MICROWAVE FILTERS, IMPEDANCE-MATCHING NETWORKS, AND COUPLING STRUCTURES
GEORGE L. MATTHAEI Department of Electrical & Computer Engineering University of California Santa Barbara, CA
LEO YOUNG Naval Research Laboratory Washington, DC
E. M. T. JONES Technology for Communications International Mountain View, CA
For a complete listing of the Artech House Microwave Lihaty, turn to the back of this book.
ARTECH HOUSE
PREFACE
The o r g a n i z a t i o n o f t h i s book h a s t h r e e g e n e r a l o b j e c t i v e s .
The
f i r s t o b j e c t i v e i s t o p r e s e n t f u n d a m e n t a l c o n c e p t s , t e c h n i q u e s , and d a t a t h a t a r e o f g e n e r a l u s e i n t h e d e s i g n o f t h e wide r a n g e o f microwave s t r u c t u r e s d i s c u s s e d i n t h i s book.
The s e c o n d o b j e c t i v e i s t o p r e s e n t
s p e c i a l i z e d d a t a i n more o r l e s s handbook form s o t h a t a d e s i g n e r c a n work o u t p r a c t i c a l d e s i g n s f o r s t r u c t u r e s h a v i n g c e r t a i n s p e c i f i c c o n f i g u r a t i o n s , without having t o r e c r e a t e t h e design theory o r t h e d e r i v a t i o n of t h e equations.
(However, t h e o p e r a t i o n o f most of t h e d e v i c e s
d i s c u s s e d h e r e i n i s s u f f i c i e n t l y c o m p l e x t h a t k n o w l e d g e o f some o f t h e b a s i c c o n c e p t s a n d t e c h n i q u e s i s u s u a l l y i m p o r t a n t . ) The t h i r d o b j e c t i v e i s t o p r e s e n t t h e t h e o r y upon w h i c h t h e v a r i o u s d e s i g n p r o c e d u r e s a r e based, so t h a t t h e designer can adapt t h e various design techniques t o new a n d u n u s u a l s i t u a t i o n s , a n d s o t h a t r e s e a r c h e r s i n t h e f i e l d o f m i c r o w a v e d e v i c e s may u s e some o f t h i s i n f o r m a t i o n a s a b a s i s f o r d e r i v ing additional techniques.
The p r e s e n t a t i o n o f t h e m a t e r i a l s o t h a t i t
c a n b e a d a p t e d t o new a n d u n u s u a l s i t u a t i o n s i s i m p o r t a n t b e c a u s e many
Copyright O 1980 ARTECH HOUSE, INC. 685 Canton Street Norwood, MA 02062
Reprint of the edition publiied by McGraw-Hill Book Company, Inc., 1964 Printed and bound in the United States of America All rights reserved. No part of this book may be reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording, or by any information rtorage,and retrieval system, without permission in writing from the publieher. Library of Congress Catalog Card Number: 80.68976 Standard Book Number 049006-099-1 Printed and bound by BookMart Press, North Bergen, NJ, USA, November, 1985.
o f t h e m i c r o w a v e f i l t e r t e c h n i q u e s d e s c r i b e d i n t h i s book a r e p o t e n t i a l l y u s e f u l f o r t h e d e s i g n of microwave d e v i c e s n o t o r d i n a r i l y t h o u g h t o f a s h a v i n g a n y t h i n g t o do w i t h f i l t e r s .
Some e x a m p l e s a r e t u b e s , p a r a m e t r i c
d e v i c e s , and a n t e n n a s , w h e r e f i l t e r s t r u c t u r e s c a n s e r v e a s e f f i c i e n t impedance-matching netwbrks f o r a c h i e v i n g broad-band o p e r a t i o n . F i l t e r s t r u c t u r e s a r e a l s o u s e f u l a s slow-wave s t r u c t u r e s o r time- d e l a y s t r u c tures.
I n a d d i t i o n , microwave f i l t e r t e c h n i q u e s can be appl i e d t o o t h e r
d e v i c e s n o t o p e r a t i n g i n t h e microwave band of f r e q u e n c i e s , a s f o r i n s t a n c e t o i n f r a r e d and o p t i c a l f i l t e r s . The t h r e e o b j e c t i v e s a b o v e a r e n o t l i s t e d i n any o r d e r o f
mportance, nor
i s t h i s book e n t i r e l y s e p a r a t e d i n t o p a r t s a c c o r d i n g t o t h e s e o b j e c t i v e s .
However, i n c e r t a i n c h a p t e r s w h e r e t h e m a t e r i a l l e n d s i t s e l f t o s u c h o r g a n i z a t i o n , t h e f i r s t s e c t i o n o r t h e f i r s t few s e c t i o n s d i s c u s s g e n e r a l p r i n c i p l e s w h i c h a d e s i g n e r s h o u l d u n d e r s t a n d i n o r d e r t o make b e s t u s e o f t h e d e s i g n d a t a i n t h e c h a p t e r , t h e n come s e c t i o n s g i v i n g d e s i g n d a t a
iii
f o r s p e c i f i c t y p e s o f s t r u c t u r e s , and t h e e n d o f t h e c h a p t e r d i s c u s s e s
a p p l i c a t i o n s , b u t a l s o f o r u s e a s p r o t o t y p e s f o r c e r t a i n t y p e s of band-
t h e d e r i v a t i o n s of t h e v a r i o u s d e s i g n equations.
p a s s o r pseudo h i g h - p a s s f i l t e r s d i s c u s s e d i n Chapter 9.
A l s o , a t numerous p l a c e s
c r o s s r e f e r e n c e s a r e made t o o t h e r p o r t i o n s o f t h e book w h e r e i n f o r m a t i o n u s e f u l f o r t h e design of t h e p a r t i c u l a r s t r u c t u r e under c o n s i d e r a t i o n can be found.
F o r e x a m p l e , C h a p t e r 11 d e s c r i b e s p r o c e d u r e s f o r m e a s u r i n g t h e
u n l o a d e d Q a n d e x t e r n a l Q o f r e s o n a t o r s , and f o r m e a s u r i n g t h e c o u p l i n g c o e f f i c i e n t s between r e s o n a t o r s .
Such p r o c e d u r e s have wide a p p l i c a t i o n
Design o f l o w - p a s s f i l t e r s and h i g h - p a s s f i l t e r s from t h e semilumped-element p o i n t o f view a r e t r e a t e d i n C h a p t e r 7 . C h a p t e r s 8 , 9 , a n d 10 d i s c u s s b a n d - p a s s o r p s e u d o - h i g h - p a s s f i l t e r d e s i g n u s i n g t h r e e Which a p p r o a c h i s b e s t d e p e n d s on t h e t y p e
d i f f e r e n t design approaches.
i n t h e p r a c t i c a l d e v e l o p m e n t o f many t y p e s o f b a n d - p a s s f i l t e r s a n d
o f f i l t e r s t r u c t u r e t o be u s e d a n d t h e b a n d w i d t h r e q u i r e d .
impedance-matching networks.
of t h e v a r i o u s f i l t e r s t r u c t u r e s d i s c u s s e d i n a l l t h r e e c h a p t e r s , a summary o f t h e p r o p e r t i e s o f t h e v a r i o u s f i l t e r s t r u c t u r e s , and t h e
C h a p t e r 1 o f t h i s book d e s c r i b e s t h e b r o a d r a n g e o f a p p l i c a t i o n s f o r which microwave f i l t e r s t r u c t u r e s a r e p o t e n t i a l l y u s e f u l .
Chapters 2
A tabulation
s e c t i o n number w h e r e d e s i g n d a t a f o r t h e v a r i o u s s t r u c t u r e s c a n b e f o u n d , a r e p r e s e n t e d a t t h e b e g i n n i n g o f C h a p t e r 8.
through 6 c o n t a i n r e f e r e n c e d a t a and background i n f o r m a t i o n f o r t h e r e s t o f t h e book.
C h a p t e r 2 summarizes v a r i o u s c o n c e p t s and e q u a t i o n s t h a t
a r e p a r t i c u l a r l y u s e f u l i n t h e a n a l y s i s of f i l t e r s t r u c t u r e s .
Although
C h a p t e r 11 d e s c r i b e s v a r i o u s a d d i t i o n a l t e c h n i q u e s which a r e u s e f u l t o t h e p r a c t i c a l development o f microwave b a n d - p a s s f i l t e r s , impedance-
t h e image p o i n t o f v i e w f o r f i l t e r d e s i g n i s made u s e o f o n l y a t c e r t a i n
matching networks, and t i m e - d e l a y networks.
p o i n t s i n t h i s b o o k , some k n o w l e d g e o f image d e s i g n m e t h o d s i s d e s i r a b l e .
g e n e r a l i n t h e i r a p p l i c a t i o n and can be used i n c o n j u n c t i o n w i t h t h e
C h a p t e r 3 g i v e s a b r i e f summary o f t h e i m a g e d e s i g n c o n c e p t s w h i c h a r e
f i l t e r s t r u c t u r e s and t e c h n i q u e s d i s c u s s e d i n C h a p t e r s 8 , 9 , a n d 1 0 , and e l s e w h e r e i n t h e book.
p a r t i c u l a r l y u s e f u l f o r t h e purposes of t h i s book.
Chapters 1 t o 3 should
b e e s p e c i a l l y h e l p f u l t o r e a d e r s whose b a c k g r o u n d f o r t h e m a t e r i a l o f t h i s book may h a v e some g a p s . Most o f t h e f i l t e r and i m p e d a n c e - m a t c h i n g n e t w o r k d e s i g n t e c h n i q u e s d e s c r i b e d l a t e r i n t h e book make u s e o f a l o w - p a s s p r o t o t y p e f i l t e r a s a basis f o r design.
Chapter 4 d i s c u s s e s v a r i o u s t y p e s o f lumped-element,
These techniques a r e q u i t e
C h a p t e r 12 d i s c u s s e s b a n d - s t o p f i l t e r s , w h i l e C h a p t e r 13 t r e a t s c e r t a i n t y p e s o f d i r e c t i o n a l c o u p l e r s . The TEM-mode, c o u p l e d - t r a n s m i s s i o n l i n e , d i r e c t i o n a l c o u p l e r s d i s c u s s e d i n C h a p t e r 13 a r e r e l a t e d t o c e r t a i n t y p e s of d i r e c t i o n a l f i l t e r s discussed i n Chapter 14, while t h e branchg u i d e d i r e c t i o n a l c o u p l e r s can be designed u s i n g t h e s t e p - t r a n s f o r m e r
low-pass, p r o t o t y p e f i l t e r s , p r e s e n t s t a b l e s o f element values f o r such
p r o t o t y p e s i n Chapter 6.
f i l t e r s , d i s c u s s e s t h e i r time-delay p r o p e r t i e s , t h e i r impedance-matching
a r e d i s c u s s e d i n C h a p t e r 1 4 , w h i l e h i g h - p o w e r f i l t e r s a r e t r e a t e d i n C h a p t e r 15.
p r o p e r t i e s , and t h e e f f e c t s o f d i s s i p a t i o n l o s s upon t h e i r r e s p o n s e s .
C h a p t e r 16 t r e a t s m u l t i p l e x e r s and d i p l e x e r s , and C h a p t e r 17 d e a l s w i t h
t i e s a r e e m p l o y e d i n t h e d e s i g n o f l ~ w - ~ a s sh ,i g h - p a s s , b a n d - p a s s , a n d
f i l t e r s t h a t c a n b e t u n e d e i t h e r m e c h a n i c a l l y o r by v a r y i n g a b i a s i n g magnetic f i e l d .
In l a t e r c h a p t e r s t h e s e low-pass p r o t o t y p e f i l t e r s and t h e i r v a r i o u s proper-
band-stop microwave f i l t e r s , and a l s o i n t h e d e s i g n of microwave impedancematching n e t w o r k s , and t i m e - d e l a y networks. Various e q u a t i o n s , g r a p h s , and t a b l e s r e l e v a n t t o t h e d e s i g n o f c o a x i a l l i n e , s t r i p - l i n e , waveguide, and a v a r i e t y o f r e s o n a t o r s , c o u p l i n g s t r u c t u r e s , and d i s c o n t i n u i t i e s , a r e summarized f o r e a s y r e f e r e n c e i n C h a p t e r 5.
C h a p t e r 6 d i s c u s s e s t h e d e s i g n o f s t e p t r a n s f o r m e r s and p r e s e n t s t a b l e s o f d e s i g n s f o r c e r t a i n c a s e s . The s t e p t r a n s f o r m e r s i n Chapter 6 a r e n o t only f o r use i n conventional im~edance-transformer
Both waveguide and s t r i p - l i n e d i r e c t i o n a l f i l t e r s
I t i s h o p e d t h a t t h i s book w i l l f i l l a n e e d ( w h i c h h a s become i n c r e a s i n g l y a p p a r e n t i n t h e l a s t few y e a r s ) f o r a r e f e r e n c e book o n d e s i g n d a t a , p r a c t i c a l development t e c h n i q u e s , and d e s i g n t h e o r y , i n a f i e l d of e n g i n e e r i n g which h a s been e v o l v i n g r a p i d l y .
ACKNOWLEDGMENTS
The p r e p a r a t i o n o f t h i s book was l a r g e l y s u p p o r t e d by t h e S i g n a l C o r p s , u n d e r C o n t r a c t DA 3 6 - 0 3 9 S C - 8 7 3 9 8 ; i t s p r e p a r a t i o n was a l s o p a r t i a l l y s u p p o r t e d by S t a n f o r d R e s e a r c h I n s t i t u t e , a n d by o f f - w o r k t i m e c o n t r i b u t e d by the authors.
Many o f t h e d e s i g n t e c h n i q u e s d e s c r i b e d i n t h i s book a r e t h e
r e s u l t o f r e s e a r c h p r o g r a m s s p o n s o r e d by t h e S i g n a l C o r p s u n d e r C o n t r a c t s DA 3 6 - 0 3 9 S C - 6 3 2 3 2 , DA 3 6 - 0 3 9 SC-64625, UA 36-039 S C - 7 4 8 6 2 , a n d DA 3 6 - 0 3 9 SC-87398. Mr. N a t h a n L i p e t z o f t h e U . S . Army E l e c t r o n i c s R e s e a r c h L a b o r a t o r y , F t . Monmouth, N. J . , b e c a u s e o f h i s b e l i e f i n t h e i m p o r t a n c e o f r e s e a r c h work i n t h e a r e a o f m i c r o w a v e f i l t e r s a n d c o u p l i n g s t r u c t u r e s , a n d i n t h e p o t e n t i a l v a l u e o f a book s u c h a s t h i s , d i d much t o - m a k e t h i s book p o s s i b l e
Mr. J o h n P. A g r i o s and Mr. W i l l i a m P. D a t t i l o , b o t h o f t h e U . S . Army E l e c t r o n i c s R e s e a r c h L a b o r a t o r y a l s o were o f g r e a t h e l p .
Mr. A g r i o s main-
t a i n e d a c l o s e i n t e r e s t i n t h i s p r o j e c t throughout t h e preparation of t h i s b o o k , a s d i d Mr. D a t t i l o , who r e v i e w e d a l l o f t h e c h a p t e r s a s t h e y were prepared.
He made n u m e r o u s s u g g e s t i o n s w h i c h h a v e r e s u l t e d i n v a l u a b l e
i m p r o v e m e n t i n t h e c l a r i t y a n d u s e f u l n e s s o f t h i s book. D r . S . B. Cohn, f o r m e r l y o f S t a n f o r d H e s e a r c h I n s t i t u t e a n d p r e s e n t l y o f t h e R a n t e c C o r p o r a t i o n , l e d t h e m i c r o w a v e f i l t e r and c o u p l i n g s t r u c t u r e s e a r c h a t S t a n f o r d Research I n s t i t u t e d u r i n g t h e e a r l i e r S i g n a l Corps f i l t e r p r o g r a m s a t S R I . I n many p l a c e s t h i s book p r e s e n t s r e s e a r c h r e s u o r r e f l e c t s p o i n t s o f v i e w , w h i c h a r e d u e t o him. The a u t h o r s ' c o l l e a g u e s a t SRI, and numerous r e s e a r c h e r s e l s e w h e r e h a v e made many v a l u a b l e c o n t r i b u t i o n s t o t h e s u b j e c t a r e a o f t h i s b o o k , a n d many r e s u l t s o f t h e i r work h a v e b e e n i n c o r p o r a t e d i n t o t h i s b o o k . The a u t h o r s t h a n k t h e v a r i o u s j o u r n a l s , book p u b l i s h e r s , a n d e l e c t r o n i c s f i r m s who h a v e g i v e n p e r m i s s i o n f o r t h e u s e o f n u m e r o u s f i g u r e s and t a b l e s .
vii
And f i n a l l y , t h e a u t h o r s t h a n k t h e m a n y p e o p l e a t SRI who t o o k a s p e c i a l Mrs. E d i t h C h a m b e r s
i n t e r e s t i n t h e huge j o b of p r e p a r i n g t h i s book.
CONTENTS
spent countless painstaking hours supervising t h e preparation of t h e s t a g g e r i n g number o f i l l u s t r a t i o n s i n t h i s b o o k , a n d h e l p e d g r e a t l y i n i n s u r i n g i l l u s t r a t i o n s o f h i g h q u a l i t y a n d c l a r i t y . M r s . Mary F. A r m s t r o n g s u p e r v i s e d t h e V a r i t y p i n g o f t h e t e x t . The a u t h o r s ' t h a n k s a l s o g o t o
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
t h e e d i t o r s , s e c r e t a r i e s , a n d r e p o r t p r o d u c t i o n s t a f f a t SRI who a l l
PREFACE
were very c o o p e r a t i v e i n t h e p r o d u c t i o n o f t h i s book.
ACKNOWLEDGMENTS
CHAPTER 1
CHAPTER 2
SOME GENERAL APPLICATIONS O F FILTER STRUCTURES I N MICROWAVE ENGINEERING. Sec. 1.01, Introduction Sec. 1.02, Use of Filters for the Separation or Summing of S i g n a l s . Sec. 1.03, Impedance-Matching Networks. Sec. 1.04, Coupling Networks for Tubes and NegativeResistance Amplifiers. Sec. 1.05, Time-Delay Networks and Slow-Wave Structures Sec. 1.06, General Use of Filter Principles in the Design of Microwave Components References.
. . . . . . . . ......... . . . . . . . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . . . . . . . . ........... . . . . . . . . . . . ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SOME USEFUL CIRCUIT CONCEPTS AND EQUATIONS. . . . . . . . . . . . . . Sec. 2.01, Introduction . . . . . . . . . . . . . . . . . . . . . . . Sec. 2.02, Complex Frequency and Poles and Zeros. . . . . . . . . . . Sec. 2.03, Natural Modes of Vibration and Their Relation to Input-Impedance Poles and Zeros . . . . . . . . . . . . Sec. 2.04, Fundamental Properties of Transfer Functions . . . . . . . Sec. 2.05, General Circuit Parameters . . . . . . . . . . . . . . . . Sec. 2.06, Open-Circuit Impedances and Short-circuit Admittances. . . . . . . . . . . . . . . . . Sec. 2.07, Relations Between General Circuit Parameters and Open- and Short-circuit Parameters . . . . . . . . . . Sec. 2.08, Incident and Reflected Waves, Reflection Coe fficients, and One Kind of Transmission Coefficient Sec. 2.09, Calculation of the Input Impedance of a Terminated, Two-Port Network Sec. 2.10, Calculation of Voltage Transfer Functions. Sec. 2.11, ,plculation of Power Transfer Functions and Attenuation". Sec. 2.12, Scattering Coefficienta. Sec. 2.13, Analysis of Ladder Circuits. References.
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CHAPTER 3
. . . .
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. . . .
. . . ... . . . ... PRINCIPLES O F THE IMAGE METHOD FOR FILTER DESIGN. . . . Sec. 3.01, Introduction . . . . . . . . . . . . . . . . . . . . . . . Sec. 3.02, Physical and Mathematical Definition of Image Impedance and Image Propagation Function . . . . . . . . . Sec. 3.03, Relation Between the Image Parameters and General Circuit Parameters, Open-Circuit Impedancea, and Short-circuit Admittances.
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viii
iii vii
1 1
1 3 6 9 13 14 15 15 15 18 20 26 29 29 34 35 36 38 42 45 48 49 49 49
52
CONTENTS
CONTENTS
. . . . . . . Networks . . . . . . . . . . . . . . . . . . . . . . . . . Sec . 3.06. Constant-k and a-Derived Filter Sections . . . . . . . . . Sec . 3.07, The Effects of Terminations Which Mismatch the Image Impedances . . . . . . . . . . . . . . . . . . . . . Sec . 3.08, Design of Matching End Sections to Improve the Response of Filters Designed on the Image Basis . . . . . . Sec . 3.09. Measurement of Image Parameters . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sec Sec
CHAPTER 4
.
3.04. Image Parameters for Some Common Structures . . 3.05, The Special Image Properties of Dissipationless
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.
Sec
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sec Sec Sec Sec
CHAPTER 7
6.08, Correction for Small Step-Discontinuity
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6.09. Approximate Design when R Is Large 6.10. Asymptotic Behavior as R Tends to Infinity 6.11, Inhomogeneous Waveguide Quarter-Wave Transformers
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LOW-PASS AND HIGH-PASS FILTERS USING SEMI-LUMPED ELEMENTS O R WAVEGUIDE CORRUGATIONS Sec 7.01, Properties o f the Filters Discussed in This Chapter Sec 7.02, Approximate Microwave Realization of Lumped Elements Sec 7.03. Low-Pass Filters Using Semi-Lumped Elements Sec 7.04. Low-Pass Corrugated-Waveguide Filter Sec 7.05, Low-Pass Waffle-lron Filtera Having Very Wide Stop Bands Sec 7.06, Low-Pasa Filters from Quarter-Wave Transformer Prototypes Sec 7.07. High-Pass Filters Using Semi-Lumped Element6 Sec 7.08, Low-Pass and High-Pass Impedance-Matching Networks Sec 7.09. Low-Pass Time-Delay Networks References
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. 5.01. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5.02. General Properties of Transmienion Lines . . . . . . . . . . 5.03. Special Properties of Coaxial Lines . . . . . . . . . . . . . 5.04. Special Properties of Strip Lines . . . . . . . . . . . . . . 5.05, Parallel-Coupled Lines and Arrays of Lines Between Ground Planes . . . . . . . . . . . . . . . . Sec . 5.06. Special Properties of Waveguidea . . . . . . . . . . . . . Sec . 5.07. Common Transmission Line Discontinuities . . . . . . . . . Sec . 5.08. Transmission Lines as Resonators . . . . . . . . . . . . .
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6.03. The Performance of Homogeneous Half-Wave Filters 6.04, Exact Tchebyscheff and Maximally Flat Solutions
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5.10. Iris-Coupled Waveguide Junctions 5.11, Resonant Frequencies and Unloaded Q of
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CHAPTER 5
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LOW-PASS PROTOTYPE FILTERS OBTAINED BY NETWORK SYNTHESIS METHODS Sec 4.01. Introduction Sec 4.02, Comparison of Image and Network Synthesis Methods for Filter Design Sec 4.03, Maximally Flat and Tchebyscheff Filter Attenuation Characteristics Sec 4.04, Definition of Circuit Parameters for LowPass Prototype Filters Sec 4.05, Doubly Terminated. Maximally Flat and Tchebyscheff Prototype Filters Sec 4.06, Singly Terminated Maximally Flat and Tchebyscheff Filters Sec 4.07. Maximally Flat Time-Delay Prototype Filters Sec 4.08, Comparison of the Time-Delay Characteristics of Various Prototype Filters Sec 4.09, Prototype. Tchebyscheff Impedance-Matching Networks Giving Minimum Reflection Sec 4.10, Computation of Prototype Impedance-Matching Networks for Specified Ripple or Minimum Reflection Sec 4.11. Prototypes for Negative-Resistance Amplifiers Sec 4.12, Conversion of Filter Prototypes to Use Impedanceor Admittance-Inverters and Only One Kind of Reactive Element Sec 4.13, Effects of Dissipative Elements in Prototypes for Low-Pass. Band-Pass. or' High-Pass Filters Sec 4.14, Approximate Calculation of Prototype Stop-Band Attenuation Sec 4.15, Prototype Representation of Dissipation Loss in Band-Stop Filters References
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CHAPTER 6
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. . . . . . . . . . . . . Waveguide Resonators . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . STEPPED-IMPEDANCE TRANSFORMERS AND FILTER PROTOTYPES. . . . . . . . . Sec . 6.01. Introduction . . . . . . . . . . . . . . . . . . . . . . . Sec . 6.02, The Performance of Homogeneous Quarter-Wave Transformers . . . . . . . . . . . . . . . . . . . . . . . .
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BANI)-PASS FILTERS (A GENERAL SUMMARY O F BAND-PASS FILTERS. AND A VERSATILE DESIGN TECHNIQUE FOR FILTERS WITH NARROW O R MODERATE BANDWIDTHS) Sec 8.01, A Summary of the Propertiea of the Band-Pass or Pseudo High-Peas Filters Treated in Chapters 8. 9. and10 Sec 8.02. General Principles of Coupled-Renonator Filters
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CONTENTS
CONTENTS
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S e c . 8 . 0 3 , P r a c t i c a l R e a l i z a t i o n o f K- and J - I n v e r t e r s . S e c . 8 . 0 4 , Use of Low-Pass t o Band-Pass M a p p i n g s . Sec. 8 . 0 5 , Capacitive-Gap-Coupled T r a n s m i s s i o n L i n e Filters. S e c . 8 . 0 6 , S h u n t - I n d u c t a n c e - C o u p l e d , Waveguide F i l t e r s . S e c . 8 . 0 7 , Narrow-Band C a v i t y R e s o n a t o r F i l t e r s C o u p l e d by S m a l l I r i s e s . Sec. 8 . 0 8 , F i l t e r s Using Two-Port, Quarter-Wavelength Resonators Sec. 8 . 0 9 , F i l t e r s with P a r a l l e l - C o u p l e d S t r i p - L i n e Resonators Sec. 8.10, F i l t e r s with Quarter-Wavelength C o u p l i n g s . S e c . 8 . 1 1 , Lumped-Element, C o u p l e d - R e s o n a t o r F i l t e r s . S e c . 8 . 1 2 , B a n d - P a s s F i l t e r s w i t h Wide S t o p B a n d s S e c . 8 . 1 3 , Comb-Line, B a n d - P a s s F i l t e r s S e c . 8 . 1 4 , C o n c e r n i n g t h e D e r i v a t i o n o f Some o f t h e Preceding Equations. References
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S e c . 1 0 . 0 7 , I n t e r d i g i t a l - L i n e F i l t e r s H a v i n g Wide Bandwidths Sec. 1 0 . 0 8 , D e r i v a t i o n of t h e Design E q u a t i o n s f o r P a r a l l e l - C o u p l e d and S t u b F i l t e r s . Sec. 1 0 . 0 9 , D e r i v a t i o n o f t h e Design E q u a t i o n s f o r Interdigital-Line Filters. S e c . 1 0 . 1 0 , S e l e c t i o n o f Mapping F u n c t i o n s References
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CHAPTER l i
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CHAPTER
9
BAND-PASS FILTERS, CONTINUED (WIDE-BAND AND NARROW-BAND BAND-PASS FILTERS CONSISTING OF TRANSMISSION LINES WITH REACTIVE DISCONTINUITIES). Sec. 9 . 0 1 , I n t r o d u c t i o n S e c . 9 . 0 2 , F i l t e r s w i t h Impedance S t e p s and I m p e d a n c e Inverters. S e c . 9 . 0 3 , S y n c h r o n o u s l y Tuned R e a c t a n c e - C o u p l e d H a l f Wave F i l t e r s S e c . 9 . 0 4 , Narrow-Band Half-Wave F i l t e r s . Sec. 9 . 0 5 , P r a c t i c a l R e a l i z a t i o n of R e a c t i v e C o u p l i n g s . S e c . 9 . 0 6 , Some S t a n d a r d i z e d P s e u d o - H i g h - P a s s F i l t e r Designs. S e c . 9 . 0 7 , An E x p e r i m e n t a l Wide-Band Waveguide F i l t e r . S e c . 9 . 0 8 , D e s i g n f o r S p e c i f i e d Band-Edges and S t o p Band A t t e n u a t i o n S e c . 9 . 0 9 , Examples o f F i l t e r s H a v i n g Medium and L a r g e Bandwidths S e c . 9 . 1 0 , D e r i v a t i o n o f t h e Data f o r B a n d w i d t h C o n t r a c t i o n and P a s s - B a n d D i s t o r t i o n Sec. 9 . 1 1 , O p t i m i z i n g t h e L i n e Impedances Sec. 9 . 1 2 , Reactance-Coupled Quarter-Wave F i l t e r s References
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BAND-PASS FILTERS, CONTINUED (BAND-PASS AND PSEUW HIGHPASS FILTERS CONSISTING OF LINES AND STUBS, OR PARALLELCOUPLED ARRAYS OF LINES) S e c . 1 0 . 0 1 , C o n c e r n i n g t h e F i l t e r s and D e s i g n P r o c e d u r e s i n t h i s Chapter. Sec. 10.02, Parallel-Coupled F i l t e r s with A0/2 Resonators. Sec. 1 0 . 0 3 , F i l t e r s with Shunt o r S e r i e s A0/4 S t u b s . S e c . 1 0 . 0 4 , F i l t e r s w i t h A 0 / 2 S t u b s and A 0 / 4 C o n n e c t i n g Lines.. S e c . 1 0 . 0 5 , F i l t e r s U s i n g b o t h S e r i e s and S h u n t S t u b s . S e c . 1 0 . 0 6 , I n t e r d i g i t a l - L i n e F i l t e r s o f Narrow o r M o d e r a t e Bandwidth
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SPECIAL PROCEDURES TO AID IN THE PRACTICAL DEVELOPMENT OF COUPLED-RESONATOR BAND-PASS FILTERS, IMPEDANCEMATCHING NETWORKS, AND TIME-DELAY NETWORKS Sec. 11.01, I n t r o d u c t i o n S e c . 1 1 . 0 2 , Measurement o f V , Q e , and QL o f a S i n g l y Loaded ~ e s o n a t o r ~ . S e c . 1 1 . 0 3 , T e s t s on S i n g l e R e s o n a t o r s w i t h L o a d i n g a t Both Ends S e c . 1 1 . 0 4 , T e s t s on S y m m e t r i c a l P a i r s o f R e s o n a t o r s . Sec. 1 1 . 0 5 , Tuning o f M u l t i p l e - R e s o n a t o r Band-Pass Filters. S e c . 1 1 . 0 6 , C a l c u l a t i o n o f t h e Midband D i s s i p a t i o n L o s s of B a n d - P a s s F i l t e r s . S e c . 1 1 . 0 7 , D e s i g n o f Narrow-Band F i l t e r s f o r Minimum Midband L o s s a n d S p e c i f i e d High A t t e n u a t i o n a t some Nearby F r e q u e n c y Sec. 1 1 . 0 8 , Design of Band-Pass Impedance-Matching Networks U s i n g t h e Methods o f C h a p t e r 8 Sec. 1 1 . 0 9 , Design of Band-Pass Impedance-Matching Networks by t h e Methods o f C h a p t e r 10. S e c . 1 1 . 1 0 , D e s i g n o f Wide-Band C o u p l i n g S t r u c t u r e s f o r Negative-Resistance Amplifiers S e c . 1 1 . 1 1 , B a n d - P a s s Time-Delay F i l t e r s References
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BAND-STOP FILTERS. Sec. 1 2 . 0 1 , I n t r o d u c t i o n S e c . 1 2 . 0 2 , Lumped-Element Band-Stop F i l t e r s from LowPass Prototypes. S e c . 1 2 . 0 3 , The E f f e c t s o f D i s s i p a t i o n L o s s on BandStop F i l t e r Performance. S e c . 1 2 . 0 4 . An A p p r o x i m a t e Deaign P r o c e d u r e f o r Microwave F i l t e r s w i t h Very Narrow S t o p Bands. Sec. 1 2 . 0 5 , P r a c t i c a l R e a l i z a t i o n of Band-Stop R e s o n a t o r s f o r Narrow-Stop-Band F i l t e r s S e c . 1 2 . 0 6 , E x p e r i m e n t a l A d j u s t m e n t o f t h e C o u p l i n g s and T u n i n g o f Band-Stop R e s o n a t o r s S e c . 1 2 . 0 7 , Example o f a S t r i p - L i n e , Narrow-Stop-Band F i l t e r Design. S e c . 1 2 . 0 8 , Narrow-Stop-Band Waveguide F i l t e r D e s i g n C o n s i d e r a t i o n s , and an Example Set. 1 2 . 0 9 , A D e s i g n P r o c e d u r e Which I s A c c u r a t e f o r F i l t e r s w i t h Wide ( a n d Narrow) Stop-Band Widths. S e c . 1 2 . 1 0 , Some Examples I l l u s t r a t i n g t h e P e r f o r m a n c e O b t a i n a b l e Using t h e E q u a t i o n s i n Sec. 12.09 References
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CHAPTER 1 3
CONTENTS
TEM.MODE. COUPLED-TRANSMISSION-LINE DIRECTIONAL COUPLERS. AND BRANCH-LINE DIRECTIONAL COUPLERS . . . . . . . . . . Sec 1 3 . 0 1 , I n t r o d u c t i o n S e c . 1 3 . 0 2 , D e s i g n R e l a t i o n s f o r TEM.Mode. C o u p l e d Transmission-Line Directional Couplers ofonesection S e c 1 3 . 0 3 D e s i g n R e l a t i o n s f o r TEM.Mode. C o u p l e d Transmission-Line Directional Couplers of Three Sections. . . . . . . . . . . . . .
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DIRECTIONAL. CHANNEL-SEPARATION FILTERS AND TRAVELINGWAVE RING.RESONATORS Sec 14.01. I n t r o d u c t i o n S e c 1 4 . 0 2 . Waveguide D i r e c t i o n a l F i l t e r s Sec 14.03, S t r i p Transmission Line D i r e c t i o n a l F i l t e r s Using Half- o r Full-Wavelength S t r i p s Sec 14.04. Traveling-Wave-Loop D i r e c t i o n a l F i l t e r Sec 14.05. Traveling-Wave Ring Resonator S e c 1 4 . 0 6 , D e r i v a t i o n o f Formulas f o r Waveguide Directional Filters Sec 14.07, D e r i v a t i o n o f Formulas f o r Traveling-Wave Loop D i r e c t i o n a l F i l t e r s a n d t h e T r a v e l i n g Wave R i n g R e s o n a t o r Sec 14.08, D e r i v a t i o n o f Formulas f o r S t r i p - T r a n s m i s s i o n Line D i r e c t i o n a l F i l t e r s Using Half-Wavelength and One-Wavelength S t r i p s References
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xiv
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTIPLEXER DESIGN . . . . . . . . . . . . . . . . . . . . . . . . Sec . 16.01. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . Sec . 16.02. M u l t i p l e x e r s Using D i r e c t i o n a l F i l t e r s . . . . . . . . S e c . 1 6 . 0 3 , Mu1 t i p l e x e r s U s i n g R e f l e c t i n g Narrow-Band F i l t e r s . w i t h G u a r d B a n d s Between C h a n n e l s . . . . . . S e c . 1 6 . 0 4 . M u l t i p l e x e r s w i t h C o n t i g u o u s P a s s Bands . . . . . . . . Sec . 1 6 . 0 5 . D i p l e x e r s w i t h C o n t i g u o u s P a s s Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References
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MECHANICALLY AND MAGNETICALLY TUNABLE MICROWAVE FILTERS Sec 17.01. I n t r o d u c t i o n Sec 1 7 . 0 2 . Theory o f I d e a l . T u n a b l e Band-Pass F i l t e r s Sec 17.03. Mechanically Tunable Coaxial Band-Pass F i l t e r s S e c 1 7 . 0 4 . Waveguide M e c h a n i c a l l y T u n a b l e B a n d - P a s s F i l t e r s Sec 17.05. P r o p e r t i e s of F e r r i m a g n e t i c Resonators S e c 1 7 . 0 6 , D e t e r m i n a t i o n o f t h e C r y s t a l Axes o f S p h e r i c a l Ferrimagnetic Resonators Sec 17.07, Design o f M a g n e t i c a l l y Tunable Band-Pass F i l t e r s w i t h 'EM-Mode I n p u t s a n d O u t p u t s . a n d a Discussion of General Design P r i n c i p l e s S e c 1 7 . 0 8 , n e s u l t s o f Some T r i a l M a g n e t i c a l l y T u n a b l e BandP a s s F i l t e r I)esigns Having S t r i p - L i n e I n p u t s and O u t p u t s Sec 17.09, Magnetically Tunable Rand-Pass F i l t e r s with Waveguide I n p u t s a n d O u t p u t s
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17.10, Magnetically Tunable D i r e c t i o n a l F i l t e r s Having C i r c u l a t o r Action Sec 1 7 . 1 1 . M a g n e t i c a l l y T u n a b l e Band-Stop F i l t e r s References
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INDEX
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889 889 895 910 921 937 952 957 961 965 965 966 968 973 991 1000 1001 1001 1006 1013 1020 1027 1040 1043
1056 1068
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1072 1078 1086
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1089
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HIGH- POWER FILTERS Sec 1 5 . 0 1 . I n t r o d u c t i o n and G e n e r a l C o n s i d e r a t i o n s Sec 1 5 . 0 2 , Power-Handling Capacity of Various Transmission-Lines Sec 15.03, T h e o r e t i c a l Pulse-Power Capacity of D i r e c t Coupled-Resonator F i l t e r s S e c 1 5 . 0 4 A High-Power T u n a b l e Narrow-Band TEOl1-Mode F i l t e r Sec 1 5 . 0 5 . High-Power W a f f l e - I r o n F i l t e r s S e c 1 5 . 0 6 . D i s s i p a t i v e Waveguide F i l t e r s Sec 15.07. D i s s i p a t i v e Coaxial-Line F i l t e r s References
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CHAPTER 1 5
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1 3 . 0 4 , R e l a t i o n s f o r TEM-Mode C o u p l e d - T r a n s m i s s i o n L i n e D i r e c t i o n a l C o u p l e r s o f F i v e a n d More Sections S e c 1 3 . 0 5 , T y p i c a l D e s i g n s f o r TEM.Mode. C o u p l e d Transmission-Line Directional Couplers of One S e c t i o n w i t h A p p r o x i m a t e l y 3 d b A v e r a g e Coupling S e c 1 3 . 0 6 , T y p i c a l D e s i g n s f o r TEM Mode. C o u p l e d Transmission-Line D i r e c t i o n a l Couplers of One S e c t i o n a n d w i t h Weak C o u p l i n g . . . . . S e c 1 3 . 0 7 , D e r i v a t i o n o f D e s i g n F o r m u l a s f o r TEM.Mode. Coupled-Transmission-Line C o u p l e r s of Onesection. . . . . . . . . . . . . . . . . Sec 13.08, Quarter-Wave F i l t e r P r o t o t y p e C i r c u i t s f o r TEM.Mode. C o u p l e d - T r a n s m i s s i o n - L i n e Directional Couplers S e c . 1 3 . 0 9 , C o n s i d e r a t i o n s and G e n e r a l F o r m u l a s f o r Branch-Line Couplers . . . . . . . . . . . . Sec 13.10, P e r i o d i c Branch-Line Couplers S e c 1 3 . 1 1 , The C h a r a c t e r i s t i c s a n d P e r f o r m a n c e o f Synchronous Branch-Line C o u p l e r s S e c 13.12 T a b l e s o f I m m i t t a n c e s o f S y n c h r o n o u s BranchLine Couplers S e c 1 3 . 1 3 , Examples I l l u s t r a t i n g t h e D e s i g n and Performance of Synchronous Branch-Line Couplers Sec 13.14, Design of an Experimental Branch-Line Coupler inwaveguide References
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775
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MICROWAVE FILTERS, IMPEDANCE-MATCHING NETWORKS, AND COUPLING STRUCTURES
CHAPTER I
SOME GENERAL APPLICATIONS OF FILTER STRUCTURES I N MICROWAVE ENGINEERING
S E C . 1 . 0 1 . INTRODUCTION Most r e a d e r s w i l l b e f a m i l i a r w i t h t h e u s e o f f i l t e r s a s d i s c u s s e d i n Sec. 1 . 0 2 below.
However, t h e p o t e n t i a l a p p l i c a t i o n s o f t h e m a t e r i a l
i n t h i s book g o e s much b e y o n d t h e s e c l a s s i c a l f i l t e r a p p l i c a t i o n s t o c o v e r many o t h e r m i c r o w a v e e n g i n e e r i n g p r o b l e m s w h i c h i n v o l v e f i l t e r - t y p e s t r u c t u r e s b u t a r e not always thought of a s being f i l t e r problems. T h u s , t h e p u r p o s e o f t h i s c h a p t e r i s t o make c l e a r t o t h e r e a d e r t h a t t h i s book i s n o t a d d r e s s e d o n l y t o f i l t e r d e s i g n s p e c i a l i s t s , b u t a l s o t o a n t e n n a e n g i n e e r s who may n e e d a b r o a d b a n d a n t e n n a f e e d , t o m i c r o w a v e t u b e e n g i n e e r s who may n e e d t o o b t a i n b r o a d b a n d i m p e d a n c e m a t c h e s i n a n d o u t o f m i c r o w a v e t u b e s , t o s y s t e m e n g i n e e r s who may n e e d a microwave t i m e - d e l a y network, and t o numerous o t h e r s h a v i n g o t h e r s p e c i a l microwave c i r c u i t d e s i g n problems. SEC.
1 . 0 2 , USE OF F I L T E R S FOR THE SEPARATION OR SUMMING OF SIGNALS The m o s t o b v i o u s a p p l i c a t i o n o f f i l t e r s t r u c t u r e s , o f c o u r s e , i s
f o r t h e r e j e c t i o n o f u n w a n t e d s i g n a l f r e q u e n c i e s w h i l e p e r m i t t i n g good t r a n s m i s s i o n o f wanted f r e q u e n c i e s .
The m o s t common f i l t e r s o f t h i s
s o r t a r e designed f o r e i t h e r low-pass, high-pass,
band-pass o r band-stop
a t t e n u a t i o n c h a r a c t e r i s t i c s s u c h a s t h o s e shown i n F i g . 1 . 0 2 - 1 .
Of c o u r s e ,
i n t h e c a s e o f p r a c t i c a l f i l t e r s f o r t h e m i c r o w a v e o r any o t h e r f r e q u e n c y range, t h e s e c h a r a c t e r i s t i c s a r e o n l y achieved approximately, s i n c e t h e r e i s a h i g h - f r e q u e n c y l i m i t f o r any g i v e n p r a c t i c a l f i l t e r s t r u c t u r e above
-
which i t s c h a r a c t e r i s t i c s w i l l d e t e r i o r a t e due t o j u n c t i o n e f f e c t s , r e s o nances within t h e elements, etc. F i l t e r s a r e a l s o commonly u s e d f o r s e p a r a t i n g f r e q u e n c i e s i n diplexers o r multiplexers.
F i g u r e 1 . 0 2 - 2 shows a m u l t i p l e x e r which
s e g r e g a t e s s i g n a l s w i t h i n t h e 2 . 0 t o 4 . 0 Gc band i n t o t h r e e s e p a r a t e
channels according t o t h e i r frequencies.
P
23
t
--
4 . 0 Gc i n p u t b a n d .
.
To a c h i e v e t h i s r e s u l t t h e i n d i v i d u a l f i l t e r s m u s t
be d e s i g n e d s p e c i a l l y f o r t h i s p u r p o s e a l o n g w i t h a s p e c i a l j u n c t i o n -
STOP BAND
matching network. C PASS BAND
0
A well designed m u l t i p l e x e r of
t h i s s o r t would h a v e v e r y low VSWR a t t h e i n p u t p o r t a c r o s s t h e 2 . 0 t o
D
+C
T h e s e m a t t e r s a r e t r e a t e d i n C h a p t e r 16.
A n o t h e r way t h a t d i p l e x e r s o r m u l t i p l e x e r s a r e o f t e n u s e d i s i n t h e summing o f s i g n a l s h a v i n g d i f f e r e n t f r e q u e n c i e s .
.
LOW -PASS FILTER
t
0
f
HIGH-PASS FILTER
Supposing t h a t t h e
s i g n a l - f l o w arrowheads i n F i g . 1 . 0 2 - 2 a r e r e v e r s e d ; i n t h i s e v e n t , s i g n a l s e n t e r i n g a t t h e v a r i o u s c h a n n e l s c a n a l l be j o i n e d t o g e t h e r w i t h n e g l i g i b l e r e f l e c t i o n o r l e a k a g e of e n e r g y s o t h a t a l l of t h e s i g n a l s w i l l be s u p e r i m p o s e d on a s i n g l e o u t p u t l i n e .
I f signals i n these various channel fre-
q u e n c y r a n g e s w e r e summed by a s i m p l e j u n c t i o n o f t r a n s m i s s i o n l i n e s ( i . e . , without a m u l t i p l e x e r ) , t h e l o s s i n energy a t t h e s i n g l e o u t p u t l i n e would, 4 PASS
PASS 4- BAND
BAND*
L
*
*
PASS BAND +'
o f c o u r s e , be c o n s i d e r a b l e , a s a r e s u l t o f r e f l e c t i o n s and o f l e a k a g e o u t of l i n e s o t h e r t h a n t h e i n t e n d e d o u t p u t l i n e .
2
BAND-PASS FILTER
1
0
BAND-STOP FILTER
S E C . 1 . 0 3 , IMPEDANCE - MATCHING NETWORKS i30de1 f i r s t showed w h a t t h e p h y s i c a l l i m i t a t i o n s were on t h e b r o a d b a n d
FIG. 1.02-1
FOUR COMMON TYPES OF FILTER CHARACTERISTICS
impedance m a t c h i n g o f l o a d s c o n s i s t i n g o f a r e a c t i v e element and a r e s i s t o r in series or i n parallel.
Later, ~ a n o ' presented the general limitations
on t h e i m p e d a n c e m a t c h i n g o f a n y l o a d .
F a n o ' s work shows t h a t e f f i c i e n c y
of t r a n s m i s s i o n and bandwidth a r e exchangeable q u a n t i t i e s i n t h e impedance matching of any l o a d having a r e a c t i v e component.
2.0-4.0
Gc
CHANNEL I
To i l l u s t r a t e t h e t h e o r e t i c a l l i m i t a t i o n s w h i c h e x i s t on b r o a d b a n d
CHANNEL 2
i m p e d a n c e m a t c h i n g , c o n s i d e r t h e e x a m p l e shown i n F i g . 1 . 0 3 - 1 w h e r e t h e 2.6-3.3 Gc
l o a d t o be m a t c h e d c o n s i s t s o f a c a p a c i t o r C1 a n d a r e sistor
R,, i n p a r a l l e l .
A
l o s s l e s s impedance-matching WSSLESS IMPEDANCE MATCHING NETWORK
-
network is t o be i n s e r t e d b e t w e e n t h e g e n e r a t o r and
FIG. 1.02-2 A THREE-CHANNEL MULTIPLEXING FILTER GROUP
t h e l o a d , and t h e r e f l e c t i o n c o e f f i c i e n t between
LOAD
Zin
A-3527-95
t h e g e n e r a t o r and t h e impedance-matching n e t work i s
-
F1.G. 1.03-1
EXAMPLE OF AN IMPEDANCE-MATCHING PROBLEM
Equation ( 1 . 0 3 - 4 ) s a y s t h a t an i d e a l impedance-matching network f o r t h e l o a d i n F i g . 1 . 0 3 - 1 would b e a b a n d - p a s s f i l t e r s t r u c t u r e w h i c h would c u t o f f s h a r p l y a t t h e edges o f t h e band o f impedance match. The c u r v e s
Irl
The work o f ~ o d e land t h a t o f ~ a n o ' shows t h a t t h e r e i s a p h y s i c a l l i m i t a -
i n F i g . 1 . 0 3 - 2 show how t h e
t i o n on w h a t I- c a n be a s a f u n c t i o n o f f r e q u e n c y . r e s u l t s a r e l i m i t e d a s i n d i c a t e d by t h e r e l a t i o n *
impedance-matching f i l t e r s might l o o k .
The best p o s s i b l e
vs o c u r v e f o r p r a c t i c a l b a n d - p a s s The c u r v e m a r k e d C a s e 1 i s f o r t h e
impedance m a t c h i n g of a g i v e n l o a d o v e r t h e r e l a t i v e l y narrow band from wa t o o b , w h i l e t h e c u r v e marked Case 2 is f o r t h e impedance m a t c h i n g o f
t h e same l o a d o v e r t h e w i d e r b a n d f r o m oc t o od u s i n g t h e same number o f e l e m e n t s i n t h e i m p e d a n c e - m a t c h i n g n e t w o r k . The r e c t a n g u l a r (rl c h a r a c t e r i s t i c i n d i c a t e d by Eq. ( 1 . 0 3 - 4 ) i s t h a t w h i c h w o u l d be a c h i e v e d by a n R e c a l l t h a t f o r a p a s s i v e c i r c u i t 0 5 1i-l 5 1 , f o r t o t a l r e f l e c t i o n = 1 , and t h a t f o r p e r f e c t t r a n s m i s s i o n = 0. Thus, t h e l a r g e r In i s t h e b e t t e r t h e t r a n s m i s s i o n w i l l b e . d u t Eq. ( 1 . 0 3 - 2 ) s a y s
Irl
optimum b a n d - p a s s m a t c h i n g f i l t e r w i t h a n i n f i n i t e number o f e l e m e n t s . *
Irl
Il/rl
t h a t t h e a r e a u n d e r t h e c u r v e o f In
Il/rl
u s o can be no g r e a t e r t h a n
n/(RoC1 ).
I f a good i m p e d a n c e m a t c h i s d e s i r e d f r o m f r e q u e n c y
t o w b , best r e s u l t s c a n be o b t a i n e d i f Ii-1 = 1 a t a l l f r e q u e n c i e s e x c e p t i n t h e band f r o m a,a t o u b . Then In ll/rl = 0 a t a l l f r e q u e n c i e s e x c e p t i n t h e m a t o w b b a n d , a n d t h e a v a i l a b l e a r e a u n d e r t h e In Il/rl c u r v e c a n a l l b e c o n c e n t r a t e d i n t h e r e g i o n where i t does t h e most good.
wc
c a t i o n , Eq. ( 1 . 0 3 - 2 ) becomes
1i-1
Irl
wo
do
I
wb
RADIAN FREQUENCY. W
FIG. 1.03-2
and i f
I
I
With t h i s s p e c i f i -
Wd A-asz7-9s
CURVES ILLUSTRATING RELATION BETWEEN BANDWIDTH AND DEGREE OF IMPEDAMCE MATCH POSSIBLE FOR A GIVEN LOAD HAVING A REACTIVE COMPONENT
i s a s s u m e d t o b e c o n s t a n t a c r o s s t h e band o f i m p e d a n c e m a t c h ,
The work o f ~ a n shows o ~ t h a t s i m i l a r c o n d i t i o n s a p p l y no m a t t e r what Thus,
a s a f u n c t i o n o f f r e q u e n c y hecomes
the n a t u r e of t h e load ( a s long a s t h e load i s not a p u r e r e s i s t a n c e ) .
f o r this v e r y f u n d a m e n t a l r e a s o n , e f f i c i e n t b r o a d b a n d i m p e d a n c e - m a t c h i n g structures are necessarily filter structures.
S e e S e c s . 4.09 t h r o u g h 4.11,
7 . 0 8 , and 1 1 . 0 8 t h r o u g h 1 1 . 1 0 f o r d i s c u s s i o n s o f how t h e f i l t e r s t r u c t u r e s t r e a t e d i n t h i s book may b e d e s i g n e d f o r u s e as impedance-matching n e t w o r k s .
*
* T h i s r e l a t i o n h o l d s i f t h e impedance matching network i s d e s i g n e d
so
that the r e f l e c t i o n c o e f -
f i c i e n t b e t w e e n Ro and t h e c i r c u i t t o t h e l e f t o f Ro i n F i g . 1 . 0 3 - 1 h a s a l l o f i t s z e r o s i n the l e f t half plane.'"
S i m p l e m a t c h i n g n e t w o r k s c a n # i v e v e r y g r e a t i m p r o v e m e n t s i n i m p e d a n c e m a t c h , and a s t h e number o f m a t c h i n g e l e m e n t s i s i n c r e a s e d t h e improvement p e r a d d i t i o n a l e l e m e n t r a p i d l y becomes s m a l l e r and s m a l l e r . For t h i s r e a s o n f a i r l y s i m p l e m a t c h i n g n e t w o r k s c a n g i v e P e r f o r m a n c e w h i c h c o m e s e l a s . t o t h e t h c a r c t i c a l l y optimum p e r f o r m a n c e f o r a n i n f i n i t e number o f i m p e d m n c e - m a t c h i n g e l e m e n t s .
S E C . 1 . 0 4 , COUPLING NETWOHKS FOH TUBES AND NEGATIVE-RESI STANCE A M P L I F I E R S A p e n t o d e vacuum t u b e c a n o f t e n b e s i m u l a t e d a t i t s o u t p u t a s a n infinite-impedance current generator with a capacitor shunted across the terminals.
and t h a t a t t h e r i g h t a s
Broadband o u t p u t c i r c u i t s f o r s u c h t u b e s can be d e s i g n e d a s
a f i l t e r t o b e d r i v e n by a n i n f i n i t e - i m p e d a n c e c u r r e n t g e n e r a t o r a t o n e end w i t h o n l y one r e s i s t o r t e r m i n a t i o n ( l o c a t e d a t t h e o t h e r end of t h e filter).
Then t h e o u t p u t c a p a c i t a n c e of t h e t u b e is u t i l i z e d a s one of
t h e e l e m e n t s r e q u i r e d f o r t h e f i l t e r , a n d i n t h i s way t h e d e l e t e r i o u s
Since t h e i n t e r v e n i n g band-pass f i l t e r c i r c u i t i s d i s s i p a t i o n l e s s ,
e f f e c t s of t h e s h u n t c a p a c i t a n c e a r e c o n t r o l l e d . 3 Data p r e s e n t e d l a t e r i n t h i s book w i l l p r o v i d e c o n v e n i e n t means f o r d e s i g n i n g microwave b r o a d band c o u p l i n g c i r c u i t s f o r p o s s i b l e microwave s i t u a t i o n s o f a s i m i l a r c h a r a c t e r w h e r e t h e d r i v i n g s o u r c e may b e r e g a r d e d a s a c u r r e n t o r v o l t a g e generator plus a reactive element. I n some c a s e s t h e i n p u t o r o u t p u t i m p e d a n c e s o f a n o s c i l l a t o r o r a n a m p l i f y i n g d e v i c e may b e r e p r e s e n t e d a s a r e s i s t a n c e a l o n g w i t h o n e o r two r e a c t i v e e l e m e n t s .
I n such c a s e s impedance-matching f i l t e r s a s d i s -
though t h e p h a s e s of
I-,
and
r3 a r e
n o t n e c e s s a r i l y t h e same.
The a v a i l a b l e
power e n t e r i n g t h e c i r c u l a t o r on t h e r i g h t i s d i r e c t e d i n t o t h e f i l t e r network, and p a r t of i t i s r e f l e c t e d back t o t h e c i r c u l a t o r where i t i s The t r a n s d u c e r g a i n f r o m t h e f i n a l l y a b s o r b e d i n t h e t e r m i n a t i o n 11'. generator t o RL i s
c u s s e d i n t h e p r e c e d i n g s e c t i o n a r e n e c e s s a r y i f optimum b r o a d b a n d p e r f o r m ance i s t o be approached. N e g a t i v e - r e s i s t a n c e a m p l i f i e r s a r e y e t a n o t h e r c l a s s o f d e v i c e s which r e q u i r e f i l t e r s t r u c t u r e s f o r optimum b r o a d b a n d o p e r a t i o n .
Consider the
c i r c u i t i n F i g . 1 . 0 4 - 1 , w h e r e we s h a l l d e f i n e t h e r e f l e c t i o n c o e f f i c i e n t a t the l e f t a s
w h e r e Psvmil i s t h e a v a i l a b l e p o w e r o f t h e g e n e r a t o r a n d P r i s t h e power r e f l e c t e d back from t h e f i l t e r n e t w o r k . I f t h e r e s i s t o r R,, on t h e l e f t i n F i g . 1 . 0 4 - 1 i s p o s i t i v e , t h e t r a n s d u c e r g a i n c h a r a c t e r i s t i c m i g h t b e a s i n d i c a t e d by t h e C a s e 1 c u r v e i n F i g . 1 . 0 4 - 2 . I n t h i s c a s e t h e ,gain i s low i n t h e p a s s b a n d o f t h e f i l t e r since = i s s m a l l t h e n . H o w e v e r , i f R o i s r e p l a c e d by a
Ir, 1
Ir31
n e g a t i v e r e s i s t a n c e Ri = - R o , t h e n t h e r e f l e c t i o n c o e f f i c i e n t a t t h e l e f t becomes
As a r e s u l t we t h e n h a v e
FIG. 1.04-1
CIRCUIT ILLUSTRATING THE USE OF FILTER STRUCTURES IN THE DESIGN OF NEGATIVE-RESISTANCE AMPLIFIERS
T h e c o u p l i n g n e t w o r k shown i n F i g . 1 . 0 4 - 1 i s i n a l u m p e d - e l e m e n t form which i s n o t v e r y p r a c t i c a l t o c o n s t r u c t a t microwave f r e q u e n c i e s . However, t e c h n i q u e s w h i c h a r e s u i t a b l e f o r d e s i g n i n g p r a c t i c a l m i c r o w a v e f i l t e r s t r u c t u r e s f o r such a p p l i c a t i o n s a r e d i s c u s s e d i n Sec. 11.10, u s i n g r e s u l t s developed i n Secs. 1 1 . 0 8 and 11.09. resistance-amplifier
Low-pass p r o t o t y p e s f o r n e g a t i v e -
coupling networks a r e t a b u l a t e d i n Sec. 4.11.
S E C . 1 . 0 5 , TIME-DELAY NETWORKS AND SLOWWAVE STRUCTURES 1 . 0 5 - l ( a ) which h a s a Consider t h e low-pass f i l t e r network i n F i g . The t r a n s m i s s i o n p h a s e i s d e f i n e d a s voltage t r a n s f e r function Eo/E6
FIG. 1.04-2 TRANSDUCER GAIN BETWEEN GENERATOR IN FIG. 1.04-1 ANDTHECIRCULATOROUTPUT Case 1 i s for Ro Positive while Case 2 is for R o Replaced by R;; = -Ro The phase delay o f t h i s n e t w o r k a t a n y g i v e n f r e q u e n c y w i s T h u s , r e p l a c i n g R,, by i t s n e g a t i v e c o r r e s p o n d s t o
/I-:
1
=
l/lr31 ,
1
b e i n g r e p l a c e d by a n d t h e t r a n s d u c e r g a i n i s a s i n d i c a t e d by t h e c u r v e marked
C a s e 2 i n F i g . 1 . 0 4 - 2 . U n d e r t h e s e c i r c u m s t a n c e s t h e o u t p u t power g r e a t l y e x c e e d s t h e a v a i l a b l e power o f t h e g e n e r a t o r f o r f r e q u e n c i e s w i t h i n t h e w h i l e i t s g r o u p delay i s
p a s s hand of t h e f i l t e r . With t h e a i d o f E q s . ( 1 . 0 4 - 1 ) a n d ( 1 . 0 4 - 6 ) c o u p l i n g n e t w o r k s f o r n e g a t i v e - r e s i s t a n c e a m p l i f i e r s a r e e a s i l y d e s i g n e d u s i n g impedance-
td
matching f i l t e r design techniques. P r a c t i c a l negative-resistance elements such a s tunnel diodes a r e not simple negative resistances, s i n c e they a l s o
where
have r e a c t i v e elements i n t h e i r e q u i v a l e n t c i r c u i t . I n t h e c a s e of t u n n e l d i o d e s t h e dominant r e a c t i v e element i s a r e l a t i v e l y l a r g e c a p a c i t a n c e i n
4
='
d dw
seconds
is i n r a d i a n s and w i s i n r a d i a n s p e r s e c o n d .
(1.05-3)
Under d i f f e r e n t
c i r c u m s t a n c e s e i t h e r p h a s e o r g r o u p d e l a y may b e i m p o r t a n t , b u t i t i s
p a r a l l e l with the negative resistance.
With t h i s l a r g e c a p a c i t a n c e p r e s e n t s a t i s f a c t o r y o p e r a t i o n i s i m p o s s i b l e a t m i c r o w a v e f r e q u e n c i e s u n l e s s some
s p e c i a l c o u p l i n g network i s used t o compensate f o r i t s e f f e c t s .
In F i g . 1 . 0 4 - 1 , C1 a n d H: on t h e l e f t c a n b e d e f i n e d a s t h e t u n n e l - d i o d e c a p a c i t a n c e and n e g a t i v e r e s i s t a n c e , and t h e remainder o f t h e band-pass f i l t e r c i r c u i t s e r v e s a s a broadband c o u p l i n g network.
w
S i m i l a r p r i n c i p l e s a l s o a p p l y i n t h e d e s i g n of broadband c o u p l i n g networks f o r m a s e r s and p a r a m e t r i c a m p l i f i e r s .
I n t h e case of p a r a m e t r i c a m p l i f i e r s , however, t h e d e s i g n o f t h e c o u p l i n g f i l t e r s i s c o m p l i c a t e d somewhat by t h e r e l a t i v e l y c o m p l e x i m p e d a n c e t r a n s f o r m i n g e f f e c t s o f t h e time-varying element
.4"
8
FIG. 1.05-1(a) LOW-PASS FILTER DISCUSSED IN SEC. 1.05
n77
td
=
-
seconds
2w 1 where n i s a g a i n t h e number o f r e a c t i v e e l e m e n t s i n t h e f i l t e r .
Of c o u r s e ,
i n some c a s e s t d may v a r y a p p r e c i a b l y w i t h i n t h e p a s s b a n d , a n d E q . ( 1 . 0 5 - 5 ) i s very approximate.* F i g u r e 1 . 0 5 - 2 ( a ) shows a f i v e - r e s o n a t o r b a n d - p a s s f i l t e r w h i l e Fig. 1 . 0 5 - 2 ( b ) shows a p o s s i b l e p h a s e c h a r a c t e r i s t i c f o r t h i s f i l t e r . I n t h i s c a s e t h e t o t a l p h a s e s h i f t from w = 0 t o w = a is nn r a d i a n s ,
IE,/E, ICHARACTERISTIC
FIG. 1.05-l(b) A POSSIBLE FOR THE FILTER IN FIG. 1.05-1(a), AND AN APPROXIMATE CORRESPONDING PHASE CHARACTERISTIC
FIG. 1.05-2(a) A BAND-PASS FILTER CORRESPONDING TO THE LOW-PASS FILTER IN FIG. 1.05-l(a)
group d e l a y which d e t e r m i n e s t h e time r e q u i r e d f o r a s i g n a l t o p a s s through a c i r c ~ i t . ~ , ~ * Low-pass l a d d e r n e t w o r k s o f t h e form i n F i g . 1 . 0 5 - l ( a ) have z e r o t r a n s m i s s i o n p h a s e f o r w = 0 , and a s o becomes l a r g e (01 U-m
nrr 2
* -
radians
w h e r e n i s t h e number o f r e a c t i v e e l e m e n t s i n t h e c i r c u i t . F i g u r e 1 . 0 5 - l ( b ) shows a p o s s i b l e characteristic for the f i l t e r in Fig. 1.05-l(a) a l o n g w i t h t h e a p p r o x i m a t e c o r r e s p o n d i n g p h a s e c h a r a c t e r i s t i c . Note t h a t
IE,,/E~
1
m o s t o f t h e p h a s e s h i f t t a k e s p l a c e w i t h i n t h e p a s s band w = 0 t o w = w l . T h i s is n o r m a l l y t h e c a s e , hence a rough e s t i m a t e of t h e group time d e l a y i n t h e p a s s band of f i l t e r s o f t h e form i n F i g . 1 . 0 5 - l ( a ) can be o b t a i n e d Eom
T h a t i s , i f t h e r e i s n o a m p l i t u d e d i s t o r t i o n and d & d o i s h o n s t a n t a c r o s s t h e f r e q u e n c y band o f t h e s i g n a l , t h e n t h e o u t p u t s i g n a l r i l l be a n e x a c t r e p l i c a o f t h e i n p u t s i g n a l b u t d i a p l a c e d i n t l m e by t d a a c o n d a .
