Control of Electrical Drives 3rd Ed

ia ie UF

s

iSd, ZSq

V

m/s

ZS

iR iRd, iRq

Zm i mR ZmS

m3 J =Ws

1/f2 = S .rl W /1I1;~ "(;

rad rad rad

oe K A H/m

rad rad Wb = Vs Wb = Vs rad/s

41ndices

rad/s m

S

xv

mL mM mp Sp

armature current exciting current field voltage stator current rotor current direct and quadrature components of stator current direct and quadrature components of rotor current magnetising current magnetising current representing rotor flux magnetising current representing stator flux load torque motor torque pull-out torque pull-out slip

xV I

Abbreviations and Symbols

Abbreviations and Symbols

Y1 ----171

., (;('aphical Symbols

4 kí~

T~~=y

_~

v

~

T

-

4l~::í~

dx

~

~k~

= G (T !EL ~

+ y)

T~~+x=Gy

PI controller

first order lag

~8t!j~

~~I~) ~ ...............

Tl~~+X=G(T2~+Y)

x

41 t--:

YI

x(t) = G y (t - T)

delay

­

= Yt!Y2

division

x

= f(y)

nonlinearity

x

=y

= x =

X

Xmin X max

for Xmin < y < for y ~ Xmin for y ~ X max

X max

limiter

x = Yl - Y2

summing point

x = Yl Y2

multiplication

•

~i lu

X

..

.

---+I-~r---

X

--_.~

x

first order lead/lag

-------..~

~O

•

xmin

G r;; r,

YI

X

integrator

Y

li

XVII

II ~I +U

A/D- or D/A­ converter

current sensor

volt age sensor

x V III R

Introduction

L

(/)1~

m

[!fi L% ~ dt

u Ri

1

!e

U

= R 2· + L dtdi + e = 'fi! ­

'fi2

T

Ir? o

'1'1 t( ~ volt age arrowS indicating volt age sources

(u, e) or volt age drops (R i, L ~) 1('pl(~Sent the differences of electrical potential, pointing from the higher to I. h( ~ lower assumed potential. Hence the volt ages in any closed mesh have zero

:: 11111 ,

L: u =

O.

Energy is the basis of any technical and industrial development. As long as only human and animal labour is available, a main prerequisite for social progress and general welfare is lacking. The energy consumption per capita in a country is thus an indicator of its state of technical development, ex­ hibiting differences of more than two orders of magnitude between highly industrialised and not yet developed countries. ln its primary form, energy is widely distributed (fossil and nuclear fuels, hydro and tidal energy, solar and wind energy, geothermal energy etc.), but it must be developed and made available at the point of consumption in suitable form, for instance chemical, mechanical or thermal, and at an acceptable cost. This creates problems of transporting the energy from the place of origin to the point of demand and of converting it into its final physical form o ln many cases, these problems are best solved with an electrical intermediate stage, Fig. 0.1, where the bold numbers refer to the European grid, because electricity can be

pri7;{J _ . ;ower

------L T;ansmiSSiOntPOWer electronics+consume~ Final I

ene~tat,on T,stobuflon

Fuelcel/s Solar (PV) 46% Fossil--..... 84% 100% "~ 38% Nuclear_ Thermal _ Mechanical _Elektrical Solar --- -~ 'o,Uo=con Biomass' Water, Wind, Waves

/

Electrical drives Control/ad electrical árives

f-

'1' UI = variable ;.>, .

< 16%,« 1% UCTE- European Grid > 350 GW, 1 800 TWh/a

i

Elsctrical ....... Mechênical >50%

Electrical energy per capita and year world wide: 0.02 - 28 MWh

_

Mechanical

Electrical _Thermal _Chemical

Fig. 0.1. From primary energy to final use, a chain of conversion processes

~

Introduction Introduction

• ~enerated from primary energy (chemical energy in fossil fuel, potential hydro energy, nuclear energy) in relatively efficient generating stations, • transported with low los ses over long distances and distributed simply and at acceptable cost, • converted into any final form at the point of destination. 'I'his ftexibility is unmatched by any other form of energy. Of particular importance is the mechanical form of energy which is needed i Il widely varying power ratings wherever physical activities take place, involv­ ill~ the transportation of goods and people or industrial production processes. I,'or this final conversion at the point of utilisation, electromechanical devices i II the form of electrical drives are well suitedj it is estimated, that about half I.he clectricity generated in an industrial country is eventually converted to Ill( ~ chanical energy. Most electrical motors are used in constant- speed drives I.hat do not need to be controlled except for starting, stopping or protection, 1)11 t there is a smaller portion, where torque and speed must be matched to I.he need of the mechanical loadj this is the topic of this book. Due to the progress of automation and with a view to energy conservation, the need for col1trol is likely to become more important in future. As an example, Fig. 0.2 ~h()ws the mechanical power needed by a centrifugal pump, when the flow is couLrolled by a variable speed drive or, still in frequent use, by throttle and !).\' pass valves.

Pm Po 1,0

l

Bypass ~t_c~,,!~ary_se,,-e~ _ _ _ __

l 'rt<í '-~---, 3' I

( )

Centrifugai pump

Bypass

• Electrical drives can be designed to operate indefinitely in alI four quad­ rants of the torque-speed-plane without requiring a special reversing gear, Fig. 0.3. During braking, i.e. when operating in quadrants 2 or 4, the drive is normally regenerating, feeding power back to the line. A comparison with combustion engines or turbines makes this feature look particularly attracti ve.

(~ J Motor)

mM

) ) () Load)

_

mL

Throtlle

"I

• Electric drives are adaptable to almost any operating conditions such as forced air ventilation or totally enclosed, submerged in liquids, exposed to explosive or radioactive environments. Since electric motors do not require hazardous fuels and do not emit exhaust fumes, electrical drives have no detrimental effect on their immediate environment. The noise leveI is low compared, for instance, with combustion engines. • Electric drives are operable at a moment's notice and can be fully loaded immediately. There is no need to refuel, nor warm-up the motor. The ser­ vice requirements are very modest, as compared with other drives. • Electrical motors have low no-Ioad losses and exhibit high efficiencYj they normally have a considerable short-time overload capacity. • Electrical drives are easily controllable. The steady state characteristics can be reshaped almost at will, so that railway traction motors do not require speed-changing gears. High dynamic performance is achieved by electronic controI.

W,

mecham"cal input power

mM mL

Thro",e

= Motor torque

=Load torque

at constant speed V-úJ

Load

variabfe speed

, O

l-.. ­

Flow

1,0

00

Fig. 0.2. Mechanical input power to a centrifugai pump IIKillg difrcrent methods of fiow controi

tM - mM

'J'hc predomil1ance of electrical drives is caused by several aspects: • I':kdric drlv( ~s aT<' available for any power, frOlIl 10° W in electronic wnt.(·II('~i 1.0 I ()H W for drivill ~ plllllpS iII hydro hl.orHI~t' lílnllt.f!, • 'I'1L< ~'y e nV I ' .' tL wid,' rI,W I',C' or I.orqlll' 11.11(1 " p"( 'd, . IO'f NIII , ror il,\I ore IHill lIud,"I' , • 10)' ' /I l1itr , C m l i (·(·l1l.riflll ( · driv(· .

3

Fil';. ().:\. 0l" ~ ral.illg ma d es of nll d('ctric drive: iII 11 11 'f"il,c1 rr,"f.,; (lI' 1.11<' l.o rqllc:j" p c't· c1 plallCi

1

Introduction

Introduction

• The rotational symmetry of electrical machines and (with most motors) the smooth torque results in quiet operation with little vibrations. Since there are no elevated temperatures causing material fatigue, long operating life can be expected. • El ectrical motors are built in a variety of designs to make them compatible with the load; they may be foot- or fiange-mounted, or the motor may have an outer rotor etc. Machine-tools which formerly had a single drive shaft and complicated mechanical internal gearing can now be driven by a multitude of individually controlled motors producing the mechanical power exactly where, when and in what form it is needed. This has removed constraints from machine tool designers . III special cases, such as machine-tools or the propulsion of tracked vehicles, linear electric drives are also available.

Signallevel

I Power levei

-+­ I

Voltages, currents

I

~

Torque, acceleration, speed, angular position etc

Signal acquisition:

Contrai: Coordination: Identification: Self tuning: A daptation: Optimisation:

A

Reconstruction of unmeasurable quantities, Elimination of sensors Oecoupling, feed - forward, limiting Coordinate transformation Estimation of variable parameters Computer aided commissioning Adjustment of contrai parameters Minimisation of objective functions

.c .-=:

~

Fig. 0.4. Digital control structure of an electrical drive

1\ s wo u lei Iw IH ' :-; lIppkllwllt, (~d

( ~x p c cted ,

this long list of remarkalJle characteristics is to by Jisa.c1va.ntages of electric drives which lirnit or preclude

U1I·ir 11 SC:

• 'I'1!t'. dqH'lIdHII("( '

011 it COJlLiIlIIOll:-l POW('I'

(' II' !II'"p"I: :i
I...

' ·III' r l , ·.\. ""

UI'

sllpply

C'ill lllPS [)I·.. hl n lll !-J

(,id,I'llilry lli lI,vall n l lk ,

/l ll

witll vdti..

I ·II'r(,rir 1:1U :I'J-( Y

1'''.'\1'<1, whi<-h ln ll il il rdl,Y IlItlllY. I .... " vy

n llel

" XIH'Il ­

ntv" ( 11 1.0' " 1"'" II,n U.' " y, I
5

or turbine, fuel- or solar cells). The lack of a suitable storage battery has so far prevented the wide-spread use of electric vehicles. The weight of a present day lead-acid battery is about 50 times that of a liquid fuel tank storing equal energy, even when taking the low efficiency of the combustion engine into account. • Due to the magnetic saturation of iron and cooling problems, electric mo­ tors are likely to have a lower power-to--weight ratio than, for instance, high pressure hydraulic drives that utilise normal instead of tangential forces. This is of importance with servo drives on-board vehicles, e.g. for position­ ing the control surfaces of aircraft. The electromechanical energy conversion in controlled drives is subject to the terminal quantities of the electrical machine which can be changed with low losses by controllable power electronic converters consisting of semicon­ ductor switches; they can produce voltages and currents of almost any wave­ form as prescribed by control which today is executed by microelectronic components. Thus semiconductor technology, combining power conversion and high speed signal processing at reasonable cost, has been the essential force behind the development of todays high performance drives; this is part of the general transition from analogue to digital control systems using mi­ crocomputers and signalprocessors. Fig. 0.4 gives an impression of how the mechanical, power electronic and control functions, combining hardware and software, are interleaved in a modern drive system.

1. Some Elementary Principies

of Mechanics

Since electrical drives are linking mechanical and electrical engineering, let us recall some basic laws of mechanics.

1.1 Newtons Law A mass M is assumed, moving on a straight horizontal track in the direction of the s-axis, Fig. 1.1 a. Let IM(t) be the driving force of the motor in the direction of the velocity v and h(t) the load force opposing the motion, then Newtons law holds d (Mv) dt

1M - h = -

=M

dv dt

dM

+ vdt' -

(1.1)

where M v is the mechanical momentum. Usually the forces are dependent on velocity v and position s, such as gravitational or frictional forces. li the mass is constant, M = Mo = const., Eq. (1.1) is simplified,

1M - h

dv

(1.2)

= Mo dt ;

with the definition of velo city v = ds I dt, this results in a second order differ­ ential equation for the displacement,

tM

M

(f(

m,mM

s

J

&

V /:1 I"il-';. J .1 . 'I'n.. ,.. hd.;oll .d Jl llcl "" ULI.;oll a l

b 11101.;011,

"r

IaIlIlP"cI llm..;S(' ~

1. Elementary PrincipIes of Mechanics

~l

fM

!L

-

d2 s = Mo dt 2

1.2 Moment of Inertia

(1.3)

'

"'I .(( \) ..

wh( ~ r e

d2 s dt

dv dt

(1.4)

a = -= 2

is I.he acceleration. If the motion is rotational, which is usualIy the case with ('h:cLrical drives, there are analogous equations, Fig. 1.1 b, d 1I/.M - mL = dt (Jw)

=J

dw dt

dJ

+ w di '

(1.5)

wil.h mM being the driving- and mL the load torque. w = 21l" n is the angular vdocity, in the folIowing called speed. J is the moment of inertia of the 1'( d.:d:iIlg mass about the axis of rotation, J w is the angular momentum. The 1,'1'111 w (dJ/ dt) is of significance with variable inertia drives such as centrifuges 01' n :eling drives, where the geometry of the load depends on speed or time, or illdllst rial robots with changing geometry. ln most cases however, the inertia C; lJI ue assumed to be constant, J = J o = const., hence dw

mM -

mL = Jo di

.

(1.6)

Wit.h E the angle of rotation and w = dE / dt the angular velocity, we have mL = Jo dt 2

(1.7)

'

(ft

i:: Ih( '

d2E dt 2

(1.8)

'

;\.I1i ~ \Ilar

acceleration. II. ~; II()\Ild be noted that mM is the internal or electrical motor torque, not id( 'llli cal with the torque available at the motor shaft. The difference between i111"rllal torque and shaft torque is the torque required for accelerating the il ... rlia of Lhe motor itself and overcoming the internal friction torque of the III< 1101'. 'I'ra nslational and rotational motions are often combined, for example in v(:hich~ propulsion, elevator- or rolling mill-drives. Fig. 1.2 shows a mechanical II1 0dd , where a constant mass M is moved with a rope and pulIey; when 1I<'I',kct.iIlg the mass of the pulley and with '/lI' M \V('

= '/' fM ,

und

mL = r fL

v

= rw

tM

vr

M

- ---=--. (

M

' úJ

-"'r

---. v

Fig. 1.2. Linking linear and rotational motion

1.2 Moment or Inertia The moment of inertia, introduced in the preceding section, may be derived as follows: A rigid body of arbitrary shape, having the mass M, rotates freely about a vertical axis orientated in the direction of gravity, Fig. 1.3. An element of the mass dM is accelerated in tangential direction by the force element dfa, which corresponds to an element dma of the accelerating torque dm = a

l'

d+ =

l'

~a

dv dw dM - = '1'2 dM - . dt dt

The total accelerating torque follows by integration ma

=

wh('n~

(lw

G

/

mL '~r

M

j dma

=j

O

(r

L

t

ma

d2E

'II/.M -

L

/

'1'2

~ dM .

(1.10)

O

Due to the assumed rigidity of the body, alI its mass elements move with the sarne angular velocity; hence M

= dWj 2dM=Jdw ma dt r dt .

(LU)

o The moment of inertia, referred to the axis of rotation, M

J =

j

r

2

dM

(1.12)

o

~ dm . a

I

Jiud wilh M = consto 'II/.M

1/I. J.

'/'

a di,

(M'II) = M

z dw

l' -

di.

'2

(1.9)

.

1\'/ .,.:' 1'1'111'1': :"111.: : til,' ('qllivilklll. 1110111('111. "r illl'l'lin, 01' I.h,' liI)(:arly UIO V illl.~ I l. x i ll (Ir U", plIlI, 'y. "I'P'""III.I V 1.1... 111:1 :::: f\;1 ( 'a,1I II<: 1.1I<1I1J ·,. hl. "I /i') 1,,,1.1( ' ,[ j,iI,! 11 1111., '" " I"" r~ 1.11" , ll'('lIlId"" '1I1 "I"!.II<' \\,11<'1,1 :, (I

p

'1

./,.

rtllI )I::, l',·t'<-n:,'d 1." 1,1...

9

Fig. 1.3. M II IIII·/I1.

"r ill.... l.ia

Fig. 1.4. MCl\wlni of inertia of (·. 'ylilHlcr

cOIlc<·"Ll'i('

1. Elementary PrincipIes of Mechanics

lO

1.3 Effect Df Gearing

i~

a three-dimensional integral. III many cases the rotating body possesses rotational symmetry; as an (')(;tlllple, consider the hollow homogeneous cylinder with mass density º,Fig. I ,ti, As volume increment dV we define a thin concentric cylinder having the r;tdius r and the thickness dr; its mass is

º dV = º 2 1f r l dr .

dM =

M

.l c

f

f

r dM

= º 21f i

3

r dr =

%º i (r~ -

(1.13)

rt) .

G = ºgl1f(r~ -

ri)

(1.14)

l"('slIlts in

r~ +d G 2 9G --2= 9 ri ,

(1.15)

is the gravitational acceleration. The quadratic mean of the radii 1

I ;;

c:t1I, ~ d

7 a

2

dr

+

o

2 r dr]

o

a) 2]

M 12 [1+3 ( 1-2y = 12

(1.17)

.

The minimum inertia is obtained, when the rod is pivoted at the centre, Fig. 1.5 b.

TI

IlIl.roclucing the weight of the cylinder

'ri

~ [J r

r dM =

o

1l"!lce the moment of inertia increases with the 4th power of the outer radius.

wl!('rc.ll

1

2

r2

2

o

J =

Another example is seen in Fig. 1.5 a, where a homogeneous thin rod of length i and mass M is pivoted around a point P, the distance of which from one end of the rod is a. With the mass element dM = (M/i) dr we find for the moment of inertia

1 =

'I'bis redu ces the volume integral to a simple integration along the radius,

11

2 (ri + r~)

(1.16)

1.3 Effect of Gearing Many applications of electrical drives call for relatively slow motion and high torque, for instance in traction or when positioning robots. Since the tangen­ tial force per rotor surface, i.e. the specific torque of the motor, is limited to some N/c:m 2 by iron saturation and heat los ses in the conductors, the direct coupling of a low speed motor with the load may result in an unnecessarily large motor. It is then often preferable to employ gears, operating the motor at a higher speed and thus increasing its power density. This also affects the inertia of the coupled rotating masses. ln Fig. 1.6 an ideal gear is shown, where two wheels are engaged at the point P without friction, backlash or slip. From Newtons law it follows for the left hand wheel, assumed to be the driving wheel,

Lhe radius of gyration; it defines the radius of a thin concentric

"ylill
J

mM1 -

rI

fI

dWl

= li dt '

(1.18)

where iI is the circumferential contact force exerted by wheel 2. lf there is no load torque applied we have, correspondingly, for wheel 2

r2 fz = lz dw2 dt .

Ml2 3

(1.19)

h is the force driving wheel 2. Since the forces at the point of contact are in balance and the two wheels move synchronously,

Ml2 12

h=h, ( ~ Iimination

O b

[I

1,'1,,;. I.r., M"I","I,

"r \ '"' 1' 1. 1"

"I' (\

,."eI, piv",-,'d

_J~

2 0111. ,,(" , " '111.,-('

of h,

1 ~mMl

rI Wl

=

= r2 W2

(1.20)

,

h, W2 results in

dW l dt

.lI -

tlWI

.II,' dI.

ri

dW2

+ r2 - 1 2 - dt =

[

II

rI

1dWI

2 lz + ( -) ­ 1'2 dt

(1.21 )

1. Elementary PrincipIes of Mechanics

12

1.4 Power and Energy

dw wmM=wmL+Jwdj'

v

13

(1.26)

where PM = w mM is the driving power, PL = w mL the load power and J w (dw / dt) the change of kinetic energy stored in the rotating masses.

~IIJII :

2r3

M3

F ig. 1.6. Effect of gearing on inertia

h + (::) 2 J 2

= J 1 + (::) 2 h + (::) 2

[J3

+ M3 r~l

'

mMl

= J 1e -d

t

W3

+ -Wl

r3g M 3

.

(1.24)

The rotational motion of the mechanical arrangement shown in Fig. 1.8 is dC!icrihcd by a first. order dilferential equation for speed Ii.w

w. /, I .I ti,

Mlll !. q,l ill d ,IIIII I,y

( 01

,vi.. ldn UI<' PClW(" 1.,11 " 11('('

)-)

mL ...... ....

PL

Fig. 1.8. Power flow of drive

By integrating Eq. (1.26) with the initial condition w (t the energy input t

WM(t) =

f f

t

o

o

= O we find

t

PLdT

+

o

f

Jw: dT

o

w

t

=

f + f

PMdT =

= O)

PL dT

J

n dn

o

(1.27)

(1.23)

1.4 Power and Energy

'III'M

J

1 2 =wL(t)+2"Jw.

illduding the equivalent inertia of the mass M3 being moved in vertical direc­ lion. Applying Newtons law, taking the load of the hoist into account, results iu dWl

5EJ

(1.22)

,

which indicates that a rotating part, moving at higher speed, contributes 1I1Ore strongly to the total moment of inertia. ln Fig. 1.7 an example of a multiple gear for a hoist drive is seen. J l , .J2 , Js are the moments of inertia of the different shafts. The total effective illertia referred to shaft 1 is

J 1e

'W

PM -­

Fig. 1. 7. Hoist drive with gear

J le is the moment of inertia effective at the axis of wheel 1; it contains a component reflected from wheel 2. ln most cases it is easier to determine the speed ratio rather than the radii, J le =

M

m

(1.25)

The last term represents the stored kinetic energy; it is analogous to 1 1 1 2 -Mv 2 - Li 2 -Cu 2 . 2 ' 2 of other energy storage devices. Since the energy content of a physical body cannot be changed instantaneously - this would require infinite power ­ the linear 01' rotational velo city of a body possessing mass must always be a continuous function of time. This is an important condition of continuity which will frequently be used. Because of the definitions ds de v= and w =­ dt dt the positional quantities s and € are also continuous functions of time, due to fiuite sp eed. This is also understandable from an energy point of view, since position lIlay be associ ated with potential energy, as seen in Fig. 1.7, where I.h, : Ill as ~; 1I{, i:-l ]>os il.iollf:(} vcrtically dC(l':lIdillf!: OH th,: angle of rotation of 1.11<' d ri Vl' sJlid'i. .

1. Elementary PrincipIes of Mechanics

14

1.5 Experimental Determination of Inertia

1.5 Experimental Determination of Inertia

ú)

The moment of inertia of a complex inhomogeneous body, such as the rotor of an electrical machine, containing iron, copper and insulating material with complicated shapes can in practice only be determined by approximation. The problem is even more difficult with mechanicalloads, the constructional details of which are normally unknown to the user. Sometimes the moment of inertia is not constant but changes periodically about a mean value, as in the case of a piston compressor with crankshaft and connecting rods. Therefore experimental tests are preferable; a very sim pIe one, called the run-out or coasting test, is described in the following. Its main advantage is that it can be conducted with the complete drive in place and operable, requiring no knowledge about details of the planto The accuracy obtainable is adequate for most applications. First the input power PM (w) of the drive under steady state conditions is measured at different angular velocities w and with the load, not contributing to the inertia, being disconnected. From Eq. (1.26),

PM

= PL + J w

dw dt '

(1.28)

the last term is omitted due to constant speed, so that the input power PM corresponds to the los ses including the remaining load, PM = PL. This power is corrected by subtracting loss components which are only present during power input, such as armature copper los ses in the motor. From this corrected power loss P~ the steady-state effective load torque m~ = p~jw is computed for different speedsj with graphical interpolation, this yields a curve m~ (w) as shown in Fig. 1.9. For the run-out test, the drive is now accelerated to some initial speed wo, where the drive power is switched off, so that the plant is decelerated by the loss torque with the speed measured as a function of time, w(t). Solving the equation of motion (1.25) for J results in '" -m~(w) J '" dw '

mM=O.

(1.29)

âf(w)

Rence the inertia can be determined from the slope of the coasting curve, as shown in Fig. 1.9. Graphical constructions, particularly when a differentiation is involved, are only of moderate accuracy. Therefore the inertia should be computed at different speeds in order to form an average. The accuracy requirements regarding inertia are modest; when designing a drive control system, an error of ± 10% iil IlsualJy Itcccptable without al1y ~criom; effect. Two spcc.j;d CiLS(': ; }PaI! to [>itrticlllarly sill1pl(~ iJlt.(:rprüt,ations: ii,) Ã n:i11 III i "I'.

