., vol. 27, pp. 93–99, Feb.1979. 8. H. J. Carlin and P. Amstutz, On optimum broadband matching, IEEE Trans. Circuits Syst., vol. 28, pp. 401–405, May 1981. 9. B. S. Yarman, Broadband Matching a Complex Generator to a Complex Load, PhD thesis, Cornell University, 1982. 10. H. J. Carlin and P. P. Civalleri, On flat gain with frequency-dependent terminations, IEEE Trans. Circuits Syst., vol. 32, pp. 827–839, August 1985. 11. B. S. Yarman, A simplified real frequency technique for broadband matching complex generator to complex loads, RCA Review, vol. 43, pp. 529–541, Sept.1982. 12. H. J. Carlin and B. S. Yarman, The double matching problem: Analytic and real frequency solutions, IEEE Trans. Circuits Syst., vol. 30, pp. 15–28, Jan. 1983. 13. B. S. Yarman and H. J. Carlin, A simplified real frequency technique applied to broadband multi-stage microwave amplifiers, IEEE Trans. Microw. Theory Tech., vol. 30, pp. 2216–2222, Dec. 1982. 14. V. Belevitch, Classical Network Theory, San Francisco: Holden Day, 1968. 15. L. Weinberg, “Network design by use of modern synthesis techniques and tables”, Proc. National Electronics Conf. vol.12; October 1956. 16. D. C. Youla, H. J. Carlin, and B. S. Yarman, Double broadband matching and the problem of reciprocal reactance 2n-port cascade decomposition, Int. J. Circuit Theory and Appl., vol.12, pp. 269–281, Febr. 1984. 17. A. Fettweis, Parametric representation of brune functions, Int. J. Circuit Theory and Appl., vol. 7, pp. 113–119, 1979. 18. J. Pandel and A. Fettweis, Broadband matching using parametric representations, IEEE Int. Symp. Circuits Syst., vol. 41, pp. 143–149, 1985. 19. B. S. Yarman and A. Fettweis, Computer-aided double matching via parametric representation of brune functions, IEEE Trans. Circuits Syst., vol. 37, pp. 212–222, Febr. 1990.

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20. B. S. Yarman, Modern approaches to broadband matching problems, Proc. IEE, vol.132, pp. 87–92, April 1985. 21. J. Pandel and A. Fettweis, Numerical solution to broadband matching based on parametric ¨ representations, Archiv Elektr. Ubertragung, vol. 41, pp. 202–209,1987. 22. B. S. Yarman, Real frequency broadband matching using linear programming, RCA Review, vol. 43, pp. 626–654, Dec. 1982. 23. W.K. Chien, A theory of broadband matching of a frequency dependent generator and load, J. Franklin Inst., vol. 298, pp.181–221, Sept. 1974. 24. W. K. Chen and T. Chaisrakeo, Explicit formulas for the synthesis of optimum bandpass butterworth and chebyshev impedance-matching networks, IEEE Trans. Circuits Syst., vol. CAS-27, no. 10, October 1980. 25. W. K. Chen, Explicit formulas for the synthesis of optimum broadband impedance matching networks, IEEE Trans. Circuits Syst., vol. 24, pp.157–169, April 1977. 26. Y. S. Zhu and W. K. Chen, Unified theory of compatibility impedances, IEEE Trans. Circuits Syst., vol. 35, no. 6, pp. 667–674, June 1988. 27. Satyanaryana and W. K. Chen, Theory of broadband matching and the problem of compatible impedances, J. Franklin Inst., vol. 309, pp. 267–280, 1980.

Chapter 9

Simplified Real Frequency Technique Design of Ultra Wideband Antenna Matching Networks Binboga Siddik Yarman

Introduction Due to the difficulties encountered during the implementation phase of the analytic gain-bandwidth theory, it is preferred to deal with computer aided design techniques (CAD) to construct antenna matching. For all CAD techniques the goal is to optimize the performance of the matched systems by means of a computer algorithm. There are excellent software tools which are commercially available for RFengineers to analyze the performance of microwave circuits and also to design circuits for pre-set performance targets. In these tools, first a circuit topology is described with initial element values. Then, the performance of the circuit is optimized by varying the element values. In all the numerical optimization algorithms, the user must define an objective function in terms of the unknown parameters of the problem. In the case of the antenna matching network design problem, the user provides the measured antenna data to the program. Then, a reasonable circuit topology for the equalizer is selected. Let these elements be {X 1 , X 2 , . . . , X n }. For the optimization, one must define the transducer power gain (TPG) as a function of these unknown parameters together with the measured antenna data which could be provided as an impedance Z L ( jω) = R L (ω) + j X L (ω), or an admittance Y L = 1/Z L or a reflectance SL = L 11 = (Z L − 1)/(Z L + 1) data. In mathematical terms, we say that Transducer Power Gain T (ω) is the function of the real frequency variable ω, measured antenna data (let it be Z L ), and the unknown matching circuit elements {X 1 , X 2 , . . . , X n } and we write T = T (ω, Z L , X 1 , X 2 , . . . , X n )

(9.1)

The target for the optimization may be defined as “to reach to a pre-fixed gain level T0 as high and as flat as possible”. In this case, we set-up an error function B.S. Yarman College of Engineering, Department of Electrical-Electronics Engineering, Istanbul University, 34320, Avcilar, Istanbul, Turkey e-mail: [email protected]

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

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ε(ω) = T (ω, R L (ω), X L (ω), X 1 , X 2 , .., X n ) − T0

(9.2)

which is to be minimized over the prescribed band of operation say from ω1 to ω2 . In the classical literature, there are many optimization tools available to minimize the error function ε = ε(ω, T, T0 ). These tools are very practical to deign narrow bandwidth problems with one or two matching elements in the circuit. However, for wideband problems, equalizers require many elements to achieve a reasonable match between the generator and the load networks. It is well established that the transducer power gain of the match systems is highly nonlinear in terms of the element values of the matching networks. In order to appreciate this statement, let us comment on the nature of the matching problem as far as its nonlinear behavior is concerned. Assume that we try to solve a matching problem using a commercially available CAD package where the designer first selects the circuit topology and then, determines its element values to optimize the transducer power gain of the matched structure. In this regard, consider a simple case where we start with a low pass LC ladder which consists of n – sections. Let us designate the element values of this ladder by X i . In other words, X i either designates a series inductance L i or a shunt capacitor Ci . Let us now derive the driving point impedance Z in ( p) in terms of the elements values X i when the back end of the matching network is terminated in unit resistance. Thus, Z in ( p) = N ( p)/D( p) is given by Z in ( p) = X 1 p +

1 X2 p +

1 X3 +

(9.3)

1 ................... 1 .. + Xn p + 1

For example if n = 1, Z in ( p) = X 1 p + 1

(9.4)

For n = 2 X 1 X 2 p2 + X 2 p + 1 X1 p + 1

(9.5)

a5 p 5 + a4 p 4 + a3 p 3 + a2 p 2 + a1 p + 1 N (5) ( p) = (4) D ( p) b4 p 4 + b3 p 3 + b2 p 2 + b1 p + 1

(9.6)

Z in ( p) = or n = 5 yields, Z in ( p) =

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where a5 = X 1 X 2 X 3 X 4 X 5 ; a4 = X 2 X 3 X 4 X 5 ; a3 = X 1 X 4 X 5 + X 2 X 4 X 5 + X 1 X 2 X 5 ; a2 = X 2 X 5 + X 4 X 5 ; a1 = X 1 + X 2 + X 3 ; and

(9.7)

b4 = X 1 X 2 X 3 X 4 ; b3 = X 1 X 4 + X 2 X 4 + X 1 X 2 ; b2 = X 1 X 4 + X 2 X 4 + X 1 X 2 ; b1 = X 2 + X 4 As far as the measure of nonlinearity is concerned, the polynomial N (1) ( p) = X 1 p + 1

(9.8)

is said to be linear in variable X 1 . The polynomial N (2) ( p) = X 1 X 2 p 2 + X 2 p + 1

(9.9)

is said quadratic in variables X 1 and X 2 . Similarly, polynomials N (5) ( p) and D (4) ( p) have degree of nonlinearity dnon = 5 and dnon = 4 in variables X i ; (i = 1, 2, 3, 4, 5) respectively. Obviously, nonlinearities double when we work with the norms of the above polynomials. For example, the even polynomial ˆ (10) (ω2 ) = N (5) ( jω)N (5) (− jω) = aˆ 5 ω10 + aˆ 4 ω8 + aˆ 3 ω6 + aˆ 2 ω4 + aˆ 1 ω2 +1 (9.10) N has degree of non-linearity dnon = 10 in variables X i since its leading coefficient is specified by aˆ 5 = −(X 1 X 2 X 3 X 4 X 5 )2 doubling the degree of nonlinearity. In short, we can say that for an n – element ladder network, degree of nonlinearity ˆ 2 ) is dnon = 2n in element values X i . of the norm function N(ω Now, let us consider the scattering parameters Fi j ; i, j = 1, 2 of the LC ladder. By proper normalization, the input scattering parameter F11 ( p) is given by F11 ( p) =

N ( p) − D( p) h( p) Z in ( p) − 1 = = Z in ( p) + 1 N ( p) + D( p) g( p)

(9.11)

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where h( p) = N ( p) − D( p) = h 0 + h 1 p + . . . + h n p n

(9.12)

and g( p) = N ( p) + D( p) The transfer scattering parameter F21 ( p) is specified as F21 ( p) =

1 g( p)

(9.13)

It should be noted that losslessness condition requires g( p)g(− p) = h( p)h(− p) + 1

(9.14)

By reciprocity we state that F12 ( p) = F21 ( p). Finally, the back-end scattering parameter F22 ( p) is F22 ( p) = −

h(− p) g( p)

(9.15)

Let us now consider the simplest hypothetical matching problem where the generator and the load networks are purely resistive1 . In this case, the transducer power gain is given by T (ω) = |F21 ( jω)|2 =

1 1 = g( jω)g(− jω) h( jω)h(− jω) + 1

(9.16)

Or equivalently the polynomial P(ω2 ) =

1 = g( jω)g(− jω) > 0 T (ω)

(9.17)

= h( jω)h(− jω) + 1 > 0 Clearly, the polynomial P(ω2 ) has degree of nonlinearity of dnon = 2n in terms of the element values X i since it is specified by the nonlinearity measure of the norm function g( jω)g(− jω) = [D( jω) − N ( jω)][D(− jω) − N (− jω)]

(9.18)

1 This is actually the well established filter design problem. We can always comprehend matching problems as a special filter design problem where complex load and generator impedances deviates from standard terminations by some complex amounts gradually.

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On the other hand, nonlinearity is always quadratic in terms of the real coefficients {h 0 , h 1 , . . . , h n } of the polynomial h( p) no matter what the total number of circuit elements are. For the matching problem under consideration, maximization of the gain function is equivalent to minimization of the polynomial P(ω2 ) in terms of the selected unknowns over the specified bandwidth. In the theory of optimization, it is well known that, if the degree of the nonlinearity of the objective function goes beyond 2 then, one can easily be trapped in local minima and perhaps ultimate convergence becomes impossible. Furthermore, non-linear optimization algorithms are triggered with good initials on the unknowns. Therefore, the end results are pretty much depends on the initial values. In other words, we can say that non-linear optimizations are highly sensitive to initial values of the unknown parameters. In this regard, the computer aided matching network design methods which start with the selection of a network topology is called the “Brute Force” design techniques due to their sensitivity to element values, non-linear behavior and the non-convergent tendencies. On the other hand, if the designer initiates the matching network design on the real coefficients of the polynomial h( p), for sure, the optimization is quadratic and convergent. This is the situation for the hypothetical problem stated above. If the load network becomes complex, then the optimization becomes even harder on the element values. Thus, the above numerical aspect of the matching problem leads us to the Simplified Real Frequency Technique to construct antenna matching networks [1, 2, 3, 4, 5, 6].

“A Scattering Approach” to Matching problems The simplified real frequency technique (SRFT) is a CAD procedure to construct matching networks for various kinds of problems [1, 2, 3, 4, 5]. It posses outstanding features compared to analytic and brute force techniques as follows. (a) Compared to analytic theory of broadband matching, SRFT neither requires a model for the load network, nor requires the selection of transfer function. Therefore, one does not need to satisfy complicated gain-bandwidth restrictions imposed by the antenna. Thus, the analytic theory is simply by passed. (b) As oppose to brute force techniques, in SRFT, lossless equalizer is described by means of its bounded real input reflection coefficient. Therefore, the objective function is expressed in terms of its numerator polynomial h( p). Thus, at the beginning of the design process, one does not need to select a circuit topology for the matching network. Furthermore, the degree of the non-linearity of the objective function is almost quadratic as explained above. Hence, the end results are not sensitive to the initials and the convergence is almost guaranteed. Moreover, the SRFT algorithm presents excellent numerical stability since all

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the computations are made on the bounded real scattering parameters of the equalizer and on the reflection coefficient of the antenna. In the literature, it has been shown that SRFT results in outstanding circuit performance in designing antenna matching networks over the analytic and brute force techniques.

SRFT to Design Antenna Matching Networks The basis for the Simplified Real Frequency Techniques (SRFT) is to describe the lossless equalizer [E] in terms of its unit normalized reflection coefficient in Belevitch form such that E 22 ( p) = h ( p) /g ( p). Therefore, it is also referred as the “Scattering Approach” to matching problems. It can be shown that for the pre-fixed transmission zeros, the complete scattering parameters of a lossless reciprocal equalizer can be generated from the numerator polynomial h ( p) of E 22 ( p), using the para-unitary condition. This idea constitutes the crux of the Simplified Real Frequency Technique. Thus, TPG of the matched system is expressed as an implicit function of h ( p). Referring to Fig. 9.1, assume that the antenna which is described by means of its impedance data Z L ( jω), is matched to a resistive generator over a wide frequency band. In this case, let L i j ; i, j = 1, 2 be unit normalized scattering parameters of the load network. Then, the transducer power gain of the matched system is given by T (ω) =

|E 21 |2 . |L 21 |2 |1 − E 22 .L 11 |2

(9.19)

By losslessness condition of [E] and [L] we have, |E 21 |2 = 1 − |E 22 |2 =

| f ( jω)|2 g( jω)g(− jω)

where g( jω)g(− jω) = h( jω)h(− jω) + f ( jω) f (− jω) and the load specified quantities are L 11 =

ZL − 1 ZL + 1

|L 21 |2 = 1 − |L 11 |2

(9.20)

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Fig. 9.1 Antenna matching problem via SRFT

Thus, from Eq. (9.20) we have2 , 9 8 [ f ( jω). f (− jω)] . 1 − |L 11 |2 T (ω) = h( jω).h(− jω). 1 + |L 11 |2 + f ( jω). f (− jω) − 2r eal [L 11 .h( jω).g(− jω)] (9.21)

or T (ω) =

h( jω).h(− jω). 1 + |L 11 |

2

W (ω) + f ( jω). f (− jω) − 2r eal [L 11 .h( jω).g(− jω)]

where 8 9 W (ω) = [ f ( jω). f (− jω)] . [1 − |L 11 |2 is regarded as a frequency dependent weight factor on the transducer power gain. Notice that for f ( p) = 1, Eq. (9.21) agrees with Eq. (9.16) as L 11 goes to zero which is the lowpass LC ladder filter case.

2

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In Eq. (9.21):

r r r

L 11 = Z L − 1/Z L + 1 is determined from the measured antenna impedance. f ( jω) is constructed on the proper transmission zeros of the equalizer which is specified by the designer. Obviously f ( p) must be free of real frequency zeros within the pass-band3 . In selecting transmission zeros of f ( p), one should pay attention to the practical implementation of the matching network. In this regard, it is well known that low-pass LC ladders yield excellent sensitivity on the element values. They are easy to manufacture. Therefore, if appropriate, the best choice for f ( p) is f ( p) = 1

(9.22)

However, for some special cases, the matching network design may require the following choice. f ( p) = p k

(9.23)

It can be shown that the above choice of f ( p) compensates some reactive swings of the antenna impedance within the band of operation by introducing either series capacitors and/or shunt inductors in the equalizer topology4 .

r

The numerator polynomial h( p) = h 0 + h 1 p + h 2 p 2 + . . . + h n p n

r

(9.24)

is selected as the unknown of the antenna matching problem. Clearly, all the unknown coefficients {h 0 , h 1 , . . . , h n } are real. The strictly Hurwitz denominator polynomial g( p) = g0 + g1 p + g2 p 2 + . . . + gn p n

(9.25)

is uniquely determined from the losslessness equations given by G(− p 2 ) = g( p)g(− p) = h( p)h(− p) + f ( p) f (− p)

(9.26)

3 Regarding antenna matching network design problems, the designer does not really much concern with the shape of the transducer power gain in the stop-band. Rather, the design efforts are concentrated on the passband to make TPG as flat and as high as possible. Therefore, we always avoid utilizing transmission zeros within the equalizer since these zeros complicate the matching network structure. 4 It should be mentioned that for ultra-wideband designs, depending on the type of antenna, if the antenna exhibits a parallel capacitive impedance at the lower edge of the band, this capacitance can be compensated with a shunt inductance at the input of the equalizer by introducing a resonance at an arbitrary point on the frequency axis which makes the shunt resonance circuit open. Similarly, a series inductance seen from the antenna side can be compensated with a series capacitance at the equalizer input creating a series resonance circuit. In this case, a good choice for f ( p) is f ( p) = p or f ( p) = p 2 .

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Obviously, Eq. (9.26) implies that if f ( p) is properly selected by the designer and the real coefficients {h 0 , h 1 , . . . , h n } are initialized then, the polynomial g( p) is constructed on the left-hand plane roots of the even polynomial G(− p 2 ). This way of thinking constitutes the skeleton of the Simplified Real Frequency Technique and may be stated as in the main theorem. Main Theorem. For properly pre-fixed transmission zeros { pzk ; k = 1, 2, . . . , nz}, a lumped element, lossless reciprocal two-port [E] can be constructed from the numerator polynomial h( p) of the real normalized input reflection coefficient E 22 ( p) = h( p)/g( p) provided that h( pzk ) = 0. Proof. Since the transmission zeros { pzk ; k = 1, 2, . . . , nz} are known, the monicnumerator polynomial f ( p) of the transfer scattering parameter E 21 ( p) = f ( p)/g( p) nz % ( p − pzk ), as described in can properly be formed on these zeros as f ( p) = k=1

Chapter 75 . Then, by Eq. (9.26), one can always generate a strictly Hurwitz polynomial g( p) on the LHP roots of G(− p 2 ), provided that h( pzk )h(− pzk ) = 0. Now, let us construct g( p) as follows. On the real frequency axis Eq. (9.26) yields that G(ω2 ) = |h( jω)|2 + | f ( jω)|2 > 0 ;

∀ω

(9.27)

Equation (9.27) is always strictly positive if and only if h( jω) and f ( jω) do not vanish simultaneously. This is warranted by the statement of the theorem6 . In this case, by setting X = ω2 , we can comfortably state that the polynomial G(X ) is always positive and therefore, it must be free of X = 0 and positive real roots7 . Hence, G(X ) only posses negative real roots and complex roots with non-zero real parts. Certainly, the complex roots must be accompanied with their conjugates since G(X ) is a real polynomial. Let a root of G(X ) is designated by X k = αk + jβk = ρk e jφk ;

αk = 0

(9.28)

5 The monic polynomial is defined as the one which has a unity coefficient for its leading term. That is, f ( p) = f nz p nz + . . . f 1 p + f 0 is monic if f nz = 1. 6 The condition h( p )h(− p ) = 0 implies that G(ω2 ) can never be zero. When f ( jω) = 0, the zk zk numerator polynomial h( jω) = 0 as stated by the theorem. Similarly, if h( jω) = 0 then, f ( jω) must be different than zero otherwise a zero of f ( jω) becomes the zero of h( jω) which contradicts with the statement of the theorem. Therefore, the coefficients {h 0 , h 1 , . . . , h n } are selected in such a way that f ( jω) and h( jω) are not simultaneously zero. Thus, G(ω2 ) is always positive for all ω. 7 It should be note that if G(− p 2 ) has a root on the jω axis at p = jω then, one would write that 0 ˜ p 2 ) or G(ω2 ) = (−ω2 + ω2 )G(ω ˜ 2 ) or G(X ) = (−X + ω2 )G(X ˜ ) which G(− p 2 ) = ( p 2 + ω02 )G(− 0 0 2 2 would make G(ω ) zero at ω = ω0 (or at X = ω0 ) and makes it negative for the values of ω > ω0 (or X > ω02 ). This contradicts with the fact that G(ω2 ) > 0. Therefore, G(X ) can never have a positive real root; rather it may have negative real roots. For example, at a given negative real root ˜ ) yielding G(− p 2 ) = ( p − ζ )( p + ζ )G(− ˜ p 2 ). X = −ζ 2 , G(X ) = (X + ζ 2 )G(X

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where ρk =

: αk2 + βk2 −1

φk = tan

βk αk

If βk = 0 then αk must be strictly negative. Then, setting p 2 = −X , roots of G(− p 2 ) is found as √ j π+φk ρk e 2 p = ∓ X k = ∓ − (α k + jβk ) =

(9.29)

or p=∓

√

ρk . cos

π + φk 2

! π + φk √ + j ρk . sin = ∓ (σk + jγk ) 2

Where σk =

√

π + φk ρk . cos 2

;

π + φk √ γk = ρk . sin 2

√ √ If βk = 0 then, p = ∓ −αk = ∓ζ such that ζ = −αk > 0 As we see from Eq. (9.29), all the roots of G(− p 2 ) have mirror symmetry (Fig. 9.2). Thus, strictly Hurwitz polynomial g( p) is constructed on the left half plane roots of G(− p 2 ) yielding the complete scattering parameters of the lossless two-port [E] in the Belevitch form as

Fig. 9.2 The roots of G(− p 2 ) exhibits mirror symmetry

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h( p) f ( p) f (− p) h(− p) , E 12 ( p) = E 21 ( p) = ∓ , E 11 ( p) = − . E 22 ( p) = g( p) g( p) f ( p) h( p)

!