FIG. 1.05-2(b)
A POSSIBLE PHASE CHARACTERISTIC FOR THE FILTER IN FIG. 1.05-2(a)
* S i n c e n o t d l o f t h e phase a h i f t o c c u r s w i t h i n t h e p a s s band, Eq. ( 1 . 0 5 - 5 ) w i l l u s u a l l y OVereatimate t h e time d e l a y , o f t e n b y a8 much 8 s a f a c t o r o f two.
11
w h e r e n i s t h e number o f r e s o n a t o r s , a n d a r o u g h e s t i m a t e o f t h e p a s s band g r o u p time d e l a y i s
*
SEC.
1 . 0 6 , GENERAL USE, OF F I L T E R P R I X C I P L E S IN THE DESIGN OF MICROWAVE COMPONEKTS As c a n h e r e a d i l y s e e n by e x t r a p o l a t i n g f r o m t h e d i s c u s s i o n s i n
td
=
n7T
---Wb
p r e c e d i n g s e c t i o n s , m i c r o w a v e f i l t e r d e s i g n t e c h n i q u e s when u s e d i n t h e i r
seconds
-
most g e n e r a i way a r e f u n d a m e n t a l t o t h e e f f i c i e n t d e s i g n of a w i d e v a r i e t y of microwave components. I n g e n e r a l , t h e s e t e c h n i q u e s a r e b a s i c t o p r e c i s i ,
w h e r e ma a n d w b a r e t h e r a d i a n f r e q u e n c i e s o f t h e p a s s - b a n d e d g e s
d e s i g n when s e l e c t i n g , r e j e c t i n g , o r c h a n n e l i n g o f e n e r g y o f d i f f e r e n t f r e -
I n l a t e r c h a p t e r s more p r e c i s e i n f o r m a t i o n on t h e t i m e d e l a y c h a r a c t e r .
q u e n c i e s i s i m p o r t a n t ; when a c h i e v i n g e n e r g y t r a n s f e r w i t h low r e f l e c t i o n
i s t i c s of f i l t e r s w i l l he p r e s e n t e d .
o v e r a w i d e h a n d i s i m p o r t a n t ; o r when a c h i e \ - i n g a c o n t r o l l e d t i m e d e l a y The p o s s i b l e s p e c i f i c p r a c t i c a l s i t u a t i o n s w h e r e s u c h c o n i s important.
general time delay p r o p e r t i e s of f i l t e r s .
s i d e r a t i o n s a r i s e a r e t o o n u m e r o u s a n d v a r i e d t o p e r m i t any a t t e m p t t o t r e a t them i n d i v i d u a l l y h e r e i n . H o w e v e r , a r e a d e r who i s f a m i l i a r w i t h
E q u a t i o n s ( 1 . 0 5 - 5 ) and ( 1 . 0 5 - 6 ) a r e introduced here simply hecause they a r e h e l p f u l f o r g i v i n g a f e e l f o r t h e S u p p o s e t h a t f o r some s y s t e m a p p l i c a t i o n i t i s d e s i r e d t o d e l a y p u l s e s o f S - b a n d e n e r g y 0 . 0 5 microseconds
a n d t h a t a n o p e r a t i n g b a n d w i d t h o f 5 0 Mc i s d e s i r e d t o a c c o m m o d a t e t h e s i g n a l s p e c t r u m a n d t o p e r m i t some v a r i a t i o n o f c a r r i e r f r e q u e n c y .
If
t h i s d e l a y were t o b e a c h i e v e d w i t h a n a i r - f i l l e d c o a x i a l l i n e , 49 f e e t o f l i n e would be r e q u i r e d . Equation (1.05-6) indicates t h a t t h i s delay c o u l d h e a c h i e v e d w i t h a f i v e - r e s o n a t o r f i l t e r h a v i n g 5 0 Mc b a n d w i d t h . An S - b a n d f i l t e r d e s i g n e d f o r t h i s p u r p o s e w o u l d t y p i c a l l y b e l e s s t h a n a f o o t i n l e n g t h a n d c o u l d b e made t o b e q u i t e l i g h t . I n slow-wave s t r u c t u r e s u s u a l l y phase
velocity
o r group v e l o c i t y
i s of i n t e r e s t , where 1 i s t h e l e n g t h of t h e s t r u c t u r e and t p and t d a r e a s d e f i n e d i n Eqs.
( 1 . 0 5 - 2 ) and ( 1 . 0 5 - 3 ) .
Not a l l s t r u c t u r e s used a s
s l o w - w a v e s t r u c t u r e s a r e f i l t e r s , h u t v e r y many o f them a r e .
Some examples of slow-wave s t r u c t u r e s which a r e b a s i c a l l y f i l t e r s t r u c t u r e s a r e waveguides p e r i o d i c a l l y loaded w i t h c a p a c i t i v e o r i n d u c t i v e i r i s e s , i n t e r d i g i t a l l i n e s , a n d comb l i n e s . The m e t h o d s o f t h i s b o o k s h o u l d b e q u i t e h e l p f u l i n t h e d e s i g n of s u c h slow-wave s t r u c t u r e s which a r e basically filters.
The u s e o f f i l t e r s a s t i m e d e l a y n e t w o r k s i s discussed f u r t h e r i n S e c s . 4 . 0 7 , 4 . 0 8 , 6 . 1 5 , 7 . 0 9 , a n d 11.11.
*
As f o r Eqa.
( 1 . 0 5 - 5 1 , t h i s e q u a t i o n w i l l u s u a l l y g i v e an o v e r e s t i m a t e o f t h e time d e l a y .
t h e p r i n c i p l e s t o b e t r e a t e d i n t h i s book w i l l u s u a l l v h a v e l i t t l e t r o u b l e i n a d a p t i n g t h e m f o r u s e i n t h e many s p e c i a l d e s i g n s i t u a t i o n s h e w i l l encounter.
CHAPTER 2
REFERENCES
SOME USEFUL CIRCUIT CONCEPTS AND EQUATIONS 1. H. W. Bode, Network A n a l y s i s and Feedback A m p l i f i e r D e s i g n , pp. 360-371 (D. Van Nostrand Co. New York, N.Y.,
1945).
2.
R. M. Fano, " ' I h e o r e t i c a l L i m i t a t i o n s on t h e Broadband Matching of A r b i t r a r y Impedances,"
3.
G. E. Valley and H. Wallman, Vacuum Tube A m p l i f i e r s , C h a p t e r s 2 and 4 (McGraw-Hill Book Co.,
J o u r n a l of t h e Frank1 i n I n s t i t u t e , Vol. 249, pp. 57-84 and 139-154 ( J a n u a r y - F e b r u a r y 1950). New York, N.Y.,
G. L. M a t t h a e i , " A Study o f t h e Optimum Design of Wide-Band P a r a m e t r i c A m p l i f i e r s and UpC o n v e r t e r s . " IRE Trans. PGMTT-9, pp. 23-38 ( J a n u a r y 1 9 6 1 ) . 5.
E. A. G u i l l e m i n , Communication Networks, Vol. 2 , pp. 99-106 and 490-498 (John Wiley and Sons, I n c . , New York, N.Y., 1 9 3 5 ) .
6.
7.
The p u r p o s e o f t h i s c h a p t e r i s t o s u m m a r i z e v a r i o u s c i r c u i t t h e o r y Tlrough
1948).
M. J . Di Toro, "Phase and Amplitude D i s t o r t i o n i n L i n e a r N e t w o r k s , " P r o c . I R E , Vol. 36.
concepts and e q u a t i o n s which a r e u s e f u l f o r t h e a n a l y s i s o f f i l t e r s .
much of t h i s m a t e r i a l w i l l b e f a m i l i a r t o many r e a d e r s , i t a p p e a r s In addition, there d e s i r a b l e t o g a t h e r i t t o g e t h e r f o r easy r e f e r e n c e .
w i l l u n d o u b t e d l y b e t o p i c s w i t h w h i c h some r e a d e r s w i l l h e u n f a m i l i a r .
pp. 24-36 ( J a n u a r y 1948).
In such c a s e s t h e d i s c u s s i o n g i v e n here should provide a b r i e f i n t r o -
E. S. Kuh and M. Fukada, "Optimum S y n t h e s i s of Wide-Band P a r a m e t r i c A m p l i f i e r s and C o n v e r t e r s , " IRE Trans. K f T - 8 , pp. 410-415 (December 1961).
d u c t i o n wh-ich s h o u l d b e a d e q u a t e f o r t h e p u r p o s e s o f t h i s b o o k .
S E C . 2 . 0 2 , COMPLEX FREQUENCY AND P O L E S A N D Z E R O S A " sinusoidal" voltage
may a l s o b e d e f i n e d i n t h e f o r m
where t i s t h e t i m e i n s e c o n d s , w i s f r e q u e n c y i n r a d i a n s p e r s e c o n d , a n d i s t h e c o m p l e x a m p l i t r l d e o f t h e v o l t a g e . The q u a n t i t y E n ,
Em =
I E ~ I ~ J ~
o f c o u r s e , i s r e l a t e d t o t h e r o o t - m e a n - s q ~ ~ a rveo l t a g e E by t h e r e l a t i o n E = E,/i% S i n u s o i d a l w a v e f o r m s a r e a s p e c i a l c a s e o f t h e more g e n e r a l w a v e f o r m e(t)
where El
=
=
I ~ ~ l e *c 'o s ( w t + p)
I E ~ i~ s a~ g aJi n ~ t h e
complex a m p l i t u d e .
In t h i s case
'1
-
i s t h e complex f r e q u e n c y .
o >w - -
In this
g e n e r a l c a s e t h e w a v e f o r m may b e a pure exponential function as i l l u s -
-4-0 _-dC
tA-3527-~i:
FIG.2.02-1(a) SHAPE OF COMPLEX-FREQUENCY WAVEFORM WHEN p = u + iO
t r a t e d i n F i g . 2 . 0 2 - l ( a ) , i t may be an e x p o n e n t i a l l y - v a r y i n g sinusoid
By f a c t o r i n g t h e n u m e r a t o r a n d d e r o m i n a t o r p o l y n o m i a l s t h i s may h e written a s
t r a t e d i n Fig. 2 . 0 2 - 1 ( b ) l o r i t may be a p u r e s i n u s o i d i f p = jw.
In linear, time-invariant c i r c u i t s such a s a r e d i s c u s s e d i n t h i s book c o m p l e x - f r e q u e n c y waveforms
A t the frequencies P = pl, p 3 , p g ,
...
e t c . , where t h e numerator polynomial
goes t o z e r o t h e impedance f u n c t i o n w i l l be z e r o ; t h e s e f r e q u e n c i e s a r e
h a v e f u n d a m e n t a l s i g n i f i c a n c e n o t s h a r e d by o t h e r t y p e s o f w a v e f o r m s .
t h u s known a s t h e z e r o s o f t h e f u n c t i o n .
T h e i r b a s i c i m p o r t a n c e i s e x e m p l i f i e d by t h e f o l l o w i n g p r o p e r t i e s o f
p6,
linear, time-invariant c i r c u i t s : (1)
(2)
I f a l ' s t e a d y - s t a t e " d r i v i n g v o l t a g e o r c u r r e n t o f complex frequency p i s a p p l i e d t o a c i r c u i t t h e s t e a d y - s t a t e r e s p o n s e s e e n a t any p o i n t i n t h e c i r c u i t * w i l l a l s o h a v e a c o m p l e x - f r e q u e n c y w a v e f o r m w i t h t h e same f r e q u e n c y p . The a m p l i t u d e a n d p h a s e a n g l e w i l l , i n g e n e r a l , b e different a t d i f f e r e n t points throughout the c i r c u i t . But a t any g i v e n p o i n t i n t h e c i r c u i t t h e r e s p o n s e a m p l i t u d e and t h e p h a s e a n g l e a r e h o t h l i n e a r f u n c t i o n s o f t h e d r i v i n g - s i g n a l a m p l i t u d e and p h a s e . The v a r i o u s p o s s i b l e n a t u r a l modes o f v i b r a t i o n o f t h e c i r c u i t w i l l have complex-frequency waveforms. (The n a t u r a l modes a r e c u r r e n t a n d v o l t a g e v i b r a t i o n s w h i c h can e x i s t a f t e r a l l d r i v i n g s i g n a l s a r e removed.)
The c o n c e p t s o f i m p e d a n c e and t r a n s f e r f u n c t i o n s r e s u l t f r o m t h e f i r s t p r o p e r t y l i s t e d a b o v e , s i n c e t h e s e two f u n c t i o n s r e p r e s e n t r a t i o s between t h e complex a m p l i t u d e s of t h e d r i v i n g s i g n a l and t h e r e s p o n s e .
At t h e f r e q u e n c i e s p = p 2 , p 4 ,
. . . , e t c . , where t h e denominator polynomial i s z e r o t h e impedance function w i l l be i n f i n i t e ; t h e s e f r e q u e n c i e s a r e known a s t h e p o l e s
of t h e f u n c t i o n .
The p o l e s a n d
zeros of a t r a n s f e r f u n c t i o n a r e
I
I .
defined i n a similar fashion. A c i r c u i t w i t h a f i n i t e numb e r of lumped, r e a c t i v e e l e m e n t s
w i l l h a v e a f i n i t e number o f p o l e s and z e r o s .
However, a c i r c u i t
distributed ( w h i c h may b e r e p r e s e n t e d a s a n
A /
I
0
FIG. 2.02-l(b) SHAPE OF COMPLEX-FREQUENCY JVAVEFORM WHEN P = rn + i~
i n f i n i t e number o f i n f i n i t e s i m a l
ANDu < 0
lumped-elements) w i l l have a n i n f i n i t e number o f p o l e s a n d z e r o s . Thus, c i r c u i t s i n v o l v i n g t r a n s m i s s i o n l i n e s w i l l have impedance f u n c t i o n s
As a r e s u l t o f P r o p e r t y ( 2 ) , t h e t r a n s i e n t r e s p o n s e o f a n e t w o r k w i l l c o n t a i n a s u p e r p o s i t i o n o f t h e c o m p l e x - f r e q u e n c y waveforms o f t h e v a r i o u s
t h a t a r e t r a n s c e n d e n t a l , i . e . , when e x p r e s s e d i n t h e form i n Eq.
n a t u r a l modes o f v i b r a t i o n o f t h e c i r c u i t .
t o a l o s s l e s s , s h o r t - c i r c u i t e d t r a n s m i s s i o n l i n e w h i c h i s o n e - q u a r t e r wave-
The i m p e d a n c e o f a c i r c u i t a s a f u n c t i o n o f c o m p l e x f r e q u e n c y p w i l l t a k e t h e form
8
Unless s t a t e d otherwise,
a
l i n e a r , time-invariant c i r c u i t r i l l be u n d e r s t o o d .
they w i l l be i n f i n i t e - p r o d u c t e x p a n s i o n s .
(2.02-7)
For example, t h e i n p u t impedance
l e n g t h l o n g a t f r e q u e n c y w o may b e e x p r e s s e d a s
w h e r e Zo i s t h e c h a r a c t e r i s t i c i m p e d a n c e o f t h e l i n e . seen t o have p o l e s a t p = + j ( 2 k
-
This c i r c u i t i s
l ) w o and z e r o s a t p = 0 and +j2kwo,
w h e r e k = 1, 2 , 3 , . . . , m. R e g a r d i n g f r e q u e n c y a s t h e more g e n e r a l p = u + jw v a r i a b l e i n s t e a d o f t h e u n n e c e s s a r i l y r e s t r i c t i v e j w v a r i a h l e p e r m i t s a much b r o a d e r p o i n t of view i n c i r c u i t a n a l y s i s and d e s i g n .
Impedance and t r a n s f e r f u n c t i o n s
become f u n c t i o n s o f a c o m p l e x v a r i a h l e ( i . e . , t h e v a r i a b l e p = cr
+
t h e y become f u n c t i o n s o f
jw) and a l l of t h e powerful t o o l s i n t h e mathematical
t h e o r y o f f u n c t i o n s o f a c o m p l e x v a r i a b l e become a v a i l a b l e .
I t becomes
very h e l p f u l t o d e f i n e t h e p r o p e r t i e s of an impedance o r t r a n s f e r f u n c t i o n i n t e r m s o f t h e l o c a t i o n s o f t h e i r p o l e s a n d z e r o s , and t h e s e p o l e s and zeros a r e o f t e n p l o t t e d i n t h e complex-frequency plane o r p-plane. p o l e s a r e i n d i c a t e d by c r o s s e s a n d t h e z e r o s hy c i r c l e s .
The
Figure 2.02-2(a)
shows s u c h a p l o t f o r t h e l o s s l e s s t r a n s m i s s i o n l i n e i n p u t impedance i n Eq. ( 2 . 0 2 - 8 ) w h i l e F i g . 2 . 0 2 - 2 ( b ) shows a s e t c h of t h e s h a p e of t h e magnitude o f t h i s f u n c t i o n f o r p = jw.
'The f i g u r e a l s o s h o w s w h a t h a p p e n s
t o t h e p o l e s and z e r o s i f t h e l i n e h a s l o s s moved
to
t h e l e f t o f t h e j w a x i s , and t h e
t h e poles and zeros a r e a l l Z(lw)
1
c h a r a c t e r i s t i c becomes
rounded o f f . The c o n c e p t s o f c o m p l e x f r e q u e n c y a n d p o l e s a n d z e r o s a r e v e r y h e l p f u l i n network a n a l y s i s and d e s i g n .
FIG. 2.02-2 T H E LOCATIONS OF T H E POLES AND ZEROS O F A SHORT-CIRCUITED TRANSMISSION L I N E WHICH IS A QUARTER-WAVELENGTH LONG WHEN P =
D i s c u s s i o n s from t h i s p o i n t of view w i l l
be found i n numerous books on network a n a l y s i s and s y n t h e s i s , i n c l u d i n g t h o s e l i s t e d i n References 1 t o 5.
iu0
The Magnitude of the Input Impedance for Frequencies p = jw is also Sketched
P o l e s and z e r o s a l s o have an e l e c t r o -
s t a t i c a n a l o g y which r e l a t e s t h e magnitude and phase of an impedance o r t r a n s f e r f u n c t i o n t o t h e p o t e n t i a l and f l u x , r e s p e c t i v e l y , o f an a n a l o g o u s e l e c t r o s t a t i c problem.
T h i s analogy i s u s e f u l both a s a t o o l f o r mathe-
word v i b r a t i o n i s u s e d t o i n c l u d e n a t u r a l modes h a v i n g e x p o n e n t i a l w a v e -
m a t i c a l r e a s o n i n g a n d a s a means f o r d e t e r m i n i n g m a g n i t u d e a n d p h a s e by
forms o f f r e q u e n c y p = u a s w e l l a s o s c i l l a t o r y w w e f o r m s o f f r e q u e n c y
m e a s u r e m e n t s on a n a n a l o g s e t u p .
P = o
R e f e r e n c e s 2 , 3 , 6 , and 7 .
Some o f t h e s e m a t t e r s a r e d i s c u s s e d i n
F u r t h e r u s e o f t h e c o n c e p t s o f complex f r e -
q u e n c y a n d p o l e s a n d z e r o s w i l l b e d i s c u s s e d i n t h e n e x t two s e c t i o n s .
+ jw.
S u p p o s e t h a t t h e i n p u t i m p e d a n c e o f a c i r c u i t i s g i v e n by t h e f u n c t i o n
S E C . 2 . 0 3 , NATURAL MODES OF VIdRATION AN0 THEIR HELATION TO INPUT-IMPEDANCE POLES AND ZEROS T h e n a t u r a l modes o f v i b r a t i o n o f a c i r c u i t a r e c o m p l e x f r e q u e n c i e s a t which t h e v o l t a g e s and c u r r e n t s i n t h e c i r c u i t c a n " v i b r a t e " c i r c u i t is disturbed.
i f the
'These v i b r a t i o n s c a n c o n t i n u e e v e n a f t e r a l l
d r i v i n g s i g n a l s have been s e t t o z e r o .
I t s h o u l d be n o t e d t h a t h e r e t h e
Hearranging Eq.
(2.03-l),
I f t h e i n p u t t e r m i n a l s o f t h i s c i r c u i t a r e o p e n - c i r c u i t e d and t h e c i r c u i t i s v i b r a t i n g a t one of i t s n a t u r a l f r e q u e n c i e s , t h e r e w i l l be a complexBy E q .
frequency v o l t a g e a c r o s s Z ( p ) even though I = 0 .
(2.03-2) it i s
s e e n t h a t t h e o n l y way i n w h i c h t h e v o l t a g e E c a n b e n o n - z e r o w h i l e I = 0 i s f o r Z ( p ) t o be i n f i n i t e .
Thus, i f Z ( p ) i s o p e n - c i r c u i t e d ,
v i b r a t i o n can be o b s e r v e d o n l y a t t h e f r e q u e n c i e s p 2 , p,, a r e t h e p o l e s of t h e i n p u t impedance f u n c t i o n Z ( p ) .
p,,
natural e t c . , which
A l s o , by a n a l o g o u s
reasoning it i s seen t h a t i f Z(p) i s s h o r t - c i r c u i t e d , t h e n a t u r a l f r e q u e n c i e s o f v i b r a t i o n w i l l be t h e f r e q u e n c i e s of t h e z e r o s o f Z ( p ) . l i x c e p t f o r s p e c i a l c a s e s w h e r e o n e o r more n a t u r a l modes may b e s t i f l e d a t c e r t a i n p o i n t s i n a c i r c u i t , i f a n y n a t u r a l modes a r e e x c i t e d i n any p a r t of t h e c i r c u i t , t h e y w i l l be o b s e r v e d i n t h e v o l t a g e s and The f r e q u e n c y p n = on + lo, o f
c u r r e n t s throughout t h e e n t i r e c i r c u i t .
e a c h n a t u r a l mode m u s t l i e i n t h e l e f t h a l f o f t h e c o m p l e x - f r e q u e n c y of e x p o n e n t i a l l y i n c r e a s i n g magnitude and e n e r g y , a c o n d i t i o n which i s S i n c e under o p e n - c i r c u i t or s h o r t -
impossible i n a passive c i r c u i t .
c i r c u i t c o n d i t i o n s t h e p o l e s o r t h e z e r o s , r e s p e c t i v e l y , of an impedance f u n c t i o n a r e n a t u r a l f r e q u e n c i e s of v i b r a t i o n , any impedance of a l i n e a r , p a s s i v e c i r c u i t must have a l l of i t s p o l e s and z e r o s i n t h e l e f t h a l f p l a n e o r o n t h e jo a x i s . FUNDAMENTAL PHOPLHi'Ib5
OF 'Tl
L e t u s d e f i n e t h e v o l t a g e a t t e n u a t i o n f u n c t i o n E /EL f o r t h e network i n Fig. 2. 04-1 as T(p)
=
g
L
C(P =
(P
-
P J (P
-
p 3 ) (P
-
PJ
p 2 ) (P - P , ) (P - p , )
... "
.
p a s s i v e c i r c u i t s i n terms of t h i s t r a n s f e r f u n c t i o n and F i g . 2 . 0 4 - 1 . The z e r o s o f T ( p ) , i . e . , p l , p 3 , p s , , . . . a r e a l l f r e q u e n c i e s o f n a t u r a l modes o f v i b r a t i o n f o r t h e c i r c u i t . T h e y a r e i n f l u e n c e d by a l l o f t h e e l e m e n t s i n t h e c i r c u i t s o t h a t , f o r example, i f t h e v a l u e f o r R o r HL were changed, g e n e r a l l y t h e f r e q u e n c i e s of a l l g t h e n a t u r a l modes w i l l c h a n g e a l s o . The p o l e s o f T ( p ) , i . e . , p 2 , p 4 , p 6 , . . . , a l o n g w i t h any p o l e s of T ( p ) a t p = 0 and p = are a l l frequencies of i n f i n i t e a t t e n u a t i o n , o r " p o l e s o f a t t e n u a t i o n . " 'They a r e p r o p e r t i e s o f t h e n e t w o r k a l o n e a n d w i l l n o t b e c h a n g e d i f R o r HL i s c h a n g e d . E x c e p t f o r c e r t a i n d e g e n e r a t e case!, i f two n e t w o r k s a r e c o n n e c t e d i n c a s c a d e , t h e r e s u l t a n t o v e r - a l l r e s p o n s e w i l l have t h e p o l e s o f a t t e n u a t i o n o f b o t h component n e t w o r k s . I n a l a d d e r network, a p o l e of a t t e n u a t i o n i s c r e a t e d when a s e r i e s b r a n c h h a s i n f i n i t e i m p e d a n c e , o r when a s h u n t branch h a s z e r o impedance. I f a t a given f r e quency, i n f i n i t e impedance occurs i n s e r i e s b r a n c h e s s i m u l t a n e o u s l y w i t h z e r o impedance i n s h u n t b r a n c h e s , a h i g h e r - o r d e r p o l e of a t t e n u a t i o n w i l l r e s u l t . I n c i r c u i t s w h e r e t h e r e a r e two o r more t r a n s m i s s i o n c h a n n e l s i n p a r a l l e l , p o l e s of a t t e n u a t i o n a r e c r e a t e d a t f r e q u e n c i e s where t h e o u t p u t s from t h e p a r a l l e l c h a n n e l s have t h e p r o p e r magnitude and p h a s e t o c a n c e l each other out. This can happen, f o r example, i n b r i d g e d - T , l a t t i c e , and p a r a l l e l - l a d d e r s t r u c t u r e s . The n a t u r a l modes [ z e r o s o f T ( p ) l m u s t l i e i n t h e l e f t h a l f o f t h e p - p l a n e ( o r on t h e j a a x i s i f t h e r e a r e no l o s s e l e m e n t s ) . The p o l e s o f a t t e n u a t i o n c a n o c c u r a n y w h e r e i n t h e p-plane.
LINEAR, TIME-INVARIANT NETWORK
FIG. 2.04-1
(2.04-1)
NETWORK DISCUSSED IN SECTION 2.04
4'e
s h a l l now b r i e f l y s u m m a r i z e some i m p o r t a n t g e n e r a l p r o p e r t i e s o f l i n e a r ,
I f t h i s were n o t s o t h e v i b r a t i o n s would be
p l a n e , o r on t h e j o a x i s .
SEC. 2 . 0 4 ,
w h e r e c i s a r e a l c o n s t a n t and p i s t h e complex-frequency v a r i a b l e .
I f Eg is a z e r o impedance v o l t a g e g e n e r a t o r , t h e z e r o s o f Z i ni n F i g . 2 . 0 4 - 1 w i l l t e t h e n a t u r a l f r e q u e n c i e s o f v i b r a t i o n o f t h e c i r c u i t . These z e r o s must t h e r e f o r e correspond t o t h e zeros of t h e a t t e n u a t i o n f u n c t i o n T(p). (Occasionally t h i s f a c t i s ohscured because i n some s p e c i a l c a s e s c a n c e l l a t i o n s c a n b e c a r r i e d o u t b e tween c o i n c i d e n t p o l e s and z e r o s o f T ( p ) o r o f Zin A s s u m i n g t h a t no s u c h c a n c e l l a t i o n s h a v e b e e n c a r r i e d o u t e v e n when t h e y a r e p o s s i b l e , t h e a b o v e s t a t e m e n t always holds. )
(8)
I f t h e z e r o impedance v o l t a g e g e n e r a t o r E were r e p l a c e d by a n i n f i n i t e i m p e d a n c e c u r r e n t g e n e r a t o : I g , t h e n t h e n a t u r a l f r e q u e n c i e s of v i b r a t i o n would c o r r e s p o n d t o t h e p o l e s of 2 . . Redefining T(p) a s T 1 ( p ) = I g / E , , t h e z e r o s o f ~ ' ( p )w o u l d i n t h i s c a s e s t i l l be t h e n a t u r a l f r e q u e n c i e s o f v i b r a t i o n b u t t h e y would i n t h i s c a s e be t h e same a s t h e p o l e s o f Z i n .
modes o f v i b r a t i o n , a n d , h e n c e t h e r e a r e a n i n f i n i t e a r r a y o f z e r o s t o T ( p ) I n a l l i m p e d a n c e a n d t r a n s f e r f u n c t i o n s t h e number o f p o l e s a n d In t h i s case there z e r o s must be e q u a l i f t h e p o i n t a t p = a is i n c l u d e d . a r e n o p o l e s o f a t t e n u a t i o n on t h e f i n i t e p l a n e ; t h e y a r e a l l c l u s t e r e d As a r e s u l t o f t h e p e r i o d i c a r r a y of z e r o s , I T ( j w ) ( h a s a n a t infinity. oscillatory behavior
L e t u s now c o n s i d e r some e x a m p l e s o f how some o f t h e c o n c e p t s i n t h e s t a t e m e n t s i t e m i z e d a b o v e may b e a p p l i e d .
S u p p o s e t h a t t h e box i n
F i g . 2 . 0 4 - 1 c o n t a i n s a l o s s l e s s t r a n s m i s s i o n l i n e which i s o n e - q u a r t e r wavelength long a t t h e frequency wo.
Let us suppose f u r t h e r t h a t
R g = R L # Z O , where Zo i s t h e c h a r a c t e r i s t i c impedance of t h e l i n e . Under t h e s e c o n d i t i o n s t h e v o l t a g e a t t e n u a t i o n f u n c t i o n T ( p ) would have a,p-plane plot a s indicated i n Fig. 2.04-2(a).
Rg
=
US
w a s indicated in Fig.
2.04-2(b).
As t h e v a l u e o f
R L i s made t o a p p r o a c h t h a t o f Z o , t h e z e r o s o f T ( p ) w i l l move t o t h e
l e f t , t h e p o l e s w i l l s t a y f i x e d a t i n f i n i t y , and t h e v a r i a t i o n s i n I T ( j w )
w i l l become s m a l l e r i n a m p l i t u d e .
1
When R g = RL = Z O , t h e z e r o s w i l l h a v e
moved t o w a r d t h e l e f t t o m i n u s i n f i n i t y , a n d t h e t r a n s f e r f u n c t i o n becomes s i m p l y
Since the transmission
l i n e i s a d i s t r i b u t e d c i r c u i t t h e r e a r e a n i n f i n i t e number o f n a t u r a l which h a s I T ( j w )
1
e q u a l t o two f o r a l l p = jw.
From t h e p r e c e d i n g e x a m p l e i t i s s e e n t h a t t h e t r a n s c e n d e n t a l f u n c t i o n ep h a s an i n f i n i t e number o f p o l e s a n d z e r o s w h i c h a r e a l l c l u s t e r e d a t infinity.
The p o l e s a r e c l u s t e r e d c l o s e s t t o t h e p = +v a x i s s o t h a t i f
we a p p r o a c h i n f i n i t y i n t h a t d i r e c t i o n e P becomes i n f i n i t e . i n f i n i t y v i a t h e p = -u a x i s eP g o e s t o z e r o . AT INFINITY
ALL p""
O
t
I f we a p p r o a c h
On t h e o t h e r h a n d , i f we
a p p r o a c h i n f i n i t y a l o n g t h e p = j w a x i s , eP w i l l a l w a y s h a v e u n i t m a g n i t u d e but i t s phase w i l l v a r y . lZWO
T h i s u n i t magnitude r e s u l t s from t h e f a c t t h a t
t h e a m p l i t u d e e f f e c t s of t h e p o l e s and z e r o s c o u n t e r b a l a n c e each o t h e r along t h e p = j w a x i s .
The i n f i n i t e c l u s t e r o f p o l e s a n d z e r o s a t i n f i n i t y
forms w h a t i s c a l l e d a n e s s e n t i a l s i n g u l a r i t y . F i g u r e 2 . 0 4 - 3 shows a b a n d - p a s s f i l t e r u s i n g t h r e e t r a n s m i s s i o n - l i n e r e s o n a t o r s which a r e a q u a r t e r - w a v e l e n g t h l o n g a t t h e frequency w o , and F i g . 2 . 0 4 - 4 ( a ) shows a t y p i c a l t r a n s f e r f u n c t i o n f o r t h i s f i l t e r .
In the
example i n F i g . 2 . 0 4 - 4 , t h e r e s p o n s e i s p e r i o d i c and h a s an i n f i n i t e number o f p o l e s a n d z e r o s .
T h e n a t u r a l modes o f v i b r a t i o n [ i . e . ,
zeros
of T(p)l a r e c l u s t e r e d n e a r t h e j w a x i s n e a r t h e f r e q u e n c i e s ] a o , j 3 w o , 1 5 a 0 , e t c . , f o r w h i c h t h e l i n e s a r e a n odd number o f q u a r t e r w a v e l e n g t h s
FIG. 2.04-2 TRANSFER FUNCTION OF THE CIRCUIT IN FIG. 2.04-1 I F THE BOX CONTAINS A LOSSLESS TRANSMISSION LINE k/4 LONG AT w,, WITH A CHARACTERISTIC IMPEDANCE Zo k Rg = RL
long.
At p = 0 , a n d t h e f r e q u e n c i e s p = j 2 a o , j 4 w o , j 6 w o , e t c . , f o r w h i c h
t h e l i n e s a r e a n e v e n number o f q u a r t e r - w a v e l e n g t h s l o n g , t h e c i r c u i t f u n c t i o n s l i k e a s h o r t - c i r c u i t , f o l l o w e d by a n o p e n - c i r c u i t , a n d t h e n another s h o r t - c i r c u i t .
I n accordance with P r o p e r t y ( 3 ) above, t h i s
c r e a t e s t h i r d - o r d e r p o l e s of a t t e n u a t i o n a s i n d i c a t e d i n F i g . 2 . 0 4 - 4 ( a ) .
The a p p r o x i m a t e s h a p e o f
IT(JU)
1
is indicated in Fig. 2.04-4(b).
I f the termination values R
RL
were changed,
and
t h e p o s i t i o n s of
t h e n a t u r a l modes
[ z e r o s of T ( p ) ]
T(P) =
Eg
EL
would s h i f t and t h e s h a p e o f t h e p a s s bands would be a l t e r e d . However, t h e p o s i t i o n s of t h e
p-
PLANE
p o l e s of a t t e n u a t i o n would be unaffected
[see Property ( 2 ) 1 .
'The c i r c u i t i n F i g . 2 . 0 4 - 3
FIG. 2.04-3
A THREE-RESONATOR, BAND-PASS FILTER USING RESONATORS CONSISTING OF AN OPEN-CIRCUITED STUB IN SERIES AND TWO SHORTClRCUlTED STUBS IN SHUNT
is n o t very p r a c t i c a l because the open-circuited series stub i n t h e middle i s d i f f i c u l t t o construct in a shielded structure. 'The f i l t e r s t r u c t u r e shown i n F i g . 2 . 0 4 - 5 i s much more common and easy t o b u i l d .
c i r c u i t e d shunt s t u b s with conn e c t i n g l i n e s , t h e s t u b s and l i n e s a l l h e i n g o n e - q u a r t e r wavelength long a t frequency wo.
ate.
,I
It uses short-
T h i s c i r c u i t h a s t h e same number o f n a t u r a l modes a s
FIG. 2.04-4
I
VOLTAGE ATTENUATION FUNCTION PROPERTIES FOR THE FILTER IN FIG. 2.04-3 The Stubs are One-Quarter Wavelength Long at Frequency wO
does t h e c i r c u i t i n F i g . 2 . 0 4 - 3 , and can g i v e s i m i l a r p a s s - b a n d r e s p o n s e s f o r f r e q u e n c i e s i n t h e v i c i n i t y o f p = J u g , ] ~ G J e~ t ,c .
However, a t
p = 0 , j2wo, j4wo, e t c . , t h e c i r c u i t o p e r a t e s l i k e t h r e e s h o r t c i r c u i t s i n p a r a l l e l (which a r e e q u i v a l e n t t o one s h o r t - c i r c u i t ) , and a s a r e s u l t t h e p o l e s of a t t e n u a t i o n a t t h e s e f r e q u e n c i e s a r e f i r s t - o r d e r p o l e s o n l y .
I t c a n t h u s be s e e n t h a t t h i s f i l t e r w i l l n o t have a s f a s t a r a t e of c u t o f f a s w i l l t h e f i l t e r i n F i g . 2 . 0 4 - 3 w h o s e p o l e s on t h e J U a x i s a r e third-order poles.
The c o n n e c t i n g l i n e s b e t w e e n s h u n t s t u b s i n t r o d u c e
poles of attenuation a l s o , but as for the case i n Fig. 2.04-2,
the poles
t h e y i n t r o d u c e a r e a l l a t i n f i n i t y w h e r e t h e y do l i t t l e g o o d a s f a r a s c r e a t i n g a f a s t r a t e o f c u t o f f is c o n c e r n e d s i n c e t h e r e a r e an e q u a l number o f z e r o s ( i . e . , n a t u r a l m o d e s ) w h i c h a r e much c l o s e r , h e n c e more influential. T h e s e e x a m p l e s g i v e b r i e f i l l u s t r a t i o n s o f how t h e n a t u r a l modes and f r e q u e n c i e s of i n f i n i t e a t t e n u a t i o n o c c u r i n f i l t e r s which i n v o l v e t r a n s mission-line elements.
Heasoning from t h e v i e w p o i n t s d i s c u s s e d above can
o f t e n be v e r y h e l p f u l i n d e d u c i n g what t h e b e h a v i o r o f a g i v e n f i l t e r structure w i l l be.
24
1-3527-7
FIG. 2.04-5
A BAND-PASS FILTER CIRCUIT USING SHORTClRCUlTED STUBS WITH CONNECTING LINES A L L OF WHICH ARE A QUARTER-WAVELENGTH LONG AT THE MIDBAND FREQUENCY wo
SEC.
2 . 0 5 , GENERAL C I R C U I T PAHAME'TEHS
repeated application of t h i s operation the general c i r c u i t parameters
I n t e r m s o f F i g . 2 . 0 5 - 1 , t h e g e n e r a l c i r c u ~ tp a r a m e t e r s a r e d e f i n e d
can be c o m p u t e d f o r a n y number o f t w o - p o r t n e t w o r k s i n c a s c a d e . F i g u r e 2 . 0 5 - 3 g i v e s t h e g e n e r a l c i r c u i t p a r a m e t e r s f o r a number o f common
by t h e e q u a t i o n s
structures. Under c e r t a i n c o n d i t i o n s t h e g e n e r a l c i r c u i t p a r a m e t e r s a r e i n t e r r e l a t e d i n t h e f o l l o w i n g s p e c i a l ways:
AD-BC
I f t h e network i s r e c i p r o c a l =
1
.
(2.05-4)
or i n matrix notation I f t h e network i s symmetrical
These parameters a r e p a r t i c u l a r l y u s e f u l i n r e l a t i n g t h e performance of
If t h e n e t w o r k i s l o s s l e s s ( i . e . , w i t h o u t d i s s i p a t i v e e l e m e n t s ) , t h e n f o r f r e q u e n c i e s p = j w , A a n d D w i l l be p u r e l y r e a l w h i l e B a n d C w i l
c a s c a d e d n e t w o r k s t o t h e i r p e r f o r m a n c e when o p e r a t e d i n d i v i d u a l l y .
be p u r e l y i m a g i n a r y .
The
g e n e r a l c i r c u i t p a r a m e t e r s f o r t h e two c a s c a d e d n e t w o r k s i n F i g . 2 . 0 5 - 2 a r e g i v e n by
I f t h e network i n F i g . 2 . 0 5 - 1 is turned around, then t h e s q u a r e matrix i n Eq.
[::] I;:I:] [:::I]
(2.05-2) i s
=
where t h e p a r a m e t e r s w i t h t s u b s c r i p t s a r e f o r t h e n e t w o r k when t u r n e d . around, and t h e p a r a m e t e r s w i t h o u t s u b s c r i p t s a r e f o r t h e network w i t h i t s original orientation.
I n b o t h c a s e s , El and I 1 a r e a t t h e t e r m i n a l s
a t t h e l e f t and E 2 and I , a r e a t t h e t e r m i n a l s a t t h e r i g h t .
Uy u s e o f E q s . ( 2 . 0 5 - 6 1 , ( 2 . 0 5 - 3 ) , and ( 2 . 0 5 - 4 ) , i f t h e p a r a m e t e r s A ' , B ' , C ' , 3' a r e f o r t h e l e f t h a l f o f a r e c i p r o c a l s y m m e t r i c a l n e t w o r k , the general c i r c u i t parameters f o r t h e e n t i r e network a r e
FIG. 2.05-1
DEFINITION OF CURRENTS AND VOLTAGES FOR TWO-PORT NETWORKS
k FIG. 2.05-2
4
1-3527-10
CASCADED TWO-PORT NETWORKS
S E C , 2 0 6 , O P E N - C I H C U I T IMPEDANCES AND S H O H T - C I H C U I T ADMITTANCES I n t e r m s o f F i g . 2 . 0 5 - 1 , t h e o p e n - c i r c u i t i m p e d a n c e s of a t w o - p o r t n e t w o r k may b e d e f i n e d by t h e e q u a t i o n s
P h y s i c a l l y , z l l i s t h e i n p u t i m p e d a n c e a t End 1 when End 2 i s o p e n c i r c u i t e d 'The q u a n t i t y z 1 2 c o u l d b e m e a s u r e d a s t h e r a t i o of t h e v o l t a g e B l i 1 2 when End 1 i s o p e n - c i r c u i t e d a n d c u r r e n t I 2 i s f l o w i n g i n End 2 .
The p a r a m e t e r s z z l a n d
Zz2
may h e i n t e r p r e t e d a n a l o g o u s l y .
I n a s i m i l a r f a s h i o n , t h e s h o r t - c i r c u i t a d m i t t a n c e s may b e d e f i n e d in terms o f F i g . 2 . 0 5 - 1 and t h e e q u a t i o n s
I n t h i s c a s e y l l i s t h e a d m i t t a n c e a t End 1 when End 2 i s s h o r t - c i r c u i t e d . when End 1 i s
'lhe p a r a m e t e r y 1 2 c o u l d b e c o m p u t e d a s t h e r a t i o I l : E 2 s h o r t - c i r c u i t e d a n d a v o l t a g e E2 i s a p p l i e d a t End 2 .
F i g u r e 2 . 0 6 - 1 shows t h e o p e n - c i r c u i t impedances and s h o r t - c i r c u i t a d m i t t a n c e s f o r a number o f common s t r u c t u r e s . A = cosh
T l , 0 = Zo sinh Yt L
z1 2 - z Z 1 and y I P = y Z 1 .
For r e c i p r o c a l networks
F o r a l o s s l e s s n e t w o r k ( i . e . , one c o m p o s e d o f
r e a c t a n c e s ) , t h e o p e n - c i r c u i t i m p e d a n c e s and t h e s h o r t - c i r c u i t a d m i t TRANSMISSION L I N E 0
tances a r e a l l purely imaginary f o r a l l p = ] w . 0
c =
sinh Y t i -
Zo
1
D = cosh Y t i
+
Yt ' a t jPt PROPAGATION CONSTANT, PER UNIT LENGTH Z o = CHARACTERISTIC IMPEDANCE
S E C . 2 . 0 7 , RELATIONS BETWEEN GENEHAL C I R C U I T PARAMETERS AND OPEN - AND SHOHT- CIHCUI'T PARAEdE'rERS 'The r e l a t i o n s h i p s b e t w e e n t h e g e n e r a l c i r c u i t p a r a m e t e r s , t h e o p e n c i r c u i t impedances, and t h e s h o r t - c i r c u i t a d m i t t a n c e s d e f i n e d i n
FIG. 2.05-3
GENERAL CIRCUIT PARAMETERS OF SOME COMMON STRUCTURES
Sets. 2 . 0 5 and 2 . 0 6 a r e a s f o l l o w s :
r,, = Zo coth y t j
, I,,=
z 0 sinh Yt{
TRANSMISSION L I N E
YII
=
(c)
coth q1
-
YI2
zo Y2,
=
-
Zo sinh Ytt coth Y t l
-I
Zoslnh Ytl
-I
9
Y22 '
zo
Yt = a , + 1Pt =PROPAGATION CONSTANT, PER UNIT L E N G T H Zo' CHARACTERISTIC IMPEDANCE 8-3527-12
FIG. 2.06-1 OPEN-CIRCUIT IMPEDANCES AND SHORT-CIRCUIT ADMITTANCES OF SOME COMMON STRUCTURES
where
t r a n s m i s s i o n t o End 2 from End 1.
I f t h e network i s r e c i p r o c a l , m L 2 = m Z 1 .
These p o l y n o m i a l s f o r a g i v e n c i r c u i t a r e i n t e r r e l a t e d by t h e e x p r e s s i o n s
1 ( f o r reciprocal networks)
=
and, a s i s d i s c u s s e d i n R e f . 8 , t h e y can y i e l d c e r t a i n l a b o r - s a v i n g a d v a n t a g e s when t h e y t h e m s e l v e s a r e u s e d a s b a s i c p a r a m e t e r s t o d e s c r i b e the performance of a c i r c u i t . As i s i n d i c a t e d i n E q s . ( 2 . 0 7 - 4 ) t o ( 2 . 0 7 - 6 ) , when t h e d e t e r m i n a n t s
A,
Az
and
Ay
a r e formed a s a f u n c t i o n of p , t h e r e s u l t i n g r a t i o n a l
function w i l l necessarily contain cancelling polynomials.
This f a c t can
be v e r i f i e d by u s e o f E q s . ( 2 . 0 7 - 1 ) t o ( 2 . 0 7 - 3 ) a l o n g w i t h ( 2 . 0 7 - 8 ) . Removal o f t h e c a n c e l l i n g p o l y n o m i a l s w i l l u s u a l l y c u t t h e d e g r e e o f t h e I f any of t h e s e v a r i o u s c i r c u i t p a r a m e t e r s a r e e x p r e s s e d a s a f u n c t i o n
polynomials i n t h e s e f u n c t i o n s roughly i n h a l f .
of complex f r e q u e n c y p , t h e y w i l l c o n s i s t of t h e r a t i o of two p o l y n o m i a l s ,
e x i s t when t h e n e t w o r k c o n t a i n s d i s t r i b u t e d e l e m e n t s , a l t h o u g h t h e
Analogous p r o p e r t i e s
e a c h o f w h i c h may b e p u t i n t h e f o r m
p o l y n o m i a l s t h e n become o f i n f i n i t e d e g r e e ( s e e S e c . 2 . 0 2 ) a n d a r e m o s t c o n v e n i e n t l y r e p r e s e n t e d by t r a n s c e n d e n t a l f u n c t i o n s s u c h a s s i n h p a n d
polynomial
=
c (p - p l ) ( p - p 2 ) ( p - p 3 ) .
..
(2.07-7)
cosh p .
For example, f o r a l o s s l e s s transmission l i n e
where c i s a r e a l c o n s t a n t and t h e p k a r e t h e r o o t s of t h e p o l y n o m i a l . As s h o u l d b e e x p e c t e d f r o m t h e d i s c u s s i o n s i n S e c s . 2 . 0 2 t o 2 . 0 4 , t h e l o c a t i o n s of t h e r o o t s 6 f t h e s e polynomials have p h y s i c a l s i g n i f i c a n c e . ,
.
I h e q u a n t i t i e s on t h e r i g h t i n E q s . ( 2 . 0 7 - 1 ) t o ( 2 . 0 7 - 6 ) h a v e b e e n i n t r o -
duced t o c l a r i f y t h i s p h y s i c a l s i g n i f i c a n c e . The s y m h o l s
n s s , noo, nos,
and
nSo
i n t h e e x p r e s s i o n s above r e p r e s e n t
p o l y n o m i a l s o f t h e f o r m i n E q . ( 2 . 0 7 - 7 ) whose r o o t s a r e n a t u r a l f r e q u e n c i e s o f v i b r a t i o n o f t h e c i r c u i t u n d e r c o n d i t i o n s i n d i c a t e d hy t h e s u b s c r i p t s . Thus, t h e r o o t s of
nSS
a r e n a t u r a l f r e q u e n c i e s when b o t h p o r t s a r e o p e n - c i r c u i t e d . nos
The r o o t s
The s y m b o l s m 1 2 a n d m Z 1 r e p r e s e n t p o l y n o m i a l s whose r o o t s a r e p o l e s o f ( s e e S e c . 2 . 0 4 ) of t h e c i r c u i t , except f o r t h o s e p o l e s of has roots corresponding t o the
p o l e s o f a t t e n u a t i o n f o r t r a n s m i s s i o n t o End 1 from End 2 i n F i g . 2 . 0 5 - 1 , while t h e polynomial m z l
nP -
noo
w h i l e t h e r i g h t p o r t i s s h o r t - c i r c u i t e d , a n d t h e i n v e r s e o b t a i n s f o r n,,.
a t t e n u a t i o n a t p = a. The p o l y n o m i a l m 1 2
Z; s i n h
b o
a r e n a t u r a l f r e q u e n c i e s when t h e l e f t p o r t i s o p e n - c i r c u i t e d
attenuation
=
a r e t h e n a t u r a l f r e q u e n c i e s of t h e c i r c u i t i n
F i g . 2 . 0 5 - 1 when b o t h p o r t s a r e s h o r t - c i r c u i t e d , w h i l e t h e r o o t s o f of
nsS
h a s r o o t s which a r e p o l e s of a t t e n u a t i o n f o r
where Z0 i s t h e c h a r a c t e r i s t i c i m p e d a n c e of t h e l i n e , a n d w o i s t h e r a d i a n frequency f o r which t h e l i n e i s o n e - q u a r t e r wavelength long. m12 =
In t h i s case,
m Z 1 i s a c o n s t a n t s i n c e a l l of t h e p o l e s of a t t e n u a t i o n a r e a t i n -
f i n i t y ( s e e S e c . 2 . 0 4 and H e f . 8 ) .
The c h o i c e o f c o n s t a n t m u l t i p l i e r s f o r
t h e s e " p o l y n o m i a l s " i s a r b i t r a r y t o a c e r t a i n e x t e n t i n t h a t any o n e m u l t i p l i e r may be c h o s e n a r h i t r a r i l ~ ,h u t t h i s t h e n f i x e s what t h e o t h e r c o n s t a n t m u l t i p l i e r s m u s t be.'
SEC. 2 . 0 8 ,
INCIDENT AND REFLECTED WAVES, REFLECTION C O E F F I C I E N T S , AND ONE KIND OF TRANSMISSION C O E F F I C I E N T
Let u s suppose t h a t i t i s d e s i r e d t o analyze t h e transmission across the terminals 2-2'
i n F i g . 2 . 0 8 - 1 f r o m t h e wave p o i n t o f v i e w .
where I i i s t h e i n c i d e n t c u r r e n t amplitude, Ir i s t h e r e f l e c t e d c u r r e n t a m p l i t u d e , and I i s t h e t r a n s m i t t e d c u r r e n t a m p l i t u d e which i s a l s o t h e The i n c i d e n t c u r r e n t i s actual c u r r e n t passing through t h e terminals 2 - 2 ' .
By
definition
w h e r e E i i s t h e a m p l i t u d e o f t h e i n c i d e n t v o l t a g e wave e m e r g i n g f r o m
where I,, i s t h e c u r r e n t w h i c h w o u l d p a s s t h r o u g h t h e t e r m i n a l s 2 - 2 '
t h e n e t w o r k , E r i s t h e r e f l e c t e d v o l t a g e wave a m p l i t u d e , a n d E i s t h e
they were s h o r t - c i r c u i t e d t o g e t h e r .
t r a n s m i t t e d v o l t a g e wave a m p l i t u d e
if
The c u r r e n t r e f l e c t i o n c o e f f i c i e n t
i s then d e f i n e d a s
(which i s a l s o t h e v o l t a g e t h a t would be measured a c r o s s t h e t e r m i n a l s 2 - 2 ' ) . I f Z S = Z L , t h e r e w i l l be no r e f l e c t i o n * s o t h a t E r = 0 a n d E , = E.
NETWORK
I
2'
zs
(0)
Replacing
t h e n e t w o r k and g e n e r a t o r a t ( a ) i n F i g . 2 . 0 8 - 1 by a T h e v e n i n e q u i v a l e A t
where Y S = l / Z s a n d YL = l / Z L I n a d d i t i o n , sometimes t h e v o l t a g e t r a n s m i s s i o n c o e f f i c i e n t
g e n e r a t o r a s shown a t ( b ) , i t i s r e a d i l y s e e n t h a t s i n c e f o r ZS Ei = E , then
=
LL,
i s used.
The c o r r e s p o n d i n g c u r r e n t t r a n s m i s s i o n c o e f f i c i e n t i s
2' (b)
n-3527-,3
w h e r e E o c i s t h e v o l t a g e w h i c h would be measured a t t e r m i n a l s 2 - 2 '
FIG' 2'08-1 CIRCU'TS DlSCUSSEDIN SEC. 2.08 FROM THE WAVEANALYSIS VIEWPOINT
wee open-crcuted
i f thev
Using Eqs.
(2.08-11
and ( 2 . 0 8 - 2 ) t h e v o l t a g e r e f l e c t i o n coefficient i s defined a s
I t w i l l be n o t e d t h a t t h e s e t r a n s m i s s i o n c o e f f i c i e n t s
T
and
Tc
are not
t h e same a s t h e t r a n s m i s s i o n c o e f f i c i e n t t d i s c u s s e d i n S e c . 2 . 1 0 . S E C . 2 . 0 9 , CALCULATION OF THE INPUT IMPEDANCE OF A T E R M I N A T E D , rwo-POH'TNETWORK The i n p u t i m p e d a n c e ( Z i n ) l d e f i n e d i n F i g . 2 . 0 9 - 1 c a n b e c o m p u t e d from Z 2 a n d a n y o f t h e c i r c u i t p a r a m e t e r s u s e d t o d e s c r i b e t h e p e r f o r m -
An a n a l o g o u s t r e a t m e n t f o r c u r r e n t w a v e s p r o c e e d s a s f o l l o w s :
ance o f a t w o - p o r t n e t w o r k .
I n terms of t h e g e n e r a l c i r c u i t p a r a m e t e r s ,
the o p e n - c i r c u i t impedances, and t h e s h o r t - c i r c u i t a d m i t t a n c e s ,
* N o t e t h a t n o r e f l e c t i o n o f t h e v o l t a g e w a v e d o e a n o t n e c e s s a r i l y i m p l y maximum p o w e r t r a n s f e r . F o r n o r a f l e c t i o n o f t h e v o l t a g e w a v e Z, = ZL, w h i l e f o r mmximum p o w e r t r a n s f e r Z Z i where Zi i a t h e complex conjugate of Z
L'
0-fI FIG. 2.09-1
I
NETWORK
2
and
DEFINITION OF INPUT IMPEDANCES COMPUTED IN SEC. 2.09
T r a n s f e r f u n c t i o n s s u c h a s t h e E g , E 2 f u n c t i o n p r e s e n t e d above a r e comm.only u s e d b u t h a v e a c e r t a i n d i s a d v a n t a g e . t h a t , d e p e n d i n g on w h a t t h e r e l a t i v e s i z e o f
H1
T n i s d i s a d v a n t a g e is a n d Hz a r e , c o m p l e t e
e n e r g y t r a n s f e r may c o r r e s p o n d t o any o f a w i d e r a n g e o f
IE
/E2 1
values.
Such c o n f u s i o n i s e l i m i n a t e d i f t h e t r a n s f e r f u n c t i o n
i s used i n s t e a d . r e s p e c t i v e l y , w h e r e Y2 = 1 / Z 2 .
S i m i l a r l y , f o r t h e impedance ( Z i , ) 2
'The q u a n t i t y
in
Fig. 2.09-1
w i l l he r e f e r r e d t o h e r e i n a s t h e u u a z l a b l e v o l t a g e , w h i c h i s t h e v o l t a g e a c r o s s A2 when t h e e n t i r e a v a i l a b l e power o f t h e g e n e r a t o r i s a b s o r b e d by R2.
Thus, f o r complete energy t r a n s f e r
I
w h e r e Y1
NETWORK
1 ( E 2 ) a Y a i l / E 21
=
2
= l/Z1
SEC. 2 . 1 0 ,
CALCULATION OF VOLTAGE TRANSYEIl FUNCTIONS
The t r a n s f e r f u n c t i o n B g / E 2 f o r t h e c i r c u i t i n F i g . 2 . 1 0 - 1 c a n be c o m p u t e d i f any o f t h e s e t s o f c i r c u i t p a r a m e t e r s d i s c u s s e d i n S e c s . 2 . 0 5 t o 2 . 0 7 a r e known f o r t h e n e t w o r k i n t h e b o x .
The a p p r o p r i a t e e q u a t i o n s a r e
FIG. 2.10-1
A CIRCUIT DISCUSSED IN SEC. 2.10
1 regardless
w i l l he u s e d i n t h i s book i n s t e a d o f i n s e r t i o n l o s s . The f u n c t i o n P p v a l l / P 2 i s known a s a t r a n s d u c e r l o s s r a t t o , w h e r e P 2 i s a g a i n t h e power d e l i v e r e d
o f t h e r e l a t i v e s i z e s o f R2 a n d R 1 . N o t e t h a t ( E 2 ) a Y a i l h a s t h e same phase a s E g .
t o R2 i n F i g . 2 . 1 0 - 1 w h i l e
In the l i t e r a t u r e a transmission coefficient
t i s commonly u s e d w h e r e
Note t h a t t h i s i s n o t t h e same a s t h e t r a n s m i s s i o n c o e f f i c i e n t s discussed i n Sec. 2.08.
T
or
Tc
The t r a n s m i s s i o n c o e f f i c i e n t t i s t h e s a m e ,
h o w e v e r , a s t h e s c a t t e r i n g c o e f f i c i e n t s S12 = S Z 1 , d i s c u s s e d ' i n S e c . 2 . 1 2 . A l s o n o t e t h a t t i s an o u t p u t / i n p u t r a t i o o f a " v o l t a g e g a i n " r a t i o ,
i s t h e a v a i l a h l e power o f t h e g e n e r a t o r composed o f E a n d t h e i n t e r n a l r e s i s t a n c e R 1 . T h u s , f o r c o m p l e t e power t r a n s f e r PaVail/P2 = 1 r e g a r d l e s s of t h e r e l a t i v e s i z e o f R1 a n d R 2 . N o t e t h a t t i n E q . ( 2 . 1 1 - 2 ) i s t h e t r a n s m i s s i o n c o e f f i c i e n t d e f i n e d by E q . ( 2 . 1 0 - 6 ) .
I t w i l l o f t e n be d e s i r a b l e t o e x p r e s s P p V a i l / P 2 i n d b s o t h a t
w h i l e t h e f u n c t i o n i n E q . ( 2 . 1 0 - 4 ) i s an i n p u t / o u t p u t r a t i o o r a " v o l t a g e attenuation" ratio. H e r e i n , when a t t e n u a t i o n i s r e f e r r e d t o , t h e t r a n s d u c e r l o s s ( i . e . ,
S E C . 2 . 1 1 , CALCULATION OF POWER TRANSFER FUNCTIONS AND " ATTENUATION"
t r a n s d u c e r a t t e n u a t i o n ) i n d b a s d e f i n e d i n Eq. ( 2 . 1 1 - 4 ) w i l l be u n d e r -
One commonly u s e d t y p e o f power t r a n s f e r f u n c t i o n i s t h e i n s e r t i o n loss function
stood, unless otherwise s p e c i f i e d . I f we d e f i n e L i = 1 0 l o g l o P2,,/P2
a s i n s e r t i o n l o s s in db, then t h e
attenuation i n db i s
w h e r e R1, R 2 , E g , and E 2 a r e d e f i n e d i n F i g . 2 . 1 0 - 1 , a n d c o m p u t e d by u s e o f E q s . ( 2 . 1 0 - 1 ) t o ( 2 . 1 0 - 3 ) .
I E ~ / E ~I
The q u a n t i t y
c a n be
P2 is t h e
power a b s o r b e d by R2 when t h e n e t w o r k i n F i g . 2 . 1 0 - 1 i s i n p l a c e , w h i l e
P2, i s t h e power i n R2 when t h e n e t w o r k i s removed and R2 i s c o n n e c t e d d i r e c t l y t o R1 a n d E . I n s e r t i o n l o s s f u n c t i o n s h a v e t h e same d i s a d v a n t a g e a s t h e E g / E 2
Note t h a t i f R1 = R 2 , t h e n i n s e r t i o n l o s s a n d t r a n s d u c e r a t t e n u a t i o n a r e t h e same. I f t h e network i n Fig. 2.11-1 c o n t a i n s d i s -
f u n c t i o n d i s c u s s e d i n S e c . 2 . 1 0 , i . e . , c o m p l e t e power t r a n s f e r may c o r -
c a n n o t be n e g l e c t e d , t h e n
r e s p o n d t o a l m o s t a n y v a l u e o f P 2 0 , / P 2 , d e p e n d i n g on t h e r e l a t i v e s i z e s of
L A may h e c o m p u t e d hy u s e
R1 a n d R 2 .
F o r t h i s r e a s o n t h e power t r a n s f e r f u n c t i o n
LOSSLESS NETWORK SEPARATED INTO TWO PARTS
s i p a t i v e e l e m e n t s which
of Eqs. ( 2 . 1 1 - 4 ) ,
(2.11-2),
and a n y o f E q s . ( 2 . 1 0 - 1 ) to (2.10-3).
However, i f
t h e n e t w o r k i n t h e box may be r e g a r d e d a s l o s s l e s s ( 1 . e . )
w i t h o u t any
FIG. 2.11-1
A NETWORK DISCUSSED IN SEC. 2.11
d i s s i p a t i v e e l e m e n t s ) , t h e n some s i m p l i f i c a t i o n s c a n b e t a k e n a d v a n t a g e of For example, a s d i s c u s s e d i n S e c . 2 . 0 5 , f o r a d i s s i p a t i o n l e s s network A and D w i l l be p u r e l y r e a l w h i l e B and C w i l l be p u r e l y i m a g i n a r y f o r f r e quencies ]w.
Because o f t h i s , t h e form and HI and ( Z i n ) l a r e a s d e f i n e d i n F i g . 2 . 1 1 - 1 .
The r e f l e c t i o n c o e f -
f i c i e n t a t t h e o t h e r end i s
becomes c o n v e n i e n t f o r c o m p u t a t i o n .
This expression applies t o dissipa-
t i o n l e s s r e c i p r o c a l n e t w o r k s and a l s o t o n o n - r e c i p r o c a l d i s s i p a t i o n l e s s n e t w o r k s f o r t h e c a s e o f t r a n s m i s s i o n from L e f t t o r i g h t .
I f we f u r t h e r
and f o r a d i s s i p a t i o n l e s s n e t w o r k
R , t h a t t h e n e t w o r k i s r e c i p r o c a l ( i . e . , AD - BC = 1 1 , a n d t h a t t h e n e t w o r k i s s y m m e t r i c a l ( i . e . , A = D ) , t h e n Eq. ( 2 . 1 1 - 6 )
s p e c i f y t h a t Rl
=
R2
=
becomes s o t h a t t h e magnitude of e i t h e r r e f l e c t i o n c o e f f i c i e n t c o u l d be u s e d i n
[ I t s h o u l d be u n d e r s t o o d i n p a s s i n g t h a t t h e phase o f
Eq. ( 2 . 1 1 - 9 ) .
i s n o t n e c e s s a r i l y t h e same a s t h a t o f F u r t h e r m o r e , i t i s c o n v e n i e n t i n s u c h c a s e s t o compute t h e g e n e r a l c i r c u i t p a r a m e t e r s A ' , B ' , C ' , D' Eqs.
f o r t h e l e f t h a l f of t h e network o n l y .
r2 e v e n
t h o u g h Eq.
rl
(2.11-12) holds.]
The r e f l e c t i o n c o e f f i c i e n t
Then by
( 2 . 0 5 - 7 ) and ( 2 . 1 1 - 7 ) , t h e t r a n s d u c e r l o s s r a t i o f o r t h e o v e r - a l l
network is 'avail - = P2
1
A'U'
I fi
C 1 3 ' R)
(2.11-8)
1
between Z a a n d Z b i n F i g . 2 . 1 1 - 1 c a n n o t be u s e d i n Eq. ( 2 . 1 1 - 9 ) i f b o t h Z a and Z b a r e c o m p l e x .
However,
i t c a n be shown t h a t
I n t h e c a s e o f d i s s i p a t i o n l e s s n e t w o r k s s u c h a s t h a t shown i n F i g . 2 . 1 1 - 1 , t h e power t r a n s m i s s i o n i s e a s i l y c o m p u t e d f r o m t h e g e n e r a t o r p a r a m e t e r s and t h e i n p u t impedance of t h e d i s s i p a t i o n l e s s network t e r m i nated i n R Z . end up i n R 2 .