1.11 (1 1" 01"1,,,(' 1,'.1 IOI'!Il t.OI·qlle' '111,'1. to !lo Il,pproxilJlat.dy (:OIlK(:i\.Ut.

Jilldl. \,d Iii>!"'" ,"I,,·, vld ,

15

ill n.

coasting curve

oo(t)

Corrected steady state load curve m'L (00)

m'

L

00,

, m'L (00 )

1

Fig. 1.9. Run - out test m~ ~ const,

for

Wj

< w < w2

,

then w(t) resembles a straight line; the inertia is determined from the slope of this line. b) li a section of the loss torque may be approximated by a straight line, m~ ~ a+bw,

for

Wj

< w < W2

.

a linear differential equation results,

dw

J dt +bw= -a.

The solution is, with

w(t) = -~

b

+(w

2

W(t2)

=

W2,

+~) e- b (t-t2)/J b

'

t

2:

t2 •

Plotting this curve on semi-Iogarithmic paper yields a straight line with the slope -bj J, from which an approximation of J is obtained.

2. Dynamics of a Mechanical Drive

2.1 Equations Describing the Motion of a Drive with Lumped Inertia The equations derived in Chapo 1

dw J dt = mM (w, dE: dt

é,

(2.1)

YM,t) - mL (w, é, YL,t) ,

(2.2)

= w,

describe the dynamic behaviour of a mechanical drive with constant inertia in steady state condition and during transients. Stiff coupling between the different parts of the drive is assumed so that all partial masses may be lumped into one common inertia. The equations are written as state equations for the continuous state variables w, é involving energy storage, e. g. [38,88]). Only mechanical transients are consideredj a more detailed description would have to take into account the electrical transients defined by additional state variables and differential equations. The sarne is true for the load torque mL which depends on dynamic effects in the load, such as a machine tool or an devator. AIso the control inputs YM, YL to the actuators on the motor and load side have to be included. Fig. 2.1 shows a block diagram, representing the iuteractions of the mechanical system in graphical formo The output variables of the two integrators are the continuous state variables, characterising the (~l1ergy state of the system at any instant. Linear transfer elements, such ;\.S integrators with fixed time constants, are depicted by blocks with single !"rames containing a figure of the step response. A block with double frame ,lCllominates a nonlinear function; if it represents an instantaneous, i. e. static, Ilonlinearity, its characteristic function is indicated. The nonlinear blocks i Il Fig. 2.1 may contain additional dynamic states described by differential ('qllations. Dcpenc\cIlce of the driving torque mM on the angle of rotation is a char­ ad(~ristic fCiI.tUfe of synchronous motors. However, the important quantity is II' d. Lhl' allgl!' of roCaLiol\ é itself but the difrc r('uc( ~ itllgle 8 against the no-load 1(( w ii ich is dd(~rIlJiu(~d hy 1.111' l.o('(pl(· . () ud('r 1.1«' loa.! 11111',1(, S iH (,oll:d.alll..

11.111,:1"

~lt.('ady statl'

condi I.io/l s

2.1 Equations Describing the Motion of a Drive with Lumped Inertia

2. Dynamics of a Mechanical Drive

li>

However, when the partial masses are coupled by flexible linkages, such as with mine hoists, where drive and cage are connected by the long winding rope or in the case of paper mill drives with many gears, drive shafts and large rotating masses particularly in the drying section, a more detailed description becomes necessary. The free oscillations may then have frequencies of a few Hertz which are well within the range of a fast controlloop. ln Fig. 2.2 an example is sketched, where the drive motor and the load having the moments of inertia J 1 , J 2 are coupled by a flexible shaft with the torsional stiffness K. The ends of the shaft, the mass of which is ignored, have the angles of rotation C1, C2 and the angular velocities W1, W2 . Assuming a linear torsionallaw for the coupling torque me,

Power

supp/y

YM

o

Motor

co

mM

mL

me Fig. 2.1. Simplified block diagram of lumped inertia drive

me

~1fl>, Motor

))K)(1 me

Cü2,E2

J2 Load

=K

(2.3)

(C1 - c2) ,

and neglecting internal friction effects, the following state equations result

ln order to gain a better insight, let us first assume that the electrical transients within the motor and the internalload transients decay consider­ ably faster than the mechanical transients of w and ci as a consequence, it r()l1ows that the motor and load torques mM, mL are algebraic, i.e. instan­ (,allcouS functions of w, c and 8. Hence, by neglecting the dynarnics of motor ;Lmlload, we arrive at a second order system that is completely described by lhe two state equations (2.1), (2.2). So [ar we have assumed that alI moving parts of the drive can be combined lo forrn a single effective inertia. However, for a more detailed analysis of dy­ n
/TIM , <Út,f,1

19

)))

mL

Jo'ilf . 2,2. I )riv!' l'1I1I:.i/<(, I.'I: 01' mot,or nlld 101\.1 '-,0111'1".1 1,.1 ""xij,k J;ltart.

dw 1 J 1 dt = mM - me = mM (Wt, C1, YM) - K (c1 - c2) ,

J2

dw

dt2 = me de 1

dt = W1, de 2

-=W2·

dt

mL

=K

(cl - c2) - mL

(W2,

c2, YL) ,

(2.4) (2.5) (2.6)

(2.7)

A graphical representation is seen in Fig. 2.3 a. Here too, the torques mL are in reality defined by additional differential equations and state vdriables. If only the speeds are of interest, the block diagram in Fig. 2.3 b "':1..)' be useful, which, containing three integrators, is described by a third lIJ'dc[' differential equation. With increasing stiffness of the shaft, the quantities C1, C2 and W1, W2 I"'collle tighter coupledi in the limit the case of lumped inertia emerges, w lU'r<: C1 = C2, W2 = W1. Ir Lhe transmission is affected by mechanical backlash, where the two "",rLias separate when the shaft torque changes sign, the linear torsional III " Il('h f{ is replaced by a nonlinear function, as shown in Fig. 2.3 c. This is III' i IlIportance with reversible drives, for instance servo drives. (lltviollSly the subdivision of the inertia may be continued indefinitely; o' Vo-J'y Lime a new partial inertia is separated, two more state variables have to 100 ' ,I, -Ii 11(:(1, t.ransforming Fig. 2.3 into a chainlike structure. A typical example, w l",I'" 1I1,tlly partial masses must be taken into account for calculating stress 111141 I'n.t.igw!, is a turbine rotor. III IV!,

20

2.2 Two Axes Drive in Polar Coordinates

2. Dynamics of a Mechanical Drive

where w = dE; / dt and v = dr / dt are the rotational and radial veloci ties . After separating the terms of acceleration in tangential and radial direction and superimposing frictional and gravitational components, Newtons law is ap­ plied in both directions, resulting in the equations for the mechanical motion of the centre point of M 2

Power supply

Power supp/y

21

fO

J r,,-_~A

(ll + h

,

dw

+M2r2) dt = Coriolis

Gravitation

,...-"'-...

,.-"'-----....

mM - 2M2 rwv-M2 gr cose:-mF -mL,

(2 .10)

de:

dt = w ,

b

a

dv M 2 dt

c~JP

dr dt = v .

F ig. 2.3. Block diagram of twin-inertia drive with flexible shaft (a) Model of mechanical plant, (b) Reduced order model,

c) Mechanical transmission with backlash

2.2 Two Axes Drive in Polar Coordinates On machine tools or robots there are normally several axes of motion, that rnust be independently driven or positioned. An example is seen in Fig. 2.4 a, where an arm, carrying a tool or workpiece , is rotated by an angle e:(t) around a horizontal axis. The radial distance r(t) from the axis to the cen­ ter of the mass M 2 represents a second degree of freedom, so that M2 can bc positioned in polar coordinates in a plane perpendicular to the axis. The rotary and radial motions are assumed to be driven by servo motors, produc­ ing a controlled driving torque mM and a driving force 1M through a rotary gear and a rotary to translational mechanical converter, for instance a lead screw. With fast current control the motors are generating nearly instanta­ \ICOIlS impressed torques, serving as control inputs to the mechanical planto For simplicity, the masses are assumed to be concentrated in th.e joints, re­ sltlt.ing in the inertias h, J 2 . The coupling terms of the motion can be derived by (~xpressillg t,he acceh~ration of the mass M 2 in complex formo dr

!l

dI

(li

,(J. )'

,11 ',\

(1' (..1. ' )

,II

. (1< i' ) ~

(2.8)

('II I :i r w) (.3 ~ ,

,, ~\

,

(/L'II d'

I_,.}

.~ ) ;I , .

. (

t' I (,

.\1 "

= 1M +

, '~~ (LI

'fi

I

'I'

IlldIU )

(' j ' '

I

(/..!J )

(2.11) C entrifugai

Gravitation

,-"--.. 2 2r w

,...-"'-...

M

-

M 2 g sine: - lF ­

h,

(2.12) (2.13)

The equations (2.10)-(2.13) are depicted in Fig. 2.4 b in the graphical form of a block diagram, containing four integrators for the state variables. Despite the simple mechanics, there are complicated interactions, which become more prominent with increasing rotary and radial velocities. The control of this mechanical structure is dealt with in a later chapter. Clearly, the two motions are nonlinearly coupled though gravitational, Coriolis- and centrifugaI effects; they are described by four nonlinear state equations. mF, lF and mL, h are due to friction and external load forces with may exhibit their own complicated geometric or dynamic dependencies, If it is important for the application to express the position of mass M 2 in cartesian coordinates, this is achieved by a polar-cartesian conversion x (t) = r cos e: ,

y(t)

=r

sine: .

(2.14)

(2.15)

Moving the arm also in the direction of the axis of rotation, so that the mass M 2 can be positioned in cylindrical coordinates, would introduce a third

dccoupled degree of freedom . The dynamic interactions for a general motion, involving six degrees of r)'(~edom (three for the position, three for the orientation of the tool) are ex­ ('cedilJgly complicated, they must be dealt with when controlling the motions 01' Jllult.i ax( ~s rouots with high dynamic performance [M53, 013, 849].

2.3 Steady State Characteristics of Motors and Loads

2. Dynamics of a Mechanical Drive

'22

variable is the load angle 0, i. e, the displacement of the shaft from its no­ load angular position. When the maximum torque is exceeded, the motor falls out of step; asynchronous operation of larger motors is not allowed for extended periods of time because of the high currents and pulsating torque, The electrical transients usually cannot be neglected with synchronous motor drives. The rigid speed of synchronous motors when supplied by a constant fre­ quency source makes them suitable for only a few applications, for exarnple large slow-speed drives for reciprocating compressors or synchronous gener­ ators operating as motors in pumped storage hydro power stationsj at the low end of the power scale are electromechanical clocks, The situation is dif­ ferent, when the synchronous motor is fed from a variable frequency inverter because then the speed of the drive can be varied freely (Chap. 14). With the progress of power electronics, these drives are now more widely used.

Friction

mLosd

r2

,

M2 V J +J2+ grcos E

M

23

gsinE

I rw 2

y/, r

fLoad

Friction

x

a

b Fig. 2.4. Two axes drive in polar coordinates (a) Mechanical plant (b) Block diagram

2.3 Steady State Characteristics of

Different Types of Motors and Loads

Consider first the steady state condition, when the torque and speed of a single axis lumped inertia drive are constant and the angle changes linearly with timej this condition is reached when mM - mL = O. With some motors, such as single phase induction motors, or loads, for example piston compres­ sors or punches, the torque is a periodic function of the angle of rotationj in this case, the steady state condition is reached, when the mean values of both torques are equal, mM - mL = O. The speed then contains periodic oscillations, which must be kept within limits by a sufficiently large inertia. The stcady state characteristics of a motor or 1030<1 are often functions given in graphical [o fi II , <:onnN:ting main varíilhl(~K, sllch as speed and torque; lh(' provisíou i:i 1.11;t\. :lllxili
li(.ld

(·II!"l"('IIt.,

Ht.iI.lIl..

I" 1' 11\ '). ,1, /1 ,1,11,1"" 1.,Ypi(,iI.I

mM Fig. 2.5. Steady state torque-speed characteristics of (li) electrical and (b) mechanical drives

The "asynchronous" or "shunt-typ" characteristic in Fig, 2.5 a is slightly clrooping; often there is also a pronounced maximum torque. The lower por­ Lioll of the curve is forbidden in steady state in view of the high losses. With I.hree- phase asynchronous motors the rotor angle has no effect on the torque i Il steady state. Motors with "series-type" characteristic show considerably larger speed drop under loadj with DC or AC commutator motors, this is achieved by ;1. suitable connection of the field winding. The main area of applications at larger ratings are traction drives because the curved characteristic resembling ;1. hyperbola facilitates load sharing on multi pie drives and permits nearly collst.ant power operation over a wide speed range without gear changej this is par\.icularly suited to a Diesel-electric or turbo- electric drive, where the 1',,11 !)()Wl'r o[ lhe thermal engine must be used. For ('olllparisoll, SOIlJC t:ypical charactcrist.ics of a Lurbine anel a Diesel ('I\/';i ll(' ai. ('ClIlHl.n,lll. flld illjc('.t.ioJl pc r Ht.I'III(' lU'" i'iC'('1\ iII l"il':. :~.S h.

25

2.3 Steady State Characteristics of Motors and Loads 2. Dynamics of a Mechanical Drive

21.

The curves in Fig. 2.5 a are "n~tural" characteristics which can be modi­ [ied at will by different control inputs, e. g. through the power supply. With dosed loop control a shunt motor could assume the behaviour of asynchronous O[ of a series type motor. As an example, typical steady state curves of a con­ l.rolled DC drive are shown in Fig. 2.6; they consist of a constant speed branch (normal operation) which is joined at both ends by constant torque sections aetivated under overload condition through current limito Figure 2.7 depicts Lhe steady state curves of the motor for driving a coiler. li the electrical power rcference is determined by the feed velo city v of the web or strip to be wound and includes the friction torque, the coiler operates with constant web force f independent of the radius r of the coil, PL = vi + PF· The steady state characteristics of mechanicalloads are of great variety; however, they are often composed of simple elements. This is seen in Fig. L.8 with the example of a hoist and a vehicle drive. The gravitational lift Lorque mL is independent of speed (Fig. 2.8 c); in the first quadrant the load is lifted, increasing its potential energy, hence the drive must operate in the Illotoring region. ln the fourth quadrant, the power flow is reversed with the load releasing some of its potential energy. Part of that power is flowing back Lo the line, the remainder is converted to heat losses. The lower half of the winding rope, seen in Fig. 2.8 a, serves to balance the torque caused by the

ú)

m = const

1 L __ _

ú)tl

r"

= r:onst m =const

Ú)2

Torque limit

mL = r 9 (MI

- M2 )

(J)

Gravitational torque

mL mL = r fL

= r9 M sin a

a

c

b

Fig. 2.8. Drives involving gravitational forces

weight of the rope; this effect can be substantial on a winder for a deep mine, tending to destabilise the drive. All mechanical motion is accompanied by frictional forces between the surfaces where relative motion exists. There are several types of friction, some of which are described in Fig. 2.9. ln bearings, gears, couplings and brakes one observes dry or Coulomb friction (a), which is nearly independent of speed; however one has to distinguish between sliding and sticking friction, the difference ofwhich may be considerable, depending at the roughness ofthe surfaces. The forces when cutting or milling material also contain Coulomb type friction. ln welllubricated bearings there is a component offrictional torque which rises proportionally with speed; it is due to laminar flow of the lubricant and is called viscous friction (b). At very low speed and without pressurised lubrication, Coulomb type friction again appears.

m Fi~. 2.6. Torque/speed curves of controlled motor

wiLlt constant speed branch and torque limit

a

ú)

/b ~

/./

ú)

/

i/ /

~

---,

/// /// /..,./

__ --:/' r/'lM

[)

lt\i,-,. ~ . 7 . ('~Iil~ll' dll\'(~ ('1) 1\~1·(" IL·H I(('~ 1 lIllel (II)

I d, f,t,k

IllIHI

/

Fi~. ('IIl"V(':1

a

~

mG

/

mL

mL

I

II Gravltatlona .. I

r----

torque

I

/

n

ú)

/ ......,c

2.U. llirrnnnL typcs of frictional Fig. 2.10. Torque/speed curves of a hniHI. d ri v('

l.o!''I''I:

26

2. Dynamics of a Mechanical Drive

2.4 Stable and Unstable Operating Points

With pumps and ventilators, where turbulent flow occurs, the torque rises with the square of speed; the drag force due to air flow around vehicles or l,!Je torque required by a centrifugaI blower pushing cooling air through an clectric motor also have this characteristic (c). ln practical drives, with the motor as well as load, all these types of friction cxist simultaneously, with one or the other component dominating. When driving a paper mill, printing presses or machine tools, Coulomb friction is usually the main constituent but with centrifugaI pumps and compressors the lorque following the square law is most important, representing the useful lllechanical output. Note that frictional torques are always opposed to the direction of relative motion. ln Fig. 2.10 various torques, acting on a crane under load, are drawn; me is the constant gravitational torque caused by the load of the crane, mF is the Cri ctional torque, resulting in the totalload torque curve (a). The speed Wo corresponds to the run- away speed with no external braking torque. When for safety reasons a self-Iocking transmission, such as a worm gear (b), is eruployed the crane must be powered even when lowering the load; this is du e to the large sticking friction.

or in normalised form,

J dLlw

k

dt

+ Llw

= O

where

k= -

â

(mL - mM)

âw

"t ,"", ", .": =='1(

00

O

--;/'

00,

mL

mL

w,

= mM (w, t) - mL (w, t) .

/

m

(2.16)

Apparently, a steady state condition with a constant rotational speed possible, if the characteristics are intersecting at that point, i. e. for mM

(Wl) - mL (WI) =

Wl

I

,1 (cl ,

.I

wc

d !\ (,) ,/I

littd 1.1\(:

o.

lI'111 nl

IIt .J

lill('aris(~d

l ,\,,) 'u

I

('quat.iotl

Ihll.J, }(4 )

I 1. 11

!\/,; ,

m

mM

L

m

k=O

stable

k
indifferent

a

m

unstable

b

c

Fig. 2.11. Stable and unstable operating points

cu

L3

cu

3/ ___...... m --J.

unstable

M

2/

~

~

L2

mL

~

mM stable

~

is

III order to test wltether this condition is stable, Eq. (2.16) can be linearised ai I.h(: o])(:ratitt /-Ç poiltt W I, assumiug a smn.ll displa(:ernent Llw. With G:J = (ti l

/

/

)

di

- - -/

W,

~

k> O

J

(2.17)

+4:t __ ~~4W<:/

mM

2 .4 Stable and Unstable Operating Points

dw

IWl

This is illustrated in Fig. 2.11 with some examples. The assumed steady state is stable for k > O; in this case a small displacement Llw, that may have been caused by a temporary disturbance, is decaying along an exponential function with the time constant T m = J/ k. k can be interpreted as the slope of the retarding torque at the operating point, as seen in Fig. 2.11. For k < O the operating point at Wl is unstable, i. e. an assumed deviation of speed incre ases with time; a new stable operating point may 01' may not be attained. The case k = O corresponds to indifferent stability; there is no definite operating point, with the speed fiuctuating due to random torque variations.

......mM

I~'y ignoring the dependence of driving and load torques on the angle of rota­ t.iOll, the corresponding interaction seen in Fig. 2.1 vanishes and so does the df'('ct of Eq. (2.2) on the static and transient behaviour of the drive. Equa­ I.joll (2 .2) then becomes an indefinite integral having no effect on the drive. I r wc also neglect the electrical transients and the dynamics of the load, the n:lllaillillg mechanical system is described by a first order, usually nonlinear, d iífcrential equation but, in view of the simplifications introduced, its validity is rcstricted to relatively slow changes of speed, when the internal transients alld interactions in the electrical machine and the load can be neglected.

27

m

m

Fig. 2.12. Induction motor with differ- Fig. 2.13. Rising torquejspeed curve as ent types ar load cause of instability

Figure 2.12 depicts the steady state characteristic of an induction motor (7I/.M) /.og d.l\(:r with some load curves (Sect. 10.2). LI could be the character­ i ~ l.i(' Or ; l vClll.il a tiug fali ; thc intersedioll ] is st.ahlf:, rOltghly corrcsponding to IICIIll i ll HI I"ad . Wil.h IJ~ 1.I11~J'(' is N.l S O ;t fi l.nl)I, · "1'<')'ni..iJlI '; pnill(.) bltt. tlw 11101.0)'

11'1

2. Dynamics of a Mechanical Drive

would be heavily overloaded. With the ideallift characteristic L3, there is an IIllstable (3) and a stable (3') operating point, where the motor would also h(' overloaded. In addition, the drive would fail to start since the load would pllll the motor in the lowering direction when the brakes are released and I.!lere is insufficient frictional torque. A particularly criticaI case is seen in Fig. 2.13. A slightly rising motor characteristic, which on a De motor could be caused by armature reaction dll( ~ to incorrect brush setting, leads to instability with most load curves n ccpt those intersecting in the shaded sector. The stability test based on a linearised differential equation does not fully (:xdude instability if electrical transients or angle-dependent torques should II:tvc been included. The condition k > is only to be understood as a Iwccssary condition, even though it is often a sufficient condition as well.

°

3. Integration of the Simplified Equation of Motion

With the assumptions introduced in the preceding section the motion of a single axis lumped inertia drive is described by a first order differential equation, Fig. 3.1,

J

dw

di = mM (w,

= ma (w,

t) - mL (w, t)

(3.1)

t) ,

which upon integration yields the mechanical transients. Several options are available for performing the integration.

mM'

úJ, ê

=E) ~

~- ~} mL

Fig. 3.1. Drive with concentrated inertia

3.1 Solution of the Linearised Equation In the linearised homogeneous equation, Eq.(2.17), d(Llw) Tm~

+ Llw

= 0,

Tm

J

is Llw a small deviation from the steady state speed

k = ii.

(3.2)

=k Wl

and

~ (mL - mM)1 âw

(3.3) w]

II 1<:
.I / J..:

1111 1'

UI("

IIW:tllilll', of a IlIcchalliraJ tilli(' rolii il.nlll..

'I'll!"

point. T m

=:

1~( ~ II( ~ ral KO!lll.ioll iK

3. Integration of the Simplified Equation of Motion

:30

L1w(t) = L1w(O) e- t / Tm

3.1 Solution of the Linearised Equation

The slope of the retarding torque is

(3.4)

where L1w(O) is the initial deviation, that may be caused by switching from one motor characteristic to another resulting in a new steady state speed. Because of the stored kinetic energy, the speed is a continuous state variable (the only one because of the simplifications introduced).

k

= -mo > O; Wo

(mL - mM)

+ L1w = O,

d(L1w) Tm~

mL

(3,5)

where

Jwo

Tm

---,mM1

;\w(O)

â = -âw

hence the differential equation is stable, it has the form

,ócu

cu

31

mo

is the mechanical time constant.The steady state operating point at the in­ tersection of the two torque curves is Wl = wo; due to the initial condition at standstill, we find L1w(O) = -Wo, which leads to the particular solution

a

m

b

Tm

F ig. 3.2. Mechanical transient of linearised drive system

L1w(t)

or wi th w = Wo

w(t) Til Fig. 3.2 a it is assumed that the motor, initially in a steady state ('(Jlldition at point 1, is switched to another supply voltage so that a new ('iJaraderistic for the driving torque mM2 is valido This causes an initial devi­ ;i.I.io ll L1w(O) with respect to the new operating point. The deviation vanishes IIlolI/,. ; LII exponential function (Fig. 3.2 b), the time constant of which is de1."l'llIill pd hy the inertia and the slope of the load- and the driving-torques. 'l'hi N is di scu::ised with the help of some simple examples.