This completes the proof. Now let us evaluate G(− p 2 ) in terms the real coefficients of h( p)

Numerical Construction of Strictly Hurwitz Polynomial g( p) In this section details of the numerical construction of the strictly Hurwitz polynomial g( p) from the given real coefficients of h( p) is presented. Let the numerator polynomial be h( p) = h 0 + h 1 p + h 2 p 2 + . . . + h n−1 p n−1 + h n p n and for the sake of simplicity, let the monic polynomial which contains transmission zeros of the equalizer be f ( p) = p k . Now, let us form the even polynomial H (− p 2 ) = h( p)h(− p): H (− p 2 ) = h( p).h(− p) = h 0 + h 1 p + h 2 p 2 + h 3 p 3 + h 4 p 4 + h 5 p 5 + . . . + h n−1 p n−1 + h n p n h 0 − h 1 p + h 2 p2 − h 3 p3 + h 4 p4 − h 5 p5 + . . . +(−1)n−1 h n−1 p n−1 + (−1)n h n p n or H (− p 2 ) = h 20 − h 0 h 1 p + h 0 h 2 p 2 − h 0 h 3 p 3 + h 0 h 4 p 4 − h 0 h 5 p 5 + . . . . . . . . . . . . . . . . . . . . . . . . . +h 0 h 1 p − h 21 p 2 + h 1 h 2 p 3 − h 1 h 3 p 4 + h 1 h 4 p 5 + . . . . . . . . . . . . . . . . . . . . +h 0 h 2 p 2 − h 1 h 2 p 3 + h 22 p 4 − h 2 h 3 p 5 + . . . . . . . . . . . . . . . . +h 0 h 3 p 3 − h 1 h 3 p 4 + h 2 h 3 p 5 + . . . . . . .. +h 0 h 4 p 4 + h 0 h 5 p 5 + . . . .

or we can write that selectfont H (− p 2 ) = h 20 + [(h 0 h 2 − h.1 h 1 + h 2 h 0 ] p 2 + [(h 0 h 4 − h 1 h 3 + h 2 h 2 − h 3 h 1 + h 4 h 0 )] p 4 + . . . +

0 2i /

1 (−1)r h r h 2i−r

p 2i + .. + (−1)n p 2n

r =0

Thus, H (− p 2 ) = h( p)g(− p) =

n / i=0

Hi p 2i

(9.30)

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where H0 = h 20 H1 = h 0 h 2 − h 1 h 1 + h 2 h 0 = −h 21 + 2h 0 h 2 H2 = h 0 h 4 + h 2 h 2 − h 1 h 3 − h 3 h 1 + h 4 h 0

2x2 2 / / r 2 r −1 = (−1) h r h 4−r = h 2 + 2 h 4 h 0 + (−1) h r −1 h 2r −2+1 r =0

r =2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

2i k / / r k 2 r −1 Hi = (−1) h r h 2i−r = (−1) h i + 2 h 2i h 0 + (−1) h r −1 h 2r −i+1 ; r =0

r =2

i ≤ (n − 1) ............................................................. Hn = (−1)2 h 2n It should be noted that in the above formulation, for any index k > n the coefficients h k must be zero since the coefficients of h( p) are only defined up to index n. In this case, G(− p 2 ) = g( p)g(− p) =

n /

G 1 p 2i = H (− p 2 ) + (−1)k p k

(9.31)

i=0

Thus, the coefficients G i is given by G 0 = H0 = h 20 ≥ 0 ...................... 2i G i = Hi = h r h 2i−r r =0

...................... 2k G k = Hk + (−1)k = h r h 2k−r + (−1)k

(9.32)

r =0

.................................. G n = (−1)n h 2n ≥ 0 It should be noted that the above lengthly operations can easily be evaluated employing the MatLab function “convolution”. This function is called by the MatLab statement “W = conv(U, V)” where, the vectors “U” and “V” are the coefficients of polynomials U (X ) and V (X ) to be multiplied and they are given by U (X ) = U1 X n + U2 X n−1 + . . . + Un X + Un+1 and V (X ) = V1 X n + V2 X n−1 + . . . + Vn X + Vn+1 Similarly, W contains the coefficients of the polynomial W (X ) = U (X ).V (X ). Equation (9.32) simply indicates that once the coefficients {h 0 , h 1 , h 2 , h 3 , . . . h n−1 , h n } are initialized, coefficients of the even polynomial G(− p 2 ) = G 0 +

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G 1 p 2 + G 2 p 4 + . . . + G n p 2n are immediately computed leading the construction of the strictly Hurwitz polynomial g( p) as follows. There are two major approaches to generate strictly Hurwitz g( p). 1. The first one is a classical method for which the mirror symmetric roots of G(− p 2 ) = G 0 + G 1 p 2 + G 2 p 4 + . . . + G n p 2n is determined by means of a well known root finding techniques. For example, MatLab’s function “pr = roots(C)” computes all the possible roots of a given polynomial C(X ) = C1 X n + C2 X n−1 + . . . + Cn X + C n+1 ; where the argument vector C contains all the coefficients such that2 C = C1 C2 . . . Cn+1 . Then, g( p) is constructed on the LHP roots of G(− p ). Let these roots be pr = −σr ∓ jγr ; σr ≥ 0. Then, g( p) =

Gn

n $

[( p + σr ) ± jγr ]

(9.33)

r =1

It should be clear that in Eq. (9.33), complex roots ( pr = −σr ∓ jγr ; σr ≥ 0) must appear with their complex-conjugates pr +1 = −σr +1 + jγr +1 = pr∗ = −σr ± jγr . If the imaginary part βr of Eq (9.28) is zero then, a single term ( p + σr ) appears in g( p) by its self. From the numerics point of view, Eq. (9.33) can easily be evaluated on MatLab8 calling the function “C = poly(p) where the vector p = p1 . . . p contains all the roots of the polynomial C(X ) and C˜ = (C˜ 1 = 1) C˜ 2 . . . C˜ n+1 ˜ ) with is a row vector contains the coefficients of a monic polynomial C(X ˜ ) in a straight forward C˜ 1 = 1. Obviously, C(X ) is directly obtained from C(X manner by multiplying C(X ) with the original leading term coefficient C1 . Thus, ˜ ). C(X ) = C1 .C(X 2. In the second approach, by equating the coefficients of g( p)g(− p) = G(− p 2 ), a quadratic equation set is solved yielding the coefficients {g0 , g1 , g2 , . . . , gn−1 , gn } of the strictly Hurwitz polynomial g( p) = g0 + g1 p + g2 p 2 + . . . + gn−1 p n−1 + gn p n as follows. g02 = G 0 > 0 − g0 g2 + g1 g1 − g2 g0 = G 1 g0 g4 − g1 g3 + g2 g2 − g3 g1 + g4 g0 = G 2 g0 g6 − g1 g5 + g2 g4 − g3 g3 + g4 g2 − g5 g1 + g6 g0 = G 3 .................................. 2i /

(−1)r gr g2i−r = G i

r =0

.................................. gn2 = (−1)n G n > 0

8

c MATLAB , “The Language of Technical Computing”, by MathWorks, Inc. USA

(9.34)

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Equation (9.34) is called the “fundamental equation set” which leads to Strictly Hurwitz factorization of g( p)g(− p) = G(− p 2 ) in n + 1 unknowns. However, among the unknowns g0 and gn are immediately found as, g0 = gn =

G0 > 0

(9.35)

|G n | > 0

Equation (9.34) can be solved employing the “Stable Factorization” method developed by Omer Egicioglu9 . This method solves the quadratic equation set by Newton’s iteration with special care. It is proved that the method is stable and convergent. Thus, the root finding techniques are by passed in the Stable Factorization method. Details are omitted here. Hence, the numerical construction of the strictly Hurwitz polynomial is completed. Now let us run some examples with the help of MatLab. Example 1. Assuming a lowpass ladder structure for the lossless equalizer with f ( p) = 1, here, we present an example for the construction of strictly Hurwitz g( p) from the given numerator polynomial h( p) = 0 + p + 2 p 2 + p 3 which means that h 0 = 0,

h 1 = 1,

h 2 = 2,

h 3 = 1.

Step-1: Firstly, we have to generate the even polynomial using Eq. (9.30). H (− p 2 ) = H0 + H1 p 2 + H2 p 4 + H3 p 6 with h 0 = 0, h 1 = 1, h 2 = 2, h 3 = 1. where H0 = h 20 = 0, H1 = h 0 h 2 − h 1 h 1 + h 0 h 2 = −1 H2 = h 0 h 4 − h 1 h 3 + h 22 − h 3 h 1 + h 4 h 0 = 0 − 1 + 4 − 1 + 0 = +2 H3 = h 0 h 6 − h 1 h 5 + h 2 h 4 − h 23 + h 4 h 2 − h 1 h 5 + h 6 h 0 = 0 − 0 + 0 − 1 + 0 − 0 + 0 = −1 Notice that in the above equations h 4 = h 5 = h 6 = 0 since h( p) runs up to 3rd term. Above computations can be completed on MatLab using the function convolution. 9 O. Egicioglu, B.S. Yarman,”Stable Factorization to Construct Strictly Hurwitz Polynomials for Real Frequency Techniques”, Special Report, Department of Computer Science, University of California Santa Barbara, CA 93106, USA, [email protected], July 20, 2005

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In this case, by flipping the order of the coefficients for MatLab we set U = [(h 3 = 1) (h 2 = 2) (h 1 = 1) (h 0 = 0)] , V = [−1 2 −1 0] and W = conv(U, V ) yields the full coefficients of H ( p) in descending order including zeros for the coefficients of odd terms. Thus we have the following output from MatLab: >> W = conv(U, V) W= −1 0 2 0

−1

0

0

Which means that W = [−1 0 2 0 − 1 0 0] with its corresponding full polynomial H ( p) = −1 p 6 + 0 p 5 + 2 p 4 + 0 p 3 − p 2 + 0 p + 0 Step 2: In this step, using Eq. (9.32) even polynomial G(− p 2 ) must be generated. Thus, G 0 = H0 + 1 = 1, G 1 = H1 = −1, G2 = 2 G 3 = −1. Or G(− p 2 ) = − p 6 + 2 p 4 − p 2 + 1 = (− p 2 )3 + 2(− p 2 )2 − p 2 + 1. By setting X = − p 2 we have G(X ) = X 3 + 2X 2 + X + 1 Step 3: Now, we have to find the roots of G(X = − p 2 ) employing the MatLab function “roots”. Then, generate the mirror symmetric roots of G(− p 2 ). (a) Computation of the roots of G(X ): Here, we set the row vector as X = [1 2 1 1] and call the function roots. Thus we have the following output from MatLab: >> X = [1 2 1 1] >> Xk = roots(X) Xk = −1.7549 −0.1226 + 0.7449i −0.1226 − 0.7449i

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As it is observed, the real root of G(X ) is negative at X 1 = α1 = −1.7549 as it should be. 2 (b) Now, √ we can compute the mirror symmetric roots of G(− p ) by setting Pk = ∓ −(X k ). Hence, the MatLab function sqrt (−Xk) yields the roots with “+” sign. Then, by introducing “−” sign we can find the mirror image symmetry of these roots. Hence, the MatLab output is given by: >> pk = sqrt(−Xk) pk = 1.3247 0.6624 − 0.5623i 0.6624 + 0.5623i The mirror symmetry of these roots are given by pkm = −pk. pkm = −1.3247 −0.6624 + 0.5623i −0.6624 − 0.5623i Step 4: Compute the monic polynomial g˜ ( p) constructed on the LHP roots pkm of G(− p 2 ) using the MatLab function “C = poly(pkm)”. Thus the MatLab output is: >> C=poly(pkm) C= 1.0000 2.6494 2.5098 1.0000 Which means that g˜ ( p) = p 3 + 2.6494 p 2 + 2.5098 p + 1 √ Step 5: Finally, the strictly Hurwitz polynomial g( p) = |G 3 |g˜ ( p) is computed. In this case, G 3 = −1. Therefore, g( p) = p 3 + 2.6494 p 2 + 2.5098 p + 1. Thus, the numerical construction of the stricty Hurwitz polynomial g( p) is completed within 5 steps. In order to shown the computational efficiency of the above algorithm, let us run the following example on MatLab and present the results without details. Example 2. Let f ( p) = 1 and h( p) = 5 p 5 + 3 p 4 + p 3 + 0.5 p 2 + p find the strictly Hurwitz polynomial g( p). This problem is little bit more complicated that that of example 1. However, using matLab functions, the solution is straight forward. Step 1: Compute H (− p 2 ) by Eq. (9.30). Set U = [5 3 1 0.5 1 0], V = [−5 3 − 1 0.5 − 1 0] and call matLab function W = conv(U, V). The MatLab output is: W = [−25.0000 0 − 1.0000 0 − 8.0000 0 − 1.7500 0 − 1.0000 0 0] Which means that H (− p 2 ) = −25 p 10 − p 8 − 8 p 6 − 1.75 p 4 − p 2 + 0

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Step 2: Compute G(− p 2 ) using Eq. (9.32). G(− p 2 ) = H (− p 2 ) + 1. Thus we have, G(− p 2 ) = −25 p 10 − p 8 − 8 p 6 − 1.75 p 4 − p 2 + 1 Step 3: Set X = − p 2 find the roots of G(X ) and also mirror sysmmetric roots of G(−p2 ): (a) G(X ) = (25)(− p 2 )5 − (− p 2 )4 + (8)(− p 2 )3 + (−1.75)(− p 2 )2 + (− p 2 ) + 1 Or G(X ) = 25X 5 − X 4 + 8X 3 − 1.75X 2 + X + 1 Let the MatLab row vector be X = [25 − 1 8 − 1.75 1 1]. Then, the roots are Xk = roots[X]: Xk = −0.1734 + 0.6366i −0.1734 − 0.6366i 0.3645 + 0.3680i 0.3645 − 0.3680i −0.3424 The real root of G(X ) is negative as expected. (b) Then, we compute the mirror symmetric roots of G(− p 2 ): pk = sqrt(−Xk): 0.6454 − 0.4932i 0.6454 + 0.4932i 0.2770 − 0.6643i 0.2770 + 0.6643i 0.5851 and >> pkm = −pk pkm = −0.6454 + 0.4932i −0.6454 − 0.4932i −0.2770 + 0.6643i −0.2770 − 0.6643i −0.5851 Step 4: Compute the monic polynomial g˜ ( p) on the LHP roots of G(− p 2 ) using MatLab function C = poly(pkm): C = [1.0000 Step 5: Generate g( p) =

2.4300 √

2.9725

|G 10 |g˜ ( p)

2.1419

0.9470

0.2000]

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Here, G 10 = −25 and the coefficients of the polynomial g( p) is given by the MatLab row vector gc in descending order: >> gc=5*C gc = 5.0000 12.1502

14.8626

10.7096

4.7349

1.0000

which means that g( p) = 5 p 5 + 12.1502 p 4 + 14.8626 p 3 + 10.7096 p 2 + 4.7349 p + 1 Example 3. In this example let us choose f ( p) = p 3 and h( p) = p 10 − 2 p 9 + p 8 − 0. p 7 + p 6 − p 5 − 1.1 p 4 + 3 p 3 + p 2 − p + 1 Find strictly Hurwitz g( p). This is highly complicated problem which results in lossless two-port with n = 10 reactive elements. Among these elements k = 3 of them is placed in the circuit either as series capacitors and/or shunt inductors. No matter, how complicate the problem is, we can immediately construct g( p) using MatLab functions. Let us summarize the numerical steps without details. Step 1: Construction of H (− p 2 ): U = [1 − 2 1 − 0 1 − 1 − 1.1 3 1 − 1 1] V = [1 2 1 + 0 1 1 − 1.1 − 3 1 1 1] Hence, W=[1.0000 0 −2.0000 0 3.0000 0 −4.2000 0 12.8000 0 −3.2000 0 11.2100 0 −11.2000 0 4.8000 0 1.0000 0 1.0000]

Or the coefficients of H (− p 2 ) is given in descending order as H =[1 −2 3 −4.2 12.8 −3.2 11.21 −11.2 4.8 1 1] Step 2: Construction of G(− p 2 ) = H (− p 2 ) + (−1)3 p 6 G =[1 −2 3 −4.2 12.8 −3.2 11.21 −12.2 4.8 1 1] Step 3: Set X = − p 2 find the roots of G(X ) and the mirror symmetric roots of G(− p 2 ) (a) Computation of the roots of G(X ) = G(− p 2 )| X =− p2 using MatLab function Xk = roots(X): Set X = [1 2 3 4.2 12.8 3.2 11.21 12.2 4.8 − 1 1] and find the roots Xk: Xk = −1.6552 + 1.2385i −1.6552 − 1.2385i 0.3922 + 1.3781i 0.3922 − 1.3781i 0.7332 + 1.0876i 0.7332 − 1.0876i

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−0.6360 + 0.3954i −0.6360 − 0.3954i 0.1659 + 0.3010i 0.1659 − 0.3010i (b) Find the roots of G(− p 2 ) using the MatLab function pk = sqrt(−Xk): >> pk = sqrt(−Xk) pk = 1.3643 − 0.4539i 1.3643 + 0.4539i 0.7213 − 0.9553i 0.7213 + 0.9553i 0.5378 − 1.0111i 0.5378 + 1.0111i 0.8321 − 0.2376i 0.8321 + 0.2376i 0.2982 − 0.5048i 0.2982 + 0.5048i Mirror image of pk is given by pkm = −pk which yields the LHP zeros of G(− p 2 ). pkm = −1.3643 + 0.4539i −1.3643 − 0.4539i −0.7213 + 0.9553i −0.7213 − 0.9553i −0.5378 + 1.0111i −0.5378 − 1.0111i −0.8321 + 0.2376i −0.8321 − 0.2376i −0.2982 + 0.5048i −0.2982 − 0.5048i Step 4: Construct the monic polynomial g˜ ( p) using the MatLab function C=poly(pkm) >> C = poly(pkm) computes the coefficients of g˜ ( p) in descending order: C =[1. 7.5075 27.1813 62.2405 99.3593 115.0207 97.6436 60.0206 25.7357 7.1043 1.]

√ Step 5: Generate√g( p) = |G 10 |g˜ ( p) In this example |G 10 | = 1. Thus, g( p) ≈ p 10 + 7 .5 p 9 + 27 .2 p 8 + 62 .24 p 7 + 99 .36 p 6 + 115 .02 p 5 + 97 .6 p 4 + 60 .02 p 3 + 25 .7 p 2 + 7 .1 p; +1

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The above examples indicate that using MatLab, strictly Hurwitz polynomial can easily be generated from the given function f ( p) and the initialized h( p) no matter what the degree of n is. Eventually, it is clear from the above examples and descriptions that for the prefixed f ( p) and initialized h( p) = h 0 +h 1 p +. . .+h n p n strictly Hurwitz polynomial is constructed in a straight forward manner, perhaps using the standard functions of MatLab summarized in the following algorithm.

Algorithm: Generation of Strictly Hurwitz Polynomial g( p) Inputs: i. initial coefficients {h 0 , h 1 , . . . , h n } for the polynomial h( p) = h 0 + h 1 p + . . . + h n pn ii. Degree k for the polynomial f ( p) = p k , 0 ≤ k ≤ n Computational Steps: Step:1 Let the MatLab row vector U = h n h n−1 . . . h 0 be of order (n + 1) and Let V = (−1)n h n (−1)n−1 h n−1 . . . h 0 . Generate H (− p 2 ) = h( p)h(− p) = H0 + H1 p+. . . Hn p n using MatLab function W = conv(U, V ) or by means of Eq. (9.30). Step 2: Generate the even polynomial G(− p2) = H (− p2) + (−1)k p 2k = G 0 + G 1 p 2 + ... + G n p 2n using Eq. (32). Step 3: (a) Store the MatLab row vector X = (−1)n G n (−1)n−1 G n−1 . . . G 0 and generate the roots of G(X ) using MatLab function X k = r oots(X ) (b) Compute the RHP and LHP roots of G(− p 2 ) by using the MatLab functions pk = sqr t(−X k) and pkm = − pk respectively. Step 4: Generate the monic polynomial g˜ ( p) using MatLab function C = poly( pkm) Step √ 5: Generate the strictly Hurwitz g( p) using the MatLab function gc = |G n | × C where gc is a MatLab row vector of order (n + 1) containing the coeffi- cients {g0 , g1 , . . . , gn } of g( p) = g0 + g1 p + . . . + gn p n as gc = gn gn−1 . . . g0 Eventually, for the given antenna data, transducer power gain can easily be computed using the Eq. (9.21) in a straight forward manner as shown in the following example. Example 4. This example exhibits the usage of MatLab to develop the necessary functions to construct lossless two-ports numerically and also to compute the transducer power gain for a given load network.

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Fig. 9.3 Antenna model with normalized element values

For the problem under consideration, the load network is specified by Fig. 9.3. It includes a parallel combination of a resistance R = 1, and a capacitor C = 4 in series with an inductor of L = 0.6708. The equalizer [E] to be constructed is specified by its back-end reflection coefficient E 22 ( p) = h( p)/g( p) for which the numerator polynomial h( p) is by h( p) = −3.2944 p 4 − 3.1010 p 3 − 4.1546 p 2 − 1.8843 p − 0.5035. Furthermore, we wish [E] to be a simple lowpass LC ladder. Therefore, we set f ( p) = 1. It should be reminded that the simplest bandpass form of f ( p) is given by f ( p) = p k where the integer k designates the degree of the DC transmission zeros. Here, we are asked to write MatLab functions together with a main program to perform the following activities. (a) Develop a MatLab function to construct the equalizer from the given f ( p) and h( p) polynomials. In this regard, strictly Hurwitz denominator polynomial g( p) must be determined. (b) Generate the transducer power gain numerically as in Eq. (9.21) over the normalized frequency band from zero to 1.5. (c) Synthesize the equalizer from the given reflectance E 22 ( p) = h( p)/g( p). Solution Let us perform the above tasks step by step. Step 1: At this step a MatLab function which is called “g = Hurwitzpoly g(h, k)” is written to compute the coefficients of the Strictly Hurwitz polynomial g( p) from h( p) and f ( p). In this function, inputs are

r

the row the coefficients of h( p) in descending order. That is vector h containing h = h n h (n−1) . . . h 1 h 0 r order of the transmission zeros “k” at DC. The output row vector “g” contains all the coefficients of g( p) as g = gn g(n−1) . . . g1 g0 ]. Thus, we have the following list for g = Hurwitzpoly g(h, k): function g=Hurwitzpoly g(h,k) % Computation of g(p)=g1pˆn+g2pˆ(n−1)+...+g(n+1)*p+g(n+1). % Inputs: Row vector h which contains coefficients of h(p) in descending order. % k which is the order of f(p)=pˆk. na=length(h);

204

% % Step 1: Generate even polynomial H(−pˆ2)=h(p)h(−p); % % Computation of even polynomial with convolution: W=conv(U,V) U=h; % Generate row vector V sign=−1; for i=1:na sign=−sign; V(na−i+1)=sign*U(na−i+1); end W=conv(U,V); % Select non-zero terms of W and form the vector H for i=1:na H(i)=W(2*i−1); end % % Step 2: Generate G(−pˆ2)=H(−pˆ2)+(−1)ˆk for i=1:na G(i)=H(i); end ka=na−k; sign=(−1)ˆ(k); G(ka)=H(ka)+sign; % % Step 3:Generate the row vector X and Compute % the roots of G(−pˆ2) % Arrange the sign of vector X. sign=−1; for i=1:na sign=−sign; X(na−i+1)=sign*G(na−i+1); End % % Compute the RHP and LHP Roots of G(−pˆ2) Xr=roots(X);pr=sqrt(−Xr);prm=−pr; % % Step 4: Generate the monic polynomial g’(p) C=poly(prm); % % Step 5: Generate Scattering Hurwitz polynomial g(p) Cof=sqrt(G(1)); for i=1:na g(i)=Cof*C(i); end

B.S. Yarman

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Thus, when the above function is run within a main program, then, the output is printed as, >> h h =[-3.2944 -3.1010 -4.1546 -1.8843 -0.5035] which is h(p)=−3.2944 p 4 − 3.1010 p 2 − 4.1546 p 3 − 1.8843 p 2 − 0.5035 And the output is >> g g =[ 3.2944 4.4538 meaning that

5.7057

3.4847

1.1196]

g( p) = 3.2944 p 4 + 4.4538 p 2 + 5.7057 p 3 + 3.487 p 2 + 1.1196 Step 2: In this step, the load data is generated as a MatLab function called “SL = reflection(W)”. This function computes the load reflection coefficient at each frequency point. The result of this function is used for the computation of Transducer power gain. This function is given below. function SL=reflection(W) % This function computes the load reflecetion coefficient SL=(ZL−1)/(ZL+1). % In this function load impdenace is given as ZL=R//C with C=4//R=1. % Generate the complex variable p=sqrt(−1)*W; % % GENERATION OF THE LOAD DATA % Reactive C=4 YS=4*p; YL=1+YS; ZL=0.6078*p+1.0/YL; SL=(ZL−1)/(ZL+1); Step 3: We are almost ready to generate transducer power gain. However, first we should construct the equalizer using the MatLab function called “Equalizer(W,h,g,k)”. This function computes the input reflection coefficient SE at a given normalized frequency W for the specified row vectors h and g and for the integer k of f ( p) = p k . function SE=Equalizer(W,h,g,k) p=sqrt(−1)*W; gval=polyval(g,p); hval=polyval(h,p); fval=pˆk; SE=hval/gval; Step 4: So far we have generated the load and the equalizer data. Then, TPG can easily be computed using the Eq. (9.21) as follows.