T h i s i s b e c a u s e any power a b s o r b e d by
must s u r e l y
The c o m p u t a t i o n s may b e c o n v e n i e n t l y made i n t e r m s o f t h e
voltage reflection coefficient
discussed i n Sec. 2.08.
I n t h e s e terms
where Z t i s t h e c o m p l e x c o n j u g a t e o f L b . T h u s , i f Z a = R a + j X a a n d Zb
=
Rb
+
j X b , by u s e o f E q s . ( 2 . 1 1 - 1 4 ) a n d ( 2 . 1 1 - 9 ) we o b t a i n
For c a s e s w h e r e network, Eq.
Za
=
Z b such a s o c c u r s a t t h e middle of a symmetrical
( 2 . 1 1 - 1 5 ) reduces t o
i
The p e r f o r m a n c e o f a n y l i n e a r t w o - p o r t n e t w o r k w i t h t e r m i n a t i o n s can be d e s c r i h e d i n t e r m s o f f o u r s c a t t e r i n g c o e f f i c i e n t s : S l l , S 1 2 , S Z 1 , and S 2 2 . With r e f e r e n c e t o t h e t w o - p o r t network i n F i g . 2 . 1 1 - 1 , A n o t h e r s i t u a t i o n w h i c h commonly o c c u r s i n f i l t e r c i r c u i t s i s f o r t h e s t r u c t u r e t o be a n t r m e t r r c a l * a b o u t i t s m i d d l e . I n s u c h c a s e s , i f
Z g and
Zb i n F i g . 2 . 1 1 - 1 a r e a t t h e middle of a a n t i m e t r i c a l network,
then f o r a l l f r e q u e n c i e s
r2
a r e s i m p l y t h e r e f l e c t i o n c o e f f i c i e n t s a t Ends 1 a n d S Z 2= Sll = and 2 , r e s p e c t i v e l y , a s d e f i n e d i n E q s . ( 2 . 1 1 - 1 0 ) a n d ( 2 . 1 1 - 1 1 ) . The scattering coefficient
Szl is simply t h e transmission c o e f f i c i e n t , t ,
f o r t r a n s m i s s i o n t o End 2 f r o m End 1 a s d e f i n e d i n E q s . ( 2 . 1 0 - 5 ) a n d (2.10-6)
The s c a t t e r i n g c o e f f i c i e n t , S 1 2 , i s l i k e w i s e t h e same a s t h e
t r a n s m i s s i o n c o e f f i c i e n t , t , f o r t r a n s m i s s i o n t o End 1 f r o m End 2 .
Of
c o u r s e , i f t h e network i s r e c i p r o c a l SZ1= S 1 2 . The r e l a t i o n s i n
where H h i s a r e a l , p o s i t i v e c o n s t a n t .
Defining Z a again a s R a + J X ~ , by E q s . ( 2 . 1 1 - 1 7 ) , ( 2 . 1 1 - 1 4 1 , a n d ( 2 . 1 1 - 9 ) ,
Sec. 2 . 1 1 i n v o l v i n g t , r l , and r 2 , of c o u r s e , a p p l y e q u a l l y w e l l t o t = S12 = S Z 1 , = S l l , a n d r2 = S Z 2 ,r e s p e c t i v e l y .
rl
Thus, i t is s e e n t h a t a s f a r a s t w o - p o r t n e t w o r k s a r e c o n c e r n e d , the s c a t t e r i n g c o e f f i c i e n t s a r e simply t h e r e f l e c t i o n c o e f f i c i e n t s or t r a n s m i s s i o n c o e f f i c i e n t s d i s c u s s e d i n S e c s . 2 . 1 0 and 2 . 1 1 .
However,
s c a t t e r i n g c o e f f i c i e n t s may b e a p p l i e d t o n e t w o r k s w i t h a n a r b i t r a r y number o f p o r t s . (2.11-18)
For example, f o r a t h r e e - p o r t network t h e r e a r e n i n e
s c a t t e r i n g c o e f f i c i e n t s , w h i c h may b e d i s p l a y e d a s t h e m a t r i x
T h e q u a n t i t y A,, i s o b t a i n e d m o s t e a s i l y by e v a l u a t i n g
R,,
=
qb = real,
positive
(2.11-19)
(2.12-1)
a t a f r e q u e n c y w h e r e Z a a n d Z b a r e b o t h known t o b e r e a l .
The maximally f l a t and T c h e b y s c h e f f l o w - p a s s p r o t o t y p e f i l t e r s t r u c t u r e whose element v a l u e s a r e l i s t e d i n T a b l e s 4 . 0 5 - l ( a ) , ( b ) and 4 . 0 5 - 2 ( a ) , ( b ) a r e s y m m e t r i c a l f o r a n odd n u m b e r n o f r e a c t i v e e l e m e n t s , a n d t h e y a r e a n t i m e t r i c a l f o r a n e v e n n u m b e r n o f r e a c t i v e e l e m e n t s . The s t e p t r a n s -
For a n n - p o r t n e t w o r k t h e r e a r e n 2 c o e f f i c i e n t s . network w i t h r e s i s t i v e t e r m i n a t i o n s ,
formers d i s c u s s e d i n C h a p t e r 6 a r e a d d i t i o n a l examples o f a n t i m e t r i c a l
(2.12-2)
circuits. S E C . 2 . 1 2 , SCATTEHING C O E F F I C I E N T S I n t h i s book t h e r e w i l l b e some o c c a s i o n t o make u s e o f s c a t t e r i n g c o e f f i c i e n t s . S c a t t e r i n g c o e f f i c i e n t s a r e u s u a l l y d e f i n e d e n t i r e l y from a wave p o i n t o f v i e w .
However, f o r t h e p u r p o s e s of t h i s book i t w i l l be
s u f f i c i e n t t o simply e x t r a p o l a t e from p r e v i o u s l y developed c o n c e p t s .
*
T h i s term
war
coined by G u i l l e m i n .
See p p .
371
a n d 467 o f Ref. 2 .
I n g e n e r a l , f o r any
i s t h e r e f l e c t i o n c o e f f i c i e n t between t h e i n p u t impedance ( Z i n ) , a t Port j and t h e t e r m i n a t i o n H I a t t h a t p o r t . F o r t h e o t h e r c o e f f i c i e n t s , analoqously t o Eqs. ( 2 . 1 0 - 5 ) and ( 2 . 1 0 - 6 1 ,
where
w h e r e P I i s t h e power d e l i v e r e d t o t h e t e r m i n a t i o n H i a t P o r t j , a n d In accord with (p,v,il)k i s t h e a v a i l a b l e p o w e r of a g e n e r a t o r a t P o r t k . E q . ( 2 . 1 1 - 4 ) t h e d b a t t e n u a t i o n f o r t r a n s m i s s i o n from P o r t k t o P o r t j
(with a l l s p e c i f i e d t e r m i n a t i o n s connected) i s The v o l t a g e E . i s t h e r e s p o n s e a c r o s s t e r m i n a t i o n i?. a t P o r t j d u e t o a g e n e r a t o r o f v o l t a g e ( E g ) k a n d i n t e r n a l i m p e d a n c e H, a t P o r t k . I n comp u t i n g t h e c o e f f i c i e n t s d e f i n e d by E q s . ( 2 . 1 2 - 2 ) t o ( 2 . 1 2 - 4 ) , a l l p o r t s a r e assumed t o always be t e r m i n a t e d i n t h e i r s p e c i f i e d t e r m i n a t i o n s R . .
F u r t h e r d i s c u s s i o n of s c a t t e r i n g c o e f f i c i e n t s w i l l be found i n Hef. 9
I f an n - p o r t network i s r e c i p r o c a l ,
SEC. 2 . 1 3 , ANALYSIS OF LADDER C I R C U I T S
By E q s . ( 2 . 1 1 - 9 ) a n d ( 2 . 1 1 - 1 2 ) f o r a d i s s i p a t i o n l e s s r e c i p r o c a l t w o - p o r t
L a d d e r c i r c u i t s o f t e n o c c u r i n f i l t e r w o r k , some e x a m p l e s b e i n g t h e l o w - p a s s p r o t o t y p e f i l t e r s d i s c u s s e d i n C h a p t e r 4 . The r o u t i n e o u t l i n e d
network
below i s p a r t i c u l a r l y c o n v e n i e n t
r-
f o r computing t h e response of
1I
such networks.
13
The f i r s t s t e p i n t h i s r o u t i n e is t o characterize each s e r i e s b r a n c h by i t s i m p e d a n c e
and
I
I
and t h e c u r r e n t f l o w i n g t h r o u g h t h e b r a n c h , and e a c h s h u n t The a n a l o g o u s r e l a t i o n f o r t h e g e n e r a l n - p o r t , d i s s i p a t i o n l e s s , r e c i p r o c a l network is
b r a n c h by i t s a d m i t t a n c e a n d t h e voltage a c r o s s t h e branch.
This FIG. 2.13-1
characterization is illustrated i n F i g . 2 . 1 3 - 1 . Then, i n g e n e r a l t e r m s we d e f i n e
A LADDER NETWORK EXAMPLE DISCUSSED IN SEC. 2.13
where [S] i s t h e s c a t t e r i n g m a t r i x of s c a t t e r i n g c o e f f i c i e n t s [as i l l u s t r a t e d i n Eq. ( 2 . 1 2 - 1 ) f o r t h e c a s e o f n = 3 1 , [SI*i s t h e same m a t r i x w i t h a l l o f i t s c o m p l e x n u m b e r s c h a n g e d t o t h e i r c o n j u g a t e s , and [ I ] i s
Fk
=
s e r i e s impedance o r s h u n t a d m i t t a n c e o f Branch k
an n t h - o r d e r u n i t m a t r i x .
Uk
=
series-branch current o r shunt-branch v o l t a g e of 8ranch k
Eq. ( 2 . 1 2 - 5 ) a p p l i e s and
S i n c e t h e n e t w o r k i s s p e c i f i e d t o be r e c i p r o c a l , its principal diagonal.
[Sl i s s y m m e t r i c a l a b o u t
F o r any network w i t h r e s i s t i v e t e r m i n a t i o n s ,
(2.13-1)
Urn =
series-branch current o r shunt-branch v o l t a g e f o r t h e l a s t b r a n c h on t h e r i g h t
(2.13-3)
b',,
current or voltage associated with the d r i v i n g g e n e r a t o r on t h e l e f t .
(2.13-4)
=
I n g e n e r a l , i f Branch 1 i s i n s h u n t , U o s h o u l d be t h e c u r r e n t o f an i n f i n i t e - i m p e d a n c e c u r r e n t g e n e r a t o r ; i f Branch 1 is i n s e r i e s , U o s h o u l d be t h e v o l t a g e o f a z e r o - i m p e d a n c e v o l t a g e g e n e r a t o r . T h e n , f o r a l l c a s e s ,
45
T h u s , A l i s t h e t r a n s f e r f u n c t i o n b e t w e e n E o a n d E 4 . The i m p e d a n c e (Zi,)l and admittance (Yin), d e f i n e d i n t h e f i g u r e a r e
T h u s Al
i s t h e t r a n s f e r f u n c t i o n from t h e g e n e r a t o r on t h e l e f t t o
B r a n c h m on t h e r i g h t .
(Fin)L
=
I f we define
impedance l o o k i n g r i g h t t h r o u g h B r a n c h k i f Branch k i s i n s e r i e s , o r admittance l o o k i n g rightthroughBranch k 1fBranchk is i n shunt,
(2.13-6)
then
To i l l u s t r a t e t h i s p r o c e d u r e c o n s i d e r t h e c a s e i n F i g . 2 . 1 3 - 1 There m = 4 and
REFERENCES CHAPTER 3 1. E. A. h i l l e m i n , I n t r o d u c t o r y C i r c u i t Theory (John Wiley and Sons, New York C i t y , 1953).
2.
E. A. h i l l e m i n , S y n t h e s i s of P a s s i v e Networks (John Wiley and Sons, New York C i t y , 1957).
3.
D. F. T u t t l e , J r . , Network S y n t h e s i s , Vol. 1 (John Wiley and Sons. New York C i t y , 1958).
4.
M. E. Van Valkenburg, Network A n a l y s i s ( P r e n t i c e - H a l l , I n c . , New York C i t y , 1955).
5.
E. S. Kuh and D. 0. Pederson, P r i n c i p l e s of C i r c u i t S y n t h e s i s (McGraw-Hill Book 6.I n c . , New York C i t y , 1959).
6.
W. W. Hansen and 0. C. Lundatrom, "Experimental Determination of Impedance Functions by t h e Use of an E l e c t r o l y t i c Tank," P r o c . IRE, 33, pp. 528-534 (August 1945).
7.
R. E. S c o t t , "Network S y n t h e s i s by t h e Use o f P o t e n t i a l Analogs."Proc. (August 1952).
8.
G. L. M a t t h a e i , "Some S i m p l i f i c a t i o n s f o r Analysis o f L i n e a r C i r c u i t s , " IRE Trans. on C i r c u i t Theory. a ; 4 , pp. 120-124 (September 1957).
9.
H. J. C a r l i n , "'Ihe S c a t t e r i n g M a t r i x i n Network 'Iheory, pp. 88-97 (June 1956).
IRE, 40. PP. 970-973
" IRE Trans. on C i r c u i t Theory. a - 3 ,
PRINCIPLES OF TEE IMAGE METHOD FOR FILTER DESIGN
SEC.
3.01,
INTROUUCTION
A l t h o u g h tile i m a g e method f o r f i l t e r d e s i g n w i l l n o t be d i s c u s s e d i n d e t a i l in t h i s b o o k , i t w i l l b e n e c e s s a r y f o r r e a d e r s t o u n d e r s t a n d t h e image m e t h o d i n o r d e r t o u n d e r s t a n d some o f t h e d e s i g n t e c h n i q u e s u s e d The o b j e c t i v e o f t h i s c h a p t e r i s t o s u p p l y t h e n k c i n l a t e r chapters. e s s a r y b a c k g r o u n d by d i s c u s s i n g t h e p h y s i c a l c o n c e p t s a s s o c i a t e d w i t h t h e image m e t h o d and b y s u m m a r i z i n g t h e most u s e f u l e q u a t i o n s a s s o c i a t e d w i t h t h i s m e t h o d . D e r i v a t i o n s w i l l b e g i v e n f o r o n l y a few e q u a t i o n s ; more c o m p l e t e d i s c u s s i o n s w i l l b e f o u n d i n t h e r e f e r e n c e s l i s t e d a t t h e end o f t h e c h a p t e r . SEC.
3.02,
PHYSICAL AND MATHEMATICAL D E F I N I T I O N OF IMAGE IMPEDANCE AND IMAGE PROPAGATION FUNCTION
T h e image v i e w p o i n t f o r t h e a n a l y s i s o f c i r c u i t s i s a wave v i e w p o i n t m u c h t h e same a s t h e wave v i e w p o i n t commonly u s e d f o r a n a l y s i s o f t r a n s m i s s i o n lines. I n f a c t , f o r t h e case of a uniform transmission l i n e t h e i m p e d a n c e o f t h e l i n e i s a l s o i t s image i m p e d a n c e , a n d i f Yt i s t h e p r o p a g a t i o n c o n s t a n t p e r u n i t l e n g t h t h e n y,l
i s t h e image
Propagation function f o r a l i n e of length I . However, t h e t e r m s image L m ~ e d a n c ae n d image p r o p a g a t i o n f u n c t i o n h a v e much more g e n e r a l m e a n i n g than t h e i r d e f i n i t i o n with r e g a r d t o a uniform t r a n s m i s s i o n l i n e a l o n e suggest. Consider
t h e c a s e of a t w o - p o r t network which can be s y m m e t r i c a l ,
b u t w h i c h , f o r t h e s a k e o f g e n e r a l i t y , w i l l b e a s s u m e d t o be u n s y m m e t r i c a l with Of
different
i m p e d a n c e c h a r a c t e r i s t i c s a t End 1 t h a n a t End 2 .
3 . 0 2 - 1 shows t h e c a s e o f a n i n f i n i t e number o f i d e n t i c a l n e t w o r k s s o r t a l l c o n n e c t e d s o t h a t a t e a c h j u n c t i o n e i t h e r End Is a r e
C o n n e c t e d t o g e t h e r o r End 2 s a r e c o n n e c t e d t o g e t h e r . S i n c e t h e c h a i n of e x t e n d s t o i n f i n i t y i n e a c h d i r e c t i o n , t h e same i m p e d a n c e Z I 1 i s
I
I j
general c i r c u i t parameters A s , B s , Cs, and D, f o r t h e two b o x e s c o n n e c t e d a s shown c a n b e c o m p u t e d by u s e o f Eq. ( 2 . 0 5 - 7 ) .
a
ETC. TO
INFINITY
4
I
211
'I'
T h e n by E q . ( 2 . 0 9 - 1 )
I
z12
ETC. TO INFINITY A-3927-I8
FIG. 3.02-1
INFINITE CHAIN OF IDENTICAL NETWORKS USED FOR DEFINING IMAGE IMPEDANCES AND THE IMAGE PROPAGATION FUNCTION
FIG. 3.02-2
CIRCUIT DISCUSSED IN SEC.
S e t t i n g Z i n = ZL = Z I 1 and s o l v i n g f o r Z I 1i n t e r m s o f A, R , C , a n d D g i v e s
s e e n l o o k i n g b o t h l e f t a n d r i g h t a t a j u n c t i o n o f t h e two End l s , w h i l e a t a j u n c t i o n o f two End 2 s a n o t h e r i m p e d a n c e Z 1 2 w i l l b e s e e n when The i m p e d a n c e s Z I 1 a n d Z 1 2 ,d e f i n e d a s i n a r e t h e i m a g e i m p e d a n c e s f o r End 1 a n d End 2 ,
looking e i t h e r l e f t o r r i g h t . d i c a t e d i n Fig. 3.02-1,
r e s p e c t i v e l y , of t h e network.
For an unsymmetrical network t h e y a r e
The same p r o c e d u r e c a r r i e d o u t w i t h r e s p e c t t o End 2 g i v e s
g e n e r a l l y unequal. N o t e t h a t b e c a u s e o f t h e way t h e i n f i n i t e c h a i n o f n e t w o r k s i n F i g . 3.02-1 a r e c o n n e c t e d , t h e impedances s e e n l o o k i n g l e f t and r i g h t a t each j u n c t i o n a r e a l w a y s e q u a l , h e n c e t h e r e i s n e v e r any r e f l e c t i o n o f a wave p a s s i n g t h r o u g h a j u n c t i o n .
T h u s , f r o m t h e wave p o i n t o f v i e w , t h e
networks i n Fig. 3 . 0 2 - 1 a r e a l l p e r f e c t l y matched.
I f a wave i s s e t t o
p r o p a g a t i n g towards t h e r i g h t , t h r o u g h t h e c h a i n o f networks, i t w i l l be
F i g u r e 3 . 0 2 - 3 shows a n e t w o r k w i t h a g e n e r a t o r whose i n t e r n a l impeda n c e i s t h e same a s t h e i m a g e i m p e d a n c e a t End 1 a n d w i t h a l o a d i m p e d a n c e on t h e r i g h t e q u a l t o t h e image i m p e d a n c e a t End 2 . With t h e t e r m i n a t i o n s m a t c h e d t o t h e image i m p e d a n c e s i n t h i s m a n n e r i t c a n be shown t h a t
a t t e n u a t e d a s determined by t h e p r o p a g a t i o n f u n c t i o n o f e a c h network, but w i l l p a s s o n from n e t w o r k t o n e t w o r k w i t h o u t r e f l e c t i o n . Note t h a t t h e image i m p e d a n c e s
Z I 1 a n d Z 1 2 a r e a c t u a l l y i m p e d a n c e o f infinite n e t w o r k s ,
and a s s u c h t h e y s h o u l d b e e x p e c t e d t o h a v e a m a t h e m a t i c a l f o r m d i f f e r e n t from t h a t o f t h e r a t i o n a l i m p e d a n c e f u n c t i o n s t h a t a r e o b t a i n e d f o r f i n i t e , lumped-element networks.
I n t h e c a s e s o f lumped-element
f i l t e r structures,
t h e image i m p e d a n c e s a r e u s u a l l y i r r a t i o n a l f u n c t i o n s ; i n t h e c a s e s o f microwave f i l t e r s t r u c t u r e s which i n v o l v e t r a n s m i s s i o n l i n e e l e m e n t s , t h e image i m p e d a n c e s a r e u s u a l l y b o t h i r r a t i o n a l a n d t r a n s c e n d e n t a l . An e q u a t i o n f o r t h e i m a g e i m p e d a n c e i s e a s i l y d e r i v e d i n t e r m s o f t h e c i r c u i t i n Fig. 3.02-2. I f Z L i s made t o b e e q u a l t o Z I 1 t h e n t h e impedanct Z i n s e e n l o o k i n g i n f r o m t h e l e f t o f t h e c i r c u i t w i l l a l s o b e e q u a l t o ZI1o Now, i f A , B , C, and D a r e t h e g e n e r a l c i r c u i t p a r a m e t e r s f o r t h e b o x o n t h e l e f t i n Fig. 3.02-2,
assuming t h a t t h e network i s r e c i p r o c a l , t h e
FIG. 3.02-3 NETWORK HAVING TERMINATIONS WHICH ARE MATCHED ON THE IMAGE BASIS
Table 3.03-1 IMAGE PARAMETERS IN TERMS OF GENERAL CIRCUIT PARAMETERS, OPEN-CIRCUIT IMPEDANCES, OR SHORT-CIRCUIT ADMITTANCES IMAGE PARAMETER
where
I N TERMS O F
I N TERMS O F A . B , C,D
' 1 1 ' z 1 2 = '21"22
I N TERMS O F
I N CONVENIENT
Yll'y12 = Y21,Y22
MIXED FORM
i s t h e zmage p r o p a g a t i o n f u n c t i o n , a i s t h e Image a t t e n u a t i o n i n n e p e r s , * a n d ,B i s t h e image p h a s e i n r a d i a n s . N o t e t h a t t h e h I 2 / Z I l f a c t o r i n Eq. ( 3 . 0 2 - 5 ) h a s t h e e f f e c t o f m a k i n g Y i n d e p e n d e n t o f t h e r e l a t i v e impedf a c t o r i n Eq. ( 2 . 1 0 - 4 ) . a n c e l e v e l s a t E n d s 1 a n d 2 , much a s d o e s t h e
ml
An a l t e r n a t i v e f o r m o f E q . ( 3 . 0 2 - 5 ) i s
where I l = E1/ZI1 and I ; = E2/Z12 a r e a s d e f i n e d i n F i g . 3 . 0 2 - 3
I t s h o u l d b e emphasized t h a t t h e image p r o p a g a t i o n f u n c t i o n d e f i n e s t h e t r a n s m i s s i o n t h r o u g h t h e c i r c u i t a s i n d i c a t e d by Eq. ( 5 . 0 2 - 4 ) , ( 3 . 0 2 - 5 ) , o r ( 3 . 0 2 - 7 ) o n l y if t h e t e r m i n a t i o n s m a t c h t h e image i m p e d a n c e s a s i n F i g . 3 . 0 2 - 3 . The e f f e c t s o f m i s m a t c h w i l l b e d i s c u s s e d i n S e c . 3 . 0 7 . F o r a r e c i p r o c a l n e t w o r k t h e i m a g e p r o p a g a t i o n f u n c t i o n i s t h e same f o r p r o p a g a t i o n i n e i t h e r d i r e c t i o n e v e n t h o u g h t h e n e t w o r k may n o t b e symmetrical. S E C . 3 . 0 3 , RELATION BETKEEN THE IMAGE PARAMETERS AND GENERAL C I R C U I T PARAMETERS, O P E N - C I R C U I T IMPEDANCES, AND
Table. 3.03-2 GENERAL CIRCUIT PARAMETERS, OPEN-CIRCUIT IMPEDANCES, AND SHORT-CIRCUIT ADMITTANCES IN TERMS OF IMAGE PARAMETERS
c = -
sinh y
D =
J";I
cosh y
SHORT-CIRCUIT ADMITTANCES The t r a n s m i s s i o n p r o p e r t i e s o f a l i n e a r t w o - p o r t n e t w o r k c a n b e d e f i n e d i n t e r m s o f i t s image p a r a m e t e r s a s w e l l a s i n t e r m s o f t h e v a r i o u s parameters discussed i n Secs. 2.05 t o 2.07. Any o f t h e s e o t h e r p a r a m e t e r s can be computed from t h e image p a r a m e t e r s and v i c e v e r s a . T h e s e v a r i o u s r e l a t i o n s h i p s a r e summarized i n T a b l e s 3.03-1 and 3.03-2. For s i m p l i c i t y , only e q u a t i o n s f o r r e c i p r o c a l networks a r e i n c l u d e d . To change nepers t o decibel. ma~ltiply nepers by 8.686.
zll
= ZIl coth y
ZZ1
=
z 12
, zZ2
= Z12 coth y
S E C . 3 . 0 4 , IMAGE PARAMETERS FOR SOME COMMON STRUCTURES The image p a r a m e t e r s o f t h e L - s e c t i o n n e t w o r k i n F i g . 3 . 0 4 - 1 a r e g i v e n by
FIG. 3.04-2 A SYMMETRICAL T-SECTION NETWC (3.04-12)
( 3 . 0 4 -3 )
FIG. 3.04-1
AN L-SECTION NETWORK
Note t h a t t h e c i r c u i t i n F i g . 3 . 0 4 - 2 c a n b e f o r m e d by two L - s e c t i o n s a s i n F i g . 3 . 0 4 - 1 p u t back t o back s o t h a t Z b i n F i g . 3 . 0 4 - 2 i s o n e - h a l f o f
Z c i n F i g . 3 . 0 4 - 1 . Then Z I 1 w i l l b e t h e s a m e f o r b o t h n e t w o r k s and y f o r the T-section i s twice t h a t f o r the L-section. For t h e n - s e c t i o n i n F i g . 3 . 0 4 - 3 t h e image a d m i t t a n c e s a r e
and
N o t e t h a t by E q s .
( 3 . 0 4 - 2 ) and ( 3 . 0 4 - 4 ) A r - s e c t i o n can a l s o be c o n s t r u c t e d from two h a l f s e c t i o n s b a c k t o back, s o t h a t Y1 = l / Z c and
YJ = 1/(2Za). For t h e symmetrical T - s e c t i o n i n F i g . 3 . 0 4 - 2
For F i g . 3 . 0 4 - 1 ,
Y 1 2 = 1 / z 1 2 w i l l t h e n b e t h e same I-3517-23
as Y12 = Y I 1 i n Fig. 3 . 0 4 - 3 , while
Y f o r Fig. 3 . 0 4 - 3 w i l l a g a i n be twice t h a t f o r Fig. 3 . 0 4 - 1 .
FIG. 3.04-3
A SYMMETRICAL n-SECTION NETWO
For a uniform t r a n s m i s s i o n l i n e o f l e n g t h 1 , c h a r a c t e r i s t i c impedance
ZO, and p r o p a g a t i o n c o n s t a n t
y,
+
= at
jPt
per unit length,
The i n v e r s e , h y p e r b o l i c c o t a n g e n t f u n c t i o n i n Eq. ( 3 . 0 5 - 6 ) i s a m u l t i v a l u e d f u n c t i o n , whose v a r i o u s p o s s i b l e v a l u e s a l l d i f f e r by . u l t i p l e s of l n . For t h i s r e a s o n , i t i s convenient t o w r i t e E q . ( 3 . 0 5 - 6 ) i n t h e form
SEC.
3 . 0 5 , THE SPECIAL IMAGE PROPERTIES OF D I S S I P A T I O N L E S S &ere
t h e i n v e r s e h y p e r b o l i c f u n c t i o n i s t o be e v a l u a t e d t o g i v e an i m a g i n a r y p a r t h a v i n g minimum m a g n i t u d e , and w h e r e t h e a p p r o p r i a t e v a l u e
NETWORKS
By T a b l e 3 . 0 3 - 1
must b e d e t e r m i n e d by e x a m i n a t i o n o f t h e c i r c u i t u n d e r
for the integer
n
consideration.
E q u a t i o n ( 3 . 0 5 - 7 ) a l s o h a s t h e e q u i v a l e n t form
while
Y = a +
jp
=
c
~
t
h
.'
~
~( 3 . 0 ~ 5-2) ~
F o r a d i s s i p a t i o n l e s a n e t w o r k , we may w r i t e f o r f r e q u e n c i e s p = j w
Two d i s t i n c t c a s e s o c c u r i n t h e e v a l u a t i o n o f Eq. ( 3 . 0 5 - 5 ) a n d Eq. ( 3 . 0 5 - 7 ) o r ( 3 . 0 5 - 8 ) d e p e n d i n g o n w h e t h e r ( X o c ) and ( X I c ) h a v e t h e 1 1 same s i g n o r o p p o s i t e s i g n s . T h e s e two c a s e s w i l l be s u m m a r i z e d separately.
and Case A , C o n d i t i o n f o r a P a s s Band-In
t h i s case (Xoc)
1
and ( X s c )
1
have o p p o s i t e s i g n s a n d Z
w h e r e j ( X o , ) i s t h e i m p e d a n c e a t End 1 o f t h e n e t w o r k w i t h End 2 o p e n 1 c i r c u i t e d , , a n d j ( X l c ) i s t h e i m p e d a n c e a t End 1 w i t h End 2 s h o r t 1 c i r c u i t e d . Then b y Eqs. ( 3 . 0 5 - 1 ) t o ( 3 . 0 5 - 4 1 , f o r d i s s i p a t i o n l e s s n e t works
and
=
d
m
=
r e a l and p o s i t i v e .
(3.05-9)
I t c a n b e shown t h a t , a t t h e same t i m e , t h e c o n d i t i o n
Z12
=
=
r e a l and p o s i t i v e .
(3.05-10)
must a l s o e x i s t , w h e r e ( X o c ) a n d ( X I , ) a r e t h e o p e n - and s h o r t - c i r c u i t 2 2 i m p e d a n c e s m e a s u r e d f r o m End 2 . Under t h e s e c o n d i t i o n s , E q s . ( 3 . 0 5 - 7 ) and ( 3 . 0 5 - 8 ) y i e l d f o r
a
a n d ,B,
I
I f ( X o c )1 < ( X l c ) 1 , Eq. ( 3 . 0 5 - 8 ) s h o u l d b e u s e d , and i t g i v e s radians
=
[tan-/M] -:
radians
.
and
,B
=
(2n
N o t e t h a t f o r t h i s p a s s - b a n d c a s e , t h e a t t e n u a t i o n i s z e r o w h i l e t h e phase
-
1)
' 2
radians
.
(3.05-18)
i s g e n e r a l l y non-zero and v a r y i n g w i t h f r e q u e n c y .
I n E q s . ( 3 . 0 5 - 1 1 ) and ( 3 . 0 5 - 1 2 ) t h e n n term h a s b e e n o m i t t e d s i n c e t h e m u l t i v a l u e d n a t u r e of
(though perhaps t h e m u l t i v a l u e d n a t u r e of i n v e r s e hyperbolic f u n c t i o n s
Note t h a t f o r t h i s s t o p - b a n d c a s e t h e image a t t e n u a t i o n i s n o n - z e r o a n d w i l l v a r y w i t h f r e q u e n c y . M e a n w h i l e , t h e image p h a s e i s c o n s t a n t u s . f r e q u e n c y a t some m u l t i p l e o f T , o r odd m u l t i p l e o f ~ / 2 . However, i t w i l l
may n o t ) .
be f o u n d t h a t t h e image p h a s e c a n make d i s c r e t e j u m p s a t p o i n t s i n t h e
these inverse trigonometric functions w i l l be familiar t o the reader
s t o p band w h e r e t h e r e a r e p o l e s o f a t t e n u a t i o n f o r f r e q u e n c i e s j w . Case B , C o n d i t i o n s f o r a S t o p Band-In t h i s c a s e (Xoc) and (Xlc) 1 1 [ a n d a l s o ( X o , ) and (Xsc 1 1 h a v e t h e same s i g n . Then 2 2
A s i m i l a r a n a l y s i s f o r d i s s i p a t i o n l e s s networks can be c a r r i e d o u t
u s i n g t h e v a r i o u s o t h e r e x p r e s s i o n s f o r t h e image p a r a m e t e r s i n S e c s . 3 . 0 3 and 3 . 0 4 . The v a r i o u s e q u a t i o n s g i v e n f o r t h e image p r o p a g a t i o n c o n s t a n t w i l l i n v o l v e i n v e r s e h y p e r b o l i c f u n c t i o n s of a p u r e l y r e a l o r ~ u r e l y i m a g i n a r y a r g u m e n t . Due t o t h e m u l t i v a l u e d n a t u r e o f t h e s e i n v e r s e h y p e r b o l i c f u n c t i o n s c a r e must be t a k e n i n e v a l u a t i n g them. T a b l e 3 . 0 5 - 1
and
should prove h e l p f u l f o r t h i s purpose.
N o t e t h a t i n some c a s e s a d i f f e r e n t
e q u a t i o n must b e used depending on whether l u l o r IvI i s g r e a t e r o r l e s s t h a n one. T h i s i s b e c a u s e , f o r e x a m p l e , c o s h - l w when t a k e n t o b e a f u n c t i o n o f a r e a l v a r i a b l e c a n n o t be e v a l u a t e d f o r w = l u l < 1 ; i f h o w e v e r , w i s a a r e both purely imaginary.
B o t h X I 1 and X12 m u s t h a v e p o s i t i v e s l o p e s us. f r e q u e n c y , i n a c c o r d w i t h F o s t e r ' s r e a c t a n c e theorem. If ( X o c )1 > ( X a c ) 1 Eq.
( 3 . 0 5 - 7 ) s h o u l d b e u s e d t o o b t a i n a and
P:
f u n c t i o n o f a c o m p l e x v a r i a b l e t h e above e x a m p l e h a s a v a l u e , n a m e l y , ~ ( C O S - l u ) .The proper v a l u e of t h e i n t e g e r n t o be used with t h e v a r i o u s e q u a t i o n s i n T a b l e 3 . 0 5 - 1 m u s t b e d e t e r m i n e d by e x a m i n a t i o n o f t h e c i r c u i t a t some f r e q u e n c y w h e r e t h e t r a n s m i s s i o n p h a s e i s e a s i l y e s t a b l i s h e d . As was d o n e i n t h e c a s e o f E q s .
nepers
been o m i t t e d f o r f o r m s i n v o l v i n g i n v e r s e t r i g o n o m e t r i c f u n c t i o n s s i n c e t h e i r m u l t i v a l u e d n a t u r e i s much more w i d e l y f a m i l i a r t h a n i s t h a t o f i n verse hyperbolic functions.
and radians
( 3 . 0 5 - 1 1 ) and ( 3 . 0 5 - 1 2 ) , t h e n n t e r m s h a v e
Table 3.05-1 EVALUATION OF SOME INVERSE HYPERBOLIC FUNCTIONS FOR PURELY REAL OR PUhELY IMAGINARY ARGUMENTS In g e n e r a l . R = U + jV, Function
cOttl-lw
=
a
+ j v , and
Caae of
w
=
w
n ia an intagar
+ I
v
=
( p o s i t i r s , u s g a t i r e or r e r o )
I
u
= coth-lu t j n n
where R' , L ' , a n d C ' a r e f o r t h e n o r m a l i z e d c i r c u i t a n d R , L , a n d C a r e c o r r e s p o n d i n g e l e m e n t s f o r t h e s c a l e d c i r c u i t . The r a t i o R o / R b d e f i n e s t h e c h a n g e i n i m p e d a n c e l e v e l w h i l e wl/w;
d e f i n e s t h e change i n frequency
scale. F i g u r e 3 . 0 6 - l ( a ) shows a n o r m a l i z e d c o n s t a n t - k f i l t e r h a l f s e c t i o n . I t s image i m p e d a n c e s a r e
z,
=
Jl- (w02
and n = even
w
=
cash-lu
ifu.1 t jnn
W = sinh-lv t j(2n
= odd i f u , < -1
= odd ifv>O
.
= even ifv
.
n
,
-
1) f
n
n
.
I t s propagation function is y
S E C . 3 . 0 6 , CONSTANT-k A N D m-DERIVED F I L T E H SECTIONS
=
a t jp
for the 0
5
w' $ 1 p a s s b a n d , a n d
for the 1
6
w'
=
0 t j sin-'w'
C o n s t a n t - k and m-derived f i l t e r s a r e c l a s s i c examples o f f i l t e r s which a r e d e s i g n e d from t h e image p o i n t o f view. T h e i r p r o p e r t i e s w i l l b e b r i e f l y s u m m a r i z e d i n o r d e r t o i l l u s t r a t e some o f t h e i m a g e p r o p e r t i e s of d i s s i p a t i o n l e s s networks d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n , and t o p r o v i d e r e f e r e n c e d a t a . T h e f i l t e r s e c t i o n s shown a r e a l l n o r m a l i z e d s o t h a t t h e i r i m a g e i m p e d a n c e i s H i = 1 ohm a t w' q u e n c y o c c u r s a t w; = 1 r a d i a n l s e c .
= 0 and t h e i r c u t o f f f r e -
However, t h e s e n o r m a l i z e d c i r c u i t s
can e a s i l y be changed t o o t h e r impedance and f r e q u e n c y s c a l e s . resistance, inductance, o r capacitance is s c a l e d using
Each
5
a s t o p band, where
Figures 3.06-l(b),
a i s i n n e p e r s and p i s i n r a d i a n s .
( c ) show s k e t c h e s o f t h e image i m p e d a n c e and
a t t e n u a t i o n c h a r a c t e r i s t i c s of t h i s s t r u c t u r e .
Note t h a t , a s d i s c u s s e d
i n S e c . 3 . 0 5 , Z I T a n d ZIT a r e p u r e l y r e a l i n t h e p a s s b a n d a n d p u r e l y i m a g i n a r y i n t h e s t o p b a n d . A l s o n o t e t h a t ci = 0 i n t h e p a s s b a n d w h i l e
,B i s c o n s t a n t i n t h e s t o p b a n d . F i g u r e 3 . 0 6 - 2 ( a ) shows a " s e r i e s , m-derived" h a l f s e c t i o n .
Its
image impedances a r e
where
N o t e t h a t E q . ( 3 . 0 6 - 8 ) i s i d e n t i c a l t o Eq. d i f f e r s from Eq.
(3.06-5).
w'
5
1 p a s s band,
i n t h e 1 = w'
2
wA s t o p b a n d , a n d
in the 0
2
FIG. 3.06-1 THE IMAGE PROPERTIES OF NORMALIZED, CONSTANT-k HALF SECTIONS
i n t h e w&
2
w'
2
r.n
( 3 . 0 6 - 4 ) , b u t Eq. ( 3 . 0 6 - 9 )
The p r o p a g a t i o n f u n c t i o n i s
s t o p band
F i g u r e s 3 . 0 6 - 2 ( b ) and ( c ) show s k e t c h e s o f t h e image i m p e d a n c e and Note t h a t i n t r o d u c i n g a
propagation c h a r a c t e r i s t i c s of t h i s s t r u c t u r e .
s e r i e s resonance i n t h e shunt branch i n Fig. 3.06-2(a) has produced a p o l e o f a t t e n u a t i o n a t t h e f r e q u e n c y w: w h e r e t h e s h u n t b r a n c h s h o r t circuits transmission.
(See discussion i n Sec.2.04.)
Note t h a t
R I n m i n t h e p a s s b a n d i n F i g . 3 . 0 6 - 2 ( b ) i s more n e a r l y c o n s t a n t T h i s p r o p e r t y o f m-derived image impedances t h a n i s RI,, i n F i g . 3 . 0 6 - l ( b ) .
ZITI
=
makes them h e l p f u l f o r i m p r o v i n g t h e i m p e d a n c e m a t c h t o r e s i s t o r terminations. The " s h u n t m - d e r i v e d " h a l f s e c t i o n i n F i g . 3 . 0 6 - 3 ( a ) i s t h e d u a l o f that i n Fig. 3.06-2(a).
The image impedances a r e
where a g a i n
I n t h i s c a s e Z I T , i n Eq. ( 3 . 0 6 - 1 4 ) d i f f e r s f r o m Z I T i n Eq. ( 3 . 0 6 - 4 ) , b u t Eqs. ( 3 . 0 6 - 1 5 ) a n d ( 3 . 0 6 - 5 ) a r e i d e n t i c a l . The image p r o p a g a t i o n f u n c t i o n f o r t h i s s e c t i o n i s t h e same a s t h a t i n E q s . ( 3 . 0 6 - 1 1 ) t o ( 3 . 0 6 - 1 3 ) .
FIG. 3.06-2 NORMALIZED, SERIES, m-DERIVED HALFSECTION CHARACTERISTICS
F i g u r e s 3 . 0 6 - 3 ( b ) and ( c ) show s k e t c h e s o f t h e image c h a r a c t e r i s t i c s o f t h i s f i l t e r s e c t i o n . I n t h i s c a s e , a p o l e of a t t e n u a t i o n is produced a t t h e f r e q u e n c y w:
where t h e s e r i e s b r a n c h h a s a p o l e o f impedance which b l o c k s
The image i m p e d a n c e Z I T . i s s e e n t o b e more n e a r l y c o n S t a n t i n t h e p a s s band t h a n was Z I T i n F i g . 3 . 0 6 - l ( b ) . T h u s , m - d e r i v e d half s e c t i o n s of t h i s t y p e a r e a l s o u s e f u l f o r improving t h e impedance match t o r e s i s t o r t e r m i n a t i o n s . a l l transmission.
F i g u r e 3 . 0 6 - 4 ( a ) and ( b ) show how c o n s t a n t - k and i n - d e r i v e d h a l f s e c t i o n s may b e ~ i e c e dt o g e t h e r t o form a s i z e a b l e f i l t e r . I n t h i s c a s e , t h r e e c o n s t a n t - k h a l f s e c t i o n s a r e used a l o n g w i t h two, s e r i e s , m - d e r i v e d ,
FIG. 3.06-4
FIG. 3.06-3
NORMALIZED, SHUNT, m-DERIVED HALF-SECTION CHARACTERISTICS
A F I L T E R P I E C E D TOGETHER FROM THREE CONSTANT-k AND TWO m-DERIVED H A L F SECTIONS The resulting imoge propagation function i s sketched at ( c )
half sections.
The two m - d e r i v e d s e c t i o n s h a v e m = 0 . 5 , which i n t r o d u c e 8
a p o l e o f a t t e n u a t i o n a t w: of t h e f i l t e r .
= 1 . 1 6 and g r e a t l y i n c r e a s e s t h e r a t e of
cutoft,
A s i n d i c a t e d i n F i g . 3 . 0 6 - 4 ( a ) t h e s e c t i o n s a r e a l l chosen i Under t h e s e c o n d i t i O p ,
s o t h a t t h e image i m p e d a n c e s match a t e a c h j u n c t i o n .
when t h e s e c t i o n s a r e a l l j o i n e d t o g e t h e r , t h e image a t t e n u a t i o n and t h e
I
image p h a s e f o r t h e e n t i r e s t r u c t u r e a r e s i m p l y t h e sum o f t h e image a t t e n . ! u a t i o n and p h a s e v a l u e s f o r t h e i n d i v i d u a l s e c t i o n s .
Likewise, with a l l o l ~
FIG. 3.07-1 NETWORK DISCUSSED IN SEC. 3.07
t h e s e c t i o n s matched t o e a c h o t h e r , t h e image impedances s e e n a t t h e ends a r e t h e same a s t h e image i m p e d a n c e s o f t h e e n d s e c t i o n s b e f o r e t h e y were connected t o t h e i n t e r i o r s e c t i o n s . The c i r c u i t i n F i g . 3 . 0 6 - 4 ( b ) would h a v e t h e t r a n s m i s s i o n c h a r a c t e r -
1
i
i s t i c s i n d i c a t e d i n F i g . 3 . 0 6 - 4 ( c ) i f i t were t e r m i n a t e d i n i t s image i m p e d a n c e s a t b o t h e n d s . However, s i n c e i n p r a c t i c e r e s i s t o r t e r m i n a t i o n s a r e g e n e r a l l y required, t h i s transmission c h a r a c t e r i s t i c w i l l be considera b l y a l t e r e d ( m a i n l y i n t h e p a s s b a n d ) due t o t h e r e f l e c t i o n s a t b o t h ends of the f i l t e r .
where
I n o r d e r t o reduce t h e magnitude of t h e s e r e f l e c t i o n s e f -
f e c t s , i t i s customary w i t h f i l t e r s of t h i s t y p e t o i n t r o d u c e m-derived h a l f - s e c t i o n s a t e a c h end o f t h e f i l t e r w i t h t h e impedance ZIT. o r ZIT. n e x t t o t h e t e r m i n a t i o n r e s i s t o r . B'ith m = 0 . 6 , t h e s e image impedances
and
a r e r e l a t i v e l y c o n s t a n t i n t h e p a s s band and i t becomes p o s s i b l e t o greatly r e d u c e t h e r e f l e c t i o n e f f e c t s o v e r much o f t h e p a s s band.
These m a t t e r s
w i l l be d i s c u s s e d f u r t h e r i n S e c s . 3 . 0 7 and 3 . 0 8 .
SEC. 3 . 0 7 , THE EFFECTS OF TERMINATIONS WHICH MISMATCH THE
a r e t h e r e f 1 e c t i o n c o e f f i c i e n t s a t Ends 1 and 2 w h i l e
IMAGE IMPEDANCES The r e s i s t a n c e t e r m i n a t i o n s u s e d on d i s s i p a t i o n l e s s f i l t e r s t r u c t u r e s c a n n o t match t h e image impedance o f t h e s t r u c t u r e e x c e p t a t d i s c r e t e f r e q u e n c i e s i n t h e p a s s band.
As a r e s u l t of t h e multiple r e f l e c t i o n s t h a t
o c c u r , t h e p e r f o r m a n c e o f t h e f i l t e r may be c o n s i d e r a b l y a l t e r e d f r o m t h a t p r e d i c t e d by t h e image p r o p a g a t i o n f u n c t i o n .
and
T h i s a l t e r a t i o n i s most severe
i n t h e p a s s band and i n t h e s t o p band n e a r c u t o f f .
F o r m u l a s which ~ C C O U D ~
f o r t h e e f f e c t s of s u c h t e r m i n a l r e f l e c t i o n s a r e summarized below. C o n s i d e r t h e c i r c u i t i n F i g . 3 . 0 7 - 1 whose image i m p e d a n c e s ,
ZI1and
Z 1 2 ,may d i f f e r c o n s i d e r a b l y from R 1 and R 2 . The v o l t a g e a t t e n u a t i o n r a t i o , E s / E 2 , may be c a l c u l a t e d from t h e image p a r a m e t e r s and t h e t e r m i nations using the equation
are t h e t r a n s m i s s i o n c o e f f i c i e n t s ( s e e Sec. 2 . 0 8 ) .
Note t h a t t h e s e r e -
f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s a r e d e f i n e d w i t h r e s p e c t t o t h e image i m p e d a n c e s r a t h e r t h a n w i t h r e s p e c t t o t h e a c t u a l i n p u t i m p e d a n c e s
,
,1
and ( Z i , ) 2
.
The actua 1 input impedance seen looking in End 1 with End 2 terminated in R 2 is
By analogy,
(Zin)in
at all frequencies, where R o is a positive, real constant. The constant-k half section in Fig. 3.06-1 is an example of an antimetrical network. The
Fig. 3.07-1 is
2
filter in Fig. 3.06-4 also satisfies the antimetry condition given by E q . (3,.07-10). For dissipationless symmetrical networks with symmetrical terminations,
Z r l = R I 1 in the pass band and the attenuation is Equations (3.07-1) to (3.07-7) apply whether the circuit has dissipation or not. For a dissipationless network at pass band frequenc ies where jn~1,,,,,~,,, Eq. (3.07-6) shows that
y = 0
*
...
while in the stop band
Z I 1 = j X I l and
Similarly for dissipationless antimetrical networks with antimetrical terminations, in the pass band
ZI1and Z12will (Zi,).
where for
be purely real.
Analogous expressions also exist
2
Equation (3.07-1) is quite general, and it can be used with Eqs. (2.11-2) and (2.11-4) for computing the attenuation o f a network.
while in the stop band E q . case.
(3.07-13) applies just as for the symmetrical
For the symmetrical case
However, simpler expressions (about to be presented) can be used if the network is dissipationless. Such expressions become especially simple if the dissipationless network is symmetrical ( i . e . , ZI1= Z12)and has symmetrical terminations ( i . e . , H1 = R 2 ) Another case of relative simplicity is that of a dissipationless antimetrical network (see Sec. 2.11) with antimetrical terminations.
Such a filter will satisfy the conditions
70
while for the antimetrical case
rrl =-rr2 . 71
For the dissipationless symmetrical case the stop-band image phase is a multiple of 77 radians, while in the dissipat ionless antimetric case it is an odd multiple of ~ / 2radians. The actual
ass-band
attenuation which will result from mismatched
image impedances is seen by Eqs. (3.07-12) and (3.07-14) to depend strongly on the image ~ h a s e ,,8. For given Z I 1 and R 1 it is easily shown that the maximum possible pass-band attenuation in a dissipationless symmetrical or antimetrical network with symmetrical or antimetrical terminations, respectively, is
where
with either definition giving the same answer. For symmetrical networks, the value given by Eq. (3.07-17) applies when R , = (2n 1)~/2 radians while L A = 0 when B , = n n radians (where n is an integer). For antimetrical
-
networks Eq. (3.07-17) applies when ,B = n r radians while L A = 0 when radians. Figure 3.07-2 shows a plot of maximum L A us. a, and also shows the corresponding input VSWR.
,8 = (2n - 1)77/2
S E C . 3 . 0 8 , D E S I G N OF M A T C H I N G E N D S E C T I O N S TO IMPROVE THE RESPONSE OF F I L T E R S D E S I G N E D O N THE IMAGE B A S I S As mentioned in Sec. 3.06, one way in which the pass-band responsc of constant-k filters can be improved is to use m-derived half sections at the ends. Experience shows that a half section with m about 0.6 will cause Z I T . or Z I T " to give the best approximation of a constant resistance in the pass band, and hence will cause the ends of the filter to give the best match to resistor terminations. As an example, Fig. 3.08-1 shows the normalized filter structure in Fig. 3.06-4(b) with matching sections added to improve the pass-band match to the one-ohm terminations shown. The matching sections also introduce poles of attenuation at w i = 1.25, which will further sharpen the cutoff characteristics of the.filter.
FIG. 3.07-2 MAXIMUM POSSIBLE PASS-BAND ATTENUATION A N D VSWR FOR DISSIPATIONLESS SYMMETRICAL NETWORKS WlTH SYMMETRICAL TERMINATIONS, O R DISSIPATIONLESS ANTIMETRICAL NETWORKS WlTH ANTIMETRICAL TERMINATIONS ,..., for the These values will apply if P = (2n - 1)(77/2)1,=,, symmetrical case or ,I3 = nnl,,,, 2,3,..., for the antimetrical case
FIG. 3.08-1 THE NORMALIZED FILTER CIRCUIT IN FIG. 3.06-4(b) WITH in-DERIVED HALF SECTIONS ADDED TO IMPROVE THE PASS-BAND IMPEDANCE MATCH TO RESISTOR TERMINATIONS
I n t h e d e s i g n o f microwave f i l t e r s t r u c t u r e s on t h e image b a s i s , i t i s o f t e n d e s i r a b l e t h a t t h e m a t c h i n g end s e c t i o n s be o f t h e same g e n e r a l
form a s t h e main p a r t o f t h e f i l t e r . C o n s i d e r t h e c a s e o f a wide band, b a n d - p a s s f i l t e r t o b e c o n s t r u c t e d u s i n g f i l t e r s e c t i o n s a s shown i n Fig. 3.08-2(a).
The f i l t e r s e c t i o n s h a v e image c h a r a c t e r i s t i c s a s shown
i n Fig. 3.08-2(b),
(c).
F i g u r e 3 . 0 8 - 3 shows t h e l e f t h a l f o f a symmetrical
f i l t e r formed from s u c h s e c t i o n s . I n t h i s f i l t e r t h e i n t e r i o r s e c t i o n s of t h e f i l t e r a r e a l l a l i k e , b u t two s e c t i o n s a t e a c h end a r e d i f f e r e n t i n o r d e r t o improve t h e p a s s - b a n d match t o t h e t e r m i n a t i o n s .
The d e s i g n of
s u c h end s e c t i o n s w i l l now be c o n s i d e r e d .
A s i s s e e n from F i g . 3 . 0 8 - 2 ( c ) , e a c h s e c t i o n o f t h e f i l t e r h a s a m i d band image p h a s e s h i f t o f
P
= n/2.
The t o t a l midband image p h a s e s h i f t f o r t h e end m a t c h i n g n e t w o r k i n F i g . 3 . 0 8 - 3 a t f o i s t h u s P = 77. A t midband, t h e n , t h e end m a t c h i n g network w i l l o p e r a t e s i m i l a r l y t o a h a l f w a v e l e n g t h t r a n s m i s s i o n l i n e , and i n F i g . 3 . 0 8 - 3
FIG. 3.08-2 A BAND-PASS FILTER SECTION USING TRANSMISSION LINES, AND ITS lMAGE CHARACTER ISTICS T h u s , i f Z I i s t h e image impedance of t h e i n t e r i o r s e c t i o n s o f t h e f i l t e r , and Z I e i s t h e image impedance o f t h e s e c t i o n s i n t h e end m a t c h i n g network, then i f
I_ END SECTIONS I FOR MATCHING [LINE IMPEDANCE(Z,)J
I INTERIOR SECTIONS [LINE IMPEDANCE zO]
---
P
FIG. 3.08-3 ONE-HALF OF A SYMMETRICAL FILTER COMPOSED O F SECTIONS OF THE TYPE IN FIG. 3.08-2
FIG. 3.08-4 RELATIVE IMAGE IMPEDANCE CHARACTERISTICS FOR THE END MATCHING NETWORK AND INTERIOR SECTIONS OF A PROPOSED FILTER OF THE FORM IN FIG. 3.08-3
a p e r f e c t match i s a s s u r e d a t f o , r e g a r d l e s s o f t h e s i z e o f ZIe a t t h a t frequency.
At p a s s - b a n d f r e q u e n c i e s f n l z
s h i f t o f t h e end matching network i s
P
=
a n d f 3 n 1 2 , w h e r e t h e i m a g e phase n / 2 and 3 n / 2 , r e s p e c t i v e l y ,
i n t e r i o r s e c t i o n s of t h e f i l t e r s o t h a t the
p
=
n / 2 a n d 3 ~ / 2p h a s e s h i f t
points w i l l occur n e a r t h e c u t o f f frequencies of t h e i n t e r i o r s e c t i o n s . The end m a t c h i n g n e t w o r k i s d e s i g n e d s o t h a t Eq. ( 3 . 0 8 - 4 ) w i l l b e s a t i s f i e d , a t l e a s t a p p r o x i m a t e l y , a t t h e s e two f r e q u e n c i e s i n o r d e r t o g i v e a good i m p e d a n c e m a t c h c l o s e t o t h e c u t o f f f r e q u e n c i e s o f t h e f i l t e r . I n s i m i l a r l y t o Eq. ( 3 . 0 7 - 9 ) .
T h u s , s e t t i n g Z i n = ZI a n d s o l v i n g f o r Z I e
gives
a s t h e c o n d i t i o n f o r a p e r f e c t i m p e d a n c e m a t c h when t h e end matching network.
P
= n / 2 o r 3n/2 for
By s u c h p r o c e d u r e s a p e r f e c t i m p e d a n c e m a t c h
c a n b e a s s u r e d when t h e e n d m a t c h i n g n e t w o r k h a s n / 2 , n , o r 3 n / 2 r a d i a n s image p h a s e . F i g u r e 3 . 0 8 - 4 shows how t h e image i m p e d a n c e o f t h e e n d m a t c h i n g n e t work m i g h t c o m p a r e w i t h t h e i m a g e i m p e d a n c e o f t h e i n t e r i o r s e c t i o n s f o r a p r a c t i c a l d e s i g n . I n t h i s c a s e R g i s made a l i t t l e l e s s t h a n ZI f o r t h e i n t e r i o r s e c t i o n s a t f,,, b u t ZI a n d Z r e a r e b o t h made t o b e e q u a l t o
R g a t f a a n d f,,,
a l i t t l e t o each s i d e of f o ,
s o t h a t a p e r f e c t match w i l l
be a c h i e v e d a t t h o s e two f r e q u e n c i e s . T h i s p r o c e d u r e w i l l r e s u l t i n a s m a l l mismatch i n t h e v i c i n i t y o f f o , b u t s h o u l d improve t h e o v e r - a l l r e sults.
T h e e n d m a t c h i n g n e t w o r k i s made t o b e more b r o a d b a n d t h a n t h e
t h i s p a r t i c u l a r example t h e r e a r e o n l y t h r e e d e g r e e s of freedom i n t h e d e s i g n o f t h e e n d m a t c h i n g n e t w o r k , namely t h e s i z e o f C e , t h e s i z e o f
( Z o ) , a n d t h e l e n g t h o f t h e t r a n s m i s s i o n , l i n e s i n t h e s e c t i o n s o f t h e end m a t c i i n g n e t w o r k . One d e g r e e o f f r e e d o m i s u s e d i n f i x i n g t h e c e n t e r f r e q u e n c y o f t h e r e s p o n s e , a n o t h e r may b e u s e d f o r s e t t i n g Z I e = R g a t f r e quency f a i n F i g . 3 . 0 8 - 4 , a n d a n o t h e r may be u s e d f o r s a t i s f y i n g Eq. ( 3 . 0 8 - 4 ) a t f n 1 2 .
Although matching c o n d i t i o n s a r e n o t s p e c i f i c a l l y i n F i g . 3 . 0 8 - 4 , t h e y w i l l be a p p r o x i f o r c e d a t f r e q u e n c i e s f b a n d f3,,,2
mately s a t i s f i e d b e c a u s e of t h e n e a r l y symmetrical n a t u r e of t h e r e s p o n s e about f o . The d e s i g n p r o c e d u r e d e s c r i b e d a b o v e p r o v i d e s a p e r f e c t i m p e d a n c e match a t c e r t a i n f r e q u e n c i e s a n d a s s u r e s t h a t t h e maximum m i s m a t c h t h r o u g h o u t t h e p a s s b a n d w i l l n o t b e l a r g e . I n a d d i t i o n i t s h o u l d be r e c a l l e d t h a t p e r f e c t t r a n s m i s s i o n w i l l r e s u l t a t pass-band f r e q u e n c i e s where t h e image p h a s e o f t h e o v e r - a l l f i l t e r s t r u c t u r e i s a m u l t i p l e o f n r a d i a n s , a s w e l l a s a t p o i n t s w h e r e t h e image i m p e d a n c e s a r e p e r f e c t l y m a t c h e d . T h e s e same p r i n c i p l e s a l s o a p p l y f o r t h e d e s i g n o f m a t c h i n g s e c t i o n s f o r other types of f i l t e r s .
77
S E C . 3.09, M E A S U R E M E N T
OF I M A G E P A R A M E T E R S
Occasionally it will be desirable to measure the image parameters of a circuit.
A general method is to measure the input impedance at one end
for open- and short-circuit terminations at the other end.
Then
and for a reciprocal network
In these equations ( Z o c ) and ( Z S c ) are impedances measured at End 1 with 1 1 End 2 open-circuited and short-circuited, respectively. Impedances ( Z o c ) 2 and ( Z a c ) are corresponding impedances measured from End 2 with End 1 2
open-circuited or short-circuited. If the network has negligible dissipation and is symmetrical, a convenient method due to Dawirs5 can be used. Using this method the network is terminated at one port in a known resistive load RL and its input impedance Z i n = B i n + ] X i ,
is measured at the other port.
impedance Z I can be computed from Z i n and
Then the image
R L by the equation5
RA-3527-200 SOURCE:
FIG. 3.09-1
which applies for both the pass and stop bands. Dawirs5 has expressed this method in terms of a very useful chart which is reproduced in Fig. 3.09-1. This chart should be thought of as being superimposed on top of a Smith chart6" with the zero "wavelengths toward generator" point coinciding with that of the Smith chart.
By c o u r t e s y of H. N. Dawirs and Proc. IRE.=
Then
DAWIRS' CHART FOR DETERMINING THE IMAGE PARAMETERS OF SYMMETRICAL, DISSIPATIONLESS NETWORKS
t o o b t a i n t h e image p a r a m e t e r s , Z i n measured a s d i s c u s s e d a b o v e , i s n o r m a l i z e d w i t h r e s p e c t t o R L . N e x t , t h e p o i n t Z i n / R L i s f i r s t p l o t t e d on a S m i t h c h a r t , and t h e n s c a l e d t o t h e same p o i n t on t h i s c h a r t by u s e o f a s c a l e and c u r s o r . I n t h e p a s s band t h e Z i n / R L p o i n t s w i l l f a l l w i t h i n e i t h e r o f t h e two heavy c i r c l e s marked " c u t o f f c i r c l e , " w h i l e i n t h e s t o p band t h e Z i n / R L p o i n t s w i l l f a l l o u t s i d e o f t h e s e c i r c l e s .
Further details o f t h e u s e o f t h e c h a r t a r e p e r h a p s b e s t i l l u s t r a t e d by e x a m p l e s . Suppose t h a t Z i n / R L = 0 . 2 0 + j 0 . 2 5 . P l o t t i n g t h i s p o i n t on a Smith c h a r t and t h e n r e s c a l i n g i t t o t h i s c h a r t g i v e s t h e p o i n t shown a t A i n The c i r c l e s i n t e r s e c t i n g t h e v e r t i c a l a x i s a t r i g h t a n g l e s Fig. 3.09-1. give
t h e image impedance w h i l e t h e n e a r l y v e r t i c a l l i n e s g i v e t h e p h a s e c o n s t a n t . F o l l o w i n g t h e c i r c l e from p o i n t A a r o u n d t o t h e v e r t i c a l a x i s g i v e s a n o r m a l i z e d image impedance v a l u e o f R I / R L = 0 . 3 5 , w h i l e t h e p h a s e constant i s seen t o be approximately 0.37 A.
T h i s c h a r t u s e s t h e term
" c h a r a c t e r i s t i c impedance" f o r image impedance and e x p r e s s e s t h e image phase i n wavelengths f o r s p e c i f i c r e f e r e n c e t o t r a n s m i s s i o n l i n e s .
How-
e v e r , t h e more g e n e r a l image impedance c o n c e p t a l s o a p p l i e s and t h e c o r r e s p o n d i n g image p h a s e i n r a d i a n s ( w i t h i n some unknown m u l t i p l e o f T ) i s s i m p l y 277 t i m e s t h e number o f w a v e l e n g t h s .
P
Thus, i n t h i s c a s e
= 0.37(27~) + n r radians.
I f Z i r l / R L gave t h e p o i n t B i n F i g . 3 . 0 9 - 1 ,
t h e f i l t e r would be c u t
o f f , h e n c e , t h e image impedance would be i m a g i n a r y and ct would b e nonzero.
I n t h i s c a s e t h e image impedance i s r e a d by f o l l o w i n g t h e l i n e t o
t h e o u t e r e d g e of t h e c h a r t t o r e a d j X I / R L = j 1 . 4 , w h i l e t h e image a t t e n u a t i o n i n db i s r e a d from t h e h o r i z o n t a l a x i s of t h e c h a r t a s b e i n g a b o u t 8 . 5 db.
S i n c e t h e n e t w o r k i s s p e c i f i e d t o be s y m m e t r i c a l , t h e s t o p -
band image p h a s e w i l l b e z e r o o r some m u l t i p l e o f
T
r a d i a n s ( s e e Sec. 3.07).
REFERENCES
T. E. Shea, Transmission Networks and Wave F i l t e r s (D. Van Nostrand C o . , New York C i t y , 1929).
E. A. Guillemin, Communication N e t m r k s , Vol. 2, Chapters 4 , and 7 t o 10 (John Wiley and Sons, New York C i t y , 1935). Harvard Radio Research Laboratory S t a f f , Very High-Frequency Techniques, Vol. 2 , C h a p t e r s 2 6 and 27 by S. B. Cohn (McGraw-Hill Book Co., I n c . , New York C i t y , 1947).