:1.1. 1 St.art of a Motor with Shunt-type Characteristic at No-Ioad

t = O, Fig. 3.3. The of the motor is assumed to be linear between no-load liI)('( 'd WI) (lIeglecting friction load) and stalled torque mo'. With normal mo­ 1.111:: or lIwdium size, the stalled torque may be 8 or 10 times nominal torque; i I. is dd.cnnined by extrapolating the drooping shunt-type characteristic to Hla lld sLill anel hence represents only a reference quantity that cannot be mea­ :: lIl'cd iII pmctice. However, in the present example we assume that a starting 1'(·::i::(.OI" lia.s been inserted, reducing the stalled torque to perhaps twice nomi­ lI a l Lorqlw alld thus extending the validity of the linear characteristic down to HL;\.lIdsLill. This, by the way, m akes the simplifying assumptions more realistic :l llIn : Lhe dectric;J.l tmn sicllts becornc faster while the m echanical transients fi. IJlolor which initially is at rest is started at

= -Wo e- t / T= + L1w

= Wo (1 -

e- t / T =) .

(3.6)

From the torquejspeed curve of the motor ~

Wo

= 1- mM

(3.7)

mo

the motor torque during the start- up is obtained (Fig. 3.4), mM(t)

= mo e- t / Tm

.

(3.8)

The discontinuity of the motor torque is caused by the omission of the elec­ trical transients. ln reality the torque is also associated with energy states and, hence, is continuous.

1 .() I'Ill\"jsp( ~c d- curve

cuo'" t~

00

Uort0

1~k--------

mM

- mL=O

=O mà m t


alT

:l lIl ldl v. dll'·:

cu mM cuo' mo

cu

I"il;. :1 .:1. SI.fIl' l.illl'.

"r

1l1ll1.0!'

ai.

110

10ft"

Tm Fil-'( . :1.'1 . St.art,iIl14 t.rall fi it~ lIl,

3. Integration of the Simplified Equation of Motion

:1:2

3.1 Solution of the Linearised Equation

:1.1.2 Starting the Motor with a Load Torque Proportional to Sp eed

w

~ WI'

mM

Wo

mo

1

1

111 Fig. 3.5 the curves of a drive with speed-proportional load torque are displayed. The steady state operating point is now at WI < Wo, hence L1w(O) =

33

m1

W1

Wo

mo

"­

úJ

"­

,

' . . . . .......m L = O

....

....

{Oo

mM

Tm1

a

Tm

t

Tm1

b

Tm

t

Fig. 3.6. Starting transient with load

(1)1

3.1.3 Loading Transient or the Motor Initially Running at N o-load Speed

m

m1

The motor is now supposed to be initially in a no-Ioad condition at the speed Wo with the starting resistor short circuited, so that the speed droop is reduced and the extrapolated stalled torque assumes its large nominal value mOn· The torque/speed-curve is then linear only in a narrow speed range around no-Ioad speed.

I"ig. 3.5. Starting of motor with linearly rising load torque

The slope of the retarding torque is

kl = -

ô

ÔW

mo

(mL - mM) = -

> O,

(3.9)

WI

úJ

t= O

wlüch leads to a reduced mechanical time constant, JWI

Tml = mo

=

WI

Wo

T

m

.

(3.10)

TIl\' speed transient is

W(t) =

WI

mL

(1 - e- t / Tm ,).

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

(3.11)

. . . . . . . . mM ................

1';lilIlination of speed from the motor characteristic Eq. (3.7) results in the. d ri vi11g torque mM(t)

= mo

(1- :J = [1- :~

= mI + (mo Ilol.h

tnl.nsi(~J1l.s af( ~

mo

mI) e- t / T ""

.

t (1- e- / Tm ,)]

........

mOn m

Fig. 3.7. Applying constant load torque to motor running at no-load (3.12)

plol.ted in Fig. 3.6, together with the no-Ioad starting 1.1';t.J1Hi(~J1(. (111./, - O) . AI. I. 0, t.he Rlopc of tltc specd curves is the sal1lC, 1)('("a.lI :ic- tlI(" ;,1<'("(,1('1 n l.illl', l.orqll(,i' ;1.1"(' idcul.ieal n.1. (v · O.

=

m1

At t = O a constant torque load without inertia is applied to the motor iloft, possibly by a mechanical brake, Fig. 3.7, causing the motor to slow dowlI [.0 Lhe steady state speed wI; hence the initial deviation is L1w(O)

Iii I

(')0

W I .

III vil'W


("ow;L;mt

load torqll(',

WI:

lilld

3.1 Solution of the Linearised Equation

3. Integration of the Simplified Equation of Motion

:11

~ (mL âw

k =

mM)

mO n

I

Tm

Wo

Wl

JwO = T mn

= mO n

·

(3.13)

Sillce mO n corresponds to the extrapolated stalled torque with short circuited sl.arting resistor, T mn is also called short circuit mechanical time constant. The solution of Eq. (3.5) is

Llw(t) = (wo - wd e­ t / Tmn 01'

w(t) =

Wl

+ (wo - wd e- t / Tmn

(3.14)

•

The driving torque is again obtained from Eq. (3.7), mM(t)

= mO n =

mI

(1-

~J

= mO n (1-

(1 - e- t / Tmn )

~:)

(1-

t Tmn e- / )

(3.15)

•

III Fig. 3.8 the transients are drawn for loading and unloading the motor.

mM ..ill..

- wo m- On 1

----1(O

___

..ill..

o

W 1~~-~

(1)0

mi

computer, can be made immune against short time disturbances on the line side, when it is supplied by a rotating motor-generator set having a fl.ywheel attached.

3.1.4 Starting or a De Motor by Sequentially Short-circuiting Starting Resistors With DC motors fed from a constant voltage bus the starting circuit shown in Fig. 3.9 may be used, where the contactors C 1 , ••• CN are closed in succession. By suitable choice of the resistors, the droop of the resulting torquejspeed characteristic is progressively reduced. ln order to achieve fast starting with­ out overloading the motor, the torque is specified to vary between the limits mI, m2 during the start, as is shown in Fig. 3.10. The armature current, being proportional to torque, then varies between the limits i I, i 2 . A similar starting procedure exists for AC induction motors having a wound rotor and starting resistors in the rotor circuito As the motor is accelerated, the torque mM decreases until the lower limit mI is reached; then the next section of the resistor is short-circuited causing the torque to jump again to the upper limit m2. This is repeated until alI the starting resistors are switched out and the motor operates on its natural torquejspeed curve. ln steady state the torque is determined by the load. The required number of sections of the starting resistor and their appropriate values can be calculated in closed form [37] .

1

IL-­

C1

1

mM

m() n

1 1_

=-=-

mOn

C2

Cn

1

1 1_ _ _ _ _ _

Co

I' 1

Tmn

35

t

vai

Re

Le

1"11';. ~L8. Loading transient

'l'l\( ~ drivillg torque apparently follows the applied load torque with a lag

Fig. 3.9. Starting circuit for a De motor

d!'l.(~rlllillcd

by T mn . The reason for this is that during the deceleration phase h Ol1W of til(' kinctic energy stored in the inertia is released and makes up for pMI. 0[' I.he loat! torque; when the drive is accelerated after disconnecting the !O, H!, I.he 1l1ÜiSill~ kinetic encrgy is restored. Thc mcchanical energy storage t.ltll S act.s HS a hl\fr(~r !>dweell lhe load a.nel lhe ek('(,ri(:al sllJ>ply fccding the Tltin dr,'d Ill ay b(: n.cn~Ilt.\1at.(:d hy iI.dtlilll~ It f1yw'llt'd lo Lhe: drive:, (II IIl'Ikl' l.tI 1'1'1",,'(' " tllt' :lllpp! y fiy Ht.(1II1 r""11I h ip,h !, 1I11 1 11111'1( " ;1 Hllrll n~i ti IO/i(> ('1I 1l 1.~ , d 1,1' 111>1'1'." 10 11 1111 ', IlIill d dvl 'rlj """VI1l'lll'l y, II /11' 11 /111. 11', , 1,,1101 , 1'''1 "',11.1111'' ' ii

I HO l.or .

From Fig. 3.10 the following relation is found rn,2 1/1,2

mI

w

(v + 1) - w (v) Wo - W(v)

Wl,idl 1" ':-:1111.::, wil.lt Ul(: ;-\.IJI,rnvia(.joll '11/.1

(3.16)

jm./,

U·

-I, iII a. recllrsiv(' forlllula

:~ (;

3.1 Solution of the Linearised Equation

3. Integration of the Simplified Equation of Motion

37

mM

w (l)(N)

m1

(()(v+ 1)

(ú(v)

(ú(1)

..... ..... ..... ,>,

I

w(O)=O jo'i~. 3.10.

m

m:z=mo{O)

1

•

mdO)

m

: - Tm(I)--: I I I I

Torque/speed-curves for different values of armature resistor

cu W

CUo

(1/ + 1) = (1 - a) Wo + aw (v) = w(l) + aw (v) .

i\SSlIlllÍug w(O)

+ a) w(l) (1 + a + a2 ) w(l)

w(:\) =

(,!( II)-CC (1

I

I

I

I

-------i------I I I I

= O, this leads to a geornetric progression

w(2) = (1

I

I

I

a

CU 1

b

+ a + ... + aV-I) w(l)

Fig. 3.11. Starting a motor by successively reducing the starting resistor I.h,~ ~;1I11l

wit.ll

1,1(11)

I

Il

w(t)

v

w(l) = (1 -

I -- a

a

V )

(3.17)

Wo .

'1'11<' Li::L uI' Lhe N switching instants is reached when the natural torque/speed "IIIV" yJ('ld~ avalue of the torque which is less than the maxirnum, m2, (!(N) -

(1- aN)wo 2: (1-

m

2

mO n

lll(l/a) I"rl 1111 pl'(~viou~

N integer.

Wo ;

I, I ( I' )

I ,I

( , I (II

I I)

e-(t-t.)/T=(v)] ,

(3.19)

,

m2 e-(t-t.)/Tm(v) .

(3.20)

as soon as mM(t) has dropped to mI, the next contactor C v +1 is closed. The mechanical time constant, valid in the interval, is (3,18)

rcslllLs iL is lmown that, when assmning linear torque/spcecl ",II varia.hk:, arc ex pO!lclItinl f1lJlctio!ls oI' \.ime, w i th tl)(~ time COl1s\.all ts 1I"Il('llIlillJ', (III 1.1\1' ;'III!,II' ~; "r 1.11(: curve:; n(, I.!t(, poiul. (Ir i\l\.(:r: I( '.('.I.Íoll, III F Il'. :\.11 I I ::1.11111111', LrnJl:;i('Jll. wil.h"lIl. 1"11.11 h ill\(\wlI; III I,h,' iIlL('l'v;d

,'II r V( '1_)

w(v)] [1 -

where tv is the instant, when C v closes. The driving torque in the sarne interval follows also an exponential func­ tion

mM(t) = )

:;"lviul'; for N, this results in N /" 11l(mo n /7n2)

= w(v) + [wo -

T m (V ) --

J Wo _ J Wo v _ T a - mn mo(v) m2

mOn

m2

v,

a,

it decreases continuously as the starting resistor is reduced .

(3.21)

3.3 Numerical and Graphical Integration

3. Integration of the Simplified Equation of Motion

:\H

:L2 Analytical Solution

t2 = J

411' Nonlinear Differential Equation

/

= J Wo [ 2 mp

dw = mM(w) - mL(w) dt

J -

I 'y scparation and integration,

f () W2

t l -

tI

=J

-

o

fi direct solution of the equation

(3.22)

mM w 1-mL () w dw

Wl

is rarely possible because there is either no analytical expression of the func­ I.Íolls mM (w) and mL (w) or, if there is such a formula, the integral cannot be

J Wo /0 ( S =-Sp + -Sp) 2mp S

dw mM

W2

39

dS

52

1

-

S~

_ Sp IS] n 2

Sp

(3.23)

•

This function is plotted in Fig. 3.12 b. Since the speed approaches the syn­ chronous speed Wo asymptotically, the definition of a starting time requires the choice of a final steady state slip S2. Computing the starting transient on the basis of the steady state torque/ speed characteristic makes sense only if the starting is sufficiently slow in order to justify ignoring the electrical transients. This condition is met when the load contributes substantial additional inertia or when the start is delayed due to reduced volt age. Otherwise the start of a smaller motor would be completed within a few periods of the line voltages, long before the electrical transients have decayed; this will be discussed in Sect. 10.3.

so!ved in general terms.

ú)o

ú)o

1 c O ')p ", O,1 0,9

1

a

3.3 Numerical and Graphical Integration

..ill..

.!Q.

,

1 0,9

1

S

The most versatile method - with regard to accuracy and applicability ­ of integrating nonlinear differential equations is the stepwise integration with the digital computer. It is known that by introducing state variables Xi (t), i = 1, 2, ... , n, a differential equation of n th order may be written as a system of n first order differential equations

j

+-t---_

dx ' d: = CPi(Xj, Yk, t),

m mp

b

Fil~'

3 .12. Starting of induction motor (11) Steady state torque/speed curve (b) Starting transient at no-load

Au exception is the no-Ioad starting of an induction motor without sta­ l.or rcsistance and skin effect in the rotor bars; the normalised steady state l.urque/speed function is

= S/Sp + Sp/S

'

1.111' i lll." )'.r·,,,\,l ,," ywlli ::

(3.24)

where Yk (t) are the independent forcing functions and t the time as vari­ able of integration. The functions CPi may be arbitrary, given as analytical expressions, curves or even discrete samples. Once the state variables Xi(t) are known, the output variables Zi(t) can be calculated from algebraic ex­ pressions, i.e. without further integration. Zi (t)

= cp i (x j, Yk , t) .

(3.25)

(v+l) Llt

wil.h S _ 1 - w/wo being the slip and Sp the pull-out slip at which maximum l.orq\l(~ 'l/1. p ocellfS. 'J'lw function which will be derived in Chapo 10 is seen in Fil'.. :~ , 12 a for SI' 0, I. Wil.h 110 \,,;ul ('111" . O) nnd tJw illil.in.! cOllllil.jolllv(t.l ' c.:. O) "-" 0,8(0)

,n,

'J'he simultaneous solution of the n state equations (3.24) is performed in s teps of length Llt

2mp rnM

i,j,k = 1, ...

=1

x;( (v

+ 1) Llt) =

Xi

(v Llt)

+

/

CPi( Xj , Yk ,

t) dt .

(3 .26)

v Llt

:/:; (/l I.\ t.)

;I:i (/I) are the results obtained in the preceding step, Xi(V + 1) is "r v lI.ll1cr; al. time (v+ 1) Llt. Da,.'i (·~ d 011 Hvproxirn ations provided hy 1II III.h"III i\ l.i ni alld wil.iI Hullici('ut.ly Hlll a ll :; I,I:p h:1l ~ l.h wc CHIl wril.l~

I.h (· IU'W ~1(' 1. 11I1I1I( ' I' k rtl

,III

3. Integration or the Simplified Equation or Motion

:Ci (V

+ 1) ~

Xi(V)

+ F['Pi(V),

'Pi(V - 1), ... ] ,

3.3 Numerical and Graphical Integration

(3.27)

w!ln"!! F is a linear combination of the functions 'Pi at previous intervals. 'I'Itere exists a large number of integration formulae which differ in complex­ il,y, i.e. computing time, sensitivity to numerical instability and accuracy [22]. 'l'lte ones best known are square-, trapezoidal- and Simpsons-rules, further­ IIl
dv

1

-1--

=M

ds dt

= V,

(t

[JM(V, t) - h(s, v, t)] ,

wll(!l'(' 1M is the internal driving force of the locomotive and h is the to­ tal loar! force, íncluding frictional and braking forces a.s well as gravitational l'ol'( 'cs Oll grades. The task of finding an acceptaiJl(' tillldable is an optimisa­ I,i(lll prohlclJI of cOIlsiderable complexity, sinc(' lIlally cOllstraiJlts and bound­ :lr ,Y cOlldi(,ioIlS llIust h( ! tõlken into aCCOllnl.. 'l'llú; indlld,':; Lhe Il S(! of the same Lr:lt'k I,y I.r:tills Itavill g diffcrc llt I\.('cdcra(.ioll :1,lId vl'lociLy lilllÍts, such a.'i in­ L('IT,iLy
41

Since it cannot be ruled out that, even today, this may have to be done occasionally, let us briefly consider the principIe of graphical integration, using a simple example. Eq. (3.1), where mM(w) and mdw) are given as empirical curves, Fig.3.13, J

dw

di = mM(w) -

mdw)

= ma(w);

(3.28)

the equation is first normalised with the help of arbitrary values J WI ~ (~) dt

mI

WI

mM _ mL _

_ -

mI

mI

-

ma

WI,

mI, (3.29)

mI '

where ma/mI corresponds to the normalised accelerating torque.

cu

mL

/

/

/

/

/

m Fig. 3.13. Torque/speed curves or motor and load

With the abbreviations JWI mI

= TI,

t

-T=r 1

W

'

WI

= X,

ma mI

=y

a nondimensional equation results

dx

dr =y(x).

(3.30)

Normalisation avoids the awkward choice of scale factors; the reference values are of no significance, even though it is recommendable to choose characteristic values such as nominal speed and torque in order to work with handy numbers . ln Fig. 3.14 the normalised curve y(x), obtained from Fig. 3.13 is shown, with t.lt p lI ol:llInlif;cd speed plotted against Ilormalised torque. The x- axis is sllbdividcd illl.o scv( ~ r;tl illt(!l'vals, JlOt, !)(!cPHsarily of c<.J. l1al lcngth; it is in fact :l.pPl'Opri:i.I,,' 1.0 /'C'dll('" 1,111' 1(!lIgLhH 01' 1. Ili' illkrv:d:; iII /;cdiolJ S wheJ'c y(:,:) iH ('Ia ii "1',1 " I', I'lI. lIidl y , 111":,,1'11 illl.(·rV:I,1 y i: : ' i.J'I" 'II ' "II :tf,(·cJ h,Y II, (·IIII I
I\~

3. Integration of the Simplified Equation of Motion

x=

4. Thermal Effects in Electrical Machines

01 01 1

x -----.......... .......

t

T =r

1

ma

y= m

r

1

Fig. 3.14. PrincipIe of graphical integration

rCJlresenting the average of y(x) in that interval. Approximating y(x) by a function has the consequence that dx j dr is assumed constant in ('
~it.airc ase

4.1 Power Losses and Temperature Restrictions 80 far our considerations were only dealing with mechanical phenomena and the pertinent steady-state and dynamic conditions, but suitable torquejspeed­ curves and adequate power are not the only criteria for designing electrical drives. Of equal importance are the thermal transients in the motor caused by the unavoidable power losses during the process of energy conversion and the ensuing heat flow from the points of origin to the cooling medium. The various materiais used in an electrical machine naturally have different temperature limits; of particular importance are the insulating materiais, which determine the temperature classes of the machine, for example class A, B, F, H,

Lh9 Lh9 Lh9 Lh9

< 60° C, < 80° C, < 105° C, < 125° C,

cotton, synthetic, paper, resin, shellac, mica, epoxy, glass fibre, silicone rubber.

..119 is the mean temperature rise above an assumed ambient temperature of 190 = 40°C. All temperatures are measured in degree Celsius (centigrades). For a given frame size and type of cooling, the output power and the allowable power losses are higher for the elevated temperature insulation classes; at the sarne time the cost of the machine increases. The main causes of power losses in an electrical machine are: a) Conductor heat loss (copper losses) in the windings, cables, brushes, slip­ rings and commutator. If there is alternating current, the effective resis­ tance of the conductor may be noticeably increased by eddy currents (skin efFect); this is accentuated when large conductors are surrounded by mag­ netic material and may have to be alleviated by the use of stranded con­ ductors . h) Irou heat losses in stationary or moving magnetic material exposed to dl
"r

t.llI' Itl n /',rrdi .. lidd .

,1,1

4.2 Heating or a Homogeneous Body

4. Thermal Effects in Electrical Machines

I' ) I,~,.ú;tion

'I'he various types of losses depend in complicated fashion on the oper­ :d.iollal state of the machine. The main factors of influence are speed and lo:ul torque, as well as voltages and currents with their RMS-values and w:wdorms. 111 order to induce heat flow from the points of origin to the cooling ::llrf'aces, a temperature gradient arises inside the machine determining the
[

lasses, including bearing-, brush- and ventilation losses.

= L119

+ (Ll-!9 max

L119) + {)o

40°C = 130°C.

-

~IA

Pj

t)

C

o ~P2

t}O Fig. 4.1. Homogeneous body with thermal storage capacity

power Pl, at the same time emitting heat power pz by convection. The am­ bient temperature is {)o, the coefficient of heat transfer a. The component of radiated heat is assumed to be negligible due to the relatively low operating temperatures (it rises with the fourth power of absolute temperature) and due to back radiation, for example at the cooling fins. Hence the equation, describing the power balance is d{)

C- =Pl -P2. WlwlI operating the machine at elevated ambient temperatures, e.g. in the I.ropies, the power rating may have to be reduced. A n accurate prediction of the heat flow and the temperature distribution iII :w dect.rical machine is exceedingly difficult; this is due to the complex I';( 'ollldrical shapes, the use of heterogeneous and anisotropic materiaIs (lam­ ill:J!,I'd irou , insulation), furthermore the complicated laws ofheat generation wiLh n'sl)(~d to space and time, as well as the different cooling conditions. A b " Lhe l)(~al. conductivity of the various materiaIs does not differ by orders II/" 1I1:J.1',lIiLlldc', as is the case with electrical or magnetic fields. As a result, ti ... 1."II'IWra(.llre distribution can only be computed in restricted sections and wli,1I (·oll.'ii(krable simplifications. TI", IISI~r (Ir dectrical drives has normally little influence on the tempera­ 1.111'" di :-;(.rihlll.ioll within the machine. He must be confident that the designer II /lS dlllSI'1I slIHiciently large conductors and cooling channels so as not to I· F I"'.! LClllpl~rature limits under specified operating conditions. With this IU il'i lllll(.Lioll iI. is usually adequate to drastically simplify the thermal model ,,( Lili'. "lllOlor" by regarding it as a homogeneous body exhibiting thermal : i L()ril-f.'.I~, whose internal transients are unknown and of no interest. Naturally, ::<,,'11 a I:rude rnodel cannot offer any detailed information on specific internal I.I11'I'IIlal I ~ollditions.

LIeating of a Homogeneous Body

'I'I,,~ :;illlplifi(~d l, hl ~ J'lllallII()dd

is

charactl'~ris('d

hy l.hc followiug aSSlIlllptiolls,

I"i'r', . ,I. I ;

fi t.lIlI'd

1""lv wil.lt tllI' :illr(",("I ' 11, 1.1 ... I'.I11'1'1 II ii. I (':Ip:u 'il'.y (), 1111':1. \lV II ' C, 1111..1 1.1 .. ' 1111'11.11 ::111 /'lI,(' I' 1."1111'('.,," 1,111" ,'I I; : 1.... \.1"'1 I"v I.hl ' illjlllL

hll I IIOI 'l "lIf'IIII II iII

(4.1)

dt

The heat transfer by convection is a linear function of the temperature dif­ ference

pz

=

with {) -

a A ({) - {)o) ; {)o

d(L1{))

=

C~

L1{)

+a

(4.2)

this results in

A L1{)

= Pl

(4.3)

or

T dL1{) {) dt

+

L1{)

= ~ a A '

(4.4)

a linear first order differential equation; T{) = C/ aA is the thermal time constant. With Pl = PlO = consto and L1{)(O) = O the solution is

L1{)(t) = L1{)(oo) (1 - e-t/T~) ,

(4.5)

where

L1{)(oo) = PlO/a A f1.~

45

(4.6)

is the steady state end temperature, at which the heat transfer by convection equals the input power. lu Fir;. 4.2 a tlwrrnal transient is seen. The shaded area between pI/a A aud Lh'l(t,) c()ITI',~p()llds to the energy stored as heat. Hence, the storage effect 1.1'11(1:: 1.11 dc'I:I,)' 1.11f' 1 . l'llll)(~l'at.lIn! chaJlg(!:;, ea.llsiJlp; the temperature to be a (""lIl.illllllll~: 1'11111'1.;1111 III' 1.;1111',

4. Thermal Effects in Electrical Machines

~ ti

\ I~

LW(oo)

4.2 Heating of a Homogeneous Body

little thermal storage capacity. The following time scales are usually valid with electrical machines:

L1~

= P10 aA

L1~(O)

electromagnetic 0.1 - 100 ms

TJJ

TJJ

cooling-off transient

a= 50t0500WtCm

Fig. 4.2. Thermal transients

The approximation incurred with this simple model is mainly due to the assumption of evenly distributed heat sources in the homogeneous body and t\w Ileglect of an internal heat flow; hence, transportation delays are not rOlltained in the simplified model. A rough estimate of a typical thermallag may be derived from the data sll('d. of an enclosed 100 kW standard induction motor cooled by an external \';\.1I (TEFC): 100 k W,

) ' /I

nominal power mass 0.92, nominal efliciency GO°C is the steady-state temperature rise due to the losses PI at

~ OO

M ,I . I,,') (' x,)

kg,

11
1'"

1'1

(~-1)

LIli' C(l lI v( ' cl.ive

11.