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B.S. Yarman

function T=gain(W,h,g,k,L11) % Define complex variable p=jW; j=sqrt(−1) p=sqrt(−1)*W; %Compute the values of h and g using MatLab function polyval(p,X) gval=polyval(g,p); hval=polyval(h,p); fval=pˆk; % Construct the input reflection coefficient of the equalizer from the % back-end. E22=hval/gval; %COMPUTATION OF˜GAIN % Weight=fval*conj(fval)*(1−L11*conj(L11)); %Compute the denominator of the gain function as in Eq.(9.21). D=hval*conj(hval)*(1+L11*conj(L11))+fval*conj(fval)− 2*real(L11*hval*conj(gval)); T=Weight/D;

Step 5: Back-end reflection coefficient can be realized as a lossless network in resistive termination. Since the equalizer is a low-pass ladder, then it can be synthesized using long division from the input impedance or admittance function such that ZL =

1 + E 22 = L1 p + 1 − E 22 C2 p +

YL =

1 − E 22 = C1 p + 1 + E 22 L2 p +

1 1 L 3 p+......+ X n 1p+1

Or similarly, 1 1 C3 p+......+ X n 1p+1

In the following we present a simple MatLab function called “[m, CV] = synthesis (h,g) which carries the long division on the input reflection coefficient10 . In this function, the inputs are the MatLab row vectors h and g which describe the back end reflection coefficient E 22 ( p)h( p)/g( p). The output quantities are m which is an integer indicating that

r r

10

a series inductor is extracted if m = 1, a shunt capacitor is extracted if m = −1. The MatLab function synthesis is provided by Dr. Metin Sengul of Kadir Has University.

9 Simplified Real Frequency Technique

Thus, here is the list for the synthesis function: function [m,CV]=synthesis (h,g) %Given polynomial h and g for Low Pass case (i.e. k=0) %Compute the element values of the matching Network %Elements are stored in vector CV. %If m=1 start with inductor % n1=length(h); k=0; for i=1:n1 h poly(i)=h(n1−i+1); g poly(i)=g(n1−i+1); end m=g poly(1)/h poly(1); n=(g poly+h poly); d=(g poly−h poly); CV=0; b=1; if m<0 while length(CV)
207

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B.S. Yarman

clear d d=n; clear n n=rr; clear q clear r end end elseif m>0 while length(CV)
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Step 6: We developed a main program to run all the above presented functions. The list of the program is given below. % Main Program: Numerical Computation of % Transducer Power Gain as in Eq (21) and the Synthesis of the Equalizer. % % Inputs: % h(p), and f(p)=pˆk % % Outputs: % Numerical Construction of the Strictly Hurwitz polynomial g(p) % Complete Scattering Parameters of the Equalizer, % Transducer Power Gain of the Matched System, % Realization of the Equalizer with its ladder network element % values, % Plot of the Gain Performance % clear % Enter or type the coefficients of h(p) as a MatLab row vector: h = [−3.2944 − 3.1010 − 4.1546 − 1.8843 − 0.5035]; % Compute the length of the row vector h: na=length(h); % Enter or type the order of the DC transmission zeros of the Equalizer: k k=0; % % Compute the Strictly Hurwitz Polynomial g(p) from % the given h(p) and f(p)=pˆk % g=Hurwitzpoly g(h,k); % % Start the frequency loop % Type the frequency band: W1=0; W2=1.5; % Type the total numer of sampling points within the % band % Nsample=200; DW=(W2−W1)/Nsample; W=0; for j=1:Nsample+2 Wa(j)=W; E22=Equalizer(W,h,g,k); % % GENERATION OF THE LOAD DATA

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% Reactive C=4 L11=reflection(W); %COMPUTATION OF GAIN % T=gain(W,h,g,k,L11); Ta(j)=T; W=W+DW; end % for i=1:na y(i)=g(na−i+1); x(i)=h(na−i+1); end % [m,CV]=synthesis(x,y); plot(Wa,Ta) Eventually, the above program is run and results are reported as follows. In Table 9.1, the transducer power gain is listed and it is depicted in Fig. 9.4 For the computer generated back-end reflection coefficient11 Table 9.1 TPG Gain performance of the equalizer constructed for Example 4 Normalized frequencies

Transducer power gain

0.0000 0.0682 0.1364 0.2045 0.2727 0.3409 0.4091 0.4773 0.5455 0.6136 0.6818 0.7500 0.8182 0.8864 0.9545 1.0227 1.0909 1.1591

0.7978 0.8000 0.8033 0.7992 0.7813 0.7518 0.7217 0.7046 0.7095 0.7367 0.7720 0.7866 0.7664 0.7459 0.7795 0.7697 0.3638 0.0967

Remember that h( p) and g( p) were given in terms of MatLab row vectors h = [−3.2944 − 3.1010 − 4.1546 − 1.8843 − 0.5035] and g = [3.2944 4.4538 5.7057 3.4847 1.1196]. 11

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Fig. 9.4 Transducer power gain plot of Example 4

−3.29 p 4 + −3.1 p 3 − 4.15 p 2 − 1.88 p − 0.5 is synthesized resulting 3.29 p 4 + 4.45 p 3 + 5.7 p 2 + 3.48 p + 1.119 in m = −112 ; CV = 4.8706 0.6550 3.8439 0.3310 0.3796]. which means that when the equalizer is synthesized from the back-end, the first element is a shunt capacitor with C1 = 4.8706, then it is followed by a series inductor L2 = 0.655; then, a shunt capacitor C3 = 3.8430, and a series inductor L4 = 0.331. Finally, the equalizer is terminated in R = 0.3796 which introduces a transformer in the circuit with turn ratio n 2t = 0.3796. Synthesized equalizer is shown in Fig. 9.5 In Fig. 9.5, the equalizer is terminated in a non-unit resistance R = C V (5) = 0.3796. Therefore, for this situation we had to use a transformer to match this nonunity resistance to a generator with unit resistance. However, if h 0 of h( p) were zero then, at DC (i.e. ω = 0) one would end up with zero value for E 22 (0) which in turn yields a unity transfer scattering parameter (i.e. |E 21 (0)| = 1). In this case, the terminating resistances at both generator and the load ends of the matched system must be the same. Thus, in order to eliminate the transformer in the equalizer design, one must fix the coefficient h 0 at zero. Thus, when dealing with LC Low-Pass E 22 ( p) =

In function synthesis m = −1 indicates that LC ladder starts with a capacitor. m = 1 corresponds to an LC ladder starting with an inductor.

12

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B.S. Yarman

Fig. 9.5 4-element equalizer design with transformer

ladder structures for the equalizer design, h 0 = 0 is selected to avoid the use of a transformer.

Optimization of the Transducer Power Gain All the computer aided design methods employ optimization algorithms to reach to the preset performance criteria. As mentioned at the beginning of this chapter, Brute force techniques involve with highly nonlinear optimization schemes. On the other hand, Simplified Real Frequency Technique presents all most second order nonlinearity which is always convergent. It has been long experienced that especially Least Square data fitting methods provides excellent solution for the real frequency matching algorithms. For this purpose, MatLab’s nonlinear Least Square data fitting algorithm yields outstanding solutions for all antenna matching problems. Therefore, in the following, utilization of MatLab nonlinear optimization routines is summarized.

The Least Squares Fitting Method with MatLab The MatLab Curve Fitting Toolbox uses the method of least squares when fitting data. The fitting process requires a model that relates the response data to the predictor data with one or more coefficients. The result of the fitting process is an estimate of the “true” but unknown coefficients of the model. To obtain the coefficient estimates, the least squares method minimizes the summed square of residuals. The residual for the ith sample frequency εi is defined as the difference between the sampled transducer power gain Ti = T (ωi , h) and the pre-set flat gain level T0 . εi = Ti − T0

(9.36)

The summed square of residuals is given by S=

nopt / i=1

εi2 =

nopt / i=1

(Ti − T0 )2

(9.37)

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213

where nopt is the number of data points selected within the frequency band of operation, included in the fit and S is the sum of squares error estimate. In Eq. (9.37) coefficients of the h( p) polynomial can be determined by an appropriate optimization algorithm. It has been verified that Levenberg-Marquad technique results in excellent solution for it [6,7,8,9]. Certainly, proper initialization of the coefficients {h n , h n−1 , . . . , h 2 , h 1 , h 0 } is quite important. However, ad hoc choice of h i = ∓1 is quite adequate. MatLab’s Least Square Nonlinear Optimization algorithm is called “lsqnonlin” yields excellent solutions for almost all the antenna matching problems. Let us now introduce lsqnonlin function of MatLab:

MatLab Function lsqnonlin The MatLab function lsqnonlin solves the nonlinear least-squares (nonlinear datafitting) problems in its classical form described below. 2 min ε12 + ε22 + ε32 + . . . .. + εnopt x

(9.38)

The way we use it in this book, function lsqnonlin is used employing the following Syntax: x = lsqnonlin(‘sopt’,x0,[ ],[ ],OPTIONS,T0,ntr,k,nopt,wlow,whigh) In the above presentation, x is the unknown vector to be determined. ‘sopt’ is the user defined function which generates transducer power gain in terms of the unknown row vector ‘x’ which contains the coefficients of the polynomial n−1 h( p) = h n p n + h n−1 p + ..h 1 p + h 0 in descending order. Therefore we set x = X 1 X 2 .. X n+1 = h n h n−1 . . . h 0 x0 is the initial value of the unknown vector x. Then, we have two empty brackets [ ], [ ]. They refer to the lower and the upper bounds of the unknown variable vector x. However, we skip these bounds since there is no restriction imposed on the real coefficients of the numerator polynomial h( p) which are the unknowns of the optimization problem. The input array OPTIONS includes optimization brackets such that OPTIONS = OPTIMSET (‘MaxFunEvals’,-number-1–,‘MaxIter’,-number-2–); which refers to maximum number of function evaluation and maximum number of iterations for the “Levenberg-Marquard technique employed utilized within lsqnonlin. For example, for numer-1 we usually select 2000, for number-2 we use 50000. In this case we set OPTIONS = OPTIMSET (‘MaxFunEvals’,2000,‘MaxIter’,5000) within the main program. Rest of the input variables are special for the Simplified Real Frequency Technique:

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T0 is the targeted flat gain level subject to minimization. The ideal value of T0 is equal to 1. Depending on the result of the optimization, the user can sweep the values of T0 to end up with an acceptable average transducer power gain variation with small fluctuations. ntr is a control flag (or integer). (a) ntr = 1 if the user allows a transformer in the equalizer. In this case k ≥ 0 is chosen. (b) ntr = 0 refers to the equalizer design without transformer. If this is the case, the equalizer structure must be a low-pass LC ladder form for which h 0 of h( p) is fixed at zero and it does not enter into optimization (i.e. h 0 = 0). k is an integer and it refers to the total number of transmission zeros at DC which is specified by f ( p) = p k . It should be noted that if the user selects k > 0 then, it is clear that all the coefficients {h n , h n−1 , h n−2 , . . . , h 2 , h 1 , h 0 } of h( p) will enter into optimization. In this case, total number of variables which enters into optimization is na = n + 1. If k = 0 chosen, ntr could be zero or 1. If ntr = 0 chosen then, the user must fix h 0 = 0. In this case, the unknown vector x must contain x = [x1 , x2 , .., xn ] = [h n , h n−1 , . . . , h 2 , h 1 ] which means that total number of unknowns is equal to the degree n of the polynomial h( p). nopt refers to the total number of sampling points within optimization as indicated by Eq (9.37). It means that within the pass-band, we take nopt number of samples to evaluate the error {εi = T (ωi , x0) − T0 ; i = 1, 2, . . . , nopt} to be minimized in the least mean square sense. wlow refers to the lower edge of the pass-band. It is a normalized angular frequency. whigh designates the upper edge of the normalized angular frequency. Now, let us run an example to optimize the transducer power gain. Example 5. Develop a MatLab program to optimize the transducer power gain of the load network specified by Example 4 over the frequency band from wlow = 0 to whigh = 1.0. Answer For this example, we have develop a computer program named SRFTWSOP which is the abbreviation of the “Simplified Real Frequency Technique With Simple OPtimization”. The inputs to this program are listed as: T0: Flat gain level ntr: Design with transformer. k: Transmission zeros ad DC nopt: Sample points for optimization wlow: Lower edge of the frequency band whigh: Upper edge of the frequency band h: initials for the h polynomial

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Here, let us choose T0 = 0.8, ntr = 1, k = 0, nopt = 20, wlow = 0, whigh = 1 and the initial h vector is selected as in pervious example. That is h = [−3.2944 − 3.1010 − 4.1546 − 1.8843 − 0.5035] chosen. For the optimization, nonlinear-Least square optimization function lsqnonlin is employed as listed in SRFTWSOP: MAIN PROGRAM % PROGRAM SRFTWSOP: Simplified Real Frequency Technique (SRFT) with Error! No table of contents entries found. % simple optimization uisng the MatlLab function lsqnonlin % % Inputs: % T0: Desired flat gain level % ntr: Control flag for equalizer design: % ntr=1; equlizer design with a transformer. % In this case k may be different than zero. % ntr=0; equlizer is constructed without transfermer. % In this case k must be zero (k=0) % and equlizer will be constructed with Low Pass LC elements % % initial values for h(p), and f(p)=pˆk % nopt: total number of sampling points within optimization. % This is the dimension of the objective function invT subject to % optimization. % wlow: Low frequency end of the optimization. % whigh: High Frequency end of the optimization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Outputs: % Construction of the antenna matching network with Least Square % Optimization using Levenberg-Marquad Technique. % Optimized Scattering Parameters of the Equalizer. % Optimized Transducer Power Gain of the Matched System % Realization of the Equalizer as an LC ladder with explicit element values. % Plot of the Gain Performance clear % Enter the inputs: T0=0.8, % Flat gain level ntr=1, %Design with transformer. k=0, %Single Transmission zero ad DC nopt=20; %optimization over 20 points wlow=0,%lower edge of the frequency band whigh=1.0, %upper edge of the transfer function h=[−3.2944 −3.1010 −4.1546 −1.8843 −0.5035];% initials for the h polynomial % %

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% na=length(h); % % Start Optimization using simplest version of the LSQNONLIN % % Type design prefernce: % ntr=1: design with transformer % ntr=0: design without transformer. % -------------------------------------------------------------% %%%%%%%%%%% Preparetion for the optimization % OPTIONS=OPTIMSET(’MaxFunEvals’,20000,’MaxIter’,50000); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Design with transformer: if ntr==1, x0=h; % Call the optimization function lsqnonlin: x=lsqnonlin(’sopt’,x0,[],[],OPTIONS,T0,ntr,k,nopt,wlow,whigh); % h=x; end %-------------------------------------------------------------------------% % Design without transformer: if ntr==0, for r=1:na−1 x0(r)=h(r); % Call the optimization function lsqnonlin: x=lsqnonlin(’sopt’,x0,[],[],OPTIONS,T0,ntr,k,nopt,wlow,whigh); end for r=1:na−1 h(r)=x(r); h(na)=0; end end % Generate strictly hurwits polynomial g(p) from optimized h(p): g=Hurwitzpoly g(h,k); % Compute the optimized transducer power gain over the frequency band: W1 and W2 W1=0; W2=1.5; Nsample=100; DW=(W2−W1)/Nsample; W=0;

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for j=1:Nsample+1 Wa(j)=W; E22=Equalizer(W,h,g,k); % GENERATION OF THE LOAD DATA % Reactive C=4 L11=reflection(W); %COMPUTATION OF GAIN T=gain(W,h,g,k,L11); Ta(j)=T; W=W+DW; end for i=1:na y(i)=g(na−i+1); x(i)=h(na−i+1); end % Syntisize the lossless equalizer from its back-end reflection % coefficient: [m,CV]=synthesis(h,g); plot(Wa,Ta)

The above main program calls the optimization routine lsqnonlin with an input function called “sopt”. As mentioned before, sopt is a user defined function and it is developed by us as listed below: function fun=sopt(x,T0,ntr,k,nopt,wlow,whigh) %Inputs: % T0: Desired gain level % ntr: Control flag for equalizer design: % ntr=1; equlizer design with a transformer. % In this case k may be different than zero. % ntr=0; equlizer is constructed without transfermer. % In this case k must be zero (k=0) % and equlizer will be constructed with Low Pass LC elements % x=Row Vector includes the unknow coefficients of the polynomial h(p). % for ntr=0 and k=0, dimension of x must be n. % for ntr=1, dimension of x must be n+1. % nopt: total number of sampling points within optimization. % This is the dimension of the objective function fun(j) % subject to optimization. % wlow: Low frequency end of the optimization. % whigh: High Frequency end of the optimization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Output:

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% fun: Objective function Vector subject to optimization. % For the design of Antenna Matching Networks fun=T−T0 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % nx=length(x); w=wlow; DW=(whigh−wlow)/nopt; % Start frequency loop to compute objective function within passband: for j=1:nopt % Prepare to generate transducer power gain: % Call load function to compute load reflection coefficient. L11=reflection(w); % Set h vector for the values of ntr: if ntr==1, h=x; end % if ntr==0, for r=1:nx h(r)=x(r); end na=nx+1; h(na)=0; end % % Generate g polynomial from h: g=Hurwitzpoly g(h,k); % Generate the Transducer power Gain % T=gain(w,h,g,k,L11); % % Generate objective function as fun(j)=T−T0; w=w+DW; end return Thus, the main program SRFTWSOP is run and we obtained the following results: Initial h vector is h = [−3.2944 − 3.1010 − 4.1546 − 1.8843 − 0.5035] Optimized h vector is h = [−1.0607 − 2.3496 − 2.7543 − 1.8602 − 0.5515] Corresponding denominator polynomial is g = [1.0607 2.7743 3.7803 3.0094 1.1420]

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Fig. 9.6a Optimized transducer power gain performance of the matched load given by Example 5

The transducer power gain performance is depicted in Fig. 9.6a In this example we see that gain response is flattened within the passband. The result is listed in Table 9.2. Referring to Fig. 9.5, the component values of the Low-Pass LC equalizer is given by CV = [C1 = 4.9944 L2 = 0.5342 C3 = 3.2523 L4 = 0.1444 RG = 0.3487] Example 6. Finally, let us design an band-pass equalizer for the load network given by Example 4 over the frequency band of wlow = 0.2 to whigh = 1.0. Table 9.2 Gain performance of the optimized matched System of Example 5 Normalized angular frequency (W)

Transducer power gain

0 0.1500 0.3000 0.4500 0.6000 0.7500 0.9000 1.0500 1.2000

0.7668 0.7649 0.7629 0.7670 0.7729 0.7749 0.7942 0.5100 0.0897

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Answer Using the program SRFTWSOP let us choose the following inputs: Fix the flat gain level at T0 = 0.80, Since we are designing a band-pass equalizer, we can choose k = 1 and the design will employ a transformer. Therefore, ntr = 1. Let us choose nopt = 20. Clearly one has to set wlow = 0.2 and whigh = 1.0 as specified by the example. For the initial start let us choose an ad hoc vector such that h = [−1 −1 −1 −1 −1]. Thus, the results obtained with SRFTWSOP is listed below. The above table is plotted in Fig. 9.6b Optimized values of h and g vectors are given by h = [−1.8884 − 1.9863 − 1.9372 − 0.6279 − 0.0468] g = [1.8884 2.6721 2.7830 1.2138 0.0468] −1.8884 p 4 − 1.9863 p 3 − 1.9372 p 2 − 0.6279 p − 0.0468 is Finally, E 22 ( p) = 1.8884 p 4 + 2.6721 p 3 + 2.783 p 2 + 1.2138 p + 0.0468 synthesized as shown in Fig.9.713

Fig. 9.6b Transducer power gain performance of Example 6

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For this example, we use a general long division synthesis function provided by Dr. Ali Kilinc of Elma Corp. of Istanbul.

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Fig. 9.7 Synthesis of the reflection coefficients E 22 ( p)

In the above examples all the crucial steps of the Simplified Real frequency Technique is completed. Therefore, all the results presented in the previous sections can be summarized in the SRFT Algorithm.

Algorithm: Design of Ultra Wide Band Matching Networks via SRFT using MatLab Computational Philosophy For single matching problems, Simplified Real Frequency Technique generates the lossless equalizer [E] in terms of its back-end reflection coefficient E 22 ( p) = h( p)/g( p). In this approach, by properly selecting the transmission zeros of [E], the complete scattering parameters of the equalizer is generated from numerator polynomial h( p). Form the engineering point of view, it is very practical to choose the transmission zeros at DC and infinity which in turn requires f ( p) = p k for the transfer scattering parameter E 21 ( p) = f ( p)/g( p). Therefore, coefficients x = h n , h n−1 , h n−2 , . . . , h 2 , h 1 , h 0 of h( p) is selected as the unknown of the single matching problem.