M. E. Van Valkenburg, Network A n a l y s i s , Chapter 1 3 ( P r e n t i c e - H a l l , I n c . , Englewood C l i f f s , N . J . , 1955). H. N. , D a y i r s , "A Chart f o r Analyzing Transmission-Line F i l t e r s from Input Impedance Chara c t e r l s t l c s , " P r o c . IRE 43, pp. 436-443 ( A p r i l 1955). P. H. Smith, " A Transmission Line C a l c u l a t o r , " E l e c t r o n i c s 12, pp. 29-31 (January 1 9 3 9 ) . E. L. Ginzton, Microwave Measurements, pp. 228-234 (McGraw-Hill Book Co., I n c . , New York C i t y , 1957).
CHAPTER 4
LOW-PASS PROTOTYPE FILTERS OBTAINED BY NETWORK SYNTHESIS METBODS
SEC. 4 . 0 1 , INTRODUCTION Many o f t h e f i l t e r d e s i g n m e t h o d s t o b e d i s c u s s e d i n l a t e r c h a p t e r s
i
i
o f t h i s book w i l l make u s e o f t h e l u m p e d - e l e m e n t ,
I
pass, o r h a n d - s t o p microwave f i l t e r s t o be d i s c u s s e d w i l l d e r i v e t h e i r
!
i m p o r t a n t t r a n s m i s s i o n c h a r a c t e r i s t i c s from t h o s e o f a l o w - p a s s p r o t o t y p e f i l t e r used i n t h e i r d e s i g n . Element v a l u e s f o r such low-pass p r o t o t y p q
f i l t e r s discussed i n t h i s chapter.
low-pass p r o t o t y p e
Most o f t h e l o w - p a s s , h i g h - p a s s ,
band-
f i l t e r s .were o r i g i n a l l y o h t a i n e d by n e t w o r k s y n t h e s i s m e t h o d s , o f D a r l i n g t o r and o t h e r s ? - 3 venient of
However, more r e c e n t l y c o n c i s e equations4-" which a r e con-
f o r computer programming have been found f o r t h e e l e m e n t v a l u e s
t h e t y p e s o f p r o t o t y p e f i l t e r s of i n t e r e s t i n t h i s book, and numerous
f i l t e r d e s i g n s have heen t a b u l a t e d .
Some o f t h e t a b l e s i n t h i s book w e r e
o b t a i n e d f r o m t h e work o f ~ e i n b e r ~w ,h ~i l e, ~o t h e r s w e r e c o m p u t e d a t S t a n f o r d R e s e a r c h I n s t i t u t e f o r t h e p u r p o s e s o f t h i s book.
No d i s c u s s i o n
of f o r m a l network s y n t h e s i s methods w i l l b e i n c l u d e d i n t h i s book s i n c e t h e s e m a t t e r s a r e d i s c u s s e d e x t e n s i v e l y e l s e w h e r e ( s e e R e f s . 1 t o 3, f o r example), and s i n c e t h e a v a i l a b i l i t y o f t a b u l a t e d d e s i g n s makes s u c h d i s cussion unnecessary.
The m a i n o b j e c t i v e s o f t h i s c h a p t e r a r e t o make c l e a r
t h e p r o p e r t i e s o f t h e t a b u l a t e d p r o t o t y p e f i l t e r s , d e l a y n e t w o r k s , and i m p e d a n c e - m a t c h i n g n e t w o r k s s o t h a t t h e y may b e u s e d i n t e l l i g e n t l y i n t h e s o l u t i o n o f a wide v a r i e t y o f microwave c i r c u i t d e s i g n problems o f t h e s o r t s d i s c u s s e d i n C h a p t e r 1.
I t s h o u l d be n o t e d t h a t t h e s t e p t r a n s f o r m e r s i n C h a p t e r 6 c a n a l s o be u s e d a s p r o t o t y p e s f o r t h e d e s i g n o f c e r t a i n t y p e s o f m i c r o w a v e f i l t e r s as i s discussed i n Chapter 9. SEC. 4 . 0 2 ,
COMPAHISON O F IMAGE AND NETWORK SYNTHESIS METMOUS bOR F I L T E R DESIGN
As was d i s c u s s e d i n C h a p t e r 3 , t h e i m a g e i m p e d a n c e a n d a t t e n u a t i o n function of a f i l t e r s e c t i o n a r e d e f i n e d i n terms of an i n f i n i t e chain of i d e n t i c a l f i l t e r s e c t i o n s connected t o g e t h e r .
Using a f i n i t e ,
1
d i s s i p a t i o n l e s s f i l t e r network with r e s i s t o r terminations w i l l permit t h e image i m p e d a n c e s t o b e m a t c h e d o n l y a t d i s c r e t e f r e q u e n c i e s , a n d t h e r e f l e c t i o n e f f e c t s can c a u s e s i z e a b l e a t t e n u a t i o n i n t h e p a s s band, a s well a s d i s t o r t i o n of the stop-hand edges. I n S e c . 3 . 0 8 p r i n c i p l e s were d i s c u s s e d f o r t h e d e s i g n o f end s e c t i o n s which r e d u c e t h e s e r e f l e c t i o n e f f e c t s . A l t h o u g h s u c h m e t h o d s w i l l d e f i n i t e l y r e d u c e t h e s i z e o f r e f l e c t i o n s i n f i l t e r s d e s i g n e d by t h e image m e t h o d , t h e y g i v e no a s s u r a n c e a s t o how l a r g e t h e p e a k r e f l e c t i o n l o s s v a l u e s may b e i n t h e p a s s b a n d . T h u s , t h o u g h t h e image m e t h o d i s c o n c e p t u a i l y s i m p l e , i t r e q u i r e s a good d e a l o f " c u t a n d t r y " o r "know how" i f a p r e c i s i o n d e s i g n w i t h low p a s s - b a n d r e f l e c t i o n l o s s and v e r y a c c u r a t e l y d e f i n e d band e d g e s i s r e q u i r e d . Network s y n t h e s i s methodslfo3 f o r f i l t e r d e s i g n g e n e r a l l y s t a r t o u t by s p e c i f y i n g a t r a n s f e r f u n c t i o n [ s u c h a s t h e t r a n s m i s s i o n c o e f f i c i e n t t , d e f i n e d by tiq.
s EL
4.03,
MAXIMALLY FLA'I' AND TCHE3YSC9EF'F FILTEH ATTENUATION C;lAHACTEnISTICS
I
j
F i g u r e 4 . 0 3 - 1 shows a t y p i c a l m a x i m a l l y f l a t , * l o w - p a s s f i l t e r a t 'The f r e q u e n c y w ; , w h e r e t h e a t t e n u a t i o n i s L A r , i s defined a s t h e pass-hand edge. This c h a r a c t e r i s t i c i s expressed
1
where
I
i
'
tenuation c h a r a c t e r i s t i c .
1 i
I
The r e s p o n s e i n F i g . 4 . 0 3 - 1 c a n b e a c h i e v e d by l o w - p a s s f i l t e r c i r c u i t s s u c h a s t h o s e d i s c u s s e d i n S e c s . 4 . 0 4 and 4 . 0 5 , and t h e p a r a m e t e r n i n
(2.10-611 a s a f u n c t i o n of complex frequency p . From t h e t r a n s f e r f u n c t i o n t h e i n p u t impedance t o t h e c i r c u i t i s found a s a function o f p . T h e n , by v a r i o u s c o n t i n u e d - f r a c t i o n o r p a r t i a l - f r a c t i o n e x p a n s i o n
Eq. ( 4 . 0 3 - 1 ) c o r r e s p o n d s t o t h e numher of r e a c t i v e e l e m e n t s r e q u i r e d i n t h e
p r o c e d u r e s , t h e i n p u t i m p e d a n c e i s e x p a n d e d t o g i v e t h e e l e m e n t v a l u e s of t h e c i r c u i t . The c i r c u i t o b t a i n e d by t h e s e p r o c e d u r e s h a s t h e same t r a n s f e r
i s t i c a c q u i r e s i t s name m a x i m a l l y f l a t
f u n c t i o n t h a t was s p e c i f i e d a t t h e o u t s e t , a n d a l l g u e s s work a n d " c u t and try" is eliminated. Image c o n c e p t s n e v e r e n t e r s u c h p r o c e d u r e s , a n d t h e e f f e c t s o f t h e t e r m i n a t i o n s a r e i n c l u d e d i n t h e i n i t i a l s p r c i f i c a t i o n s of
has ( 2 n
circuit.
This attenuation character-
from t h e f a c t t h a t t h e q u a n t i t y w i t h i n t h e s q u a r e b r a c k e t s i n Eq. ( 4 . 0 3 - 1 )
-
1 ) z e r o d e r i v a t i v e s a t w' = 0.
I n most c a s e s w; f o r maximally
the transfer function.
f l a t f i l t e r s i s d e f i n e d a s t h e 3-db I n g e n e r a l , a l o w - p a s s f i l t e r d e s i g n e d by t h e image m e t h o d a n d an
hand-edge p o i n t .
F i g u r e 4 . 0 3 - 2 shows
a n a l o g o u s f i l t e r d e s i g n e d f o r t h e same a p p l i c a t i o n by n e t w o r k s y n t h e s i s
p l o t s of t h e s t o p - b a n d a t t e n u a t i o n
m e t h o d s w i l l b e q u i t e s i m i l a r . However, t h e f i l t e r d e s i g n e d b y n e t w o r k it s y n t h e s i s m e t h o d s w i l l h a v e somewhat d i f f e r e n t e l e m e n t v a l u e s , t o
c h a r a c t e r i s t i c s o f maximally f l a t f i l t e r s where L A r = 3 d b , f o r n = 1 t o 1 5 .
the specified response.
Note t h a t f o r c o n v e n i e n c e i n p l o t t i n g
The T c h e b y s c h e f f a n d m a x i m a l l y f l a t t r a n s f e r f u n c t i o n s d i s c u s s e d i n
the d a t a Iwt/w;
LL]
1
w'-
FIG.4.03-l
rodlonr
A--21-
92
AMAXIMALLYFLATLOWPASS ATTENUATION CHARACTERISTIC
- 1 was u s e d f o r t h e
'The m a g n i t u d e s i g n i s u s e d
t h e n e x t s e c t i o n a r e o f t e n s p e c i f i e d f o r f i l t e r a p p l i c a t i o n s . The f i l t e r s whose e l e m e n t v a l u e s a r e t a b u l a t e d i n S e c . 4 . 0 5 w i l l p r o d u c e r e s p o n s e s
abscissa.
discussed i n Sec. 4.03 e x a c t l y .
Pass o r b a n d - s t o p mappings t o be d i s c u s s e d i n l a t e r c h a p t e r s can y i e l d
iIowever, i n d e s i g n i n g m i c r o w a v e f i 1 t e r s from l o w - p a s s , lumped-element p r o t o t y p e s a p p r o x i m a t i o n s w i l l b e i n v o l v e d .
N e v e r t h e l e s s , t h e a p p r o x i m a t i o n s w i l l g e n e r a l l y h e v e r y good o v e r s i z e a b l e frequency r a n g e s , and t h e u s e of such p r o t o t y p e s i n d e t e r m i n i n g t h e p a r a m e t e r s o f t h e m i c r o w a v e f i l t e r w i l l e l i m i n a t e t h e g u e s s work i n h e r e n t i n t h e c l a s s i c a l image method.
on
ol/cl);
because t h e low-pass t o band-
n e g a t i v e v a l u e s of w l / w ; f o r which t h e a t t e n u a t i o n i s i n t e r p r e t e d t o be t h e same a s f o r p o s i t i v e v a l u e s o f ~ ' / w ; . A n o t h e r commonly u s e d a t t e n u a t i o n c h a r a c t e r i s t i c i s t h e T c h e b y s c h e f f In this case Lar o r " e q u a l - r i p p l e " c h a r a c t e r i s t i c ahown i n ~ ' i g . 4 . 0 3 - 3 . 8
T h i s c h a r a c t e r i s t i c is a l s o k n o w n a s a
Butternorth
f i l t e r characteristic.
T h i s t y p e o f c h a r a c t e r i s t i c c a n a l s o be a c h i e v e d by t h e f i l t e r s t r u c t u r e s d e s c r i b e d i n S e c s . 4 . 0 4 a n d 4 . 0 5 , and t h e p a r a m e t e r n i n E q s . ( 4 . 0 3 - 3 ) and ( 4 . 0 3 - 4 ) i s a g a i n t h e number o f r e a c t i v e e l e m e n t s i n t h e c i r c u i t . I f n i s e v e n t h e r e w i l l b e n/2
f r e q u e n c i e s where L A = 0 f o r a l o w - p a s s
T c h e b y s c h e f f r e s p o n s e , w h i l e i f n i s odd t h e r e w i l l b e ( n + 1 ) / 2 s u c h f r e q u e n c i e s . F i g u r e s 4 . 0 3 - 4 t o 4 . 0 3 - 1 0 show t h e s t o p - b a n d a t t e n u a t i o n c h a r a c t e r i s t i c s o f Tchebyscheff f i l t e r s having L A r = 0 . 0 1 , 0.10, 0 . 2 0 , 0.50, 1 . 0 0 , 2 . 0 0 , and 3 . 0 0 db pass-band r i p p l e .
Again,
Iw'/w;
I -
1is
used a s t h e a b s c i s s a .
I t i s i n t e r e s t i n g t o compare t h e maximally f l a t a t t e n u a t i o n c h a r a c t e r i s t i c s i n F i g . 4.03-2 with t h e 'Tchebyscheff c h a r a c t e r i s t i c s i n F i g s . 4.03-4
I t w i l l be s e e n t h a t f o r a given pass-band a t t e n u a t i o n t o l e r -
t o 4.03-10.
a n c e , L A r , a n d number o f r e a c t i v e e l e m e n t s , n , t h a t a T c h e b y s c h e f f f i l t e r w i l l g i v e a much s h a r p e r r a t e o f c u t o f f . F o r e x a m p l e , t h e m a x i m a l l y f l a t c h a r a c t e r i s t i c s i n F i g . 4.03-2 and t h e Tchebyscheff c h a r a c t e r i s t i c s i n Fig. 4 . 0 3 - 1 0 b o t h have L A r = 3 d b . For t h e n = 1 5 maximally f l a t c a s e ,
FIG. 4.03-2 ATTENUATION CHARACTERISTICS OF MAXIMALLY F L A T FILTERS The Frequency u,' i s the 3-db Band-Edge Point
70 d b a t t e n u a t i o n i s r e a c h e d a t w' = 1.7 w;; for the n = 15 Tcheb~scheff c a s e , 70 db a t t e n u a t i o n i s r e a c h e d a t Because of t h e i r s h a r p
w1 = 1 . 1 8 w ; .
i s a g a i n t h e maximum d b a t t e n u a t i o n i n t h e p a s s b a n d , w h i l e w ; i s t h e e q u a l - r i p p l e band e d g e . A t t e n u a t i o n c h a r a c t e r i s t i c s o f t h e form i n
cutoff, Tchebyscheff c h a r a c t e r i s t i c s
F i g . 4 . 0 3 - 3 may b e s p e c i f i e d m a t h e m a t i c a l l y a s
s i b l e c h a r a c t e r i s t i c s ; however, i f
are often preferred over other posthe r e a c t i v e elements o f a f i l t e r have a p p r e c i a b L e d i s s i p a t i o n l o s s t h e shape of t h e pass-band r e s p o n s e o f any t y p e o f f i l t e r w i l l b e a l t e r e d a s
and
compared w i t h t h e l o s s l e s s c a s e , a n d the e f f e c t s w i l l be p a r t i c u l a r l y l a r g e i n a 'l'chehyschef f f i l t e r . These m a t t e r s w i l l b e d i s c u s s e d i n
Set. 4 . 1 3 .
hlaximally f l a t f i l t e r s
have o f t e n b e e n r e p u t e d t o h a v e l e s s delay d i s t o r t i o n than Tchebyscheff
FIG. 4.03-3
A TCHEBYSCHEFF LOW-
PASS CHARACTERISTIC
FIG. 4.03-4 0.01-db-RIPPLE TCHEBYSCHEFF FILTER CHARACTERISTICS
FIG. 4.03-5
0.10-db-RIPPLE TCHEBYSCHFFF FILTER CHARACTERISTICS
FIG, 4.03-6
0.20-db-RIPPLE TCHEBYSCHEFF F I L T E R CHARACTERISTICS
FIG. 4.03-7 0.50-db-RIPPLE TCHEBYSCHEFF FILTER CHARACTERISTICS
FIG. 4.03-8
1.00-db-RIPPLE TCHEBYSCHEFF FILTER CHARACTERISTICS
FIG. 4.03-9
2.00-db-RIPPLE TCHEBYSCHEFF FILTER CHARACTERISTICS
f i l t e r s ; h o w e v e r , a s d i s c u s s e d i n S e c . 4 . 0 8 , t h i s may n o t b e t r u e , d e p e n d i n g on t h e s i z e o f L A r . The m a x i m a l l y f l a t a n d T c h e b y s c h e f f c h a r a c t e r i s t i c s i n F i g s . 4 . 0 3 - 1 and 4 . 0 3 - 3 a r e n o t t h e o n l y p o s s i b l e c h a r a c t e r i s t i c s o f t h i s t y p e .
For
example, t h e T c h e b y s c h e f f c h a r a c t e r i s t i c s o f t h e impedance-matchingn e t w o r k p r o t o t y p e s t o b e d i s c u s s e d i n S e c s . 4 . 0 9 and 4 . 1 0 w i l l b e s i m i l a r in shape, but L A w i l l n o t touch zero a t t h e bottom of t h e r i p p l e s .
Some-
t i m e s T c h e b y s c h e f f f i l t e r s a r e d e s i g n e d t o h a v e b o t h an e q u a l - r i p p l e c h a r a c t e r i s t i c i n t h e p a s s band, and an " e q u a l - r i p p l e " a s p e c i f i e d a t t e n u a t i o n l e v e l i n t h e s t o p band.
a p p r o x i m a t i o n of
Although such f i l t e r s
a r e used a t low f r e q u e n c i e s , t h e y a r e very d i f f i c u l t t o d e s i g n p r e c i s e l y f o r u s e a t m i c r o w a v e f r e q u e n c i e s . One p o s s i b l e e x c e p t i o n i s t h e t y p e of microwave f i l t e r d i s c u s s e d i n Sec. 7 . 0 3 .
SEC. 4 . 0 4 ,
D E F I N I T I O N OF C I R C U I T PARAMETEHS FOR LOW-PASS PROTOTYPE FILTERS
The e l e m e n t v a l u e s g o , g l , g 2 ,
. . . , g,,, g n t l o f
t h e low-pass p r o t o t y p e
f i l t e r s d i s c u s s e d i n t h i s c h a p t e r a r e defined a s s h o w n i n F i g . 4.04-1.
FIG. 4.03-10
3.00-db-RIPPLE TCHEBYSCHEFF FILTER CHARACTERISTICS
FIG. 4.04-1
DEFINITION OF PROTOTYPE FILTER PARAMETERS 901 91, 92,
"'I
g,,
9,+1
A prototype circuit i s shown at (a) ond its dual i s shown a t (b). Either form w i l l give the same response.
One p o s s i b l e form o f a p r o t o t y p e f i l t e r i s shown a t ( a ) w h i l e i t s d u a l i s shown a t ( b ) . E i t h e r f o r m may b e u s e d , s i n c e b o t h g i v e i d e n t i c a l r e s p o n s e s . S i n c e t h e n e t w o r k s a r e r e c i p r o c a l , e i t h e r t h e r e s i s t o r on
And, f o r c a p a c i t a n c e s ,
t h e l e f t o r t h e o n e o n t h e r i g h t may b e d e f i n e d a s t h e g e n e r a t o r i n t e r n a l impedance.
I t s h o u l d b e n o t e d t h a t i n F i g . 4 . 0 4 - 1 t h e f o l l o w i n g convent i o n s a r e observed: the inductance of a s e r i e s coil, g 4 = l t o n
go
=
=
{ {
In t h e s e e q u a t i o n s t h e p r i m e d q u a n t i t i e s a r e f o r t h e n o r m a l i z e d p r o t o t y p e and t h e u n p r i m e d q u a n t i t i e s a r e f o r t h e c o r r e s p o n d i n g s c a l e d c i r c u i t . As i n d i c a t e d from t h e p r e c e d i n g d i s c u s s i o n , f o r t h e p r o t o t y p e s i n t h i s & a p t e r , g o = Rh = 1 o r g o = Gh = 1.
or the capacitance of a shunt capacitor t h e g e n e r a t o r r e s i s t a n c e Rh i f g l = C;
A s an e x a m p l e o f how t h i s s c a l i n g 1 s a c c o m p l i s h e d , s u p p o s e t h a t we
,but is
(4.04-1) d e f i n e d a s t h e g e n e r a t o r c o n d u c t a n c e GA i f g l = L ; the load resistance Ritl
gnt1
/
i f gn =
LA
The r e a s o n f o r u s i n g t h e s e c o n v e n t i o n s i s t h a t t h e y l e a d t o e q u a t i o n s o f i d e n t i c a l form w h e t h e r a g i v e n c i r c u i t o r i t s d u a l i s u s e d .
T c h e b y s c h e f f f i l t e r w l t h 0 . 1 0 - d b r i p p l e and an e q u a l - r i p p l e b a n d e d g e [ s e e t h e c a s e o f 0 . 1 0 - d b r i p p l e and n = 2 i n of w ; = 1 r a d i a n . Assuming t h a t i t i s d e s i r e d t o s c a l e t h i s p r o t o t y p e Table 4 . 0 5 - 2 ( a ) . ] so t h a t Ro = 5 0 ohms and s o t h a t t h e e q u a l - r i p p l e band e d g e o c c u r s a t
i f g n = CA, b u t i s
d e f i n e d a s t h e l o a d c o n d u c t a n c e G:+l
have a l o w - p a s s p r o t o t y p e w i t h Rb = 1 . 0 0 0 ohm, C; = 0 . 8 4 3 0 f a r a d , These e l e m e n t v a l u e s a r e f o r a
L; = 0 . 6 2 2 0 h e n r y , and Gi = 1 . 3 5 5 4 mho.
Besides the
f l = 1 0 0 0 Mc, t h e n ( R o / R h ) = 5 0 , a n d (w;/wl) = l / ( 2 7 1 0 9 ) = 0 . 1 5 9 X N e x t , by E q s . ( 4 . 0 4 - 2 ) t o ( 4 . 0 4 - 4 ) , Ro = 50 ohm, C1 = ( 1 L 5 0 ) ( 0 . 1 5 9
X
(0.8430) = 2.68 X lo-'' f a r a d , L z = 50 ( 0 . 1 5 9 X l o a 9 ) ( 0 . 6 2 2 0 ) = 4 . 9 4 X 1 0 - l o h e n r y , a n d G j = ( 1 / 5 0 ) ( 1 . 3 5 5 4 ) = 0 . 0 2 7 1 mho.
c i r c u i t e l e m e n t v a l u e s , g k , a n a d d i t i o n a l prototype parameter, o;,w i l l a l s o be u s e d .
The p a r a m e t e r o; i s t h e r a d i a n f r e q u e n c y o f t h e p a s s - b a n d e d g e , which i s d e f i n e d i n F i g s . 4 . 0 3 - 1 and 4 . 0 3 - 3 f o r maximally f l a t and Tchebyscheff f i l t e r s of t h e s o r t d i s c u s s e d h e r e . I t s d e f i n i t i o n i n t h e case of maximally f l a t time-delay f i l t e r s i s d i s c u s s e d i n Sec. 4.07.
3 KC. 4 . 0 5 , DOUBLY TERMINATED, MAXIMALLY FLAT AND TCHEBYSCHEFF PROTOTYPE FILTERS For maximally f l a t f i l t e r s h a v i n g r e s i s t o r t e r m i n a t i o n s a t b o t h ends a r e s p o n s e o f t h e form o f t h a t i n F i g . 4 . 0 3 - 1 w i t h L A r = 3 d b , g o = 1 , and w ; = 1 , t h e e l e m e n t v a l u e s may b e computed a s
The e l e m e n t v a l u e s o f t h e p r o t o t y p e f i l t e r s d i s c u s s e d i n t h i s c h a p t e r
follow^:^
a r e a l l n o r m a l i z e d t o make g o = 1 a n d w ; = 1. T h e s e p r o t o t y p e s a r e e a s i l y c h a n g e d t o o t h e r i m p e d a n c e l e v e l s a n d f r e q u e n c y s c a l e s by t h e f o l l o w i n g transformations applied t o the c i r c u i t elements. conductances,
For,resistances o r
'cable 4 . 0 5 - l ( a ) g i v e s e l e m e n t v a l u e s f o r s u c h f i l t e r s h a v i n g For i n d u c t a n c e s ,
n =
1 to
10 r e a c t i v e e l e m e n t s , w h i l e T a h l e 4 . 0 5 - l ( b ) p r e s e n t s c o r r e s p o n d i n g f i l t e r s with
n =
11 t o 1 5 r e a c t i v e e l e m e n t s .
For Tchebyscheff f i l t e r s h a v i n g r e s i s t o r t e r m i n a t i o n s a t b o t h e n d s , r e s p o n s e s o f t h e f o r m shown i n F i g . 4 . 0 3 - 3 h a v i n g ripple, go Table 4 . 0 5 - l ( a )
1 , and
=
GJ;
=
L A r db pass-band
1 , t h e e l e m e n t v a l u e s may b e c o m p u t e d a s f 0 1 l o w s : ~ J
f i r s t compute
ELEMENT VALUES FOR FILTERS WITH MAXIMALLY FLAT ATTENUATION HAVING go = 1 , o; = 1 , and n = 1 t o 1 0 The r e s p o n s e s a r e o,f t h e form i n Fig. 4 . 0 3 - 1 w i t h L A r = 3 db
y
($)
=
rinh
gl
=
2a1 Y
tl
=
1 for
n
=
coth2
(+) f o r
then c o m p u t e
Table 4 . 0 5 - l ( b )
ELEMENT VALUES FOR FILTERS WITH MAXIMALLY FLAT ATTENUATION HAVING go = 1 , w; = 1, and n = 1 1 t o 15 The r e s p o n s e s a r e o f t h e form i n Fig. 4 . 0 3 - 1 w i t h L A r = 3 db
g,
odd n
even
Table 4 . 0 5 - 2 ( a ) g i v e s element v a l u e s f o r such f i l t e r s f o r v a r i o u s
L A r and
1 t o 10 r e a c t i v e elements. Table 4.05-2(b) g i v e s corresponding d a t a f o r f i l t e r s h a v i n g n = 11 t o 1 5 r e a c t i v e e l e m e n t s . =
I t w i l l be noted t h a t a l l o f t h e f i l t e r prototypes discussed i n t h i s n i s odd. I f n i s even, they have t h e property o f a n t i m e t r y m e n t i o n e d i n S e c s . 2 . 1 1 and 3 . 0 7 . U n d e r t h i s c o n d i t i o n o n e half of t h e network i s t h e r e c i p r o c a l of t h e o t h e r h a l f of t h e network w i t h r e s p e c t t o a p o s i t i v e r e a l c o n s t a n t R h , w h e r e R h may b e d e f i n e d a s section a r e symmetrical i f
Table 4.05-2(a) Table 4.05-2(a) C o n c l u d e d
ELEMENT VALUES FOR TCHEBYSCHEFF FILTERS HAVING g o = 1, o; = 1, AND RESPONSES OF THE FORM IN FIG. 4.03-3 WITH VARIOUS db RIPPLE Cases of n = 1 to 10
VALUE OF n
1'
2'
g3
'4
85
6'
7'
8'
'9
'10
'11
0.01 db r i p p l e
-.-
VALUE OF n
1'
2'
63
'4
8s
86
1.0 d b r i o o l e
L. lot
1.5386 1.2275 1.9761 2.1660 2.2394 2 2756 212963 2.3093 2.3181
1.0000 0.8468 1.3370 1.4555 1.5001 1.5217 1.5340 1.5417
1.5386 1.3394 2.0974 2 2756 213413 2.3728 2.3904
1 0000 0'8838 1'3781 1'4925 1'5340 1:5536
1 5386 1'3722 2:1349 2 3093 2:3720
1 0000 0'8972 1 5386 1'3938 1'3860 1 0000 1:5066 211514 0:9034 1.5386
1 9841 1'7372 2:5093 2 6678 2:7231
1 0000 0'8796 1 9841 1.2690 1'7504 1 0000 1:3485 2:5239 0:8842 1.9841
0.5 d b r i p p l e
0.6986 1.4029 1.5963 1.6703 1.7058 1.7254 1.7372 1.7451 1.7504 1.7543
1.0000 0.7071 1 0967 1'1926 1:2296 1 2479 1:2583 1.2647 1.2690 1.2721
1.9841 1 5963 2'3661 2:5408 2 6064 2:6381 2 6564 216678 2.6754
1 0000 0:8419 1:2296 1.3137 1.3444 1.3590 1.3673 1.3725
1 9841 1'7058 2'4758 2:6381 2 6964 217239 2.7392
1 0000 0'8696 1'2583 1'3389 1'3673 1:3806
Table 4.05-2(b) ELEMENT VALUES FOR TCHEBYSCHEFF FILTERS HAVING g o = 1 ,
a n d w h e r e R; a n d R:+l w;
= 1, AND RESPONSES
OF THE FORM IN F I G . 4 . 0 3 - 3 h I 1 H VARIOUS d h RIPPLE.
of the f i l t e r .
a r e t h e r i s i s t a n c e s of t h e terminations a t t h e ends
I f Z; i s t h e i m b e d a n c e o f o n e b r a n c h o f t h e f i l t e r l a d d e r \
network, t h e n
I
C a s e s o f n = 11 t o 1 5
RZ,
\s
A'
\
+ 1'7A
=
-
Zi
(4.05-4)
w h e r e Z,',+l-b i s t h e d u a l b r a n c h a t t h e o t h e r e n d o f t h e f i l t e r . By Eq. ( 4 . 0 5 - 4 ) i t w i l l be s e e n t h a t t h e i n d u c t i v e r e a c t a n c e s a t o n e end of t h e f i l t e r a r e r e l a t e d t o t h e c a p a c i t i v e s u s c e p t a n c e s a t t h e o t h e r e n d by
A1 s o ,
-
s o t h a t i t i s p o s s i b l e t o o h t a i n t h e element v a l u e s of t h e second h a l f of t h e f i l t e r from t h o s e o f t h e f i r s t h a l f i f t h e f i l t e r i s a n t i m e t r i c a l , ( a s w e l l a s when t h e f i l t e r i s s y m m e t r i c a l ) .
I t w i l l be found t h a t t h e symmetry and a n t i m e t r y p r o p e r t i e s d i s c u s s e d above w i l l o c c u r i n maximally f l a t and TcheLyscheff f i l t e r s o f t h e form i n Fig. 4.04-1 having terminations a t both ends, provided t h a t t h e f i l t e r i s d e s i g n e d s o t h a t L A = 0 a t o n e o r more f r e q u e n c i e s i n t h e p a s s band a s shown i n F i g s . 4 . 0 3 - 1 a n d 4 . 0 3 - 3 .
The m a x i m a l l y f l a t a n d T c h e b y s c h e f f
f i l t e r s d i s c u s s e d i n S e c s . 4 . 0 6 , 4 . 0 9 , and 4 . 1 0 do n o t have t h i s p r o p e r t y . The m a x i m a l l y f l a t t i m e - d e l a y f i l t e r s i n S e c . 4 . 0 7 a r e n o t s y m m e t r i c a l o r a n t i m e t r i c a l , even though L A = 0 a t w' = 0. I n some r a r e c a s e s d e s i g n s w i t h n g r e a t e r t h a n 1 5 may b e d e s i r e d . I n s u c h c a s e s g o o d a p p r o x i m a t e d e s i g n s c a n b e o b t a i n e d by a u g m e n t i n g a n n = 1 4 o r n = 1 5 d e s i g n b y r e p e a t i n g t h e two m i d d l e e l e m e n t s o f t h e
Thus, s u p p o s e t h a t an n = 1 8 d e s i g n i s d e s i r e d .
filter.
An n = 1 4 d e s i g n c a n be
a u g m e n t e d t o n = 1 8 hy b r e a k i n g t h e c i r c u i t i m m e d i a t e l y f o l l o w i n g t h e g7 ? l e m e n t , r e p e a t i n g e l e m e n t s g 6 a n d g, t w i c e , a n d t h e n c o n t i n u i n g on w i t h element gg and t h e r e s t o f t h e e l e m e n t s .
Thus, l e t t i n g primed g ' s i n d i c a t e
element v a l u e s f o r t h e n = 1 8 I i l t e r , and unprimed g ' s i n d i c a t e e l e m e n t v a l u e s from t h e n values
=
14 design, the
n =
1 8 d e s i g n would have t h e element,
Thus i n t h i s c a s e i t i s c o n v e n i e n ,t t o u s e t h e v o l t a g e a t t e n u a t i o n f u n c t i o n
T h i s i s , o f c o u r s e , a n a p p r o x i m a t e p r o c e d u r e , b u t i t i s b a s e d on t h e F i g u r e 4 . 0 6 - 2 shows t h e d u a l c k s e t o t h a t i n F i g . 4 . 0 6 - 1 .
f a c t t h a t f o r a given Tchebyscheff r i p p l e t h e element values i n a design
In this
change v e r y l i t t l e a s n i s v a r i e d , once n i s a r o u n d 1 0 o r more. T h i s i s r e a d i l y s e e n by c o m p a r i n g t h e e l e m e n t v a l u e s f o r d i f f e r e n t v a l u e s o f n ,
l a t t e r c a s e t h e c i r c u i t i s d r i v e n by a n i n f i n i t e - i m p e d a n c e c u r r e n t g e n e r a t o r and i t i s convenien t t o u s e t h e c u r r e n t a t t e n u a t i o n f u n c t i o n
down t h e c o l u m n s a t t h e l e f t i n 'Table 4 . 0 5 - 2 ( b ) .
defined as
SEC. 4 . 0 6 ,
LI
SINGLY TERMINATED MAXIMALLY FLAT AND TCHEBYSCHEFF F I L T E R S
A l l of t h e p r o t o t y p e f i l t e r s d i s c u s s e d i n Sec. 4 . 0 5 have r e s i s t o r t e r m i n a t i o n s a t b o t h e n d s . H o w e v e r , i n some c a s e s i t i s d e s i r a b l e t o u s e f i l t e r s with a r e s i s t o r t e r m i n a t i o n a t o n e end o n l y . F i g u r e 4 . 0 6 - 1 shows a n e x a m p l e
where
and a z e r o i n t e r n a l impedance v o l t a g e g e n e r a t o r on t h e r i g h t to drive the circuit.
FIG. 4.06-1
AN n a 5 REACTIVE ELEMENT SINGLY TERMINATED FILTER DRIVEN BY A ZERO-IMPEDANCE VOLTAGE GENE RATOR
In this
case t h e attenuation LA defined by Eq. ( 2 . 1 1 - 4 ) d o e s n o t a p p l y , s i n c e a z e r o i n t e r n a l impedance voltage generator has i n f i n i t e a v a i l a b l e power.
The power
a b s o r b e d by t h e c i r c u i t i s
20 l o g 1 ,
I g , I , R , a n d Z[,, a r e a s d e f i n e d i n F i g . 4 . 0 6 - 2 .
I f L A and L A r
i n S e c . 4 . 0 3 a r e r e p l a c e d by a n a l o g o u s q u a n t i t i e s L E a n d L E r , o r L I a n d L I r , a l l o f t h e e q u a t i o n s and c h a r t s i n S e c . 4 . 0 3 a p p l y t o t h e s i n g l y t e r m i n a t e d maximally f l a t o r 'Tchebyscheff f i l t e r s o f t h i s s e c t i o n a s well a s t o t h e doubly terminated f i l t e r s i n Sec. 4.05.
of such a f i l t e r with a res i s t o r t e r m i n a t i o n on t h e l e f t
=
Equation ( 4 . 0 6 - 1 ) h o w s t h a t f o r a given g e n e r a t o r v o l t a g e , E g , t h e power t r a n s m i s s i o n t h r o u g h t h e f i l t e r i s c o n t r o l l e d e n t i r e l y by Re Y:,. Thus, i f t h e f i l t e r i n F i g . 4 . 0 6 - 1 i s t o have a maximally f l a t o r T c h e b y s c h e f f t r a n s m i s s i o n c h a r a c t e r i s t i c , Re Y l , m u s t a l s o h a v e s u c h a c h a r a c t e r i s t i c . F i g u r e 4 . 0 6 - 3 shows t h e a p p r o x i m a t e s h a p e o f He Y f , and
Y i n f o r t h e c i r c u i t i n Fig. 4 . 0 6 - 1 i f designed t o g i v e a 'Tchebyscheff t r a n s m i s s i o n c h a r a c t e r i s t i c . 'The c u r v e s i n F i g . 4 . 0 6 - 3 a l s o a p p l y t o t h e c i r c u i t i n F i g . 4 . 0 6 - 2 i f Y l n i s r e p l a c e d by L i n . As w i l l b e d i s -
Im
c u s s e d i n C h a p t e r 1 6 , t h i s p r o p e r t y o f Re Y i n o r 3 e 21,
for singly loaded
f i l t e r s makes them q u i t e u s e f u l i n t h e d e s i g n o f d i p l e x e r s and m u l t i p l e x e r s . Prototypes of t h i s s o r t w i l l a l s o be u s e f u l f o r t h e d e s i g n of f i l t e r s t o b e d r i v e n by e n e r g y s o u r c e s t h a t l o o k a p p r o x i m a t e l y l i k e a
where Y i n and E g a r e d e f i n e d i n t h e F i g . 4 . 0 6 - 1 . m u s t h e a b s o r b e d i n G;,
S i n c e a l l o f t h e power
zero-impedance v o l t a g e g e n e r a t o r o r an i n f i n i t e - i m p e d a n c e c u r r e n t g e n e r a t o r . A t y p i c a l example is a p e n t o d e t u b e which, from i t s p l a t e c i r c u i t may r e s e m b l e a c u r r e n t g e n e r a t o r w i t h a c a p a c i t o r i n p a r a l l e l . I n s u c h c a s e s a broadhand response can be o b t a i n e d i f t h e s h u n t c a p a c i t a n c e i s used a s t h e f i r s t element of a s i n g l y terminated f i l t e r . Orchards g i v e s formulas f o r s i n g l y t e r m i n a t e d maximally f l a t f i l t e r s n o r m a l i z e d s o t h a t g o = 1 , and w ; = 1 a t t h e b a n d - e d g e p o i n t w h e r e L , = L I r o r L E = L E r i s 3 d b . They may be w r i t t e n a s f o l l o w s :
\
I 1
~
\'
\
with the element values
g1
=
a*
gn
=
FIG. 4.06-2 THE DUAL CIRCUIT TO THAT IN FIG. 4.06-1 In this case the generator is an infinite-impedance current generator.
[check:
,
ngl]
I
where t h e g k d e f i n e d above a r e t o be i n t e r p r e t e d a s i n Fig. 4 . 0 4 - l ( a ) and ( b ) . T a b l e 4 . 0 6 - 1 g i v e s e l e m e n t v a l u e s f o r s u c h f i l t e r s f o r t h e
I
cases of
I
I
i
n =
1 to
=n -
10. Table 4.06-1
ELEMENT VALUES FOR SINGLY TERMINATED MAXIMALLY FLAT FILTERS HAVING g o = 1, gncl = m, AND o; = 1
I
FIG. 4.06-3 THE APPROXIMATE FORM OF THE INPUT ADMITTANCE Y,: IN FIG. 4.06-1 FOR AN n = 5 REACTIVE-ELEMENT, SINGLY TERMINATED TCHEBYSCHEFF FILTER
VALUE OF n
g1
g2
64
g5
86
g10
gg
g7 I
I
I
I
9 Note:
Data b y c o u r t e s y of ..I
Wainberg a n d t h e J o u r n a l o f the F r a n k l i n I n s t i t u t e
F'or singly l o a d e d T c h e b y s c h e f f f i l t e r s h a v i n g g o = 1 , w ; = 1 , and
L I r o r L E r db pass-band r i p p l e , O r c h a r d ' s e q u a t i o n s C g i v e
Table 4.06-2
7
=
sinh
ELEMENT VALUES FOR SINGLY TERMINATED TCHEBYSCHEFF FILTERS HAVING = m, AND = 1 go = 1 , B,+l
(---)P VALUE OF^
gl
(4.06-7) with element v a l u e s
0.50 d b r i p p l e
Table4.06-2 p r e s e n t s element v a l u e s f o r s i n g l y terminated f i l t e r s f o r v a r i o u s amounts of Tchebyscheff r i p p l e .
1 2 3 4 5 6 7 8 9 10
m 0.3493 m 0.70140.9403 0.7981 1.3001 1.3465 0.8352 1.3916 1.7279 0.8529 1.4291 1.8142 0.8627 1.4483 1.8494 0.8686 1.4596 1.8675 0.8725 1.4666 1.8750 0.8752 1.4714 a 8 5 6 0.8771 1.4748 1.8905
1 2 3 4 5 6 7 8 9 10
0.7648 1.2441 1.3553 1.3962 1.4155 1.4261 1.4328 1.4366 1.4395 1.4416
m
1.3138 1.4426 1 . 5 3 8 8 1.7101 1.9018 1.7371 1.9712 1.7508 1.9980 1.7591 2.0116 1.7645 2.0197
m m 1.4042 1.7254 1.5982 1 . 7 8 3 8 1.9571 1 . 8 0 5 5 2.0203 1.8165 2.0432
m
1.4379 1.7571 1.8119
1.5238 1.9816
1.4539
1.9553 2.3794
1.2353
SEC. 4 . 0 7 . MAXIMALLY FLAT TIME- DELAY PROTOTYPE F I L T E R S The v o l t a g e a t t e n u a t i o n r a t i o ( E 2 ) a v a i , / E 2 ( s e e S e e . 2 . 1 0 ) f o r a n o r m a l i z e d , m a x i m a l l y f l a t , t i m e - d e l a y f i l t e r may b e d e f i n e d a s m s 9 2.00 d b r i p p l e
where p '
= c'
+ jo' i s t h e n o r m a l i z e d c o m p l e x - f r e q u e n c y v a r i a b l e ,
a r e a l , p o s i t i v e c o n s t a n t , and
c is
NOTE:
m
0.9766 i.2740 1.3389 1.3640 1.3765 1.3836 1.3881 1.3911 1.3932
m
1.7717 2.2169 2.3049 2.3383 2.3551 2.3h45 2.3707 2.3748
m
1.1727 1.4468 1.4974 1.5159 1.5251 1.5304 1.5337
1.9004 2.3304 2.4063 2.4332 2.4463 2.4538
m
1.2137 1.4836 1.5298 1.5495 1.5536
m
1.9379 2.3646 2.4386 2.4607
m
1.2284 1.4959 1.5419
M o s t o f t h e d a t a i n t h i s t a b l e w e r 5 o b t a i n e d by c o u r t e a y o f J o u r n a l o f the Franklin Institute.
L.
Weinberg and t h e
i s a B e s s e l p o l y n o m i a l f u n c t i o n o f I,$'. E q u a t i o n s 4 0 7 - 1 ) and ( 4 . 0 7 - 2 ) r e d u c e t o a s i m p l e p o l y n o m i a l o f t h e form
and f o r i n c r e a s i n g n t h e a t b e n u a t i o n a p p r o a c h e s t h e G a u s s i a n f ~ r r n ' ~ ~ ' ~ - ~
Let
2
For n
3 t h e 3 db bandwidth i s n e a r l y
(4.07-4)
T h e n , a s was d i s c u s s e d i n S e c . 1 . 0 5 , t h e t i m e d e l a y ( i . e . , g r o u p d e l a y ) i s
t>
=
d4' -
W e i n b e r g g h a s p r e p a r e d t a b l e s o f e l e m e n t v a l u e s f o r n o r m a l i z e d maxi-
secs
do'
m a l l y f l a t t i m e - d e l a y f i l t e r s , and t h e element v a l u e s i n T a b l e 4. 0 7 - 1 a r e These element v a l u e s a r e normalized s o t h a t t L o = l / w ; = f r o m h i s work.
w h e r e a' i s i n r a d i a n s p e r s e c o n d . The t r a n s f e r f u n c t i o n , d e f i n e d by E . 4 . 0 7 - and ( 4 . 0 7 - 2 ) h a s t h e p r o p e r t y t h a t i t s g r o u p d e l a y , t ; , h a s t h e maximum p o s s i b l e n u m b e r o f z e r o d e r i v a t i v e s w i t h r e s p e c t t o o' at '
= 0.
1 s e c o n d , a n d g o = 1.
I n order t o obtain a d i f f e r e n t time delay, t d O ,
t h e f r e q u e n c y s c a l e mus-t b e c h a n g e d b y t h e f a c t o r 1
w h i c h i s why i t i s s a i d t o h a v e m a x i m a l l y f l a t t i m e d e l a y .
The t i m e d e l a y , t ; , may b e e x p r e s s e d asiOvg
t:,
1
=
(4.07-6)
using t h e s c a l i n g procedure d i s c u s s e d i n Sec. 4.04.
Weinberg a l s o p r e -
s e n t s some c o m p u t e d d a t a s h o w i n g t i m e d e l a y a n d a t t e n u a t i o n i n t h e Table 4.07-1 ELEMENT VALUES FOR MAXIMALLY ?.AT TIME DELAY FILTERS HAVING go = 1 and ol = l / t d O = 1
w h e r e J - n - X ( o i / ~ ; )a n d J n + ( o l o ) a r e B e s s e l f u n c t i o n s o f r r l / o ; , a n d VALUE OF n
is t h e g r o u p d e l a y a s
W'
0.
61
62
83
g4
85
g6
87
6s
I
89
1
810
The magnitude of ( E 2 ) a v a i , / E 2 is
Note:
Data by courteay o f L. Weinberg and the Journal o f t h e Franklin ~ n a t i t u t a . '
111
I
gll
I
g12
v i c i n i t y o f t h e p a s s band f o r f i l t e r s w i t h n = 1 t o 11. H i s d a t a h a v e been p l o t t e d i n F i g s . 4 . 0 7 - 1 a n d 4 . 0 7 - 2 , and c u r v e s h a v e been d r a w n i n t o a i d i n i n t e r p o l a t i n g bbtween d a t a p o i n t s .
Although t h e t i m e - d e l a y
c h a r a c t e r i s t i c s a r e very konstant i n t h e pass-band region, these f i l t e r s
w i l l b e s e e n t o have low-pass f i l t e r a t t e n u a t i o n c h a r a c t e r i s t i c s which a r e g e n e r a l l y i n f e r i o r t o c h o s e of o r d i n a r y maximally f l a t a t t e n u a t i o n o r T c h e b y s c h e f f f i l t e r s h a v i n g t h e same number o f r e a c t i v e e l e m e n t s . SEC.
4.08,
COMPARISON OF THE TIME- DELAY*CHARACTERISTICS OF VARIOUS PROTOTYPE F I L T E R S
I f t h e terminations of a prototype f i l t e r a r e equal o r a r e n o t too g r e a t l y d i f f e r e n t , t h e g r o u p t i m e d e l a y a s w' ' 0 c a n b e computed f r o m
44
the r e l a t i o n t
A-un-u
NOTE: Plotted from data prepared hy L. Weinb g and published in the /ournal of the Fmnklin Institute.Y
FIG. 4.07-1 TIME-DELAY CHARACTERISTICS OF MAXIMALLY FLAT TIME-DELAY FILTERS where
g l ,
g2,
...,
g,, a r e t h e p r o t o t y p e e l e m e n t v a l u e s a s d e f i n e d i n
Fig. 4.04-1. A l s o i n T a b l e 4 . 1 3 - 1 a n d F i g . 4 . 1 3 - 2 a c o e f f i c i e n t C,, i s t a b u l a t e d f o r maximally f l a t and Tchebyscheff p r o t o t y p e f i l t e r s where tko
=
seconds
Cn
(4.08-2)
which i s k x a c t . I f t h e frequency s c a l e of a low-pass p r o t o t y p e is a l t e r e d s o t h a t OJ;
becomes
wl,
then t h e time s c a l e is a l t e r e d s o t h a t a s
U +
0 the
delay is
tdo =
4
tLo -
seconds
.
(4.08-3)
1
I f a b a n d - p a s s f i l t e r is d e s i g n e d from a low-pass p r o t o t y p e , t h e n t h e
wyw; s-YI-U
midband t i m e d e l a y i s ( a t l e a s t f o r n a r r o w - b a n d c a s e s ) $
NOTE: Plotted from date prepared by L. Weinherg and published in the Journal of the Franklin Institute.9
FIG. 4.07-2
PASS-BAND ATTENUATION CHARACTERISTICS OF MAXIMALLY FLAT TIME-DELAY FILTERS
*
''
The time d e l a y c h a r a c t e r i s t i c s of some a d d i t i o n a l structures w i l l be found i n Sec. 6.15. This equation is due t o S. B. Cohn and can he derived by use of Eqs. ( 4 . 1 3 - 9 ) and (4.13-11) t o f o l l o w . Thia is the approximate d e l a y for a lumped-clement band-pass f i l t e r c m r i s t i n g of r ladder of a e r i e s If transmission l i n e c i r c u i t s are used there may be additional t i w delay due and shunt resonator.. t o the physical length of the f i l t e r .
w h e r e u l a n d w2 a r e t h e p a s s - b a n d e d g e s o f t h e b a n d - p a s s r e s p o n s e c o r r e s p o n d i n g t o w; f o r t h e l o w - p a s s r e s p o n s e .
j
In order t o determine t h e time delay a t o t h e r f r e q u e n c i e s it i s n e c e s s a r y t o work f r o m t h e t r a n s f e r f u n c t i o n s .
For a l l o f t h e p r o t o t y p e f i l t e r s d i s c u s s e d i n t h i s c h a p t e r t h e v o l t a g e a t t e n u a t i o n r a t i o (E,)a,,i,/fi
d e f i n e d i n S e c . 2 . 1 0 c a n b e r e p r e s e n t e d by a p o l y n o m i a l P n ( p l ) s o t h a t
and c i s a g a i n a r e a l c o n s t a n t . T h e c o n s t a n t s c i n E q s . ( 4 . 0 8 - 5 ) t o ( 4 . 0 8 - 9 ) a r e t o b e e v a l u a t e d s o a s t o f i x t h e minimum a t t e n u a t i o n o f t h e
where p i = u ' + j w ' i s t h e c o m p l e x f r e q u e n c y v a r i a b l e . In t h e case of prototype f i l t e r s with maximally f l a t a t t e n u a t i o n , n r e a c t i v e elements, w ; = 1 , and L A r = 3 d b ( s e e F i g ,
response.
For example, f o r t h e Tchebyscheff r e s p o n s e i n Fig. 4 . 0 3 - 3 ,
c
would b e e v a l u a t e d s o a s t o make, L A = 20 l o g l O ( E 2 ) a r a i l / E 2 = 0 a t t h e
4 . 0 3 - l ) , P n ( p l ) is f o r n even
bottom o f t h e p a s s - b a n d r i p p l e s .
However, f o r t h e T c h e b y s c h e f f r e s p o n s e
i n t h e impedance-matching f i l t e r response t o be presented i n Fig. 4.09-2 a d i f f e r e n t v a l u e o f c would b e r e q u i r e d s i n c e L A n e v e r g o e s t o z e r o i n t h i s l a t t e r case.
Both c a s e s would, however, have i d e n t i c a l phase s h i f t
and t i m e d e l a y c h a r a c t e r i s t i c s . and f o r n o d d The p h a s e s h i f t and g r o u p t i m e d e l a y f o r f i l t e r s w i t h m a x i m a l l y f l a t o r T c h e b y s c h e f f a t t e n u a t i o n c h a r a c t e r i s t i c s c a n b e computed by u s e o f ~ohn* Eqs. ( 4 . 0 8 ' 5 ) t o ( 4 . 0 8 - 9 ) a b o v e a n d Eqs. ( 4 . 0 7 - 4 ) a n d ( 4 . 0 7 - 5 ) .
(n-1)/2 Pn(pl)
=
c(p
+
1)
77 a=
[(Pi)'
+ (2
nm
cos -)p'
t 11
1
h a s computed t h e p h a s e and t i m e d e l a y c h a r a c t e r i s t i c s f o r v a r i o u s p r o t o t y p e f i l t e r s w i t h n = 5 r e a c t i v e e l e m e n t s i n o r d e r t o compare t h e i r
I
(4.08-6)
r e l a t i v e m e r i t i n s i t u a t i o n s where t i m e - d e l a y c h a r a c t e r i s t i c s a r e i m -
where c i s a r e a l c o n s t a n t .
portant.
For T c h e b ~ s c h e f f p r o t o t y p e f i l t e r s h a v i n g n r e a c t i v e e l e m e n t s , and
LAr
db r i p p l e ( s e e Fig.
4.03-3), P n ( P ' ) is f o r n even
2n
1,
H i s r e s u l t s a r e shown i n F i g . 4 . 0 8 - 1 t o 4 . 0 8 - 3 .
F i g u r e 4 . 0 8 - 1 shows t h e p h a s e c h a r a c t e r i s t i c s o f T c h e b ~ s c h e f f f i l t e r s h a v i n g 0 . 0 1 - d b a n d 0 . 5 - d b r i p p l e w i t h w ; = 1, and a m a x i m a l l y f l a t a t t e n u a t i o n f i l t e r w i t h i t s 3-db ~ o i n ta t w; = 1. T h e 3-db p o i n t s o f t h e Tchebyscheff f i l t e r s a r e a l s o i n d i c a t e d . Note t h a t t h e 0.5-db r i p p l e f i l t e r 1 a s c o n s i d e r a b l y more c u r v a t u r e i n i t s p h a s e c h a r a c t e r i s t i c t h a n It w i l l e i t h e r t h e 0.01-db r i p p l e o r maximally f l a t a t t e n u a t i o n f i l t e r s . be found t h a t i n g e n e r a l t h e l a r g e r t h e r i p p l e of a Tchebyscheff f i l t e r t h e l a r g e r t h e c u r v a t u r e of t h e phase character;st,ic w i l l be i n t h e As a r e s u l t , t h e l a r g e r t h e r i p p l e , t h e more t h e d e l a y v i c i n i t y of w;. d i s t o r t i o n w i l l be near cutoff.
0.5 d b RIPPLE
-
- MAXIMALLY FLAT ATTENUATION
1
SOURCE: Final Report, Contract DA 36-039 SC-74862, Stanford Research Institute, reprinted in The Microwave Journal ( s e e Ref. 13 by S. B. Cohn).
FIG. 4.08-1
PHASE-SHIFT CHARACTERISTICS OF FILTERS WITH MAXIMALLY FLAT OR T C H E B Y S C H ~ ATTENUATION RESPONSES AND n = 5
SOURCE: Final Report, Contract DA 36-039 SC-74862, Stanford Research Institute, reprinted in The Microwave Journal ( s e e Ref. 13 by S. B. Cohn).
FIG. 4.08-2
NORMALIZED TIME DELAY vs. w f / w i d b FOR VARIOUS PROTOTYPE FILTERS
I
1
1
0 . 8 8 wl/w;
,ttenuation a t t h a t frequency.
1
(See Fig. 4.07-2.)
Thus, i t i s s e e n t h a t
the c o s t of a l e s s c o n s t a n t a t t e n u a t i o n c h a r a c t e r i s t i c .
!
i
w h i l e t h e maximally f l a t time d e l a y f i l t e r has about 2 . 2 db
m a x i m a l l y f l a t t i m e - d e l a y f i l t e r s a c h i e v e a more c o n s t a n t t i m e d e l a y a t
I n some c a s e s a band o f low l o s s a n d low d i s t o r t i o n i s d e s i r e d u p
! 1
dh
t o a c e r t a i n f r e q u e n c y and t h e n a s p e c i f i e d h i g h a t t e n u a t i o n i s d e s i r e d a t an a d j a c e n t h i g h e r f r e q u e n c y . F i g u r e 4 . 0 8 - 3 shows t h e t i m e - d e l a y c h a r a c t e r i s t i c s o f v a r i o u s pro,,totype f i l t e r s w i t h t h e f r e q u e n c y s c a l e n o r m a l i z e d t o t h e 6 0 - d b a t t e n u a t i o n f r e q u e n c y uLo d b f o r e a c h f i l t e r .
For
I
a f l percent tolerance-on t d , a 0.1-db-ripple
I
t.o 0 . 1 0 6 wLo
I
r i p p l e f i l t e r i s u s a b l e t o 0.154 w i o d b w h i l e t h e maximally f l a t a t t e n u a t i o n f i l t e r i s u s a h l e o n l y t o 0 . 1 1 6 w L O d b . The m a x i m a l l y f l a t t i m e - d e l a y f i l t e r
' 1
f i l t e r is found t o h e u s a b l e
w h i l e a maximally f l a t a t t e n u a t i o n f i l t e r i s w i t h i n t h i s t o l e r a n c e o n l y t o 0 . 0 4 0 m i o d b . F o r a f 1 0 p e r c e n t t o l e r a n c e on t d a 0 . 5 - d b db
a g a i n h a s by f a r t h e b r o a d e s t u s a b l e b a n d f o r a g i v e n t i m e - d e l a y t o l e r a n c e ; For
however, i t s r e f l e c t i o n l o s s w i l l a g a i n be an i m p o r t a n t c o n s i d e r a t i o n . e x a m p l e , f o r w' SOURCE: Final Report, Contract DA 36-039 SC-74862, Stanford Research Institute, reprinted in The Microwave lournal ( s e e Ref. 13 by S. B. Cohn).
FIG. 4.08-3
NORMALIZED TIME DELAY PROTOTYPE FILTERS
..-,.(.-.,.-,."
i s 3 d b f o r w'
= 0 . 1 m i Od b i t s a t t e n u a t i o n i s 1 . 2 5 db and i t s a t t e n u a t i o n = 0 . 1 5 wL0 d b .
In contrast the 0.1-db-ripple prototype f i l t e r
h a s 0 . 1 d b a t t e n u a t i o n o r l e s s o u t t o w' = 0 . 2 9 4 m i o d b . The c h o i c e b e t w e e n t h e s e v a r i o u s t y p e s o f f i l t e r s w i l l d e p e n d o n t h e a p p l i c a t i o n under consideration.
vs. ~ ' / w g FOR ' ~ VARIOUS ~ ~
I n most c a s e s where t i m e d e l a y i s o f
i n t e r e s t i n microwave f i l t e r s , t h e f i , l t e r s u s e d w i l l f i l t e r s of narrow o r moderate bandwidth.
rob ably b e b a n d - p a s s
S u c h f i l t e r s c a n be d e s i g n e d
from p r o t o t y p e f i l t e r s o r s t e p t r a n s f o r m e r s by m e t h o d s d i s c u s s e d i n F i g u r e 4 . 0 8 - 2 shows t h e t i m e d e l a y c h a r a c t e r i s t i c s o f 0 . 1 - and 0.5-db r i p p l e Tchebyscheff f i l t e r s , o f a maximally f l a t a t t e n u a t i o n f i l t e r , a l o n g with t h a t of a maximally f l a t t i m e d e l a y f i l t e r . The 0, s c a l e o f t d i s n o r m a l i z e d t o t h e t i m e d e l a y t d O , o b t a i n e d a s w' +
a n d t h e f r e q u e n c y s c a l e i s n o r m a l i z e d t o t h e f r e q u e n c y a; d,,
where
L A = 3 db f o r each case. Note t h a t t h e t i m e - d e l a y c h a f a c t e r i s t i c of t h e 0.5-db r i p p l e f i l t e r i s q u i t e e r r a t i c , b u t t h a t d e l a y c h a r a c t e r i s t i c s f o r t h e 0.1-db r i p p l e f i l t e r a r e s u p e r i o r t o t h o s e of t h e maximally f l a t a t t e n u a t i o n f i l t e r . The 0 . 1 - d b - r i p p l e c u r v e is c o n s t a n t w i t h i n 51 p e r c e n t f o r w l / a ; d b 5 0 . 3 1 w h i l e t h e m a x i m a l l y f l a t f i l t e r i s w i t h i n t h i s tolerance only f o r w'/w;
d,,
2
0.16.
The m a x i m a l l y f l a t t i m e - d e l a y f i l t e r i s s e e n t o h a v e by f a r t h e most c o n s t a n t t i m e d e l a y o f a l l . Howe v e r , t h e e q u a l - r i p p l e band f o r t h e 0 . 1 - d b - r i p p l e f i l t e r e x t e n d s t o
C h a p t e r s 8 , 9 , and 1 0 . '
For c a s e s where t h e s p e c t r u m o f a s i g n a l b e i n g
t r a n s m i t t e d i s a p p r e c i a b l e a s compared w i t h t h e bandwidth o f t h e f i l t e r , v a r i a t i o n s i n e i t h e r time delay o r pass-band a t t e n u a t i o n within t h e s i g n a l spectrum w i l l cause s i g n a l distortion.13
However, f o r e x a m p l e , a m a x i m a l l y
f l a t t i m e - d e l a y f i l t e r which h a s v e r y l i t t l e d e l a y d i s t o r t i o n a n d a monot o n i c a l l y i n c r e a s i n g a t t e n u a t i o n w i l l tend t o round a p u l s e out w i t h o u t overshoot o r r i n g i n g , while a f i l t e r with a s h a r p c u t o f f (such a s a T c h e b y s c h e f f f i l t e r ) w i l l t e n d t o c a u s e r i n g i n g . 1 3 The t r a n s i e n t r e s p o n s e requirements f o r t h e given a p p l i c a t i o n w i l l be dominant c o n s i d e r a t i o n s when c h o o s i n g a f i l t e r t y p e f o r s u c h c a s e s w h e r e t h e s i g n a l s p e c t r u m a n d f i l t e r p a s s band a r e o f s i m i l a r bandwidth.
* A S 1s d i a c u a s s d I n S e c . 1 . 0 5 , m o s t m ~ c r o r a v s f i l t e r s w i l l h a v e e x t r a t l n e d e l a y o v e r t h a t o f their p r o t o t y p e s b e c a u s e o f t h e e l e c t r i c a l l e n g t h o f t h e l r physical s t r u c t u r e s .
I n o t h e r s i t u a t i o n s t h e s i g n a l s p e c t r u m may be n a r r o w c o m p a r e d w i t h t h e bandwidth o f t h e f i l t e r s o t h a t t h e s p e c t r a l components o f a g i v e n
good m a t c h i s r e q u i r e d .
Thus, t h e s h a r p e r t h e c u t o f f o f a properly
designed matching network, t h e b e t t e r i t s performance can be.
s i g n a l s e e e s s e n t i a l l y c o n s t a n t a t t e n u a t i o n a n d d e l a y f o r a n y common f i l t e r r e s p o n s e , a n d d i s t o r t i o n o f t h e s i g n a l s h a p e may t h u s b e n e g l i g i b l e , I n s u c h c a s e s a c h o i c e o f f i l t e r r e s p o n s e t y p e s may d e p e n d o n c o n s i d e r a t i o n s of allowable time d e l a y t o l e r a n c e over t h e range o f p o s s i b l e f r e quencies, allowable variation of attenuation i n the c a r r i e r operating band, and r e q u i r e d r a t e o f c u t o f f . For example, i f time-delay constancy was o f m a j o r i m p o r t a n c e a n d i t d i d n ' t m a t t e r w h e t h e r s i g n a l s w i t h d i f f e r e n t c a r r i e r f r e q u e n c i e s s u f f e r e d d i f f e r e n t amounts o f a t t e n u a t i o n , a
i f t h e l o a d h a s a r e a c t i v e p a r t , p e r f e c t power t r a n s m i s s i o n t o t h e l o a d i s p o s s i b l e o n l y a t d i s c r e t e f r e q u e n c i e s , and n o t o v e r a b a n d o f f r e F u r t h e r m o r e , i t w i l l u s u a l l y be found t h a t t h e o v e r - a l l
quencies.
t r a n s m i s s i o n c a n b e i m p r o v e d i f a t l e a s t a s m a l l amount o f power i s r e f l e c t e d a t a l l f r e q u e n c i e s . T h i s i s i l l u s t r a t e d i n F i g . 4 . 0 9 - 2 , where i t w i l l b e a s s u m e d t h a t t h e d e s i g n e r ' s o b j e c t i v e i s t o k e e p ( L A ) m a xa s
s m a l l a s p o s s i b l e f r o m w'
m a x i m a l l y f l a t t i m e - d e l a y f i l t e r would b e t h e b e s t c h o i c e . SEC. 4 . 0 9 ,
Another important p r o p e r t y o f impedance-matching networks is t h a t
=
0 t o w' = w ; , w h e r e t h e d b a t t e n u a t i o n L A
r e f e r s t o t h e a t t e n u a t i o n o f t h e power r e c e i v e d by t h e l o a d w i t h r e s p e c t I f ( L A ) m i ni s t o t h e a v a i l a b l e power o f t h e g e n e r a t o r ( s e e S e c . 2 . 1 1 ) .