-:"c

that the motor consists of solid iron having the specific heat capacity is

~ =

A

j

2

= 2.5 to 25 kW m ;

the dependence of the heat transfer on the velo city of the cooling air has the consequence that self-cooled motors exhibit at standstill a cooling time constant that is much longer than the thermal time constant of the motor when running. Larger motors operating at variable speed and load are usually provided with separate forced cooling, causing the motor speed to have little effect on the cooling conditions. To show that the heat transfer by radiation can normally be neglected, a rough estimate of the power flow by radiation is given for comparison. A "black body" emits to the environment power by radiation according to Boltzmanns law,

= uT 4

,

u = 5.710- 8 W jK 4 m 2 CFe,

'I'his yidds the thermal time constant

;'}

2

where

L119(oo)

c = Cl"e M. "

P2c

A

J\ h S lllllill~ I. !tI , hea.L

thermal transients 10 - 60 min

at Ll79 = 50°C this results in a power flow by convection of

PR

heat conductivity is

PI

Ir

mechanical 10 ms - 10 s

Travelling wave phenomena in windings are not considered; they are faster yet by several orders of magnitude. The large difference of more than a factor of 100 between the time scales of mechanical and thermal transients usually permits a separate treatment which results in considerable simplification. The heat transfer by convection from a ventilated surface depends strongly on the velocity of the cooling air; the usual range of the transfer coefficient is

Change of stored thermal energy

hea ting transient

47

C/"e

M Ll19(oo)

=

.!.::.'L p 11.. 71

n 11.

'l'y pirall y , Lhcl'lllal

Lrll.llsj('lI(.s ar(:

1'1\', '1'1.::

()I' ((1111 ::" ,

1.111'\'"

111 :.1. 111"

Wl ll d i JlI',

0.48 kWs j kgOC 800 kg 50°C ~ 34 mino 9 kW

IIl1lch I;!ow(:r I.hall d( ,ct. rj(·:t! 01' J11(·('hallica.! di/l l( 1II(1(.tll·H, wlll'l'! ' 1.111' nr­ Irl I' Ji ui<'d d i ,.,·(·l.Iy (III !III I lI li lllll,I,I II P, d lll lt I UIfI IJl IIII H'III II ':I V('I 'y 11.1 '(' (·:x('(:p\.j(lIl H :1111'11 III '

is the coefficient ofradiation and T the absolute temperature. With 19 = 70°C, i.e. T = 343 K, the power flow due to radiation is PR

A

= 0.77kWjm 2 .

This relatively small amount is further reduced by inverse radiation. AIso, the r acliation coefficieIlt of a motor surface is much lower than that of a "black hod y" .

is the extrapolated end temperature during overload. Hence

4.3 Different Modes of Operation

Pls

III contrast to internal combustion engines, most electrical drives possess (", Hlsiderable overload-capacity, the limit of which is determined by the power losses rising with the load. Since the temperature follows the power loss with :I. slIhstantiallag, a temporary overload is admissible without exceeding the 1."llIperature limits, provided that the motor is initially below the nominal 1.' ~ llIperature. This effect may be utilised when selecting a drive for a particular :tpplication. Let us look at some modes of operation frequently occurring in jll'actice. il.:~.l

49

4.3 Different Modes of Operation

4. Thermal Effects in Electrical Machines

I\{\

-

Pln

::; 1

-

1

T{)

-t

-t '

e 1 IT~ ~

1

t1

« T{) ,

(4.9)

is the maximum relative power loss during the time tI. lncreasing the output power while simultaneously reducing the time h of operation is of course only feasible within the available torque range; this is illustrated in Fig. 4.4. At the left, a typical curve relating torque to power loss is drawn, while the other curve depicts Eq. (4.9). This indicates that the power loss rises much faster than the torque (assuming constant speed). The shaded contour corresponds to the admissible short time overload range.

Continuous Duty PIS

TI! is lIIode of operation exists if the load torque is constant during an ex­ LCllIl(~d period of time, corresponding to a multiple of the thermal time con­

SL;\lIt, so that the temperature reaches its steady- state value, Fig. 4.2 a. 'l'his is the normal continuous duty, e.g. with paper mill drives or boiler feed p11111p5. The motor must be so chosen that its nominal output power equals "I' ('xceeds the continuous load. With traction drives, other definitions are also ill use, such as a one-hour- or five-minutes-rating of the drive motors. tl. :.L 2 Short Time Intermittent Duty

III t.hi s

Lhe tillle of operation is considerably less than the thermal time :wc! t1w motor is allowed to cool off before a new load cycle be­ 1'111::, 'l'1,i ~i dllLy apilhes for example to some crane drives or small motors in j"'II::,'III>I
I i (,i()lIS. ln Fig. 4.3 the temperature transient is drawn, following a short time pmv,:r lo:{s 1'1. which may be considerably in excess of the nominal power lo:;s 111",' The temperature rises rapidly towards an extrapolated steady-state vai 11(' L119 j when L119 1 , is reached, the motor is disconnected, giving it the 00 npport.lluity to cool. ln the example it is assumed that the motor is self­ mole!! C
, ' , 1111

V;t.!IIC ,

Tlw Illaximum t.cmperature L119 1 , at the time of disconnection is

A" ~' I1 Q()

[1

-

<

C - t';T~] _

A" .L..l'f!rna :c ,

(4.7)

wlwrl' I

\n ,. ,

I

\1'/

IIJ" , 111 "

1I I I I

I

(J! .r\)

,119 - PIO

! ~ - aA

I I I I I I

-4~rJ-t-: o

o

, ' , II 1: , !.;utL

A 'I I w'U

,119

ti

TjJ

Fig. 4.3. Thermal transient during short time duty

m

ti

mn

Ti}

Fig. 4.4. Short time overload limits

With motors exhibiting a pull-out effect there is an absolute torque limit, whereas the losses continue to rise up to the stalled condition; on commutator machines, the commutator is usually the weakest link. To realise high torque pulses which exceed even the short time overload capacity described by Fig. 4.4, a mechanical energy storage device, for in­ stance a fiywheel, may be used; of course, the inertia must be accelerated to sufficient speed before the load can be applied. This is the usual solution with highly pulsating loads such as punching presses. 4.3.3 Periodic intermittent duty

With some types of load, for example elevators, automatic machine tools, mine hoists or rolling mill drives typicalload cycles exist which are repeated periodiea.lly. A very simpie case is seen in Fig. 4.5, where the power loss Pli follow s a rectallGular periodic pattern. This n:slllLs iII t.( ~ lll])('rature fluctuation s, tlte rnean of which rises until a :' I.(·ady :.;1,;1.1,( ' ("')\ldiLioll iI; l"(·ac1l<'d. Wit.lt !.lI< : (,IWl'lll;d I',Ínw ("on stanls TI 7'~ valid dllrill/" (hl (11111 ()I!' l,iIlH' S /' 1, t 'l. , !.II(' rollowi ll l', rd:t.l.ioll s Itnld: I

~i()

4. Thermal Effects in Electrical Machines

T1

,1t'}+

i

I.

,1t'}

1-

00 . ...

5. Separately Excited

I .1

l --...

T2

De Machine

.1

I

---l

I

1

~~(1) i

1,11J(2)

I

/ //

//+-­

/

/Ll1J'(1)

: -- ;1t'}'(2) I

I

t1

o

2(t 1+t2)

t 1+t2

5.1 Introduction

Fig. 4.5. Periodic intermittent duty cycle

L1t9'(v + 1) = L1t9(v) e- t2 / T2 , L1t9(v + 1) = L1t9'(v + 1) e-tl/Tl l';liminating L119' (v

L1t9(v

+ 1)

+ 1)

= L1t9(0)

(4.10)

+ L1t9 oo [l -

e-tI/Tl] .

yields

+ a L1t9(v)

(4.11)

,

lN lt( ~ lT

(4.12)

L1'/?(O) = [1 -

e-tl/Tl] L1t9 oo . a = e-(tI/Tl +t2 / T 2) 1.

<

(1.11) represents a linear recursion between two successive peak 01' l.lw tcmperature. The solution is a geometrical progression

1':qll:tI,ioll Vío.lll. '~:

,

/\/1(11) =

1- a IJ +I

l-a

L1t9(0)

=

1 - e-h / Tl 1 _ e-(tI/Tl+t2 / T 2) [1- e-(IJ+I)(tl / T l +t 2/T2)j L119 00 .

+ v (h a time constant

I kC;lllse of t = tI liavillg

'L'

. ,'I

~

tl

t2 ITh1 + + t2 IT2

+ t2) .

wlt

(4.13)

the peak values lie on an exponential function

h

TI < T{) < T 2·

(4.14)

'I'IIC st.(~a.dy state peak temperature follows from Eq. (4.13)

Ll1~( (0) = Wit.h

1_

1 - e-h/Tl

rorc<~d v"IlLilal.ioll ,

'1',

'{'.

'('i I

(4.15)

(~- (td'J\+t2/T2\ L1'Ooo . n sJlccial

('.as(~

(' X

ist.:; wil,h

Direct current (DC) motors have been dominating the field of adjustable speed drives for over a centuryj they are still the most common choice if a controlled electrical drive operating over a wide speed range is specified. This is due to their excellent operational properties and control characteristicsj the only essential disadvantage is the mechanical commutator which restricts the power and speed of the motor, increases the inertia and the axiallength and requires periodic maintenance. With alternating current (AC) motors, fed by variable frequency static power converters, the commutator is eliminated, however at the cost of increased complexity of control. This is one of the reasons why controlled AC drives could not immediately supplant DC drives, once the semiconductor-technology had sufficiently advanced. The principie of a DC machine operating in steady state is assumed to be known [4,64], but let us recall some basic facts. ln Fig. 5.1a a schematic cross-section of a two-pole DC machine is shown, containing the fixed stator S and the cylindrical rotor, called armature A. While rotor and pole shoes are always laminated to reduce the iron los ses caused by the varying magnetic field, the rest of the stator is laminated only in larger machines, when the motor is required to operate with rapidly varying torque and sp eed or when a static power converter with highly distorted voltages and currents is employed as the power supply. The main poles MP are fitted with the field windings, carrying the field current ie which drives the main flux Pe through stator and rotor. A closed armature winding is placed in the axial slots of the rotor and connected to the commutator barsj it is supplied through the brushes and the commutator with the armature current ia. This creates a distributed ampereturns (m.m.f.) wave, fixed in space and orientated in the direction of the quadrature axis, orthogonal to the main field axis, so that maximum torque for a given armature current is produced. ln view of the large airgap in the quadrature direction the resulting ar­ mature flux P" is much smaller than the main flux Pe. It can be reduced CV(~\l furtll er hy placing compensating windings CW in axial slots on the pote sh()( ~s Hlld ( ' ()lIlJ(~dillg "Item iu ::led es with the armature. Their opposing 11,1111)('1,\,1 ,111'11 1-1 11 11 1'111.

(, ;\.IlI'd

til<'

qlladral,lIl'< '

li"'d ('x eiL(,,1 hy

I.IJ( ~ a rtnal,lIlT

u.II
5.1 Introduetion

5. Separately Exeited DC Maehine

;,2

ar ·

QUadratture axlS

Neutral Lone

I I I I J J J J

JJ

~ff /

/ /

ie

.y~// / V

/

/

/

/

ia{t)t fii t+M )

iit)

ia , .

b

i 1/

Ua

/

/

/

a

a

ar ·

I I I I I J J

G

ie

li

Commutating coils

Direct axis

ia

a

53

:

tf

+­

Commutator

iit+M )

Current in

ne7gnoourTng ­ p---­ conductor I:

Jo'i~~. G.lo (a) Cross-seetion and (b) sehematie eireuit of a DC maehine

b

----'VrCIIIOV(: t.he undesired armature reaction, which otherwise tends to distort the I'V('II di,~t.ribution of the main field under the poles along the circumference (Ir !.II(' rot.or. Compensating windings are common only on larger machines 1>1 ' ('()llvcrt.(:r-fcd motors for heavy duty applications such as traction or steel­ Illdl driv(:s . Compensated DC machines can withstand higher overloads than IlIJ('UIII\)(:ll satc(1 machinesj also the armature current may rise much faster 11.11< I larl~( T current harmonics are acceptable without detrimental effects on l'Ollllllllt.at.ioll, i. e. sparking of brushes. This is of particular importance if the 111:\.('i1iIlC is supplied by a static converter. 'I'!t (: co lI1mutating poles CP, placed between the main poles and also rarryillg t.hc a rm ature current, have the task of locally modifying the field in I.h(' ll Cllt.ral zone, to achieve rapid and sparkfree commutation. This is done I,,Y illdll Cillg a suitable voltage in the armature coil tempora rily shorted by 1.lte hrllshes. ' l'lw priuciple of commutation is illustrated in Fig. 5.2, where the c10sed n l'lll a l.lln' wiuding of a two- pole DC motor is schCIllatically drawn , showing I.hl ' hrllsll j)osit.iulls a t two cO!lseclltive ius tallt.H 0(' t.illl<:. Clcarly, when the r( 'c
--!?- ­ - - - - J :

/l.

Current in an

armature conductor

Fig. 5.2. Commutation of eurrent in an armature eoil, (a) Winding and (b) Cur­ rents in two adjaeent armature eonduetors

commutator acts like a multi-phase electromechanical switching converter which is synchronised by the rotor angle. While the current is being inverted, the commutating coil is temporarily short-circuited by the brushes overlaping adjacent commutator bars. Since each coil, embedded in slots and surrounded by iron , exhibits some induc­ tance, the commutation is a continuous process which requires a finite timej this limits the speed at which the motor can operate without excessively sparking brushes. For a two pole motor with 100 commutator bars operating at 3000 rev /min, the overlap takes about 200 f..Ls. If, due to excessive speed , the short circuit of the commutating coil is opened before the commutation is completed, a voltage is induced which causes sparking at the trailing edges of the brushes. TlI c) e[pc:\.rical torque mM is exerted at the rotor surface; more precisely, hy t,a.ll ~(~lIt.i a) llJagllctic forces acting on the sides of the slots. It is propor­ t.iOllil.I Ln t. IJ( ~ prodlict. 01' Illa.in f1l1 x and ill'llla(;lIrc CllITCnt. The volt.ages in the 1l.1'I11 i1.l.lliT ('i r('lIil. (·t1l1 siKt. 01' ali ill
\

5. Separately Excited

~ ', Ij

De Machine

5.2 Mathematical Model of the

IIllx and speed, in addition to resistive and inductive volt age drops in the commutating- and compensating windings as well as the arma­ 1. 11 r< ~ connections and brushes. The main flux is thus of prime importance for opcrating and controlling the De machine; greatest flexibility exists, when Lhe field current is supplied by an independent source so that it can be varied ;d, will. ln Abb. 5.1 b a simplified scheme of a separately excited De machine is shown, where the resulting magnetic fluxes are concentrated into lumped illclllctances which are orientated in parallel and orthogonal to the main field a x is. The total armature volt age Uo. thus consists of the volt age e, induced hy rotation, the voltage of self-induction Lo. dio./ dt and the ohmic drop Raia, whcre La, ROo are the resulting parameters in the armature circuito With la,rgcr machines, Raio. « Uo. is valid at normal speed and load. AIso, the p()wcr required for excitation, i.e. the losses in the field winding, amount to nlll y a few percent of the power converted in the armature. With motors having permanent magnet excitation the power losses in the lidd winding and the need for a separate field power supply are avoided at Lhe cost of operational flexibility. Servomotors for machine tool feed drives ;m! llsually of the permanent magnet type. Thc De machine as a dynamic system, including the interactions of the d(!('tromagnetic and the mechanical effects, is dealt with in the next section.

ia

; ~rlllature-,

;1.:2 Mathematical Model of the

De

Machine

'1'11<- (!quivalcnt circuit of the separately excited De machine can be repre­

schematic form by Fig. 5.3 containing concentrated components 11.11<1 :i l!owillg no longer a geometrical resemblance to the actual system. Til(! lH!rtinent differential equations are

::( ' 111.('<1 iII

Il" ('

'111M

=

=

,

'I".

'}', ti,

tlt

La ~ ctt + e = Uo.

-I-

Newtons law, concentrated mass

mM - mL ('2


at

N d
electrical torque

= Ue

field circuit

J(i e}

static magnetising curve, omitting hysteresis angular velo city.

=w 1 ':lilllillaLill/~
("i

armature circuit

La

ueQ

Ua

55

eJ J

Fig. 5.3. Equivalent circuit of a separately excited

De machine

where e ia is the instantaneous electrical power converted in the armature to the mechanical power mM W. Both must be identical, hence CI = C2 = C. Normalisation with the nominal values Wo


= eo = c

20.0

= ROo

mo UeO

= c
nominal speed at no-Ioad (base speed), nominal flux nominal armature voltage extr. current at standstill with nominal voltage (8-10 times nominal current)

extrapolated torque at standstill

nominal field voltage

results in the following non-dimensional state equations T

d o. dt

d T eo dt

(ii) _Uo. io.o - Uo.O -

ia W
~-fe (~:) = UeO (~:)

,

Lo. To. = ROo;

T

eo

= N e
(5.1)

(5.2)

induced volt age (e.m.f.)

P" w

/:1

llw ,j Ilt

}l"

I

'I,,,

Ra

De Machine

"I

H

'J

III ~ I I "

froul I

t.h(~

S('<:()\1<1 H.u
Tm

:t (:) (ê) =

d TE; dt

êO

ia
w

Wo

mL

mo

T _ êo E; - Wo

T

_ Jwo • mO '

m -

(5.3)

(5.4)

ie/ieo = fe(
5.3 Steady State Characteristics with Armature and Field Control

S. Separately Excited DC Machine

!dl

=0, ( ~) PeO .

~-fe

mL3

mo

UeO

57

(5.7)

mu mo~

~~

_ mL

PeO

mo

iaO E

'.!iL.

Eo

" ,' ,~

(5.8)

The angle of rotation e changes linearly with time; it is of no consequence for the time being. For the following discussion the normalised flux b=

~<1

PeO ­

which is limited by saturation of the iron core, is assumed as an independent control input, instead of ue/ueO. This results in

11;,0


Ue

Ueo

Jo'i~.

=0.

W _


Wo

ia

5.4. Block diagram of a separately excited DC machine

iao

L" i.ll( ~

larger airgap in the quadrature axis and the possible presence of a com­ 1H'll sal;ing winding Ta « T eo holds. Magnetic saturation in the quadrature ti I !('ctiO!l is usually negligi ble. Tb(~ dynamic system, described by Eqs. (5.1)-(5.4), is represented as a hlock diagram in Fig. 5.4, containing linear integrators with their time con­ ,~L;wts and nonlinear algebraic functional blocks; it is assumed that the load Loj'(l'w consists of an independent component mLl acting as a disturbance ;1.11<1 lWI) components as nonlinear functions of speed and position respec­ I.ivdy; wlwn the load torque shows no such dependence, the corresponding 1'1 I II cl.i OIlS are zero.

. - - - + f L2 ( -W ) + f L3

'/II./, ._ nlLl ~ W'll

mo

Wo

(e) eo ­

.

(5.5)

1

1

Ua

- b UaO

-

mL

b2 mo '

= ~ mL b mo

(5.9)

(5.10)

Hence the steady state behaviour of the controlled motor,

~=h

Ua mL) ( uao' mo

~=h

(:~)

Wo

iao

'

is characterised by linear functions, connecting speed, torque and armature current. Depending on whether U a or Pe , i.e. Ue, serves as control input, the term armature- or field-control applies. 5.3.1 Armature ControI

;•. :~ Steady State Characteristics wil.h Armature and Field Control 'I'lw

~t(~:tdy :;t,
i,; ()hl,;ÜII(~d

Assuming b = Pe/PeO = 1, the nonlinear effect of the two multiplications in I"ig. 5.4 vanishes and a particularly sim pIe set of linear control curves results,

cOIldiLiol\ , valid with constant input quantities ?La, u(., hy sdLilll~ til(' d( ~rivalive~ iu Eqs. (5.1)- (5.3) ('<juallo 'l;ero,

'U 4 ,

'/.4.

UI/I)

1 .. 11

( tJ

,I'.

{"u 1/'1'11

II,

'In/,

u,·(')

W

Ua

mL

Wo

'Uao

mo

'I.'L

UI. /.

"/II)

'III.()

(5.11)

(5.12)

r.H

5.3 Steady State Characteristics with Armature and Field ControI

5. Separately Excited DC Machine

.!L

(j)

1 1 1

iaO

59

the motor designer would not have fulIy utilised the potential of the motor ­ hence it is only feasible to weaken the field,

1

-1 ~ b = ip e / ip eO ~ 1 .

1 1 1

10,5 1 1

I

-1

Normal operating range

a Fi~.

b

:,illcc th c armature voltage U a is referred to its nominal value Uao, only t.I)(' Jalll~(~ -1 ~ ua/uaO ~ 1 is of interest. Excessive voltage would cause :11'11.1"11 i 11!', of t.he brushes and possibly commutation failure by formation of an

::iJ()rt.· cin:uiting the brushes (flashover).

'l'lw al"luature current is proportional to the torque, independent of volt­ ' 1) \" :1.1141 s p(~ed. Armature current and torque are normalised with their ex­ I.ra» olal.cd values at standstill. Therefore the normal operating region in Abb. ~I.~J I, is (:olltained in a narrow band, e.g.

-0.2

7nL

<-

rno

i = ~ ~ 0.2 , ZaO

whcll allowing a temporary overload of about twice nominal torque. ()liI.:, ide t,his range, the curves are likely to be distorted by armature reaction IUld I.h c !"c ll1ay be commutation problems, part.i<:lllar1y ou lllachines without ;1. ("oll1pt:ll~;al.illg winding.

, 'V,' II

. J. :I. ~

Fi(' ld COlll.rol

mL

standstill:

W

= O,

WNL

Wo

I )n;v

(,\I

1llll. llrl d,LlIlI 'I',.

(· llllllol.

I,,·

I, y ,'1. /1111'.111/ ; Ir,lI e IJl1I.iu flu x 111111 1'1 1 1" .' V' '11d 'I', fi "1.",,rw i ~I( ·

IU"(." " \"

'·I U' I" ' II t1(·d,

= O,

mst

mo

=~

b'

=b .