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In this regard, the MatLab main program called SRFTWOP utilizes the Least Mean Square Optimization function named lsqnonlin with proper inputs as described in Example 6. lsqnonlin requires a user defined function which generates the error function fun as { f un = εi = T (ωi , h) − T0 ; i = 1, 2, 3, . . . nopt}. In the main program SRFTWSOP, the optimization routine lsqnonlin is called by x = sopt(x, T0, ntr, k, nopt, wlow, whigh). The name ‘sopt’ is abbreviated from the expression “Simple Optimization”. The last line of this function is the sampled error called fun = T − T0; where T is the transducer power gain which is generated as a function the unknown numerator polynomial h( p). Therefore, lsqnonlin determines the unknown coefficients of h( p) by minimizing the sum of the squared errors employing the Levenberg-Marquard Algorithm. The result is returned to the unknown vector x. Then, the coefficient vector h = [] of h( p) is generated by properly setting h = x. Thus, the optimized performance of the matched system is computed by generating the scattering parameters of the equalizers and the transducer power gain of the matched system. Eventually, unit normalized reflection coefficient E 22 ( p) = h( p)/g( p) or E 11 ( p) = (−1)k+1 h(− p)/g( p) of the equalizer is synthesized by a long division algorithm. Hence, inputs and the computational steps of the MatLab based SRFT algorithm can be summarized as follows: Inputs of the Main Program SRFTWOP: It should be noted that the input parameters given below directly typed into the program as SRFTWOP listed for Example 6.

r r r

r r

r

Enter flat gain level T0. Enter the transmission zeros of the equalizer to be design. Usually, we choose f ( p) = p k . Therefore, assuming ladder structures for the equalizer with all the transmission zeros at DC and infinity, enter the value of integer k. Enter your desire for the equalizer to be designed with transformer or without transformer. Accordingly select the value of the integer ntr. For the main program SRFTSOP. Set ntr = 1 for designs with transformers, or it is set ntr = 0 for designs without transformer. It should be noted that if k > 0 then, the single matching equalizer must include a transformer. In this case, the user must set ntr = 1. Enter the total number of sample points nopt to generate the error function {εi = T (ωi , h) − T0 ; i = 1, 2, 3, . . . nopt} subject to minimization in pass-band. Enter initial value of the unknown coefficients of h( p) = h n p n + h n−1 p n−1 + h n−2 p n−2 +. . .+ h 1 p + h 0 as a matLabrow vector h = [h(1), h(2), h(3), ..h(n), h (n + 1)] = h n , h n−1 , h n−2 , . . . , h 1 , h 0 . At this point we should note that vector h includes na = n + 1 entries which in turn specifies the total number of reactive elements “n” to be employed within the lossless matching network. Load reflection coefficient L11 must be supplied to SRFTWSOP as a user defined MatLab function as was practiced for Examples 4, 5 and 6. The function must start with the first line SL = function reflection(W) where SL is the reflection coefficient of the load and W is the normalized angular frequency. The last line of the function must be SL = . . . . . ..etc.

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Computational Steps and outputs: Step 1: Develop a user define function to call the optimization function “lsqnonlin”. Let this function be function fun = sopt(x, T0, ntr, k, nopt, wlow, whigh). In this function compute the followings. i. Set the unknown vector x = [.] to the coefficient vector h = x properly. ii. Generate the strictly Hurwitz polynomial by calling the function g = Hurwitzpoly g(h, k) iii. Generate the complex value of the load reflection coefficient L11(jW) by calling the function L11 = reflection(W). iv. Compute the transducer power gain by calling the function T = gain(w, h, g, k, L11). v. Compute the error function fun(j) = T − T0 at the selected sample frequency W Step 2: Call the Levenberg-Marquard minimization routine lsqnonlin to determine the unknown coefficients of the numerator polynomial h( p) via minimization of the error “fun = T − T0” at sample frequencies. At this step, lsqnonlin returns optimized vector x = lsqnonlin(‘sopt’,x0,[],[],OPTIONS,T0,ntr,k,nopt,wlow,whigh) Step 3: Properly set h = x and generate the strictly Hurwitz polynomial by calling g = Hurwitzpoly g(h, k) Step 4: This step is devoted to the visual generation of the transducer power gain function over the pre-set frequencies and it is pictured by calling the function plot(Wa,Ta) where Wa and Ta designates the array forms of the normalized angular frequency and the transducer power gain. It should be noted that at this step, one may wish to generate back-end reflection coefficient E 22 ( jωi ) just for the sake of visual comprehension by calling the function E22 = Equalizer(W, h, g, k). Step 5: Finally, analytic form of E 22 ( p) = h( p)/g( p) or E 11 ( p) = (−1)k+1 h(− p)/ g( p) is synthesized by long division. For this step, we have developed a robust MatLab function called [m, CV] = synthesis (h,g) which work only for k = 0 case. In this function, at the output the integer “m” specifies the first element of the Low-pass LC ladder. If m = −1 the equalizer starts with a shunt capacitor with the component value of CV(1). If m = +1 then, the ladder starts with a series inductor of CV(1). The final component value CV(n + 1) should be the resistive termination of the equalizer. It can come either as a resistance or a conductance value; depending on the degree n of h( p) = h n p n + . . . + h 1 p + h 0 . Thus, the outputs of the main program SRFTWSOP is listed as follows. Outputs:

r r r

The optimized values of vectors h and g generated by Step 3, Transducer Power Gain T and the back-end reflection coefficient E22 as computed by Step 4. Component Values CV of the lossless equalizer determined by synthesizing the back-end reflection coefficient using a long-division algorithm as in Step 5.

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Referencess 1. Yarman, B. S. A simplified real frequency technique for broadband matching complex generator to complex loads, RCA Review, Vol. 43, pp. 529–541, Sept.1982. 2. Yarman, B. S. and H. J. Carlin, A simplified real frequency technique applied to broadband multi-stage microwave amplifiers, IEEE Trans. Microwave Theory and Tech., Vol. 30, pp. 2216–2222, Dec. 1982. 3. Yarman, B. S. A dynamic CAD technique for designing broad-band microwave amplifiers, RCA Rewiev, 1983. 4. Yarman, B. S. Modern approaches to broadband matching problems, Proc. IEE, Vol.132, pp. 87–92, April 1985. 5. Yarman B.S., “Broadband Network”, Wiley Encyclopedia of Electrical and Electronics Engineering John G.Webster, Editor, Vol 2, pp.589–605, 1999, John Wiley& Sons corp. 6. Levenberg, K., “A Method for the Solution of Certain Problems in Least Squares,” Quart. Appl. Math. Vol. 2, pp 164–168, 1944. 7. Madsen, K. and H. Schjaer-Jacobsen, “Algorithms for Worst Case Tolerance Optimization,” IEEE Transactions of Circuits and Systems, Vol. CAS-26, Sept. 1979. 8. Marquardt, D., “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” SIAM J. Appl. Math. Vol. 11, pp 431–441, 1963. 9. Mor´e, J.J., “The Levenberg-Marquardt Algorithm: Implementation and Theory,” Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp 105– 116, 1977.

Chapter 10

Applications Design Examples via SRFT Binboga Siddik Yarman

Design of Wideband Antenna Equalizers Employing SRFT In this chapter, we present two typical real life examples to design antenna matching networks employing the Simplified Real Frequency Technique. First example covers the design of an equalizer for a given short monopole antenna which is described in terms of its measured impedance data up FM band. The second example includes the design of a “multi-band” equalizer for a PIFA antenna which can be utilized for all the existing commercial cellular communication bands.

Equalizer Design for a Short Monopole Antenna The antenna under consideration is designed for emergency mobile terrestrial communications over a wide frequency band up to 100 MHz. Applications could be military as well as commercial. For the sake of conceptual understanding, in this example, we are only is interested in the best electrical performance which can be squeezed out of the measured impedance data. The antenna impedance data is measured from 20 to 100 MHz. Then, it is modeled as a Darlington 2-port as shown in Fig. 10.1. The imaginary part of this impedance is shown in Fig. 10.3. The antenna was built for experimental purpose to investigate the possible usable frequency band. Assume that we wish to design an antenna matching network over the frequency band of B:30M H z–70M H z with gain fluctuations less than ΔT = 0.05. In view of broadband matching, one needs to answer the following essential questions:

B.S. Yarman College of Engineering, Department of Electrical-Electronics Engineering, Istanbul University, 34320 Avcilar, Istanbul, Turkey e-mail: [email protected]

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

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B.S. Yarman L4 = 0.1 uH L2 = 0.4 uH C4 = 19.8 pF L3 = 0.01 uH L1 = 0.225 uH

Circuit Model for the short monopole antenna

R0 = 50 ohm

Short Monopole Antenna

C3 = 11.4 pF Antenna Impedance ZL = RL + jXL

Fig. 10.1 Monopole Antenna Model built on the measured impedance data specified over frequency band of (20–100) MHz. The real part of the antenna impedance is depicted in Fig. 10.2

◦ Question 1: Let T0 be the ideal flat gain level over the prescribed frequency band B. For this band (e.g. 30–70 MHz), what is the maximum attainable average value of the transducer power gain Tav ≈ T0 with minimum fluctuations ΔT ? In this regard, it may be appropriate to express the upper and the lower bounds of the gain function in terms of Tav and ΔT such that the upper and lower bounds are Tmax = Tav + ΔT and Tmin = Tav − ΔT respectively; or equivalently min within band of interest. For most practical applications, it is Tav = Tmax +T 2 demanded that Tmin ≥ 0.5 which specifies the “half power bandwidth” and it is preferable to have ΔT ≤ 0.1. Obviously, Tmin and ΔT are the two major

Fig. 10.2 Real Part of the antenna impedance

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Fig. 10.3 Imaginary part of the antenna impedance

parameters to measure the quality of the matched antenna system. For example, if Tmin is less than 0.5 then, one should consider to replace the antenna with a better one. The answer of this question is called the “optimum performance of the equalizer” to be designed for the antenna under consideration. ◦ Question 2: Having known the optimum performance, how can we obtain the lossless equalizer network which yields almost optimum performance? Literally, the first question can be answered using the analytic gain bandwidth theory if it is accessible. Now, let us attempt to implement the theory with the following steps. First of all, we have to choose a proper transfer function with desired shape. where ε is Per say, let us choose a Chebyshev function with T (ω) = 1+εT2max Tn2 (ω) the ripple factor and Tn (ω) = cos n cos−1 (ω) is the n th order Chebyshev polynomial. The degree n specifies the total number of reactive elements in the equalizer topology if the matching problem were a lowpass design. However, in the present case, we have to use lowpass to bandpass transformation to shift the frequency band of the prototype to the band of interest (e.g. to 30–70 MHz). In order to do so, it is appropriate to normalize the actual frequency band with respect to f 0 = 100M H z. In this case, our normalized frequency band will be running from f 30M H z (e.g. from ωlow−end = 100M = 0.3) to ωhigh−end = high−end ωlow−end = flow−end f0 Hz f0

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Fig. 10.4 Plot of Chebyshev transfer functions with Tmax = 1 for various values of degree n 70M H z (e.g. ωhigh−end = 100M = 0.70). The Low-pass (or the proto-type) form of the Hz Chebyshev function results in an equal ripple monotone-roll-off characteristics over the frequency band −1 ≤ B(ω) ≤ +1 as shown in Fig. 10.4. For bandpass problems described by the frequency band B: ωlow−end ≤ B(ω) ≤ ωhigh−end , first, the frequency band dictated by the lowpass transfer function must be normalized to yield the desired bandwidth over −ωc ≤ B(ω) ≤ +ωc such that ωc = ω −ω B = high−end 2 low−end and then, it is shifted by an amount of B2 + ωlow−end to obtain 2 the required shape of the TPG in the frequency interval specified by ωlow−end ≤ . This process replaces the normalized angular frequency ω by B(ω) ≤ ωhigh−end ωc . ω ⇒ ω − B2 + ωlow−end The lowpass based Chebyshev transfer function T (ω) takes its maximum value Tmax at the zeros of the Chebyshev polynomial Tn (ω). It is minimum when Tn (ω) = min = Tmax2ΔT and the 1. Obviously, the ripple factor ε is specified by ε2 = TmaxT−T −2ΔT min Tmax +Tmin . average flat gain level is determined as T0 = 2 Then, we have to insert proper all pass functions on it to be able to extract the load network given by Fig. 10.1. Afterwards, complicated gain-bandwidth restrictions must be satisfied which in turn yields the flat gain level T0 .1 For the problem under consideration, the real part of the antenna impedance has zeros

1

At this point, it should be noted that by satisfying the gain-bandwidth restrictions for a preselected ε2 = Tmax2ΔT , we can determine the maximum gain level Tmax which in turns yields the −2ΔT desired average gain level T0 .

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◦ at DC of order 4 due to the series capacitor C4 = 19.8 p F and the shunt inductor L 1 = 0.225μH , ◦ at a finite actual frequency f z1 = 2π √1L C ≈ 471M H z due to the shunt arm 3 3 constructed by the inductor L 3 = 0.01μH in series with the capacitor C3 = 11.4 p F. ◦ and at infinity of order 2 due to series connections of the inductors L 4 = 0.1μH and L 2 = 0.4μH forming a degeneracy as the frequency approaches to infinity. These zeros must appear as the transmission zeros of the matched system in the transfer function T (ω). Lowpass to bandpass transformation can easily introduce zeros at DC and preserves the zeros at infinity. However, there is no way to introduce finite frequency zeros of transmission on the monotone roll-off functions. Therefore, our choice of monotone roll-off Chebyshev function at the beginning was not appropriate. In this case, we may try to utilize elliptic functions to introduce finite transmission zeros on the transfer function. Unfortunately, this process is tedious and the form of the elliptic transfer function is not explicit. Therefore, we will not attempt to go through this tedious and difficult process which we know by heart that it can not yield better solutions over those of real frequency techniques. At this point it is also crucial to mentioned that any kind of analytic transfer function obtained via lowpass to bandpass transformation yields a bulk lossless 2-port with parallel and series tank circuits from which it may not be possible to extract the load network. Thus, we conclude that there is no way to employ the analytic theory of matching to match the short monopole antenna to a 50 ohm resistive generator. On the other hand, the Simplified Real Frequency Technique can easily be used to answer the above questions and eventually we can construct the desired equalizer as detailed in the following sections.

Determination of the Gain Bandwidth Limits of a Given Complex Load via SRFT Gain Bandwidth limit of a given complex load may be determined in an add-hoc manner using the simplified real frequency technique as outlined within the following rule of thumbs. ◦ From the analytic theory, we know that for a fixed frequency band, the ideal solution can be reached by increasing the total number equalizer elements. Therefore, the SRFT algorithm can be run for the matching problem under consideration, starting with total number of element values for n = 2, 3, 4, 5, .. etc.2 ◦ It is always preferable to deal with a practical equalizers consist of series inductors and shunt capacitors. In this case, we can start running the SRFT algorithm 2 In the MatLab main program SRFTWSOP, the total number of equalizer elements or equivalently n (n−1) degree of the numerator polynomial + . . . h 1 p + h 0 is extracted from h( p) = h n p + h (n−1) p the given initial row vector h = h n h n−1 . . . h 1 h 0 .

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by setting the integer k = 0 of the transfer scattering parameter E 21 ( p) = g(pp) . However, some complicated loads such as our short monopole antenna may insist to employ zeros of transmissions at DC within the equalizer. In this case, SRFT algorithm can be run for various values of the integer k = 0, 1, 2, ..etc. ◦ The results of the above trails must be compared with each other subject to a pre-selected gain level T0 which automatically reveals the fluctuation ΔT as the result of optimizations. Therefore, we should also run SRFT algorithm for various flat gain levels T0 perhaps starting from its ideal value T0 = 1 then, come down to T0 = 0.5 level with sweeping steps of δT = 0.05. ◦ Obviously, selected initials on the coefficients of h = [h n h n−1 .. h 1 h 0 ] are extremely important to end up with successful optimization. However, since SRFT is almost quadratic in terms of the unknown coefficients of h = [h n h n−1 .. h1 h 0 ], it may be sufficient to start with an ad-hoc choice of h = ±1 ∓1 .. ±1 ∓1 and improve the optimization results on this choice. Thus, the above trails can be stopped at a point when we reach to an acceptable average gain level 0 ≤ T0 ≤ 0.5 with desired fluctuation. In this regard, employing the SRFT algorithm with simple optimization (SRFTWSOP),3 we can determine the maximum attainable gain level over the frequency band of 30–70 MHz.4 Eventually, choosing n = 4 and k = 1, we swept the flat gain level from T0 = 1 down to T0 = 0.75 and using the ad-hoc initial values of h = [−1 1 − 1 1 − 1] then, continuing with the following iterations; Iteration 1 : h = [−1.7385 Iteration 2 : h = [−1.7386

1.9746 1.9750

0.4949 0.4946

0.7287 0.7286

− 0.1274] − 0.1273]

Iteration 3 : h = [−1.7391 Iteration 4 : h = [−1.7393

1.9752 1.9758

0.4948 0.4944

0.7286 0.7286

− 0.1273] − 0.1273]

final value of the unknown h vector is obtained as: h = [ − 1.7396

1.9759

0.4946

0.7287 − 0.1274]

and the corresponding strictly Hurwitz vector g is found: g = [1.7396

4.0936

3.1996

1.5722

0.1274]

which yields the transducer power gain as in Table 10.1.5 3

The main SRFTWOP is operated with the load model shown in Fig. 10.1. In order to bias the optimization at the high-end of the passband, we choose whigh = 0.75 instead of 0.7 which helps to jack the gain up at ω = 0.7. This is the regular trick always pulled during the optimizations of the transducer power gain to improve system performance of the matched antenna. 5 At this stage, while we run SWRFWSOP, output of each run manually fed into the main program. This process continues until we end up with reasonable optimization results. 4

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Table 10.1 Transducer power gain variation of the matched short-monople antenna Actual frequency (MHz)

Transducer Power Gain

5.0000 10.0000 15.0000 20.0000 25.0000 30.0000 Low-end 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70.0000 High-end 75.0000 80.0000 85.0000 90.0000 95.0000 100.0000

0.0000 0.0003 0.0047 0.0514 0.3656 0.7031 = Tmin 0.7185 0.7274 0.7309 0.7167 0.7057 0.7210 0.7533 = Tmax 0.7037 0.4732 0.2363 0.1098 0.0535 0.0280 0.0157

As it seen from the above table, the maximum gain Tmax = 0.7533 is obtained at the 65 MHz then comes down to Tmin = 0.7037 at the high-end of the passband which is 70 MHz. p) is synthesized Eventaully, the back-end reflection coefficient E 22 ( p) = h( g( p) yielding the desired lossless equalizer as shown in Fig. 10.6. Thus, the average gain is found as Tav = 0.728 ± 0.024 (i.e. the fluctuation is ΔT = 0.024). The gain plot is given by Fig. 10.5. In the following, the main program SRFTWSOP and the Monopole antenna reflection coefficient is listed as they are used to produce the above results.

MatLab List for Program SRFTWSOP to Design Lossless Equalizer for the Short Monopole Antenna % PROGRAM:SRFT Monopole: Design of a Monopole Antenna Equalizer % using Simplified real Frequency Technique with simple optimization % function lsqnonlin % In this program the objective function is selected as the inverse of the TPG % Inputs: % T0: Desired flat gain level % ntr: Control flag for equalizer design: % ntr = 1; equlizer design with a transformer.

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Fig. 10.5 Transducer Power Gain of the Matched monopole antenna

Fig. 10.6 Lossless Equalizer to Match the Monopole Antenna to Resistive Generator over the frequency band of 30–70 MHz

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% In this case k may be different than zero. % ntr = 0; equlizer is constructed without transfermer. % In this case k must be zero (k = 0) % and equlizer will be constructed with Low Pass LC elements % % initial values for h(p), and f(p) = pˆk % nopt: total number of sampling points within optimization. % This is the dimension of the objective function invT subject to optimization. % wlow: Low frequency end of the optimization. % whigh: High Frequency end of the optimization % cccccccccccccccccccccccccccccccccccccccccccccccccccccc % Outputs: % Construction of the antenna matching network with Least Square % Optimization using Levenberg-Marquad Technique. % Optimized Scattering Parameters of the Equalizer. % Optimized Transducer Power Gain of the Matched System % Realization of the Equalizer as an LC ladder with explicit element values. % Plot of the Gain Performance clear % ccccccccccccccccccccccccccccccccccccccccccccccccccccc T0 = 0.75 ntr = 1; % Design without Transformer k = 1; %Numer of transmission zeros at DC; k=0 Lowpass Ladder Structure nopt = 30; %Total number of sample point within optimization routine wlow = 0.3; %Low-end of the passband whigh = 0.73; %High-end of the passband % % Enter coefficients of h(p) as a MatLab row vector. %h=[ −1 1 1 1 −1]; %(Note that if ntr=0; n=na, if ntr=1 then n=na−1) %h =[−1.7385 1.9746 0.4949 0.7287 −0.1274] %h =[−1.7386 1.9750 0.4946 0.7286 −0.1273] %h =[−1.7391 1.9752 0.4948 0.7286 −0.1273] h =[−1.7393 1.9758 0.4944 0.7286 −0.1273] %optimization na=length(h); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Start Optimization using simplest version of the LSQNONLIN % % Type design prefernce: % ntr=1: design with transformer % ntr=0: design without transformer. % -------------------------------------------------------------% %%%%%%%%%%% Preparetion for the optimization %%%%%%%%%%%%%

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OPTIONS=OPTIMSET(‘MaxFunEvals’, 20000, ‘MaxIter’, 50000); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %Design with transformer: if ntr==1, x0=h; % Call optimization function lsqnonlin: x=lsqnonlin(‘sopt’,x0,[],[],OPTIONS,T0,ntr,k,nopt,wlow,whigh); % h=x; end %-------------------------------------------------------------------------% % Design without transformer: if ntr==0, for r=1:na-1 x0(r)=h(r); %Call optimization function lsqnonlin: x=lsqnonlin(‘sopt’,x0,[],[],OPTIONS,T0,ntr,k,nopt,wlow,whigh); end h=x h(na)=0; end % Generate strictly hurwits polynomial g(p) from optimized h(p): g=Hurwitzpoly˙g(h,k); % Compute the optimized transducer power gain W1=0; W2=1.5; Nsample=30; DW=(W2−W1)/Nsample; f0=100; W=0; for j=1:Nsample+1 Wa(j)=f0*W; E22=Equalizer(W,h,g,k); % GENERATION OF THE LOAD DATA % Reactive C=4 L11=reflection(W); SRL(j)=real(L11); SXL=imag(L11); hR=polyval(h,W); hRa(j)=hR; ZL=(1+L11)/(1-L11); RLa(j)=real(ZL); XLa(j)=imag(ZL); % COMPUTATION OF GAIN T=gain(W,h,g,k,L11);

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Ta(j)=T; TdB(j)=10*log10(Ta(j)); W=W+DW; end % plot(Wa,Ta) CV=general synthesis(h,g)

MatLab List for the Computation of the Antenna Reflection Coefficient function SL=reflection(w) % function reflection computes the reflection coefficient of the load % network: As it is, here we have Short Monopole Antenna Model % This is a user defined function for matlab. % Input: % Angular frequency w %Output: % SL: Load reflection coefficient of the load. % SL is a complex quantity. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p=w*sqrt(−1); %%%%%%%%%%%%%%%%%%%%%%%%%%% % % Short Monope Antena 20 MHz to 100 MHz % pi=4*atan(1.0); % Normalization Numbers f0=100e6; w0=2*pi*f0; R0=50; % L1=(0.225e−6)*w0/R0; Y1=1.0/(L1*p)+1.0; % L2=(0.4e−6)*w0/R0; % Z2=(L2*p)+1/Y1; Y2=1/Z2; % L3=(0.01e−6)*w0/R0; C3=(11.4e−12)*w0*R0; % Z3a=L3*p+1/(C3*p);

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Y3=1/Z3a+Y2; Z3=1/Y3; % L4=(0.1e−6)*w0/R0; C4=(19.8e−12)*w0*R0; % Z4a=L4*p+1/(C4*p); Z4=Z4a+Z3; % ZL=Z4; SL=(ZL−1)/(ZL+1); return In addition to “function reflection”, the following MatLab functions are also used to run the main Program SRFTWSOP as in Chapter 9:

function SE=Equalizer(W,h,g,k) function T=gain(W,h,g,k,L11) function CV=general synthesis(h,g) function g=Hurwitzpoly g(h,k)

Investigation of the Half Power Bandwidth of the Short Monopole Antenna In this section we will empirically determine the half power bandwidth of the matched antenna system which corresponds to “−3 dB” gain drop from zero 0 dB gain level. This description approximately yields the average gain T0 ≈ Tav ≈ 0.5 in absolute terms.6 Obviously, at 0 dB gain level the transducer power gain is unity. At this point, it is appropriate to carefully examine the antenna impedance variation to develop some meaningful understanding of the problem. For this purpose let us list the measured antenna impedance as in Table 10.2. The above table reveals that the minimum value Rmin ≈ 0.3 of the real part is at 20 MHz and it reaches to its maximum of Rmax = 19.08 at 70 MHz. This is a huge variation (Rmax /Rmin = 63.6) which makes the investigation extremely difficult. However, from the previous section we found that from 30 to 70 MHz, we are able to obtain an average gain of Tav = 0.728 ∓ 0.027. In this frequency band, the real part variation is R(70M H z)/R(30M H z) ≈ 19.89/0.6783 = 28.1. Actually, this variation is more-less preserved in the frequency band of 30–100 MHz. In this case,

In decibel (dB), the transducer power gain is given by T (d B) = 10. log10 (T ). Based on this definition, T = 0.5 and T = 1 corresponds to the values of −3 dB and zero dB respectively.