PROTOTYPE, TCHEBYSCHEFF IMPEDANCE-MATCHING NETWOHKS GIVING MINIMUM REFLECTION
made v e r y s m a l l s o a s t o g i v e v e r y e f f i c i e n t t r a n s m i s s i o n a t t h e b o t t o m s
I n t h i s s e c t i o n t h e l o w - p a s s impedance m a t c h i n g o f l o a d s r e p r e s e n t a b l e a s a r e s i s t a n c e a n d i n d u c t a n c e i n s e r i e s , and o f l o a d s r e p r e -
o f t h e p a s s - b a n d r i p p l e s , t h e e x c e s s i v e l y good t r a n s m i s s i o n a t t h e s e p o i n t s m u s t b e c o m p e n s a t e d f o r by e x c e s s i v e l y p o o r t r a n s m i s s i o n a t t h e On t h e c r e s t s o f t h e r i p p l e s , and a s a r e s u l t , ( L A ) n a xw i l l i n c r e a s e .
s e n t a b l e a s a r e s i s t o r and L{'o\
--
L&
.
- -
--
I
-
-
-
c a p a c i t a n c e i n p a r a l l e l w i l l be d i s c u s s e d . A l o a d o f t h e former
f+g5 -
type with a matching network of
-
t h e s o r t t o b e t r e a t e d i s shown
MATCWINQ NETWORK
FIG. 4.09-1
A-Bw-.~
i n F'ig. 4 . 0 9 - 1 . In general, the e l e m e n t s go, and g l i n t h e c i r c u i t s i n Fig. 4 . 0 4 - l ( a ) ,
A LOAD WITH A LOW-PASS IMPEDANCE-MATCHING NETWORK (Case of n = 4)
( b ) may
be r e g a r d e d a s l o a d s , a n d ,.he remainder of t h e r e a c t i v e e l e ments regarded a s impedancematching networks.
For convenience
i t w i l l b e a s s u m e d t h a t t h e impedance l e v e l o f t h e l o a d t o b e matched h a s been normalized s o t h a t t h e r e s i s t o r o r conductance i s e q u a l t o one, and t h a t t h e f r e q u e n c y s c a l e h a s b e e n n o r m a l i z e d s o t h a t t h e e d g e o f t h e d e s i r e d b a n d o f good i m p e d a n c e match i s w ; = 1. As was d i s c u s s e d i n S e c . 1 . 0 3 , i f a n i m p e d a n c e h a v i n g a r e a c t i v e p a r t i s t o b e m a t c h e d o v e r a band o f f r e q u e n c i e s , a n optimum i m p e d a n c e - m a t c h i n g
n e t w o r k m u s t n e c e s s a r i l y h a v e a f i l t e r - l i k e c h a r a c t e r i s t i c . Any d e g r e e o f i m p e d a n c e m a t c h i n f r e q u e n c y r e g i o n s o t h e r t h a n t h a t f o r w h i c h a g o o d match i s r e q u i r e d w i l l d e t r a c t from t h e p e r f o r m a n c e p o s s i b l e i n t h e b a n d w h e r e
FIG. 4.09-2
DEFINITION OF (LA)ma,AND (LA)min FOR TCHEBYSCHEFF IMPEDANCE MATCHING NETWORKS DISCUSSED HEREIN
o t h e r h a n d i f ( L A ) m i n i s s p e c i f i e d t o b e n e a r l y e q u a l t o ( L A ) m m x t, h e small pass-band r i p p l e w i l l r e s u l t i n a reduced r a t e of c u t o f f f o r the f i l t e r ; a s i n d i c a t e d above, t h i s reduced r a t e o f c u t o f f w i l l degrade the to increase. Thus, i t i s s e e n t h a t
performance and a l s o c a u s e
f o r a g i v e n l o a d , a g i v e n number o f i m p e d a n c e - m a t c h i n g e l e m e n t s , and a g i v e n i m p e d a n c e - m a t c h i n g b a n d w i d t h , t h e r e i s some d e f i n i t e v a l u e o f T c h e b y s c h e f f p a s s - b a n d r i p p l e ( L A ) m a x- ( L A ) m i n t h a t g o e s w i t h a minimum value of
The p r o t o t y p e i m p e d a n c e - m a t c h i n g n e t w o r k s d i s c u s s e d i n
t h i s s e c t i o n a r e optimum i n t h i s s e n s e , ~ . e . , t h e y d o m i n i m i z e ( L A ) m a x f o r a l o a d and i m p e d a n c e - m a t c h i n g n e t w o r k o f t h e form i n F i g . 4 . 0 9 - 1 o r i t s g e n e r a l i z a t i o n i n terms of F i g s . 4.04-l(a)., ( b ) .
I t i s c o n v e n i e n t t o c h a r a c t e r i z e t h e l o a d s u n d e r c o n s i d e r a t i o n by t h e i r d e c r e m e n t , which i s d e f i n e d a s
where t h e v a r i o u s q u a n t i t i e s i n t h i s e q u a t i o n a r e a s i n d i c a t e d i n F i g s . 4 . 0 3 - 1 , 4 . 0 9 - 2 , and 4 . 0 4 - l ( a ) , of t h e
Q
(h).
Note t h a t 6 is t h e r e c i p r o c a l
o f t h e l o a d e v a l u a t e d a t t h e edge o f t h e impedance-matching
Land and t h a t 6 e v a l u a t e d f o r t h e u n - n o r m a l i z e d l o a d i s t h e sane as t h a t f o r the normalized load.
F i g u r e 4 . 0 9 - 3 shows t h e minimum v a l u e o f (L.A)m,x vs 6 f o r c i r c u i t s h a v i n g n = 1 t o n = 4 r e a c t i v e e l e m e n t s ( a l s o f o r c a s e
of n =
S i n c e one of t h e r e a c t i v e e l e m e n t s i n each c a s e i s p a r t o f t h e l o a d , t h e n = 1 c a s e i n v o l v e s no L o r C i m p e d a n c e - m a t c h i n g e l e m e n t s ,
t h e optimum r e s u l t b e i n g d e t e r m i n e d o n l y by optimum c h o i c e o f d r i v i n g g e n e r a t o r i n t e r n a l impedance.
Note t h a t f o r a g i v e n ' v a l u e o f 6 , i s d e c r e a s e d by u s i n g more c o m p l e x m a t c h i n g n e t w o r k s ( i . e . , l a r g e r v a l u e s o f n ) . However, a p o i n t o f d i m i n i s h i n g r e t u r n s i s r a p i d l y r e a c h e d s o t h a t i t i s u s u a l l y n o t w o r t h w h i l e t o go b e y o n d n = 3 o r 4. n = IX i s n o t g r e a t l y b e t t e r t h a n n = 4 .
Note t h a t
F i g u r e 4 . 0 9 - 4 sh.ows t h e d b T c h e b y s c h e f f r i p p l e us 6 f o r minimum ( L A ) m a x . Once a g a i n , g o i n g t o l a r g e r v a l u e s o f n w i l l g i v e b e t t e r r e s u l t s 1 s i n c e when n i s i n c r e a s e d , t h e s i z e o f t h e r i p p l e i s r e d u c e d f o r a g i v e n 8 . For n = t h e r i p p l e goes t o z e r o .
FIG. 4.09-3
( L A ) m a xV S . 6 FOR THE IMPEDANCE-MATCHING NETWORKS WHOSE ELEMENT VALUES ARE GIVEN IN FIGS. 4.09-5 TO -8
F i g u r e s 4 . 0 9 - 5 t o 4 . 0 9 - 8 show c h a r t s o f e l e m e n t v a l u e s optimum T c h e b y s c h e f f m a t c h i n g n e t w o r k s . i l l u s t r a t e d by a n e x a m p l e .
6 for
us
Their use i s probably best
Suppose t h a t an impedance match i s d e s i r e d
t o a l o a d w h i c h c a n b e r e p r e s e n t e d a p p r o x i m a t e l y by a 50-ohm r e s i s t o r (Go = 0 . 0 2 0 mho) i n s e r i e s w i t h an i n d u c t a n c e L1
=
3.98
X
lo-'
henry,
and t h a t good i m p e d a n c e m a t c h i s t o e x t e n d u p t o f l = 1 Gc s o t h a t wl = 2 n f l = 6 . 2 8 1/(0.020
X
6.28
X X
lo9. 10'
X
Then t h e d e c r e m e n t i s 6 3.98
lo-')
X
Figs. 4 . 0 9 - 3 and 4.09-4 f o r 6
=
= 0.20.
=
l/(GowlL,)
=
After consulting
0 . 2 0 l e t us suppose t h a t n = 4 i s chosen
w h i c h c a l l s f o r ( L A ) m l x = 1 . 9 d b a n d a r i p p l e o f a b o u t 0 . 2 5 d b . T h e n by F i g . 4 . 0 9 - 8 ( w h i c h i s f o r n = 4 ) we o b t a i n f o r g o = 1 , w ; = 1 , a n d
6
= 0.20:
g l / 1 0 = 0 . 5 0 , g, = 0 . 4 4 5 , g , / l 0 = 0 . 5 4 , g 4 = 0 . 2 0 5 , a n d This corresponds t o the c i r c u i t i n Fig. 4.09-1 with
g5/10 = 0.39. go
Gb = 1, g 1 = 5 . 0 0 = L ; , g , = 0 . 4 4 5 = C;, g , = 5 . 4 0 = L j , g 4 = 0 . 2 0 5 U n - n o r m a l i z i n g t h i s by u s e o f E q s . ( 4 . 0 4 - 2 ) t o , and g 5 = 3 . 9 0 = R;. =
. 0 4 - 4 ) w i t h ( G o / G b ) = 0 . 0 2 0 / 1 a n d w;/wl ves:
Go = 0 . 0 2 0 mho, L1 = 3 . 9 8
= 1/(6.28
X
10')
= 1.59
X
=
10-lo
l o - ' h e n r y , C, = 1 . 4 1 5 x l o - ' , f a r a d , f a r a d , and R5 = 195 ohms. Note = 4 . 2 9 X l o - ' h e n r y , C4 = 6 . 5 2 X 1 0 - I . h a t Go a n d L1 a r e t h e o r i g i n a l e l e m e n t s g i v e n f o r t h e l o a d . The p h y s i c a l r e a l i z a t i o n o f m i c r o w a v e s t r u c t u r e s f o r s u c h a n a p p l i c a t i o n c a n b e accomp l i s h e d u s i n g techniques discussed i n Chapter 7. X
I t i s i n t e r e s t i n g t o n o t e how much t h e i m p e d a n c e - m a t c h i n g n e t w o r k d e s i g n d i s c u s s e d a b o v e a c t u a l l y i m p r o v e s t h e power t r a n s f e r t o t h e l o a d . I f t h e A-L l o a d t r e a t e d a b o v e w e r e d r i v e n d i r e c t l y by a g e n e r a t o r w i t h a 50-ohm i n t e r n a l i m p e d a n c e , t h e l o s s w o u l d a p p r o a c h 0 d b a s f i t w o u l d b e 8 . 6 d b a t f l = 1 Gc.
-
0, but
t h e optimum
1 d e s i g n f o r t h i s c a s e would c a l l f o r t h e g e n e r a t o r i n t e r n a l impeda n c e t o b e a b o u t 256 ohms, w h i c h w o u l d g i v e a b o u t 2 . 6 d b l o s s a s f 0 n
FIG. 4.09-4 TCHEBYSCHEFF RIPPLE IN db vs. 6 FOR THE IMPEDANCE-MATCHING NETWORKS WHOSE ELEMENT VALUES ARE GIVEN IN FIGS. 4.09-5 TO 4.09-8
By F i g s . 4 . 0 9 - 3 t o 4 . 0 9 - 5 ,
=
+
and 5 . 9 d b l o s s a t 1 Gc ( a r e d u c t i o n o f 2 . 7 d b f r o m t h e p r e c e d i n g c a s e ) . T h u s , t h e n = 4 d e s i g n w i t h o n l y 1 . 9 d b maximum l o s s a n d a b o u t 0 . 2 5 d b v a r i a t i o n a c r o s s t h e o p e r a t i n g band i s seen t o r e p r e s e n t a major improvement i n p e r f o r m a n c e .
Going t o l a r g e r v a l u e s o f n would g i v e s t i l l g r e a t e r
i m p r o v e m e n t , b u t e v e n w i t h n = a, ( L A ) m a xw o u l d s t i l l b e a b o u t 1 . 4 6 d b . I n most microwave c a s e s band-pass r a t h e r than low-pass impedance matching networks a r e d e s i r e d . i n Secs.
The d e s i g n of such networks i s d i s c u s s e d
1 1 . 0 8 and 1 1 . 0 9 working from t h e d a t a i n t h i s s e c t i o n .
Aspecial
f e a t u r e of band-pass impedance-matching networks i s t h a t they a r e e a s i l y d e s i g n e d t o p e r m i t any d e s i r e d v a l u e o f g e n e r a t o r i n t e r n a l r e s i s t a n c e ,
FIG. 4.09-5
ELEMENT VALUES vs. 6 FOR TCHEBYSCHEFF IMPEDANCE-MATCHING NETWORKS THAT MINIMIZE ( L A ) m a x
FIG. 4.09-6
ELEMENT VALUES vs. S FOR TCHEBYSCHEFF IMPEDANCE-MATCHING NETWORKS THAT MINIMIZE ( L A I m a x
FIG. 4.09-7
ELEMENT VALUES vs. 6 FOR TCH EBYSCH E F F IMPEDANCE-MATCHING NETWORKS THAT MINIMIZE ( L A ) m o x
FIG. 4.09-8
ELEMENT VALUES v s . 6 FOR TCHEBYSCHEFF IMPEDANCE-MATCHING NETWORKS THAT MINIMIZE ( L A ) m a x
whereas l o w - p a s s m a t c h i n g networks must h a v e a s p e c i f i e d g e n e r a t o r w h e r e n i s t h e number o f r e a c t i v e e l e m e n t s i n t h e p r o t o t y p e .
i n t e r n a l r e s i s t a n c e f o r optimum d e s i g n . The a t t e n u a t i o n c h a r a c t e r i s t i c s o f t h e i m p e d a n c e - m a t c h i n g n e t w o r k s
Next
compute
d i s c u s s e d i n t h i s s e c t i o n a n d i n S e c . 4 . 1 0 may b e computed by
e
=
d
-
28 s i n
(E)
and t h e maximum, p a s s - h a n d r e f l e c t i o n c o e f f i c i e n t v a l u e w h e r e L;
i s t h e a t t e n u a t i o n o f t h e i m p e d a n c e - m a t c h i n g n e t w o r k a n d LA i s
Irlmax=
o b t a i n e d by E q s . ( 4 . 0 3 - 3 ) t o ( 4 . 0 3 - 5 1 , o r by F i g s . 4 . 0 3 - 4 t o 4 . 0 3 - 1 0 f o r t h e a p p r o p r i a t e db Tchebyscheff r i p p l e LAr = (LA)max (LA)min.
-
I n t h e n e x t s e c t i o n t h e c a l c u l a t i o n o f p r o t o t y p e impedance-matching
cosh (n sinh-I
e)
c o s h (n s i n h - l d )
Then t h e ( L A ) m a xv a l u e w h i c h m u s t b e a c c e p t e d i s
networks s o a s t o g i v e a s p e c i f i e d Tchebyscheff r i p p l e [ a t t h e c o s t of a larger w i l l b e d i s c u s s e d . The m e t h o d by w h i c h F i g s . 4 . 0 9 - 3 t o 4.09-8 were p r e p a r e d w i l l a l s o be o u t l i n e d . i
S EC.
4.10,
COMPUTATION O F PROTOTYPE IMPEDANCE- MATCHING NETWORKS FOR S P E C I F I E D R I P P L E OR MINIMUY REFLECTION
F i g u r e 4 . 1 0 - 1 shows a p l o t o f ( L A ) m B vXs 8 f o r v a r i o u s v a l u e s o f n
I
The n e t w o r k s d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n w e r i s p e c i f i e d s o that
was t o b e a s s m a l l a s p o s s i b l e . U n d e r t h a t c o n d i t i o n , i t was n e c e s s a r y t o a c c e p t w h a t e v e r p a s s - b a n d T c h e b y s c h e f f r i p p l e t h e c h a r t s
and v a r i o u s a m o u n t s o f T c h e b ~ s c h e f f r i p p l e a m p l i t u d e [ ( L A ) m a xThis Suppose t h a t 8 = 0 . 1 0 and 0 . 1 0 - d b r i p p l e i s d e s i r e d w i t h n = 2. c h a r t shows t h a t ( L A ) m a xw i l l t h e n be 5 . 9 d b . By F i g s . 4 . 0 9 - 3 a n d 4 . 0 9 - 4 i t i s s e e n t h a t f o r t h e same 6 , when ( L A ) m a xi s m i n i m i z e d ,
4 . 8 db w h i l e t h e r i p p l e i s 0 . 9 8 d b
might c a l l f o r i n t h e c a s e o f any g i v e n d e s i g n .
A l t e r n a t i v e l y , we may s p e c i f y t h e pass-hand Tchebyscheff r i p p l e and a c c e p t whatever value o f
(LA)mPX = Thus, t h e p r i c e f o r reducing t h e
r i p p l e from 0 . 9 8 d b t o 0 . 1 0 d b i s a n i n c r e a s e i n ( L A ) m a xo f a b o u t 1.1 d b .
( L A ) n a xmay r e s u l t . S i n c e i n some c a s e s k e e p i n g t h e p a s s - h a n d a t t e n u a t i o n c o n s t a n t may b e t h e m a j o r c o n s i d e r a t i o n , c o m p u t a t i o n o f p r o t o t y p e
the element v a l u e s .
matching-network element v a l u e s f o r a s p e c i f i e d Tchebyscheff r i p p l e w i l l
c h a p t e r , we o b t a i n
G r e e n ' s work6,' a p p e a r s t o p r o v i d e t h e e a s i e s t means f o r d e t e r m i n i n g Using h i s e q u a t i o n s a l t e r e d t o t h e n o t a t i o n of t h i s
be b r i e f l y o u t l i n e d . P r o t o t y p e c i r c u i t s f o r s p e c i f i e d decrement 6 = l/(goglw;) r i p p l e may b e o b t a i n e d a s f o l l o w s .
ff
=
antilogl
( d b Tchebyschef f r i p p l e ) 10
and
and db
F i r s t compute14
where t h e g j 7 s a r e a s d e f i n e d i n F i g . 4 . 0 4 - 1 . t h e n computed by u s e o f t h e e q u a t i o n s
The e l e m e n t v a l u e s a r e
where t h e k 1 - 1 8 1
a r e c o u p l i n g c o e f f i c i e n t s t o b e e v a l u a t e d a s shown b e l o w .
Green's equations for the
FIG. 4.10-1
( L A ) m aVS. x 8 FOR IMPEDANCE-MATCHING NETWORKS HAVING A SPECIFIED TCHEBYSCHEFF RIPPLE
where
k J - l ,
are6,'
Also, f o r n arbitrary,
A c o m p u t e r p r o g r a m was s e t u p t o f i n d v a l u e s o f a a n d b t h a t s a t i s f y Eq. ( 4 . 1 0 - 1 7 ) u n d e r t h e c o n s t r a i n t g i v e n by E q . ( 4 . 1 0 - 2 0 ) . From t h e s e a and b v a l u e s f o r v a r i o u s 6 , v a l u e s f o r d and e w e r e o b t a i n e d by d = s i n h a
s i n Z r B c o s z r B + ( c o s 2 r B + D~ s i n 2 r e ) ( s i n z 8 ) s 2 kr, ,+l
=
s i n (2r - 1)B s i n (2r + 1)B -16)
a n d e = s i n h b . When v a l u e s o f d a n d e h a d b e e n o b t a i n e d f o r v a r i o u s 6 , t h e e l e m e n t v a l u e s f o r t h e n e t w o r k s were computed u s i n g Eqs. ( 4 . 1 0 - 6 ) t o (4.10-15).
where
T h e d a t a f o r t h e c h a r t s i n F i g . 4 . 0 9 - 3 w e r e o b t a i n e d by u s i n g t h e v a l u e s o f a a n d b u s 6 o b t a i n e d a b o v e , a n d t h e n c o m p u t i n g ( L A ) n a x by u s e
It is usually convenient t o normalize the prototype design s o t h a t g o = 1 and w; = 1 , a s h a s b e e n d o n e w i t h t h e t a b u l a t e d d e s i g n s i n t h i s
of Eqs.
(4.10-18),
(4.10-19), (4.10-4),
and ( 4 . 1 0 - 5 ) .
The d a t a i n F i g . 4 . 0 9 - 4 w e r e o b t a i n e d by s o l v i n g E q s . ( 4 . 1 0 - 1 8 1 , ( 4 . 1 0 - 1 9 ) , ( 4 . 1 0 - 1 )
chapter.
and ( 4 . 1 0 - 2 ) f o r t h e db r i p p l e a s a f u n c t i o n o f a and b .
The e l e m e n t v a l u e s f o r t h e p r o t o t y p e m a t c h i n g n e t w o r k s d i s c u s s e d i n
L o s s l e s s i m p e d a n c e m a t c h i n g n e t w o r k s f o r some more g e n e r a l f o r m s o f
Sec. 4 . 0 9 and p l o t t e d i n F i g s . 4.09-5 t o 4.09-8 c o u l d have been o b t a i n e d using Green's charts7 of coupling c o e f f i c i e n t s and D values along with
l o a d s a r e d i s c u s s e d i n R e f s . 1 4 , 1 6 , 1 7 , a n d 1 8 . However, much w o r k r e m a i n s t o b e d o n e on t h e p r a c t i c a l , m i c r o w a v e r e a l i z a t i o n o f t h e m o r e com p l i c a t e d forms o f matching networks c a l l e d f o r i n such c a s e s . At t h e
Eqs. (4.10-7) t o ( 4 . 1 0 - 9 ) . *
However, i n o r d e r t o e n s u r e h i g h a c c u r a c y , t o add t h e n = 1 c a s e , a n d t o c o v e r a somewhat w i d e r r a n g e decrements
01
t h a n was t r e a t e d by G r e e n , t h e c o m p u t a t i o n s f o r t h e c h a r t s i n S e c . 4 . 0 9 w e r e c a r r i e d o u t f r o m t h e b e g i n n i n g . The p r o c e d u r e u s e d wasI t h a t d e 1 s c r i b e d below. ~ a n o hl a~s shown t h a t , f o r l o w - p a s s n e t w o r k s o f t h e t /'y p e u n d e r c o n sideration,
(LA)max
w i l l be a s s m a l l a s p o s s i b l e i f t a n h nu
t a n h nb
cosh a
cosh b
-
=
-
p r e s e n t t i m e t h e p r o t o t y p e n e t w o r k s i n S e c . 4 . 0 9 and t h i s s e c t i o n a p p e a r t o have t h e w i d e s t range of u s e f u l n e s s i n t h e design of low-pass, h i g h p a s s , and band-pass microwave impedance matching networks i n t h e forms d i s c u s s e d i n C h a p t e r s 7 , a n d 11. S E C . 4 . 1 1 , PKOTOTYPES FOR NEGATIVE-RESISTANCE AMPLIFIERS As was d i s c u s s e d i n S e c . 1 . 0 4 , i f a d i s s i p a t i o n l e s s f i l t e r w i t h r e s i s t o r t e r m i n a t i o n s h a s o n e t e r m i n a t i o n r e p l a c e d by a n e g a t i v e r e s i s t a n c e o f t h e same m a g n i t u d e , t h e c i r c u i t c a n become a n e g a t i v e - r e s i s t a n c e a m p l i f i e r . I t was n o t e d t h a t , i f r l ( p ) i s t h e r e f l e c t i o n c o e f f i c i e n t b e t w e e n
where
a p o s i t i v e r e s i s t a n c e R o a n d t h e f i l t e r , when Ho i s r e p l a c e d by Rg = - R o , t h e r e f l e c t i o n c o e f f i c i e n t a t t h a t e n d o f t h e f i l t e r becomes
and d and e a r e a s i n d i c a t e d i n Eqs. ( 4 . 1 0 - 2 ) a n d ( 4 . 1 0 - 3 ) . Eqs. (4.10-181, ( 4 . 1 0 - 1 9 1 , and ( 4 . 1 0 - 3 1 ,
By w h e r e p = r~ + j w i s t h e c o m p l e x f r e q u e n c y v a r i a b l e . T h e n , r e f e r r i n g t o F i g s . 1 . 0 4 - 1 and 1 . 0 4 - 2 , t h e g a i n of t h e a m p l i f i e r a s measured a t a c i r c u l a t o r w i l l he li
Barton ( s e e Ref. 1 5 ) has i n d e p e n d e n t l y a l s o computed c h a r t s e q u i v a l e n t t o the c o u p l i n g c o e f f i c i e n t c h a r t s of Green. B a r t o n , however, include^ t h e maximally f l a t c a s e i n a d d i t i o n .
w h e r e P r i s t h e power r e f l e c t e d i n t o t h e c i r c u l a t o r by t h e n e g a t i v e resistance amplifier.
I f LA i s t h e a t t e n u a t i o n i n d b ( i . e . , t r a n s d u c e r l o s s , a s d e f i n e d i n S e c . 2 . 1 1 ) between R g and R o i n Fig. 1 . 0 4 - 1 f o r t h e
d i s s i p a t i o n l e s s f i l t e r w i t h p o s i t i v e t e r m i n a t i o n s , t h e n t h e s q u a r e d magnit u d e o f t h e r e f l e c t i o n c o e f f i c i e n t f o r p o s i t i v e t e r m i n a t i o n s i s g i v e n by
When R , o n t h e l e f t i n F i g . 1 . 0 4 - 1 i s r e p l a c e d by i t s n e g a t i v e , R i . = t h e n Eq.
-R
( 4 . 1 1 - 3 ) c a n b e r e p l a c e d by
0'
0.005 0.004 0
I
I
5
I
IS
10
TRANSDUCER OAIN, Gt
I
1
20
25
- db
. 30 A-1517-80
I
--
antilogl
antilog,,
LA
10
I
FIG. 4.11-1
/ db
/
1
(4.11-4)
LA
-- 1
ATTENUATION OF A FILTER WITH POSITIVE-RESISTANCE TERMINATIONS vs. TRANSDUCER GAIN OF THE CORRESPONDING NEGATIVE-RESISTANCE AMPLIFIER USING A CIRCULATOR
10
Thus, i n choosing a f i l t e r
where G, i s t h e t r a n s d u c e r g a i n a t t h e c i r c u l a t o r , i n db. F i g u r e 4.11-1 shows t h e a t t e n u a t i o n f o r a f i l t e r w i t h p o s i t i v e - r e s i s t a n c e t e r m i n a t i o n s
r i ( p ) for the negative-resistance amplifier.
u s . t h e db t r a n s d u c e r g a i n o f t h e c o r r e s p o n d i n g n e g a t i v e - r e s i s t a n c e ampli-
r l ( p ) h a v e i t s z e r o s i n t h e l e f t h a l f p l a n e s i n c e i f t h e y a r e n o t , when
f i e r w i t h a c i r c u l a t o r , a s d e t e r m i n e d u s i n g t h e above r e l a t i o n s .
t h e s e z e r o s become p o l e s o f r : ( p )
as a prototype f o r a negative-resistance amplifier, it is important that
T h e p r o t o t y p e i m p e d a n c e - m a t c h i n g f i l t e r s d i s c u s s e d i n S e c s . 4 . 0 9 and 4.10 can a l s o be u s e d a s p r o t o t y p e s f o r n e g a t i v e - r e s i s t a n c e a m p l i f i e r s . With r e g a r d t o t h e i r u s e , some c o n s i d e r a t i o n m u s t b e g i v e n t o t h e m a t t e r o f s t a b i l i t y . L e t u s d e f i n e r l ( p ) a s t h e r e f l e c t i o n c o e f f i c i e n t between any o f t h e f i l t e r s i n F i g . 4.04-1 and t h e t e r m i n a t i o n g o = Ro o r Go a t t h e l e f t a n d r n ( ~ a) s t h e r e f l e c t i o n c o e f f i c i e n t a t t h e o t h e r e n d . I t c a n b e shown t h a t t h e p o l e s o f a r e f l e c t i o n c o e f f i c i e n t f u n c t i o n a r e t h e frequencies of n a t u r a l v i b r a t i o n of t h e c i r c u i t ( s e e Secs. 2.02 t o 2.04); hence, t h e y must l i e i n t h e l e f t h a l f o f t h e complex-frequency p l a n e i f t h e c i r c u i t i s p a s s i v e . However, t h e z e r o s o f r l ( p ) , o r o f r n ( p ) , can l i e i n e i t h e r the l e f t o r r i g h t h a l f o f t h e p-plane. l/rl(p),
t h e zeros of r J p )
S i n c e r;'.(p) = f o r t h e p a s s i v e f i l t e r become t h e p o l e s o f
f o r the negative-resistance amplifier
t h e y w i l l c a u s e e x p o n e n t i a l l y i n c r e a s i n g o s c i l l a t i o n s ( i . e . , u n t i l some non-linearity i n the c i r c u i t l i m i t s the amplitude).
-
The m a t h e m a t i c a l d a t a g i v e n i n S e c s . 4 . 0 9 a n d 4 . 1 0 f o r f i l t e r p r o t o t y p e s of t h e v a r i o u s forms i n F i g . 4.04-1 a r e such t h a t t h e r e f l e c t i o n c o e f f i c i e n t r l ( p ) i n v o l v i n g t h e t e r m i n a t i o n g o on t h e l e f t w i l l have a l l of i t s z e r o s i n t h e l e f t h a l f of t h e p - p l a n e , w h i l e t h e r e f l e c t i o n c o e f f i c i e n t r , ( p ) i n v o l v i n g t h e t e r m i n a t i o n g n + l on t h e r i g h t w i l l h a v e a l l For t h i s r e a s o n i t i s s e e n t h a t of i t s zeros i n t h e r i g h t h a l f p l a n e . ' t h e t e r m i n a t i o n g o a t t h e l e f t must be t h e o n e which i s r e p l a c e d by i t s negative, never t h e t e r m i n a t i o n g,+l
*
An e x c e p t i o n t o t h i s o c c u r s w h e n Then t h e z e r o s o f
rib)
and
e
a t the right.
= 0 i n Eq. (4.10-3) w h i c h l a a d s
rnb)a r c
to
( L A ) m i n = 0 xn F i g . 4 . 0 9 - 2 .
a l l o n t h e p = jo a x i s o f the p - p l a n e .
By u s e o f E q s . ( 4 . 1 1 - 1 ) t o ( 4 . 1 1 - 4 ) a l o n g w i t h t h e e q u a t i o n s i n Sec. 4 . 1 0 i t i s p o s s i b l e t o compute p r o t o t y p e impedance-matching networks f o r n e g a t i v e - r e s i s t a n c e a m p l i f i e r s s o a s t o g i v e a s p e c i f i e d g a i n and pass-band Tchebyscheff r i p p l e .
However, ~ e t s i n ~ e rh "a s p r e p a r e d e x t e n s i v e
t a b l e s o f p r o t o t y p e e l e m e n t v a l u e s f o r n e g a t i v e - r e s i s t a n c e a m p l i f i e r s , and h i s t a b l e s a r e reproduced i n Tables 4.11-1 t o 4.11-3 herein.
F i g u r e 4.11-2 shows how t h e e l e m e n t v a l u e s i n T a b l e s 4 . 1 1 - 1 t o 4 . 1 1 - 3 s h o u l d b e i n t e r -
preted.
The c o n d u c t a n c e o r r e s i s t a n c e o n t h e l e f t i s a l w a y s n e g a t i v e , and i t i s s c a l e d s o a s t o a l w a y s be e q u a l t o minus one. Note t h a t an i d e a l i z e d
c i r c u l a t o r i s i n c l u d e d on t h e r i g h t s i d e o f e a c h c i r c u i t . shows a t y p i c a l g a i n c h a r a c t e r i s t i c f o r t h e s e c i r c u i t s .
Figure 4.11-3
I t s h o u l d be noted
t h a t t h e frequency s c a l e i s normalized s o t h a t t h e e q u a l - r i p p l e band-edge f r e q u e n c y mi i s e q u a l t o o n e . With t h e n o r m a l i z a t i o n s i n d i c a t e d a b o v e , t h e d e c r e m e n t s f o r t h e normalized
p r o t o t y p e c i r c u i t s i n F i g . 4.11-2 and T a b l e s 4.11-1 t o 4 . 1 1 - 3 a r e
g i v e n by
i
T h i s i s t h e same d e f i n i t i o n o f d e c r e m e n t a s w a s u s e d ih S e c . 4 . 0 9 , b u t , o f c o u r s e , s i n c e t h e r e s i s t a n c e i s n e g a t i v e t h e decrement i s n e g a t i v e . P r a c t i c a l n e g a t i v e - r e s i s t a n c e devices have a p a r a s i t i c r e a c t i v e i m pedance component a s w e l l a s n e g a t i v e r e s i s t a n c e . I f a negative-resistance d e v i c e ( s u c h a s a t u n n e l d i o d e ) may b e r e p r e s e n t e d by a n e g a t i v e - r e s i s t a n c e
R,, i n p a r a l l e l w i t h a s h u n t c a p a c i t a n c e C1, o r by a n e g a t i v e c o n d u c t a n c e Go i n s e r i e s w i t h a n i n d u c t a n c e L1, t h e n i n e i t h e r c a s e i f a n o p e r a t i n g r a n g e f r o m w = 0 t o wl i s d e s i r e d , t h e d e c r e m e n t o f t h e d e v i c e i s O O C O O O O C O O O C O O O O C O
000000000
0-NmOmC.nO
o"N"?omOmo
0-NmOmOmO
O.-.N"?O'n3'nO
dd&dAL&Nn
&dddAAA&&
d&ddAA&&&
ddddAi&&&
OOOCCCOOO OOOCOOCOO
O C O O C O O O C
000000000
O C O O C O C O O
8 88282288 Z 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 m8888S8888 0 0 C 0 0 0 0 0 m 0 0 0 C c C 0 0 ~ 0 0 0 0 0 0 0 0m 0 0 c 0 C 0 0 0
T h i s decrement i s unchanged i f t h e impedance l e v e l o r f r e q u e n c y s c a l e o f t h e c i r c u i t i s changed, s o t h e d e s i r e d e q u i v a l e n t element v a l u e s f o r t h e d e v i c e c a n b e o b t a i n e d by s e l e c t i n g a d e s i g n f r o m T a b l e s 4 . 1 1 - 1 t o 4 . 1 1 - 3 which h a s t h e r e q u i r e d decrement v a l u e , and t h e n a l t e r i n g t h e impedance l e v e l and f r e q u e n c y s c a l e o f t h e d e s i g n a s d e s c r i b e d i n S e c . 4 . 0 4 s o a s t o make t h e p r o t o t y p e c i r c u i t c o r r e s p o n d t o t h e d e s i r e d d e s i g n . By u s e
o f Eq. ( 4 . 1 1 - 5 ) , i f t h e r e q u i r e d n e g a t i v e d e c r e m e n t 6 h a s b e e n d e t e r m i n e d ,
n= ODD
. n
i
t h e d e s i g n s i n T a b l e s 4 . 1 1 - 1 t h r o u g h 4 . 1 1 - 3 which c a n be used a r e t h o s e
I
t h a t have a v a l u e of g l e q u a l t o o r l a r g e r t h a n
DAD
= EVEN
A v a l u e o f g l g r e a t e r t h a n t h e v a l u e g i v e n by E q . ( 4 . 1 1 - 7 ) i s a c c e p t a b l e
(0)
b e c a u s e t o a c h i e v e t h i s v a l u e i t i s m e r e l y n e c e s s a r y t o a d d more s h u n t c a p a c i t a n c e ( o r more s e r i e s i n d u c t a n c e ) t o t h e n e g a t i v e - r e s i s t a n c e d e v i c e , However, s i n c e 6 was a s s u m e d t o h a v e
and t h i s i s a l w a y s p o s s i b l e t o d o .
been computed from t h e d e v i c e i n i t s fundamental form, from which no c a p a c i t a n c e o r i n d u c t a n c e c a n b e r e m o v e d , i t i s n o t p o s s i b l e t o make u s e of d e s i g n s c a l l i n g f o r v a l u e s o f g l t h a t a r e s m a l l e r t h a n t h a t i n d i c a t e d To o b t a i n t h e maximum g a i n c o r r e s p o n d i n g t o a g i v e n by Eq. ( 4 . 1 1 - 7 ) . b a n d w i d t h , c h o o s e a d e s i g n f r o m t h e t a b l e ( o r by i n t e r p o l a t i o n f r o m t h e
n = EVEN 11.-IUO-I
FIG. 4.1 1-2 DUAL INTERPRETATIONS OF THE LOW-PASS, NEGATIVE-~ESISTANCE AMPLIFIER DESIGNS IN TABLES 4.11-1 TO 4.11-3 !
t a b l e ) s o t h a t g l i s e x a c t l y t h e v a l u e g i v e n by Eq.
(4.11-7).
S i n c e t h e d e s i g n p r o c e d u r e h e r e i n assumes t h e u s e of a c i r c u l a t o r , and s i n c e p r a c t i c a l c i r c u l a t o r s h a v e l i m i t e d b a n d w i d t h , t h e d a t a i n t h i s s e c t i o n i s o f most i n t e r e s t a s a s o u r c e of l o w - p a s s p r o t o t y p e d e s i g n s t o be u s e d i n t h e d e s i g n o f b a n d - p a s s a m p l i i i e r s a s d e s c r i b e d i n S e c . 1 1 . 1 0 . However, i n o r d e r t o f u r t h e r c l a r i f y t h e s e l e c t i o n o f p r o t o t y p e d e s i g n s , l e t u s c o n s i d e r a h y p o t h e t i c a l l o w - p a s s example f o r u s e w i t h an i d e a l i z e d c i r c u l a t o r of unlimited bandwidth. L e t u s s u p p o s e t h a t a n e g a t i v e - r e s i s t a n c e d e v i c e can be r e p r e s e n t e d by a n e g a t i v e r e s i s t a n c e o f - 5 0 ohms i n p a r a l l e l w i t h a c a p a c i t a n c e o f
5 p p f , and t h a t i t i s d e s i r e d t o u s e t h i s d e v i c e i n an a m p l i f i e r which i s t o h a v e minimum p a s s - b a n d g a i n (Gt),i, o f 1 5 d b , a n d a T c h e b y s c h e f f r i p p l e of G t r = 1 db. needed, s o t h a t an n
Let u s suppose t h a t very high performance i s =
4 c i r c u i t from T a b l e 4 . 1 1 - 3 i s chosen i n p r e f e r -
e n c e t o t h e s i m p l e r and l o w e r p e r f o r m a n c e c i r c u i t s i n T a b l e s 4 . 1 1 - 1 and 4.11-2.
From T a b l e 4 . 1 1 - 3 t h e d e s i g n w i t h t h e d e s i r e d g a i n p a r a m e t e r s
has gl = 1.4548, g2 = 1.0081, g 3 = 1.7668, g4
FIG. 4.11-3
A TYPICAL TCHEBYSCHEFF TRANSDUCER GAIN CHARACTERISTIC FOR THE LOW-PASS PROTOTYPE NEGATIVE-RESISTANCE AMPLIFIERS IN TABLES 4.11-1 TO 4.11-3 The number of pass-band ripples shown corresponds to the case of n = 3
By E q s .
( 4 . 1 1 - 5 ) and ( 4 . 1 1 - 6 )
=
0 . 4 2 8 2 , and g 5 = 1 . 4 3 2 6 .
T h u s , t h i s d e s i g n w i l l g i v e t h e d e s i r e d 1 5 - d b minimum g a i n w i t h 1 - d b T c h e b y s c h e f f r i p p l e up t o f l = (wl/2.rr) t h e form i n F i g . 4 . 1 1 - 2 ( b )
As i n d i c a t e d i n F i g . 4 . 1 2 - 1 , a n
= 9 2 6 Mc.
The p r o t o t y p e c i r c u i t h a s ( f o r n e v e n ) w i t h R i = -1, C; = 1 . 4 5 4 8 , a n d w ' = 1.
i n v e r t e r may h a v e an i m a g e p h a s e
1
U s i n g t h e m e t h o d s o f S e c . 4 . 0 4 , t h e i m p e d a n c e o f t h i s c i r c u i t i s n e x t s c a l e d up by a f a c t o r R,,/R;
= (-50)/(-1)
-
= 5 0 , w h i l e t h e f r e q u e n c y s c a l e i s a l t e r e d by
t h e f a c t o r o l / w ; = ( 5 . 8 2 ) ( l o 9 ) / l . T h e s e s c a l e c h a n g e s make R i = -1 become Ro = -50, a n d C; = 1 . 4 5 4 8 become C1 = 5 X 1 0 - 1 2 , a n d m a k e t h e o t h e r e l e m e n t s t a k e
IMPEDANCE INVERTER
corresponding values.
and
L4 c o m p r i s e t h e e l e m e n t s o f t h e c i r c u i t r e q u i r e d t o
l i k e a s h u n t c a p a c i t a n c e from i t s exterior terminals.
The p h y s i c a l s t r u c t u r e o f t h e l o w - p a s s f i l t e r s t r u c t u r e f o r t h e a b o v e ex-
on b o t h s i d e s l o o k s l i k e a s e r i e s
However, a s mentioned above, t h i s l o w - p a s s a m p l i f i e r example i s r a t h e r h y p o t h e t i c a l be-
i n d u c t a n c e from i t s e x t e r n a l t e r -
cause ofthebandwidth limitations o f c i r c u l a t o r s , andthe r e s u l t s o f t h i s
ADMITTANCE INVERTER
s e c t i o n a r e o f m o s t p r a c t i c a l i n t e r e s t f o r u s e i n t h e d e s i g n o f b a n d - p a s s amp l i f i e r s a s described i n Sec. 11.10. SOURCE:
CONVERSION OF F I L T E R PROTOTYPES TO USE IMPEDANCE - OR ADMITTANCE - INVERTERS AND ONLY ONE KIND OF REACTIVE ELEMENT
,1
1
I I
Final Report, Contract DA 36-039 SC-74862. Stanford Reaearch Institute, reprinted in IRE Trans.. PGMTT (ace Ref. 1 of Chapter 10. by G . L. Matthaei).
FIG. 4.12-1
I
I n d e r i v i n g d e s i g n e q u a t i o n s f o r c e r t a i n t y p e s o f b a n d l p a s s a n d bands t o p f i l t e r s i t i s d e s i r a b l e t o c o n v e r t t h e p r o t o t y p e s i n Fitg. 4 . 0 4 - 1 which u s e b o t h i n d u c t a n c e s and c a p a c i t a n c e s t o e q u i v a l e n t f o r m s w h i c h u s e o n l y i n d u c t a n c e s o r o n l y c a p a c i t a n c e s . T h i s c a n be done with t h e a i d of t h e i d e a l i z e d i n v e r t e r s which a r e s y m b o l i z e d i n F i g . 4.12-1. An i d e a l i z e d impedance i n v e r t e r o p e r a t e s l i k e a q u a r t e r - w a v e l e n g t h l i n e o f c h a r a c t e r i s t i c impedance K a t a l l f r e q u e n c i e s . T h e r e f o r e , i f i t i s t e r m i n a t e d i n an i m p e d a n c e
Zbon o n e e n d , t h e i m p e d a n c e Zas e e n l o o k i n g i n a t t h e o t h e r e n d i s
Likewise, a
shunt capacitance with an i n v e r t e r
ample c o u l d b e d e s i g n e d u s i n g m e t h o d s d e s c r i b e d i n c h a p t e r 5.
4.12,
a c t i o n i n d i c a t e d by E q s . ( 4 . 1 2 - 1 ) w i t h a n i n v e r t e r on e a c h s i d e l o o k s
g i v e t h e d e s i r e d b r o a d - b a n d r e s p o n s e s h a p e u p t o 9 2 6 Mc.
SEC.
Because of t h e i n v e r t i n g and ( 4 . 1 2 - 2 ) a s e r i e s i n d u c t a n c e
Then i n t h e s c a l e d c i r c u i t , R o a n d C l o n t h e l e f t [ s e e F i g . 4 . 1 1 - 2 ( b ) l a r e r e p l a c e d by t h e n e g a t i v e - r e s i s t a n c e d e v i c e w h i c h i s t o be u s e d , w h i l e L 2 , C,,
s h i f t of e i t h e r +90 d e g r e e s o r an odd m u l t i p l e t h e r e o f .
DEFINITION OF IMPEDANCE INVERTERS AND ADMITTANCE INVERTERS
minals.
Making u s e o f t h i s p r o p -
erty, the prototype c i r c u i t s i n Fig. 4.04-1 can be converted t o e i t h e r of t h e e q u i v a l e n t forms i n F i g . 4 . 1 2 - 2 which have i d e n t i c a l transmission characteristics t o those prototypes i n Fig. 4.04-1. As c a n b e s e e n f r o m E q s . ( 4 . 1 2 - 1 ) and ( 4 . 1 2 - 2 ) , i n v e r t e r s have t h e
a b i l i t y t o s h i f t i m p e d a n c e o r a d m i t t a n c e l e v e l s d e p e n d i n g on t h e c h o i c e For t h i s reason i n F i g . 4 . 1 2 - 2 ( a ) t h e s i z e s of t h e K o r J parameters. of RA, R
a n d t h e i n d u c t a n c e s L a k may b e c h o s e n a r b i t r a r i l y a n d t h e
response w i l l be i d e n t i c a l t o t h a t o f t h e o r i g i n a l p r o t o t y p e a s i n are specified F i g . 4.04-1 p r o v i d e d t h a t t h e i n v e r t e r parameters K k ,k,l The same holds for the a s i n d i c a t e d by t h e e q u a t i o n s i n F i g . 4 . 1 2 - 2 ( a ) . Note t h a t t h e g k v a l u e s c i r c u i t i n F i g . 4 . 1 2 - 2 ( b ) o n l y on t h e d u a l b a s i s . r e f e r r e d t o i n t h e e q u a t i o n s i n F i g . 4.12-2 a r e t h e prototype element values a s defined i n Fig. 4.04-1.
Anidealizedadmittance i n v e r t e r a s d e f i n e d h e r e i n i s t h e admittance represencat i o n o f t h e same t h i n g , i . e . , i t o p e r a t e s l i k e a q u a r t e r - w a v e l e n g t h l i n e o f c h a r a c t e r i s t i c admittance J a t a l l frequencies. attached a t o n e e n d , the admittance
T h u s , i f an a d m i t t a n c e Y b i s
Ya s e e n l o o k i n g i n t h e o t h e r e n d i s
can be d e r i v e d A way t h a t t h e e q u a t i o n s f o r t h e K k , k + l a n d J k , k + l w i l l now be b r i e f l y c o n s i d e r e d . A f u n d a m e n t a l way o f l o o k i n g a t t h e ( b ) and t h e r e l a t i o n hetween t h e p r o t o t y p e c i r c u i t s i n F i g s . 4 . 0 4 - l ( a ) , c o r r e s p o n d i n g c i r c u i t i n , s a y , F i g . 4 . 1 2 - 2 ( a ) makes u s e o f t h e c o n c e p t A g i v e n c i r c u i t a s s e e n through an impedance i n v e r t e r looks l i k e t h e d u a l o f t h a t given c i r c u i t . Thus, t h e impedances s e e n from
of duality.
i n d u c t o r L a l i n F i g . 4 . 1 2 - 2 ( a ) a r e t h e same a s t h o s e s e e n f r o m i n d u c t a n c e L ; i n F i g . 4 . 0 4 - l ( b ) , e x c e p t f o r a n i m p e d a n c e s c a l e f a c t o r . The impeda n c e s seen from i n d u c t o r L a 2 i n F i g . 4 . 1 2 - 2 ( a ) a r e i d e n t i c a l t o t h o s e s e e n from i n d u c t a n c e L ; i n F i g . 4 . 0 4 - l ( a ) , e x c e p t f o r a ~ o s s i b l eimpedance s c a l e change. I n t h i s manner t h e impedances i n any p o i n t o f t h e c i r c u i t i n F i g . 4 . 1 2 - 2 ( a ) may b e q u a n t i t a t i v e l y r e l a t e d t o t h e c o r r e s p o n d i n g impedances i n t h e c i r c u i t s i n F i g . 4 . 0 4 - l ( a ) ,
(b).
F i g u r e 4 . 1 2 - 3 ( a ) shows a p o r t i o n o f a l o w - p a s s p r o t o t y p e c i r c u i t t h a t h a s been o p e n - c i r c u i t e d j u s t beyond t h e c a p a c i t o r C k + l .
The d u a l
c i r c u i t i s shown a t ( b ) , w h e r e i t s h o u l d b e n o t e d t h a t t h e o p e n c i r c u i t
(a) MODIFIED PROTOTYPE USING IMPEDANCE INVERTERS
FIG. 4.12-3
SOME CIRCUITS DISCUSSED IN SEC. 4.12 A ladder circuit i s shown at (a), and its dual is shown at (b). The analogous K-inverter form of these two circuits i s shown a t (c).
shown a t ( a ) becomes a s h o r t c i r c u i t i n t h e d u a l c a s e .
The c o r r e s p o n d i n g
c i r c u i t u s i n g a l l s e r i e s i n d u c t o r s and K i n v e r t e r s i s shown a t ( c ) .
The
c i r c u i t s i n Fig. 4.12-3 w i l l he convenient f o r d e r i v i n g t h e formula f o r i n terms of L a k , L a k t l , and t h e p r o t o t y p e element v a l u e s g k and
Kk,k+l &'h+l.
The o p e n - and s h o r t - c i r c u i t s a r e i n t r o d u c e d m e r e l y t o s i m p l i f y
the equations. R e f e r r i n g t o Fig. 4.12-3,
( b ) MODIFIED PROTOTYPE USlNG ADMITTANCE tNVERTERS RB-2316-T7-ITZR
SOURCE:
Final Report, Contract DA 36-039 SC-14862, S tanford Research Institute, reprinted in IRE Tmns., PGMTT (see Ref. 1 of Chapter 10, by G. L. Matthasi).
FIG. 4.12-2
LOW-PASS PROTOTYPES MODIFIED TO INCLUDE IMPEDANCE INVERTERS OR ADMITTANCE INVERTERS The go, gl, .,., gn are obtained from the original prototype as i n Fig. 4.04-1, while the RA; La,, La,, and R B or the GA, Ca,, Can and GB may be chosen as desired.
...,
...,
Meanwhile i n t h e c i r c u i t a t ( c )
i n the c i r c u i t a t ( a ) ,
Now 2; m u s t b e i d e n t i c a l t o Zk e x c e p t f o r a n i m p e d a n c e s c a l e c h a n g e o f Lak/Lk. Therefore
E q u a t i n g t h e s e c o n d t e r m s o f E q s . ( 4 . 1 2 - 8 ) and ( 4 . 1 2 - 9 ) l e a d s t o t h e result Equating t h e second terms i n Eqs. (4.12-4) and (4.12-5) some r e a r r a n g e m e n t ,
gives, a f t e r
S u b s t i t u t i n g gn and g n t l f o r Ln and f o r K n , n + l shown i n F i g . 4 . 1 2 - 2 .
Gntl,
respectively, gives the equation
The d e r i v a t i o n o f t h e e q u a t i o n s f o r t h e S i n c e Lk = gk and
Eq. ( 4 . 1 2 - 6 ) i s e q u i v a l e n t t o t h e e q u a t i o n f o r K k , k + l g i v e n i n F i g . 4 . 1 2 - 2 ( a ) . I t i s e a s i l y s e e n t h a t by m o v i n g t h e p o s i t i o n s o f t h e open- and s h o r t Ck+l
=
gk+l,
c i r c u i t points correspondingly,
the
S EC
same p r o c e d u r e w o u l d a p p l y f o r c a l c u -
JL,k+l
parameters i n
F i g . 4 . 1 2 - 2 ( b ) may he c a r r i e d o u t i n l i k e manner o n t h e a d m i t t a n c e ( i . e . , dual basis. 4 . 1 3 , EFFECTS OF' D I S S I P A T I V E ELEMEN'I'S I N PROTOTYPES FOH LOW-PASS, dAND-PASS, OR HIGH-PASS F I L T E R S
lation of the K's for a l l the inAny p r a c t i c a l m i c r o w a v e f i l t e r w i l l h a v e e l e m e n t s w i t h f i n i t e Q ' s ,
v e r t e r s except t h o s e a t t h e ends. and
H e n c e , Eq. ( 4 . 1 2 - 6 ) a p p l i e s f o r k = 1, 2,
...,
n many p r a c t i c a l s i t u a t i o n s i t i s i m p o r t a n t t o be a b l e t o e s t i m a t e
t h e e f f e c t o f t h e s e f i n i t e e l e m e n t Q.'s on p a s s - b a n d a t t e n u a t i o n .
n - 1.
When a
f i l t e r h a s b e e n d e s i g n e d from a l o w - p a s s p r o t o t y p e f i l t e r i t i s c o n v e n i e b t Next c o n s i d e r F i g . 4 . 1 2 - 4 . At ( a ) i s shown t h e l a s t two e l e m e n t s o f
t o r e l a t e t h e microwave f i l t e r element Q's t o d i s s i p a t i v e elements i n t h e p r o t o t y p e f i l t e r and t h e n d e t e r m i n e t h e ' e f f e c t s o f t h e d i s s i p a t i v e e l e m e n t s on t h e p r o t o t y p e f i l t e r r e s p o n s e . Then t h e i n c r e a s e i n p a s s - b a n d a t t e n u -
a p r o t o t y p e c i r c u i t and a t ( b ) i s shown a c o r r e s p o n d i n g f o r m w i t h a K inverter.
a t i o n o f t h e p r o t o t y p e f i l t e r d u e t o t h e d i s s i p a t i v e e l e m e n t s w i l l be t h e same a s t h e i n c r e a s e i n p a s s - b a n d a t t e n u a t i o n ( a t t h e c o r r e s p o n d i n g f r e -
In the circuit a t (a)
quency) o f t h e microwave f i l t e r due t o t h e f i n i t e element Q ' s . The e l e m e n t Q ' s r e f e r r e d t o b e l o w a r e t h o s e o f t h e e l e m e n t s o f a l o w - p a s s f i l t e r a t i t s c u t o f f f r e q u e n c y wl and a r e d e f i n e d a s while a t (b)
S i n c e Z i m u s t e q u a l Zn w i t h i n a s c a l e factor
Lan/Ln,
FIG.
4.12-4
ADDITIONAL CIRCUITS DISCUSSED IN SEC. 4.12 The end portion of a prototype circuit i s shown at (a) while at (b) i s shown the corresponding end portion of a circuit with K-inverters.
where R k i s t h e p a r a s i t i c r e s i s t a n c e o f t h e i n d u c t a n c e L k , a n d G k i s t h e p a r a s i t i c conductance of t h e c a p a c i t a n c e C k . * I n t h e case of a band-pass
* Here,
t h e u n p r i m e d L k , R k , C,,
C k , a n d wl
whether i t i s a n o r m a l i z e d p r o t o t y p e
v a l u e s a r e meant t o a p p l y t o a n y l o w - p a s s L a t e r i n t h i s s e c t i o n p r i m e s w i l l be
or n o t .
t o aid i n d i s t i n g u i s h i n g between the low-pass sponding bend-pasa o r h i g h - p a s s f i l t e r .
prototype
149
p a r a m e t e r s and t h o a e
of
filter, introduced
t h e COrrt-
f i l t e r w h i c h i s d e s i g n e d f r o m a l o w - p a s s p r o t o t y p e , i f ( Q B p I k i s t h e midband u n l o a d e d Q o f t h e k t h r e s o n a t o r o f t h e band-pass f i l t e r , t h e n t h e corresponding Q of t h e kth r e a c t i v e element of t h e prototype is
A s p e c i a l c a s e o f c o n s i d e r a b l e p r a c t i c a l i n t e r e s t i s t h a t where t h e Q ' s o f a l l t h e e l e m e n t s a r e t h e same s o t h a t d k = d f o r k = 1 t o n . Then, a s c a n b e s e e n from E q s . ( 4 . 1 3 - 4 ) and ( 4 . 1 3 - 5 ) , t h e e f f e c t s o f d i s s i p a t i o n c a n b e a c c o u n t e d f o r by s i m p l y r e p l a c i n g t h e f r e q u e n c y v a r i a b l e jo' f o r t h e l o s s l e s s c i r c u i t by ( j w ' + d ) t o i n c l u d e t h e l o s s e s . F o r e x a m p l e , t h i s s u b s t i t u t i o n c a n b e made d i r e c t l y i n t h e t r a n s f e r f u n c t i o n s i n
I n t h i s equation w is t h e f r a c t i o n a l bandwidth o f t h e band-pass f i l t e r a s m e a s u r e d t o i t s p a s s - b a n d e d g e s w h i c h c o r r e s p o n d t o t h e w; p a s s -band e d g e o f t h e l o w - p a s s p r o t o t y p e ( s e e C h a p t e r 8 ) . The u n l o a d e d Q o f t h e r e s o n a t o r s c a n b e e s t i m a t e d by u s e o f t h e d a t a i n C h a p t e r 5 , o r i t ca b e d e t e r m i n e d by m e a s u r e m e n t s a s
Eqs. ( 4 . 0 7 - I ) , ( 4 . 0 8 - 5 ) t o ( 4 . 0 8 - 8 ) i n o r d e r t o compute t h e t r a n s f e r c h a r a c t e r i s t i c s w i t h p a r a s i t i c d i s s i p a t i o n i n c l u d e d . At DC t h e f u n c t i o n (jo'
+ d ) becomes s i m p l y d , s o t h a t i f
I
i n Sec. 11.02. In the case of a high-pass f i l t e r d e s i g n e d from a low-pass
+02
~:Q-G;:-
prototype, t h e element Q's of t h e p r o t o t y p e s h o u l d b e made t o
f o r a d i s s i p a t i o n l e s s p r o t o t y p e , t h e DC l o s s f o r a p r o t o t y p e w i t h u n i f o r m d i s s i p a t i o n d i s f o r Q' = 0
---
b e t h e same a s t h e Q ' s o f t h e corresponding elements o f t h e high-pass f i l t e r a t its cutoff
f4>d2o2
A-SOZT-S~
FIG. 4.13-1
frequency.
LOW-PASS PROTOTYPE FILTER WITH DISSIPATIVE ELEMENTS ADDED
F i g u r e 4 . 1 3 - 1 shows a p o r t i o n o f a low-pass p r o t o t y p e
where (E2ivail/E2 i s a s d e f i n e d i n Sec. 2 . 1 0 . U s u a l l y d i s s m a l l s o t h a t o n l y t h e l a s t two t e r m s o f E q . ( 4 . 1 3 - 7 ) a r e s i g n i f i c a n t . Then i t i s e a s i l y shown t h a t
f i l t e r w i t h p a r a s i t i c l o s s elements i n t r o d u c e d . Note t h a t t h e p a r a s i t i c l o s s e l e m e n t t o go w i t h r e a c t i v e e l e m e n t g k i s d e s i g n a t e d a s d k g k , w h e r e
(MA)
=
20 log,,
[C"d
+
11
db
Using t h i s notad k w i l l be r e f e r r e d t o h e r e i n a s a d i s s i p a t i o n f a c t o r . t i o n E q . ( 4 . 1 3 - 1 ) becomes Qk = w ; g k / ( d k g h ) = w i l d k w h e r e w ; i s t h e c u t o f f frequency o f t h e low-pass p r o t o t y p e .
Thus, w h e r e ( M A ) O i s t h e d b i n c r e a s e i n a t t e n u a t i o n a t w' = 0 when d i s f i n i t e , o v e r t h e a t t e n u a t i o n when d = 0 ( i . e . , when t h e r e i s n o d i s s i p a t i o n l o s s ) . ' The c o e f f i c i e n t Cn = a l / a o w h e r e a l a n d a.
a r e from polynomial P n ( j o l )
i n Eq. ( 4 . 1 3 - 6 ) . Then f o r a s e r i e s b r a n c h o f a p r o t o t y p e f i l t e r I n t h e c a s e o f low-pass p r o t o t y p e s f o r band-pass f i l t e r s ,
( a A i)s, ,
a l s o t h e i n c r e a s e i n t h e midband l o s s o f t h e c o r r e s p o n d i n g band-pass F o r h i g h - p a s s f i l t e r s designee f i l t e r a s a r e s u l t of f i n i t e resonator Q's. and f o r a s h u n t branch
t
For e x a m p l e , a d i s s i p a t i o n l e s s , 0 . 5 - d b r i p p l e T c h e b y s c h e f f f i l t e r w i t h n = 4 would h a v e L A = If u n i f o r m d i s s ~ ~ a t i oisn i n t r o d u c e d t h e a t t e n u a t i o n f o r o' = 0 r i l l become 0. 0 . 5 d b f a r w'
LA = 0.5 +
( " L A ) 0d b .
151
from l o w - p a s s p r o t o t y p e s ,
r e l a t e s t o t h e a t t e n u a t i o n a s o -a.
Equation (4.13-8) a p p l i e s both f o r prototypes such a s those i n Sec. 4.05 which f o r t h e c a s e o f no d i s s i p a t i o n l o s s h a v e p o i n t s where L A i s z e r o , and a l s o f o r t h e impedance-matching network p r o t o t y p e s i n S e c s . 4 . 0 9 and 4.10 which even f o r t h e c a s e o f n o d i s s i p a t i o n h a v e n o n - z e r o L A a t a l l frequencies. T a b l e 4 . 1 3 - 1 i s a t a b u l a t i o n o f t h e c o e f f i c i e n t s Cn f o r p r o t o t y p e f i l t e r s h a v i n g maximally f l a t a t t e n u a t i o n w i t h t h e i r 3-db p o i n t a t = 1. F i g- u r e 4 . 1 3 - 2 shows t h e C c o e f f i c i e n t s f o r Tchebyscheff f i l t e r s p l o t t e d u s d b
w:
pass-band r i p p l e .
Table 4 . 1 3 - 1 MAXIMALLY FLAT ATTENUATION FILTER COEFFICIENTS C- FOR USE IN EQ. ( 4 . 1 3 - 8 : These c o e f f i c i e n t s are f o r f i l t e r s with t h e i r 3-db point a t o; = 1 and a r e equal tb the group time delay i n seconds a s LI''approaches zerp b e e Eq. ( 4 . 0 8 - 2 ) J
I n t h i s case the equal-
r i p p l e band e d g e i s w; = 1.
Note t h a t above
about 0.3 db-ripple, t h e curves f a l l f o r n even a n d r i s e f o r n o d d . T h i s phenomenon i s r e l a t e d t o t h e f a c t t h a t a Tchebyscheff prot o t y p e f i l t e r with n even h a s a r i p p l e maximum a t w' = 0 , w h i l e a c o r r e s p o n d i n g f i l t e r w i t h n odd h a s a r i p p l e minimum a t t h a t f r e quency.
T h e r e i s a -p-p a r e n t l y a t e n d e n c y f o r
1
1
1.00
t h e e f f e c t s o f d i s s i p a t i o n t o be most p r o nounced a t r i p p l e minima. Bodelg g i v e s a n e q u a t i o n f o r M A , t h e increase i n a t t e n u a t i o n due t o uniform d i s -
I / 8
5.13
1
lo
1
5.76 6.39
1
I :: I ::::1I
s i p a t i o n , a s a f u n c t i o n o f t h e a t t e n u a t i o n p h a s e s l o p e and t h e d i s s i p a t i o n f a c t o r , d. B o d e ' s e q u a t i o n may b e e x p r e s s e d i n t h e form
Db TCHEBYSCHEFF RIPPLE
FIG. 4.13-2
where
and i n t h i s c a
t h e i n c r e a s e i n a t t e n u a t i o n a t , fa', t h e frequency, Thus, t h i s e q u a t i o n provides a convenient a t which d4/dw1 i s e v a l u a t e d . '
I t can be aeen from Eqs.
( 4 . 1 3 - 8 ) and ( 4 . 1 3 - 9 ) t h a t the Cn c o e f f i c i e n t s i n Table 4 . 1 3 - 1 and
F i g . 4 . 1 3 - 2 a r e e q u a l t o the group t i m e d e l a y i n s e c o n d s a s o' approaches z e r o .
152
B-wl-m.