With reduced flux , the no-Ioad speed rises, while the (extrapolated) stalIed torque is reduced. Hence the slope of the curve is varying with 1/b2 ; this inherently "soft" load behaviour is undesirable with drives to be controlIed for constant speed. As is seen in Fig. 5.6 b, the armature current and the armature losses in­ crease when the motor is operated at a given torque with reduced field; there­ fore, as already mentioned, field weakening makes only sense if the desired operating point in the torque-speed plane cannot be reached by armature voltage control at fulI field or if tbe armature voltage is fixed. At less than maximum armature voltage, operation at reduced field is uncommon, except in special circumstances, for instance when several motors are supplied from a common armature bus voltage and only minor speed variations are needed as on some paper mill drive systems. Neither is field weakening a solution for the start-up of motors since at standstill the armature current is not affected by the main fluxo Fig. 5.6 a indicates also that a reduction of the field does not necessari1y lead to higher speed; because of the steeper torque/speed curves the result of field-weakening may even be reduced speed at higher armature current which is of course most undesirable. The maximum speed at a given Lorque is obtained by differentiating Eq. (5.9) with the condition U a = UaO,

ri W'" W

Til" 14" \""[[4 1 ''111.1011 fo r .." n Ll",,[,jill 'll I.hll
no load:

5.5. Steady state curves of DC motor with armature contrai, Pc = PeO

!lu:y
11.r ..

If the armature power supply permits operation in alI four quadrants of the ia-plane, it is sufficient to restrict field control to positive values, bmín < b ~ l. The purpose of field weakening is to raise the speed at light load, as seen from Eq. (5.9), accepting the disadvantage of increasing the armature current for a given torque, Eq. (5.10). Therefore it is normally not appropriate to employ field weakening unless the possibilities of armature control have already been fulIy exploited. Inserting ua/uaO = ±1 into Eq. (5.9) yields a new set of steady state curves with the normalised field factor b as control parameter; the character­ istics are again straight lines, however with increased slope (Fig. 5.6). The speed/torque curves intersect with the axes at

Ua,

tlll

I ,'/tI ,

= _ _1 ('()ll"( ' ,) ,

whi .. h 1" 1\.,[ ,1 1. ..

II.

b2

+ _2

mL

= O

b3 mo

p rllcl.i cnl lowcr Iilllil

1'0)'

Lh (~

fit:ld

1;11

5. Separately Excited DC Machine

(U

5.3 Steady State Characteristics with Armature and Field Control

..!L

(f )O

torque but high rolling speeds are desired for saving time. A similar situa­ tion exists with coilers where high speed and low torque are needed when the diameter of the coil is small, while the reverse is true when a large coil diameter has built up. After eliminating the field factor b from Eqs. (5.9), (5.10) with the con­ dition u a /uaO = I, an expression for the mechanical power is derived

iaO

.7

1

1

0,5

w mL

Wo mo

!!!..L

1

mo

!!!..L mo

operating range

a

b

Fil-'; . 5.6. Steady- state curves of DC motor in field control range, mL

--, I',",i" -- 2 mo

Ua

= UaO (5.13)

w lll'J'(' Lhe undesirable speed reduction begins. Note that mo is the ex­ f.liI.polak
I !j

'/1/.0 I

II 'I I L I,

(5.14)

", I,i,·11 1"'IH'('SCllts ali envelope for the family of torque/speed-curves with pa­ I ; t.lII'·1,1"I' I, . lVIaximum power is obtained at the point, where the straight line i:: 1. :1.1'1';"111. 1.0 the hyperbola. To avoid the situation mentioned before, the

sl!oltld lIever be operated at the right hand si de of the tangent point.

11101,01

~

IIlachillcs without compensating windings show a pronounced arma­ in the field weakening range because the field distribution in 1.1 1<' a i I'f ';ap i,~ 110 longer stabilised by saturation. Therefore the validity of the l.orqlll:/sl)(!('d -cnrves is restricted to reduced values of the normalised arma­ 1111'(' CIIIT(!lll. ; higher currents have detrimental effects on commutation and 1I1:1,,\' ('('slIlt iII sparking of the brushes. This calls for a rcduction of the max­ illllllll :i)'] II ature current at reduced field and hente a [low(!r derating. I )c::piL(! tlt esc drawbacks, field wcalccnill[.\ is n valoald(! lllcallS for increas­ i 11/', UI/: sl )('(;d ill t.Jw low torque r egiOllj typi(:al ll.pplil"it.l,io ll !-l an: spilldle drives illl Jllitchill(; l.oo1 s to allow cl\t.til\l~ at, fllll pOlVt'I , i'Vi'lI wil.lt a slllal! diallH'Ler W Clrli pi ( '( ' (' III' I.oo!. Ot.!w(' ('X illllpks lU< ' rt'Vi'), N h lj ~ I "I II II/', IIIIII " .!"III' l'o
1111'"

("( ;a("(.i O Il

61

ia iao

(1-~)

(5.15)

iao'

which only depends on the normalised armature current. Rence, with lim­ ited armature current, the motor can be operated at variable speed with a given maximum power , indicating that field weakening has a similar effect as a variable gear for increasing the speed of the load. For the reasons al­ ready mentioned and because of the mechanical stresses, the maximum speed achieved by field weakening is sei dom higher than two or three times base speed depending on the design and the size of the machine. Rowever, there are exceptions, such as on vehicle or servo drives, where a much higher speed ratio may be necessary. The similarity of a speed increase by field control and by a variable gear is corroborated by their effects on the mechanical time constant. At nominal field, Pe = Peo, the nominal time constant is Tm = Jwo/mo, where Wo is the base (no load) speed and mo the extrapolated stalled torque at nominal armature voltage and field . With reduced field, the no load speed is raised to b-lwo , while the stalled torque is lowered to b mo. Rence the mechanical time constant is a quadratic function of the field factor,

Tm (b ~ 1) = J Wo b- 2 == Tm b- 2

•

mo

A similar effect is observed if the load is coupled to the motor through a step-up gear having the speed ratio b- I ; as explained in Sect. 1.3, this would increase the effective inertia of the load, as seen from the motor side, by b- 2 • 5.3.3 Combined Armature and Field Control

From the preceding paragraph it is seen that armature and field-control of De motors each have their merits but that their normal applications exclude each other. Below base speed Wo the main :fl.ux is best kept at nominal value PeO while the speed is varied through the armature voltage U a . This is called the base speed- or armature control range. When the nominal armature voltage bl a O is reached, a further speed increase is only possible by lowering the l\laill flll X, thus creating separate field-control regions which extend the base ;) I w l ~ d r a ll g (~ iII cadl (lirectioll of rotation . Thi s is depicted in Fig. 5.7. li. i ~: fI(' I'1I I.ltal.
5.3 Steady State Characteristics with Armature and Field Control

5. Separately Excited DC Machine

(i'2.

e UaO

4>e 1

- - - __ I I'

1

e = const

I

UaO

,, "

,,

e = const (J)

,,

--UaO

roo I

_

' L ____/ . : "__ _

"

1

,

Field weakening ' I Armature range ----~,~~I· contrai range ~

- -rr-.- -

W

Pe PeO -

(5.16)

-=--<1'

4>eo

,/

I ,

I / Field ~~/~---

'I'

t----~

Wo

63

,

this causes the desired variation of the flux in the field weakening range (see also Sect . 7.3). The operating regions with armature and field control in the torque/speed plane are displayed in Fig. 5.8. The curves ia =const. are represented by parallel straight lines in the armature control regimej in the field weakening region, the torque at a given armature current is reduced, because of mM ~ Peia. For the sarne reason, the curves for U a =const. become steeper when the flux is reduced. ln the upper right hand comer a typical commutation limit is indicated belonging to a reduced armature current limit when the field is weakened.

weakening

range

-'l"eO

Fif.';. 5.7. Control ranges of separately excited DC motor in steady state

(J)

Field contrai u a == u aO 4>e< 4>eo Armature control

wil.l! the sarne polarity of the armature volt age by reversing the field (dashed cllrves); at the sarne time the torque would be reversed so that the motor olwrat.es in the third quadrant. However, this method of "field reversal" which wa~; frequently used in the past for cost reasons is rarely employed today :I,S i l. calls for the de-excitation and subsequent re-excitation of the motor; I.hie; n~qlli.r es time in view of the large magnetic energy stored in the field .. i)'nlil.. rI' lligh forcing volt age is applied to the field winding to accelerate the !III X n ~ v(~rsa.l, considerable voltages are also induced in the armature winding 1'11' 1;1.llf~( ~rillg the commutator. AIso, the armature current must be blocked dllrill!', I.he time of flux reversal causing an interruption of the torque. With 1.11/',1')' !IIotOl'S, the transition time with zero torque may be up to Is which i:: IIII!. acccptable for a fast reversing drive. It is for these reasons that drives wlticl! are cxpected to operate with both directions oftorque and speed ("four qllOldr;wt drive"), are today usually equipped with an armature power supply pl'Oviding both polarities of volt age and current ("four-quadrant DC power :,lIpply") and capable of operation in all quadrants of the ualia - plane, Sect. ~) . I; l.lIi s ha.s the advantage that the polarity of the main flux can remain Illldl
,

• 2- Troo

I I

,

/

:::r-­

I .ji I

§I

~

0., I

-uaO < ua < uaO I

I II I __t1J I

__

II I 0_t1J 1

4>e= 4>eO

-2'71:

07 -1

Field contrai Ua == -UaO 4>e< 4>eO

Commutation limit

I

I

I

Ua = UaO

§I

I

0- I

II_I ....:u· • .'t1J' 2 ~

._ ~

,

\ili \

\

\

.

\

1-2

'

'

/

I

,

mM

ua- O mn Ua = -uaO

I

,I

Fig. 5.8. Operating regions of separately excited DC motor in torquejspeed plane

The figure clearly shows that the DC motor represents a linear control plant only in the armature control regionj beyond base speed, there are con­ siderable non-linearities even in steady state condition. Simultaneous field- and armature-control is sometimes proposed as a means of reducing motor losses through diminished copper losses in the field windings and iron losses in the armature; of course, this is only applicable at very light load when the increased conduction los ses in the armature are not masking this effect. AIso, this may cause an objectionable delay in building up motor torque, should an unexpected load surge occur, because the flux would first have to be raised to full value. Hence, this practice can only be a lIIurr;inal option in few applications, such as battery supplied electric car dri v(~s, w II\~ rI ' (~I\( ~J',.;y conservation is of overriding importance and where the dyll;Ullil' drawll1l.('h :: ;I.r<: 1I0t. ohjcdionahko

5. Separately Excited DC Machine

(iI!

5.4 Dynamic Behaviour of DC Motor with Constant Flux

;:).4 Dynamic Behaviour of DC Motor with Constant Flux

I ()=F()U () a S

III order to have a clearer view of the transient behaviour of a De machine the IJlock diagram in Fig. 5.4 is simplified by assuming the armature voltage to be :ln impressed and independently variable input quantity, the load torque mL ;\.s an independent disturbance and the flux Pe a suitably chosen parameter. This results in Fig. 5.9; the angle of rotation is again omitted for the time Iwing, as it does not affect the operation of the motor.

a S

3 S

+

F()M ()= TmsUa(s)+bML(s) 1 S LS T m S (Ta S + 1) + b2

65

(5.18)

Ua(s) is the independent input or actuating variable, ML(S) is a disturbance, also assumed to be impressed. il(s) and Ia(s) stand for output variables, which appear in the frequency domain as linear combinations of the input variables Ua(s) and Mds). All partial transfer functions of the multivariable plant have the sarne denominator, N(s) = T m S (Ta S

+ 1) + b2 =

T m Ta (s - sd (s - S2) ,

(5.19)

mL

mOn

Ua

which is identical with the characteristic polynomial of the pertinent linear differential equation. The zeros of N(s) are the eigenvalues of the system,

li)

LlaQ

LlaO

Io'i~.

2~a [-1 ± )1- (2b)2Ta/Tm]

S1 ,2 =

e

the natural frequency is 5.9. Linear block diagram of DC motor

W

Assuming the main flux Pe to be a constant parameter, a linear sys­ I.em rcsults which may be described by transfer functions; with the Laplace­ I.r;I,IIs[orms

!

e=

b

~= .JTaTm '

/'(:r(/.))

=

D = - Re(sl) = ~

ISll

x(t)

dt

e-Bt

= X (s),

(5.21)

and the damping ratio

00

2b

rr;;:

(5.22)

VTa .

Field weakening (b < 1) causes the motor to respond more sluggishly to control or disturbance inputs as is manifested by a lower natural frequency

S=r7+jw,

o

,)1' !.II<' IIo11dimensional variables

I,

'1/,,, )

(

-

u"o

{, ( ,1. (1 )

= Ua(S) ,

= Ia(s) ,

1',,0 ;\.11.1

l.h( ~

L(:) L

r-2~a

= il(s) , b max = 1

(:~) = ML(S)

jw

o

1s, 1 1

À I I I

llonnalised field parameter

---7<-~--~---X


bmin

(/Jco

[,II<'

(5.20)

I

• O'

b min

I I

l'oll()wi])l~ (~'lllil, l.iolls aH'

t

olJlaill<'d

I

J1.( ,'/ )

1"1 (.~) II,, (.~ ) I I",{~) tvI/,H

/1/1"( "'1

('1:,,'/ I. I) M,.( s)

'/'", II ('/ : ,

II

blll/l x = 1 X S,0

I I) I ,,~ (!) I '1)

Pi ,r. r. . IO.

1()1)1.

1,)(',,::

"r

1)(:

IIlOl.or

for dilr"I'I'''1. li,'ld facl ,or>< " .

5.4 Dynamic Behaviour of DC Motor with Constant Flux

:.. Separately Excited DC Machine

lili

lilld illcreased damping ratio. Fig. 5.10 shows the root lo cus, i.e. the lo cus of 1.1... eigenvalues in the complex plane, for different values of the parameter b. lte;d ('igenvalues, corresponding to asymptotic transients (D > 1) exist for 'l~"

/'~~-:-1-~0--=:;;---i\7a 2' - ,2 I"

000

/

:

//_.....

--- ""-_______________

./'" f----.

> (2b)2 Ta .

1.11 Fig. 5.10 it is assumed that the motor exhibits below base speed (h",,,,,, = 1) a damped oscillatory response. This may in practice be the case

ir

00

,, 1 0=1 b =."'F,ff;'; ,2

, \b= l' 1

i.J\(~

inertia of the drive is deliberately kept smalI as with servo drives or 1'011 iIIg-mill reversing drives but in most cases the eigenvalues of a drive are ITal Ilccause of the added inertia of the load. III Fig. 5.11 the step responses of the motor speed and the armature cur­ )('111. are depicted when the motor, initially at standstill, is started without I()ad (mL = O) at t = O by switching on the armature voltage U a = 0.1 Uao. i\{1.(~r attaining its steady state speed, load is applied at h in the form of con­ s!.allt torque, mL = 0.02 mo, corresponding approximately to 0.2 of nominal I.orq Il e. The transients in Fig. 5.11 have been computed for three different values (II" II, which result in the damping ratios D = 1/"fi, 1, "fi. It is seen that field w(~akcning makes the motor much more sensitive to loadj at higher torque 1.1t(~ speed under load would even drop below the value at full flux (b = 1), ;(.s indicat.ed in Fig. 5.6 a. However, the initial slope of the speed at tI is UI<' saltJ( ! in alI cases being solely dependent on the nominal mechanical time CIlII :il.allt.

li

d'

Tm

,

( 1.<.1) I Wo

= _ mL

~ I- f(b) .

Tili:: i:; illlllwdiately understood from inspection of Fig. 5.9, as the current I" , i.('. I.h( , lllotor torque, is delayed by the armature inductance La. TI\(' sl.( ~ ady-state value of the armature current for no-Ioad condition (ne­ I',i<-dill/'; I"rictioll and windage) is zerOj this is apparent from the factor s in I. IH · 1IllIllcra!.or of the transfer function F3, Eq. (5.18), which indicates a dif­ r( ·J( ·III.i;d.illg dfect due to the cancellation of the applied armature voltage U a 11'y LlI(, illdu ct!d voltage e at no load. II" Lhe loatl torque contains a component rising with speed, Fig. 5.12 a, ·III. /, '111'11

k

'T nLI

-

W

Wo

hloek diagram in Fig. 5,12 b results; the integra!'o}' JlOW possesses pro­ p()rl.ional ne(~at.ive feedback, assuming the prop( ~ r!. ks ()I" it lag d emento The diJr(·J'('llliat.ing efI'ect ill Eq, (5 ,] 8) disapJl( ,a.rH ill I.lJl' pr(';'; ('lI('.(' of loa
I"

'"

j

II

'"

II

iI

~

l . I, I,·', ~ l rI -' ~ I', , I ('t' 'IJ'J, ~ ~ -., ''~ I h

-

1

i2 -

........

, \)a '

---,

_____

n

I

ti

..!.L iaO ,",

; " :/'\ '. 0,05

\ ,

\

.,.,~ ...

b

"

mL -

,

" '..

". ......

-

°

02

---_._­

.....

......:.:----------­

mo~' .~~~------------~-p-~

",

-"- ...... .... -

11/ 1 ' ti

b

with k L > _b 2 and kL > - T m/Ta required for stability, The damping ratio DI may be referred to the damping ratio D at no load

mu

00

Wo

arctan kL

!!2L .!!L

­

uao

!!2L mo

Lhe

(I" ( II)

0-

- . --.----­

"

I

a

(5,23)

- +'L -

rno

I

Fig. 5.11. Starting and load transients of DC motor for different field factor b ~ 1

mo T m

t,

'. ....... _

(1

' I"

'I

0,1

67

U) ~ I I)

e

uao

a F il!!; - r.,\:l_ 1)( :

b 1I>tIi.ol'

wit.ll speed- dcpendcnt load

mo

w

5. Separately Excited De Machine

fiK

n1

= D

1 ~ 1+(W)2b

r:-:-;;;:

(5.25)

6. De Motor with Series Field Winding

yl+V drives are subject to disturbances by periodic components in the supply ,11.a.ge or the load torque, causing continuous oscillations of armature current alld speed around their steady state valuesj the transfer function permits a :: i IIlple computation. When a torque consists of a constant and a sinusoidally varying component ,'-)O IIIC V(

'ln r-

mLl - - = --

mo

'11/, 0

~ +mo

coswmt

('ae h variable of the linear drive will in steady state contain a periodic com­ plllll'llt of the sarne frequency that may be computed by linear superposition. Wil.h the complex expression 7nL

mo wc

COSW m

t

~ JWm . . = --[e t + e-JWm t] 2mo

lilld for instance from Eq. (5.18) the peak value of the alternating current

COlllpOllent

/h " 'i" ,, ' -

(kL = O)

IT

m Ta (j w m )2

b

+ Tmj W m + b2

I ..1mL mo

(5.26)

;; ;"Iil;(r! y 1./11 ' plta.se of the oscillatory current may be determined. If the fre­

lwriodic disturbance lies near an eigenvalue of a poorly damped condition could exist causing large amplitudes of the oscil­ l:d,II I" Y ('III'I"C ' II(. all(l speed deviations. '1'11 1' dYII
'111( ' '' (''y .. r ;\.

"I I v(·)

ii

rc 'sollallt

This motor differs from the one previously discussed only by the design and the connection of the field winding which, according to Fig. 6.1, carries now part or all of the armature current. Ral is the armature resistance, possibly increased by an external resistor, Rp is an adjustable shunt resistor for field weakening. Because of their operating characteristics, series-wound motors of larger rating are restricted to traction drives. ln urban transportation, such as trol­ ley buses or subways, motors up to about 200 kW are used but for main line locomotives motors of 1000 kW are common. ln principie, series wound motors can be fed with either direct or alternating current but the construc­ tional design would be quite differentj for example, a laminated stator and special provisions with regard to commutation are required for the AC mo­ tor. When AC traction was developed at the begin of the century, the current induced by transformer action in the armature coil temporarily shorted by the brushes during commutation, Fig. 5.2, caused problems, which were then solved by choosing a lower supply frequencyj this is the reason for the 16 2 la Hz power supply used on central European railways until today. Since com­ pact and rugged power electronic components have become available, main line locomotives with AC supply were often equipped with rectifier- fed DC motors. ln future, all these schemes are likely to be supplanted by AC motors fed from variable frequency inverters.

ia-ie

mM,W,E

ie Re

cI>e

Ua

Rn-R,11

Fi)·:. O. I. 1)( ' "" .<. .... wi<.h ,,, 'ri "H fi..!d Willdillf(

'lI!

6.1 Block Diagram of a Series-wound Motor

I)C Motor with Series Field Winding

(j,

~ ; lIlitll

ltigh-speed AC series motors, so called universal motors, have a wid(' fidd of application in household appliances and power tools. 'l'lw typical operating characteristics of a series-wound motor could also 1)(' rcalised with a suitably controlled separately excited motor; this can even 1)(, prcferable in view of the added flexibility, for example when a traction 1II0l,or is supposed to serve as a regenerative brake.

Tal

d dt

(La

l

Ral

T

1

e

!i (~) dt
1'(jllaLions describing the dynamics of the series-wound motor are similar c" LlI()SC derived in Chapo 5. The field and the armature windings are orthog­ (III:'" Lo (~ach other and hence are magnetically decoupled. Due to the large air!·,i l.p iII the armature axis, saturation is neglected and the magnetic circuit i" (1'ladrature direction is assumed to be linear. R' a ta

-

R p ('ta

,)

- te

-

e,

(6.1)

" dPe ,TV,, - = - R' e te

+ Rp (.ta -

dt

e = c
=

lnM

.I

, ), te

(6.2)

T

c
Re Ral Re

+

ml

d dt

(w) = WI


T

ie

Ra Ral

+ Re

-

_

La Ral

+ Re '

aI -

i al

~+~ i al

Ral

ial

(6.8) '

mL mi

Tml

=

JWl mi

(6.9)

.

:t

(Ne
+ L a ia) = Ua - e -

(Ra

+ R e ) ia

(6.10)

.

(6.5)

- = 'f/I,M - mL . dI.

1) is defined at nominal armature volt age I. wit.JIIIIIL lidd w( ~akcning (R p --+ 00) and with no external armature resis­ I
mL

1/ "

1",Jd "

mI

..!!iL Ual

(' ,

C '/'",

"' II I

'l ' I

'/ II, I

I:

(

w, =

= (1 -

J.L

'T}l Ual

Ual

'T}t)

iao

e

uao

-;:;----+ Re ai

iPt: 1 i al .

(', / 11" I i~ tlte electrical efficiency at the nominal operating point (not for iron- and friction losses). These quantities are used for nor­ III:di s llq'; L1l( ~ Eqs. (6.1- 6.5).

'II

1 J( " '''11I1LiJl !~

'I." 'f'ul

Pp. )

Jo: (
(6.7)

ia - i e

+ Re

Normalisation, as before, yields

Ilw

ia i al

A graphical representation of these normalised equations is seen in Fig. 6.2; the continuous state variables appear again at the outputs of integrators. The effect of the parallel resistor R p in coupling the two magnetic circuits is clearly recognisable. Without field weakening, i.e. Rp --+ 00, ie = ia follows; armature and field winding are then forming part of a common electrical circuit, hence

(6.3) (6.4)

,

+Re

w)


'T}l
ia -

N e
'1'1",

di = U a -

1 ( Ua = 1 - 'T}l Ual Rp

H.l Dlock Diagram of a Series-wound Motor

.Ia dia

)

71

ia I cPe

(6.6)

CPeO

I:: !.I", lIul.glldisation C1lfve in the longitudinal axis oi' the machine. As i I, r('s llll" t.ltrc~(' (liuH'Ils ionless (eXC!lpt Liul(') fir s l. onkr s Latc cqllatio lls r,,1' LIli' 1.111'<'1' elll'I'!';.\' ~; L(lral~c va riahlcs 'i ,,, 'P" , I." :1.11' o!.l.ailu "I:· 1"11-\ . n.~ . 111", '1, " " 'I'/ IUII

(Ir n 1Wl'i,'!j- W
1)( ~

IIllIl,"f

.,

DC Motor with Series Field Winding

fi.

d (Pe

"

dt

.fel

Pel

La i a1 ia) i a1

+ N e Pel

= 1-

1

6.2 Steady State Characteristics

( "h

Ua U al

-

7]1

Pe Pel

w)

6.2 Steady State Characteristics

Wl

Ra + Re ia Ral + Re i a1 . 1kc,lIlse of the larger airgap in the quadrature axis, La i al

(/ = N e Pe1

_ !lial - !lie1

«

73

The steady state behaviour of the motor is again found by putting the right hand sides of Eqs. (6.8 - 6.9) equal zerOj with Eq. (6.12), a linear magnetic circuit is assumed in the longitudinal direction of the machine. Elimination of ie/ial' results in

1.