6

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237 Table 10.2 Short monopole antenna impedance variation

Normalized antenna impedance (with Respect to R0 = 50⍀) Z Antenna = R + j X Actual frequency 20.0000 25.0000 30.0000 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000 90.0000 95.0000 100.0000

R

X

0.3009 = Rmin 0.4662 0.6783 0.9599 1.3558 1.9561 2.9537 4.7977 8.5654 15.6953 19.0892 = Rmax 11.3747 5.6458 3.0460 1.8146 1.1676 0.7957

−6.1944 −4.0875 −2.4685 −1.0783 0.2391 1.6055 3.1335 4.8865 6.4790 4.8482 −5.8020 −12.1575 −11.1522 −9.1796 −7.5723 −6.3525 −5.4118

it is meaningful to sweep the flat gain level T0 starting from 0.7 down to 0.5 within this band. In the course of initialization of the optimization, we can let k = 0 just to release the gain roll-off at the lower edge of the band. Thus, setting the inputs of SRFTWOP as T0 = 0.62 ntr = 1; k = 0; nopt = 30; wlow = 0.2; Whigh = 1.; and playing with the initial vector h in an ad-hoc manner, we have the following result. The numerator h( p) and the denominator polynomials g( p) of the input reflection coefficient of the equalizer are found as h = [ −1.0406e −005 0.6929 0.1166 4.4546 2.1914 3.2635 0.4316] g = [1.0406e −005 0.6929 0.8794 5.0029 3.9505 4.1670 1.0891] yielding an average transducer power gain Tav ≈ 0.59 ± 0.07. Resulting gain plot is shown in Fig. 10.7. Complete list of the TPG is given in Table 10.3.

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Fig. 10.7 Gain variation of the short monopole antenna over 30–100 MHz bandwidth

Table 10.3 List for the transducer power gain vs frequency upto 100 MHz Actual frequency (MHz)

Transducer power gain

20.0000 25.0000 30.0000 35.0000 40.0000 45.0000 50.0000 55.0000 60.0000 65.0000 70.0000 75.0000 80.0000 85.0000 90.0000 95.0000 100.0000

0.1071 0.3443 0.5867 0.6063 0.5673 0.5535 0.5694 0.6047 0.6419 0.6606 = Tmax 0.6502 0.6219 0.5987 0.6006 0.6333 0.6523 0.5250 = Tmin

Eventually, input reflection coefficient of the lossless equalizer is synthesized as a low-pass ladder network in R = 2.31⍀2.31 ohm termination as depicted in Fig. 10.8. Obviously, using a transformer termination resistance can be level to unity. The above gain variation may be summarized in dB as follows. Tmin = 10 log(0.52) = −2.8 dB with the average gain Tav ≈ −2.32 ± 0.517 dB. It should be mentioned that we tried to stretch the band down to 20 MHz. However, we were not able end-up with a successful optimization. Magnified examination of the gain variation between 20 and 25 MHz is shown in Table 10.4.

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Fig. 10.8 Synthesis of the lossless equalizer to match short monopole antenna to a resistive generator yielding a Tav = 0.59 ± 0.07 gain variation within 30–100 MHz

Based on Table 10.4, we can conclude that −3 dB gain-bandwidth of the short monopole antenna can safely be stretch from 26 to 100 MHz. The resulting lossless equalizer is given in Fig. 10.8. The half power bandwidth may be defined with respect to the maximum value of the gain. This may be named as the relative half power bandwidth of the matched Table 10.4 Magnified gain variation over 20–25 MHz Frequency (MHZ)

Gain (dB)

20.00 20.50 21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00

−9.7022 −9.1117 −8.5356 −7.9752 −7.4318 −6.9071 −6.4029 −5.9212 −5.4639 −5.0330 −4.630

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system. In this case, maximum value of the gain is 10. log 10(0.66) = −1.8d B at 65 MHz. Then, we can determine the edge of the band frequencies at the gain point where T ≥ −4.8d B. Table 10.4 reveals that the lower edge of the relative half power passband is placed in the frequency interval of 24.5–25 MHz.

Dual Band Antenna Equalizer Realized by Utilizing 0.18 ␮m Si-Processing Technology for a Pifa-900 With the development of mobile communication systems and miniaturization of hand sets, requirements for compact antennas are rapidly growing. The Planar Inverted F Antenna (PIFA) has found widespread internal utilization within wireless terminals due to its small size for both single [1] and dual band applications [2, 3]. The PIFA has many known mechanical advantages, e.g. ease of fabrication, low manufacturing cost, ground plane compatibility and conformity with complex geometries [4]. Concerning with the electrical properties, it can be matched to any impedance (by choosing the location of the short circuit) without matching network, and the design concept is easily extended to dual band functionality [5]. Finally, internal PIFA/patch elements have lower SAR (specific absorption rate) values [6] and are less affected (in terms of resonance frequency shift and efficiency reduction) by the presence of the users head in talk position compared to other antennas, e.g. whips/monopoles [7]. On the downside, it typically suffers from a narrow impedance bandwidth, which excludes it from a variety of applications. For example, modern transceiver front-ends are available [8] with quad band support (GSM850/900/1800/1900) for global roaming. Unfortunately, this bandwidth is extremely difficult to achieve with a PIFA within the volume typically allocated for the antenna element. However, it should be kept in mind that without designing a matching network, one would never know what the true realizable bandwidth of the antenna is. Therefore, in this section, we present the design of a single matching network to extend the usage of PIFA over the commercial cellular frequency bands of 850/900/1800/1900 MHZ (i.e. 824–960 & 1710–1990 MHz). Manufacturing layout details are given in [9, 10, 11, 12, 13]. In these papers, we introduced the implementation of a dual band PIFA antenna together with its matching network to asses the achievable antenna performance employing 0.18μ Si- processing technology. Most research reported on terminal antennas has so far focused on modifications of the radiating element for wide-band [14] and/or multi-band performance [15], as opposed to applying an external matching network. In [16], a microstrip transmission line resonator was integrated with a single-band antenna 900 MHz (patch) element for impedance matching. The concept was extended in [17], where a dual band 900/1800 MHz terminal antenna of similar size as in this paper was matched using an open microstrip stub for extension of the lower band for GSM850/ 900 coverage. However, the effect on the upper frequency band was a substantial bandwidth reduction. In this section we will demonstrate the use of SRFT tool to investigate the gain bandwidth limit of the PIFA mentioned above [18, 19, 20, 21, 22]. By designing

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Fig. 10.9 PIFA 900 Layout

proper antenna equalizer, the bandwidth of the matched antenna has been extended to fully cover the commercial bands for GSM850/900/1800/1900 or more specifically (824–960 & 1710–1990 MHz).

Description of the PIFA -900 The antenna configuration is shown in Fig. 10.9. A flat PIFA element is located in the topmost section 8 mm above a typically sized 35 × 100 mm2 ground-plane, with feeding and short-circuit connected at the top-right positions for maximum impedance bandwidth [19]. Dual-band functionality is implemented by using two unequally sized galvanically coupled resonant patches [5]. A short patch is resonant at 1800 MHz and a longer, bent patch is resonant at 900 MHz. No substrate is used for the PIFA carrier. The chassis is manufactured using single-sided 0.8 mm FR-4 Table 10.5 Measured reflectance SL = ρ L ejϕ L of a PIFA over the optimization frequencies Frequency (GHz)

Amplitude ρ L

Phase ϕ L in degree

0.8000 0.8500 0.9000 0.9500 1.0000 1.7000 1.7500 1.8000 1.8500 1.9000 1.9500 2.0000

0.9166 0.7572 0.2897 0.3618 0.6710 0.5736 0.3644 0.2890 0.4477 0.6076 0.7159 0.7851

33.9069 9.9996 −39.9291 130.3831 98.0474 −44.7173 −86.9403 −165.9859 137.3146 109.8189 93.2402 81.9757

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(or = 4.44 and tan d = 0.02) PCB, with the bottom side metallized. The measured input reflectance data is given in Table 10.5. Close examination of this data indicates that bandwidth (BW) of the antenna without matching network is limited to dual-band coverage of GSM900/1800 or GSM850/1900 cellular bands, as is typical for internal antennas of similar size [20, 15]. However, in the following section, we will show that BW is increased more than 25% in the neighborhood of center frequencies 900 and 1800 MHz covering all the commercial cellular bands available over the frequencies 824–960 MHz and 1710–1990 MHz just utilizing a single matching network or so called antenna equalizer.

Equalizer Design The impedance matching network is designed using the Simplified Real Frequency Technique (SRFT) [18, 21, 22, 9]. The advantages of this technique are listed as follows.

r

r

As oppose to analytic theory, it neither requires antenna model nor the selection of the analytic form of a transfer function. SRFT directly utilizes the measured reflectance data and optimizes the transducer power gain of the matched system as flat and as high as possible which in turn yields the analytic form of the driving point reflectance of the equalizer. As oppose to brute force computer design techniques, in SRFT circuit topology is not selected in advance. Rather it is automatically obtained as the result of the synthesis of the input reflection coefficient with desired element values.

At this point, it is important to assess the theoretical gain bandwidth limit of the antenna. Unfortunately, analytic theory is incapable beyond simple cases. However, Real Frequency Line Segment Technique of Carlin [22] or SRFT provides an insight to matching problems with an excellent estimate for the upper level of the flat transducer power gain, or equivalently the minimum return loss, over the selected frequency bands. Therefore, the antenna matching network is directly designed using SRFT. Now, let us proceed with design process employing the SRFT tool specifically developed for this problem. The MatLab Main program is called “srft Ch10 PIFA.m” which stands for SRFT to design PIFA Matching network for Chapter 10. In this program, measured data is selected in two bands, namely B1: 824–960 Hz and B2: 1710–1990 MHz to optimize the transducer power gain, as shown in Table 10.5. Measured amplitude and phase reflectance of the antenna are depicted in Figs. 10.10 and 10.11 respectively. Furthermore, un matched antenna gain is shown in Fig. 10.12. Close examination of these figures reveals that this PIFA is good for only 900 MHz and 1800 MHz usage. However as it will be shown below, bandwidth of the antenna

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Fig. 10.10 Measured Amplitude Data plot of PIFA-900

will be expanded about 25% in both lower and upper bands by designing a single matching network.

First Trail: Design of an Equalizer with 2 Elements Main Program: srft Ch10 PIFA.m Optimization function: sopt10 Here, we initiate srft ch10.m with h0 = [1 1 0]. The last entry of h0 is taken as zero just to avoid the use of a transformer in the matching network. Other inputs to the program are listed below. n = 2; (Two-element design which corresponds h0) k = 0; (LC-Lowpass ladder structure for the equalizer for easy implementation) ntr = 0; (Design without transformer to make practical implementation easier) T0 = 0.9 (Flat gain level subject to optimization. Ideally it is unity. Since we are trying to expand the band we should lower it to gain bandwidth)

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Fig. 10.11 Measured Phase Data plot of PIFA-900

Antenna data subject to optimization is loaded to the program from a file called “annb opt”. This data is listed in Table 10.5. However, full measured data is loaded from the text file called “annb.txt” which is plotted on Figs. 10.10, 10.11, and 10.12. Result of optimization is given as follows: h = [−0.0883 − 0.0958 0] g = [0.0883 0.4311 1.0000] which corresponds to the input reflectance of the equalizer −0.0833 p 2 − 0.0958 p to be synthesized as a lossless two0.0833 p 2 + 0.4311 p + 1 port in unit termination as shown in Fig. 10.13. Matched and un-matched gains are plotted in Fig. 10.14. Detailed gain data is given by Table 10.6. As it is clearly seen from Fig. 10.14 and Table 10.6, bandwidth of PIFA is expanded however, gain at 850 MHz is less than −3 dB which is not sufficient for the band expansion. Optimization results can be improved by increasing the total number elements in the equalizer. In order to exhibit the power of SRFT, let us E 11(or

22) ( p)

=

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Fig. 10.12 Un matched Antenna Gain of PIFA-900

design the equalizer with 8-elements and observe the improvements on the bandwidth. At this point, one should mention that to design an 8-element equalizer with known classical methods is almost impossible. However, as we will see now, with ad-hoc choice of initial guess on the coefficients of h( p), SRFT yields an excellent solution.

Design of a Single Equalizer with 8 Elements for Double Band PIFA In this case, PIFA data is loaded to the main program as before, band width is selected over B1: 824–960 MHz; B2: 1710–1990 MHz as specified within data file: “annb opt.txt”. Program srft ch10.m is initiated as follows. Run Program: “srft Ch10 PIFA.m” with optimization function “sopt10” (see list below) Initials to the Pragram: n = 8; (Total Number of reactive elements in the equalizer)

246

Fig. 10.13 2-Element Equalizer for PIFA

Fig. 10.14 2-element Matched and un-matched gains of PIFA

B.S. Yarman

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Table 10.6 Transducer power gain performance of matched PIFA with 2-element equalizer Frequency(GHz)

Gain (dB)

0.7000

−14.6973

0.7500

−11.2129

0.8000

−7.4173

0.8500

−3.4587

0.9000

−0.5471

0.9500

−0.4740

1.0000

−1.9700

1.0500

−3.5452

1.1000

−5.0446

1.1500

−6.7668

1.2000

−9.1339

1.2500

−12.3544

1.3000

−15.1627

1.3500

−14.6046

1.4000

−12.2715

1.4500

−10.1283

1.5000

−8.3448

1.5500

−6.7963

1.6000

−5.3911

1.6500

−4.0929

1.7000

−2.9158

1.7500

−1.9207

1.8000

−1.2021

1.8500

−0.8472

1.9000

−0.8788

1.9500

−1.2342

2.0000

−1.8027

k = 0; (Lowpass LC Ladder) ntr = 0; (Design without transformer) T0 = 0.9 (Flat Gain Level) Initial for h0: h0 = [1 − 1 1 − 1 1 − 1 1 − 1 0] Result of optimization yields, h = [0.0714 − 0.1275 0.5844 − 0.8698 1.5825 − 1.5220 1.3661 − 0.1745 0] g = [0.0714 0.2924 1.0692 2.4466 4.4338 5.8071 5.6037 3.3523 1.0000]

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Table 10.7 Comparison of bandwidths among un-matched, matched with 2-elements and matched with 8 elements PIFAT Un matched PIFA

Matched with 2-elements PIFA

Matched with 8-elements PIFA

B1:860–1000 MHz; ΔB = 140

B1:840–1000 MHz; ΔB = 160

B1:820–1075 MHZ; ΔB = 225

B2:1710–1920 MHz; ΔB = 210

B2:1690–2000 MHz; ΔB = 310

B2:1640–2000 MHz; ΔB = 360

Optimized transducer power gain performance is detailed in Table 10.8 and it is depicted in Fig. 10.15. Close examination of Table 10.8 and Fig. 10.15 reveals that TPG and thus the bandwidth were sharply improved. Just for the comparison purpose, in Table 10.8 we also show the performances of the un-matched, matched with 2-element PIFA for comparison purpose. For examTable 10.8 Comparison of the TPG performance of the un-matched, Matched with 2 and matched with 8 element equalizers of PIFA Frequency

Un-matched TPG (dB)

Matched with 2element

Matched with 8-elements

0.7000 0.7500 0.8000 0.8500 0.9000 0.9500 1.0000 1.0500 1.1000 1.1500 1.2000 1.2500 1.3000 1.3500 1.4000 1.4500 1.5000 1.5500 1.6000 1.6500 1.7000 1.7500 1.8000 1.8500 1.9000 1.9500 2.0000

−15.3479 −11.8570 −7.9632 −3.6995 −0.3807 −0.6094 −2.5982 −4.4671 −6.1493 −8.0036 −10.4643 −13.7260 −16.5103 −15.8654 −13.3871 −11.0367 −8.9725 −7.0556 −5.1893 −3.3680 −1.7327 −0.6188 −0.3787 −0.9717 −2.0010 −3.1198 −4.1614

−14.6973 −11.2129 −7.4173 −3.4587 −0.5471 −0.4740 −1.9700 −3.5452 −5.0446 −6.7668 −9.1339 −12.3544 −15.1627 −14.6046 −12.2715 −10.1283 −8.3448 −6.7963 −5.3911 −4.0929 −2.9158 −1.9207 −1.2021 −0.8472 −0.8788 −1.2342 −1.8027

−15.2486 −10.0179 −4.3831 −1.6544 −1.8651 −1.5477 −0.6056 −1.0464 −4.0638 −8.0480 −12.3698 −17.0301 −20.7289 −20.5433 −18.1057 −15.3886 −12.5309 −9.3696 −5.8995 −2.7126 −1.0658 −0.8524 −0.9077 −0.9502 −1.0536 −0.8648 −0.9378

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Fig. 10.15 Matched with 8-element equalizer and unmatched gain of PIFA

ple, minimum gain of un matched PIFA is −3.69 dB while 2-element matched PIFA yields slightly improved gain of −3.458 dB and 8-element matched PIFA gives us sharply improved gain of −1.65 dB gain. On the other hand −3 dB band of unmatched, Matched with 2 and matched with 8 element PIFA yields as shown in Table 10.7. Thus, we see that matched PIFA yields improved bandwidth of 12.5% with 2elements and 37.7%, with 8-element equalizers in B1. Similarly, in B2 we have 32% and 61% improvements with 2 and 8 element matching networks respectively. Finally, 8-element equalizer with normalized element values (with respect to fnorn = 1 Ghz and R0 = 50 Ohmis) is depicted in Fig. 10.16 and actual element are given as:7 L(1) = 2.7064 nH, C(2) = 2.5420 pF; L(3) = 8.328 nH C(4) = 2.3654 pF; L(5) = 9.6989 nH; C(6) = 3.5622 pF L(7) = 4.5550 nH; C(8) = 2.7566 pF; R9 = 50 ohm. List of the MatLab Main Program: srft ch10 PIFA.m 7

Note that actual element values are given as L(i) Ci /R0 (2π f nor m).

=

L i .R0 /(2π. f nor m); C(i)

=

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B.S. Yarman

Fig. 10.16 8-Element equalizer for PIFA-900

% PROGRAM FOR Chapter 10: DESIGN OF EQUALIZER FOR PIFA-900 % Design of an Equalizer for PIFA antenna % In this program load data for the antenna is read from a file called % annb opt.txt clear n=8; k=0; ntr=0; T0=0.9 %..............Laod Data for PIFA antenna subject to optimization from text %file called annb opt load annb˙opt.txt j=sqrt(−1); Nopt=length(annb˙opt); for i=1:Nopt % Read the selected data for Frequency, Amplitudeand Phase Wopt(i)=annb˙opt(i,1); mopt(i)=annb˙opt(i,2);

10 Applications

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m21opt(i)=1−mopt(i)*mopt(i); pa(i)=annb opt(i,3); % Convert data to cartezian form: SRLopt(i)=mopt(i)*cos(pa(i)*pi/180); SXLopt(i)=mopt(i)*sin(pa(i)*pi/180); SLopt(i)=SRLopt(i)+j*SXLopt(i); end % Initialize the coeffecients for h h0=[1 −1 1 −1 1 −1 1 −1 0] %h0=[1 1 0]; % na=n+1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Start Optimization using simplest version of the LSQNONLIN % % Type design prefernce: % ntr=1: design with transformer % ntr=0: design without transformer. % -------------------------------------------------------------% %%%%%%%%%%% Preparetion for the optimization %%%%%%%%%%%%% OPTIONS=OPTIMSET(‘MaxFunEvals’,20000,‘MaxIter’,50000); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Design with transformer: if ntr==1, x0=h0; % Call optimization function lsqnonlin: x=lsqnonlin(‘sopt9’,x0,[],[],OPTIONS,T0,ntr,k,Wa,SRLopt,SXLopt); % h=x; end %-------------------------------------------------------------------------% Design without transformer: if ntr==0, for r=1:na−1 x0(r)=h0(r); end % Call optimization function lsqnonlin: x=lsqnonlin(‘sopt9’,x0,[],[],OPTIONS,T0,ntr,k,Wopt,SRLopt,SXLopt); for r=1:na−1 h(r)=x(r); end

252

h(na)=0; end % Generate strictly hurwits polynomial g(p) from optimized h(p): g=Hurwitzpoly˙g(h,k); % Compute the optimized transducer power gain % Read full Data load annb.txt j=sqrt(−1); Nb=length(annb); for i=1:Nb % Read the selected data for Frequency, Amplitudeand Phase Wa(i)=annb(i,1); ma(i)=annb(i,2); ma21(i)=1−ma(i)*ma(i); pa(i)=annb(i,3); % Convert data to cartezian form: SRLa(i)=ma(i)*cos(pa(i)*pi/180); SXLa(i)=ma(i)*sin(pa(i)*pi/180); SLa(i)=SRLa(i)+j*SXLa(i); end Ndata=length(Wa); for j=1:Ndata w=Wa(j) SR=SRLa(j); SX=SXLa(j); L11=complex(SR,SX); % COMPUTATION OF GAIN g=Hurwitzpoly g(h,k); T=gain(w,h,g,k,L11); Ta(j)=T; TdB(j)=10*log10(T); rloss(j)=10*log(ma21(j)); end na=n+1 for i=1:na y(i)=g(na−i+1); x(i)=h(na−i+1); end % plot(Wa,TdB) hold plot(Wa,rloss) CV=general˙synthesis(h,g); List of function sopt9 function fun=sopt9(x,T0,ntr,k,Wa,SRLa,SXLa)