PROTOTYPE TCHEBYSCHEFF FILTER COEFFICIENTS C, vs. db TCHEBYSCHEFF RIPPLE, FOR PROTOTYPES WITH n REACTIVE ELEMENTS AND w,' = 1 These coefficients are for use in Eq. (4.13-8). They are also the group time delay i n seconds as a ' approaches zero (see Eq. 4.08-2).
means f o r e s t i m a t i n g t h e e f f e c t s o f u n i f o r m d i s s i p a t i o n a t a n y f r e q u e n c y . B o d e ' s d i s c u ~ s i o ni n~d ~i c a t e s t h a t f o r c a s e s w h e r e a l l t h e i n d u c t a n c e s
u n e q u a l t e r m i n a t i o n s ; h e n c e , Eq.
h a v e a g i v e n Q, Q L , a n d a l l o f t h e c a p a c i t a n c e s h a v e a n o t h e r Q, Q c , good
i f t h e pass-band r i p p l e s a r e l a r g e .
r e s u l t s c a n b e o b t a i n e d by c o m p u t i n g d a s
accuracy i f t h e v a l u e o f Q i s v e r y low. T h i s happens a s a r e s u l t o f u s i n g o n l y t h e l a s t two t e r m s i n Eq. ( 4 . 1 3 - 7 ) . The a c t u a l v a l u e o f
2a;/(QL + Q,).
p r e s e n t e d a n o t h e r formula which i s convenient f o r e s t i -
cohn2'has
m a t i n g t h e e f f e c t s o f d i s s i p a t i o n l o s s o f l o w - p a s s p r o t o t y p e s f o r w'
= 0.
H i s f o r m u l a may b e e x p r . e s s e d i n t h e f o r m
( 4 . 1 3 - 1 1 ) h a s r e l a t i v e l y low a c c u r a c y Equation (4.13-8) gives reduced
( a A ) Owas c o m p u t e d by u s i n g a s many t e r m s i n Eq. ( 4 . 1 3 - 7 ) a s was r e q u i r e d i n o r d e r t o o b t a i n h i g h a c c u r a c y . The v a l u e s computed u s i n g Eq. ( 4 . 1 3 - 9 ) w e r e o b t a i n e d by c o m p u t i n g p h a s e s l o p e from F i g . 4 . 0 8 - 1 . The Q = 1 0 v a l u e s i n c l u d e d i n
N o t e t h a t t h e r e s u l t s a r e q u i t e good.
Table 4.13-2 a r e o f p r a c t i c a l i n t e r e s t s i n c e i n t h e c a s e o f low-pass w h e r e t h e d k a r e g i v e n by E q .
(4.13-3), and t h e p r o t o t y p e element values g k a r e s p e c i f i c a l l y assumed t o have been n o r m a l i z e d s o t h a t g o = 1 ( a s
has been done f o r a l l o f t h e p r o t o t y p e s d i s c u s s e d i n t h i s c h a p t e r ) .
prototypes o f band-pass f i l t e r s t h e element Q's f o r t h e low-pass p r o t o t y p e c a n become q u i t e low i f t h e f r a c t i o n a l b a n d w i d t h w o f t h e b a n d - p a s s f i l t e r i s s m a l l [ s e e Eq. ( 4 . 1 3 - 2 ) l .
Note
t h a t t h i s formula d o e s n o t r e q u i r e t h a t t h e d i s s i p a t i o n be uniform.
The a b o v e d i s c u s s i o n t r e a t s t h e e f f e c t s o f p a r a s i t i c d i s s i p a t i o n a t w'
E q u a t i o n ( 4 . 1 3 - 1 1 ) was d e r i v e d by a s s u m i n g t h a t t h e l o a d a n d s o u r c e r e -
= 0, and t h e important q u e s t i o n a r i s e s a s t o what t h e l o s s w i l l be
s i s t a n c e s e r e b o t h o n e ohm, a n d t h a t t h e e f f e c t o f e a c h R ; o r G; i n
E q u a t i o n ( 4 . 1 3 - 9 ) p r o v i d e s c o n v e n i e n t means f o r o b t a i n i n g an approximate answer t o t h i s q u e s t i o n . S i n c e i t s a y s t h a t
F i g . 4 . 1 3 - 1 a t o' = 0 i s t o a c t a s v o l t a g e o r c u r r e n t d i v i d e r w i t h r e s p e c t t o o n e ohm?' As a r e s u l t , Eq. ( 4 . 1 3 - 1 1 ) c a n l e a d t o a p p r e c i a b l e
DLA a t any f r e q u e n c y i s p r o p o r t i o n a l t o t h e a t t e n u a t i o n p h a s e s l o p e ( i . e . , t h e g r o u p t i m e d e l a y ) a t t h a t f r e q u e n c y , we c a n e s t i m a t e LAa c r o s s t h e
e r r o r i f t h e l o a d and s o u r c e r e s i s t a n c e s a r e s i z e a b l y d i f f e r e n t , though i t g e n e r a l l y g i v e s very good r e s u l t s i f t h e t e r m i n a t i o n s a r e e q u a l o r a t
p a s s band by e x a m i n i n g t h e p h a s e s l o p e a c r o s s t h a t b a n d .
least not very greatly different.'
e l s e w h e r e i n t h e p a s s band.
As s e e n f r o m
t h e examples i n F i g . 4.08-1, t h e phase s l o p e i n t y p i c a l c a s e s i s g r e a t e s t I n F i g . 4.08-1 t h e s l o p e n e a r c u t o f f i s 2.66,
near t h e c u t o f f frequency.
T a b l e 4 . 1 3 - 2 c o m p a r e s t h e a c c u r a c y o f E q s . ( 4 . 1 3 - 8 ) , ( 4 . 1 3 - 9 1 , and (4.13-11) f o r v a r i o u s Tchebyscheff f i l t e r s h a v i n g uniform d i s s i p a t i o n .
1 . 7 3 , a n d 1 . 4 9 t i m e s t h e s l o p e a t w' = 0 f o r t h e c a s e s o f 0 . 5 - d b r i p p l e , 0 . 0 1 - d b r i p p l e , and m a x i m a l l y f l a t r e s p o n s e s , r e s p e c t i v e l y . T h u s
Cases 1 t o 3, which a r e f o r f i l t e r s w i t h n = 4 r e a c t i v e e l e m e n t s , have
n e a r c u t o f f w i l l b e g r e a t e r t h a n ( M A ) o a t w'
T a b l e 4.13-2
(OL.4)0
2.39 0.5 0.5
I*
Equation
(4.13-11)
(4.13-8)
E'?.
0.232 0.214 1.95 0.357 3.05
EQ.
(4.13-9)
+
R,+1)2
S EC.
BY EQ.
--
--0.35 3.5
where R o and Rntl
(4.13-11)
formula i s d e r i v e d u s i n g t h e assumption t h a t t h e r e a c t a n c e s o f t h e s e r i e s
0.264 0.346 3.46 0.365 3.65
i n d u c t a n c e s a r e v e r y l a r g e compared t o t h e r e a c t a n c e s o f t h e s h u n t c a p a c i t o r s . When t h i s c o n d i t i o n h o l d s , t h e v o l t a g e a t o n e n o d e o f t h e f i l t e r may b e c o m p u t e d w i t h good a c c u r a c y f r o m t h a t a t t h e p r e c e d i n g n o d e u s i n g a s i m p l e v o l t a g e d i v i d e r c o m p u t a t i o n . P Cohn f u r t h e r s i m p l i f i e s h i s by
a r e t h e raaimtanoea
T h i s c a n be a e e n from t h e m l t a r n a t e p o i n t o f view i n S e c .
6.14.
4 . 1 4 , APPROXIMATE CALCULATION OF PHOTOTYPE STOP- BAND ATTENUATION
cohnZ0 h a s d e r i v e d a c o n v e n i e n t f o r m u l a f o r c o m p u t i n g t h e a t t e n u a t i o n of low-pass f i l t e r s a t f r e q u e n c i e s w e l l i n t o t h e i r s t o p bands. T h i s
(&A)o
can be made t o be more a c c u r a t e f o r t h e c a s e o f unequal t e r m i n a t i o n .
m u l t i p l ~ i n ei t s r i g h t - h a n d s i d e by 4RORntl/(R0 of t h e t e r m i n a t i o n . .
(&A '0 BY
BY VALUE
= 0 by a b o u t t h e s e f a c t o r s .
These r e s u l t s a r e t y p i c a l and a r e u s e f u l i n o b t a i n i n g an e s t i m a t e o f what t o e x p e c t i n p r a c t i c a l s i t u a t i o n s .
COMPARISON OF ACCURACY OF EQS. (4.13-8), (4.13-9), AND (4.13-11) FOR COMPUTING (A,), FOR VARIOUS TCHEBYSCHEFF FILTERS
RIPPLE
uA
-
f o r m u l a by u s e o f t h e a s s u m p t i o n t h a t ( U ~ L ~ C , , ~1) = w z L k ~ k + l
C o h n ' s f o r m u l a , when p u t i n t h e n o t a t i o n o f t h e l o w - p a s s p r o t o t y p e
than i n t h e p a s s band.
f i l t e r s i n this chapter, is
The s t o p band u s u a l l y h a s one o r more f r e -
quencies where, i f t h e f i l t e r had no d i s s i p a t i o n l o s s , t h e a t t e n u a t i o n would b e i n f i n i t e . H o w e v e r , d i s s i p a t i o n l o s s i n t h e r e s o n a t o r s w i l l p r e v e n t t h e a t t e n u a t i o n from g o i n g t o i n f i n i t y a n d i n some c a s e s may r e d u c e t h e maximum s t o p - b a n d a t t e n u a t i o n t o a n u n a c c e p t a b l y low v a l u e . I f a b a n d - s t o p f i l t e r i s d e s i g n e d from a l o w - p a s s p r o t o t y p e , i t i s q u i t e e a s y t o c o m p u t e t h e e f f e c t s o f f i n i t e r e s o n a t o r Q ' s o n t h e maximum s t o p band a t t e n u a t i o n .
. ..,
w h e r e go, g , , Fig. 4 . 0 4 - l ( a ) ,
g n t l a r e t h e prototype element values defined i n
I
( b ) a n d w' i s t h e p r o t o t y p e r a d i a n f r e q u e n c y v a r i a b l e .
F o r t h i s f o r m u l a t o h a v e h i g h a c c u r a c y , w' s h o u l d b e a number o f t i m e s a s l a r g e a s w;, t h e f i l t e r c u t o f f f r e q u e n c y .
i
The s o l i d l i n e s i n F i g . 4 . 1 5 - 1 shows a T c h e b y s c h e f f l o w - p a s s p r o t o t y p e r e s p o n s e a l o n g w i t h t h e r e s p o n s e o f a b a n d - s t o p f i l t e r d e s i g n e d from
I
As a n e x a m p l e , c o n s i d e r a T c h e b y s c h e f f f i l t e r w i t h n = 4 r e a c t i v e By T a b l e 4 . 0 5 - 2 ( a ) , g o = 1 , g, - 1 . 3 0 2 8 , gZ = 1 . 2 8 4 4 , g 3 = 1 . 9 7 6 1 , g 4 = 0 . 8 4 6 8 , g S = 1 . 5 3 8 6 , a n d t h e c u t o f f f r e elements and 0 . 2 d b r i p p l e .
q u e n c y i s w;
= 1.
LA
=
By E q . ( 4 . 1 4 - 1 1 , 20 loglo
t o s l i d e - r u l e accuracy
[(w'14(4.29)1
-
1 0 log,,
6.15
.
(4.14-2)
E v a l u a t i n g Eq. ( 4 . 1 4 - 2 ) f o r w' = 3 g i v e s L A = 4 3 . 1 d b . By F i g . 4 . 0 3 - 6 we f i n d t h a t t h e a c t u a l a t t e n u a t i o n i s 42 d b . R e p e a t i n g t h e c a l c u l a t i o n f o r w' = 2 g i v e s L A = 2 8 . 8 d b a s c o m p a r e d t o 2 6 . 5 d b by F i g . 4 . 0 3 - 6 . Thus i t a p p e a r s t h a t e v e n f o r v a l u e s o f w'/w; a s s m a l l a s 2 , Eq. ( 4 . 1 4 - 1 ) g i v e s f a i r l y g o o d r e s u l t s . The e r r o r was + 2 . 3 d b f o r w' = 2 a n d t1.1 d b f o r w' = 3 . Equation (4.14-1) n e g l e c t s t h e e f f e c t s o f d i s s i p a t i o n i n t h e c i r c u i t . T h i s i s v a l i d a s l o n g a s t h e d i s s i p a t i v e e l e m e n t s i n t h e p r o t o t y p e can b e a s s u m e d t o be a r r a n g e d a s a r e t h o s e i n F i g . 4 . 1 3 - 1 . This arrangement o f d i s s i p a t i v e e l e m e n t s i s u s u a l l y a p p r o p r i a t e f o r p r o t o t y p e s f o r lowp a s s , b a n d - p a s s , and h i g h - p a s s f i l t e r s .
However, i n t h e c a s e o f p r o t o -
types f o r band-stop f i l t e r s , t h e d i f f e r e n t arrangement of d i s s i p a t i v e elem e n t s d i s c u s s e d i n S e c . 4 . 1 5 s h o u l d b e a s s u m e d . F o r t h a t c a s e Eq. ( 4 . 1 4 - 1 )
w i l l b e q u i t e i n a c c u r a t e i n some p a r t s o f t h e s t o p b a n d . SEC.
4.15,
PROTOTYPE REPRESENTATION OF D I S S I P A T I O N LOSS I N BAND- STOP F I L T E H S
I n the case of band-stop f i l t e r s , t h e e f f e c t s of p a r a s i t i c dissipat i o n i n t h e f i l t e r e l e m e n t s a r e u s u a l l y more s e r i o u s i n t h e s t o p b a n d
156
F I G . 4.15-1
A LOW-PASS P R O T O T Y P E RESPONSE IS SHOWN A T (a), AND T H E CORRESPONDING BAND-STOP F I L T E R RESPONSE IS SHOWN A T (b) T h e dashed lines show the effects of dissipation loss.
t h i s p r o t o t y p e , both f o r t h e tion.
where
L;=P;
c a s e o f no i n c i d e n t a l d i s s i p a -
r??
Rh,,
For a t y p i c a l band-stop
+~,
f i l t e r the resonators a r e reso-
-
nant a t the center of the s t o p
~ 1 . 0 ~ 0 ~
G;.D,oI
Go3
=% ;%
---
band ( i n s t e a d o f a t t h e c e n t e r
and w a ,u b , and w o a r e a s d e f i n e d i n F i g . 4 . 1 5 - l ( b ) .
By Eq. ( 4 . 1 5 - 2 ) .
L-3517-Sl
o f t h e p a s s band a s i s t h e c a s e f o r a t y p i c a l band-pass f i l t e r ) , and a s a r e s u l t t h e l o s s e f f e c t s a r e most s e v e r e a t t h e c e n t e r o f t h e s t o p band.
The d a s h e d l i n e i n F i g . 4 . 1 5 - l ( b ) shows how d i s -
FIG. 4.15-2 LOW-PASS PROTOTYPE FILTER WITH DISSIPATIVE ELEMENTS ADDED AS REQUIRED FOR COMPUTING PEAK STOP-BAND ATTENUATION OF CORRESPONDING BAND-STOP FILTERS
a n d a s shown i n F i g . 4 . 1 5 - 2 ,
i
sipation loss i n the resonators I
w i l l round o f f t h e a t t e n u a t i o n c h a r a c t e r i s t i c o f a band-stop
where t h e gk a r e t h e p r o t o t y p e f i l t e r e l e m e n t s a s d e f i n e d i n F i g . 4 . 0 4 - 1
filter.
The d a s h e d l i n e i n F i g . 4 . 1 5 - l ( a ) shows t h e c o r r e s p o n d i n g e f f e c t i n a low-pass p r o t o t y p e f i l t e r .
I t is e a s i l y seen t h a t i n o r d e r f o r r e s i s t o r elements t o a f f e c t t h e a t t e n u a t i o n o f a p r o t o t y p e f i l t e r a s shown by t h e d a s h e d l i n e i n Fig. 4.15-1,
they should be introduced i n t o t h e prototype c i r c u i t a s
shown i n F i g . 4 . 1 5 - 2 .
a, t h e r e a c t i v e N o t e t h a t i n t h i s c a s e a s w' e l e m e n t s h a v e n e g l i g i b l e i n f l u e n c e a n d t h e c i r c u i t o p e r a t e s i n t h e same
way a s a l a d d e r n e t w o r k o f r e s i s t o r s . r e a c t i v e e l e m e n t i s g i v e n by*
+
As p r e v i o u s l y m e n t i o n e d , when w'
-03
the reactive elements i n
F i g . 4 . 1 5 - 2 may b e n e g l e c t e d , a n d t h e a t t e n u a t i o n c a n b e c o m p u t e d f r o m In typical cases, resistances of t h e remaining network of r e s i s t o r s . t h e s e r i e s b r a n c h e s w i l l be v e r y l a r g e compared t o t h e r e s i s t a n c e s o f t h e s h u n t b r a n c h e s , and C o h n ' s m e t h o d f o r c o m p u t i n g t h e s t o p - b a n d a t t e n u The a t i o n of low-pass f i l t e r s z 0 can be adapted t o c o v e r t h i s c a s e a l s o . resulting equation is
I n Fig. 4.15-2 the Q of t h e k t h
w h i c h i s a n a l o g o u s t o Eq. ( 4 . 1 4 - 1 ) f o r t h e r e a c t i v e a t t e n u a t i o n o f a low-pass f i l t e r . As a n e x a m p l e , l e t u s s u p p o s e t h a t a b a n d - s t o p f i l t e r i s d e s i r e d w h e r e a; i s t h e c u t o f f f r e q u e n c y i n F i g . 4 . 1 5 - l ( a ) .
Q
The u n l o a d e d Q,
k o f t h e k t h r e s o n a t o r o f t h e b a n d - s t o p f i l t e r i s r e l a t e d t o Qk
o f t h e p r o t o t y p e ( a t f r e q u e n c y a ; ) by
with a f r a c t i o n a l s t o p - b a n d w i d t h o f w = 0 . 0 2 ( r e f e r r e d t o t h e 3 db p o i n t s ) , and t h a t maximally f l a t p a s s bands a r e d e s i r e d . L e t u s assume f u r t h e r t h a t t h e r e s o n a t o r Q ' s a t t h e m i d - s t o p - b a n d f r e q u e n c y a r e 700 a n d t h a t t h e maximum s t o p - b a n d a t t e n u a t i o n i s t o b e c o m p u t e d . By Eq. ( 4 . 1 5 - 3 ) Q, = Qz = 0 . 0 2 ( 7 0 0 ) = 1 4 . By T a b l e 4 . 0 5 - l ( a ) t h e e l e m e n t s values of t h e d e s i r e d
*
Note t h a t t h e s e uauaual d e f i n i t i o n s o f Q r e s u l t from t h e manner i n which the d i a e i p a t i r e e l e m e n t s a r e i n t r o d u c e d i n each branch of the f i l t e r .
n =
g2 = 1 . 4 1 4 , a n d g 3 = 1 .
2 l o w - p a s s p r o t o t y p e a r e g o = 1, g , = 1 . 4 1 4 , A l s o , w ; which i n t h i s c a s e i s t h e 3-db hand-edge
!
f r e q u e n c y , i s e q u a l t o u n i t y . By Eq. ( 4 . 1 5 - 5 ) , Dl = D 2 = 1 4 , a n d , t o s l i d e - r u l e a c c u r a c y , Eq. ( 4 . 1 5 - 7 ) g i v e s ( L A ) , = 4 5 . 8 d b . I n c o m p a r i s o n , u s i n g t h e method o f S e c . 2 . 1 3 t o compute t h e a t t e n u a t i o n from t h e l a d d e r of r e s i s t o r s g i v e s (LA), = 46.7 db. As i s s u g g e s t e d by t h e d a s h e d l i n e s i n F i g . 4 . 1 5 - 1 ,
/
I
the e f f e c t s of
I
d i s s i p a t i o n i n t h e p a s s band a r e f o r t h i s c a s e most s e v e r e a t t h e p a s s -
f o r Eq. ( 4 . 1 5 - 8 ) - t h e
The i n c r e a s e i n l o s s d u e t o
Thus t o s u m m a r i z e t h e b a s i s
e q u a t i o n a s i t s t a n d s g i v e s a rough e s t i m a t e o f
t h e a t t e n u a t i o n d u e t o d i s s i p a t i o n a t band e d g e f o r t h e s i t u a t i o n w h e r e We t h e d i s s i p a t i v e e l e m e n t s a r e i n t r o d u c e d a s shown i n F i g . 4 . 1 3 - 1 .
d i s s i p a t i o n a t t h e b a n d - e d g e f r e q u e n c y c a n b e e s t i m a t e d by u s e o f t h e formula
I t i s r e a d i l y s e e n t h a t t h i s i s a good
approximation even f o r Q's a s low a s 1 0 .
band e d g e , a n d t h e y d e c r e a s e t o z e r o a s t h e f r e q u e n c y moves away f r o m t h e pass-band edge ( w i t h i n t h e p a s s band).
r e p r e s e n t s a good a p p r o x i m a t i o n .
*
I
I
j u s t i f y t h e u s e o f t h i s same e q u a t i o n f o r t h e c a s e o f d i s s i p a t i v e e l e m e n t s a r r a n g e d a s i n F i g . 4 . 1 5 - 2 on t h e b a s i s of t h e a p p r o x i m a t i o n i n I t shows t h a t a s l o n g a s t h e r e a c t i v e e l e m e n t s a r e t h e Eq. ( 4 . 1 5 - 9 ) . s a m e , and t h e e l e m e n t Q ' s a r e t h e s a m e , a n d a r o u n d 1 0 o r h i g h e r , i t d o e s n ' t make much d i f f e r e n c e w h i c h way t h e d i s s i p a t i v e e l e m e n t s a r e c o n n e c t e d a s f a r a s t h e i r e f f e c t on t r a n s m i s s i o n l o s s i s c o n c e r n e d .
This formula r e p r e s e n t s o n l y an e s t i m a t e , b u t should be reasonably accurate f o r c a s e s s u c h a s when a n n = 5 , 0 . 1 - d b r i p p l e p r o t o t y p e i s u s e d . F o r c a s e s where very l a r g e Tchebyscheff r i p p l e s a r e used t h i s e q u a t i o n w i l l u n d e r e s t i m a t e t h e l o s s ; when v e r y s m a l l r i p p l e s a r e u s e d i t w i l l o v e r e s t i m a t e t h e l o s s . For 0.1-db r i p p l e , i f n were reduced t o 2 o r 1 ,
I
Eq. ( 4 . 1 5 - 8 ) would t e n d t o o v e r e s t i m a t e t h e b a n d - e d g e l o s s . F o r t y p i c a l p r a c t i c a l c a s e s , Eq. ( 4 . 1 5 - 8 ) s h o u l d n e v e r h a v e a n e r r o r a s g r e a t a s a f a c t o r of 2. E q u a t i o n ( 4 . 1 5 - 8 ) was o b t a i n e d f r o m E q . ( 4 . 1 3 - 1 1 ) by t h e u s e o f two a p p r o x i m a t i o n s . The f i r s t i s t h a t f o r t h e a r r a n g e m e n t o f d i s s i p a t i v e e l e m e n t s shown i n F i g . 4 . 1 3 - 1 , t h e a d d e d l o s s M A d u e t o d i d s i p a t i o n a t t h e b a n d e d g e w; i s r o u g h l y t w i c e t h e v a l u e ( M A ) o o f t h e l o s s d u e t o d i s s i p a t i o n when o' = 0 .
T h i s was shown b y e x a m p l e s i n S e c . 4 . 1 3 t o b e
a r e a s o n a b l y good a p p r o x i m a t i o n f o r t y p i c a l l o w - p a s s p r o t o t y p e f i l t e r s , though i t c o u l d be markedly l a r g e r i f very l a r g e pass-band r i p p l e s a r e used.
The s e c o n d a p p r o x i m a t i o n a s s u m e s t h a t a f i l t e r w i t h d i s s i p a t i v e
.
e l e m e n t s a s shown i n F i g . 4 . 1 5 - 2 c a n b e a p p r o x i m a t e d a t t h e f r e q u e n c y o; by t h e c o r r e s p o n d i n g c i r c u i t i n F i g . 4 . 1 3 - 1 .
The r e a c t i v e e l e m e n t v a l u e s
g k a r e assumed t o have been unchanged, and a l s o t h e Q ' s o f t h e i n d i v i d u a l
r e a c t i v e element a r e assumed t o be unchanged; however, the d i s s i p a t i o n i s introduced has been changed.
t h e manner i n w h i c h
This approxlmation is
valid to the extent that
*
This formula im baaed on Eq. ( 4 . 1 3 - 1 1 ) which anaumem t h a t t h e prototypa element raluam h a r e been normmlized so t h a t go = 1 .
I
REFERENCES
CHAPTER 5
PROPERTIES OF SOME COMMON MICROWAVE FILTER ELEMENTS 1. S. Darlington, "Synthesis of Reactance 4-Poles Which Produce Prescribed Insertion Loss Characteristics," Jour. Math. and Phys., Vol. 18, pp. 257-353 (September 1939). 2. E. A. Guillemin, Synthesis of Passive Networks (John Wiley and Sons, Inc., New York, 1957). 3. M. E. Van Valkenburg, Introduction to Modern Netuork Synthesis (John Wiley and Sons, New York, 1960). 4. V. Belevitch, "Tchebyscheff Filters and Amplifier Networks," Wireless Engineer, Vol. 29, pp. 106-110 (April 1952). 5. H. J. Orchard, "Formula for Ladder Filters," Wireless Engineer, Vol. 30, pp. 3-5 (January 1953)
.
6. E. Green, "Synthesis of Ladder Networka toGive Butterworth or Chebyshev Response in the Pass Band," Proc. IEE (London) Part IV, Monograph No. 88 (1954). 7.
SEC. 5 . 0 1 ,
INTRODUCTION
P r e v i o u s c h a p t e r s h a v e s u m m a r i z e d a number o f i m p o r t a n t c o n c e p t s necessary f o r t h e d e s i g n of microwave f i l t e r s and have o u t l i n e d v a r i o u s p r o c e d u r e s f o r l a t e r u s e i n d e s i g n i n g f i l t e r s f r o m t h e image v i e w p o i n t In order t o construct f i l t e r s and from t h e i n s e r t i o n - l o s s v i e w p o i n t . t h a t w i l l h a v e m e a s u r e d c h a r a c t e r i s t i c s a s p r e d i c t e d by t h e s e t h e o r i e s ,
E. Green, A w l itude-Frequency Characteristics of Ladder Networks, pp. 62-78. Marconi' s Wireless Telegraph Co., Ltd., Chelmsford, Essex, England (1954).
i t i s n e c e s s a r y t o r e l a t e t h e d e s i g n p a r a m e t e r s t o t h e dimensions and p r o p e r t i e s o f t h e s t r u c t u r e s u s e d i n s u c h f i l t e r s . Much i n f o r m a t i o n o f
L. Weinberg, "Network Design by Use of Modern Synthesis Techniques and Tables," Proc. of Nat. Elec. Conf., Vol. 12 (1956).
t h i s type is available i n the l i t e r a t u r e .
9. L. Weinberg, "Additions1 Tables for Design of Optimum Ladder Networks," Parts I and 11, Journal of the Franklin Institute, Vol. 264, pp. 7-23 and 127-138 (July and August 1957).
The p r e s e n t c h a p t e r w i l l a t t e m p t
t o summarize i n f o r m a t i o n f o r c o a x i a l l i n e s , s t r i p l i n e s , and waveguides t h a t i s m o s t o f t e n n e e d e d i n f i l t e r d e s i g n . No p r e t e n s e o f c o m p l e t e n e s s
10. L. Storch, "Synthesis of Constant-Time-blsy Ladder Networks Using Bessel Polynomials," Proc. IRE 42, pp. 1666-1675 (November 1954).
i s made, s i n c e a c o m p l e t e c o m p i l a t i o n o f s u c h d a t a w o u l d f i l l s e v e r a l
11. W. E. Thomson, "Networks with Maximally Flat Delay," Wireless Engineer, Vol. 29, pp. 255-263 (October 1952).
I t i s hoped t h a t t h e r e f e r e n c e s i n c l u d e d w i l l d i r e c t t h e i n t e r volumes. e s t e d r e a d e r t o s o u r c e s of more d e t a i l e d i n f o r m a t i o n o n p a r t i c u l a r s u b j e c t s
12. S. B. Cohn, "Phase-Shift and Time-Delay Response of Microwave Narrow-Band Filters," The Microwave Journal, Vol. 3, pp. 47-51 (October 1960). 13. M. J. Di Toro, "Phase and Amplitude Distortion in Linear Networks,"Proc. IRE 36,pp. 24-36 (January 1948). 14. R. M. Fano, "Theoretical Limitations on the Broadband Matching of Arbitrary Impedances," J. Franklin Inst., Vol. 249, pp. 57-83 and 139-154 (January and February 1950). 15. B. F. Barton, "Design of Efficient Coupling Networks," Technical Report 44, Contract DA 36-039-SC-63203,Electronic Defense Group, University of Michigan, Ann Arbor, Michigan (March 1955). 16. H. J. Carlin, "Gain-Bandwidth Limitations on Equalizers and Matching Networks,"Proc. IRE 42, pp. 1676-1685 (November 1954). G. L. Matthaei, "Synthesis of Tchebyscheff Impedance-Matching Networks, Filters, and Interstages," IRE Trans. PCCT 3, pp. 162-172 (September 1956).
H. J. Carlin, "Synthesis Techniques for Gain-Bandwidth Optimization in Passive Transducers," Proc. IRE 48,pp. 1705-1714 (October 1960). 19. H. W. Bode, Network Analysis and Feedback Amplifier Design pp. 216-222 (D. Van Nostrand Co., Inc., New York. 1945). 20. S. B. Cohn, "Dissipation Loss in Multiple-Coupled-Resonator Filters,"Proc. IRE 47, pp. 1342-1348 (August 1959). 21. W. J. Getsinger, "Prototypes for Use in Broadbanding Reflection Amplifiers," scheduled to appear in the November 1963 issue of the IEEE Trans. PTGMTT.
SEC. 5 . 0 2 , GENERAL PROPERTIES OF TEM-MODE TRANSMISSION L I N E S T r a n s m i s s i o n l i n e s composed o f two c o n d u c t o r s o p e r a t i n g i n t h e t r a n s v e r s e e l e c t r o m a g n e t i c (TEM) mode a r e v e r y u s e f u l a s e l e m e n t s o f m i c r o w a v e f i l t e r s . L o s s l e s s l i n e s of t h i s t y p e have a c h a r a c t e r i s t i c o r image i m p e d a n c e Z o , w h i c h i s i n d e p e n d e n t o f f r e q u e n c y f , . a n d w a v e s on t h e s e l i n e s a r e propagated a t a v e l o c i t y , v , equal t o t h e v e l o c i t y of l i g h t i n t h e d i e l e c t r i c f i l l i n g t h e l i n e . D e f i n i n g R , L , G, a n d C a s t h e r e s i s t a n c e , i n d u c t a n c e , c o n d u c t a n c e and c a p a c i t a n c e p e r u n i t l e n g t h f o r s u c h a l i n e , i t i s f o u n d t h a t Z o a n d t h e p r o p a g a t i o n c o n s t a n t y t a r e g i v e n by
When t h e l i n e i s l o s s l e s s , a t i s z e r o a n d
where w = 2 r f .
&=am
radians/unit
length
(5.02-3)
The TEM modes c a n a l s o p r o p a g a t e on s t r u c t u r e s c o n t a i n i n g m o r e t h a n two 0
vL ohms
vc
conductors.
.
E x a m p l e s o f s u c h s t r u c t u r e s w i t h two c o n d u c t o r s c o n t a i n e d w i t h -
i n an o u t e r s h i e l d a r e d e s c r i b e d i n S e c . 5 . 0 5 . e x i s t on such two-conductor s t r u c t u r e s :
I n p r a c t i c e a l i n e w i l l h a v e some f i n i t e amount o f a t t e n u a t i o n
Two p r i n c i p a l modes c a n
a n e v e n mode i n w h i c h t h e c u r r e n t s
i n t h e two c o n d u c t o r s f l o w i n t h e same d i r e c t i o n , a n d an odd mode i n which t h e c u r r e n t s on t h e c o n d u c t o r s flow i n o p p o s i t e d i r e c t i o n s .
The v e l o c i t y o f
p r o p a g a t i o n o f e a c h o f t h e s e modes i n t h e l o s s l e s s c a s e i s e q u a l t o t h e w h e r e a , i s t h e a t t e n u a t i o n d u e t o c o n d u c t o r l o s s a n d ad t h e a t t e n u a t i o n
v e l o c i t y o f l i g h t i n t h e d i e l e c t r i c medium s u r r o u n d i n g t h e c o n d u c t o r s . Howe v e r , t h e c h a r a c t e r i s t i c i m p e d a n c e o f t h e e v e n mode i s d i f f e r e n t f r o m t h a t
due t o l o s s i n t h e d i e l e c t r i c .
o f t h e odd mode.
For s m a l l a t t e n u a t i o n s
SEC.
5 . 0 3 , S P E C I A L PHOPERTIES OF COAXIAL L I N E S The c h a r a c t e r i s t i c i m p e d a n c e Z 0 o f a c o a x i a l l i n e o f o u t e r d i a m e t e r
b and i n n e r d i a m e t e r d , f i l l e d w i t h a d i e l e c t r i c m a t e r i a l o f r e l a t i v e ad
=
Pt
G
-
2Yo
5
-
Pt
=
2Qd
2 tan S
dielectric constant nepers
E
~
i, s
(5.02-8) 60 -
=
w h e r e Qe = d / R , Q d = d / G , a n d t a n S i s t h e l o s s t a n g e n t o f t h e d i e l e c t r i c m a t e r i a l f i l l i n g t h e l i n e . The t o t a l Q o f t h e t r a n s m i s s i o n l i n e u s e d a s a r e s o n a t o r i s g i v e n by
b
ohms
This e x p r e s s i o n i s p l o t t e d i n F i g . 5.03-1.
The a t t e n u a t i o n a c o f a c o p p e r
c o a x i a l l i n e d u e t o ohmic l o s s e s i n t h e c o p p e r i s
db/unit length (5.03-2) These d e f i n i t i o n s a r e i n agreement w i t h t h o s e g i v e n i n terms o f t h e r e s o n a t o r r e a c t a n c e and s u s c e p t a n c e s l o p e p a r a m e t e r s i n S e c . 5.08.
For
where
fCe
is measured i n g i g a c y c l e s .
a s l i g h t l y l o s s y l i n e t h e c h a r a c t e r i s t i c impedance and p r o p a g a t i o n
very smooth and c o r r o s i o n - f r e e . )
c o n s t a n t become
of 3 . 6 c o r r e s p o n d i n g t o
K z Oo f
(Here t h e c o p p e r i s assumed t o be
The a t t e n u a t i o n i s a minimum f o r b / d 77 ohms.
The a t t e n u a t i o n ad o f t h e c o a x i a l l i n e ( o r a n y o t h e r TEM l i n e ) d u e to losses i n the dielectric i s 27.3 TO convert nepers t o d e c i b e l . ,
multiply by 8.686.
164
ad
=
< A
tan 6 db/unit length
FIG. 5.03-1 COAXIAL-LINE CHARACTERISTIC IMPEDANCE
where t a n 8 i s t h e l o s s t a n g e n t o f t h e d i e l e c t r i c , and A i s t h e f r e e - s p a c e wavelength. The t o t a l a t t e n u a t i o n a t i s t h e sum o f a c and a d . The a t t e n u a t i o n o f a c o a x i a l l i n e d u e t o ohmic l o s s e s i n t h e c o p p e r i s shown i n F i g . 5.03-2. The Q o f a d i e l e c t r i c - f i l l e d c o a x i a l l i n e may be e x p r e s s e d a s
FIG. 5.03-2 COAXIAL-LINE ATTENUATION AND Q
where Qc =
< / M ~ d e p e n d s o n l y on t h e c o n d u c t o r l o s s and Qd d e p e n d s only
on t h e d i e l e c t r i c l o s s . The Q c o f a d i e l e c t r i c - f i l l e d c o a x i a l l i n e i s i n d e p e n d e n t o f E , and i s g i v e n by t h e e x p r e s s i o n
where b and d a r e measured i n i n c h e s . l i n e o r any
TEM l i n e
The v a l u e o f Qd f o r t h e c o a x i a l
is
1 Q
The v a l u e s o f
Qc
d
=
-
(5.03-6)
.
tan 6
f o r a copper coaxial l i n e a r e p l o t t e d i n Fig. 5.03-2
Breakdown w i l l o c c u r i n a n a i r - f i l l e d c o a x i a l l i n e a t a t m o s p h e r i c p r e s s u r e when t h e maximum e l e c t r i c f i e l d E m r e a c h e s a v a l u e o f a p p r o x i m a t e l y 2 . 9 x l o 4 v o l t s p e r cm. The a v e r a g e power P t h a t c a n b e t r a n s m i t t e d on a matched c o a x i a l l i n e under t h e s e c o n d i t i o n s i s
p
=
ES
- b 2 480
In b/d
watts
(b/d)2
When t h e o u t e r d i a m e t e r b i s f i x e d , t h e maximum power c a n b e t r a n s m i t t e d when b / d i s 1 . 6 5 , c o r r e s p o n d i n g t o Z, o f 30 ohms. The f i r s t h i g h e r - o r d e r TE mode i n a c o a x i a l l i n e w i l l p r o p a g a t e wher t h e , a v e r a g e c i r c u m f e r e n c e o f t h e l i n e i s a p p r o x i m a t e l y e q u a l t o t h e waveSOURCE:
l e n g t h i n t h e medium f i l l i n g t h e l i n e . f , ( i n g i g a c y c l e s ) , o f t h i s mode i s
The a p p r o x i m a t e c u t o f f f r e q u e n c y ,
Final Report, Contract DA 3 6 0 3 9 SC-63232. SRI; reprinted in IRE Trsns., PCMTT (see Ref. 2. by S. B. Cohn).
FIG. 5.04-1
values of t/b. w h e r e b and d a r e m e a s u r e d i n i n c h e s . S E C . 5 . 0 4 , SPECIAL PROPERTIES OF S T R I P LINES The c h a r a c t e r i s t i c i m p e d a n c e o f s t r i p l i n e c a n b e c a l c u l a t e d by conformal mapping t e c h n i q u e s ; however, t h e r e s u l t i n g f o r m u l a s a r e r a t h e r c o m p l e x . F i g u r e 5 . 0 4 - 1 s h o w s t h e c h a r a c t e r i s t i c i m p e d a n c e , Z,, of a common t y p e o f s t r i p l i n e w i t h a r e c t a n g u l a r c e n t e r for v a r i o u s v a l u e s o f t / b ( 0 . 2 5 , and 0 . 1 ( w/b ( 4 . 0 . The v a l u e s shown a r e e x a c t f o r t / b = 0 and a r e a c c u r a t e t o w i t h i n a b o u t 1 p e r c e n t f o r o t h e r
t/b,
GRAPH OF
ZOvs. w/b FOR VARIOUS VALUES OF t/b
F i g u r e 5 . 0 4 - 2 shows e x a c t v a l u e s o f Z,, f o r a l l v a l u e s o f
and w/b ( 1 . 6 . ~
The t h e o r e t i c a l a t t e n u a t i o n a , d u e t o ohmic l o s s e s i n a c o p p e r s t r i p is line f i l l e d with a d i e l e c t r i c o f relative d i e l e c t r i c constant E , , shown i n F i g . 5 . 0 4 - 3 . The a t t e n u a t i o n a d d u e t o t h e d i e l e c t r i c l o s s i s As i n t h e c a s e o f t h e c o a x i a l . l i n e , t h e t o t a l g i v e n by E q . ( 5 . 0 3 - 3 ) . a t t e n u a t i o n a t i s t h e sum o f a , a n d a d . The Q o f a d i e l e c t r i c - f i l l e d s t r i p l i n e i s g i v e n by E q .
(5.03-4)
The Q c o f a d i e l e c t r i c - f i l l e d l i n e i s shown p l o t t e d i n F i g . 5 . 0 4 - 4 . ' '
AS i n t h e c a s e o f t h e c o a x i a l l i n e , Qd i s t h e r e c i p r o c a l o f t a n 6 .
0.0016 0.0015 0.0014 0.0013 0.0012 0.0011 0.0010
d b PER UNIT LENGTH
0.0009 0.0008 0.0007 0.0006 0.0005 0.0004
I .-3S*?-tOS
SOURCE:
IRE Trans. PGHTT (nee R e f . 3 , b y R . H. T. Baten).
FIG. 5.04-2
GRAPH O F ZO vs. w / b FOR VARIOUS VALUES O F t/b
SOURCE:
Final Report. Contract DA 36-039 SC-63232. SRI; reprinted in IRE Trans.. PGMTT b a e R e f . 2, by S . B. Cohn).
FIG. 5.04-3 THEORETICAL ATTENUATION OF COPPER-SHIELDED STRIP LINE IN A DIELECTRIC MEDIUM E,
SOURCE: F i n d Repon. Contract DA 36-039 SC-63232.SRI; reprinted in IRE Trans.. PCMTT (see Ref. 2, by S. B. Cohn).
FIG. 5.04-4 THEORETICAL Q OF COPPER-SHIELDED STRIP LlNE IN A DIELECTRIC MEDIUM E,
The a v e r a g e p o w e r , P ( m e a s u r e d i n k w ) , t h a t c a n b e t r a n s m i t t e d a l o n g a matched s t r i p l i n e h a v i n g an i n n e r c o n d u c t o r w i t h rounded c o r n e r s i s I n t h i s f i g u r e t h e ground p l a n e s p a c i n g b is p l o t t e d i n Fig. 5.04-5. m e a s u r e d i n i n c h e s , and t h e b r e a k d o w n s t r e n g t h o f a i r i s t a k e n a s An a p p r o x i m a t e v a l u e of Z, c a n b e o b t a i n e d f r o m 2.9 X l o 4 volts/cm. F i g s . 5 . 0 4 - 1 and 5.04-2. The f i r s t h i g h e r - o r d e r mode t h a t c a n e x i s t i n a s t r i p l i n e , i n w h i c h t h e two g r o u n d p l a n e s h a v e t h e same p o t e n t i a l , h a s z e r o e l e c t r i c - f i e l d s t r e n g t h on t h e l o n g i t u d i n a l p l a n e c o n t a i n i n g t h e c e n t e r l i n e o f t h e s t r i p , and t h e e l e c t r i c f i e l d i s o r i e n t e d p e r p e n d i c u l a r t o t h e s t r i p and ground plane.
T h e f r e e - s p a c e c u t o f f w a v e l e n g t h , he
o f t h i s mode i s
FIG. 5.04-5 THEORETICAL BREAKDOWN POWER OF AIR-DIELECTRIC ROUNDED-STRIP TRANSMISSION LlNE
where d i s a f u n c t i o n o f t h e c r o s s s e c t i o n of t h e s t r i p l i n e .
Table 5.04-1
I f t / b = 0 and
'IRE QUANTITY 4 d / b us. b / h + FOR * / b 2 0.35 AND t / b = 6
unctinn nf
SEC.
5 . 0 5 , PARALLEL- COUPLED LINES AND ARRAYS OF LINES BETWEEN GROUND PLANES
f
A number of s t r i p - l i n e components
(a)
u t i l i z e t h e n a t u r a l c o u p l i n g e x i s t i n g between p a r a l l e l c o n d u c t o r s . Examples o f s u c h components a r e d i r e c t i o n a l c o u p l e r s , f i l t e r s , b a l u n s ,
A number o f e x a m p l e s o f The ( a ) , ( b ) , and ( c ) c o n f i g u r a t i o n s shown a r e p r i m a r i l y u s e f u l i n a p p l i c a t i o n s where weak c o u p l i n g between t h e l i n e s i s d e s i r e d . The ( d ) , ( e ) , ( f ) , and ( g ) c o n f i g u r a t i o n s a r e u s e f u l where s t r o n g c o u p l i n g between t h e l i n e s i s desired. and d e l a y l i n e s s u c h a s i n t e r d i g i t a l l i n e s .
p a r a l l e l - c o u p l e d l i n e s a r e shown i n F i g . 5 . 0 5 - 1 .
The c h a r a c t e r i s t i c s o f t h e s e c o u p l e d l i n e s c a n be s p e c i f i e d i n t e r m s o f Z o e and Z o o ,t h e i r e v e n and odd i m p e d a n c e s , r e s p e c t i v e l y .
Zoe
i s d e f i n e d a s t h e c h a r a c t e r i s t i c impedance o f one l i n e t o g r o u n d when
e q u a l c u r r e n t s a r e f l o w i n g i n t h e two l i n e s . Z o o i s d e f i n e d a s t h e c h a r a c t e r i s t i c impedance o f one l i n e t o g r o u n d when e q u a l and o p p o s i t e c u r r e n t s a r e f l o w i n g i n t h e two l i n e s .
F i g u r e 5.05-2 i l l u s t r a t e s t h e
e l e c t r i c f i e l d c o n f i g u r a t i o n o v e r t h e c r o s s s e c t i o n o f t h e l i n e s shown DISTANCE d NEGATIVE
i n F i g . 5 . 0 5 - l ( a ) when t h e y a r e e x c i t e d i n t h e even and odd modes. Thin S t r i p Lines--The
(f)
e x a c t even-mode c h a r a c t e r i s t i c impedance of
t h e i n f i n i t e s i m a l l y t h i n s t r i p c o n f i g u r a t i o n of Fig. 5 . 0 5 - l ( a ) i s 4
FIG. 5.05-1
where
k
=
tan,
(f.
3.
tanh
(t . --;--) w
t
s
CROSS SECTIONS OF VARIOUS COUPLEDTRANSMISSION-LINE CONFIGURATIONS
L/
AXIS OF E V E N SYMMETRY
and K i s t h e c o m p l e t e e l l i p t i c i n t e g r a l o f t h e f i r s t k i n d .
Convenient
t a b l e s o f K ( k l ) / K ( k ) h a v e b e e n c o m p i l e d by O b e r h e t t i n g e r and Magnus.' Nomographs g i v i n g t h e e v e n - a n d odd-mode c h a r a c t e r i s t i c i m p e d a n c e s a r e presented i n Figs.
5 . 0 5 - 3 ( a ) and ( b ) .
Thin L i n e s Coupled Through a Slot-The EVEN-MODE
ELECTRIC
FIELD
AXIS
t h i n - s t r i p configuration
shown i n F i g . 5 . 0 5 - l ( b ) w i t h a t h i n w a l l s e p a r a t i n g t h e two l i n e s h a s a
(01
DISTRIBUTION
OF 0 0 0 SYMMETRY
value of Zoo =
w h i c h i s t h e c h a r a c t e r i s t i c i m p e d a n c e o f an u n c o u p l e d The even-mode c h a r a c t e r i s t i c i m p e d a n c e Z o e l i n e a s given i n Sec. 5.04. Zo,
i s g i v e n a p p r o x i m a t e l y by
where I bl 000-MODE
ELECTRIC
FIELD
DISTRIBUTION
SOURCE: Final Report. Contract DA 3 6 0 3 9 SC-63232. SRI; reprinted in IRE Trans., PGMTT Gee Ref. 4, by S. B. Cohn).
FIG. 5.05-2
FIELD DISTRIBUTIONS OF THE EVEN AND ODD MODES IN COUPLED STRIP LINE
a n d E~ i s t h e r e l a t i v e d i e l e c t r i c c o n s t a n t o f t h e medium o f p r o p a g a t i o n . The e x a c t odd-mode i m p e d a n c e i n t h e same c a s e i s 1
Z"
-
I
-3 .0 x-
K(k:) K(ko)
cosh
--
7
Round Wires-The e v e n - a n d odd-mode c h a r a c t e r i s t i c i m p e d a n c e s o f r o u n d l i n e s p l a c e d ' m i d w a y b e t w e e n g r o u n d p l a n e s a s shown i n F i g . 5 . 0 5 - l ( c ) a r e g i v e n a p p r o x i m a t e l y by
ohms
where
T h e s e s h o u l d g i v e good r e s u l t s , a t l e a s t when d / b < 0 . 5 5 and s/b > 2 d/b.
300
-Z,
2-=
CHARACTERISTIC IMPEDANCE OF ONE STRIP TO GROUND WITH EQUAL CURRENTS I N SAME DIRECTION.
Zoo=
CHARACTERISTIC IMPEDANCE OF ONE STRIP TO GROUND w m EQUAL CURRENTS I N OPPOSITE DIRECTION.
--
200
.
CHARACTERISTIC IMPEDANCE ff O M STRIP TO GROUND WITH EWAL CURRElS IN SAMI DIRECTION.
180 160-140 I20
'JTz o o
---
--so
100
80-TO --
I 60
--
50
--
40
--
30
--
as
--
A - 3 5 2 7 - 159
SOURCE:
Fiml Report, Contract DA 36-039 SC-63232, SRI; reprinted in IRE Trana., PCMTT ( m e Ref. 4, by S. B. Cohn).
FIG. 5.05-%a)
NOMOGRAM GIVING d b AS A FUNCTION OF Zoe AND Zoo IN COUPLED STRIP LlNE
SOURCE:
Final Report, Contract DA 36-039 SC-63232, SRI; reprinted in IRE Trans., PGMTT (ass Ref. 4 , by S. B. Cohn).
FIG. 5.05-3(b) NOMOGRAM GIVING w/b AS A FUNCTION OF Z,, AND Zoo IN COUPLED STRIP LINE
T h i n L i n e s V e r t i c a l t o t h e Ground P l a n e s - T h e
e v e n - a n d odd-mode
c h a r a c t e r i s t i c impedances of t h e t h i n c o u p l e d l i n e s s h o w n i n F i g . 5.05-1(6)
The c a p a c i t a n c e C;,
a r e g i v e n a p p r o x i m a t e l y by t h e f o r m u l a s 6
a t each edge o f each s t r i p t o t h e p a r a l l e l p l a t e c a p a c i t a n c e , s o t h a t the t o t a l c a p a c i t a n c e t o g r o u n d f o r t h e e v e n mode w i l l b e c o r r e c t . C i 0 i s
i s t h e c a p a c i t a n c e p e r u n i t l e n g t h t h a t m u s t be added
t h e c o r r e s p o n d i n g q u a n t i t y f o r t h e odd mode and E , , i s t h e r e l a t i v e d i e l e c t r i c c o n s t a n t . The e v e n - and odd-mode f r i n g i n g c a p a c i t a n c e s a r e plotted i n Fig. 5.05-4.
I n t h e s e f o r m u l a s k'
is a parameter equal t o
p l e t e e l l i p t i c i n t e g r a l of the f i r s t kind.
m,a n d K
i s t h e com-
The r a t i o w/b i s g i v e n by
The i n v e r s e c o s i n e a n d t a n g e n t f u n c t i o n s a r e e v a l u a t e d i n r a d i a n s b e t w e e n 0 a n d n/2.
Z
To f i n d t h e d i m e n s i o n s o f t h e l i n e s f o r p a r t i c u l a r v a l u e s o f and Z o o , o n e f i r s t d e t e r m i n e s t h e v a l u e o f t h e k from E q . ( 5 . 0 5 - 1 3 )
k i n Ref. 5 . Then b/s i s d e t e r m i n e d and f i n a l l y w/b i s d e t e r m i n e d f r o m E q . ( 5 . 0 5 - 1 5 ) .
and t h e t a b l e s of K ( k ) / K ( k l ) u s . f r o m Eq.
(5.05-14)
Equations (5.05-13) through (5.05-15) and s / b ,
a r e a c c u r a t e f o r a l l v a l u e s o f w/b
a s l o n g a s w/s i s g r e a t e r t h a n a b o u t 1 . 0 .
Thin L i n e s Superimposed-The
f o r m u l a s f o r t h e even- and odd-mode
,
c h a r a c t e r i s t i c i m p e d a n c e s o f t h e c o u p l e d l i n e s shown i n F i g . r e d u c e t o f a i r l y s i m p l e e x p r e s s i o n s when ( w / b ) / ( l found t h a t
-
s/b)
5.05-l(e)
0.35.6
It i s SOURCE: F i n a l Report, C o n t r a c t DA-36-039SC-74862, SRI; reprinted in IRE T r a n s . , PGMTT ( s e e Ref. 5, by S. B. Cohn).
FIG. 5.05-4
EVEN- AND ODD-MODE FRINGING CAPACITANCES FOR BROADSIDE-COUPLEC VERY T H I N STRIPS P A R A L L E L TO THE GROUND PLANES
The e v e n - a n d odd-mode c h a r a c t e r i s t i c i m p e d a n c e s o f t h e c o u p l e d l i n e s shown i n F i g s . 5 . 0 5 - l ( a ) , ( d ) , and ( e ) a r e m o d i f i e d s l i g h t l y when t h e s t r i p s have a f i n i t e t h i c k n e s s . Correction terms t h a t account f o r the e f f e c t s o f f i n i t e t h i c k n e s s h a v e b e e n d e r i v e d by Cohn.'
I n t e r l e a v e d Thin Lines--The
c o n f i g u r a t i o n of coupled s t r i p l i n e s
i l l u s t r a t e d i n F i g . 5 . 0 5 - l ( f ) , i n which t h e two l i n e s o f w i d t h c a r e always o p e r a t e d a t t h e same p o t e n t i a l , i s p a r t i c u l a r l y u s e f u l when i t i s d e s i r e d t o o b t a i n t i g h t c o u p l i n g w i t h t h i n s t r i p s t h a t a r e s u p p o r t e d by a homogeneous d i e l e c t r i c , o f r e l a t i v e d i e l e c t r i c c o n s t a n t e r , t h a t comp l e t e l y f i l l s t h e r e g i o n between s h e g r o u n d p l a n e s . = The d i m e n s i o n s o f t h e s t r i p s f o r p a r t i c u l a r v a l u e s o f Z o , and Z o o c a n be d e t e r m i n e d w i t h t h e a i d of F i g s . 5.05-5 through 5.05-8. definitions that
For t h i s purpose one needs t h e
where C ~ i a n dCoo a r e t h e t o t a l c a p a c i t i e s t o g r o u n d p e r u n i t l e n g t h o f t h e s t r i p s o f w i d t h c o r t h e s t r i p pf w i d t h a , when t h e l i n e s a r e e x c i t e d i n t h e e v e n and odd modes, r e s p e c t i v e l y .
The a b s o l u t e d i e l e c t r i c c o n s t a n t i s e q u a l t o 0..225 e r p f p e r i n c h . U s i n g t h e v a l u e s o f Z o e and Z o o which a r e assumed t o be known, one t h e n computes AC/E from
Values o f b and g a r e t h e n s e l e c t e d and d / g i s d e t e r m i n e d from F i g . 5 . 0 5 - 5 . Next, v a l u e s of CLm/6 and C:./E a r e r e a d from F i g s . 5 . 0 5 - 6 and 5 . 0 5 - 7 . These q u a n t i t i e s . t o g e t h e r w i t h t h e v a l u e o f C0,/6 from E q . a r e t h e n s u b s t i t u t e d i n Eq. ( 5 . 0 5 - 2 1 ) t o g i v e c / b :
(5.05-18).
F i n a l l y , C : e / ~ i s found from F i g . 5 . 0 5 - 8 and s u b s t i t u t e d i n Eq. ( 5 . 0 5 - 2 2 ) t o g i v e a/b: (5.05-22a) b
.
Thus a l l t h e p h y s i c a l d i m e n s i o n s a r e d e t e r m i n e d .
SOURCE: F i n a l Report. Contract DA 3 6 0 3 9 SC-74862, SRI; reprinted in IRE Trans., PGMTT b e e Ref. 32, by W. J. Getainger).
FIG. 5.05-6 FRINGING CAPACITANCE OF OFFSET THIN STRIP IN FIG. 5.05-l(f)
T h e s e f o r m u l a s a r e e x a c t i n t h e l i m i t o f a a n d c >> b s o t h a t f r i n g i n g f i e l d s a t o p p o s i t e edges o f t h e s t r i p s d o n o t i n t e r a c t . w i t h i n 1 . 2 4 p e r c e n t when a / b
2
0 . 3 5 and [ ( c / b ) / ( l
-
g/b)l
They a r e a c c u r a t e t o 2 0.35. I f these
c o n d i t i o n s a r e n o t s a t i s f i e d , i t i s p o s s i b l e tomake approximate c o r r e c t i o n s b a s e d o n i n c r e a s i n g t h e p a r a l l e l p l a t e c a p a c i t a n c e t o compensate f o r t h e l o s s of f r i n g i n g c a p a c i t a n c e due t o i n t e r a c t i o n of t h e f r i n g i n g f i e l d s .
If a n
i n i t i a l v a l u e a l / b i s f o u n d t o b e l e s s t h a n 0 . 3 5 , a new v a l u e , a 2 / b , c a n b e u s e d where
provided 0 . 1 < a2/b < 0.35.
A s i m i l a r formula f o r c o r r e c t i n g an i n i t i a l
v a l u e c l / b g i v e s a new v a l u e c 2 / b , a s
provided g/b i s f a i r l y s m a l l and 0 . 1 < ( c , / b ) / ( l
-
g/b).
When t h e s t r i p o f w i d t h a i s i n s e r t e d s o f a r b e t w e e n t h e s t r i p s o f w i d t h c t h a t d l g > 1 . 0 t h e e v e n - m o d e v a l u e s C : , / ~ a n d c L e / e , d o n o t change from t h e i r v a l u e s a t d / g ~ 1 . 0 . H o w e v e r t h e v a l u e o f AC/E d o e s c h a n g e and i t c a n b e found s i m p l y by a d d i n g 4 ( d / g
- 1) to the
v a l u e o f AC/E a t d / g = 1 . 0 .
For s p a c i n g be-
tween t h e s t r i p s o f w i d t h c g r e a t e r t h a n g l b = 0 . 5 , o r f o r a s e p a r a t i o n d / g < -2.0,
some o f t h e c o n f i g u r a t i o n s shown i n F i g . 5 . 0 5 - l ( a ) ,
(b), o r (c) are
p r o b a b l y more s u i t a b l e . Thick R e c t a n g u l a r Bars-The
t h i c k r e c t a n g u l a r b a r c o n f i g u r a t i o n of
coupled t r a n s m i s s i o n l i n e s , i l l u s t r a t e d i n F i g . 5 . 0 5 - l ( g ) can a l s o be c o n v e n i e n t l y u s e d w h e r e t i g h t c o u p l i n g b e t w e e n l i n e s i s d e s i r e d . 3 3 The d i m e n s i o n s o f t h e s t r i p s f o r p a r t i c u l a r v a l u e s o f Z o e and Z o o c a n b e d e t e r m i n e d w i t h t h e a i d o f F i g s . 5 . 0 5 - 9 and 5 . 0 5 - l O ( a ) , ( b ) . procedure f o r using t h e curves is a s follows.
A convenient
F i r s t o n e d e t e r m i n e s AC/E
from E q .
( 5 . 0 5 - 2 0 ) , u s i n g t h e s p e c i f i e d v a l u e s o f Z o e and Z o o . N e x t a c o n v e n i e n t v a l u e o f t / b i s s e l e c t e d and t h e v a l u e o f s / b i s d e t e r m i n e d from F i g . 5 . 0 5 - 9 .
The v a l u e o f w/b i s t h e n d e t e r m i n e d from t h e e q u a t i o n
and t h e t o t a l e v e n mode c a p a c i t a n c e o f a b a r i s
p a r a l l e l pl a t e capacitance = 0.2256, pf p e r i n c h .
The n o r m a l i z e d p e r - u n i t - l e n g t h Cp/e = 2w/(b
-
t ) , and
SOURCE:
E
Quarterly Report 2, Contract DA 36-039 SC-87398, SRI; reprinted in IRE Trans., PGMTT (see Ref. 33, by W. J. Getsinger).
FIG. 5.05-11 COUPLED RECTANGULAR BARS CENTERED BETWEEN PARALLEL PLATES ILLUSTRATING THE VARIOUS FRINGING AND PARALLEL PLATE CAPACITIES SOURCE:
Quarterly Report 2, Contract DA 36-039 SC-87398, SRI; reprinted in IRE Tmw.. PCMTT ( s e e Ref. 33, by W. J. Getsingsr).
The e v e n - and odd-mode f r i n g i n g c a p a c i t a n c e s C i e / e and C;,/E
FIG. 5.05-10(b) NORMALIZED FRINGING CAPACITANCE FOR AN ISOLATED RECTANGULAR BAR The v a l u e o f C o e t o u s e i s d e t e r m i n e d from t h e s p e c i f i e d v a l u e o f Z O e The f r i n g i n g c a p a c i t a n c e C i e f o r t h e e v e n mode c a n u s i n g Eq. (5.05-18). h e r e a d f r o m F i g . 5 . 0 5 - 9 , a n d C; c a n b e d e t e r m i n e d from F i g . 5 . 0 5 - 1 0 ( b ) . The c u r v e s i n F i g . 5 . 0 5 - 1 0 ( a ) a l l o w one t o d e t e r m i n e C;o d i r e c t l y . The v a r i o u s f r i n g i n g a n d p a r a l l e l - p l a t e c a p a c i t a n c e s u s e d i n t h e above d i s c u s s i o n a r e i l l u s t r a t e d i n F i g . 5.05-11. fringing capacitances
Cia
N o t e t h a t t h e odd-mode
c o r r e s p o n d t o t h e f r i n g i n g c a p a c i t a n c e s between
t h e i n n e r e d g e s o f t h e b a r s and a m e t a l l i c w a l l h a l f w a y b e t w e e n t h e b a r s . I t i s s e e n t h a t t h e t o t a l o d d mode c a p a c i t a n c e o f a b a r i s
were
d e r i v e d by c o n f o r m a l m a p p i n g t e c h n i q u e s and a r e e x a c t i n l i m i t s o f [w/b/(l
-
t/b)]
+
a.
I t i s b e l i e v e d t h a t when [ w / b / ( l
-
t/b)I > 0.35
t h e i n t e r a c t i o n h e t w e e n t h e f r i n g i n g f i e l d s i s s m a l l enough s o t h a t t h e v a l u e s o f C o o / € a n d G o , / € d e t e r m i n e d from E q s . ( 5 . 0 5 - 2 4 ) and ( 5 . 0 5 - 2 5 ) a r e r e d u c e d by a maximum o f 1 . 2 4 p e r c e n t o f t h e i r t r u e v a l u e s . I n s i t u a t i o n s w h e r e a n i n i t i a l v a l u e , w/b i s f o u n d from Eq.
(5.05-23)
t o be l e s s t h a n 0 . 3 5 [I - ( t l b ) ] s o t h a t t h e f r i n g i n g f i e l d s i n t e r a c t , a new v a l u e o f w l / b c a n be u s e d w h e r e
Table 5.05-1
RELATIONS BETWEEN LINE ADMITTANCES, IMPEDANCES, AND CAPACITANCES PER UNIT LENGTH OF UNSYMMETRICAL PARALLEL-COUPLED LINES
provided 0 . 1 < (w1/b)/[l
-
(t/b)l
< 0.35.
v
Unsymmetrical P a r a l l e l - C o u p l e d L i n e s - F i g u r e
=
v e l o c i t y o f l i g h t i n media of p r o p a g a t i o n
5 . 0 5 - 1 2 shows a n un=
symmetrical p a i r o f p a r a l l e l - c o u p l e d l i n e s and v a r i o u s l i n e c a p a c i t a n c e s .
To =
i n t r i n s i c impedance of f r e e s p a c e
=
376.7 ohms
N o t e t h a t Ca i s t h e c a p a c i t a n c e p e r u n i t l e n g t h b e t w e e n L i n e a a n d g r o u n d , C o b i s t h e c a p a c i t a n c e p e r u n i t l e n g t h between L i n e a and L i n e b , w h i l e C b
i s t h e c a p a c i t a n c e p e r u n i t l e n g t h b e t w e e n L i n e b and g r o u n d .
When Ca i s
n o t e q u a l t o C b , t h e two l i n e s w i l l h a v e d i f f e r e n t odd- and even-mode adm i t t a n c e s a s i s i n d i c a t e d by Eqs. even-mode c a p a c i t a n c e s , C:o
=
( 1 ) i n T a b l e 5.05-1.
I n t e r m s o f odd- and
f o r Line a Ca+2Cab,
Cze
=
Ca
(5.05-27)
while f o r Line b
LINE a
LINE b
FIG. 5.05-12 AN UNSYMMETRICAL PAIR O F PARALLEL-COUPLED LINES C,, Cab, and C b are line capacitances per unit length.