-U

a

U al

=

[w 7]1 b -

Wl

+ (1

-

7]1)

r

]

.ia Zal

(6.13)

WiLh Eq. (6.6), noting ie = ia, Eq. (6.10) is transformed into

where

w)

( -Pe )] = -1- ( -U a - 7]1 -Pe P el Wl Pel 1 - 7]1 U al Ra + Re ia (6.11) fl al + Re i a1 The associated block diagram is drawn in Fig. 6.3 for q « 1. A character­ isLic feature of the series-wound motor is the dependence of the flux on the ;U'JIlature current, which causes it to vary with the load. Since most of the ali 11 ature voltage is balanced by the induced voltage, U a ~ e = cP e W, high ".pccd is to be expected at light loadj this is reflected by the steady state CII rvcs discussed next. To gain a better insight, saturation of the iron may be ignored which in I"ig,~. 6.2 and 6.3 is achieved by omitting the nonlinear functional block, Eq. T e1 -d [Pe dt Pel

+ q fe

(lU.), ,i "_. = "." I

f (~) Pel

Pe

(6.12)

Pe1

t\ I. h il ';lI cunent, when saturation cannot be neglected, the motor character­ d .ícs
b= ie = ~ ia Re + Rp is a factor characterising field weakening and

Ra + bRe R a1 + Re

r=-----...:.

,-

- -----

,I

ia i a1 =

..!..!.L.

mL

m..

1(1/1

11" ,

()

11. 11

','IH . lI . lI . l!l,,, '" dlil !", li 11/1 " I rll·r h·./l

IVlllwd

II< :

Il u d ," , '

-V IU"," ', II" L
W

R a1

+ Re

(6.15)

(lmL

Vb -:;;; .

(6.16)

Eliminating ia/ial from Eqs. (6.13), (6.16) finally yields the torque/speed curves of the unsaturated series-wound motor ~ tLal

_

Wl -

ia1

R+~ a Re+Rp

is the ratio of the effective armature circuit resistance to the nominal re­ sistance without parallel and series resistors. Eq. (6.13) indicates that the armature circuit of an unsaturated series-wound motor appears like a resis­ tive circuit, whose effective resistance depends on speedj this is of course due to the induced armature volt age e which is proportional to armature current and speed. Likewise we find from Eqs. (6.5,6.9) with mM = mL

W

J.L

(6.14)

7]1

1-

7]1

r

Jb!!!:k - ----:;;;- b ' m,

(6.17)

where the normalised armature voltage U a /U a l may be continuously vari­ able whereas field weakening (b) and armature resistance (r) are subject to discontinuous control actionsj the load torque mL/ml represents again a dis­ turbance. A set of these curves is shown in Fig. 6.4, calculated for 7]1 = 0.9. As ex­ pected, the torque-speed curves are very steep at light load. This is desirable for vehiclp. rlrives, because the motor can be operated over a wide speed range anel sharill('; Lhe load on multi-motor drives is no problem. The similarity of t.ltc CII)'V(' S L() I.II( ~
6. DC Motor with Series Field Winding

7'1

6.2 Steady State Characteristics

'l'lte speed rise at no-Ioad could of course be a source of danger in other than I.raction applications. The armature current/torque curves, also depicted in Fig. 6.4, show a silllilar increase at field weakening as was observed with the separately excited IlIotOr. ln view of the higher copper losses in the armature and the more nitical commutation at reduced field, the amount of field weakening must be I(~stricted here too. On street-cars, trolley buses and subways, where normally a constant di­ rcd. voltage is used for supply, as well as with battery operated vehicles, :iwitched armature circuit resistors may be employed for controI. ln recent )'('<\rs electronic choppers have been introduced which permit continuous vari­ ;\!.ion of the mean armature voltage. At higher speed the field may be weak­ ('lIcd by parallel resistors. ln addition, if several drive motors are present, they (";111 be connected in series or parallel, thus providing a simple way of changing I.he voltage in a few steps. With city trains or main line locomotives having an /\ C catenary the armature voltage can be varied in fine steps by transformer I.a!> changer or continuously by power electronic devices; rectification may or 111;\'y not be included. To reverse the torque, which is proportional to Pe ia, a reversal of the field willding is required; the machine can so be operated in the motoring, i.e. the lirsl; anel third quadrants of the torque/speed plane. When operating in the ti('coml and fourth, i.e. the generating quadrants, an unstable condition exists wliic" is manifested by a horizontal asymptote of the torque/speed curve at n ("lil.i(";\1 speed, Eq. (6.17), Lv /" ( 1. '1

III

l-r/l~
This constitutes a state of self-excitation, where the effective armature resis­ tance, contained in the brackets in Eq. (6.13) becomes zero which means that the armature, being driven in the second or fourth quadrant by an active load torque, is supplying the los ses of the armature circuito The result is a steep increase of the braking torque produced by the machine. The critical speed Wc is a function of the total armature circuit resistance and field weakening but is independent of the applied armature volt age. The effect of self-excitation is seen in Fig. 6.5 showing a calculated tran­ sient with an unsaturated machine. At first, the motor is accelerated in the first quadrant at no load with a small value of the armature volt age U a until time t l , when a constant load torque mL is applied which decelerates and eventually reverses the motor into the fourth quadrant. When the critical speed Wc < O is reached, self-excitation in the form of a nonlinear transient sets in with speed oscillations and high peak torques which in practice could endanger the motor shaft. Since this effect does not depend on the supply voltage U a , it can be utilised as a supply-independent electric brake in vehi­ eles; this is described in Fig. 6.6, [B2] . mM. W

mI '

Wl

mL _ mI

!!!.b...(t) m1

\\

I'

I

V'

'>

Acceleration

at no load

(6.18)

-~ b

Wc

f4\ ia -.::..;/_ia1

Wl

- !b- ! - !r -R a

UUal

- RI

-

ie ia

(J)

®

® 8)

Load applied

mL mI

=0,25 =const

Nominal operation

Fig. 6.5. Transient of self-excitation of a series-wound motor 0,5

1 1

I

4 1

I

0,5

p

- p; =const 1111

"" t< ' liA .

75

:-1 1,' ·II .~ y .'I l.Id." .. l. ar l"·C," ill l,i" r! I,r 11 11' \1' 1" 11

\V""""'

1)1 :

,,,,,1.,,,'

It is assumed that the field winding has been reversed and the circuit is elosed through a braking resistor. The motor is running with W > O, so that braking calls for operation in the second quadrant of the torque/speed plane; it is controlled by the effective armature circuit resistance, Eq. 6.15. A detailed analysis of the braking transient requires the inclusion of iron saturation, but the principIe may be deduced qualitatively from the curves in Fig. G.a h, WIICl'C e(ia) is the induced armature voltage at a given overcriticaI :·qH'd nlld U. '':" i:; t.IlC voltage across the total braking resistor including the 1'(·:1i: ;1.1\1I("(· 11'." 1 ,"I'!.lll' arlllat\l),(~ itsdf. III Lhe initinl pha..se, the current is elriven II,Y 1.1", 1" '11 ' fi 111'11' ,. 1'1111.11.1',(' ('II, wil,ltolll. ,'x·I.('J'lIal V()It. nl ~(' lwillf~ appli(~d . Tlw

7(;

6. DC Motor with Sedes Field Winding

e, Ria

e(ia , (O) (O

R


= const

(O

e-R,'a--N d
-- el a)

b)

dt

ia

Fig. 6.6. Short circuit electric brake with series-wound DC motor. (II) Circuitj (b) Characteristics

ral.( ~

of rise and the eventual stable operating point Sare determined by the

(I iIkrcntial equation dPe N .e -

dt

= e ('za, W ) -

7. ContraI of a SeparateIy Excited DC Machine

R'za,

;11':aill llcglecting the self-inductance of the armature compared with that of I.h(' lidd winding. ln the steady-state condition the machine then operates :t.; :I. 1': ( ' II( ~ rator, converting mechanical power to heat in the predominantly ,' xl.('1I1;d hraking resistor R. (>II Il laill lilH~ locomotives, the drive motors are often reconnected as sep­ dl .d.('l y (· xcil.c( l generators during braking operation, in order to have more d i I(., I. (o/llrol. The auxiliary field power may be taken from a battery, if :: lIl'pl ,\' illdq ) ( ~ lId e nt braking is specified. 1\11 I:x: l.lllplc of controlling a DC series motor is discussed in Sect. 9.2.

7.1 Introduction ln Chapo 5 the steady state and dynamic behaviour of a separately excited DC machine with adjustable armature and field voltage has been explained; this discussion is now extended by considering the machine as part of a feedback control system. The reason for this is that in practice the choice of a DC drive is normally motivated by the possibility of operating over a wide range of the torque/ speed plane with low losses and matching the behaviour of the motor to the need s of the mechanicalload. To achieve the desired operating char­ acteristics in the presence of supply- and load-disturbances, feedback control is usually necessary. Another reason why DC drives are normally contained in feedback loops is that the armature of a larger motor represents a very small impedance which - when supplied with nominal voltage - would result in an excessive current of up to 10 times nominal value. Under normal oper­ ating conditions this is prevented by the induced armature voltage e, which cancels most of the applied voltage U a so that only the difference is driving the armature current ia; these two quantities are performing the actual elec­ tromechanical energy conversion. ln transient operation, for example while accelerating or braking, there is always the danger of excessive current due to the rapidly changing armature voltage or sp eed; the same is true with a steady state overload on the motor. It is therefore important to provide a fast current or torque limit to protect the motor, the power supply and the load; this is b est realised by feedback control establishing an effective safeguard against electrical or mechanical stresses. At the same time an unequivocal criterion is gained for distinguishing tolerable temporary overcurrents from currents caused by a functional fault which calI for immediate clearance by a breaker or by fuses. ln most cases the user of a controlIed drive wants to b e able to select a reference speed wRef which the motor should maintain as -long as it is not overloaded. When overload occurs, the motor should produce maximum torque ill th c pr e~ crib e d direction; twice nominal torque is often specified as a shorl. Lcrlll lillliL This results in stea.dy st at,(, eharacteristics of th e tyP( ~ Slt O WII iII I"il ~ .

7. 1.

111 1l. 1l 'y I\. jljll i.-: tI.ioll S, sHch as (:!<: viI,l.()r H, 1.111':-:(' i'l jl('c ilicHI.i o IlH ar(' ill1mJ' II('i" 111. I",\ ·H.IiI II· 1.11"1'<' 11 1'(' "I)I'I·a l. iI !J ( ,·ollel il.ioll il Wh" 11 I.h,' 11.,...,·I'·I'il.l. i"lI (II ' 1.11<' 1,'01'

7. Control of a Separately Excited DC Machine

'7il

7.2 Cascade Control of DC Motor in the Armature Control Region Supply

Torque limit

ú)

cRef roRef

normal operation

79

aRef ~

ú)Ref

Ye

Rotating exciter or converter

ie

!Ue

I IRe

~

C>

-mn

mn

2m n

m

liamaxll

~

c::

C> ()

- -Ú)2 - , - - - - - - - - - - 11 1

"'i~.

c,rol

lia

lUa

Rotating generator ar converter

I lU a

7.1. Steady-state torquejspeed curves of a controlled drive with torque limit

angular position must be controlIed according to prescribed references. AIso l11achine tools or robots, the feed drives in the different axes must accu­ ral.dy follow prescribed angular commands as functions of time, in order to 1I10V(' the tool or workpiece along a desired spatial trajectory. Ali these requirements can be fulfilIed with the general scheme shown in Fig. 7.2 where the armature and field of the motor are fed from separate con­ I,mllaillc power supplies. Rotating generators in the form of the welI known Wal'd Leonard scheme have been the common choice in the past but, after l, ri,;I" 111(.('rllldes by magnetic amplifiers and mercury-arc converters, power ,'I,~,·(.r,,"ic COllvcrters employing solid state semiconductor switching elements (\';lli"II:: I.ypes o[ thyristors or power transistors) are now the standard solu­

()Il

I. i" II,

W iI, h a "r()1If quadrant" armature supply allowing both polarities of volt­

1\.,"1 Cllrl'(~lIt, the power flow to the motor can be reversed, with the III III 'I,ill" "perating in alI four quadrants of the torquejspeed plane. This is 1.111' (':1."', ir a rotating DC generator driven by a line-fed AC motor serves as n, P"WI'I' :-iIlPply; the sarne can be achieved with switched converters. With 1I11I/',II('l,i.: alllplifiers, containing only diodes instead of switches, regeneration WIL; : 11,,1. possible. Of course, there is always the option of electrical braking I',Y "laóllg resistors in the armature circuito 'I'he :-;tcady state power required for the field winding is only a few per­ ,','11(. of I.ltat for the armature; a "single quadrant" rectifier, controlIed or 1II1'·oll'l.rolkd is adeqllate for feeding the field winding. li"i/',. '1.2 contains a block "control equipment" which is controlIed by ref­ "I"~I((," ~;i/';Ilal:::; én,'!, wne!, CXRe!, 'ia mal" as welI as feedback signals é, w, a, I.,,, a~ i 11,:,'']('<1 for sp" cifie control taslcs. The sp eeu signal is normalIy obtailled f (' OIlI II, Hll lHll pCl'lI mllcll t 11l
Fig. 7.2. General schematic of DC drive control

each angular incremento The frequency of the pulse train is then proportional to speed and counting the individual pulses provides a measure for the angu­ lar position change; the sign of the rotation is determined from the sequence of pulses generated by a dual track sensor with the two signals being 90° out of phase, Chapo 15.2. The controlIers transform these input signals into the actuating variables Ya, Ye which serve as control inputs for the two adjustable power supplies. ln the folIowing paragraphs some proven control procedures for DC drives are discussed, making no specific reference to the type of power supply used, since this topic is dealt with in later sections.

7.2 Cascade ControI of DC Motor in the Armature ControI Region Let us first look at the case when the motor operates below base speed, -Wo :::; w :::; Wo. As explained in Chapo 5, the main flux is kept at the nominal value PeO to have the lowest possible armature current for a given torque and to guard against unexpected load surges. ln Fig. 7.3 the schematic diagram of the armature circuit is drawn; e a is the internal volt age source of the power supply. This synthetic voltage which is controlled by y" differs from the armature volt age U a due to the internal impcd all cc / li , / ,; 01' Lhe s1lpply circuit.; this m1lst. be taken into account. when llorJlIalisill/'; I.h(' 11I()I."r p;l.rnlJld.(~rH,

7. Contrai of a Separately Excited DC Machine

I)()

7.2 Cascade Contrai of DC Motor in the Armature Contrai Region

1-----------­ I I

-1:I

R·I

L·I

I

c:=Jt--I_---­

I~

Ya

I

~el

ua

a e

J

La

ia Ra

00----­

~-----

______ I

Power supp/y

(actuator)

F ig. 7.3. DC motor with armature power supply

La +Li

eaO

i"o = Ra

+ Ri'

Ta

= Ra +Ri

Witlt the results of 8ect. 5.3 and b = I, the block diagram of the motor a SSlllllCS

the form shown in Fig. 7.4 a, where the controllable volt age source

('" is represented by a first order lag element having the voltage gaín G as and Lh( '

81

The armature time constant Ta usually has values between 10 ms and 100 ms; it is determined by the impedance of the complete armature circuit including possibly a smoothing choke that may be needed for reducing the current ripple caused by switching convertersj with small servo motors hav­ ing an iron-free rotor the armature time constant could be less than 1 ms. The mechanical time constant T mn comprises the total inertia of the drive (assuming rigid couplíng) j therefore it covers a wide range from a few mil­ liseconds (servo motors and reversing rolling miU dri ves) to several seconds (paper míll drives and mine hoists). There is general agreement that the most effective control scheme for drives is a cascaded or nested structure with a fast inner controlloop, límiting torque or, on a De motor, armature current, to which an outer speed control loop is superimposed [K24, L79, 885]. This multi-Ioop system proves very flexible when the control plant possesses a chain like structurej for example, in order to control position, it can be extended by a position loop superimposed on the existing speed control loop. AIso, if acceleration is of importance, a corresponding inner controlloop may be added.

tiUle constant Tas, which strongly depends on the type of power supply.

Whell a rotating generator is used, Tas corresponds to its field time constant

;uld may be in the range of 0.1 to 1.0 s, whereas the corresponding value WOJdd \)e 1 to 5 ms, if the residual delay of a líne-commutated converter is 1. (1 III: Jll()(klled. Naturally this difference greatly affects the dynamics of the I ' , 111I,rol systern; it must be taken into account, when designing the controllers. I"i,... '(, /1 b i s ;LIl equivalent structure, redrawn for noninteracting access to the ÓI,rlll i d ,IIIT

c urrc llt .

Actu­ ator

/'(Iwnr :al/'/'/Y VII

Supp/y

mL

ia

mo

mM

OJ

~I;;;=mo

E

ú)

~I _ _ _ _ _UaO

n

Fig. 7.5. Cascade contrai structure of drive including position-, speed-, and current contrai with feed-forward from a reference model m I..

mo

I­

;a mM ~t;;;;= mo

(I)

The idea of cascade control is exemplífied by Fig. 7.5, where a lumped inertia drive is cquipped with most of the controlloops mentionedj it is seen t,h at the SCqUCll CC: torque-acceleration-speed-position is a natural one as it cO llforIJlS to Lltr. strn d nre of the plaJlt. Tll( ~ torque (or annature current.) loop, f'ur;llIilll ( I,!il; illlW)'Illost (:oul.ro[ fllllcl.io ll l\my lw regankd as appro Ki­ Ill a l.r'ly nl 'l d, i ll ,~ :UI illl[)J '( ': ;:a'
V,, ­

I, I" i ~( •

[k ~,.1, 1''llli 'l! d l"nl l1 J1l f' ct t 1)(:

1111

ai.II I'

rl l,1.

"..

1I 111 1lI 11 11 1I 1'" \'\11' 1

II l'pl Y, 'I',

'/' , II

('(lIll.lolln 11:1:: 1.(, <1" 11.1 pri lll n rily w'il.lt 1,111'
1.111' pow\'r li lll'l'l ,v Il.Ild

K:l

7. Control of a Separately Excited DC Machine

Lhc armature, cancelling the effect of the induced armature voltage; thus it provides the fastest available control action. By limiting the reference mM ReJ (or 'ia ReJ), the inner controlloop also assumes a protective function. An ac­ ("d(~ration controlloop which is not shown in Fig. 7.5 could be desirable with I! iglt dynamic performance drives, such as servo or elevator drives; its main I"l('pose would be to counteract the effects of load torque mL as well as cl!;lllges of inertia, by generating - within the admissible range - a suitable ("llrn~nt reference ia ReJ. The hierarchical structure is then continued with ;1, spccd controller which controls the closed inner loop in conjunction with LI[(' subsequent integrator; the same idea is once more extended when going Lo ;l position control loop with the position controller generating the speed ,'('I(~r(,Ilce.

()f course, this multi-Ioop control structure can only function under the ;I,s:mmption that the bandwidth of the control increases towards the inner loops with the current loop being the fastest and the position loop the slowest; ("\('arly, the position controller can only perform well if the speed loop is (jllickly executing the commands it is given, etc. Following this reasoning, the dcsign of the controllers proceeds in the direction from the lower to the higher ('()lIl.rol leveis, at every stage approximating by a simple model whatever has I)(~(~n a.chieved before. Obviously, the design of the control system is greatly :lÍlllplincd by such an iterative procedure, where only a part of the complete pLtlIl, is dcalt with at a time. The fact that each feedback variable can be lirllil.(~d by damping the pertinent reference is a major advantage of cascade 1'11111,]'01; a.l~o) ficld work is considerably simplified, since the controlloops can I,,' plll. ird,o op<,ration and tested one after another. II, rll:t,y IJc aq,;ued that a control system containing several inner loops is I"",Iy 1.11 IT:;pOlld slower to changes of the reference than an equivalent single 1,,"1' ("olll.rol (provided that it can be stabilised). Therefore, when dynamic 1"',lmIJI;\lIc,; is illlportant, e.g. with position controlled servo drives, the inner 1""1':: ,':lllIldd be activated in parallel by feed-forward reference signals derived 1111111 :\.11 "xt,cmal reference generator as indicated in Fig. 7.5; as long as the I('I"I('II('C I\(~llcrator is outside the feedback loops, the stability of the control :;y::L"1I1 ('clIlains unaffected. At the sarne time ali disadvantages with regard to dYll
83

7.2 Cascade Control of DC Motor in the Armature Control Region

after the cost of control amplifiers was dramatically reduced by transistor operational amplifiers and electronic integrated circuitry. When comparing the structure of the plant in Fig.7.5 with the block diagram of a DC machine in Fig. 7.4 a it is seen that the machine-internal feedback of the induced volt age is not immediately compatible with cascade control because it represents a deviation from the nested loop structure of the plant. However, by redrawing Fig. 7.4 a in the form of Fig. 7.4 b, a non-interacting structure is obtained; the fact that the load torque is now entering the plant in two places is no problem since mL is assumed to be an independent disturbance. The variables not specified in Fig. 7.4 b have no immediate physical significance. The part ofthe plant that lies between ea/uaO and ia/iao may be described by the transfer function

~(8) Ea

=

Tm

Tm Ta 8 2

(7.1)

8

+ Tm 8 + 1

corresponding to F3 (8) in Eq. (5.18) for b = 1. The differentiating effect vanishes with speed dependent load, as was seen from Eq. (5.24). The design of the current controller is explained with the help of Fig. 7.6 assuming no inherent damping by the load. The feedback signal derived from the current transducer is passed through a low-pass filter to reduce possible ripple; a filter time constant of 2 to 5 ms is usually sufficient. mL

mo -----1 Motor I Current controller

Actuator

I I I I

Tms

Tm Ta 8 2tTm 8t1

I.

'

-f'ª-­ = mM mo {aO

Sensor

Fig. 7.6. Design of the current controller, Pc = p.o

If a POW(T dcctronic converter is clllploy(~(! as armature supply, a pro.. port.ioualilll.(·I':ral cO!ll;rollcr (PI) i::; t\
1\1

7. Control of a Separately Excited DC Machine

7.2 Cascade Control of DC Motor in the Armature Control Region

Zener díodes

and the applied voltage rise linearly with time; with other words, generation of a ramp function with a controller containing a single integration calls for a constant control error. For the practical application it is normally of no consequence whether the current controller admits a steady state control error during acceleration, because current control represents an inner (auxiliary) function , being part of the superimposed speed loop. The main purpose of the current controller is to linearise the plant and prevent overloading the motor; at constant speed, this is accomplished by a controller with a single integration. It would in principIe be possible to use a current controller performing double integration, e.g. a dual PI-controller having the transfer function

Clamping circuit Feedback circuit

7]C4 Reference signal

~ Rs

RI

X1i

Y

Actuating

signal

1

Fb,i(S) = GC,i T l S + 1 T 2 S + 1 Tl S T2 S

wíth

X21

Rs « R4 «R3' C3 « C4

Feedback signal

y

(T I s+1)(T2 s+1) F (s)=-(s)""G c XI c T I s(T '2 s + 1) R3

,

Gc ""-, T I ", R3 C3 , T2 ""R4 C4 , T 2 "" RS C4 RI

Fig. 7.7. Circuit of analogue PID-controller employing an operational amplifier tlw controller function by an analog operational amplifier, this is a minor ildditi()\l t o the RC-feedback network of the amplifier. A typical example of /1.11 all;tl() ~ \le PID-controller is shown in Fig. 7.7. Wil.h tb ese assumptions the open loop transfer function of the current (' ()) tI ,]" () I bccomes PI

PDT

r---A--. r---A--.