B.S. Yarman

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% Inputs: % T0: Desired gain level % ntr: Control flag for equalizer design: % ntr=1; equlizer design with a transformer. % In this case k may be different than zero. % ntr=0; equlizer is constructed without transfermer. % In this case k must be zero (k=0) % and equlizer will be constructed with Low Pass LC elements % x=Row Vector includes the unknow coefficients of the polynomial h(p). % for ntr=0 and k=0, dimension of x must be n. % for ntr=1, dimension of x must be n+1. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Output: % fun: Objective function Vector subject to optimization. % For the design of Antenna Matching Networks fun=T−T0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nopt=length(Wa); nx=length(x); for j=1:nopt % Prepare to generate transducer power gain: % Call load function to compute load reflection coefficient. w=Wa(j); SR=SRLa(j); SX=SXLa(j); L11=complex(SR,SX); % Set h vector for the values of ntr: if ntr==1, h=x; end % if ntr==0, for r=1:nx h(r)=x(r); end na=nx+1; h(na)=0; end % % Generate g polynomial from h: g=Hurwitzpoly g(h,k); % Generate the Transducer power Gain T=gain(w,h,g,k,L11); % Generate objective function as fun(j)=T−T0; end return

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Referencess 1. T. Taga and K. Tsunekawa. Performance analysis of a built-in planar inverted F antenna for 800 MHz band portable radio units. IEEE Journal on Selected Areas in Communications, SAC-5:921–929, June 1987. 2. C.R. Rowell and R.D. Murch. A compact PIFA suitable for dual-frequency 900/1800-MHz operation. IEEE Transactions on Antennas and Propagation, 46(4):596–598, April 1998. 3. K-L. Wong, Y-C. Lin, and T-C. Tseng. Thin internal GSM/DCS patch antenna for a portable mobile terminal. IEEE Transactions on Antennas and Propagation, 54(1):238–242, January 2006. 4. S. Yarasi, G.R. Kadambi, and T. Hebron. Conformal shaped PIFAs for mobile communication applications. In Proc. Antennas and Propagation Society International Symposium, volume 3, pages 82–85, June 2003. 5. Z.D. Liu, P.S. Hall, and D. Wake. Dual-frequency planar inverted-F antenna. IEEE Transactions on Antennas and Propagation, 45(10):1451–1458, October 1997. 6. G. Lazzi, J. Johnson, S.S. Pattnaik, and O.P Gandhi. Experimental study on compact, high-gain, low SAR single-and dual-band patch antenna for cellular telephones. In Proc. Antennas and Propagation Society International Symposium, volume 1, pages 130–133, June 1998. 7. K.H. Chan, K.M. Chow, L.C. Fung, and S.W. Leung. SAR of internal antenna in mobile-phone applications. Microwave and Optical Technologt Letters, 45(4):286–290, May 2005. 8. R. Magoon, A. Molnar, J. Zachan, G. Hatcher, and W. Rhee. A single-chip quad-band (850/900/1800/1900 MHz) direct conversion GSM/GPRS RF transceiver with integrated VCOs and fractional-n synthesizer. IEEE Journal of Solid-State Circuits, 37(12):1710–1720, December 2002. 9. http://rfic.eecs.berkeley.edu/∼niknejad/asitic.html 10. P. Lindberg, M. S¸eng¨ul, E. C ¸ imen, B.S. Yarman, A. Rydberg, A. Aksen. A Single Matching Network Design for a Dual Band Pifa Antenna via Simplified Real Frequency Technique, The first European Conference on Antennas and Propagation (EuCAP 2006), 6–10 November 2006 Nice, France. 11. B.S. Yarman et al. A Single Matching Network Design for a Double Band PIFA Antenna via Simplified Real Frequency Technique, Asia Pacific Microwave Conference, December 13–15, 2006, Yokohama, Japan. 12. Unno, T. Fujita, P. Lindberg, B.S. Yarman, N. Fujii. Dual Band Equalizer Realized by Utilizing 0.18 μm Si-Processing Technology for PIFA-900 Antenna. IEEE International Symposium on Circuits and Systems, to be held in New Orleans, USA, May 27–30, 2007, Proceeding ISBN No: 1-4244-0921-7, pp. 3908–3911. 13. Yarman et al. Design of Broadband Matching Networks, ECT January 24–27, 2007, Okinawa, Japan. (Invited Talk) 14. K.R. Boyle. Differentially slotted and differentially filled PIFAs. Electronics Letters, 39(1):9–10, January 2003. 15. H-T Chen, K-L. Wong, and T-W. Chiou. Pifa with a meandered and folded patch for the dual- band mobile phone application. IEEE Transactions on Antennas and Propagation, 51(9):2468–2471, September 2003. 16. N. Fayyaz, E. Shin, and S. Safavi-Naeini. A novel dual band patch antenna for GSM band. In Proc. IEEE AP-S Conference on Antennas and Propagation for Wireless Communications, pages 1 56–160, November 1998. 17. J. Ollikainen, O. Kivekas, C. Ichein, and P. Vainikainen. Internal multiband handset antenna realized with an integrated matching circuit. In Proc. Antennas and Propagation Society International Symposium, volume 2, pages 629–632, March 2003. 18. B.S. Yarman and H.J. Carlin. A simplified real frequency technique applied to broad-band multistage microwave amplifiers. IEEE Transactions on Microwave Theory and Techniques, 82:2216–2222, December 1982.

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19. O. Kiveks, J. Ollikainen, T. Lehtiniemi, and P. Vainikainen. Bandwidth, SAR, and efficiency of internal mobile phone antennas. IEEE Transactions on Electromagnetic Compatibility, 46(1):71– 86, February 2004. 20. P-L. Teng, S-T. Fang, and K-L. Wong. Pifa with a bent, meandered radiating arm for gsm/dcs dual-band operation. In Proc. Antennas and Propagation Society International Symposium, volume 3, pages 107–110, June 2003. 21. B.S. Yarman. Broadband Networks Wiley Encyclopedia of Electrical and Electronics Engineering, volume 2. Wiley, New York, 1999. 22. H.J. Carlin. A new approach to gain-bandwidth problems. IEEE Transactions on Circuits and Systems, CAS-24(4):170–175, April 1977.

Chapter 11

Initialization of Simplified Real Frequency Technique Perfect Match and Analytic methods Binboga Siddik Yarman

Initialization of Simplified Real Frequency Technique Nonlinear optimization of any objective function requires reasonable initial guess. In this regard, for the simplified real frequency technique, program SRFTWSOP may be initialized on the numerator polynomial h( p) = h n p n + h n−1 p n−1 + . . . + h 1 p + h 0 by various methods. The easiest way is to assume perfect match at the load-end. Another method is to roughly approximate the driving point impedance of the equalizer using some what analytic approach. Let us elaborate on these methods in the following sections.

Initialization of h( p) with Perfect Match Assumption For single matching problems, let SL ( jω) = S R L (ω) + j S X L (ω) be the specified data for the real (or unit) normalized load reflection coefficient. In terms of the actual load impedance Zˆ L ( jω), SL is given by SL =

Zˆ L − R0 Zˆ L + R0

(11.1)

or SL =

ZL − 1 ; ZL + 1

ZL =

Zˆ L R0

where Z L is the normalized load impedance with respect to normalization resistance R0 . For most of the communication systems, R0 is chosen as 50 ohms.

B.S. Yarman College of Engineering, Department of Electrical-Electronics Engineering, Istanbul University, 34320 Avcilar, Istanbul, Turkey e-mail: [email protected]

B.S. Yarman, Design of Ultra Wideband Antenna Matching Networks, C Springer Science+Business Media B.V. 2008

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At the back-end of the lossless equalizer perfect match condition requires that E 22 ( jω) = SL∗ = S R L (ω) − j S X L (ω)

(11.2)

where “∗ ” designates the complex conjugate of a complex number SL ( jω). Notice that the on the complex axis jω E 22 ( jω) may be given as E 22 ( jω) =

h R (ω) + j h X (ω) h( jω) = g( jω) g R (ω) + jg X (ω)

(11.3)

Equating Eqs. (11.2) and (11.3) we found that S R L (ω) =

h R g R + h x gx g 2R + gx2

−S X L (ω) =

h x g R − h R gx g 2R + gx2

(11.4)

or h R (ω) = [g R (ω)S R L (ω) + g X (ω)S X L (ω)] h X (ω) = − [g X (ω)S R L (ω) − g R (ω)S X L (ω)]

(11.5)

On the other hand, polynomial form of h( jω) reveals that h( jω) = h 0 − h 2 ω2 + h 4 ω4 − h 6 ω6 − h 8 ω8 + . . . + j h 1 ω − h 3 ω3 + h 5 ω5 − h 7 ω7 − h 9 ω9 + . . .

(11.6)

More explicitly, h X (ω) = (h 1 ω − h 3 ω3 + h 5 ω5 − h 7 ω7 − h 9 ω9 + . . .) 2 4 6 8 h R (ω) = h 0 − h 2 ω + h 4 ω − h 6 ω − h 8 ω + . . . or h X (ω) = ω(h 1 − h 3 ω2 + h 5 ω4 − h 7 ω6 + h 9 ω8 + . . .) (11.7)

Let A(ω2 ) = h R (ω)

(11.8)

and B(ω2 ) =

h X (ω) ω

then, h( jω) = h R (ω) + j h X (ω) = A(ω2 ) + jωB(ω2 )

(11.9)

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259

By setting x = ω2 we can write If n = even then, setting ma = n/2 A(x) = A1 x ma + A2 x ma−1 + A3 x ma−2 + . . . + Ama x + Ama+1 B(x) = B1 x ma−1 + B2 x ma−2 + B3 x ma−3 + . . . + Bma−1 x + Bma

(11.10)

with A j = (−1)(ma− j+1) h 2(ma− j) ; j = 1, 2, .., ma B j = (−1)(ma− j) h 2(ma− j)+1 ; j = 1, 2, . . . , ma

and If n = odd then, setting ma = (n − 1)/2 A(x) = A1 x ma−1 + A2 x ma−2 + A3 x ma−3 + . . . + Ama−1 x + Ama B(x) = B1 x ma + B2 x ma−1 + B3 x ma−2 + . . . + Bma x + Bma+1

(11.11)

with A j = (−1)(ma− j) h 2(ma− j) ; j = 1, 2, . . . , ma B j = (−1)(ma− j+1) h 2(ma− j)+1 ; j = 1, 2, . . . , ma Replacing x by ω2 and setting p = jω and − p 2 = ω2 , it is straight forward to express h( p) in terms of the polynomials A(− p 2 ) and B(− p 2 ) as h( p) = A(− p 2 ) + p B(− p 2 )

(11.12)

It is interesting to note that if g( p) or equivalently g( jω) = g R (ω) + jg X (ω) is known, one can readily generate the data for h R (ω) and h X (ω) by means of Eq. (11.4). Actually, this can be done since SL ( jω) is known and by perfect match assumption which is made at the beginning of the optimization, 1 − |E 22 |2 = 1 − |SL |2 yielding |E 21 ( jω)|2 = 1 − |SL ( jω)|2 = 1 − (S R2 + S X2 ). On the other hand, the designer is free to choose the equalizer structure by properly selecting the transmission zeros of E 21 ( p) = f ( p)/g( p). As indicated several times, for most practical situations, it is preferable to set f ( p) = p k ; 0 ≤ k ≤ n. In this case, one can express |E 21 ( jω)|2 as |E 21 |2 =

ω2k = 1 − |SL |2 G(ω2 )

(11.13)

where k is the total number of DC transmission zeros of the equalizer which dictated by the designer and

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G(ω2 ) = g( jω)g(− jω) > 0 is an even non-negative polynomial. Thus, data for G(ω2 ) = G n ω2n + G n−1 ωn−1 + . . . G 1 ω2 + G 0 =

ω2k > 0; ∀ω 1 − |SL |2

(11.14)

is generated and strictly Hurwitz polynomial g( p) shall be constructed numerically as detailed in the following section.

Initialization of the Denominator Polynomial g( p) One can employ several numerical techniques to generate strictly Hurwitz polynomial g( p). A straight forward method is to use linear regression to determine the real coefficients {G 0 , G 1 , . . . , G n }. In this case, MatLab’s function “polyfit” can be utilized. However, this method does not in general guarantee the positivenss of Eq. (11.11). Another method is to utilize an auxiliary even polynomial Pa (ω2 ) = a0 + a1 ω2 + . . . + a na ω2na =

ωk 1 − |SL |2

= 0; ∀ω

(11.15)

Then, we can easily determine the coefficients a j ; j = 0, 1, ..na by means of linear regression. In this case, we set 2 G(ω2 ) = Pa (ω2 ) > 0; ∀ω

(11.16)

with n = 2na Thus, positiveness of G(ω2 ) is guaranteed with the expense of degree doubling which yields n = 2na. In this method, coefficients {G 0 , G 1 , . . . , G n } can directly be computed using MatLab function “convl”. Once G(− p 2 ) = G(ω2 )|ω2 =− p2 is computed then, initial form of the strictly Hurwitz polynomial g( p) is determined which in turn yields g( jω) = g R (ω) + jg X (ω). Hence data for h R (ω) and h X (ω) is generated by means of Eq. (11.5). In this case, we can generated polynomials A(ω2 ) and B(ω2 ) as in Eq. (11.8) employing simple polynomial curve fitting algorithm. In MatLab, function polyfit immediately

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261

reveals the desired coefficients of the polynomials A(ω2 ) and B(ω2 ) as A = polyﬁt(W, h R , na) and B = polyﬁt(W, h X , nb) where W = ω1 ω2 . . . ωns−1 ωns are the sampled normalized angular frequencies over which the reflection coefficient SL ( jω) = S R + j S X is specified. Integers na and nb are determined from the given degree of n of h( p). Eventually, we can initialize the numerator polynomial h( p) as in Eq. (11.12) in terms of the real coefficients. Let us clarify the above derivations by means of examples. Example 1. Let the degree of the numerator polynomial h( p) be n = 5. What should be the degrees na and nb of the polynomials A(x) and B(x) respectively? Answer Since na is the degree of the even polynomial A(x) = h R (ω2 )|ω2 =x it will be doubled in h( p). Therefore, 2na must be less than or equal to n (i.e 2na ≤ n). Thus, na ≤ n/2 or it must be na = (n − 1)/2 = 4/2 = 2. Similarly, nb is the degree of the even polynomial B(x) = ω1 h X (ω) = B(ω2 )|ω2 =x satisfying the relation h( p) = A( p) + p B( p). Therefore, 2nb + 1 ≤ n. For the present case, it must be nb = (n − 1)/2 = 2. Here, it should be noted that if n = even then, na = n/2 and nb = (n − 2)/2; if n = odd then, na = nb = (n − 1)/2. Example 2. Let the degree n of the numerator polynomial h( p) be n = 5 with MatLab polynomials A(x) = 4x 2 + 2x + 1 and B(x) = 5x 2 + 3x + 1. Generate h( p) = A(− p 2 ) + p B( p 2 ) in full coefficient form by writing a MatLab function. Answer By inspection, the answer is straight forward: A(− p 2 ) = 4(−ω2 )2 + 2(−ω2 ) + 1 = 4ω4 − 2ω2 + 1. Similarly, B(− p 2 ) = −5ω4 + 3ω2 + 1. Then, h( p) = 5 p 5 + 4 p 4 − 3 p 3 − 2 p 2 + p + 1. In order to perform the operation described by h( p) = A(− p 2 ) + p B( p 2 ) in full coefficient form, we developed a simple MatLab function called h = ABtoh(A, B, n) which stands for the abbreviation “from polynomial A(x) and B(x) to polynomial h(p). So, the inputs of the MatLab function are the coefficients A = [A1 A2 . . . Ana Ana+1 ] and B = [B1 B2 . . . Bnb Bnb+1 ] of the polynomials A(x) = A1 x na + A2 x na−1 +. . .+ Ana x+ Ana+1 , B(x) = B1 x nb +B2 x nb−1 +. . .+Bnb x+Nnb+1 respectively and the degree n.

function h=ABtoh(n,A,B) % In this function it is assummed that h(p) = A(−p2)+pB(−p2) as in Chapter % 10. % Given n, A(−p2), B(−p2) such that h(p) = A(−p2)+pB(−p2) % Find the coefficients of h(p) as in: % h(p)=h1pˆn+h2pˆ(n−1)+h3pˆ(n−2)+...+hnp+h(n+1). % % Determine if n=even ma=n/2;

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m=fix(ma); delm=abs(ma−m); if delm==0; % n=even case % loop without sign consideration for i=1:m k=2*i−1; H(k)=A(i); H(k+1)=B(i); end H(n+1)=A(m+1); % for n=even case change the signs of A(w2) and B(w2)to obtain h(p)=A(−p2)+pB(−p2) sigma=−1; for i=1:m k=2*i−1; h(n−k+2)=−sigma*H(n−k+2); h(n−(k+1)+2)=−sigma*H(n−(k+1)+2); sigma=−sigma; end h(1)=−sigma*H(1); end % if delm > 0 % n=odd case m=fix((n−1)/2); % loop without sign consideration for i=1:m+1 k=2*i−1; H(k)=B(i); H(k+1)=A(i); end H(n+1)=A(m+1); % for n=odd case change the signs of A(w2) and B(w2)to obtain h(p)=A(−p2)+pB(−p2) sigma=−1; for i=1:m+1 k=2*i−1; h(n−k+2)=−sigma*H(n−k+2); h(n−(k+1)+2)=−sigma*H(n−(k+1)+2); sigma=−sigma; end h(1)=sigma*H(1); end % return

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Hence, on the Matlab command window, we can simply type: >> n = 5 >> A = [4 2 1] >> B = [5 3 1 1] Then, determine the coefficients of the MatLab polynomial h(p) as the array of h by typing: >> h = ABtoh(n,A,B) Thus the result is h= 5 4 −3 −21 1 which agrees with the solution of h( p) = 5 p 5 + 4 p 4 − 3 p 3 − 2 p 2 + p + 1. Example 3. Let the degree of the numerator polynomial be n = 4. Let the normalized angular frequencies be given as a MatLab array Wa: Wa = [0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000] Let the real part of h( jω) be given as a MatLab array hRa: hRa = [−0.4022 −0.3508 −0.3071 −0.3343 −0.5208 −0.9802 −1.8514 −3.2986 −5.5110 −8.7033]

Let the imaginary part of h( jω) be given as MatLab array hXa: hXa = [0.2456 0.4514 0.5776 0.5842 0.4316 0.0798 −0.5110 −1.3806 −2.5688 −4.1155]

Find the polynomials A(x), B(x) utilizing MatLab function C = polyfit(Xd, Yd, n) where C is a row vector which includes all the coefficients of polynomial y(x) = C1 x n + C2 x 2 + . . . + Cn x + Cn+1 . Xd and yd are the row vectors including the given data for x and y respectively. Answer Since n = 4 is an even integer, then na = n/2 = 2, and nb = (n−2)/2 = 1. First of all let us see the open forms for the polynomials A(x) and B(x): A(x) = A1 x 2 + A2 x + A3 and B(x) = B1 x + B2 . Data for polynomial A(x) is specified by Ad = hRa. On the other hand, data for polynomial B(x) must be generated as a row vector Bd = h X /ω. In this regards, Bd is found as: Bd = [2.4556 2.2565 1.9245 1.4597 0.8622 0.1318 − 0.7313 − 1.7273 − 2.8560 − 4.1175]

Thus, setting x = ω2 or on the MatLab command window setting Xd = W.∗ W , and Ad = hR, we generate the coefficients of A(x) and B(x) as >> A = polyfit(Xd,Ad,na) A = [−10.5211 2.2414 − 0.4236] >> B = polyfit(Xd,Bd,nb) B = [−6.6396 2.522] Then, the coefficients of h( p) is found by calling the function “ABtoh” >> n = 4 >> h = ABtoh(n, A, B).

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Hence, the result is h = [−10.5211 6.6396 − 2.2414 2.5220 − 0.4236] Example 4. Let the real part (Wa versus SRLa) and the imaginary part (Wa versus SXLa) of the load reflectance SL be specified as Wa=[0.2000 0.2790 0.3580 0.4370 0.5160 0.5950 0.6740 0.7530 0.8320 0.9110 0.9900] SRLa=[0.9351 0.7400 0.1619 0.3987 0.7322 0.8522 0.8998 0.918 0.9207 0.9119 0.8927] SXLa=[−0.3092 −0.5109 −0.3604 0.2688 0.2225 0.1040 0.0033 −0.0840 −0.1645 −0.2430 −0.3226]

Assuming a lowpass ladder structure for the equalizer, that is to say, selecting k = 0, generate the initial coefficients of the denominator polynomial g( p) of degree n = 4. Answer Data for G(ω2 ) = g( jω)g(− jω) is given by Eqs. (11.14–11.16). In this case, setting x = ω2 = − p 2 , g( p) is constructed by explicit factorization of G(− p 2 ). Let us now, reach to the solution step by step. Step 1: Generation of array Xd to construct auxiliary polynomial Pa (x). First of all we should define the array Xd for the regression of the auxiliary polynomial Pa (x) by setting x = ω2 . Hence we have, Xd = [0.0400 0.0778 0.1282 0.1910 0.2663 0.3540 0.4543 0.5670 0.6922 0.8299 0.9801]

Step 2: Generation of Data points Pd for Pa (x). From Eq. (11.15), data for Pd is obtanined as Pa (ω) = :

ωk

⇒ Pd = [. . . ..]. 1 − S R2 L + S R2 X

Thus, using the data provided for this example we have, Pd = [5.7690 2.2857 1.0886 1.1405 1.5536 1.9505 2.2921 2.5819 2.8248 3.0236 3.1794]

In the following steps, we shall set x = ω2 for regression purpose. Using a simple MatLab routine polynomial Pa is obtained:

Step 3: Polynomial fitting on the data given by Xd versus Pd for the selected degree nr = n/2 MatLab function polyfit generates the polynomial Pa (x). In this regard, the degree of the regression must be nr = n/2 so that the degree of g( p) results is n = 4 as required. Hence, we directly type the following commands on the MatLab command window: >>Pa = polyfit(Xd,Pd,2) results in the coefficients of Pa (x) = P1 x 2 + P2 x + P3 such that Pa = [7.1550 − 6.5538 3.3516]. Step 4: Compute G(x) = Pa2 (x) > 0 MatLab function G = conv(Pa, Pa) generates the strictly positive polynomial G(x) = Pa2 (x) > 0.