F o r s y m m e t r i c a l p a r a l l e l - c o u p l e d l i n e s t h e odd-mode i m p e d a n c e s a r e s i m p l y t h e r e c i p r o c a l s o f t h e odd-mode a d m i t t a n c e s , and a n a l o g o u s l y f o r t h e even-mode impedances and a d m i t t a n c e s . However, a s can be d e m o n s t r a t e d from Eqs. ( 2 ) i n T a b l e 5 . 0 5 - 1 , parallel-coupled lines.
t h i s i s not t h e case f o r unsymmetrical F o r u n s y m m e t r i c a l l i n e s , t h e o d d - a n d even-mode
i m p e d a n c e s a r e n o t s i m p l y t h e r e c i p r o c a l s o f t h e o d d - and even-mode
E
= dielectric constant
=
0.225
Er
@/inch
admittances.
The r e a s o n f o r t h i s l i e s i n t h e f a c t t h a t when t h e o d d - and
even-mode a d m i t t a n c e s a r e c o m p u t e d t h e b a s i c d e f i n i t i o n o f t h e s e a d m i t t a n c e s assumes t h a t t h e l i n e s a r e b e i n g d r i v e n w i t h v o l t a g e s of i d e n t i c a l magnitude w i t h e q u a l o r o p p o s i t e p h a s e , w h i l e t h e c u r r e n t s i n t h e l i n e s may b e o f d i f f e r e n t m a g n i t u d e s .
To d e s i g n a p a i r of l i n e s s u c h a s t h o s e i n F i g . 5 . 0 5 - 1 3 s o a s t o h a v e s p e c i f i e d odd- a n d even-mode a d m i t t a n c e s o r i m p e d a n c e s , f i r s t u s e E q s . ( 3 ) o r ( 4 ) i n T a b l e 5 . 0 5 - 1 t o compute C,/E, a convenient v a l u e f o r t / b , and n o t i n g t h a t
C a b / € , and C b / &
Select
When t h e o d d - a n d even-mode
i m p e d a n c e s a r e c o m p u t e d , t h e b a s i c d e f i n i t i o n o f t h e s e i m p e d a n c e s assumes t h a t t h e l i n e s a r e b e i n g d r i v e n by c u r r e n t s of i d e n t i c a l m a g n i t u d e w i t h e q u a l o r o p p o s i t e p h a s e s , w h i l e m a g n i t u d e s o f t h e v o l t a g e s o n t h e two l i n e s may b e d i f f e r e n t .
T h e s e two d i f f e r e n t s e t s o f b o u n d a r y c o n d i t i o n s
c a n be s e e n t o l e a d t o d i f f e r e n t v o l t a g e - c u r r e n t r a t i o s i f t h e l i n e s a r e
use Fig. 5.05-9 t o determine s / b , F i g . 5 . 0 5 - 1 0 ( b ) d e t e r m i n e C;/E,
unsymmetrical.
and a l s o C;,/E
Using t / b and
and t h e n compute
Some u n s y m m e t r i c a l p a r a l l e l - c o u p l e d l i n e s w h i c h a r e q u i t e e a s y t o d e s i g n a r e shown i n F i g . 5 . 0 5 - 1 3 .
B o t h b a r s h a v e t h e same h e i g h t , a n d
b o t h a r e assumed t o be wide enough s o t h a t t h e i n t e r a c t i o n s between t h e
ELECTRIC WALL FOR ODD MODE MAGNETIC WALL FOR EVEN MODE
Knowing t h e g r o u n d - p l a n e s p a c i n g b , t h e r e q u i r e d b a r w i d t h s w a a n d w b a r e then determined. where t / b = 0 .
T h i s procedure a l s o works f o r t h e t h i n - s t r i p c a s e
I f e i t h e r wa/b o r wb/b i s l e s s t h a n 0 . 3 5 [ 1 - t / b I ,
E q . ( 5 . 0 5 - 2 6 ) s h o u l d be a p p l i e d t o o h t a i n c o r r e c t e d v a l u e s .
A r r a y s of P a r a 1 l e 1 - C o u p l e d L r n e s - - F i g u r e
5 . 0 5 - 1 4 shows a n a r r a y o f
p a r a l l e l - c o u p l e d l i n e s such a s i s used i n t h e i n t e r d i g i t a l - l i n e f i l t e r s
FIG. 5.05-13 CROSS SECTION OF UNSYMMETRICAL, RECTANGULAR-BAR PARALLELCOUPLED LINES
discussed i n Chapt. 10.
I n t h e s t r u c t u r e shown, a l l of t h e b a r s have t h e
same t / b r a t i o a n d t h e o t h e r d i m e n s i o n s o f t h e b a r s a r e e a s i l y o b t a i n e d
f r i n g i n g f i e l d s a t t h e r i g h t and l e f t s i d e s o f each b a r a r e n e g l i g i b l e , o r a t l e a s t s m a l l e n o u g h t o b e c o r r e c t e d f o r by u s e o f Eq. ( 5 . 0 5 - 2 6 ) . On t h i s b a s i s t h e f r i n g i n g f i e l d s a r e t h e same f o r b o t h b a r s , a n d t h e i r d i f f e r e n t c a p a c i t a n c e s Co and C b t o ground a r e due e n t i r e l y t o d i f f e r e n t p a r a l l e l - p l a t e c a p a c i t a n c e s C i a n d c:.
F o r t h e s t r u c t u r e shown SOURCE:
Quarterly Progress Report 4, Contract DA 36-039 SC-87398, SRI; reprinted in the IRE Trans. PGMTT (see Ref. 3 of Chapter 10, by G . L. Matthaei)
FIG. 5.05-14 CROSS SECTION OF AN ARRAY OF PARALLEL-COUPLED LINES BETWEEN GROUND PLANES
-
by g e n e r a l i z i n g t h e p r o c e d u r e d e s c r i b e d f o r d e s i g n i n g t h e u n s y m m e t r i c a l
I f wk/b < 0 . 3 5 [ 1
p a r a l l e l - c o u p l e d l i n e s i n F i g . 5.05-13.
i n Eq. ( 5 . 0 5 - 2 6 ) s h o u l d b e a p p l i e d t o t h o s e b a r s w h e r e t h i s c o n d i t i o n
I n t h e s t r u c t u r e i n F i g . 5.05-14
t h e e l e c t r i c a l p r o p e r t i e s o f t h e s t r u c t u r e a r e characterized i n terms of
t / b ] f o r any of t h e b a r s , t h e w i d t h c o r r e c t i o n given
exists.
t h e s e l f - c a p a c i t a n c e s Ck p e r u n i t l e n g t h of each bar w i t h r e s p e c t t o ground, and t h e mutual c a p a c i t a n c e s C k n k t l p e r u n i t l e n g t h between adj a c e n t b a r s k and k + 1. T h i s r e p r e s e n t a t i o n i s n o t n e c e s s a r i l y a l w a y s h i g h l y a c c u r a t e b e c a u s e t h e r e c a n c o n c e i v a b l y b e a s i g n i f i c a n t amount o f f r i n g i n g c a p a c i t a n c e i n some c a s e s b e t w e e n a g i v e n l i n e e l e m e n t , a n d , f o r example, t h e l i n e e l e m e n t beyond t h e n e a r e s t n e i g h b o r .
However, a t l e a s t
f o r g e o m e t r i e s s u c h a s t h a t s h o w n , e x p e r i e n c e h a s shown t h i s r e p r e s e n t a t i o n t o have s a t i s f a c t o r y a c c u r a c y f o r a p p l i c a t i o n s s u c h a s i n t e r d i g i t a l
S E C . 5 . 0 6 , S P E C I A L PROPERTIES OF \~AVECUILlES A waveguide c o n s i s t i n g of a s i n g l e hollow conductor t h a t can propa-
g a t e e l e c t r o m a g n e t i c energy above a c e r t a i n c u t o f f frequency,
fc,
is also
a v e r y u s e f u l e l e m e n t i n m i c r o w a v e f i l t e r s . A w a v e g u i d e c a n p r o p a g a t e an i n f i n i t e number o f m o d e s , w h i c h c a n b e c h a r a c t e r i z e d a s b e i n g e i t h e r TE ( t r a n s v e r s e e l e c t r i c ) o r TM ( t r a n s v e r s e m a g n e t i c ) .
The TE modes h a v e a
magnetic f i e l d b u t no e l e c t r i c f i e l d i n t h e d i r e c t i o n of propagation,
f i l t e r design. For d e s i g n o f t h e p a r a l l e l - c o u p l e d a r r a y s t r u c t u r e s d i s c u s s e d i n
w h i l e TM modes h a v e a n e l e c t r i c f i e l d b u t no m a g n e t i c f i e l d i n t h e d i r e c t i o n o f p r o p a g a t i o n . U s u a l l y a waveguide i s o p e r a t e d s o t h a t i t propa-
t h i s book, e q u a t i o n s w i l l be g i v e n f o r t h e normalized s e l f and mutual
g a t e s e n e r g y i n a s i n g l e mode, and u n d e r t h i s c o n d i t i o n it c a n b e d e s c r i b e d
c a p a c i t a n c e s C k / e a n d C k ,k + l
a s a t r a n s m i s s i o n l i n e w i t h a p r o p a g a t i o n c o n s t a n t y , and a c h a r a c t e r i s t i c
structure.
/E
per u n i t length for a l l the l i n e s i n the
Then t h e c r o s s - s e c t i o n a l d i m e n s i o n s o f t h e b a r s a n d s p a c i n g s
between them a r e d e t e r m i n e d a s follows. b.
F i r s t , choose v a l u e s f o r
t
and
Then, s i n c e
i m p e d a n c e Z,,.
The p r o p a g a t i o n c o n s t a n t f o r a w a v e g u i d e i s u n i q u e l y d e -
fined.
The c h a r a c t e r i s t i c impedance of a waveguide can be c o n s i d e r e d t o be t h e wave i m p e d a n c e o f t h e g u i d e , Z w ( ~ . e . , t h e r a t i o o f t h e t r a n s v e r s e e l e c t r i c t o t h e t r a n s v e r s e m a g n e t i c f i e l d i n t h e g u i d e ) , m u l t i p l i e d by a constant.
F i g . 5.05-9 can be used t o d e t e r m i n e s k ,k + l / b . spacings s
~
I n t h i s manner, t h e , b~e t w+e e ~ n a l l t h e b a r s a r e obtained. Also, u s i n g t h e n o r m a l i z e d f r i n g i n g c a p a c i t a n c e s (c;.) k , k + l / ~ a s s o c i a t e d
Fig. 5.05-9, with t h e gaps s
~
, b~e t w+e e ~n b a r s a r e o b t a i n e d .
Then t h e n o r m a l i z e d
T h e v a l u e o f t h e c o n s t a n t d e p e n d s o n what d e f i n i t i o n o f c h a r a c -
t e r i s t i c impedance i s employed ( i . e . , v o l t a g e - c u r r e n t , v o l t a g e - p o w e r , o r current-power). Thus i t i s s e e n t h a t t h e c h a r a c t e r i s t i c impedance o f a w a v e g u i d e i s n o t a u n i q u e q u a n t i t y , a s i t i s i n t h e c a s e o f a TEM t r a n s mission l i n e .
However, t h i s l a c k o f u n i q u e n e s s t u r n s o u t t o b e u n i m p o r t a n t
i n waveguide f i l t e r c a l c u l a t i o n s because one can always n o r m a l i z e a l l waveguide e q u i v a l e n t c i r c u i t e l e m e n t s t o t h e c h a r a c t e r i s t i c impedance of
width of t h e k t h bar i s
the guide. I n a l o s s l e s s waveguide f i l l e d w i t h d i e l e c t r i c of r e l a t i v e d i e l e c t r i c constant
E,.,
t h e g u i d e wavelength As, f r e e - s p a c e wavelength A , wavelength
i n t h e d i e l e c t r i c A l , and c u t o f f wavelength h e , a r e r e l a t e d a s I n t h e c a s e of t h e b a r a t t h e end of t h e a r r a y ( t h e b a r a t t h e f a r l e f t i n Fig. 5.05-14),
C;,/E
f o r t h e e d g e o f t h e b a r which h a s no n e i g h b o r
m u s t b e r e p l a c e d by C ; / E w h i c h i s d e t e r m i n e d f r o m F i g . 5 . 0 5 - 1 0 ( b ) .
Thus,
f o r example, f o r Bar 0 i n F i g . 5.05-14, The c h a r a c t e r i s t i c i m p e d a n c e t h a t we s h a l l a s s u m e f o r c o n v e n i e n c e t o e q u a l t h e wave i m p e d a n c e i s
and f o r t h e T E i l mode i n c i r c u l a r g u i d e
The p r o p a g a t i o n p h a s e c o n s t a n t
P,
=
27T -
P,
is
radians/unit length
.
where f i s measured i n g i g a c y c l e s . (5.06-3)
These values of a t t e n u a t i o n a r e
plotted i n Fig. 5.06-1. The a t t e n u a t i o n c a u s e d by l o s s e s i n t h e d i e l e c t r i c i n a n y waveguid
The m o s t common f o r m o f w a v e g u i d e f o r u s e i n m i c r o w a v e f i l t e r s i s a
mode i s
r e c t a n g u l a r w a v e g u i d e o f w i d t h a a n d h e i g h t b o p e r a t i n g i n a TElo mode. TE,,
modes h a v e c u t o f f w a v e l e n g t h s
where t a n 6 i s t h e l o s s t a n g e n t o f t h e d i e l e c t r i c .
The u n l o a d e d
0, o f
a waveguide* i s
The i n d e x a e q u a l s t h e n u m b e r o f h a l f - w a v e s o f v a r i a t i o n o f t h e e l e c t r i c f i e l d a c r o s s t h e w i d t h , a , o f t h e g u i d e . The c u t o f f f r e q u e n c y f c (measured i n g i g a c y c l e s ) i s r e l a t e d t o t h e c u t o f f wavelength i n i n c h e s a s
where
Qd
d e p e n d s o n l y on l o s s e s i n t h e d i e l e c t r i c a n d i s g i v e n by
1
The d o m i n a n t mode, t h a t i s , t h e o n e w i t h t h e l o w e s t c u t o f f f r e q u e n c y , i s Qd
t h e TEl
mode.
The d o m i n a n t mode i n c i r c u l a r w a v e g u i d e o f d i a m e t e r D i s t h e mode.
TEll
T h e c u t o f f w a v e l e n g t h o f t h e T E l l mode i s 1 . 7 0 6 0 .
1
i
=
(5.06-10)
tan 6
and Q c i s a f u n c t i o n o f t h e ohmic l o s s e s i n t h e w a v e g u i d e w a l l s and t h e r a t i o f/f,, a n d i s g i v e n by
The a t t e n u a t i o n o f t h e s e modes d u e t o l o s s e s i n t h e c o p p e r c o n d u c t o r s a r e f o r TEmo modes i n r e c t a n g u l a r g u i d e
A d d i t i o n a l d i s c u s s i o n r e l e v a n t t o the use of waveguides Sec. 5.08.
199
as
r e s o n a t o r s w i l l b e found i n
F o r r e c t a n g u l a r c o p p e r w a v e g u i d e s o p e r a t i n g i n t h e T E m o mode, we h a v e
where u and b a r e measured in i n c h e s , and f i n g i g a c y c l e s . c u l a r waveguide o p e r a t i n g i n t h e
TEll
where D i s m e a s u r e d i n i n c h e s a n d f i n g i g a c y c l e s . for
Qc
For a c i r -
mode, we h a v e
These e x p r e s s i o n s
a r e p l o t t e d i n Fig. 5.06-2.
The p u l s e - p o w e r c a p a c i t y P m a xo f a i r - f i l l e d g u i d e s , * a t a t m o s p h e r i c p r e s s u r e , a s s u m i n g a b r e a k d o w n s t r e n g t h o f 2 9 kv/cm,
f o r t h e T E m o mode i n
rectangular guide i s
(TE.~)
'nsx
=
3 . 6 ab
A -
,
megawatts
(5.06-14)
AB
and f o r t h e
TEll
mode i n c i r c u l a r g u i d e
where t h e d i m e n s i o n s a r e i n i n c h e s . I n a r e c t a n g u l a r w a v e g u i d e operating i n t h e
l'bl0
mode, w i t h an a s p e c t
r a t i o b l a o f 0 . 5 o r 0 . 4 5 , t h e n e x t ! l i p h e r - o r d e r mode i s t h e
o f f w a v e l e n g t h Ac
=
a.
Y e x t come t h e
TEll
t h e same c u t o f f w a v e l e n g t h , A c = 2 a b / d a 2 t h e n e x t h i g h e r - o r d e r mode i s t h e
*
TMol
or
+ bZ.
TMll
TEZ0
with c u t -
modes e a c h o f which h a s
I n t h e c i r c u l a r waveguide,
n o d e , w h i c h h a s Ac = 1 . 3 0 5 3 .
See S e c . 1 5 02 f a r f u r t h e r d i s c u s s i o n o f t h i s s u b j e c t
This section presents formulas and curves for some of the common discontinuities in transmission lines. to be found in the literature.
Other more complete results are
v10-u,121213
Changes in Diameter of Coazial Lines-When
a change is made in the
diameter of either the inner or outer conductor of a coaxial line, or in both conductors simultaneously, the equivalent circuits can be represented as shown in Fig. 5.07-l.lO." The equivalent shunt capacity, C d , for each of these cases is given in Fig. 5.07-2. These equivalent circuits apply when the operating frequency is appreciably below the cutoff frequency of the next higher-order propagating mode. Changes in Width of Center Conductor of a Strip Line-The
change in
width of the center conductor of a strip line introduces an inductive reactance in series with the line.' is small and can be neglected.
In most situations this reactance
The approximate equivalent circuit for
this situation is shown in Fig. 5.07-3. Compensated Hight-Angle Corner in Strip Line-A
low-VSWR right-angle
corner can be made in strip line if the outside edge of the strip is beveled.
Figure 5.07-4 shows the dimensions of some matched right-angle
corners for a plate-spacing-to-wavelength ratio, b / h , of 0.0845.
These
data were obtained for a center strip conductor having negligible thickness; however, the data should apply with acceptable accuracy for strips of moderate thickness. Fringing Capacitance for Semi-Infinite Plate Centered Between Para1 le 1 Ground Planes-The
exact fringing capacitance,
C:, from one
corner of a semi-infinite plate centered between parallel ground planes
where
E
=
0 . 2 2 5 e r micromicrofarads per inch and
E?
is the relative
dielectric constant of the material between the semi-infinite plate and the ground planes. Dats for
and 9.05.
Fringing capacitance, C;,
is plotted in Fig. 5.07-5
discontinuity configurations rill b e found in Sees. 8.05, 8.06,
8.08,
8.12.
i ' "
T LONGITUDINAL SECTION
T
EQUIVALENT CIRCUIT
( a ) STEP I N INNER CONDUCTOR
1 T
T LONGITUDINAL SECTION
EQUIVALENT CIRCUIT
( b ) STEP I N OUTER CONDUCTOR
SOURCE:
T LONGITUDINAL SECTION
s
In
b a
In
d
C
EQUIVALENT CIRCUIT
FIG. 5.07-2 COAXIAL-LINE-STEP FRINGING CAPACITIES Cd
=
21rcC;~
+
2 ~ a C & ~
( c ) STEP I N INNER AND OUTER CONDUCTORS A-3627-139
FIG. 5.07-1
Proc. IRE (see Ref. 10 and 11 by J. R. Whinnery and H. W. Jamisson).
COAXIAL-LINE DISCONTINUITIES
TOP VIEW
EQUIVALENT CIRCUIT
X
=
-
"
I n csc
I:;
--
SOURCE: IRE Trans., PCYTT G e e Ref. 12. by A. A. Oliner).
FIG. 5.07-3 STRIP-LINE STEP EQUIVALENT CIRCUIT
SOURCE: Final Report Contract DA 36-039 SC-63232, SRI; reprinted in IRE Trams., PGMTT (see Ref. 2, by S. B. Cohn).
FIG. 5.07-4
MATCHED STRIP-LINE CORNER The parameter 8 i s the effective length around the corner.
FIG. 5.07-5
EXACT FRINGING CAPACITANCE FOR A SEMI-INFINITE PLATE CENTERED BETWEEN PARALLELGROUNDPLANES
Strip-Line T-Junctions-A
symmetrical strip-line T-junction of the
type illustrated in Fig. 5.07-6(a) can be represented by the equivalent circuit shown in Fig. 5.07-6(b). A short-circuit placed in turn in each of the three arms, at distances equal to multiples of one-half wavelength from the corresponding reference planes labeled P I and P P , will block transmission between the other two arms of the junction. Measured values obtained for the equivalent circuit parameters of sixteen different strip-line T-junctions are shown in Figs. 5.07-7, 5.07-8, and 5.07-9. The thickness, t , of the strips used in these measurements was 0.020 inch, while the ground-plane spacing was 0.500 inch. The widths of the strips having 35, 50, 75, and 100 ohms characteristic impedance were 1.050, 0.663, 0.405, and 0.210 inches, respectively. Measurements carried out in the frequency band extending from 2 to 5 Gc, corresponding to values of b/A varying from 0.085 to 0.212. It was found
(b'
FIG. 5.07-6 EQUIVALENT CIRCUIT
A STRIP-LINE T-JUNCTION
that the reference plane pasitions were almost independent of frequency for all sixteen 7-junctions, and therefore only the values corresponding to b/A of 0.127 are shown in Fig. 5.07-7. It is seen from an inspection of Fig. 5.07-8 that A , the equivalent transformer turns ratio squared, is sensitive to frequency and has a value approximately equal to unity for b/A very small, and decreases considerably for larger values of b/X. The values of the discontinuity susceptance, B d , vary considerably from one junction to another, and in some instances are quite frequency-sensitive. It is believed that Ed is essentially capacitive in nature. Thus positive values of B d correspond to an excess of capacitance at the junction, while negative values correspond to a deficiency. Although the data presented in Figs. 5.07-7, 5.07-8, and 5.07-9 are for T-junctions with air-filled cross section and with the ratio t/b = 0.040, these data may be applied to other cross sections. For instance, it is expected that these data should hold for any strip-thickness ratio, t/b, up to at least 0.125 if the same characteristic impedances are maintained. In the case of a dielectric-filled section, E~ > 1, the data are expected to apply with good accuracy if one divides the characteristic impedances Zoland Z,,
by
<
and multiples b/A and B d / Y o by
<.
Change in Height of a Rectangular Waveguides-The equivalent circuit of the junction of two waveguides of different height but the same width, which are both operating in the TElo mode can be represented as shown in
FIG. 5.07-7 REFERENCE-PLANE LOCATIONS vs. ZO2
.-*Iz.-.?"-
,...
FIG. 5.07-8 MEASURED TRANSFORMER-TURNS RATIO SQUARED FOR SIXTEEN STRIP-LINE T-JUNCTIONS
+
I
0 N
0 O
N
t---a---i CROSS SECTIONAL VlEW
SIDE VIEW
EQUIVALENT CIRCUIT
( a ) SYMMETRIC JUNCTION
b' b
r a l CROSS SECTIONAL VlEW
SIDE VlEW
EQUIVALENT CIRCUIT
SOURCE:
A-3511-149
Wavesuide Handbook ( s e e Ref. 8 , edited by N. ~ a r c u v i t z ) .
FIG. 5.07-11 SHUNT SUSCEPTANCE FOR CHANGE IN HEIGHT OF RECTANGULAR GUIDE
(b) ASYMMETRIC JUNCTION SOURCE:
Waveguide Handbook ( s e e Ref. 8 , edited by N. Marcuvitz).
FIG. 5.07-10
EQUIVALENT CIRCUIT FOR CHANGE IN HEIGHT OF RECTANGULAR WAVEGUIDE
Fig. 5.07-10. The normalized susceptance BAg/Yob is plotted in Fig. 5.07-11 for various values of b/Ag, and is accurate to about 1 percent for b/Ag 5 1. SEC. 5 . 0 8 , T R A N S M I S S I O N L I N E S A S R E S O N A T O R S In many microwave filter designs, a length of transmission line terminated in either an open-circuit or a short-circuit is often used as a resonator. Figure 5.08-1 illustrates four resonators of this type, together with their lumped-constant equivalent circuits. It is to be noted that the resonators in Fig. 5.08-l(a) and 5.08-l(b) each have lengths which are multiples of one-half guide wavelength, and that the lumped-constant equivalent circuit of the transmission line which is
'in
-
1 T
short-circuited at one end is the dual of the equivalent circuit of the transmission line with an open-circuit termination. Similarly, the resonators in Fig. 5.08-l(c) and 5.08-l(d) have lengths which are odd multiples of one-quarter guide wavelength, and their lumped constant equivalent circuits are also duals of one another. The quantities a , , A g o and A. are the attenuation of the transmission line in nepers per unit length, the guide wavelength at the resonant frequency, and the plane-wave wavelength at the resonant frequency, respectively, in the dielectric medium filling the resonator. The equivalence between the lumped constant circuits and the microwave circuits shown was established in the following fashion. The values of the resistance, R, and conductance, G, in the lumped-constant equivalent circuits were determined as the values of these quantities for the various lines at the resonance angular frequency, w o . The reactive elements in the lumped-constant equivalent circuits were determined by equating the slope parameters (defined below) of the lumped-element circuits to those of the transmission-line circuits which exhibited the same type of resonance. The general definition of the reactance slope parameter 6 , which applies to circuits that exhibit a series type of resonance., is
where X is the reactance portion of the input impedance to the circuit. The susceptance slope parameter A , which appl ies to circuits that exhib a parallel type of resonance, is
214
(a)
(b)
FIG. 5.08-1 SOME TRANSMISSION LINE RESONATORS
A-31P7-152
4
=
Wo
dE
mhos Wo
where B is the susceptance component of the input admittance of the circuit. T h e above general definitions for slope parameters provide a convenient means for relating the resonance properties of any circuit to a simple lumped equivalent circuit such a s those in Fig. 5.08-1.
The
reactance slope parameter x given by Eq. (5.08-1) is seen to be equal to 0
L
= l/(woC)
for the equivalent, series, lumped-element circuit, while
the susceptance slope parameter & is equal to
woC = l/(uOL) for the
equivalent, parallel, lumped-element circuit.
Considerable use will be
made of these parameters in later chapters dealing with band-pass and band-stop microwave filters. It should be noted i n Fig. 5 . 0 8 - 1 that the use of reactance o r susceptance slope parameters also leads to convenient expressions for
Q,
and for the input impedance or admittance o f the circuit in the vicinity of resonance.
For narrow-band microwave applications, the approximate
equivalence
is often convenient for use in the expressions for input impedance or admittance. SEC. 5.09, COUPLED-STHIP-TRANSMISSION-LINE
FILTER SECTIONS
T h e natural electromagnetic coupling that exists between parallel transmission lines can be used to advantage in the design of filters and
FIG. 5.08-1
Concluded
directional c o u p l e r s . 1 4 * 1 5 ~ 1 6 ~ 1 7 ~ 1 8In~ 1this 9 ~ m section, formulas are given for filter sections constructed of parallel-coupled lines of the types illustrated in Fig. 5 . 0 5 - 1 . Several cases involving unsymmetrical Parallel-coupled lines a s r n Figs. 5 . 0 5 - 1 2 and 5 . 0 5 - 1 3 are also considered. T h e ten coupling arrangements that can be obtained from a pair o f Symmetrical, coupled transmission lines by placing open- or short-circuits On various terminal pairs, or by connecting ends of the lines together,
are illustrated in Fig. 5.09-1. In this figure, schematic diagrams of single sections of each type are shown, together with their image parameters and either their open-circuit impedances or their short-circuit admittances.
In addition, equivalent open-wire transmission-line
circuits for eight of the coupled transmission line sections are shown SCHEMATIC AND EQUIVALENT CIRCUIT
beneath the corresponding schematic diagram. In the schematic diagrams of the coupled-transmission-line sections
zll
in Fig. 5.09-1, the input and output ports are designated by small open circles.
-
212
=
j
The image impedance seen looking into each of these ports is
also indicated near each port.
Open-circuited ports of the coupled lines
with the standard grounding symbol. circuits shown beside tation is used.
zoo
+
Zoe
' -j
Z12
are shown with no connection, while short-circuited ports are designated
0
2 ~ o e z o o
Zoe
'oe
+
tan
2
zoo csc ze0
B
In the equivalent transmission-line
the schematic diagrams, a two-wire line represenz22
In each case, the characteristic impedance or admittance
-
-
zll
=
(Zoe z o o ) 2 J 2(Zoe + Zoo) tan B
of the lengths of transmission line is shown, together with the electrical TWO-PORT CIRCUIT PARAMETERS
length, 8. The equivalence between the parallel-coupled line sections and the non-parallel-coupled line sections shown is exact. Figure 5.09-2 shows the same parallel-coupled sections as appear in Figs. 5.09-l(b),
z~I
(c), (d), but for cases where the strip transmission
lines have unsymmetrical cross sections.;
The line capacitances G o , C a b ,
and Cb per unit length are as defined in Fig. 5.05-12.
It is interesting
to note that in the case of Fig. 5.09-2(a) the line capacitances per unit length for the left and right shunt stub i n the equivalent open-wire representation are the same as the corresponding capacitances per unit length between Line a and ground, and Line b and ground, respectively. Meanwhile, the capacitance per unit length for the connecting line in the open-wire circuit is the same as the capacitance per unit length b e t w e e n Lines n and b of the parallel-coupled representation.
cosh ( a
In Fig. 5.09-2(b)
+ jp)
the dual situation holds, where L a and L b are the self-inductances per unit length of Lines a and b in the parallel-coupled representation, while
2 ZI,
L a b is the mutual inductance per unit length between the parallel-coupled
= [-(Zoe
lines.
Since the line capacitances are more convenient to deal with, the The
quantity v indicated in Vig. 5.09-2 is the velocity of light in the medium of propagation. The r e s u l t s i n F i g . 5 . 0 9 - 2 and a l s o t h o s e i n F i g s . 5 . 0 9 - 3 and 5 . 0 9 - 4 r e r e o b t a i n e d b y e x t e n s i o n o f t h e r e s u l t s in R e f s . 1 9 and 2 0 .
+
cos 0
(Zo,
+
Zoo)' c o s 2 B]%
IMAGE PARAMETERS
line impedances of the equivalent open-wire circuit are also given in terms o f G o , C a b , and C b , for all three cases in Fig. 5.09-2.
-
zoeLoo
(0) SOURCE:
Low
PASS
A-3527-A162
Adapted from f i g u r e s in F i n a l R e p o r t , Contract DA36-039 SC-64625, SRI; which w e r e reprinted in IRE Trans. PCMTT ( s e e R e f . 19 by E. M. T. J o n e s and J. T. Bolljahn).
FIG. 5.09-1
SOME PARALLEL-COUPLED TRANSMISSION-LINE FILTER SECTIONS (continued
on p .
220)
-6__3
Zr.
I
SCHEMATIC AND EQUIVALENT CIRCUIT SCHEMATIC AND EQUIVALENT CIRCUIT
-
y12
(Yo.
-j
-
"0.)
e
C S ~
TWO-PORT ClRCLJlT PARAMETERS TWO-PORT CIRCUIT PARAMETERS
cosh ( a
+
jp)
=
cosh ( a +
jp)
IMAGE PARAMETERS IMAGE PARAMETERS ( b ) BAND PASS
FIG. 5.09-1 Continued
A - ~ W ~ - B I ~ Z
( C ) BAND PASS
FIG. 5.09-1 Continued
SCHEMATIC AND EQUIVALENT CIRCUIT SCHEMATIC
TWO-PORT CIRCUIT PARAMETERS
IMAGE PARAMETERS TWO-PORT CIRCUIT PARAMETERS
( e ) A L L PASS
SCHEMATIC AND EQUIVALENT CIRCUIT
TWO-PORT CIRCUIT PARAMETERS
cosh ( a
+ jp)
-
.1
1 sin
-
'Zo.
\Ioe
Zoo
)1
- zoo +
% cOsz
81
IMAGE PARAMETERS
IMAGE PARAMETERS
( d 1 BAND PASS
A-3527-0162
FIG. 5.09-1 Continued FIG. 5.09-1 Continued
SCHEMATIC SCHEMATIC AND EOUIVALENT CIRCUIT
TWO-PORT CIRCUIT PARAMETERS
TWO-PORT
cos
p
-
ClRCUlT PARAMETERS IMAGE PARAMETERS
zo tan2 8 zoe Zoe +
,
tanp
2/zoe
=
(Z.,. c o t 8
' tan2 8
Zoo
- Zoo tan
8)
-
z ~ l
( i ) ALL STOP
K x
zoo
IMAGE PARAMETERS
( g A L L PASS
SCHEMATIC AND EOUIVALENT CIRCUIT
SCHEMATIC AND EQUIVALENT CIRCUIT
TWO-PORT CIRCUIT PARAMETERS
TWO-PORT CIRCUIT PARAMETERS
ZI1
=
-
j
m
cot
e
C O S ~
a
Zoe =
Zo*
+
Zoo
- zoo
IMAGE PARAMETERS IMAGE PARAMETERS
( j ) A L L STOP
( h ) A L L STOP
FIG. 5.09-1 Concluded FIG. 5.09-1
Continued
A - ~ s ~ ~ - G I ~ z
I n t h e c a s e s o f t h e c i r c u i t s i n F i g s . 5 . 0 9 - 2 ( a ) , and ( b ) , i f t h e parallel-coupled sections a r e properly terminated, t h e i r equivalent o p e n - w i r e l i n e c i r c u i t s i m p l i f i e s i n a v e r y i n t e r e s t i n g a n d use>-d-way. T h i s i s i l l u s t r a t e d i n F i g . 5 . 0 9 - 3 ( a ) and ( b ) .
N o t e t h a t when t h e i n d i -
c a t e d c o n s t r a i n t s a r e a p p l i e d , t h e e q u i v a l e n t open-wire c i r c u i t reduces t o s i m p l y an i d e a l t r a n s f o r m e r and a s i n g l e s t u b . I n s p i t e of t h e cons t r a i n t e q u a t i o n s which a r e e n f o r c e d i n t h e s e c i r c u i t s , t h e r e a r e s t i l l s u f f i c i e n t d e g r e e s o f f r e e d o m s o t h a t f o r s p e c i f i e d Ys a n d G T o r R T , a w i d e r a n g e o f Y , o r Z A , r e s p e c t i v e l y , c a n h e accommodated.
Z S and
For
t h i s r e a s o n t h e s e two s t r u c t u r e s w i l l p r o v e q u i t e u s e f u l f o r u s e w i t h c e r t a i n t y p e s o f band-pass f i l t e r s f o r t h e purpose of e f f e c t i v e l y r e a l i z i n g a s e r i e s - o r s h u n t - s t u b r e s o n a t o r , a l o n g w i t h o b t a i n i n g an impedance t r a n s f o r m a t i o n w h i c h w i l l accommodate some d e s i r e d t e r m i n a t i n g i m p e d a n c e . I n a somewhat m o r e c o m p l e x way, t h e c i r c u i t i n F i g . 5 . 0 9 - 2 ( c ) w i l l a l s o prove u s e f u l f o r s i m i l a r purposes. F i g u r e 5 . 0 9 - 4 shows t h e p a r a l l e l - c o u p l e d s e c t i o n i n F i g . 5 . 0 9 - l ( i ) g e n e r a l i z e d t o c o v e r t h e c a s e w h e r e t h e two s t r i p l i n e s may b e o f d i f ferent widths.
At ( a ) i s shown t h e s t r u c t u r e u n d e r c o n s i d e r a t i o n , w h i l e
a t ( b ) a n d ( c ) a r e shown two o p e n - w i r e l i n e s t r u c t u r e s w h i c h a r e i d e n t i c a l l y e q u i v a l e n t e l e c t r i c a l l y t o t h e s t r i p - l i n e s t r u c t u r e a t ( a ) . As previously i n d i c a t e d , p a r a l l e l - c o u p l e d s t r u c t u r e s of t h i s s o r t a r e a l l s t o p s t r u c t u r e s a s t h e y s t a n d , b u t when p r o p e r l y u s e d w i t h lumped c a p a c i t a n c e s , t h e y become t h e b a s i s f o r t h e c o m b - l i n e form o f f i l t e r discussed i n Sec. 8.13. SEC. 5 . 1 0 ,
I R I S - C O U P L E D WAVEGUIDE JUNCTIONS
~ e t h e " - *h a~s ~d e~v e~l o~p e d a g e n e r a l p e r t u r b a t i o n t e c h n i q u e f o r c a l c u l a t i n g t h e s c a t t e r i n g o f power by s m a l l i r i s e s c o n n e c t i n g o n e t r a n s mission l i n e with another.
T h e t h e o r y i s a p p l i c a b l e e v e n t h o u g h t h e two
t r a n s m i s s i o n l i n e s have d i f f e r e n t c r o s s s e c t i o n s and o p e r a t e i n d i f f e r e n t modes; h o w e v e r , i t a p p l i e s r i g o r o u s l y o n l y t o i n f i n i t e s i m a l l y t h i n i r i s e s whose d i m e n s i o n s a r e s m a l l i n t e r m s o f t h e o p e r a t i n g w a v e l e n g t h .
These
i r i s e s s h o u l d b e l o c a t e d f a r from any c o r n e r s , i n a t r a n s m i s s i o n - l i n e w a l l whose r a d i u s o f c u r v a t u r e i s l a r g e i n t e r m s o f w a v e l e n g t h . 0-3517-474
FIG. 5.09-4 A PARALLEL-COUPLED SECTION AND TWO OPEN-WIRE-LINE CIRCUITS WHICH ARE EXACTLY EQUIVALENT The Cat Cab, and Cb are as indicated in Fig. 5.05-12.
In
p r a c t i c e i t i s f o u n d t h a t t h e t h e o r y h o l d s r e a s o n a b l y w e l l e v e n when t h e i r i s e s a r e located r e l a t i v e l y close t o sharp corners i n transmission-line w a l l s of f a i r l y small r a d i i of curvature.
For i r i s e s of f i n i t e t h i c k n e s s ,
i t i s found t h a t B e t h e ' s t h e o r y i s s t i l l a p p l i c a b l e e x c e p t t h a t t h e
t r a n s m i s s i o n t h r o u g h t h e i r i s i s r e d u c e d . 2 5 I n many i n s t a n c e s i t i s p o s s i b l e t o u s e C o h n ' s f r e q u e n c y c o r r e c t i ~ nw h~e~r e t h e i r i s d i m e n s i o n s
F i g u r e 5.10-2 i l l u s t r a t e s t h e magnetic f i e l d c o u p l i n g through an Again t h e d i s i r i s c o n n e c t i n g two p a r a l l e l - p l a n e t r a n s m i s s i o n l i n e s . -- ._-_
a r e not n e g l i g i b l y small with r e s p e c t t o a wavelength.
t o r t e d m a g n e t i c f i e l d w i t h i n t h e i r i s can be c o n s i d e r e d t o a r i s e from
S e t h e ' s o r i g i n a l d e r i v a t i o n ~ ' , ~ , ~ a p p e a r ei dn a s e r i e s o f MIT R a d i a t i o n Laboratory R e p o r t s , c o p i e s of which a r e q u i t e d i f f i c u l t t o
two m a g n e t i c d i p o l e moments e a c h o f s t r e n g t h m , i n d u c e d i n t h e i r i s by The m a g n e t i c d i p o l e moment t h e e x c i t i n g t a n g e n t i a l magnetic f i e l d , H o t .
obtain.
i n t h e upper l i n e is d i r e c t e d a n t i - p a r a l l e l t o H o t , while t h a t i n t h e
R e c e n t l y C o l l i n z h a s d e r i v e d some o f D e t h e ' s r e s u l t s u s i n g a d i f f e r e n t approach, and t h e s e r e s u l t s a r e r e a d i l y a v a i l a b l e . ~ a r c u v i t z ~ r e c a s t much o f B e t h e ' s work a n d d e r i v e d many e q u i v a l e n t c i r c u i t s f o r
lower l i n e i s o p p o s i t e l y d i r e c t e d and p a r a l l e l t o H o t .
i r i s - c o u p l e d t r a n s m i s s i o n l i n e s , many o f w h i c h a r e p r e s e n t e d i n t h e Waveguide H a n d b o ~ k . ~A p a p e r by O l i n e r " c o n t a i n s some a d d i t i o n a l c i r c u i t s for iris-coupled lines. B e t h e ' s c a l c u l a t i o n o f t h e s c a t t e r i n g o f power by s m a l l i r i s e s a c t u a l l y c o n s i s t s o f two d i s t i n c t s t e p s . T h e f i r s t s t e p i s t h e comput a t i o n o f t h e e l e c t r i c d i p o l e moment, p , a n d t h e m a g n e t i c d i p o l e moment,
-
i n d u c e d i n t h e i r i s by t h e e x c i t i n g f i e l d s . The n e x t s t e p i s t h e c a l c u l a t i o n o f t h e f i e l d s r a d i a t e d by t h e e l e c t r i c a n d m a g n e t i c d i p o l e
m,
FIG. 5.10-2 MAGNETIC DIPOLE MOMENTS INDUCED IN AN IRIS BY A MAGNETIC FIELD TANGENTIAL TO THE PLANE OF THE IRIS
moments. F i g u r e 5 . 1 0 - 1 i l l u s t r a t e s two p a r a l l e l - p l a n e t r a n s m i s s i o n l i n e s c o n n e c t e d by a s m a l l i r i s . T h e e l e c t r i c f i e l d . E o n , i n t h e b o t t o m l i n e w i l l c o u p l e t h r o u g h t h e i r i s i n t h e m a n n e r shown i n F i g . 5 . 1 0 - l ( a ) . To a f i r s t - o r d e r approximation, t h e d i s t o r t e d f i e l d within t h e i r i s can be c o n s i d e r e d t o a r i s e from t w o e l e c t r i c d i p o l e moments.
each o f s t r e n g t h
b,
i n d u c e d i n t h e i r i s by t h e e x c i t i n g e l e c t r i c f i e l d E o n a s shown i n
T h e s t r e n g t h o f t h e e l e c t r i c d i p o l e moment, p , i s p w p o r t i o n a l t o t h e p r o d u c t o f t h e e l e c t r i c p o l a r i z a b i l i t y P of t h e i r i s and t h e e x c i t i n g f i e l d , E o n . I t s v a l u e i n mks u n i t s i s
Fig. 5.10-l(b). The e l e c t r i c d i p o l e moment i n t h e u p p e r l i n e i s p a r a l l e l t o E o n . w h i l e t h e e l e c t r i c d i p o l e moment i n t h e l o w e r l i n e i s o p p o s i t e l y
-
directed.
w h e r e r,, = 8 . 8 5 4 x 1 0 - l 2 f a r a d s / m e t e r , a n d n i s a u n i t v e c t o r d i r e c t e d away f r o m t h e i r i s on t h e s i d e o p p o s i t e f r o m t h e e x c i t i n g f i e l d . T h e s t r e n g t h o f t h e m a g n e t i c d i p o l e moment i s p r o p o r t i o n a l t o t h e p r o d u c t o f t h e m a g n e t i c ~ o l a r i z a b i l i t y ,M , o f t h e i r i s a n d e x c i t i n g t a n g e n t i a l m a g n e t i c f i e l d A o t . F o r t h e u s u a l t y p e o f i r i s t h a t h a s axe. o f s y m m e t r y , t h e m a g n e t i c d i p o l e moment i s , i n mks u n i t s ,
FIG. 5.10-1
ELECTRIC DIPOLE MOMENTS INDUCED IN AN IRIS BY AN ELECTRIC FIELD NORMAL TO THE PLANE OF THE IRIS
I n t h i s e x p r e s s i o n t h e u n i t r e c t o r s u and
;l i e
i n t h e plane of t h e i r i s
a l o n g t h e a x e s o f s y m m e t r y , M I a n d M2 a r e t h e m a g n e t i c p o l a r i z a b i l i t i e ~ ,
and Hey and H o w t h e e x c i t i n g m a g n e t i c f i e l d s a l o n g t h e
u
and
axes,
respectively.
-
The e l e c t r i c d i p o l e moment, p ,
s e t up i n an i r i s by a n e x c i t i n g
e l e c t r i c f i e l d , w i l l r a d i a t e power i n t o a g i v e n mode i n t h e s e c o n d a r y waveguide o n l y when t h e e l e c t r i c f i e l d o f t h e mode t o b e e x c i t e d h a s a
-
component p a r a l l e l t o t h e d i p o l e moment, p . ,
Similarly the magnetic
d i p o l e moment s e t u p i n t h e a p e r t u r e by a n e x c i t i n g m a g n e t i c f i e l d w i l l r a d i a t e power i n t o a g i v e n mode i n t h e s e c o n d a r y waveguide o n l y when t h e m a g n e t i c f i e l d o f t h e mode t o b e e x c i t e d h a s a component p a r a l l e l t o t h e m a g n e t i c d i p o l e moment G. I n o r d e r t o be a b l e t o apply Bethe's theory, i t i s necessary t o know t h e e l e c t r i c p o l a r i z a b i l i t y P and t h e m a g n e t i c p o l a r i z a b i l i t i e s
MI and M 2 o f t h e i r i s . T h e o r e t i c a l v a l u e s o f t h e p o l a r i z a b i l i t i e s c a n only be obtained f o r i r i s e s o f simple shapes. F o r example, a c i r c u l a r i r i s of diameter d has a v a l u e of
MI
=
M2
= d3/6 and P = d3/12.
narrow i r i s o f l e n g t h 1 and w i d t h r h a s P =
M2
a
(7d16)
A long,
1w2, i f t h e ex-
c i t i n g magnetic f i e l d i s p a r a l l e l t o t h e narrow dimension of t h e s l i t
-
( t h e v d i r e c t i o n i n t h i s c a s e ) , and t h e e x c i t i n g e l e c t r i c f i e l d i s p e r p e n d i c u l a r t o t h e p l a n e o f t h e s l i t . The p o l a r i z a b i l i t i e s o f e l l i p t i c a l i r i s e s h a v e a l s o b e e n computed.
In addition, the polarizabilities of
i r i s e s o f o t h e r s h a p e s t h a t a r e t o o d i f f i c u l t t o c a l c u l a t e h a v e been measured by ~ o h n i n ~ an ~ ' e l~e c t r o l y t i c t a n k .
The measured v a l u e s o f t h e p o l a r i z a b i l i t y o f a number o f i r i s e s a r e shown i n F i g s . 5 . 1 0 - 3 and 5 . 1 0 - 4 ( a ) , ( b ) , t o g e t h e r wkth t h e t h e o r e t i c a l v a l u e s f o r e l l i p t i c a l i r i s e s . C i r c u l a r i r i s e s a r e t h e e a s i e s t t o p a c h i n e , b u t sometimes e l o n g a t e d i r i s e ~ a r e r e q u i r e d i n o r d e r t o o b t a i n a d e q u a t e c o u p l i n g between r e c t a n g u l a r waveguides. F o r many a p p l i c a t i o n s t h e e q u i v a l e n t - c i r c u i t r e p r e s e n t a t i o n o f i r i s c o u p l e d t r a n s m i s s i o n l i n e s i s more c o n v e n i e n t t h a n t h e s c a t t e r i n g r e p r e sentation.
F i g u r e s 5.10-5 t o 5.10-12 c o n t a i n t h e e q y i v a l e n t - c i r c u i t
r e p r e s e n t a t i o n s o f s e v e r a l two- and t h r e e - p o r t waveguide j u n c t i o n s c o u p l e d by i n f i n i t e s i m a l l y t h i n i r i s e s . self-explanatory.
Most o f t h e i n f o r m a t i o n i n t h e f i g u r e s i s
It i s t o be noted t h a t i n each case t h e r e f e r e n c e
planes f o r the equivalent c i r c u i t s are a t the center of gravity of the iris.
The symbol K u s e d i n some c i r c u i t s s t a n d s f o r an impedance i n v e r t e r
a s d e f i n e d i n Sec. 4.12.
A l s o i n c l u d e d i n e a c h f i g u r e i s t h e power t r a n s -
mission c o e f f i c i e n t through t h e i r i s , expressed a s t h e square of t h e magnitude o f t h e s c a t t e r i n g c o e f f i c i e n t .
(Sec.
2.12).
FIG. 5.10-3 MEASURED ELECTRIC POLARIZABILITIES OF RECTANGULAR, ROUNDED, CROSS- AND DUMBBELL-SHAPED SLOTS
SOURCE: Proc. IRE (see Ref. 29. by S. B. C O ~ U ) .
FIG. 5.10-4(b) MAGNETIC POLARlZABlLlTlES OF H-, CROSS-, AND DUMBBELL-SHAPED APERTURES
JRCE: Proc. IRE (see Ref. 29, by S. B. Cohn).
FIG. 5.10-4(a) MAGNETIC POLARlZABlLlTlES OF RECTANGULAR, ROUNDED-END, AND ELLIPTICAL SLOTS
T E , , MODE
CROSS SECTIONAL VlEW
SlDE VlEW
EQUIVALENT CIRCUIT CROSS SECTIONAL VIEW
Isl2l- =
-
bI2
SIDE VIEW
EQUIVALENT CIRCUIT
=
.-15z7-144
.2b2~2
Adapted from the Waveguide Handbook ( s e e Ref. 8 edited by N. Marcuvitz)
FIG. 5.10-5
-
IRIS CONNECTING RECTANGULAR WAVEGUIDES OF THE SAME CROSS SECTION
TEln MODE
If,
CROSS SECTIONAL VIEW
Is1J2
FIG. 5.10-7
- -.-
Y;)
1BI2
"0
.
64n2~' sin2
A-3527-14e
IRIS CONNECTING CIRCULAR WAVEGUIDES OF THE SAME RADIUS
TE,, MODE
SIDE VIEW
4~02
Adapted from the Waveguide Handbook ( s e e R e f . 8 edited by N. Marcuvitz)
EQUIVALENT CIRCUIT nx' nx sin2 a a
aa' b b ' ~ ~ ~ ;
CROSS SECTIONAL VIEW
SIDE VIEW
EQUIVALENT CIRCUIT
Adapted from the Waveguide Handbook ( s e e Ref. 8 edited by N. Marcuvitz)
FIG. 5.10-6
IRIS CONNECTING RECTANGULAR WAVEGUIDES OF DIFFERENT CROSS SECTIONS
Adapted from the Waveguide Handbook (see R e f . 8 edited by N. Marcuvitz)
FIG. 5.10-8
IRIS CONNECTING CIRCULAR WAVEGUIDES OF DIFFERENT RADII
MODES
SlDE VlEW
GENERAL VlEW
Ej_
SlDE VlEW
GENERAL VlEW
EQUIVALENT CIRCUIT
EQUIVALENT CIRCUIT
M'
FIG. 5.10-9
=
M I sin2
6 +
Mz cos2
M
4
IRIS-COUPLED SHUNT T-JUNCTION IN RECTANGULAR GUIDE E-PLANE
=
M1 c o s 2 q5
+ M2
sin2
6
B-3517-14.
Adapted from the Waveguide Handbook (see Ref. 8 edited by N. Marcuvitz)
FIG. 5.10-10 IRIS-COUPLED T-JUNCTION OF RECTANGULAR AND CIRCULAR GUIDES
239
1
G E N E R A L VlEW
SIDE VlEW
GENERAL VIEW
T,
T
EQUIVALENT CIRCUIT E Q U I V A L E N T CIRCUIT
--
lsZ3l2
Y;,
Yo
II
=
M1 c o s 2
4 + M2 s i n 2
$
Y: -
1
~
~
aa'bb'A6Ab
-
M1 s i n 2 q5 + M2 c o s 2 $
Adapted from the Waveguide H ~ d b o o k( s e e Ref. 8 edited by N. Marcuvitz)
FIG. 5.10-11
nx ~ a
bb12
M'
6 s i n 2~
A-3W.7-16S
IRIS-COUPLED SERIES T-JUNCTION IN RECTANGULAR GUIDE, E-PLANE
4-3327-150
Adapted from the Waveguide Handbook ( s e e Ref. 8 edited by N. Marcuvitz)
FIG. 5.10-12 IRIS-COUPLED SHUNT T-JUNCTION IN RECTANGULAR GUIDE, H-PLANE
When t h e i r i s e s a r e n o t s m a l l w i t h r e s p e c t t o f r e e - s p a c e w a v e l e n g t h , i t i s found t h a t t h e e q u i v a l e n t c i r c u i t s o f F i g s .
5 . 1 0 - 5 t o 5.10-12 a p p l y w i t h good a c c u r a c y i f t h e s t a t i c m a g n e t i c p o l a r i z a b i l i t y M1 g i v e n i n
Fig. for
5 . 1 0 - 4 i s r e p l a c e d by t h e m a g n e t i c p o l a r i z a b i l i t y M i .
Mi
The e x p r e s s i o n
is
SEC. 5 . 1 1 , RESONANT FREQUENCIES AND UNLOADED OF WAVEGUIDE RESONATORS ',1
Q
Two i m p o r t a n t c h a r a c t e r i s t i c s o f a w i v e g u i d e r e s o n a t o r t h a t a r e where kc i s t h e f r e e - s p a c e w a v e l e n g t h a t t h e c u t o f f f r e q u e n c y f o r t h e l o w e s t - o r d e r mode i n a waveguide h a v i n g t h e same c r o s s s e c t i o n a s t h e
u s e f u l i n t h e d e s i g n of waveguide f i l t e r s a r e t h e r e s o n a n t f r e q u e n c y o f t h e r e s o n a t o r and i t s u n l o a d e d Q , Q U . T h i s s e c t i o n p r e s e n t s c u r v e s and formulas y i e l d i n g these q u a n t i t i e s f o r completely closed c a v i t i e s o f t h e When a s m a l l c o u p l i n g i r i s i s c u t
i r i s , and h i s t h e f r e e - s p a c e w a v e l e n g t h a t t h e f r e q u e n c y o f o p e r a t i o n .
r e c t a n g u l a r and c y l i n d r i c a l v a r i e t i e s .
For l o n g , t h i n i r i s e s o f l e n g t h 1 , hc i s a p p r o x i m a t e l y e q u a l t o 21.
i n a c a v i t y i t s r e s o n a n t f r e q u e n c y and
t , o f an i r i s r e d u c e s t h e t r a n s m i s s i o n through It i s found t h a t t h e t o t a l a t t e n u a t i o n a o f a t h i c k i r i s i s p r e -
d i c t e d w i t h r e a s o n a b l e a p p r o x i m a t i o n a s t h e sum o f t h e a t t e n u a t i o n a. o f a t h i n i r i s and t h e a t t e n u a t i o n
w i l l b e n e a r l y t h e same a s
t h o s e of t h e u n p e r t u r b e d c a v i t y .
The f i n i t e t h i c k n e s s ,
it.
Qu
U1
of a l e n g t h o f t r a n s m i s s i o n l i n e having
a length equal t o t h e i r i s t h i c k n e s s .
R e c t a n g u l a r Waveguide R e s o n a t o r s - R e c t a n g u l a r
resonators are
p r o b a b l y u s e d more o f t e n i n waveguide f i l t e r s t h a n any o t h e r t y p e . An example o f s u c h a r e s o n a t o r i s i l l u s t r a t e d i n F i g . 5 . 1 1 - l ( a ) . The modes t h a t c a n e x i s t i n t h i s r e s o n a t o r a r e c o n v e n i e n t l y d i v i d e d i n t o two s e t s ,
Thus,
t h e t r a n s v e r s e e l e c t r i c TE-modes and t h e t r a n s v e r s e - m a g n e t i c TM-modes. The TE-modes h a v e no e l e c t r i c f i e l d components, E , a l o n g t h e z a x i s and t h e TM-modes h a v e no m a g n e t i c f i e l d components, H , a l o n g t h e z a x i s . The two t y p e s o f modes a r e f u r t h e r s p e c i f i e d i n t e r m s of t h e i n t e g e r s and
1, m,
and n.
These a r e defined a s
I = number o f h a l f - p e r i o d v a r i a t i o n s o f E and H a l o n g x E and H a l o n g y n = number o f h a l f - p e r i o d v a r i a t i o n s of E and H a l o n g 2 . m = number o f h a l f - p e r i o d v a r i a t i o n s o f
where A i s an e m p i r i c a l l y d e t e r m i n e d c o n s t a n t a p p r o x i m a t e l y e q u a l t o one f o r a round hole.%
F o r an e l o n g a t e d s l o t o f l e n g t h
I
i n a wall t t h i c k ,
A i s a b o u t 3 i f t < 0 . 0 2 1, b u t A d e c r e a s e s i n s i z e a s t becomes l a r g e r . "
F o r a g i v e n s e t o f i n t e g e r s a mode i s c o m p l e t e l y s p e c i f i e d , and t h e modes a r e d e s i g n a t e d a s e i t h e r TE1,, ,, , o r m l , ,
,,.
The r e s o n a n t f r e q u e n c i e s a r e g i v e n by t h e e q u a t i o n
The i n f o r m a t i o n i n Eqs. ( 5 . 1 0 - 3 1 , ( 5 . 1 0 - 4 ) , and ( 5 . 1 0 - 5 ) c a n b e comb i n e d t o y i e l d an e q u i v a l e n t p o l a r i z a b i l i t y M i , f o r a t h i c k i r i s whose c r o s s - s e c t i o n a l dimensions a r e n o t small i n terms of a wavelength. expression i s
The where A , B , and L a r e measured i n i n c h e s , and f i s e x p r e s s e d i n g i g a cycles.
F i g u r e 5 . 1 1 ( a ) a l s o c o n t a i n s a mode c h a r t i n which f 2 A 2 i s
p l o t t e d a s a f u n c t i o n of A 2 / L 2 f o r a11 o f t h e TE- and TM-modes h a v i n g
I
I n t h i s f i g u r e , a l l dimensions
1 , n, n _1 2 i n a c a v i t y i n w h i c h B / A = 1 / 2 . /'
a r e i n i n c h e s , and f r e q u e n c y i s measured i n g i g a c y c l e s .
I
The u n l o a d e d Q o f a c a v i t y i s most c o n v e n i e n t l y t a b u l a t e d i n t h e d i m e n s i o n l e s s form QU8/A where 8 i s t h e s k i n d e p t h and A i s t h e f r e e s p a c e T a b l e 5 . 1 1 - 1 p r e s e n t s b\ a l u e s o f 8 / h f o r v a r i o u s m e t a l s h a v i n g
wavelength. I
\
polished, corrosion-free s u r f aces.
'\ Table 5 . 1 1 - 1
4
i
VALUES OF $/A FOR V ~ O U SMETALS The Values Given Are For P o l i s h e d C o r r o s i o n - F r e e S u r f a c e s . The Frequency f C , I s 1; G i g a c y c l e s .
I
Silver,
For TE-modes we f i n d t h a t Qu(S/A) i s g i v e n by:'
f o r ( 1 and n ) > 0 ;
FIG. 5.11-l(a)
MODE CHART FOR RECTANGULAR WAVEGUIDE RESONATOR WITH B/A = 1/2 The dimensions A, 0, and L are measured in inches and frequency f is measured in gigacycles
S
Q
-
U h
=
( q * + r 2 ) 3/2
-ABL . 2
rq2L(B
ABL - ' 2
p 2 ~ ( ~2 B )
+
,
for 1
=
0 ;
(5.11-3)
(pa + r 2 ) 3 , + r2A(L + 28)
for n
=
0
(5.11-4)
2A)
+
r 2 B ( L + 2.4)
and S
Q,-=
h where p =
+
1 / A , q = n/B, r = n / L .
F i g u r e 5 . 1 1 - l ( b ) shows a c h a r t of Qu(S/A)
v e r s u s AIL f o r v a r i o u s a s p e c t r a t i o s k = A/B f o r t h e TElolmode.
Right-Circular-Cylinder Resonators-Cylindrical t y p e i l l u s t r a t e d i n F&.
resonators of the
5 . 1 1 - 2 a l s o h a v e n o r m a l modes t h a t c a n b e
c h a r a c t e r i z e d a s TE-modes when t h e r e a r e n o e l e c t r i c f i e l d c o m p o n e n t s , E , a l o n g t h e z a x i s , a n d a s TM-modes when t h e r e a r e n o m a g n e t i c f i e l d coma l o n g t h e z a x i s . The i n d i v i d u a l TE- a n d TM-modes a r e f u r t h e r
p o n e n t s , H,
i d e n t i f i e d b y means o f t h e t h r . e e i n t e g e r s 1 , a s follows:
L
m,
and n, which a r e d e f i n e d
%\
= number o f f u l l - p e r i o d v \ i r i a t i o n s o f
E , with r e s p e c t t o 0
\
F I G 5 l ( b ) CHART FOR ESTIMATING THE UNLOADED Q OF TE,,,-MODE RECTANGULAR WAVEGUIDE RESONATORS
6.
ABL 4
(pZ+q2) (P2+q2+r2)*
,
6
-
ABL - *
for n
>0
, D ~ B ( A + L ) + ~ ~ A ( B + L )
(p2 + q2)
*
P ~ B ( A + S L )+ q 2 ~ ( ~ +
n = number o f h a l f - p e r i o d v a r i a t i o n s o f
Er
SOURCE:
with respect t o z
Technique of Microwave Measurementa, see Ref. 31 by C. G. Monteornery
RESONATOR
and
Q"h
E8 w i t h r e s p e c t t o r
FIG. 5.11-2 RIGHT-CIRCULAR-CYLINDER
F o r t h e TM-modes we f i n d t h a t QU i s g i v e n by:'
Quh = -.
m = number o f h a l f - p e r i o d v a r i k g t i o n s o f
;
where Er and Eg a r e t h e f i e l d components i n t h e r and 8 d i r e c t i o n s .
As
i n t h e c a s e o f t h e r e c t a n g u l a r c a v i t y modes t h e r i g h ~c i r c u l a r c y l i n d e r The r e s o n a n t f r e q u e n c i e o r TMl, ,,
modes a r e a l s o d e s i g n a t e d a s TEl , , , ,
, 2 ~
f e r n - 0 )
o f t h e s e modes a r e g i v e n b y t h e e x p r e s s i o n 3
I n t h i s e x p r e s s i o n f i s measured i n g i g a c y c l e s , t h e d i m e n s i o n s D and L a r e measured i n inches. The q u a n t i t i e s x l , a a r e x,,
=
n t h r o o t o f J;(x)
=
0 f o r t h e TE-modes
X l , .
=
mth r o o t o f J ~ ( x ) =
0 f o r t h e TM-modes
.
(5.11-8)
V a l u e s o f a few o f t h e s e r o o t s a r e g i v e n i n T a b l e 5 . 1 1 - 2 .
Table 5.11-2 ROOTS OF J , ( . r ) AM) J ; ( z ) TY- mode
Oln Iln 2l n 02n 31n 12n 41n 22n 03n 5 1n 32n 61n 1311
Sourca:
T e c h n i q u e of M i c r o r o v e M e s a u r e a e n r ~ , r o e Ref.
b y C.
31.
G. Montgomery.
FIG. 5.11-3 F i g u r e 5 . 1 1 - 3 i s a mode c h a r t i n which f 2 ~ 2i s p l o t t e d a s a f u n c t i o n o f D ' / L ~ , f o r s e v e r a l o f t h e l o w e r - o r d e r TE- a n d TM-modes. In this figure a l l dimensions a r e i n i n c h e s and frequency i s measured i n g i g a c y c l e s . V a l u e s of
Qy f o r r i g h t - c i r c u l a r - c y l i n d e r c o p p e r r e s o n a t o r s a r e
p l o t t e d f o r TE-modes i n F i g s . 5 . 1 1 - 4 and 5 . 1 1 - 5 , Fig. 5.11-6
a n d f o r TM-modes i n
MODE CHART FOR RIGHT-CIRCULAR-CYLINDER RESONATOR The diameter D and length L are measured i n inches and the frequency f i s measured in gigacycles
SOURCE:
FIG. 5.11-4
THEORETICAL UNLOADED Q OF SEVERAL TEo-MODES I N A RIGHT-CIRCULARCYLINDER COPPER RESONATOR Frequency i s measured in gigacycles
Technique of Microwave Measwements. see Ref. 31 by C . G. Montgomery
FIG. 5.11-5 THEORETICAL UNLOADED Q OF SEVERAL TE-MODES IN A RIGHT-CIRCULARCYLINDER COPPER RESONATOR Frequency is measured in gigacycles
/'
REFERENCES
1.
S. B. Cohn, "Characteristic Ilrpedance of the Shielded-Strip Transmission Line,'' IRE Trans., PCtWl'-2. pp. 52-72 (July 1954).