, 1'11/' ,; (8) = GCi

T l S + 1 T2 S + 1 Gas T l S T~ S + 1 Tas S + 1

'-------.,,,...------' ' - v - - '

PID-controller, T~ ~ T2 Supply

,

Tms T m Ta s2 +Tms v

Armature

85

1

+ 1 Tfs + 1

(7.2)'

'''-v--' Sensor

lkcau:ic of the factor s in the numerator of the armature transfer func­ a jJroportional-acting cJosed loop results, lim Fods) = FodO) finite, s-+O (, V( ~ II I.h()lIgh the controll cr itseJf comprises an integral term; as a result th e c!()s(,d ('.IIITI,'I11. r ()lIl,rol loup will exhilJit a steady state erro r. The reason is III' f'(l\ u r~ (\ I.hl' ,(' .11 I. f. r(,,'dhadc which rcmaim, dfcctive also ir; t.h( ~ l,r: tll li f; )l'IIJ( ~d I,l"d!, cl l(l.,~f·'~ 1 11 "r I" I I ~ '(, II h. C l{~arly, wlll'lI :; lIpplyill!,: (·()II Hl.al1 l. 1' ( 11'1" ' \11. I.() UI(' IlH ' l l d.l\ II' 1,1' II UI : 111 (11.... ' 11. 1. "" lClfv l, 1.1 .... "",1."1' /1 )""'11, 1.1i. , illdl W I'cI VI .!!.;II',"

(7.3) '

for avoiding a steady state control error during acceleration. However this is not normally done in view of the reduced margin of stability. If maintaining a prescribed acceleration is a necessary condition, as may be the case with machine tool- or elevator-drives, it would be easier to include a separate acceleration controlloop as will be shown in Fig. 15.9. The choice of parameters for the current controller according to Eq. (7.2) foJlows the usual lines for the design of cascade control systems [K25, 38] . ln the case of a Ward Leonard scheme containing a rotating generator and with aperiodic damping of the drive (Tm > 4 Ta) the lead time constants T l , T 2 of the PID-controller could be tuned to Ta and one of the emerging lag time constants of the drive, while the rest can be dealt with by approximation on the basis of the principie of "equivalent lag" , e.g. [38]. The closed current controlloop is now inserted into the speed loop form­ ing the next higher levei of control, as is shown in Fig. 7.8. Provided the current loop is well damped, it is acceptable to replace its closed loop trans­ fer function by a proportionallag element having the gain Gc L,i and the time constant Tc L,i' Since this equivalent lag, being a first order, low frequency approximation of a much more complicated transfer function, cannot be sim­ plified by lead time constants, a PI-controller for speed is the appropriate choice. Hence the transfer function of the open speed loop is approximated by

FOL,w( S) ~ G ow

1 Tc,w s + 1 GCL,i T c ,w S TCL , iS + 1 T m S

(7.4)

'-v-'~

PI· Controller

l.i()lJ,

Current control

The pa rameters G c , w , Tc, w may be chosen in accordance with the "symmetri­ cal Opt.illIHIlI" IX 2:i,L33,38], which is a standard design procedure for transfer fllJldioll s cU Ilt.a i II i II g a double integratioll (indllding the controller). '1'1", ll lll,i ll i el" II, ln I.() c11()W, (~ t.h(~ cross-()vc'r frc'CjIl('IWY at t\w gC'ornetri c IIlca n III' 1.11<' I.w" ,'""," (','

r... ''I ""ll(' i('H, 1.0

ohl.;I.ill 111 11 Si 11111111 ph lls,' IIIi1rl ~ llI 'l/I,{ whi ch iII

7, Control of a Separately Excited DC Machine

r((\

7.2 Cascade Control of DC Motor in the Armature Control Region

Limiter

From the cross-over condition mL

mo

.!!!1..

(7.7)

!FOL,w(j wd)1 = 1

mo

the gain of the speed controller is obtained with Eqs.(7.5) ,(7.6)

(Ollol

("0

1 Tm G c,w = aGcL ,i T CL,i Speed control/er

"'i ~ .

Current controlloop (Approximation)

ctJe = ctJeO

7.8. Speed control with approximate representation of current controlloop

" úJd

- 1

/" 1I

(J)

FoOw)

1 I

II

a;

1±j

J (a; 1 1-

1) '

(7.9)

~ Pha se margin

b

1/

(7.10)

is the damping ratio of the oscillatory part of the response. Assuming a slightly underdamped response, D = ljV2, we have a = 1 + V2 : : ,; 2.4l. When aperiodic transients are specified, D = 1, leading to a = 3, a triple real pole at s = -Wd results. Due to the double integrating term in the open loop transfer function, the closed speed loop exhibits zero control area or, synonymously, zero velocity error [38,88]. This means that the step response is characteúsed by consider­ able overshoot eventhough the transients are well damped. To eliminate this effect which is caused by the lead term of the Pl-controller, a corresponding lag term

Re

I I Cross-over frequency

'PI)

D = a-I 2

jlm

' ""--2

I I

".' ~ w, [-

/

W

1

1

~ -w"

Hence

aWd

I........

(7.8)

The poles of the resulting transfer function of the closed speed control loop, i.e. the eigenvalues of the speed control loop, can be ca1culated explicitely. They are, assuming a conjugate complex pair, positioned on a circle in the s-plane with the radius Wd,

I' " I

II

87

"' ll\· 7.!.1. Bode-diagram (a) and frequency response locus (b)

1 T c ,w s + 1

n""ord i llg to "symmetrical optimum"

t lll"ll wiU result ín optimum damping of the speed loop. The name stems chc Dode-diagram that shows symmetry with respect to the cross-over Ilt'qll ( ~ ll cy, Fig. 7.9. Il f llorrnalising with

1'1"1111

'/:.)(1)

Ll1I'

(1,'2

'1 (,' I"

i ,

('ro S~H )V( ' r f'y qlll'llry

/";1

) '1'I

, i"

'I' f

' I} , i

a > 1,

(7.5)

is n '/~A' l , . ,

(7.1i)

may be inserted into the reference channel of the speed controller. ln practice this can be done by connecting a suitable capacitor Cl between a center tap of the input resistor and ground, as is indicated in Fig. 7.7. The speed controller shown in Fig. 7.8 contains a nonlinear feedback block which Jimits the current reference to the range -ia ma:c :S iRe! :S ia ma:c' This serves as a n effective protection for the power supply and the drive, pro­ dUeillg tll P desired steady state torque-speed characteristics shown in Fig. 7,1Ü; of (,()lIr s ( ~ , a :;)Iort overshoot of the current , for example following a load disl.llrhall(,(· , C l lI llPI. h(~ rnled out hy dalllpi,'1!; t11C eurrent. rdcrenee. If the l'It '('j,f(lIti. · cOlrl,I'"II,,!' i,: lI'a li s(·d wit.ll t!te IJI'lp /)1' nll opt'r:üiollal a'llplifi( ~ r , 1.11(:

HH

7. Control of a Separately Excited DC Machine O)

Speed control/er clamped

0)0

O)Ref

0)0

o

_TI

iam~ICD

I" fIItJX

('o

iaO

7.2 Cascade Control of DC Motor in the Armature Control Region

transients, is seen in Fig. 7.11; it restricts the occurrence of current limit to load surges, for instance when a reversing rolling mill is mechanically jammed because of a wrong screw- down setting calling for an excessive reduction of the billet thickness. On the other hand, current limit is not reached just be­ cause an impatient or careless operator rapidly changes the manual speed command.

dJ.L

J.L+Ta ~ ,I

"

ia Ref

iaO

'aO

0,5 "'ig. 7.10. Steady-state torque- speed characteristics of speed control loop with 1.1)"'1I1' ~

lilllit

0,4

dt

/~

ea --

//.

I I

,

UaO/

I

0,3

td / W.f/u l 1rJ1)

rl~

:I

I ... r--;~--F--

I ... úJRef II --/ ,I

' I ,

úJo

0,2

=

0, 1

° °V

0,5

___

I 'uao

_______ _

/

I, ­ '"

cPe =cPeO

UaO

. Start at no load

0,2

ea -­

,I ,I

I1 II

/

..1;\lIlpiug can be achieved by placing Zener-diodes in the feedback branch, I"il~· 7.7. Limiting the current reference removes in principIe alI restrictions "II I.he rate of change of the speed reference, which is important if it is pro­ dllC(~d lllanually. Whenever the speed controller, due to a rapidly changing "')(lIlllal1<1, reaches clamped condition, the speed loop is disconnected while Uu ' Cllrrent control loop remains operative with clamped current reference. fi:; S ()OI1 as the speed has caught up with the command, the speed controller I"I'I'I'I.S 1.0 linear operation and resumes controI. Ilow('ver, with larger motors under manual control, a rate-of-change lim­ it" ·,, is ()rt.( ~ 1l illserted in the reference channel ofthe speed controller to ensure 1.1, :11, 1.11<' (,111J'(~llt limit, which may correspond to twice nominal current, is not "'./1111'.1 1IIIIICn~ssa.rily. A block diagram ofthis device, together with typical

89

~

J.L

I'

iao

"

1,0 1,5 Load applied

~ s

· '. -iamax .-=0,2 LImlt. 'aO

"

ia Ref.J"

0,1

lo' a ,

,

úJRef

úJo

I

°

I

°

I

II

'" ._ 0,5\1

1,0

1,5

s

Fig. 7.12. Computed transients of DC motor with currentjspeed cascade control iIITorqet ü)o

\

úJRef \ /

o

úJ

~

I'

"' 1,.,; . 7 . i I . 't ll l., ' ,01 d" " ilv' ',," l l.n

Computed transients of a DC drive with currentjspeed cascade control are seen in Fig. 7.12. The parameters assumed correspond to those of a thyristor­ fed drive with a six-pulse-converter, Chapo 8. The controllers have been de­ signed as outlined in the preceding paragraph; if a rotating motor-generator were used instead of the power electronic converter, the response would nat­ urally lw llllldl slower, particular1y in the current loop. AI. Jirsl., a sLarl,illg (,ransient is displayed, caused by a step of the speed ref­ ("ICll(',(~ 1'J'll/1I '/,!'l'n Lo 0 .:\ Wo. Tltc p{r( ~d of CIl1T<'lIt, limit dllring the acccleration ph;I::" ;,: : \\".Ii /Iii /.I", dyll:l.Illic ("olll.!'()l erro!' are c1(~ar-ly :;('(~II. '1'11 i:; I.rall::icIlL

!l()

7. Control of a SeparateIy Excited DC Machine

7.3 Cascade Control of DC Motor in the FieId-weakening Region

corresponds roughly to the transition of the operating point in Fig. 7.10 from () lo 3 via 1, 2. At time t = 1 s the drive, having attained the no-load point (:1) , is loaded with nominal torque. This causes a transient speed drop but the drive remains in the linear control regime, eventually reaching the operating poillt 4. Prom Fig. 7.12 it is seen that the source voltage e a rises with the induced vo ltage, which is proportional to speed; in addition there is the resistive and luductive voltage drop in the armature circuito The principie of cascade control has proven its usefulness over the years III ill11umerable applications; as mentioned before, the two leveis of control III ay be extended in both directions, for example by adding acceleration or position controi. Since cascade control is by no means restricted to DC drives, W( ~ will come across more examples where this exceedingly fl.exible method of mlltrol can be used to great advantage, see also Chapo 12.

7.~

Cascade Control of DC Motor iII the Field-weakening Region III S(~ct.. 5.3 it was shown that by reducing the main field, a DC motor can II(" ()p(~rated above base speed. For good utilisation it is important that the

111"tor should be supplied either with maximum field- and reduced armature voll,ag<' or with maximum armature voltage and reduced field voltage; as 111('lll.iOIl(~d uefore there can be some overlap for convenience of controi. Si11('(' Lhe motor may alternately operate in either one of these regions it is :"l'proJlrial.c to choose a control strategy which fulfills both requirements and I "'1"111 iI.s a continuous and automatic transition from one operating regime to

with a constant reference, e.g. lema", I ~ 0.9 Uao; the e-controller then attempts to remove the difference by adjusting the field voltage. Since, however, the reference voltage is out of reach as long as the motor operates below base speed, the controller containing an integral term will be saturated and the field power supply produces maximum field voltage which is the aim in the armature control range. When the speed is rising, the induced voltage lei approaches the reference value lema",l; after a balance has been reached, the e-controller leaves saturation and begins to lower the field voltage to limit the induced voltage, thus automatically weakening the field as desired. Of course, the initiative for a change in field voltage must always come from the armature because this is the direct channel of the speed controller, working through the armature current loop. The arrangement drawn in Fig. 7.13 represents a complex and highly nonlinear control system, confounding all general design methods; however, considerable simplification results if one succeeds in speeding up the e-controI loop, active in the field weakening range only, so that e may be approximated in that operating region by a constant impressed voltage. Such an approxima­ tion is not entirely unrealistic if the field power is supplied from a converter having a sufficiently large ceiling voltage (only one polarity of the field cur­ rent is required). If the stator and the field poles are laminated, the "control plant" may be represented by a simple lag element, having a speed-dependent gain, and the fast dynamics of the actuator; no serious stability problems are mL

mo

( '

11. "

H.. i"

di"

(7.1 I)

" " -/ ll.. ;

III .'I.II II,I"!',II' · (.,,[('1 III I,/ II/" "i l'('flit, 1101. fl hoWII III FI'" ./ 1:1, 1I 111.'y 1)(' 1'(11 1'"rJ" . " I ' l d tl fl ll'. tHd "II I,'.'VI·I'/dll/,. dl l VtlI'I l ill. l 'I 11111, 111111 ,(,1,011

11I1I'!!

foI' 1.1 IiI-<

Iii l 'IIIII JllIl','d

Fi/ter

ú)

%

1.11<' "l.IwJ'.

/\ (', nll.rol scheme that has proved useful in practice, is shown in Fig. 7.13 III ,: illlplificd form [G20, V9]. It contains the currentjspeed cascade control 11<1. i I1/'; through the armature voltage, as discussed before; in addition, there III 11,11 ;l.Il xiliary controlloop with the field voltage as actuating variable, which Ims I.IIC ~ purpose of limiting the induced armature voltage as the speed rises I)(,y olld base speed. The adjustable voltage source supplying the field winding, iei IIS II;dly realísed by a thyristor converter, which is represented in Fig. 7.13 I'y a la!,!; cl(~lnent. The nonlinear model of the magnctic circuit corresponds !.o Io'i/(. ;).4. /\ r(~cdklclc signal for the induced volt age CilU II(' n !c()l1structed from the 111( ';I.SlIl'cd annature voltage and current accoJ'( lill t.( 1,<) 1,11<: vo lta,gc equa tion

91

Ú)Re'

Wõ

Limiting

1el UaO

I emr1. 1--o-..t



ullo

- control/er.

1"IK. 7 . 1:1. ( :,,111,,'," "i' I )(: II "'Lo I' iII (.lU ! IIfIlLl.LlIre a.lld fidd wP',d{(:ning

regiolls

7.4 Supplying a DC Motor from a Rotating Generator

7. Control of a Separately Excited DC Machine

~) '2

.!!!...j mM

be expected even at high controUer gain. Neither is the nonlinearity of I.he magnetic circuit of great concern since the e-controUer operates predomi­ lI alltly at reduced flux where saturation is not pronounced. For these reasons a PI-controUer is quite adequatej its lead time may be tuned to the field I.i Ille constant of the motor while the residual delays are taken care of by the r<'llIaining I-term of the controUer. Assuming fast e-control in the field weakening range, the simplified equiv­ al( 'llt block diagram in Fig.7.14 results. There the effect of the e-controlloop (:; :l)lproximated by a constant volt age le max I acting like an external distur­ h;l.Il ce and the flux approximately becomes a nonlinear algebraic function of 1.0

~W 1,010,~ ~ Wõ

_W

i'';;';/J

II

I l

1

mo

WRef

L/Wa __

_____ >

~--

ia -ç

lPe ~eo

1,0 0,5

-I

Wo

'[>eO

r-

I'

lPe lP ea ~

mo

úJo

: :p('('d,

1> e

for

1:01> 1,

allccLing the mechanical time constant. ln principie, this undesirable change

"r pa rameter could be compensated by dividing the speed error by Pe/PeO I.he input of the speed controUer, as is indicated in Fig. 7.14j however, in III ()S(. cases this is omitted and practical results prove that the correction is II s llaIJy not necessary. Some consideration must of course be given that the :, pe('c! COlltroUer yields acceptable results in the armature as weU as the field rl/lIl.rol regime. As an example, Fig. 7.15 shows a simulated no-Ioad reversing I.r; LIIs i(~J\t between ±1.5 Wo that was computed on the basis of the system :: I! " WII in Fig. 7.13. The effect of the e-control loop is clearly recognisable 1', ()1I1 III<' ">,.- and the e-traces. 1I. h ; I,,~ IJC(m mentioned that the use of a fast e-control loop with high r"r" il lj ': voll.age across the field winding calls for a laminated motor frame. W 11.1, b"I':' ~ Illotors it may be necessary to employ a more accurate model of LI ... li,'ld ,'jrclli t. for the design ofthe e-controlloop, taking eddy currents in the II "" iII!." :ICCOllut [G20] . AIso the high voltages induced by transformer action III LlI<' ;1.1"I1I:t!.nre winding affecting the commutator must be considered. To

:,!I- --

:d.

u. M()

~ iao

.!!§...

{1 1, r,.,

93

!!!.b.. mo

- - -ç ia' - -­

'r­

~Reversal of

ea

e

:

speed reference

uao' UaO

1,0

--,

~

~:-"'~uao , :-...

, :-...

;::... ~

uao

;::...

;::...

;::...

:-...

Fig. 7.15. Computed reversing transient of speed control system in Fig. 7.13

eliminate the effects of winding temperature and supply voltage fluctuations on the main flux in the base speed region, an inner field current controlloop is often included which maintains a prescribed maximum field current in the armature control range independent of a changing resistance of the field winding and helps to speed up flux contro!.

~,:\~}

(110

,.

«/'"

<1111/ 1

( 1./111

I 't'l ll 1( '11<. 1/'

emexl . (ü) l-uao 'slgn "'o 1\ / ÍlllilO!

7.4 Supplying a De Motor

from a Rotating Generator

,

((uel a ui II/ui cI

cOl/lrol

III /lul< I WII. 1/" " li" li I "U;OIl

I" IH . 7 . 1'1. Sl lIq dW"d " "1>1 " ' /(' 11 1.' 1.1.;1111 "l' nl', \(,,1 t'" " (.If .I I, ,, ,1' ill l i"loI WI\lI l1l' " I'''I ', "/l1l11"

lu HlI~ prpc(~c(lilll~ sections was mentioned that any adjustable voltage source (".ol1ld :,\('\'V( ' ; \:; p(Jw( ~r :->npply for the armatllre of the De motor. The classical ::('11('1111' (Ir I )( ~ III ' d.or ('o\l!.rol (kvdopcd hy W:I,rd T.(~ollard [L21, 1,22, Y 4, Y51 ,

H,I

7.4 Supplying a DC Motor from a Rotating Generator

7. Control of a Separately Excited DC Machine

1'111 pi oys for this purpose a rotating D C generator, the armature volt age of wllidl is contro11ed through the field. Combined with up-to-date control, d ri VI~S of this type are still in use today. The basic circuit is seen in Fig. '/ . 1(;; Lbe separately excited DC motor (2) driving the load, for instance a rullillg mill or an elevator, is fed from a separately excited DC generator, J"lllIlIillg at approximately constant speed. While a rolling mill or a gear-Iess "I"vaLor drive would norma11y operate at very low speed, for example between I 100 .L/min, resulting in high torque and a relatively large frame size, the 1'.'·llI'raLor being of similar power rating as the motor can run at any conve­ Uí' ·lll.ly higher speed Wl, with the aim ofreducing its size and cost. lnstead of :1. I l(; gCllerator, an AC generator with rectifier could be used for supplying Lili' I)C-link; of course, this would rule out regeneration. 'I 'lte prime mover of the rotating converter is most frequently an AC induc­ Lil III I.llotor (1) with cage or wound rotor, Chapo 10, requiring no continuous C"IlIII.rol. The power fiow of the drive is then reversible with the contro11ed Illut,or being able to operate in a11 quadrants of the torque/speed plane. ln ("as. ~ of a ship- or vehicle-propulsion scheme a turbine or a Diesel engine could bc sllhstituted as prime mover (1) (Turbo- or Diesel-electric drive), while the drive lllotor (2) would most likely be a series type motor. Resistive braking cUllld he employed instead of regeneration which is of course not possible wil.lt a llllidirectional prime mover. W iLll largc rolling mill- or mine hoist-drives the total drive is often split IIII iIILI I several machines; for instance, on a large reversing mill, two twin 111()l.ur~ Illay drive the upper and lower ro11s without direct mechanical con­ 11 ...·l.iUII :l.pnrLiC1llar interest with weak supply systems but its importance re­ cl'dceI ;,IS LlII~ slIpplips !lave become stronger anel more closely interconnected. (:ollvI ·rsl'i.v, ;1, J'()l.nl.illl~ COllvcrtc~r can servI: for protl~(".t.ion of sensitivc loads, i i i Ir" ;1:: [l il.pl'l' IllilLdriVI':i, agaillSI. Short-l.IIIlC lillO di sturhallccs. Silllilar argll­ 1111 \111..'1 Ilrn vl tI,,1 wlll'll 1I111111.1'ITllpl.ildl· [lOWI'J' rol' hllplirl.alll loat!s is 1'(~'111i['(.'d,

95

for example computers, elevators or operating rooms in hospitals; with an intermediate motor-generator-set supplying these loads, there is more time for switching to an alternative supply in case of a line disturbance. The main drawback of the Ward-Leonard-drive is of course that, besides the adjustable speed drive, two additional machines of about the sarne power rating are needed, Even though they may be smaller in size because of higher speed, large machines will still require heavy foundations and continuous service of bearings and commutators. It is for these reasons that static power converters, particularly of the modern semiconductor type have supplanted rotating motor-generator-sets in new insta11ations. If the field of a rotating generator is fed from a rotating exciter, both responding relatively slowly, it is with some care quite possible to control the drive motor manually and without any automatic equipment. However, the machine cannot be fu11y utilised because of the danger of overcurrent; feed­ back control with its much faster response then offers considerable advantages making it possible to take various side conditions into account so that the AG supp/y

Rotating converter set

•

DG- generator Induction I Motor 1

DG-motor

Load

III I!