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Thus, we have G = [51.1936 − 93.7841 90.9129 − 43.9310 11.2331]. Step 5: Factorization of G(x) = Pa2 (x) > 0 2 As in Chapter 8, one can easily find the roots √ of G(x) = Pa (x) > 0 by MatLab function Xr = roots(G); then, setting pr = −X r and we can select the LHP roots to construct g( p) using MatLab function pr = sqrt(−Xr) which generates the roots in RHP. Then setting prm = −pr we can find the LHP roots. Thus we have, >>Xr=roots(G) Xr= 0.4584 + 0.5090i 0.4584 − 0.5090i 0.4575 + 0.5082i 0.4575 − 0.5082i >>pr=sqrt(−Xr) pr= 0.3366 − 0.7561i 0.3366 + 0.7561i 0.3364 − 0.7554i 0.3364 + 0.7554i >>prm = −pr prm= −0.3366 + 0.7561i −0.3366 − 0.7561i −0.3364 + 0.7554i −0.3364 − 0.7554i Step 6: Construction of the monic polynomial C(p) on the LHP roots of prm. Using MatLab function C = poly(prm) monic polynomial C( p) is constructed on the LHP roots of g( p). Hence, we have >>C=poly(prm) C= 1.0000 1.3459 1.8217 0.9212 0.4684 Finally, g( p) is obtained as g( p) =

√

|G(1)| C(x). Thus,

>> g=sqrt(abs(G(1)))*C g=[7.1550 9.6299 13.0342 6.5908 3.3516]

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This completes the generation of g( p). All the above steps are gathered in a MatLab function called “initial g0” as listed below: function g0=initial g0(n,k,Wa,SRLa,SXLa) %-----------------Inputs: % n=Degree of g(p). Note that n must be even % integer. % k=DC transmission zeros of the equalizer % Wa=[....] normalized angular frequency row vector % SRLa=[....] real part of the load reflectance as a % row vector % SXLa=[...] imaginary part of the load reflectance %------------------Output: % g0=[...] initila value of the denominator % polynomial g(p)=g1pˆng2pˆ9n-1)+...+gnp+g(n+1) N=length(Wa); for j=1:N W=Wa(j); p=(sqrt(−1)*W); % Step 1: Generate row vector Xd for linear regression of the Auxiliary polynomial. Xd(j)=W*W; %----------------------------------------------------% Step 2: Generate data for the auxiliary polinomial Pd SRL=SRLa(j); SXL=SXLa(j); % L21=sqrt(1−SRL*SRL-SXL*SXL); Pd(j)=(Wˆk)/L21; end %----------------------------------------------------%Step 3: % Generate Auxiliary polynomial Pa(x) where x=wˆ2 np=n/2 Pa=polyfit(Xd,Pd,np); %-----------------------------------------------------%Step 4: % Generate Even positive polynomial G(x) where x=−pˆ2 G=conv(Pa,Pa); % Find the roots of G(x) %------------------------------------------------------% Step 5:Find the roots of G(x) Xr=roots(G);

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%------------------------------------------------------% Step 6: % select LHP roots in p domain pr=sqrt(−Xr);prm=−pr; % Form normalized g(p) polynomial. C=poly(prm); % Consruct initial g polynomial. Cof=sqrt(G(1)); for i=1:n+1 g0(i)=Cof*C(i); end return

Example 5. Let the unit normalized load reflectance of an antenna be specified as; Wa = [0.3000 0.3300 0.3600 0.3900 0.4200 0.4500 0.4800 0.5100 0.5400 0.5700 0.6000] SRLa = [0.6233 0.3869 0.1504 0.1168 0.2850 0.4775 0.6208 0.7172 0.7818 0.8259 0.8567] SXLa = [−0.5541 −0.5401 −0.3407 −0.0096 0.2157 0.2838 0.2715 0.2316 0.1854 0.1399 0.0971]

Determine the initial coefficients for the numerator polynomial h( p). Answer For this example we developed a MatLab function to generate initial coefficients for the numerator polynomial h( p). In this function, first we generate the initial denominator polynomial g0 as described in Example 4. Then, Eq. (11.4) is programmed to generate data for h R (ω) and h X (ω). Thereafter, we generate the polynomials A(x) and B(x) as in Example 3 using the MatLab function [A, B] = polyfitAB(n, Xd, Ad, Bd). Finally, h0 is generated employing the MatLab function h0 = ABtoh(n, A, B) as in Example 2. Now, let us summarize the result step by step: Step 1: Generate row vector Xd by setting x = ω2 . Xd=[0.0900 0.1089 0.1296 0.1521 0.1764 0.2025 0.2304 0.2601 0.2916 0.3249 0.3600]

Step 2: Generate initial denominator polynomial g0 as in Example 4 and compute g0(jω) = gR + jgX g0 = [31.6902 25.7336 22.2788 6.9245 2.2946] Step 3: Employing Eq. (5), generate array vectors Ad and Bd. Ad = [−0.4256 − 0.6402 − 0.4493 − 0.0534 0.0310 − 0.2193 − 0.5880 − 0.9454 − 1.2364 − 1.4348 − 1.5230] Bd = [−3.8812 − 1.9948 − 0.4825 − 0.3426 − 1.0132 − 1.3967 − 1.2719 − 0.7818 − 0.0646 0.7939 1.7423]

Step 4: Determine polynomials A(x) and B(x) in similar manner to that of Example 3. A = [−33.9689 10.2984 − 1.0892] B = [13.1764 − 3.5771]

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Step 5: Finally, generate the coefficients of h polynomial in a similar manner to that of Example 2. Thus, h0 = [−33.9689 − 13.1764 − 10.2984 − 3.5771 − 1.0892]. All the above steps are gathered in the MatLab function called [Xd,Ad,Bd,A,B,g0,h0] = initial h0(n,k,Wa,SRLa,SXLa) which is listed as below. function [Xd,Ad,Bd,A,B,g0,h0]=initial h0(n,k,Wa,SRLa,SXLa) % This function generates the initial guess for the numerator polynomial % h(p). %--------------------Inputs: % Wa: Normalized angular frequency which includes % sample points for the load reflectance over the % pass band. % SRLa: Row vector. Real part of the load reflectance % SXLa: Row vector. Imaginary part of the load reflectance %---------------------Output: % h0=[...] row vector. Initial for the % polynomial h(p). N=length(Wa); g0=initial g0(n,k,Wa,SRLa,SXLa); for j=1:N W=Wa(j); p=(sqrt(−1)*W); % Step 1: Generate row vector Xd for linear regression of the Auxiliary polynomial. Xd(j)=W*W; %----------------------------------------------------% Step 2: Generate gR and gX from g0(p) SRL=SRLa(j); SXL=SXLa(j); gval=polyval(g0,p); gR=real(gval); gX=imag(gval); %-----------------------------------------------------% % Step 3: Generate array vectors Ad and Bd to generate polynomilas A(x) and B(x) % Note that A9x)=hR and B(x)=hX/w Ad(j)=gR*SRL+gX*SXL; Bd(j)=−(gX*SRL−gR*SXL)/W; end % % Step 4: Find the polynomials A(x) and B(x) by polynomial curve fitting: ng=length(g0); n=ng−1; [A,B]=polyfitAB(n,Xd,Ad,Bd);

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% % Step 5: Determine the initial coefficients h0=[...] h0=ABtoh(n,A,B); return Example 6. For a short monopole antenna the measured reflectance data is given over the frequency band of 30–60 MHz as Wa = [0.3000 0.3300 0.3600 0.3900 0.4200 0.4500 0.4800 0.5100 0.5400 0.5700 0.6000] SRLa = [0.6233 0.3869 0.1504 0.1168 0.2850 0.4775 0.6208 0.7172 0.7818 0.8259 0.8567] SXLa = [−0.5541 − 0.5401 − 0.3407 − 0.0096 0.2157 0.2838 0.2715 0.2316 0.1854 0.1399 0.0971]

Design a Lowpass ladder type equalizer with 4 elements over the normalized frequency band of ωlow = 0.3 to ωhigh = 0.6. Select the gain level T0 by trail and error to end-up with a reasonable level as high as possible with small fluctuations less than ΔT = ±0.05. Answer This is a single matching problem over an octave bandwidth (30–60 MHZ). Since we will deal with a low pass ladder structure the, we set k = 0. Design will be made with 4 elements. In this case, we set n = 4. For this problem, we wrote a simple optimization program which directly works on the measured data as specified in this example. Therefore, the optimization function of the program SRFTWSOP was modified accordingly. We omit details here. The list of the main program “srft ch11.m” and the optimization function “sopt11.m” is listed below. % PROGRAM FOR CHAPTER 11: Initialization for Example 6 clear %--------------------------------inputs:--------------------------------------n=4; k=0; ntr=1; T0=0.84 %----------Measured antenna data normalized with respect to f0=100MHz, and R0=50ohm. Wa=[0.3000 0.3300 0.3600 0.3900 0.4200 0.4500 0.4800 0.5100 0.5400 0.5700 0.6000] SRLa =[0.6233 0.3869 0.1504 0.1168 0.2850 0.4775 0.6208 0.7172 0.7818 0.8259 0.8567] SXLa =[−0.5541 −0.5401 −0.3407 −0.0096 0.2157 0.2838 0.2715 0.2316 0.1854 0.1399 0.0971] %---------------Generation of Initials---------------[Xd,Ad,Bd,A,B,g0,h0]=initial h0(n,k,Wa,SRLa,SXLa); %--------------------------------------------------------% na=n+1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Start Optimization using simplest version of the LSQNONLIN % % Type design prefernce: % ntr=1: design with transformer % ntr=0: design without transformer. % -------------------------------------------------------------%

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OPTIONS=OPTIMSET(‘MaxFunEvals’,20000,‘MaxIter’,50000); % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Design with transformer: if ntr==1, x0=h0; % Call optimization function lsqnonlin: x=lsqnonlin(‘sopt11’,x0,[],[],OPTIONS,T0,ntr,k,Wa,SRLa,SXLa); % h=x; end %-------------------------------------------------------------------------% Design without transformer: if ntr==0, for r=1:na−1 x0(r)=h0(r); end %Call optimization function lsqnonlin: x=lsqnonlin(‘sopt11’,x0,[],[],OPTIONS,T0,ntr,k,Wa,SRLa,SXLa); for r=1:na−1 h(r)=x(r); end h(na)=0; end % Generate strictly hurwits polynomial g(p) from optimized h(p): g=Hurwitzpoly g(h,k); % Compute the optimized transducer power gain nopt=length(Wa); for j=1:nopt w=Wa(j); SR=SRLa(j); SX=SXLa(j); L11=complex(SR,SX); %COMPUTATION OF GAIN g=Hurwitzpoly g(h,k); T=gain(w,h,g,k,L11); Ta(j)=T; end na=n+1 for i=1:na y(i)=g(na−i+1); x(i)=h(na−i+1); end % if k==0 % Syntisize the lossless equalizer from its back-end reflection % coefficient: [m CV]=synthesis(x,y); end plot(Wa,Ta) CV=general synthesis(h,g);

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function fun=sopt11(x,T0,ntr,k,Wa,SRLa,SXLa) % Inputs: % T0: Desired gain level % ntr: Control flag for equalizer design: % ntr=1; equlizer design with a transformer. % In this case k may be different than zero. % ntr=0; equlizer is constructed without transfermer. % In this case k must be zero (k=0) % and equlizer will be constructed with Low Pass LC elements % x=Row Vector includes the unknow coefficients of the polynomial h(p). % for ntr=0 and k=0, dimension of x must be n. % for ntr=1, dimension of x must be n+1. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Output: % fun: Objective function Vector subject to optimization. % For the design of Antenna Matching Networks fun=T−T0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nopt=length(Wa); nx=length(x); for j=1:nopt % Prepare to generate transducer power gain: % Call load function to compute load reflection coefficient. w=Wa(j); SR=SRLa(j); SX=SXLa(j); L11=complex(SR,SX); % Set h vector for the values of ntr: if ntr==1, h=x; end % if ntr==0, for r=1:nx h(r)=x(r); end na=nx+1; h(na)=0; end % % Generate g polynomial from h: g=Hurwitzpoly g(h,k); % Generate the Transducer power Gain T=gain(w,h,g,k,L11); % Generate objective function as

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fun(j)=T−T0; end return It was found that T0 = 0.84 results in a reasonable gain variation: Result of optimization is summarized as follows: As matLab command window outputs, initial coefficients for polynomials g0 and h0 are >> g0 g0 = 31.6978 25.7371 22.2820 6.9250 2.2948 >> h0 h0 = −33.9679 − 13.1796 − 10.2977 − 3.5776 − 1.0892 Optimized values for h and g are given as >> h h= −17.3671 8.6043 − 5.0936 2.7294 − 0.7398 >> g g= 17.3671 19.2051 13.5810 5.8052 1.2439 Resulting transducer power gain is listed as >> [Wa Ta ] ans = 0.3000 0.7632 0.3300 0.8517 0.3600 0.7868 0.3900 0.7811 0.4200 0.8215 0.4500 0.8531 0.4800 0.8391 0.5100 0.8046 0.5400 0.8044 0.5700 0.8601 0.6000 0.8206 The above list indicates that minimum of TPG is Tmin = 0.7632 at 30 MHz and its maximum Tmax = 0.8601 is reached at 57 MHz. Thus, the gain is T = T 0±ΔT = 0.811±0.0485. Plot of the transducer power gain is shown in Fig. 11.1. Eventually, input reflectance of the equalizer E = h/g is synthesized as shown in Fig. 11.2.

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Fig. 11.1 Plot of the transducer power gain for Example 6

Initialization on the Back-End Driving Point Impedance Z Q In Chapter 7, it is shown that realizable back-end driving point impedance of the equalizer is

Z Q = 2η L

RL RL − ZL − Z L = 2η DL η L − F22 η DL − S

(11.17)

where Z L ( p) = N L ( p)/D L ( p) = Ev(− p 2 ) + Od( p) = A L ( p)A L (− p)/D L ( p) D L (− p) + Od( p) is load impedance, η L ( p) = A L (− p)/A L ( p).D L ( p)/D L ( p) is called the “load all pass function” constructed on the finite transmission zeros of the load, specified by A L (− p) as well as the zeros of the denominator polynomial D L ( p), η DL ( p) = D L (− p)/D L ( p), F22 ( p) = η AL ( p).η DL ( p)Z Q ( p)− Z L ( p)/Z Q ( p) − Z Q (− p) = η AL ( p)S( p) is the real normalized realizable scattering parameter of cascade connection of the lossless equalizer [E] and the lossless load networks [L]. On the real frequency axis, it is straight forward to shown that Z Q ( jω) = R Q (ω) + j X Q (ω) with

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Fig. 11.2 Synthesis of the input reflectance of the Matching network design for a short monopole antenna over 30–60 MHz

R Q (ω) =

1 1 − |S( jω)|2 Z Q ( jω) + Z Q (− jω) = R L (ω) ≥ 0 ; ∀ω 2 |η DL ( jω) − S( jω)|2 (11.18)

and 1 2R L (ω) Z Q ( jω) − Z Q (− jω) = Im {η DL (− jω)S( jω)} − X L (ω) 2j |η DL − S|2 (11.19) Transducer power gain of the matched system is given by

X Q (ω) =

T = 1 − |F22 ( jω)|2 = 1 − |S( jω)|2

(11.20)

As you remember from Chapter 7, S( p) is the Youla’s complex normalized, regular back-end reflection coefficient and it is given by S( p) = η DL ( p)

Z Q ( p) − Z L (− p) Z Q ( p) + Z L ( p)

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Ideally, we wish to target a flat gain level T0 over the band of operation. Therefore, it may be appropriate to set |S( jω)|2 ≈ 1 − T0

(11.21)

which is purely real; or S ≈ μ 1 − T 0;

μ = ±1

In this case, Eqs. (11.15) and (11.16) becomes, R L (ω)T0 R Q (ω) ≈ ≥ 0; √ η DL ( jω) − μ 1 − T0 2

∀ωand

μ = ±1

(11.22)

and < ; 2R L (ω) X Q (ω) ≈ 2 Im η DL (− jω)μ 1 − T0 − X L (ω); μ = ±1 √ η DL − μ 1 − T0 (11.23) In all the above equations the sign “≈” means that equations are approximately equal. Obviously, for T0 = 1 Eqs. (11.19) and (11.20) yield R Q = R L and X Q = −X L . Once Z Q ( jω) is determined as set of data points then, it is modeled as a unit normalized reflection coefficient as described in the previous section. In this case, Eq. (11.3) is re-written as E 22 ( jω) =

h R (ω) + j h X (ω) ZQ − 1 h( jω) = SR Q + j SQ X = = ZQ + 1 g( jω) g R (ω) + jg X (ω)

(11.24)

and Eq. (11.4) becomes h R g R + h x gx g 2R + gx2 h x g R − h R gx S X Q (ω) = g 2R + gx2 S R Q (ω) =

(11.25)

Finally, Eq. (11.5) is revised as, h R (ω) = [g R (ω)S R Q (ω) − g X (ω)S X Q (ω)] h X (ω) = −[g X (ω)S R Q (ω) + g R (ω)S X Q (ω)]

(11.26)

It should be noted that for the present case, initial form of the denominator polynomial g0 ( p) is obtained as in Eqs. (11.14–11.16) where S L ∗ is replaced by E 22 ( jω) given by Eq. (11.24). That is,

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G(ω2 ) = G n ω2n + G n−1 ωn−1 + . . . G 1 ω2 + G 0 = Pa (ω2 ) = a0 + a1 ω2 + . . . + a na ω2na =

ω2k > 0; ∀ω 1 − |E 22 |2

ωk 1 − |E 22 |2

= 0; ∀ω

(11.27) (11.28)

and 2 G(ω2 ) = Pa (ω2 ) > 0; ∀ω

(11.29)

is modeled as a strictly Hiurwitz polynomial. Thus, by Eqs. (11.24–11.29), one can easily be programmed to generate initial polynomials g0 ( p) and h 0 ( p) as in MatLab program “srft-Ch11.m.” It is noted that the above initialization requires a load model and it may be trouble some to make it. However, load model can be generated using linear interpolation or scattering techniques as described by [1, 2, 3, 4]. At this point we should emphasize that Simplified Real Frequency Algorithms can always be initialized in ad-hoc manner such that h = [1 − 1 1 . . . 1..]. By trail and error one can come up with acceptable initials aas described in Example 7. Example 7. The purpose of this example is to demonstrate ad-hoc use of initialization for the the simplified real frequency technique. Here, we use the MatLab Main Program “srft-ch11-Example-7.m” to design a matching network for the short monopole antenna. Measured data for the antenna is given as in Example 6. That is, Frequency versus real and imaginary parts of the load reflectance are given as MatLab row vectors Wa, SRLa and SXLa respectively. Equalizer will include 4elelemnts in a low pass ladder configuration (i.e. n = 4, k = 0). We will try to hit flat gain level of T0 = 0.84. Design will be made with a transformer (i.e. ntr = 1). Program is initiated with ad-hoc coefficients h = [1 1 1 1 1]: Hence, inputs to the main program “srft-ch11-Example-7.m”are given as copied from the program: % Measured antenna data normalized with respect to f0=100 MHz and R0=50 Ohms: % Wa=[0.3000 0.3300 0.3600 0.3900 0.4200 0.4500 0.4800 0.5100 0.5400 0.5700 0.6000] SRLa=[0.6233 0.3869 0.1504 0.1168 0.2850 0.4775 0.6208 0.7172 0.7818 0.8259 0.8567] SXLa=[−0.5541 − 0.5401 − 0.3407 − 0.0096 0.2157 0.2838 0.2715 0.2316 0.1854 0.1399 0.0971] %-------------------------------------------------------------------------% n=4; % Total number of elements in the equalizer k=0; % Low pass Ladder Structure ntr=1; % Design with transformer T0=0.84; % Flat Gain Level h=[1 1 1 1 1] % Ad-hoc Initials:

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First run of the program results the following: h= 0.1613 0.7912 − 0.9635 1.7231 0.0254 g= 0.1613 1.2850 2.2137 2.7289 1.0003 Wa Ta ] ans = 0.3000 0.6110 0.3300 0.8111 0.3600 0.8355 0.3900 0.7884 0.4200 0.7512 0.4500 0.7457 0.4800 0.7717 0.5100 0.8197 0.5400 0.8639 0.5700 0.8563 0.6000 0.7551 Well gain is 0.611 at 30 MHz. It may be improved by changing the initials. In this case, we take new initials by alternating the signs of the first initials as h0 == [−1 1 − 1 1 1]. Thus we find, h= −16.9390 8.4109 − 4.9062 2.7006 − 0.7201 g= 16.9390 18.9441 13.4113 5.7689 1.2323 Gain = 0.3000 0.7596 0.3300 0.8508 0.3600 0.7882 0.3900 0.7814 0.4200 0.8197 0.4500 0.8510 0.4800 0.8393 0.5100 0.8069 0.5400 0.8061 0.5700 0.8581 0.6000 0.8216 Referring to Fig. 11.2, element values of the ladder is found as C1 = 1.2385, L2 = 3.4943, C3 = 4.7519, L4 = 0.8438, R = 0.2624 (or transformer ratio n 2t = 0.2624)

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Complete list of the main program is listed as below: % PROGRAM FOR CHAPTER 11: Ad-hoc Initialization for Example 7 clear %--------------INPUTS---------------------------% % Measured antenna data normalized with respect to f0=100 MHz and R0=50 Ohms: % Wa=[0.3000 0.3300 0.3600 0.3900 0.4200 0.4500 0.4800 0.5100 0.5400 0.5700 0.6000]; SRLa =[0.6233 0.3869 0.1504 0.1168 0.2850 0.4775 0.6208 0.7172 0.7818 0.8259 0.8567]; SXLa =[−0.5541 −0.5401 −0.3407 −0.0096 0.2157 0.2838 0.2715 0.2316 0.1854 0.1399 0.0971]; %-------------------------------------------------------------------------n=4; % Total number of elements in the equalizer k=0; % Low pass Ladder Structure ntr=1; % Design with transformer T0=0.84; % Flat Gain Level %h=[1 1 1 1 1] % First ad-hoc Initials: h0 =[−1 1 −1 1 1]; %Second initials % na=n+1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Start Optimization using simplest version of the LSQNONLIN % % Type design prefernce: % ntr=1: design with transformer % ntr=0: design without transformer. % -------------------------------------------------------------% %---------------Preparetion for the optimization ---OPTIONS=OPTIMSET(‘MaxFunEvals’,20000,‘MaxIter’,50000); %%%%%%%%%%%%%%%%% %Design with transformer: if ntr==1, x0=h0; % Call optimization function lsqnonlin: x=lsqnonlin(‘sopt11’,x0,[],[],OPTIONS,T0,ntr,k,Wa,SRLa,SXLa); % h=x; end %-------------------------------------------------------------------------% Design without transformer: if ntr==0, for r=1:na−1 x0(r)=h0(r); end %Call optimization function lsqnonlin: x=lsqnonlin(‘sopt10’,x0,[],[],OPTIONS,T0,ntr,k,Wa,SRLa,SXLa); for r=1:na−1 h(r)=x(r); end h(na)=0; end % Generate strictly hurwits polynomial g(p) from optimized h(p): g=Hurwitzpoly g(h,k);

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% Compute the optimized transducer power gain nopt=length(Wa); for j=1:nopt w=Wa(j); SR=SRLa(j); SX=SXLa(j); L11=complex(SR,SX); %COMPUTATION OF GAIN g=Hurwitzpoly g(h,k); T=gain(w,h,g,k,L11); Ta(j)=T; end na=n+1 for i=1:na y(i)=g(na-i+1); x(i)=h(na-i+1); end % if k==0 % Syntisize the lossless equalizer from its back-end reflection % coefficient: [m CV]=synthesis(x,y); end plot(Wa,Ta) CV=general synthesis(h,g); h, g, Gain=[Wa Ta ]

Referencess 1. A. Kılınc¸, Novel data modeling procedures: impedance and scattering approaches, Dissertation, ˙Istanbul, ˙Istanbul University, 1995. 2. B.S. Yarman, A. Kılınc¸, A. Aksen, “Immitance data modeling via linear interpolation techniques: a classical circuit theory approach,” Int J Circuit Theory Appl, vol. 32(6), pp. 537–563, 2004. 3. B.S. Yarman, A. Aksen, A. Kılınc¸, “An immitance based tool for modeling passive one-port ¨ vol. 55(6), pp. devices by means of Darlington equivalents,” Int J Electron Commun (AEU), 443–451, 2001. 4. B.S Yarman, M. Sengul, A. Kilinc, “Design of Practical Matching Networks with Lumped Elements Via Modeling”, IEEE Trans. CAS-I, vol. 54, No. 8, August 2007, pp. 1829–1837.