2.
S. B. Cohn, "Problems in Strip Transmission Lines," IRE Trans., PCM'IT-3, 2, pp. 119-126
3.
R. H. T. Bates, "The Characteristic Inpedance-.ofShielded Slab Line," IRE Trans., PCUlT-4, pp. 28-33 (January 1956).
4.
S. B. Cohn, "Shielded Coupled-Strip Transmission Lines," IRE Trans.. PGMIT-3, pp. 29-38
(March 1955).
(October 1955).
5.
F. Oberhettinger and W. Magnus, Awendung der El 1 i~tischenFunctionen in Physick and Technik, (Springer-Verlag, Berlin, 1949).
6.
S. B. Cohn, "Characteristic Inpedances of Broadside-Coupled Strip Transmission Lines," IRE Trans., PGMIT-8, 6, pp. 633-637 (Ebvember 1960).
7.
S. B. Cohn, "Thickness Corrections for Capacitive Obstacles and Strip Conductors," IRE T~Ms., PG(IIJT-8, 6, pp. 638-644 (November 1960).
8.
N. Marcuvitz, Waveguide Handbook, MIT Radiation Laboratory Series, Vol. 10 (McGraw Hill Book Co., Inc., New York City, 1951).
9.
SOURCE: Technique of Yicmwave Meoarucmantr. by C. G. Mantromery
FIG. 5.11-6
see
Ref. 31
THEORETICAL UNLOADED Q OF SEVERAL TM-MODES IN A RIGHTCIRCULAR-CYLINDER RESONATOR Frequency i s measured in gigacycles
T. Moreno, Microwave
Transmission Design Data (Dover Publications Inc., New York City, 1958).
10.
J. R. Whinnery and H. W. Jamieson, "Equivalent Circuits for Discontinuities in Transmission Lines," Proc. IRE 32, 2, pp. 98-114 (February 1944).
11.
J. R. Whinnery, H. W. Jamieson and T. E. Robbins, "Coaxial-Line Discontinuities," Proc. IRE 32, 11, pp. 695-709 (November 1944).
12.
A. A. Oliner, "Equivalent Circuits for Discontinuities in Balanced Strip Transmission Line," IRE Trans., PGW?T-3, 2, pp. 134-143 (March 1955).
13.
H. M. Altschuler and A. $ Oliner, "Discontinuities in the Center Conductor of S y m t r i c Strip Transmission Line, IRE Trans., PCWT-8, 3, pp. 328-339 (May 1960).
14.
A. Alford, "Coupled Networks in Radio-Frequency Circuits," Proc. IRE 29, pp. 55-70 (February 1941).
15.
J. J. Karakash and D. E. Mode, "A Coupled Coaxial Transmission-Line Band-Pass Filter," Proc. IRE 38, pp. 48-52 (January 1950).
16.
W. L. Firestone, "Analysis of Transmission Line Directional Guplers," Proc. IRE 42, pp. 1529- 1538 (October 1954).
17.
B. M. Oliver, "Directional Electromagnetic Couplers," Proc. IRE, Vol. 42, pp. 1686-1692
18.
R. C. Knechtli, "Further Analysis of Transmission-Line Directional Couplers," Proc. IRE 43, pp. 867-869 (July 1955).
19.
E. M. T. Jones and J. T. Bolljahn, "Coupled-Strip-Transmission-LineFilters and ~irectional Couplers," IRE Trans., PCM'l7'-4, 2, pp. 75-81 (April 1956).
20.
H. Ozaki and J. Ishii, "Synthesis of a Class of Strip-Line Filters," IRE Trans., PCCT-5, pp. 104-109 (June 1958).
21.
H. A. &the, "Lumped Constants for Small Irises," Report 43-22, M.I.T. Cambridge. Massachusetts (March 1943).
(November 1954).
Radiation Laboratory,
22.
H A Bethe "Theory of Side Windows in Wave Gbo;atory,'~ambridge,
23.
Massachusetts (April
f%fl)' " Fkport
43-27,
M. I.T. Radiation
H. A. &thes "Formal Theory of Waveguides of Arbitrary Cross Section," Report 43-26, M. I.T. Radiation Laboratory, Cambridge, Massachusetts (March 1943).
24. 25. 26.
CHAPTER 6 I
H. A. &the, "Theory of Diffraction by Small Holes," Phys. Rev. Vol. 66, pp. 163-182 (1944). S. B. Cohn, "Microwave Coupling by Large Apertures, * Proc. IRE 40, pp. 696-699 (June 1952). R. E. Collin, Field neory of Guided Waves, Sec. 7.3 (McGraw Hill Book Co., Inc., New York City, 1960).
27.
1
STEPPED-IMPEDANCE TRANSFORMERS AND FILTER PROTOTYPES
1 SEC. 6 . 0 1 , INTRODUCTION
N. Marcuvitz, "Waveguide Circuit Theory: Coupling of Waveguides by Small Apertures, " Report No. R-157-47, Microwave Research Institute, Polytechnic Institute of Brooklyn (1947) PIB-106.
28.
A. A. Oliner, "Equivalent Circuits for Small Synmmtrical Longitudinal Apertures and Cbstaclea," IRE Trans., PGU'IT-8 1, pp. 72-80 (January 1960).
29.
S. B. Cohn, "Datermination of Aperture Parameters by Electrolytic Tank Measurements," Proc. IRE 39, pp. 1416-1421 (November 1951).
30.
S. B. C o b , "The Electric Polarizability of Apertures of Arbitrary Shape," Proc. IRE 40, pp. 1069-1071 (September 1952).
I
.- design equations and The objective of this chapter is to present numerical data for the design of quarter-wave transformers, with two applications in mind: the first application is as an impedance-matching device or, literally, transformer; the second is as a prototype circuit, which shall serve as the basis for the design of various band-pass and low-pass filters.
31. C. G. Montgomery, Technique of Microrave Meoourements, Secs. 5.4 and 5.5 (McGraw-Hill Book Co., New York City, N.Y., 32.
1947).
W. J. Getsinger "A Coupled Strip-Line Configuration Using hinted-Circuit Construction that Allows Very Gwling," LRE P-., RMT-9, pp. 535-544 (NW& 1x1). 33. W. J. Gtsinger, "Coupled Rectangular Bars &meen Parallel Plat-,'' LRE Pm.,m - 1 0 , pp. 65-72 (January 1x2).
This chapter is organized into fifteen sections, with the following purpose and content: Section 6.01 is introductory. It also discusses applications, and gives a number of definitions, Sections 6.02 and 6.03 deal with the performance characteristics of quarter-wave transformers and half-wave filters. In these parts the designer will find what can be done, not how to do it. Sections 6.04 to 6.10 tell and half-wave filters. If available, and solvable by rule, these sections would
how to design quarter-wave transformers simple, general design formulas were nothing more complicated than a slidebe much shorter.
Section 6.04 gives exact formulas and tables of complete designs for Tchebyscheff and maximally flat transformers of up to four sections. Section 6.05 gives tables of designs for maximally flat (but not Tchebyscheff) transformers of up to eight seccions. Section 6.06 gives a first-order theory for Tchebyscheff and maximally flat transformers of up to eight sections, with explicit formulas and numerical tables. It also gives a general first-order formula, and refers to existing numerical tables published elsewhere which are suitable for up to 39 sections, and for relatively wide (but not narrow) bandwidths. Section 6.07 presents a modified fimt-order theory, accurate for larger transformer ratios than can be designed by the (unmodified) first-order theory of Sec. 6.06.
S e c t i o n 6.08 d e a l s w i t h t h e d i s c o n t i n u i t y e f f e c t s o f non-ideal j u n c t i o n s , and f i r s t - o r d e r c o r r e c t i o n s t o c o m p e n s a t e f o r them.
I
S e c t i o n s 6.09 and 6 . 1 0 apply p r i m a r i l y t o p r o t o t y p e s f o r f i l t e r s , s i n c e they a r e concerned with l a r g e impedance s t e p s . They become e x a c t o n l y i n t h e l i m i t a s t h e o u t p u t - t o - i n p u t i m p e d a n c e r a t i o , R, t e n d s t o i n f i n i t y . Simple f o r m u l a s a r e g i v e n f o r any number o f s e c t i o n s , a n d n u m e r i c a l t a b l e s o n l u m p e d - c o n s t a n t f i l t e r s are referred to. Note:
S e c t i o n s 6 . 0 9 and 6 . 1 0 complement S e c s . 6 . 0 6 a n d 6 . 0 7 , w h i c h g i v e exact r e s u l t s only i n t h e l i m i t a s R tends t o zero. It is p o i n t e d o u t t h a t t h e d i v i d i n g l i n e b e t w e e n " s m a l l R" a n d " l a r g e R" i s i n t h e o r d e r
o f [ 2 / ( q u a r t e r - w a v e t r a n s f o r m e r b a n d w i d t h ) 12n, w h e r e n i s t h e number o f sections. and 6 . 0 7 ,
T h i s determines whether t h e f i r s t - o r d e r theory o f Secs. 6.06 o r t h e formulas o f S e c s . 6.09 and 6 . 1 0 a r e t o b e used.
e x a m p l e ( E x a m p l e 3 o f Sec. 6 . 0 9 ) w h e r e is s o l v e d by b o t h t h e "small
( S e c . 7 . 0 6 ) ; b r a n c h - g u i d e d i r e c t i o n a l c o u p l e r s lo ( C h a p t . 1 3 ) ; a s w e l l a s o p t i c a l m u l t i - l a y e r f i l t e r s a n d t r a n s f o r m e r s , Y*12 and a c o u s t i c a l transformers.
1
i
13*14
'.__
-,
The a t t e n u a t i o n f u n c t i o n s c o n s i d e r e d h e r e a r e a l l f o r m a x i m a l l y I t i s of i n t e r e s t t o f l a t o r T c h e b y s c h e f f r e s p o n s e i n t h e p a s s band. n o t e t h a t o c c a s i o n a l l y o t h e r r e s p o n s e s h a p e s may b e d e s i r a b l e . T h u s TEM-mode c o u p l e d - t r a n s m i s s i o n - l i n e d i r e c t i o n a l c o u p l e r s a r e a n a l y t i c a l l y e q u i v a l e n t t o quarter-wave t r a n s f o r m e r s (Chapt. 1 3 ) , but r e q u i r e f u n c t i o n s w i t h maximally f l a t o r e q u a l - r i p p l e c h a r a c t e r i s t i c s i n t h e O t h e r a t t e n u a t i o n f u n c t i o n s may b e c o n v e n i e n t f o r o t h e r s t o p band.
An
R is i n t h i s borderline region,
a p p l i c a t i o n s , but w i l l not be considered here.
R" and t h e " l a r g e R" a p p r o x i m a t i o n s , a n d
both methods g i v e t o l e r a b l y good r e s u l t s f o r most purposes.
C i r c u i t s t h a t can be d e s i g n e d u s i n g q u a r t e r - w a v e t r a n s f o r m e r s a s a i m p e d a n c e transformer^'^ ( a s i n t h i s c h a p t e r ) ; prototype include: reactance-coupled f i l t e r s 7 + (Chapt. 9 ) ; s h o r t - l i n e low-pass f i l t e r s
I
As i n t h e d e s i g n o f a l l m i c r o w a v e c i r c u i t s , o n e must d i s t i n g u i s h
S e c t i o n s 6 . 1 1 and 6 . 1 2 d e a l w i t h " i n h o m o g e n e o u s " t r a n s f o r m e r s , which a r e n o t u n i f o r m l y d i s p e r s i v e , s i n c e t h e c u t o f f wavelength changes a t each s t e p .
between t h e i d e a l c i r c u i t s a n a l y z e d , and t h e a c t u a l c i r c u i t s t h a t
S e c t i o n 6 . 1 3 d e s c r i b e a a p a r t i c u l a r t r a n s f o r m e r whose p e r f o r m a n c e and o v e r - a l l l e n g t h a r e s i m i l a r t o t h o s e o f a s i n g l e - s e c t i o n q u a r t e r - w a v e t r a n s f o r m e r , b u t which r e q u i r e s o n l y m a t c h i n g s e c t i o n s whose c h a r a c t e r i s t i c i m p e d a n c e s a r e e q u a l t o t h e i n p u t a n d o u t p u t impedances.
d e f i n i t i o n s : l5
S e c t i o n 6.14 c o n s i d e r s d i s s i p a t i o n l o s s e s . f o r m u l a f o r t h e midband d i s s i p a t i o n l o s s .
It gives a general
h a v e p r o m p t e d t h e a n a l y s i s and w h i c h a r e t h e d e s i r e d end p r o d u c t . To b r i n g t h i s o u t e x p l i c i t l y , w e s h a l l s t a r t w i t h a l i s t o f
Homogeneous t r a n s f o r m e r - a t r a n s f o r m e r i n which t h e r a t i o s of i n t e r n a l w a v e l e n g t h s and c h a r a c t e r i s t i c i m p e d a n c e s a t d i f f e r e n t p o s i t i o n s along t h e d i r e c t i o n of propagation a r e independent of frequency.
I
1
Inhomogeneous transformer-a t r a n s f o r m e r i n which t h e r a t i o s o f i n t e r n a l w a v e l e n g t h s and c h a r a c t e r i s t i c impedances a t d i f f e r e n t p o s i t i o n s a l o n g t h e d i r e c t i o n o f p r o p a g a t i o n may c h a n g e w i t h frequency.
S e c t i o n 6.15 r e l a t e s group d e l a y t o d i s s i p a t i o n l o s s i n t h e p a s s b a n d , and p r e s e n t s n u m e r i c a l d a t a i n a s e t o f u n i v e r s a l c u r v e s . Q u a r t e r - w a v e t r a n s f o r m e r s h a v e numerous a p p l i c a t i o n s b e s i d e s b e i n g i m p e d a n c e t r a n s f o r m e r s ; an u n d e r s t a n d i n g o f t h e i r b e h a v i o r g i v e s i n s i g h t
1
Quarter-wave transformer-a cascade of s e c t i o n s of l o s s l e s s , uniform* t r a n s m i s s i o n l i n e s o r media, each s e c t i o n b e i n g o n e - q u a r t e r ( i n t e r n a l ) w a v e l e n g t h l o n g a t a common f r e q u e n c y .
i n t o many o t h e r p h y s i c a l s i t u a t i o n s n o t o b v i o u s l y c o n n e c t e d w i t h impedance t r a n s f o r m a t i o n s .
The d e s i g n e q u a t i o n s and n u m e r i c a l t a b l e s have, moreover, been developed t o t h e p o i n t where t h e y can b e used c o n v e n i e n t l y f o r t h e s y n t h e s i s o f c i r c u i t s , many o f w h i c h w e r e previously d i f f i c u l t t o design.
1 * A uniform tranamisaion l i n e , medium, c t c . , is here defined as one i n which the physical and e l e c t r i c a l w i t h diatancc along the direction of propagation. l l i s i a a g e n e r a l i ~ a t i o n c h a r a c t e r i s t i c s do not 16). o f the IRE d e f i n i t i o n o f uniform waveguide ( s e e Ref.
Note:
Homogeneous and inhomogeneous q u a r t e r - w a v e t r a n s f o r m e r s a r e now
d e f i n e d by a c o m b i n a t i o n o f t h e above d e f i n i t i o n s .
For i n s t a n c e , a n inhomogeneous q u a r t e r - w a v e t r a n s f o r m e r i s a q u a r t e r - w a v e t r a n s f o r m e r i n
which t h e r a t i o s o f i n t e r n a l w a v e l e n g t h s and c h a r a c t e r i s t i c i m p e d a n c e s t a k e n between d i f f e r e n t s e c t i o n s , may c h a n g e w i t h f r e q u e n c y .
IMPEDANCE INVERTER
IMPEDANCE STEP
Ideal junction-the c o n n e c t i o n between two impedances o r t r a n s m i s s i o n l i n e s , when t h e e l e c t r i c a l e f f e c t s o f t h e c o n n e c t i n g w i r e s , o r t h e j u n c t i o n d i s c o n t i n u i t i e s , c a n be n e g l e c t e d . (The j u n c t i o n e f f e c t s may l a t e r b e r e p r e s e n t e d by e q u i v a l e n t r e a c t a n c e s and t r a n s f o r m e r s , o r by p o s i t i v e and n e g a t i v e l i n e l e n g t h s , e t c . ) I d e a l quarter-wave transformer-a quarter-wave transformer i n which a l l o f t h e j u n c t i o n s ( o f g u i d e s o r media h a v i n g d i f f e r e n t c h a r a c t e r i s t i c i m p e d a n c e s ) may be t r e a t e d a s i d e a l j u n c t i o n s .
LINE CHARACTERISTIC IMPEDANCES * ZI .Z2
LINE CHARACTERISTIC IMPEDANCES=
IMPEDANCE RATIO OR JUNCTION VSWR: V = 2 2 / 2 1 OR 2 1 / 2 2 , WHICHEVER > I
IMPEDANCE OF INVERTER' K
ELECTRICAL LENGTH'O AT ALL FREQUENCIES
ELECTRICAL LENGTH. 90. AT ALL FREQUENCIES
FOR SAME COUPLING: JUNCTION VSWR, V z (+o)r2
>1 A-,,2?-2S,
Half-wave f i l t e r - a cascade of s e c t i o n s o f l o s s l e s s uniform t r a n s m i s s i o n l i n e s o r m e d i a , each s e c t i o n b e i n g o n e - h a l f ( i n t e r n a l ) w a v e l e n g t h l o n g a t a common f r e q u e n c y . Synchronous t u n i n g condition-a f i l t e r consisting of a s e r i e s of d i s c o n t i n u i t i e s spaced along a transmission l i n e is synchronously t u n e d i f , a t some f i x e d f r e q u e n c y i n t h e p a s s band, t h e r e f l e c t i o n s from any p a i r o f s u c c e s s i v e d i s c o n t i n u i t i e s a r e p h a s e d t o g i v e t h e maximum c a n c e l l a t i o n . (A q u a r t e r - w a v e t r a n s f o r m e r i s a s y n c h r o n o u s l y t u n e d c i r c u i t i f i t s i m p e d a n c e s form a monotone sequence. A half-wave f i l t e r i s a synchronously tuned c i r c u i t i f i t s impedances a l t e r n a t e l y i n c r e a s e and d e c r e a s e a t e a c h s t e p along i t s length.) Synchronous frequency-the "fixed frequency" r e f e r r e d t o i n t h e previous d e f i n i t i o n w i l l be c a l l e d t h e synchronous frequency. ( I n t h e case of quarter-wave transformers, a l l s e c t i o n s a r e one-quarter wavelength long a t t h e synchronous frequency; i n t h e c a s e of half-wave f i l t e r s , a l l s e c t i o n s a r e one-half wavelength long a t t h e synchronous frequency. S h o r t - l i n e , low-pass f i l t e r s may a l s o b e d e r i v e d from h a l f - w a v e f i l t e r s , w i t h t h e s y n c h r o n o u s frequency being thought of a s zero frequency.)
SOURCE: Quarterly Progress Report 4, Contract DA 36-039 SC-87398, SRIi reprinted in IRE Trans. PGMTT (See Ref. 36 by L. Young)
FIG. 6.01-1
SEC.
6.02,
CONNECTION BETWEEN IMPEDANCE STEP AND IMPEDANCE INVERTER
THE PERFORMANCE OF HOMOGENEOUS QUARTER-WAVE TRANSFORMERS
T h i s s e c t i o n summarizes t h e r e l a t i o n s h i p s between t h e p a s s - b a n d and s t o p - b a n d a t t e n u a t i o n , t h e f r a c t i o n a l b a n d w i d t h , w q , and t h e Although t h e e x p r e s s i o n s o b t a i n e d number o f s e c t i o n s o r r e s o n a t o r s , n. hold e x a c t l y only f o r i d e a l quarter-wave t r a n s f o r m e r s , they h o l d r e l a t i v e l y a c c u r a t e l y f o r r e a l p h y s i c a l q u a r t e r - w a v e t r a n s f o r m e r s and f o r c e r t a i n f i l t e r s , e i t h e r without modification o r a f t e r simple c o r r e c t i o n s h a v e been a p p l i e d t o a c c o u n t f o r j u n c t i o n e f f e c t s , e t c .
A q u a r t e r - w a v e t r a n s f o r m e r i s d e p i c t e d i n F i g . 6.02-1. The r e a l i z a t i o n o f t r a n s m i s s i o n - l i n e d i s c o n t i n u i t i e s by impedance s t e p s i s e q u i v a l e n t t o t h e i r r e a l i z a t i o n by means o f i d e a l impedance i n v e r t e r s ( S e c . 4 . 1 2 ) . The main d i f f e r e n c e i s t h a t w h i l e impedance s t e p s can be p h y s i c a l l y r e a l i z e d o v e r a wide band o f f r e q u e n c i e s ( a t l e a s t f o r s m a l l s t e p s ) , i d e a l impedance i n v e r t e r s c a n b e a p p r o x i m a t e d o v e r o n l y l i m i t e d bandwidths.
A s f a r a s using e i t h e r c i r c u i t a s a mathematical model, o r p r o t o t y p e c i r c u i t , i s c o n c e r n e d , t h e y g i v e e q u i v a l e n t r e s u l t s , a s c a n b e s e e n from F i g . 6 . 0 1 - 1 .
t h e q u a r t e r - w a v e t r a n s f o r m e r f r a c t i o n a l b a n d w i d t h , w q , by
Define
,+'t"t"l
ELECTRICAL LENGTHS :
L
=
9
2(h1
+
2
A0 - 4
where f s t a n d s f o r frequency. is defined a s t h e r a t i o of actually t h e a v a i l a b l e g e n e r a t o r p o w e r , t o P L ,_ t h e power d e l i v e r e d t o t h e l o a d . T h e " e x c e s s l o s s , " € , i s h e r e i n d e f i n e d by
T h e t r a n s d u c e r l o s s r a t i o ( S e c . 2-11)
-__
NORMALIZED IMPEDANCES : Z0= I
2,
JUNCTION VSWR's: V,
z,
z2
2,
---
v,
v,
v,
-- -
r,
r
r,
---
Zn
Z",i-R
",+I
For t h e maximally f l a t quarter-wave t r a n s f o r m e r of n s e c t i o n s and o v e r - a l l impedance r a t i o R ( F i g . 6.02-1)
E i s g i v e n by
REFLECTION COEFFICIENTS:
r
rn+, where
SOURCE: Quarterly Regress Report 4. Contract DA 36-039 SC-87398, SRI; reprinted in IRE Tmnr. PCMTT (See Ref. 36 by L. Young)
FIG. 6.02-1 QUARTER-WAVE TRANSFORMER NOTATION
'go
b e i n g t h e g u i d e wavelength a t band c e n t e r ,
when 6' = rr/2;
and where
where A
and Ag2 a r e t h e l o n g e s t and s h o r t e s t g u i d e w a v e l e n g t h s , 81 r e s p e c t i v e l y , i n t h e p a s s band o f t h e q u a r t e r - w a v e t r a n s f o r m e r . The
l e n g t h , L, of each s e c t i o n (Fig. 6.02-1)
is nominally one-quarter
wavelength a t c e n t e r f r e q u e n c y and i s g i v e n b y
is t h e g r e a t e s t excess l o s s possible.
( I t o c c u r s when 6 i s a n i n t e g r a l
m u l t i p l e o f T , s i n c e t h e s e c t i o n s t h e n a r e an i n t e g r a l number o f h a l f wavelengths long. ) The 3 - d b f r a c t i o n a l b a n d w i d t h o f t h e m a x i m a l l y f l a t q u a r t e r - w a v e t r a n s f o r m e r i s g i v e n by
where t h e c e n t e r f r e q u e n c y i s d e f i n e d a s t h a t f r e q u e n c y a t which t h e guide wavelength A
is equal t o A
80'
When t h e t r a n s m i s s i o n l i n e i s n o n - d i s p e r s i v e , l e n g t h A may b e u s e d i n E q s .
and
t h e f r e e - s p a c e wave-
( 6 . 0 2 - 1 ) a n d ( 6 . 0 2 - 2 ) , w h i c h t h e n become
The f r a c t i o n a l bandwidth o f t h e maximally f l a t q u a r t e r - w a v e t r a n s f o r m e r b e t w e e n t h e p o i n t s o f x-db a t t e n u a t i o n i s g i v e n by
For t h e T c h e b y s c h e f f t r a n s f o r m e r o f f r a c t i o n a l bandwidth w q .
-
NORMALIZED FREQUENCY. f 0R.NORMALIZED RECIPROCALGUIDE I WAVELENGTH . XgO/&
where p
Tn
=
sin
(7) ,
i s a T c h e b y s c h e f f p o l y n o m i a l ( o f t h e f i r s t k i n d ) o f o r d e r n, and
(b) TCHEBYSCHEFF
I
where t h e q u a n t i t y
i s t h e maximum e x c e s s l o s s i n t h e p a s s band.
below.]
[Compare a l s o Eq.
(6.02-18
The s h a p e o f t h e s e r e s p o n s e c u r v e s f o r maximally f l a t and
T c h e b y s c h e f f q u a r t e r - w a v e t r a n s f o r m e r s i s shown i n F i g . 6 . 0 2 - 2 .
Notice t h a t t h e peak t r a n s d u c e r l o s s r a t i o f o r any q u a r t e r - w a v e t r a n s f o r m e r i s Parai 1 I
PL
eatl
=
(R +
112
4R
FIG. 6.02-2 QUARTER-WAVE TRANSFORMER CHARACTERISTICS (6.02-14)
and i s d e t e r m i n e d s o l e l y by t h e o u t p u t - t o - i n p u t impedance r a t i o , R. For t h e maximally f l a t t r a n s f o r m e r , t h e 3-db f r a c t i o n a l b a n d w i d t h , wq,Jdb, i s p l o t t e d a g a i n s t l o g R f o r n = 2 t o n = 1 5 i n F i g . 6.02-3. The a t t e n u a t i o n g i v e n by Eq.
(6.02-6) can a l s o b e determined from t h e corresponding lumped-constant, low-pass, p r o t o t y p e f i l t e r (Sec. 4.03). I f a' i s t h e f r e q u e n c y v a r i a b l e o f t h e m a x i m a l l y f l a t , l u m p e d - c o n s t a n t ,
l o w - p a s s p r o t o t y p e , and a; i s i t s band e d g e , t h e n
SOURCE: Quarterly Progreaa Report 4, Contract DA 36-039 SC-87398. SRI; reprinted in IRE Tmns. PGMTT (See Ref. 36 by L. Young)
where po i s d e f i n e d by Eq. ( 6 . 0 2 - 1 2 ) . and w q (which o c c u r s i n t h e d e f i n i t i o n o f p,,)
i s t h e f r a c t i o n a l bandwidth o f t h e maximally f l a t
q u a r t e r - w a v e t r a n s f o r m e r between p o i n t s o f t h e same a t t e n u a t i o n a s 1' t h e a t t e n u a t i o n o f t h e maximally f l a t l o w - p a s s f i l t e r a t w' = a'
This enables one t o t u r n t h e graph of a t t e n u a t i o n versus w ' / w ; i n i n t o a graph o f a t t e n u a t i o n v e r s u s c o s 6 o f t h e q u a r t e r F i g . 4.03-2 wave t r a n s f o r m e r , u s i n g Eq. ( 6 . 0 2 - 1 5 ) . For t h e T c h e b y s c h e f f t r a n s f o r m e r ,
i
Table 6 . 0 2 - 1
SOURCE: Q u ~ t e r l yRogreas Report 4. Contract DA 36-039 SC-87398. SRU r e w n t e d in IRE Trans. PGMTT (See Ref. 36 by L. Young)
Table 6.02-1 ( c o n c l u d e d )
FIG. 6.02-3 3-db BANDWIDTHS OF MAXIMALLY FLAT TRANSFORMERS
w h e r e IU i s t h u s d e f i n e d a s a f u n c t i o n o f t h e number o f s e c t i o n s , n, and t h e b a n d w i d t h , w q . I t s h o w s how much t h e p a s s - b a n d t o l e r a n c e i n c r e a s e s when i t i s d e s i r e d t o i m p r o v e t h e p e a k r e j e c t i o n . T h e f u n c t i o n M i n Eq. ( 6 . 0 2 - 1 5 ) i s g i v e n i n T a b l e 6 . 0 2 - 1 f o r a l l f r a c t i o n a l b a n d w i d t h s , w q , i n s t e p s o f 1 0 p e r c e n t , f o r n = 2 t o n = 15. T h e s m a l l e s t f r a c t i ~ o n a lb a n d w i d t h i n T a b l e 6 . 0 2 - 1 i s w q = 0 . 1 . For s m a l l bandwidths,
* 4 means " m u l t i p l y by lo4," and no on. SOURCE:
Quarterly Progress Report 4 . Contract DA 3 6 - 0 3 9 SC-87398. S R I ; reprinted i n IRE T r a n s . PGMTT ( s e e Ref. 3 6 by L. Young)
Equation (6.02-17) i s a c c u r a t e t o b e t t e r than about 1 percent f o r w l e s s t h a n 0.1.
Table 6.02-2
Table 6.02-3
MAXIMUM VSWR FOR SINGLE-SECTION QUARTER- WAVE TRANSFORMERS
MAXIMUM VSWR FOR TWO-SECTION TCHEBYSCHEFF QUARTER-WAVE TRANSFORMERS*
R
0.2
1.25 1.50 1.75 2.00 2.50
The a t t e n u a t i o n g i v e n by Eq. (6.02-11) f o r t h e T c h e b y s c h e f f q u a r t e r - w a v e t r a n s f o r m e r c a n a l s o b e d e t e r m i n e d from t h e g r a p h s i n F i g s . 4 . 0 3 - 4 t o 4.03- 10 f o r t h e c o r r e s p o n d i n g l u m p e d - c o n s t a n t , l o w - p a s s , p r o t o t y p e f i l t e r [ a s a l r e a d y e x p l a i n e d f o r t h e maximally f l a t c a s e i n
-
BANDWIDTSI, m q
IMPEDANCE RATIO,
1.03 1.06 1.09 1.12 1.16
MPEDANCE RATIO, R
/
0.4
0.6
0.8
1.0
1.2
1.07 1.13 1.19 1.24 1.34
1.11 1.20 1.30 1.38 1.53
1.14 1.27 1.39 1.51 1.73
1.17 1.33 1.49 1.64 1.93
1.20 1.39 1.57 1.76 2.12
BANDWIDTH, 0.2
1
0.4
1 0.61
0.8
c o n n e c t i o n w i t h Eq. ( 6 . 0 2 - 1 5 ) ] by u s i n g t h e same Eq. ( 6 . 0 2 - 1 5 ) e x c e p t t h a t now i s t h e T c h e b y s c h e f f ( e q u a l - r i p p l e ) band edge o f t h e low-pass f i l t e r . I n t h e d e s i g n of t r a n s f o r m e r s a s such, one i s i n t e r e s t e d o n l y i n t h e p a s s - b a n d p e r f o r m a n c e f o r s m a l l R ( u s u a l l y l e s s t h a n l o o ) , and t h i s i s e x p r e s s e d i n t e r m s o f maximum VSWR r a t h e r t h a n maximum a t t e n u a t i o n . Tables 6.02-2 through 6.02-5 VSWR i n s i d e t h e p a s s band f o r t r a n s f o r m e r s impedance r a t i o s , R, o f l e s s t h a n 100, and up t o 1 2 0 p e r c e n t , f o r t r a n s f o r m e r s o f n =
g i v e d i r e c t l y t h e maximum with output-to-input f r a c t i o n a l bandwidths, w 9' 1, 2 , 3 , a n d 4 a e c t i o n s . 4
For a l l o t h e r c a s e s , t h e maximum VSWR may b e worked o u t from T a b l e 6 . 0 2 - 1 , using the r e l a t i o n
SOURCE:
APEDANCE RATIO, R
BANDWIDTH, u c
0.2
0.4
0.6
0.8
3.00 4.00 5.00 6.00 8.00
t h e minimum number o f s e c t i o n s f o r a t r a n s f o r m e r o f impedance r a t i o R = 1 0 0 t o h a v e a VSWR o f l e s s t h a n 1 . 1 5
(6.02-81,
BANDWIDTH, w
1.25 1.50 1.75 2.00 2.50
Example I - D e t e r m i n e
and from Eq.
m (me. R f . 4 by L. Young)
Table 6.02-5
MAXIMUM VSWR FOR THREE-SECTION TCHEBYSCHEFF QUARTER- WAVE TF~ANSFORMERS*
where V r i s t h e r i p p l e VSWR (maximum VSWR i n t h e p a s s b a n d ) , t o g e t h e r w i t h Eqs. ( 6 . 0 2 - 8 ) and ( 6 . 0 2 - 1 6 ) .
From Eq. ( 6 . 0 2 - 1 8 1 ,
P
MAXIMUM VSWR FOR FOUR-SECTION TCHEBYSCHEFF QUARTER-WAVE TRANSFORMERS*
R
=
Trans.
Table 6.02-4
IPEDANCE RATIO,
over a 100-percent bandwidth ( w
IRE
1.0).
10.00 12.50 15.00 17.50 20.00
f o r Vr = 1 . 1 5 ,
25.00 30.00 40.00 50.00 60.00
f o r R = 100,
80.00 100.00 SOURCE:
SOURCE: IRE Pans. PWTT ( s e e Ref. 4 by L. Young)
*
IRE Trans. PCWIT ( s e e Asf. 4 by L. Young)
Corresponding Tshebyacheff transformer d e s i g n 8 are presented on pp. 275-278.
267
H e n c e , Eq.
(6.02-16) gives and t h e l e n g t h
/F Eq.
L' o f e a c h s e c t i o n [compar
( 6 . 0\2 - 2 ) l
is
'-. i n t h e column w q = 1 . 0 , i t i s s e e n t h a t t h i s v a l u e
From T a b l e 6 . 0 2 - 1 ,
o f M ( n , w q ) f a l l s b e t w e e n n = 5 and n = 6 . T h e r e f o r e , t h e t r a n s f o r m e r must h a v e a t l e a s t s i x s e c t i o n s . ( S e e a l s o Example 1 o f S e c . 6 . 0 7 ) . SEC. 6 . 0 3 ,
THE PERFORMANCE OF HOMOGENEOUS HALF- WAVE F I L T E R S
w h e r e h g l and h g 2 a r e t h e l o n g e s t a n d s h o r t e s t w a v e l e n g t h s , r e s p e c t i v e l y , T h i s can be s i m p l i f i e d f o r i n t h e p a s s band of the half-wave f i l t e r . n o n - d i s p e r s i v e l i n e s by d r o p p i n g t h e s u f f i x l ' g , " a s i n Eqs. ( 6 . 0 2 - 3 ) A h a l f - w a v e f i l t e r w i t h t h e same j u n c t i o n VSlYRs V i and ( 6 . 0 2 - 4 ) . ( F i g s . 6.02-1 and 6.03-1) w
a s a quarter-wave transformer of bandwidth
h a s a bandwidth
The h a l f - w a v e f i l t e r was d e f i n e d i n S e c . 6 . 0 1 . I t i s shown i n Fig. 6.03-1. I t s f r a c t i o n a l b a n d w i d t h w,, i s d e f i n e d [compare Eq. ( 6 . 0 2 - 1 1 1 by s i n c e i t s s e c t i o n s a r e t w i c e a s l o n g and t h e r e f o r e t w i c e a s f r e q u e n c y s e n s i t i v e . The performance o f a half-wave f i l t e r g e n e r a l l y can b e d e t e r m i n e d d i r e c t l y from t h e p e r f o r m a n c e o f t h e q u a r t e r - w a v e t r a n s f o r m e r w i t h t h e same n u m b e r o f s e c t i o n s , n , and j u n c t i o n VSWRs
ELECTRICAL LENGTHS :
F8'-ct)-@Y
V i , by a l i n e a r s c a l i n g o f t h e f r e q u e n c y a x i s by a s c a l e - f a c t o r o f 2.
PHYSICAL LENGTHS:
kL'-LtCL1d
Compare F i g s . 6 . 0 3 - 2 and 6 . 0 2 - 2 . The q u a r t e r - w a v e t r a n s f o r m e r w i t h t h e same n a n d Vi a s t h e h a l f - w a v e f i l t e r i s h e r e i n c a l l e d i t s prototype c i r c u i t .
NORMALIZE0 IMPEDANCES:
Z& I
zh
Z;
Z;
I n t h e c a s e o f t h e h a l f - w a v e f i l t e r , R i s t h e maximum VSWR, w h i c h i s no l o n g e r t h e o u t p u t - t o - i n p u t impedance r a t i o , a s f o r t h e q u a r t e r wave t r a n s f o r m e r , b u t may g e n e r a l l y b e d e f i n e d a s t h e p r o d u c t o f t h e
---
G+,
2,:
JUNCTION V S W R ~ :
v,
v2
v3
~r~
tr3
---
vn
REFLECTION COEFFICIENTS:
r
- - - frn
junction
VSWRs :
""+I
I
Frn+l
SOURCE: Quarterly Regress Report 4. Contract DA 36-039 SC-87398, SRI; reprinted in IRE Tmns. PCMTT (See Ref. 36 by L. Young)
FIG. 6.03-1 HALF-WAVE FILTER NOTATION
T h i s d e f i n i t i o n a p p l i e s t o b o t h t h e q u a r t e r - w a v e t r a n s f o r m e r and t h e h a l f - w a v e f i l t e r , a s w e l l a s t o f i l t e r s whose p r o t o t y p e c i r c u i t s t h e y are. ( I n t h e l a t t e r c a s e , t h e Vi a r e t h e i n d i v i d u a l d i s c o n t i n u i t y VSWRs a s i n C h a p t e r 9 . ) T h e e q u a t i o n s c o r r e s p o n d i n g t o Eqs.
(6.02-6)
through (6.02-18) w i l l
now b e r e s t a t e d , w h e r e v e r t h e y d i f f e r , f o r t h e h a l f - w a v e f i l t e r .
ifyyy
and t h e bandwidth between t h e p o i n t s of x-db a t t e n u a t i o n i s
(a) MAXIMALLY FLAT VSWR
1
I
I
I
I
I
I
I
I
I
Wq.xdb
I
I
0 112 I I l/Z 2 NORMALIZED FREQUENCY, f 0R.NORMALIZED RECIPROCALGUIDE WAVELENGTH XgO/Ag
Wh,zdb
=
'7,
(6.03-8)
', , w h i c h c a n b e o b t a i n e d f r o m E q s . ( 6 . 0 2 - 9 ) and ( 6 . 0 2 - 1 0 ) . F o r t h e T c h e b ~ s c h e f fh a l f - w a v e f i l t e r ,
&
=
- 1) 2
(R
T:(~in "/p0)
vswR 4R
T: (
)
(b) TCHEBYSCHEFF
I
I
I I I
where
I I
1 I
I I PO =
0 112 I 1112 2 NORMALIZED FREQUENCY t.OR. NORMALIZED RECIPROCAL GUIDE WAVELENCTH AgO/Ag
SOURCE: Quarterly Propeas Report 4, Contract DA 36-039 SC-87398, SRI; reprinted in IRE Trans. PCMTT (See Ref. 36 by L. Young)
FIG. 6.03-2
sin
(7) (7) sin
=
The q u a n t i t i e s &,, &,, and t h e maximum t r a n s d u c e r l o s s r a t i o a r e s t i l l g i v e n by E q s . ( 6 . 0 2 - a ) , ( 6 . 0 2 - 1 3 ) , a n d ( 6 . 0 2 - 1 4 ) . F o r m a x i m a l l y f l a t half-wave f i l t e r s , t h e graph of F i g . 6 . 0 2 - 3 can a g a i n be used,
HALF-WAVE FILTER CHARACTERISTICS
but with t h e right-hand s c a l e . F o r t h e maxi.mally f l a t h a l f - w a v e f i l t e r o f n s e c t i o n s ,
The l u m p e d - c o n s t a n t ,
low-pass, p r o t o t y p e f i l t e r graphs i n
F i g s . 4 . 0 3 - 2 a n d 4 . 0 3 - 4 t o 4 . 0 3 - 1 0 may a g a i n b e u s e d f o r b o t h t h e m a x i m a l l y f l a t a n d T c h e b y s c h e f f h a l f - w a v e f i l t e r s by s u b s t i t u t i n g
-
W'
s i n 8'
4
0
- =
where
f o r Eq. ( 6 . 0 2 - 1 5 ) , w h e r e p o i s g i v e n by Eq.
(6.03-10).
E q u a t i o n ( 6 . 0 2 - 1 6 ) and T a b l e 6 . 0 2 - 1 s t i l l a p p l y , u s i n g Eq. ( 6 . 0 3 - 3 ) i n s t e a d o f Eq. band c e n t e r . filter i s
(6.02-71,
s o t h a t 8 ' = rr ( i n s t e a d o f 8 = r / 2 )
at
t o convert between
wp
and w , , .
The 3 - d h b a n d w i d t h o f t h e m a x i m a l l y f l a t h a l f - w a v e Example 1 - F i n d
87 f o r a h a l f - w a v e f i l t e r o f s i x s e c t i o n s h a v i n g a
T c h e b y s c h e f f f r a c t i o n a l b a n d w i d t h o f 60 p e r c e n t w i t h a p a s s - b a n d r i p p l e o f 1 db
\
H e r e , w,, = 0 . 6 , o r w q = 1 . 2
.
From Eq. ( 6 . 0 2 - 1 3 ) ,
where
and from T a b l e 6 . 0 2 - 1 f o r w q = 1 . 2 ,
For T c h e b y s c h e f f T r a n s f o r m e r s w i t h n =
Hence, R = 850. SEC. 6 . 0 4 ,
EXACT TCHEBYSCHEFF AND MAXIMALLY FLAT SOLUTIONS FOR UP TO FOUR SECTIONS
Enough e x a c t s o l u t i o n s w i l l b e p r e s e n t e d t o p e r m i t t h e s o l u t i o n o f a l l i n t e r m e d i a t e c a s e s by i n t e r p o l a t i o n f o r T c h e b y s c h e f f and maximally f l a t t r a n s f o r m e r s and f i l t e r s h a v i n g up t o f o u r s e c t i o n s . The s o l u t i o n s were o b t a i n e d from C o l l i n ' s f o r m u l a s . ' With t h e n o t a t i o n o f F i g . 6 . 0 2 - 1 , t h e y c a n be r e d u c e d t o t h e e x p r e s s i o n s g i v e n below.
The e q u a t i o n s a r e f i r s t g i v e n f o r maximally f l a t t r a n s f o r m e r s and t h e n f o r T c h e b y s c h e f f t r a n s f o r m e r s . F o r maximally f l a t t r a n s f o r m e r s w i t h n = 2, 3, and 4:
where
2 , 3, and 4:
where A 2
=
Table 6.04-1
, (6.04-6)
and
Z l FOR TWO-SECTION TCHEBYSCHEFF WAmER-WAVE TRANSFORMERS*
o or
(contd. )
Z1
= Z2 =
BANDWIDTH,
IMPEDANCE RATIO. R
2
= 2.0,
wq
0.0
1.00000 1.05737 1.10668 1.15016
1.00 1.25 1.50 1.75
1.00000 1.05810 1.10808 1.15218
1.00000 1.06034 1.11236 1.15837
Wq
0.6
0.8
1.0
1.2
1.4
1.6
1.8
1.00000 1.06418 1.11973 1.16904
1.00000 1.06979 1.13051 1.18469
1.00000 1.07725 1.14495 1.20572
1.00000 1.08650 1.16292 1.23199
1.000 1.096 1.183 1.261
1.000 1.107 1.203 1.291
1.000 1.115 1.218 1.314
0.4
0.2
A d i f f e r e n c e between t y p i c a l q u a r t e r - w a v e t r a n s f o r m e r s , and h a l f wave f i l t e r s s u i t a b l e f o r u s e a s p r o t o t y p e s f o r microwave f i l t e r s , i s t h a t , f o r t h e former, R i s r e l a t i v e l y small ( u s u a l l y l e s s than 100) and o n l y t h e p a s s - b a n d p e r f o r m a n c e i s o f i n t e r e s t ; f o r t h e l a t t e r , R i s r e l a t i v e l y l a r g e , and t h e p e r f o r m a n c e i n b o t h p a s s band and s t o n
band i s i m p o r t a n t .
Two s e t s o f t a b l e s a r e p r e s e n t e d f o r n = 2, 3, The f i r s t s e t ( T a b l e s 6 . 0 4 - 1 t o 6 . 0 4 - 4 ) c o v e r R from 1 t o 100. S i n c e t h e s e t a b l e s a r e most l i k e l y t o be u s e d i n t h e d e s i g n o f and 4.
t r a n s f o r m e r s , t h e impedances
Z1 and Z2 ( F i g . 6 . 0 2 - 1 ) a r e t a b u l a t e d ;
t h e r e m a i n i n g impedances a r e o b t a i n e d from t h e symmetry r e l a t i o n , which c a n b e w r i t t e n ( f o r any n )
*
Z
2
i. g i v e n b y
z2 =
R/Z]
The maximum pass-hand VSWR's of t h e s e design. are tabulated on p. 267.
SOURCE:
-
'6)
IRE Trans. PGW'IT ( s e e Ref. 4 b y L. Young)
f Table 6 . 0 4 - 3 Table 6.04-2 Z , FOR THREE-SECTIONTCHEBYSCHEFF QUARTER-WAVE ~RANSFORMERS* (For rp = 2.0, Z1 = -
=
Z3 =
BANDWIDTH,
R
0.0
1.00 1.25 1.50 1.75 2.00 2.50 3.00 4.00 5.00 6.00 8.00 10.00 12.50 15.00 17.50 20.00 25.00 30.00 40.00 50.00 60.00 80.00 100.00
1.00000 1.02829 1.05202 1.07255 1.09068 1.12177 1.14793 1.19071 1.22524 1.25439 1.30219 1.34089 1.38110 L.41512 1.46675 L.47108 L. 51650 L. 55498 ..61832 66978 .71340 .78522 .84359
..
0.2
0.4
0.6
0.8
--IMPEDANCE RATIO. R
rg
1.0
1.00000 1.00000 1.00000 1.00000 1.00000 1.02883 1.03051 1.03356 1.03839 1.04567 1.05303 1.05616 1.06186 1.07092 1.08465 1.07396 1.07839 1.08646 1.09933 1.11892 1.09247 1.09808 1.10830 1.12466 1.14966 1.12422 1.13192 1.14600 1.16862 1.20344 1.15096 1.16050 1.17799 1.20621 1.24988 1.19474 1.20746 1.23087 1.26891 1.32837 1.23013 1.24557 1.27412 1.32078 1.39428 1.26003 1.27790 1.31105 1.36551 1.45187 1.30916 1.33128 1.37253 1.44091 1.55057 1.34900 1.37482 1.42320 1.50397 1.63471 1.39048 1.42039 1.47674 1.57157 1.72651 1.42564 1.45924 1.52282 1.63055 1.80797 1.45630 1.49328 1.56355 1.68331 1.88193 1.48359 1.52371 1.60023 1.73135 1.95013 1.53075 1.57661 1.66464 1.81693 2.07364 1.57080 1.62184 1.72040.1.89229 2.18447 1.63691 1.69719 1.81471 2.02249 2.38028 1.69080 1.75922 1.88378 2.13434 2.55256 1.73661 1.81246 1.96266 2.23376 2.70860 1.81232 1.90144 2.08004 2.40750 2.98700 1.87411 1.97500 2.17928 2.55856 3.23420
1.2
SOURCE:
1.6
1.8
1.00 1.25 1. 50 1.75 2.00 2.50 3.00 4.00 5.00 6.00 8.00 10.00 12.50 15.00 17.50 20.00 25.00 30.00 40.00 50.00 60.00 80.00 100.00
* =
fi
Z3 = A/Z, The maxim-
1.4
1.00000 1.000 1.000 1.00( 1.05636 1.071 1.091 1. lo! 1.10495 1.134 1.170 1.ZOi 1.14805 1.189 1.243 1.29t 1.18702 1.240 1.310 1.38: 1.25594 1.332 1.434 1.53! 1.31621 1.413 1.543 1.672 1.41972 1.556 1.736 1.911 1. 50824 1.679 1.907 2.139 1.58676 1.790 2.060 2.329 1.72383 1.985 2.333 2.677 1.84304 2.159 2.577 2.984 1.97543 2.354 2.849 3.329 2.09480 2.532 3.098 3.640 2.20457 2.698 3.325 3.924 2.30687 2.848 3.541 4.191 2.49446 3.129 3.934 4.678 2.66499 3.384 4.288 5.124 2.97034 3.845 4.920 5.909 3.24219 4.249 5.480 6.600 3.49018 4.616 5.987 7.226 3.93524 5.286 6.896 8.338 4.33178 5.870 7.700 9.318
Z2 and Zj a r e g i v e n b y
z2
pas*-band V S W s of theme demigns are t d u l a t e d on p. 267.
IRE Tran.. PCV7T b e e Ref. 4 by L. Young)
FOUR-SECTICN T C ~ B Y S C ~ FQUARTER-WAVE F TRANSFCR~ERS* (For u = 2 . 0 , Zl = Z 2 = Z 3 = Z 4
=A)
&)
--
IMPEDANCI RATIO.
*
Z2
z1 FCR
BANDWIDTH. 0.0
0.2
0.4
0.6
0.8
i
"q 1.0
1.2
'1.4
1.6
1.8
1.00000 1.000 1.000 1.000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.03560 1.050 1.073 1.102 1.02662 1.01405 1.01440 1.01553 1.01761 1.02106 1.06576 1.093 1.137 1.193 1.04898 1.03866 1.03227 1.02842 1.02635 1.02570 1.131 1.194 1.277 1.09214 1,06838 1.05385 1.04488 1.03568 1.03659 1.03949 1.165 1.247 1.354 1.04444 1.04558 1.04921 1.05598 1.06726 1.08559 1.11571 1.226 1.342 1.495 1.15681 1.11531 1.09026 1.07494 1.06577 1.05933 1.06088 1.280 1.426 1.622 1.07176 1.07364 1.07963 1.09086 1.10967 1.14059 1.19218 1.371 1.574 1.847 1.25182 1.18259 1.14159 1.11685 1.10216 1.09190 1.09435 1.703 2.045 1.450 1.30184 1.21721 1.10801 1.11093 1.12026 1.13784 1.16759 1.520 1.820 2.225 1.34555 1.24702 1.18974 1.15559 1.13549 1.12486 1.12153 1.642 2.028 2.545 1.14356 1.14758 1.16043 1. 18482 1.22654 1.29722 1.42054 1.749 2.213 2.828 1.16129 1.16588 1.18060 1.20863 1.25683 1.33920 1.48458 1.869 2.420 3.146 1.17961 1.18483 1.20156 1.23353 1.28883 1.38421 1.55461 1.977 2.609 3.433 1.19506 1.20082 1.21931 1.25475 1.31638 1.42350 1.61690 2.784 3.699 2.077 1.67357 1.45869 1.34074 1.27335 1.23478 1.20847 1.21471 2.170 2.948 3.946 1.22035 1.22703 1.24854 1.28998 1.36269 1.49074 1.72593 2.342 3.249 4.399 1.24078 1.24824 1.27232 1.31891 1.40125 1.54791 1.82099 2.498 3.524 4.809 1.90654 1.59831 1.43467 1.25803 1.26618 1.29251 1.34367 2.780 4.015 5.538 1.28632 1.29564 1.32587 1.38498 1.49127 1.68552 2.05820 3.031 4.451 6.182 1.30920 1.31953 1.35308 1.41905 1.53879 1.76055 2.19214 4.848 6.765 1.32853 1.33974 1.37624 1.44833 1.58022 1.82732 2.31378 3.261 5.556 7.801 3.674 2.53156 1.94412 1.65091 1.49736 1.41455 1.36025 1.37297 4.043 6.183 8.715 1.38591 1.39992 1.44587 1.53798 1.71073 2.04579 2.72559
Sac F o o t n o t e , T a b l e 6 . 0 4 - 4
SOURCE: IRE Trans. P G W ( s e e Ref. 4 by L. Young)
The c h a r a c t e r i s t i c i m p e d a n c e s , Z , , a r e o b t a i n e d - f r s t r r - t h e j u n c t i o n t r a n s f o r m e r and VSWRs, V,, u s i n g F i g . 6 . 0 2 - 1 f o r t h e q u a/r' t e r - w a v e F i g . 6 . 0 3 - 1 f o r t h e h a l f - w a v e f i l t e r . I t 1s c o n v e n i e n t t o n o r m a l i z e
...
given w i t h r e s p e c t t o Z O , a n d a s a r e s u l t , t h e v a l u e s o f Z1, Z 2 , i n t h e t a b l e s a r e f o r Z o = 1. The t a b l e s g i v i n g t h e Z, a l l r e f e r t o q u a r t e r - w a v e t r a n s f o r m e r s . To o b t a i n t h e Z: o f h a l f - w a v e f i l t e r s ,
T a b l e 6.04-4
z,
FOR FOUR-SECTION TCHEBYSCHEFF QUARTER -WAVE TRANSFORMERS* (FOP
u p = 2.0,
z1 = z2 = z3 = z4 = 6 )
IMPEDANCE RATIO,
1.00 1.25 1.50 1.75 2.00 2.50 3.00 4.00 5.00 6.00 8.00 10.00 12.50 15.00 17.50 20.00 25.00 30.00 40.00 50.00 60.00 80.00 100.00
*
BANDWIDTH, r q 0.0
R
1.00000 1.07223 1.13512 1.19120 1.24206 1.33204 1.41051 1.54417 1.65686 1.75529 1.92323 2.06509 2.21803 2.35186 2.47169 2.58072 2.77447 2.94423 3.23492 3.48136 3.69752 4.06810 4.38263
0.2
0.4
0.6
1.00000 1.00000 1.00000 1.07260 1.07371 1.07559 1.13584 1.13799 1.14162 1. I9224 1.19537 1.20065 1.24340 1.24745 1.25431 1.33396 1.33974 1.34954 1.41296 1.42036 1.43290 1.54760 1.55795 1.57553 1.66118 1.67423 1.69642 1.76043 1.77600 1.80248 1.92990 1.95009 1.98446 2.07315 2.09756 2.13915 2.22770 2.25698 2.30691 2.36303 2.39686 2.45455 2.48426 2.52237 2.58739 2.59463 2.63681 2.70880 2.79089 2.84069 2.92575 2.96299 3.01989 3.11712 3.25798 3.32792 3.44754 3.50835 3.59021 3.73029 3.72816 3.82111 3.98025 4.10544 4.21877 4.41293 4.42610 4.55802 4.78420
Z1 i s g i v e n i n Table 6.04-3.
0.8
1.0
1.2
1.4
1.00000 1.00000 1.00000 1.000 1.07830 1.08195 1.08683 1.093 1.14685 1. I5394 1.16342 1.176 1.20827 1.21861 1.23248 1.251 1.26420 1.27764 1.29572 1.320 1.36370 1.38300 1.40907 1.445 1.45105 1.47583 1.50943 1.556 1.60102 1.63596 1.68360 1.750 1.72864 1.77292 1.83358 1.918 1.84098 1.89401 1.96694 2.069 2.03453 2.10376 2.19954 2.335 2.19984 2.28397 2.40096 2.568 2.37988 2.48134 2.62317 2.826 2.53898 2.65667 2.82190 3.059 2.68264 2.81570 3.00321 3.273 2.81433 2.96208 3.17095 3.472 3.05065 3.22609 3.47548 3.836 3.26008 3.46148 3.74905 4.165 3.62377 3.87328 4.23198 4.750 3.93704 4.23091 4.65555 5.266 4.21547 4.55096 5.03760 5.734 4.7063 5.11329 5.71502 6.568 5.12003 5.60394 6.31175 7.304
Z3 and Z4 a r e g i v e n by
1.6
1.8
1.000 1.102 1.193 1.277 1.354 1.494 1.620 1.842 2.037 2.212 2.524 2.798 3.105 3.383 3.639 3.878 4.315 4.711 5.415 6.038 6.601 7.603 8.487
1.000 1.112 1.214 1.307 1.393 1.551 1.694 1.947 2.170 2.371 2.730 3.046 3.399 3.719 4.014 4.288 4.789 5.243 6.049 6.759 7.401 8.543 9.548
o b t a i n t h e V , f r o m F i g . 6 . 0 2 - 1 , and u s e t h e s e V i t o o b t a i n t h e Z: T h i s g i v e s t h e h a l f - w a v e f i l t e r w i t h t h e same from F i g . 6.03-1. but a t t e n u a t i o n c h a r a c t e r i s t i c s a s t h e quarter-wave transformer, h a v i n g a b a n d w i d t h w h = % w q . (Compare F i g s . 6 . 0 2 - 2 a n d 6 . 0 3 - 2 . ) The s o l u t i o n s of Eqs. ( 6 . 0 4 - 1 )
t o (6.04-6) f o r l a r g e r v a l u e s o f
R a r e p r e s e n t e d i n t h e second s e t of t a b l e s ( T a b l e s 6.04-5 t o 6.04-8). a n d 4. The r e m a i n i n g They g i v e t h e v a l u e s o f V 2 a n d V3 f o r n = 2, 3 , v a l u e s o f V a r e o b t a i n e d f r o m Eq. ( 6 . 0 4 - 8 ) a n d
which, f o r even n, r e d u c e s t o
and f o r o d d n , r e d u c e s t o
Equations (6.04-7) t o (6.04-12) hold f o r a l l v a l u e s of n
.
T a b l e s 6 . 0 4 - 5 t o 6 . 0 4 - 8 g i v e t h e s t e p VSWRs f o r R f r o m 1 0 t o i n m u l t i p l e s o f 10. N o t e t h a t f o r T c h e b y s c h e f f t r a n s f o r m e r s V z , V,,
. ..
Vn a n d v ~ / ( R ) %= V n + l / ( ~ ) %t e n d t o w a r d f i n i t e l i m i t s a s R t e n d s t o w a r d
The maximum pas.-band SOURCE:
VSIVR's
of these designs are tabulated on 267.
IRE Trans. P W V (nee Ref. 4 by L. Young)
i n f i n i t y , a s c a n b e s e e n f r o m E q s . 6 . 0 4 - 1 t o 6 . 0 4 - 6 f o r n up t o 4 , by (For l i m i t i n g v a l u e s a s R t e n d s l e t t i n g R t e n d toward i n f i n i t y . The t a b l e s g i v e f r a c t i o n a l toward i n f i n i t y and n > 4 , s e e S e c . 6.10.) he g r e a t e s t bandwidths, w q , from 0 t o 2 . 0 0 i n s t e p s of 0.20. p o s s i b l e bandwidth i s Eq. ( 6 . 0 2 - 1 ) .
1
wq =
2 . 0 0 , by d e f i n i t i o n , a s c a n b e s e e n f r o m
Table 6.04-5 V2 FOR TWO-SECTION QUARTER-WAVE TRANSMRMERS
h)
-
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1
2
3
4
5
6
7
8
9
3.1622 3.1070 2.9446 2.6872 2.3592 2.0000 1.6569 1.3708 1.1635 1.0405 1.0000
10.0000 9.4056 7.8214 5.8168 4.0155 2.7937 1.9939 1.5000 1.2055 1.0491 1.0000
31.6227 26.0349 15.2942 8.1357 4.6882 2.9763 2.0491 1.5172 1.2105 1.0500 1.0000
100.0000 55.6931 19.2101 8.6395 4.7783 2.9976 2.0550 1.5190 1.2110 1.0501 1.0000
316.2278 76.0577 19.8657 8.6971 4.7878 2.9997 2.0556 1.5192 1.2111 1.0502 1.0000
0
0
80.2075 19.9363 8.7030 4.7887 3.0000 2.0557 1.5192 1.2111 1.0502 1.0000
80.6742 19.9434 8.7036 4.7888 3.0000 2.0557 1.5192 1.2111 1.0502 1.0000
80.7215 19.9441 8.7037 4.7889 3.0000 2.0557 1.5192 1.2111 1.0502 1.0000
80.7263 19.9442 8.7037 4.7889 3.0000 2.0557 1.5192 1.2111 1.0502 1.0000
80.7267 19.9443 8.7037 4.7889 3.0000 2.0557 1.5192 1.2111 1.0502 1.0000
10
m m
80.7267 19.9443 8.7037 4.7889 3.0000 2.0557 1.5192 1.2111 1.0502 1.0000
0
0
h a r t e r l y Progrea. Report 4. G n t r a c t M i6-039 SC-87398, SRI; reprinted i n I R E Trans. PCY17. (Bee Ref. 36 by L. Yomug)
Table 6.04-6 V2 FOR 'IHREE-SECTION QUARTER-WAVE TRANSFORMERS
Table 6.04-7 V2 FOR FOUR-SECTION QUARTER-WAVE TRANSFORMERS
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1
2
1.7782 1.7781 1.7766 1.7698 1.7503 1.7054 1.6172 1.4676 1.2645 1.0768 1.0000
3.1622 3.1616 3.1524 3.1107 2.9928 2.7392 2.3157 1.8065 1.3725 1.0956 1.0000
m:Quarterly Progress
3
5.6234 5.6200 5.5677 5.3364 4.7320 3.7126 2.6617 1.8903 1.3886 1.0978 1.0000
4
5
6
10.0000 9.9808 9.6904 8.4944 6.1880 4.0665 2.7224 1.9004 1.3904 1.9080 1.0000
17.7827 17.6752 16.1089 11.2447 6.6331 4.1171 2.7290 1.9014 1.3905 1.0980 1.0000
31.6227 31.0220 23.4997 12.1344 6.6915 4.1225 2.7297 1.9016 1.3906 1.0980 1.0000
7
56.2341 52.9321 27.3837 12.2543 6.6975 4.1230 2.7298 1.9016 1.3906 1.0980 1.0000
a
9
10
m
100.0000 82.9105 28.0996 12.2667 6.6981 4.1230 2.7298 1.9016 1.3906 1.0980 1.0000
177.8279 106.5498 28.1787 12.2680 6.6982 4.1231 2.7298 1.9016 1.3906 1.0980 1.0000
316.2277 113.2186 28.1867 12.2681 6.6982 4.1231 2.7298 1.9016 1.3906 1.0980 1.0000
114.1604 28.1877 12.2682 6.6982 4.1231 2.7298 1.9016 1.3906 1.0980 1.0000
Report 4, Contract M 36-039 SC-87398. SRI; reprinted i n I R E Trona. PGKIT (nee Ref. 36 by L. Young)
Table 6.04-8 V 3 FOR FOUR-SECTION QUARTER-WAVE TRANSFORMERS
SOW:
2
3
4
5
6
7
8
5.2063 5.1045 4.8133 4.3689 3.8146 3.1842 2.5101 1.8739 1.3880 1.0967 1.0000
10.7833 10.3536 9.1941 7.698 5.8599 4.1904 2.8558 1.9587 1.4045 1.0989 1.0000
21.1024 19.5265 15.6771 11.1871 7.2224 4.5121 2.9136 1.9688 1.4062 1.0991 1.0000
39.7366 34.4781 23.5245 13.5788 7.5865 4.5564 2.9199 1.9699 1.4064 1.0992 1.0000
73.0562 56.7950 30.2006 14.2420 7.6326 4.5610 2.9205 1.9700 1.4064 1.0992 1.0000
132.4190 85.7685 33.0054 14.3279 7.6374 4.5615 2.9206 1.9700 1.4064 1.0992 1.0000
238.0478 114.9996 33.4806 14.3367 7.6379 4.5615 2.9206 1.9700 1.4064 1.0992 1.0000
Quarterly Progress Report 4, Contract M 36-039 SC 87398, SRI; reprinted i n I R E Trans. PCYIT (ace Ref. 36 by 1. Young)
m
When i n t e r p o l a t i n g , i t i s g e n e r a l l y s u f f i c i e n t t o u s e o n l y t h e two n e a r e s t v a l u e s o f V o r Z. I n that case, a linear interpolation on a l o g V o r l o g Z a g a i n s t l o g R s c a l e i s p r e f e r a b l e . Such i n t e r p o l a t i o n s , u s i n g o n l y f i r s t d i f f e r e n c e s , a r e most a c c u r a t e f o r
log R
--. .--
.
-
Interpolating linearly,
s m a l l R a n d f o r l a r g e R, a n d a r e l e a s t a c c u r a t e i n t h e n e i g h b o r h o o d
I n t h i s r e g i o n , s e c o n d - o r h i g h e r - o r d e r d i f f e r e n c e s may b e u s e d ( o r a g r a p h i c a l i n t e r p o l a t i o n may b e more c o n v e n i