Chapter 12

Analysis and Optimization of Matching Networks-I Getting Started with ADS Metin Sengul

Introduction Once the ideal matching network design is completed using SRFT, the resulting network shall be manufactured either with discrete components or employing the well established IC techniques as dictated by the needs. In this regards, one may wish to utilize Microwave Monolithic Integrated Circuit (MMIC), MEMS or Silicon based VLSI technologies. Obviously, before we proceed with mass production, a physical layout of the circuit must be drawn in advance. Then, it should be properly analyzed to asses the final electrical performance of the matched system to be manufactured. A practical approach may be “the extraction of an equivalent circuit from the ideal matching network”. In this case, one should concern with possible losses of the ideal circuit elements, interconnections, coupling between the elements, and parasitic due discontinues of the physical layout. All these issues can be taken care of within a well organized computer Aided Analysis and Simulation tool which works on the selected equivalent circuits. Furthermore, the system performance of the matched system can be improved by re-optimization of the circuit elements. On the market place there are several S/W tools which perform the above mentioned operations. Among these tools, ADS of Agilent Technologies and Microwave Office of Advanced Wave Research (AWR) Inc are mostly used by microwave design engineers. Therefore, the next two chapters are allocated to introduce ADS and Microwave Office at the beginners level the way we use to analyze and re-optimize SRFT designs. ADS is a sophisticated circuit simulator [1]. It will take a significant amount of time to learn all its complex features. But simply, ADS has a similar functionality as other SPICE programs like PSPICE. There is a graphical user interfaceto draw the

M. Sengul Kadir Has University, Engineering Faculty Electronics Engineering Department, 34083, Cibali-Fatih, ˙Istanbul, Turkey e-mail: [email protected]

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circuit schematics. Like many commercial SPICE programs, ADS has significant number of predefined libraries. Since the focus of ADS is RF and microwave design, the majority of the components in the library are RF and microwave devices. However, there are a few low frequency FETs and BJTs. If the simulation of power electronic circuits is desired, a more appropriate package must be used. ADS can perform several different simulations (DC analysis, AC analysis, S-Parameter analysis, Transient analysis, etc.). Traditional SPICE simulators can be used for some of them. Since the focus of ADS is RF and microwave design, S-Parameter Analysis is the most used simulation type. S-Parameter Analysis: Basically, this is the microwave equivalent of AC analysis. This analysis is frequently used in the projects and microwave circuit design. When ADS starts up several windows appear. Only two windows are important: the quick start dialog box (Fig. 12.1) and the main ADS window (Fig. 12.2). The other windows are usually for passing information to the user. From the quick start window, a new project, an existing project or an Agilent supplied example file can be opened. The quick start menu is the quickest way to get started. The main window performs the same functionality but has significantly more features most of which are only useful for the advanced user. Circuit schematics (networks), and simulation results (data) are organised into a project. Each project is self contained piece of work. Physically a project is a

Fig. 12.1 The quick start dialog box

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Fig. 12.2 The main ADS window

directory which contains several other directories. Your schematics are stored in a directory called “Networks”. Each schematic is stored in a separate file with the extension “dsn”. The results from your simulations are stored in the directory data with the extension “ds”. The other directories and files store information required for ADS to correctly work and should not be deleted or edited. By default ADS adds “ PRJ” to the end of the project name.

Circuit Simulation In this part, we will take you through several stages of a simple S-parameter simulation of a 2nd order Butterworth filter [2]. 1. Open a project file by selecting “file > new project” which opens a dialog box to select the project name. The dialog box also allows you to select the default length unit. My personal choice is always a millimeter. In the Fig. 12.3, I have used the filename Butterworthfilter and so the project directory will be called Butterworthfilter PRJ. On pressing OK several other windows open. You get a empty unnamed schematic sheet and a schematic wizard. My advice is to ignore the wizard for the time being.

Fig. 12.3 Project name and unit window

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Fig. 12.4 Schematic window

2. On the left of the window (Fig. 12.4), there are a set of components. These components can be placed on the schematic by drag and drop. The actual type of components that are available can be changed by selecting the type in the drop down menu above. In the screen shot shown we have the “Lumped Components”. On the top of the schematic we have a variety of quick access buttons. If you move your mouse over the boxes, the action pops up in a window. 3. The first step is to insert inductors and capacitors of the filter network. We will use a component already in the library rather than setting up our own. Select inductor and capacitor from the “Lumped Components” section and place them on the schematic by drag and drop (Fig. 12.5). 4. Then by double clicking on any component, open “Component Parameter Box” and enter the component value (Fig. 12.6). 5. By using the “Insert Wire” and “Insert Ground” quick access buttons on the top of the schematic window, wire up the components and add “Ground” components (Fig. 12.7). 6. The next step is to tell the simulator what simulation we want to perform. This is achieved by placing on the schematic simulation components. If more than one component is placed on a schematic, all these simulations are performed. In this case, we want to do a simple S-parameter simulation. So we will place the S-parameter simulation component on the schematic by selecting the type in the drop down menu from “Lumped-Components” to “Simulation-S Param”, and place port terminations and S-Parameter simulation box by drag and drop, and then enter simulation frequencies in the simulation box (Fig. 12.8). 7. Our network is ready to simulate. Press “Simulate” quick access button on the top of the schematic window. A data window called Filter to be opened automatically (Fig. 12.9). The data window allows you to display the data in a

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Fig. 12.5 Components on schematic window

Fig. 12.6 Component parameter window

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Fig. 12.7 Connections and ground components

variety of ways including Smith Charts. The type of graph can be selected from the lefthand toolbar. These include polar, Smith charts, tables as well as standard XY plots. 8. In our example, we want only a simple XY plot which is top right. This can be placed on the data display window by drag and drop. This will automatically open up a dialog box allowing us to change the settings of the plot include the individual traces. To add a trace, we select the parameter and then click on add.

Fig. 12.8 S-Parameter simulation box and terminations

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Fig. 12.9 Graph window

Fig. 12.10 Plot trace dialog box

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Fig. 12.11 Data window with the selected simulation graph

ADS takes care of the X-axis automatically. In the dialog box, we have selected S(1,1) parameter in dB (Fig. 12.10). 9. On the data window, simulation result of dB (S(1,1)) can be seen (Fig. 12.11). On the same data window or on the same graph, another parameter can be selected, for instance the magnitude, real or imaginary part of S(1,2), S(2,2) or S(2,1). So we have completed the Chebyshev filter simulation example. In a similar manner, other simulations (DC, AC or transient analysis) can be realized easily. Now, let us analyze and optimize the matching network over the frequency band of 30 MHz–70 MHz designed in Chapter 10 for the short monopole antenna. We will use the antenna model given in Fig. 10.1 in Chapter 10 as seen in Fig. 12.12. In Fig. 12.13, antenna model and the designed matching network are given. Let us draw the transducer power gain curve of the matched system. Press “Simulate” button, a graph window will appear. Then select a rectangular plot, and write transducer power gain expression as “1-abs(S(1,1))∗ abs(S(1,1))”. In Fig. 12.14, transducer power gain of the matched system is given. Now, let us optimize the component values in the matching network to get better transducer power gain level. First by double clicking on the component, open component properties dialog box. Then select components value, and press “Tune/Opt/Stat/DOE Setup” button. Select “Optimization”, and enable optimization status (Fig. 12.15).

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Fig. 12.12 Monopole antenna model

Change “Lumped Components” selection to “Optim/Stat/Yield/DOE”. then drag and drop “Optim” and “Goal” components. In the “Goal” box, write optimization expression as “1-abs(S(1,1))∗ abs(S(1,1))”, minimum level as 0.75, and maximum level as 0.85, as seen in Fig. 12.16. Then press “Simulate” button. After optimization, final transducer power gain curve and the one obtained from SRFT are drawn on the same grapth to be able to compare. As seen from Fig. 12.17, there is no big difference, so as a result we can conclude that the designed network via SRFT has extremelly good element values.

Fig. 12.13 Matching network and monopole antenna model

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Fig. 12.14 Transducer power gain of the matched system

Fig. 12.15 Optimization status

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Fig. 12.16 Optimization parameter box

Fig. 12.17 Initial and optimized gain curves

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Referencess 1. ADS of Agilent Techologies, www.home.agilent.com 2. Pozar, D.M. Microwave Engineering, 3rd Ed., John Wiley & Sons, Inc. (2005)

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Chapter 13

Analysis and Optimization of Matching Networks-II Getting Started with Microwave Office Metin Sengul

Introduction We oftenly use Microwave Office, Circuit Simulation and Optimization tool to check the performance of the Matching Networks designed emplying SRFT. Therefore, in this chapter breifly introduce Microwave Office S/W tool for the beginers. The following instructions allow the user to use some of the facilities of microwave office [1]. As an example, 2nd order Butterworth lowpass filter is simulated [2].

Circuit Simulation The first part is to simulate the frequency response of this filter. The initial screen shows an explorer bar to the left and a blank window on the right. There is an array of buttons and a menu at the top. The explorer window has four tabs at the bottom labelled project, elements, variables and layout. Note there is a good help section by using the menu at the top (Fig. 13.1). 1. From the top pull down menu, select “File-Save project as”, and give the name as “Butterworthfilter”. Microwave Office (MO) will add the extensin “emp”. Then select “Project-Add schematic-New schematic”, a small dialog box will appear. Write the name of the filter network as “Filter”. A schematic window now opens in the space to the right. In this window, we now enter the circuit components. (Fig. 13.2). 2. Select “Elements” on the tabs at the bottom of the explorer bar. Select “Lumped elements” from the list of elements at the top of the window on the left,and

M. Sengul Kadir Has University, Engineering Faculty Electronics Engineering Department, 34083, Cibali-Fatih, ˙Istanbul, Turkey e-mail: [email protected]

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Fig. 13.1 Initial screen

select inductor and capacitor, then place them on the window by drag and drop (Fig. 13.3). 3. Select “Elements” on the tabs at the bottom of the explorer bar, then select “Project options” to change simulation frequencies and units. Select “Global units” to change the unit as seen in Fig. 13.4.

Fig. 13.2 Schematic window

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Fig. 13.3 Components on the schematic window

Fig. 13.4 Global units

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Fig. 13.5 Simulation frequencies dialog box

4. 5.

6. 7.

8.

9. 10.

Then, select “Frequency values” to enter simulation frequencies and step size (Fig. 13.5). To change the element value, double click on the component, a dialog box opens. Enter the component values as seen in Fig. 13.6. To wire up the components, move the cursor to the end of a components and click, then select the end point end clik again. Make all the connections in the network in the same manner, and then add ground elements from the quick start buttons (Fig. 13.7). By using quick start buttons, connect the input and output ports of the filter. Now we have completed the schematic of the filter network (Fig. 13.8). Now set up an output graph. Right click on on the “Graphs” and click “Add graph”. Name the graph as “Lowpass” (Fig. 13.9). After clicking OK, a graph should now have appeared (Fig. 13.10). We now need to define what is on it. On the left, right click on “Lowpass” under “Graphs”, and select “Add measurements”, then a dialog box will appear. Select the desired simulation type, ports, network (Fig. 13.11). We are now ready to simulate the circuit. By using the “Analyze” quick start button, run the simulation of the filter (Fig. 13.12). Another parameters of the filter can be measured on the same graph or on another graph in the same manner.

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Fig. 13.6 Element options dialog box

We have now successfully simulated the low pass filter circuit. Now, let us analyze and optimize the matching network over the frequency band of 30 MHz–70 MHz designed in Chapter 10 for the short monopole antenna. We will use the antenna model given in Fig. 10.1 in Chapter 10 as seen in Fig. 13.13. In Fig. 13.14, antenna model and the designed matching network are given.

Fig. 13.7 Wired up and grounded components

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Fig. 13.8 Completed filter network

Let us draw the transducer power gain curve of the matched system. First of all, from “Graphs”, open a new graps window and call it as “Transducer Power Gain”, then from “Add Measurement”, select “Linear Gain” and “GT” options. After setting graph window, select frequency band as 20 MHz–100 MHz, and press

Fig. 13.9 Graph name dialog box

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Fig. 13.10 Graph window

“Analyze” button. In Fig. 13.15, transducer power gain of the matched system is given. Now, let us optimize the component values in the matching network to get better transducer power gain level. First select the components to optimize by checking the box in the “Element Options” window as seen in Fig. 13.16.

Fig. 13.11 Simulation parameters

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Fig. 13.12 Simulation graph of the filter

Then define an optimization goal: From “Optimizer Goals”, select “Add Opt Goal”, then define frequency band, desired gain level as seen in Fig. 13.17. We have selected frequency band as 30 MHz–100 MHz, and desired gain level as 0.8.

Fig. 13.13 Monopole antenna model

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Fig. 13.14 Matching network and monopole antenna model

Then from “Simulate” button, select “Optimize”, and an optimize window will appear (Fig. 13.18). Select optimization method and press “Start” button. After optimization, final transducer power gain curve and the one obtained from SRFT are drawn on the same grapth to be able to compare. As seen from Fig. 13.19, the designed network via SRFT has extremelly good element values.

Fig. 13.15 Transducer power gain of the matched system

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Fig. 13.16 Check box for optimization

Fig. 13.17 Optimization parameter box

Fig. 13.18 Optimize box

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Fig. 13.19 Initial and optimized gain curves

Referencess 1. Microwave Office of Applied Wave Research Inc. (AWR), www.appwave.com 2. Pozar, D.M. Microwave Engineering, 3rd Ed., John Wiley & Sons, Inc. (2005)

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Index

A The actual element values, 151 Actual power, 107, 110 Adaptive Impedance, 58 Adaptive and Smart Antennas, 35 ADS, 281–283, 288 All Pass, 123 Ampere’s Law, 10–12 Analytic Approaches to Antenna Matching Problems, 139 Analytic Theory of Broadband Matching, 5, 144, 174 Antenna Bandwidth, 32 Antenna fundamentals, 9 Antennas on Chip, 33 Antennas on Packages, 33 Aperture antennas, 16 Applied Wave Research Inc, 303 Available power, 141 AWR, 281

Challenges in mobile phone antenna development, 45 Chassis, 41 Chassis Influence on Impedance Bandwidth, 55 Chebyshev, 167 Chebyshev function, 227, 229 Circuit Simulation, 283, 293 Classical Filter Problem, 143 Complex Normalized-Regulized Reflection Coefficient, 168 Computer aided design (CAD) solutions, 144 Crosstalk, 48 Current consumption, 42 Current Standing Wave Pattern, 16

B Balun, 93, 96 Bandwidth, 46, 51, 54, 57 Beam Area, 28 Beam Efficiency, 28 Beam Solid Angle, 28 Beam Width and Side Lobe Level, 28 Belevitch, 121, 132, 135 Blashke Products, 123 Bode, 145, 172 Bounded-Real (BR) function, 110 Bounded-Real Rational functions, 121 Broadband antenna Design Techniques, 83 Butterworth, 148, 149, 167, 178

D Darlington Representation, 145 Darlington Synthesis, 113 Darlington Theorem, 112, 145 Degeneracy, 173, 178, 179 Degree of nonlinearity, 185 Denominator polynomial, 190, 218 Design Examples via SRFT, 225 Design of Ultra Wideband Antenna Matching Networks, 183 Design of Wideband Antenna Equalizers Employing SRFT, 225 Dielectric antennas, 18 Digital Phase Shifters for Antenna Arrays, 8 Direct Computational Technique (DCT), 3–4 Directive Gain, 26 Directivity, 26 Double Matching, 142, 145 DVB-H, 49

C Carlin, 144, 145, 161, 168, 242 Cassegrain antenna, 21

E Effective Aperture, 30 Effective Area, 30

305

306 Effective dielectric constant, 69 Effective Height, 31 Effective Isotropic Radiated Power (EIRP), 28 Efficiency, 60 Electric and Magnetic Field, 9 Electromagnetic Spectrum, 10 Electromagnetic Waves, 9, 12, 15, 16 Equalizer Design with Transformer, 212 Even polynomial, 185, 191, 193, 194, 196, 197, 202, 204 Extreme High Frequency (EHF), 20 Extreme Low Frequency (ELF), 20 F Factorization, 119, 121, 122, 136 Fano, 145, 172 Faraday, 12 FICA, 71 Fictitious function, 162 Filter Design, 143, 148, 153 Filter or insertion loss problem, 142 Finite Difference Time Domain (FDTD), 34 Finite Elements Method (FEM), 34 Folded Inverted Conformal Antenna (FICA), 71 Form factors, 61 Fractal Antennas, 35, 36 Frequency Tuning, 57 Function g=Hurwitzpoly g(h, k), 203, 236 Function lsqnonlin, 213, 215 Fundamental equation set, 196

Index Initialization of Simplified Real Frequency Technique, 257–279 Input impedance, 69, 73 Input Multiple Output (MIMO), 36 Input reflectance, 105, 108, 109 Is the input impedance, 106 L LAN, 40 Laplace Transformation, 102 LHP roots, 150, 155, 191 Linear antennas, 16 Line Segment Technique (LST), 3, 54 Log periodic dipole, 20 Lossless Ladders, 130 Lossless two ports, 101 Lossless Two-ports, 116 Low Frequency (LF), 20 Low pass LC ladder, 184

H HAC, 46 Half power bandwidth, 226, 236, 239 High Frequency (HF), 20 Historical Review, 1 Hurwitz polynomial, 109, 176

M Magnetic Field Patterns, 14 Main program, 215 Main Theorem, 191 Major constraints of broadband matching, 167 MatLab, 194–203, 205, 206, 210, 212–214, 221–223 Maximum attainable average, 226 Maximum power transfer, 139, 140 Maxwell, 9, 11, 13 Maxwell-Heavyside equations, 13 Measure of nonlinearity, 185 Medium Frequency (MF), 20 Method of Moments (MOM), 34 Microstrip Antenna, 33, 34 Microwave Office, 293 Mirror sysmmetric roots, 199 Mobile wireless communication, 39 Modeling via Real Frequency Techniques, 7 Monic polynomial, 191, 193, 194, 198, 199, 201, 202, 204 Monolithic Microwave Integrated Circuits (MMICs), 18 Monopole/IFA antenna, 70 Multi-band/Broadband Antenna, 83

I Immitance function, 112 Impedance Bandwidth, 32 Impedance Matching, 54 Impedance measurement, 96 Incident power, 106, 110, 111 Incident and reflected waves, 107

N Nanotube Antennas, 35, 37 Nonlinear, 184, 212, 213, 215 Nonlinearities, 185 Nonlinearity, 185–187, 212 Norton’s Equivalent Circuit, 23–24 Notch antenna, 73

G Gain, 60 Gain Bandwidth limits, 229 Gain measurement, 98 Graph window, 287 GSM, 39

Index Numerator polynomial, 187, 188, 190, 191, 193, 196, 213, 221, 222, 223 Numerical Construction of Strictly Hurwitz Polynomial, 193 O Optimization goal, 300 Optimization parameter box, 291, 302 Optimization status, 288, 290 Optimization of the Transducer Power Gain, 212 P Para-conjugate, 147, 164 Parametric Approach, 2, 3, 4 Parasitic resonators, 87 Pattern Bandwidth, 32 Permeability, 14 Permittivity, 14 Peter Lindberg, 45 Photonic Bandgap Structures, 35 PIFA, 21 PIFA Antenna, 21 Planar Inverted-F Antenna (PIFA), 67 Power Gain, 27 Power Transfer Ratio, 106, 111, 112 Practical limitations, 40 Price, 42 Propagation of Electromagnetic Waves, 12 Proper factorization, 148, 180 Proper Polynomial, 123

307 Scattering parameters, 101, 102, 114, 116, 117, 123, 131, 132 Short Monopole Antenna, 225, 231, 236, 239 Simple lumped element building blocks, 117 Simplified Real Frequency Technique(SRFT), 2, 3, 4, 6, 183, 187, 191, 211–215, 221 Single Matching, 142, 153, 158 Skin depth, 93 Slot antenna, 79 Small Antennas, 54 Smart Antennas, 35 S-Parameter Analysis, 282 Specific Absorption Rate (SAR), 59 SPICE, 281, 282 SRFT, 187–189, 221, 225 SRFT with mixed lumped and distributed elements, 6 SRFTWSOP, 214, 215, 218, 220, 222, 223, 229, 231–236, 269 Standard dipole, 15 Strictly Hurwitz, 109, 119, 122, 123, 190–193, 195, 196, 198, 200, 202, 223 Strictly Hurwitz Polynomial, 202 Superconducting Antennas, 35 Super High Frequency (SHF), 20 Synthesis, 150, 151

R Radiation Intensity, 25 Radiation Pattern, 24 Radiation and Propagation, 9 Reconfigurable Frequency Tuning, 57 Reflectance Definition, 105 Reflected power, 106, 107, 110, 111, 116 Reflected waves, 102–104, 107, 114 Reflection Coefficient, 31, 97 Resonant frequency, 74 Return Loss, 31 RF Chokes, 88 RF-ID, 40

T Talk Position, 58 Tandem Connection, 136, 145 Tera hertz and optical Frequencies, 20 Terminal antenna measurements, 93 Thevenin equivalent, 139, 140 Thevenin’s Equivalent Circuit, 23 T IS, 46–48 TPG, 148–150, 152, 153, 157, 158, 165, 167, 175 Transducer power gain, 116, 121, 183, 184, 186, 188–190, 202, 203, 205, 210, 212–216, 219, 222, 223 Transducer Power Gain of the Matched monopole antenna, 232 Transfer Function, 149 Transmission zeros, 126, 127, 148, 153, 162, 167, 168, 172, 173, 180 Triple band 900/1800/1900MHz PIFA, 87 T RP, 46

S SAR, 43, 46, 59 Scattering Approach to Matching problems, 187 Scattering Matrix, 104, 120, 132, 136

U Ultra High Frequency (UHF), 20 UMTS, 39 Unction lsqnonlin, 213 UWB, 40

Q QWERTY, 62

308 V Very High Frequency (VHF), 20 Very Low Frequency (VLF), 20 Voltage controlled oscillator (VCO), 94 W Wave Quantities, 102, 105 WCDMA, 39, 40 Weight, 42

Index WiMAX, 40 WLAN, 40 Y Yagi-Uda, 20 Yarman, 101, 125, 139, 145, 161, 166, 168, 196, 225, 257 Youla, 145, 168, 169, 172–174