'

Starting resistor


~~

-

.o~.~~,--------------

---t

Fi,l. 7.111. (~I)III.I' 1)1

e max

lil'ill'llIl: ()[

WMel L(~OIlIl.rd · drivc:

!II'

7. ContraI of a SeparateIy Excited DC Machine

drive assumes well defined operating characteristics and may be inserted into n,lIl.olllatic production processes. By exciting the generator and the drive mo­ 1,,1 I.ltrough static converters with high ceiling volt age (2 to 3 times steady ::Iill.(' voltage) the speed of response of the drive may be sufficiently improved 1m Illeding most dynamic specifications. 'l'lw control structure shown in Fig. 7.16 is essentially in line with that in I"il': · 7.13. The only difference is the possible introduction of an inner control loop ror the armature voltage U a of the generator [876]. 8ince the pertinent 1'()Jlioll of the control plant comprises only the field time constant of the 1',('ll<'ral,or and the residual delay of the electronic field power supply, the re­ ::pOIISC of this control loop can be made quite fasto Its advantages are the lill('ari,sation of the inner part of the plant, the reduction of the effective ar­ 111:\llIr(' impedance of the generator and the possibility of precisely limiting 1.11<' ;tIInature voltage. This is important with large machines where the per­ III issi ble limits of commutation must be fully used without running the risk 01" cx('cssively sparking brushes. For the superimposed current controller the c!os('d voltage controlloop looks like a linear actuator with constant gain and (·oll.si(lcrably reduced lago

8. Static Converter as a Power Actuator for DC Drives

8.1 Electronic Switching Devices A characteristic feature of the rotating converter discussed in 8ect. 7.4 is the consecutive power conversion from electrical (constant AC line volt age) to mechanical (speed of motor-generator) and back to electrical form (variable direct voltage) from which it is eventually transformed by the drive motor into controlled mechanical power. These conversions constitute the advantages of the control scheme (separate electrical circuits, decoupling by rotating masses) as well as its draw backs (cost of machines, foundations, power losses, servicing, limited speed of response). This naturally gave rise to early ideas for eliminating the intermediate mechanical stage by supplying the DC drive motor directly from the AC grid through static electronic devices. That such possibilities exist in the form of static switching converters has been known for a long time [A15,W29,46]. ln fact, switching devices are the only available means for high power converters because any other circuit components would result in prohibitive power losses. The basic element of a power converter is an electronic switch having the properties of a controllable rectifierj the idealised steady state characteristics of different switches are seen in Fig. 8.1. An uncontrolled diode rectifier, Fig. 8.1 a, has two distinct states: With reverse volt age (u < O), there is only negligible leakage current, whereas in forward direction (i > O) a low forward resistance combined with a threshold voltage exists. Controlled electronic switches have at least three different states: A thyristor, Fig. 8.1 b, is blocking reverse current like a diode, but in forward direction (u, i > O) two possible states exist: the device may either be "conducting" 01' "blocking". The first state is again associated with a low conducting resistance, typically 10- 3 [l, while in blocking condition the device exhibits a very high blocking resistance, for example 10 5 [l, similar to that in the reverse direction. Hence the thyristor valve may be approximately described by a mechanical switch in series with an uncontrolled rectifier (diode). The main difference between a thyristor and a diodc is tlw existence of the forward blocking state. 'flw I.rall:iitioll frorn "blocking" to "conducting" may be initiated by a slIlall 1'::\,1<' ('111'1'<:111. plllse -ii" while the inverse transition fl"om "conducing" lo "lllo('liilll'," (' il,II:.i for a :/'('1"0 inl.c:rscci.ioll of UI(' 1I1aill cnl"n:llt, "Tnl"ll-0fF aI. ('111'1', '111 '/,(\("" " ', I. hyr l::I,llr:, a r, ' wid('l y 11:; ('<1 ror ('olil,rollillJ': U (:
'!U

as a Power Actuator for DC Dríves

H. ,':iUtLic Converter

8.1 Electronic Switching Devices

n~cent

development are Gate-Turn-Off thyristors (GTO), where electronic blocking is also possible, Fig.8.1 Cj they are mainly Il ro , -c! III rorcc-commutated converter circuits with a DC link, to be discussed III ( :lld,P: ;. ~).2 and 11. Other types of electronic switches which are of interest ',,1 ::,,,,ci;d applications are reverse conducting thyristors or voltage controlled 1.1 1I. 1I :IIKl.ors with internal diodes, where the two forward states are combined IV iI,11 I('V" ['sc conductionj an example is the "lnsulated gate bipolar transis­ 1., 'I" (I ( : I ~T), seen in Fig. 8.1 dj there are also applications for symmetrical 11 11' 11."1 il' S, l"ig. 8.1 e, as wiU be shown in Chapo 11.4. fi

11101('

1',:11,,' '''Idrolled

forward conducting

conducting

,

u

IUverse Iilucking ,I)

conducting

!./ ",electranic rückwarts sperrend

I

\ control

forward u blocking

b)

",

/ ' " electronic \ control

!ig

u

o

Oiode

! f>r

blocking

reverse blocking c)

. I

o

o •

-u

~

!ig

f>(

o

-u

Control/ed rectifier Thyristor

Gate tum·of( Thyristor (GTO)

conducting

/ ~. JI , VO / !Jt l

" " '/Iludiu!}

"J

, conducting

~"

electronic \ control

forWârdU blocking

u

u

Insulaled gale bipolar Transistor (IGBT)

... l,~ .

H.

\

e)

electronic contraI

:. . ... .'. . .. .

blocking u

electronic \ control"-.., conducting

-1 \f.

~

,

blocking

1

i

LfJt

Ug2

--.J \.

/

L

~ u Symmetrical switch (IGBT)

I. I)II COII trolleel anel controlJed electroni c switdlC::;

III I.he pax(. Lhe fllllct.ion of controlled d(~cl.J'()lIi (' x w;Ldws lias been realised with vm;"lIs dl'VÚ'('S Ill,tkiug \l*~ o[ diJkre eurly Ilrtllll,ioll: ; nll' 1I1l/l11kf,(' (.odny. Siul't: l'I(\('.(.I'OIl Llrl1oc 'lI wít.1t Itl.'i t!.l:d· cal.hodl~s cOllld 1I"t. IllIpply "II"II I \ IJ l ' IIIT"IIt. 1,0 drivo IIIOt.OHI II JltI \\11 ' 1< ' pl nl~ll<'d wil,lr lri((1r voll,­ 'I '," cll"I" I " 111 ri I,! I,!', III 1""11 d ll d l'll" Y, 1.111',1' W "I CI! , III""IIIIt'C("d I,,Y J'~ 1U 1 lilll ..,c)I.II I'('I!

99

with heated cathode, called thyratronsj however, power rating, life time and reliability were also rather limited so that they were only useful for small drives. High power in the MW-range, such as needed for rolling mills or mine hoists, could only be supplied by mercury-arc valves having a liquid mercury cathode. These electronic valves were developed to high perfection by the 1950's but the liquid Hg-cathode limited their use in vehicles and remained a disadvantage, as well as the large size and the relatively high voltage drop across the electric arc connecting cathode and anode (20-25 V). Outside a narrow operating temperature range, controlled mercury-arc valves showed a tendency to occasional spontaneous arc-backs, i.e. conduction in reverse di­ rection resulting in short circuits and high over- currents. This required close temperature control by heating or cooling the steel vessels. AIso, mercury­ arc converters needed relatively long intervaIs for deionisation which made them unsuitable for operation at higher frequency. After a brief interlude with magnetic amplifiers where magnetic materiaIs with an approximately "square" hysteresis loop were used as "switches", alI these difficulties were completely overcome with semiconductor switching devi ces (Transistor 1948, Thyristor 1957) which have become available at steadily increasing power and with improved performance since about 1960. They have replaced mercury­ arc valves even at the highest power ratings in high volt age DC transmissions (HVDC). The physics of a thyristor are extensively dealt with in the literature, e.g. [17,24], a simple description may suffice here. ln Fig. 8.2 a a small slice of highly purified monocrystalline silicon having four electrons in the valence band is shown, which is doped with materiaIs of valence three or five, for instance Boron or Phosphorus, so that three P - n junctions are created, separating the layers of P - and n - conducting material. The inner layers are relatively weakly doped while the outer ones, forming the anode and cathode terminaIs, contain more doping materialj the inner P2 layer is also connected to a gating electrode supplied by a low impedance control circuito The following steady state operating conditions exist: • With negative voltage between anode and cathode, u < O, the diode junc­ tion PI - nl is reversely biased; this corresponds to the reverse blocking state seen in Fig. 8.1 b. • If positive voltage is applied (u > O) but without gate current (u g < O), the diode nl - P2 has reverse bias which constitutes the state of forward blocking. Again, only negligible current flows in the load circuito • With positive voltage, u > O, and with a sufficiently large gate current, ig > O, the junction nl - P2 becomes conducting, with the load current i satnrating the region nl from both sides with charge carriers. Thus the lhyri::; tor assumes very low resistance in forward direction. When i has l')( ('e(~(kd ;t cl'/'I.ailllatching threshold iL, the conducting state persists even afl.e/' Lh e' (',;tl,( ' (', lIl'rmlt: is removed .

I1I1I

H,

8.1 Electronic Switching Devices

Sl.atic Converter as a Power Actuator for DC Drives

(hll .\' Ily )'('dllcing the load current i below a lower holding threshold iH can 1.11<' I.hyrist.or again revert to blocking statej hence it represents a switch that ,'11. 11 1>(' ('Ios(~d by a low power control signal, whereas opening it requires the 111 11 <1 ('lllr(~l\t to become very small or zero.

u 1\

­ A77

200A ,1,5VI conducting

ig

u

K

"

101

i

b

.

/u9 K

iH

-1kV

= 100mA

Ig

forward blocking

reverse blocking

G

1kV U

Fil,(. H.2. Thyristor model and symbol

Fig, 8.3. Static and dynamic characteristics of thyristor

:-;illlplified characteristics of a typical power thyristor are shown in Fig. ,"{ .:\; Lhe scales of the curves are different to emphasise the details. While the kak iW;I' cnrrent in both directions could be below 100 mA, the load current in l'lllldllcting state may be several thousand A. The voltage drop of the (,()lIdlld.illg thyl'Ístor is about 1.5 V and the maximum volt age in blocking ::I.:Ii.I 'S call Iw lllore than 8 kV. W!II'II tllC lll O) the effects are 111111" 1'lllIljllt-X, Whcn the voltage reaches a breakover levei, which depends 1111 1.11<' I',ii 1.1' (,\Irr(~lll., the thyristor switches spontaneously to the conducting : ,1.:1i." 11,,\V('vI')' , 1.11I'l1-OIl by increased forward blocking volt age is not a suitable 1111"1,, "I' (' IIliI.)'o! for a power thyristor because the "firing instant" (a relict I" 'III pla::lll i(, discharge valves) is not sufficiently well defined and there may I", 111(':11 Ilvel'iwal.iug of the semiconductor elemento lnstead, the switching L" LI", ()rI " sl.ill.<: should always be effected by a gate current pulse having a ::1" ,rI. 11:;1' I.ílll(' which has the additional advantage that the thyristor fires at a 1"1 '(' itl,,1y ddiued instant, independent of the individual thyristor parameters. 11i)(' 1.0 lhe small dimensions of the active semiconductor wafer and its III \V 1.I1I~l'Il1a! capacity, the safe use of thyristors calls for observing some re­ ::l.lldiIlIlS; apart, from limits on temperature, volt age alld current there are IlIil,Xi11IIIlll values for the rate of the current rise (di/ dt) while switching On, as \1'1 ,II i l.~; UIC voltage rate (du/ dt) during and aft( ~r tu rll· OIf. This requires the ill('III: :il)ll 01' prol.cctivl; circuitry such as a seri.::; illdll('I.i\.IlC( ~ iUld a parallel ca­ JI ,)('iL( II' (.~llIlbl)(~r), COllsi(kring al:-;o tlte swil.chil'i ~ Hllq';<':, t.llat. i\.l'e llu
recommendations by the supplier are observed, the thyristor has proved to be a reliable and very robust switching element that can stand high short time over-Ioad current, such as a 10 fold surge for one cyde. Persistent overload must be prevented by a fast acting closed loop control and in the limit by fuselinks. An important design parameter of thyristors, particularly when applied in force commutated converters, is the recovery time tr· It indicates the time that must elaps e from the instant when the load current has reached zero, until the thyristor has attained full blocking capability in forward direction (u > O). The purpose of this interval is to gi ve the residual charge carriers accumulated in region nl during the conduction period time to decay through diffusion and recombinationj depending on the type of thyristor, reverse volt­ age and junction temperature, the recovery time ranges from a few Jls for a high frequency thyristor to several hundred Jls for a high power thyristor designed for low voltage drop. t> tr / /

sfored charge

I, .... _ /

I I

I

r::.:J u

/u(t)

o

i

\.

a Fig. liA. '1'lIrlld lfr I.mllHil~nt of thyristor (n) (:1111"111. i'.lId v"It.II.I~" (h) I'roLeclive circllil

I-

b

...I'~_I .. I

o

I I ): ~

K.

8.2 Line-commutated Converter in Single-phase Bridge Connection

SI.;tLic Converter as a Power Actuator for DC Drives

fi Lypi cal transient of the volt age u(t) and the current i(t) of a thyristor dlllilll ': Lllrn-Off is depicted in Fig. 8.4 a. As the load current i reaches zero, 1,11\" I.hyri stor remains conducting for a short period with reverse current, thus (' 1 ("; Llill ~ the charge-carriers from the junction. After this charge has disap­ P( '; II ('
Line-commutated Converter iII Single-phase Bridge Connection

H.2

'I'ilyrisl.ors, which are available in a variety of voltage and current ratings, ;1./ (' e'Jlllhillecl to converter circuits to serve as highly efficient electronic power :: lIl'l'li('s for a wide power range. Of particular importance for the supply of 1)( : IIIOl.orS are the so called line-commutated converter circuits fed from a ::iUI',11' o)' Lhree-phase AC supply of constant volt age and frequency. They are '1I Iil.:d,I" ror sllpplying adjustable direct voltage and current to the armature 1.1111 li ('ld willcliIlg of the motor, either directly or through a transformer.

LA

a

.--o

2'

I

io

Uo l' fi

iI

F l,( . tl .li .

(I\')

d ' ~ ' III I,

:-; ilfl \~ "

!o

iA

UAi

I

~ tJ

111;1,1,,:\' l, i. l w" "" II I1H!lt,at. ~· d ("O IlV I' rf,I11' (I.) l'II I,,'I:IIQ01 1',, 1 I''I" 'I, v,.I,' 1I 1.

io ."

i Uo

103

The single-phase circuits discussed first are of interest, apart from railway traction drives, for low power applications up to a few kW. The circuit shown in Fig. 8.5 a contains four thyristors, which are made conducting or "fired" in diagonal pairs during every alternate half cyele of the line voltage UA· Provided there is a continuous direct current iD, the function of the circuit may be described by a mechanical commutating switch, connecting the DC circuit with alternate polarity to the line voltage

UA = UA sinT,

wt =

(8.1)

T.

When assuming at first that the direct current is smooth, iD = ID, being maintained by a fictitious current source, and that the impedance of the AC circuit is negligible, with UA coming from an ideal volt age source, the volt age UD(T) between the DC terminals consists of periodic sections of the AC voltage or its inverse; this is seen in Fig. 8.6 for different firing angles a, which determine the instant at which the other diagonal pair of thyristors is made conducting. The firing angle represents the switching delay with respect to a "natural" firing instant that would be valid for a circuit consisting of diodes; for the thyristors 1, l' the "natural" firing instants are T = O, 27r, ... when the volt age UA (T) and - with thyristors 2, 2' conducting - the voltages UTl (T) , UTI' (T) become positive. The maximum firing delay is ama", = 7r; for a > 7r, the thyristors 1, l' cannot be made conducting since they are reversely biased. This is also seen from the curve of the volt age across the thyristor, UTI (T), drawn in Fig. 8.6 c for different firing angles a. Later the range of the control angle will have to be further narrowed. The trace of the thyristor current iTl (T), also shown in Fig. 8.6 c, elearly indicates the switching action of the thyristors; at any time at least one of the quantities UTl (T), iTl (T) is zero. ln steady state condition with a = consto the volt ages and currents in thyristors 2, 2' pertaining to the opposite diagonal pair are identical, being shifted by a half period. Finally, in Fig. 8.6 d the input current iA(T) for the three different values of the firing angle a is drawn. As a consequence of the periodic commuta­ tion between the two pairs of switches the impressed continuous current ID appears at the line side of the converter as a square wave alternating current iA(T), whose phase lag ep with respect to the line volt age UA(T) is prescribed by the firing delay, ep = a. The active power (mean of the instantaneous power) at the line side is - assuming sinusoidalline volt age ­ determined by the fundamental component iAl (T) of the line current

PAC

= UA IAI

cosep

2V2 =- UA ID 7r

cosa,

(8.2)

whe['(~ (J II, I II' are H.MS-values of the line voltage and the fundamental cur­ n:lll. ('.OIIlj>OIlC·I\I ,. For LI < a < ~ , the power flows from the AC - to the DC ­ :;id", wilil., ("r :: . 11 ' n""",,, (./1I~ (lO WN f10w is ITv( ~ rs( : d.

104

8. Static Converter as a Power Actuator for DC Drives

8.2 Line-commutated Converter in Single-phase Bridge Connection

105

rot

a

Inverter I Rectifier

IX

1

=

lX.2= 90°

30°

a=7C

(X3=150°

2 Fig. 8.7. Phasor diagram of a single phase converter

rot Uo

blocking voltage V,T > O is created for a < O, during which the thyristor can be fired. Forced commutation calls for additional components and thyristors with short recovery time which involves increased cost and complexity; this wiU be discussed later.

2.

rot

Üo

-u 11: Reclifier

c lX.2

«( I

'II I

I-----

.

~ ~ /II

_'AI \

/

I

- , 1\ Ir

,

I

I

rot

I

~ ~

I

7C

I

\ I

\ ~

rol

~Tr

I

'_I

,. ~Al . ----T-J IA

I

I

\

I

I

iA \

I

I

a

a3

,~

Inverter

\

t7' \

/

d 1"'1(' H.G. V()ltages and currents of an ideal single phase converter

'I'II(~

first mode corresponds to controlled rectifier-, the other to inverter

"llCral.iof! . 111 Fig. 8.7 a phasor diagram is showing the phasors of the line volt­ !l~" I J A

aud of the fundamental component of the line current,

I Al .

Clearly,

a cllul.roUcd converter appears as an inductive load at the line side, drawing Ja.j':I':ill p; J'( ~ a("t.iv(! cunent; this is due to the control principIe allowing only ,kla.y ( ~ d lirill,~.

Fig. 8.8. Control curve of phase controlled converter

The possibility of reversing the power flow by electronic control is of importance if 'the converter is to suppIy a DC motor since it permits regen­ eration. It is pointed out, however, that firing controI affects not onIy the active but also the reactive power flow at the line side. At a = ~ the reactive power assumes a maximum value (for I D = const.) while the active power is zero; this is the case when the motor produces torque at standstill. Identical results are obtained when examining the DC side of the con­ verter, where the mean voltage is

Firiup; adv;mce is uot possible with line-commutatcd convcrt­

1H".ctUl s(' Lhe jiriug plllses would bc appli(~d (.0 tltyristors having rcver s (~ Il ta!-l volL i'lt'í< ' whirll i'f('v('uLH tlH:lll [('IIJH (·OJldlld.illg; advaúC(~ firillg i R ollly !,ow d ld(' wil.h· r"I" '" ,·o)[)l ll u(.n.I.('d '~OIlV .. [ I.'· lii , wI"," : illI iul.<'rval wil,h rorw
!'I ii

........ ­

\

!

o+rr

tt u

1

71'

'It A

sin T

2

dT

= -7r í/, A

cos a

2)2

= ­ 7r-

UA cos a

= UDO

COS

a

UU)

fi I U I I

I

,~

C f\

8. Static Converter as a Power Actuator for DC Drives

1.06

8.2 Line-commutated Converter in Single-phase Bridge Connection

This is described by the control characteristic UD(Ú) seen in Fig. 8.8; for reasons discussed later the curve is only valid in the interval O < Ú < ú max , with Ú max ~ 150°. Because of i D = I D = const., the mean of the output jJower is F DC

2..)2

= UD I D = -7r- UA I D

cos Ú

(8.4)

,

which is in agreement with Eq. (8.2). This is so because the idealised switch­ iii! ; converter consumes no power; it functions by shifting the 180 0 -blocks

()r Lh e direct

current with respect to the line voltage. The transferred mean changes sign in accordance with the direct voltage as the firing angle II" exceeds i. Since the current cannot change sign because of the unidirec­ Liolla[ thyristors, the line-commutated converter circuit in Fig. 8.5 represents ;t cO l1 trolled power supply operable in the two quadrants of the ID, uD-plane, ; lS seen in Fig. 8.9. The control is performed by electronically delaying the li rill f.!; pulses to the thyristors by the firing angle ú. p.. wcr

'.

107

Controlling a converter by electronic phase shift of firing pulses requires little control power and involves only very short delay. This is seen in Fig. 8.10 showing the output voltage uD(r) with the firing angle being switched between Ú = 30° and Ú = 150°. Since only one diagonal pair of thyristors can be fired in each half period, there is a maximum delay of 10 ms with a 50 Hz supply until the converter output responds to a command calling for a chang e in firing angle. Hence a static converter represents an actuator of high dynamic performance combining - high power gain - large output power and low los ses - very fast response . These features may yet be enhanced by employing multi-phase circuits, as will be shown in the next section.

(Ut = r

Uo

"

'/1' (//1

a=O

Fig. 8.10. Output volt age of single phase converter with changing firing angle

Rectifier

lO Inverter

a= amax /I

1//1

I"il~ · 1'1.0. ' I'wo

quadrant operation of line-commutated converter

(;Icarly, a switched converter causes interactions with the line, involving reactive power as well as current harmonics. Because of the supply­ / 1,11(1 1.11(' Lrausformer leakage impedances this causes voltage fluctuations and hlu'IJlollic ([i stortion which must be taken into consideration as the rating of 1,1", ('()Ilv(~rt(']" inereases. While the harmonics problem may be relieved by I',(JIII, ~ t,o Illlllt,i-phasp. circuit,s, the reactive power is a characteristic feature of alll ill( '-cOllllllllt,al( :d CO llverter::; with ddayccl firill g control; it is independent of LI", 11111111)('1' 01' pha~.,\,s alld call ollly 1)(' (:(lllllLt'racl,cd hy reac tive compou:mtioJl, AI. ili!'.h pmv"1 II()i.h ,,11" '1'1.:1 IIln.v J'( ~qll, ij'(' C"l'I'l'djoll I,y hi\.J'IIl()I;icJill,, · r~i havillJ>; II, I"'" dill!,, IIII'IIL ,""V"I' 1'111'1 ,01' u.t. lill( ' 1''' ''111 1\11' '.1' I;I./',I ~ illg

One drawback, though of little practical importance, is the fact that the steady-state and the dynamic behaviour of the converter is highly nonlinear, corresponding to a nonlinear sampled data system with one firing transient in each half period, From Fig. 8.10 it is apparent that, following a changing firing command, the result may not become effective immediately because there are variable waiting intervals. This is particularly pronounced when the firing angle is increased for inverter operation because it is necessary to allow for the relatively "slow" reversal of the line voltage. When going to rectifier operation, i.e. reducing the firing angle, the converter may re­ spond instantaneously. An accurate analysis of dynamic converter operation is quite complex; for small perturbations linearised difference equations can be derived but large signal dynamics are governed by non-linear difference equations for which there is no analytical solution [F12,L34,P46]. Fortunately the converter is usually followed by an inductive load having low-pass transfer characteristics, such as the armature circuit of a DC machine which smoothes the current; therefore the converter dynamics may be approximated by very simple models when designing the inner current controlloop. Anot,lwr Jloint where high power converters may occasionally cause prob­ kws is a djj'(~ct. collscquence of its advantages mentioned before; the power HIIJ)pli, :cI Lo LIli' lontl IIlw;t, iustallt arwollsly COIIIO frOITl the AC líne. There is II I) d' ·(O""plllll'. 1'1(" '( I. t,.y illt,I ' l'llaJl y sl,( IITt! ( ' IlC'1') ~'v hl·I ':\.II S(· ali id('al C() ll v(~ r" ,(~ r

lOS

("ontains only loss-free switches. As was mentioned before, the importance of Lhese effects has receded as the capacity of the interconnected supply systems Itas increased and is more closely meshed. The control of the converter is normally performed by the output signal I)r an electronic controller influencing the phase of the firing pulses through ii, firing control device. Its basic function is explained with the help of Fig. !:UI. The output signal Yl (t) of the innermost controller, usually the current coutroller, is compared with a saw-tooth-shaped periodic ramp function YT which is synchronised by the zero intersections of the line voltage UA. When­ eVI'r the difference becomes positive for the first time in each half period, lhe diagonal pair of thyristors, Fig. 8.5, having forward bias voltage is fired. '!'/te selection is done by alternately activating one of two monostable trigger circllits MT 1 , MT 2 . One ofthe trigger circuits produces a short pulse which, ;IJt,I~r amplification, is passed through a small isolating transformer to the 1~"Lcs of the thyristors to be fired (only one of the output windings is drawn). ILL high voltage applications, such as high voltage DC-transmission or static n:- Now a I.ltyriHL()r pai r 1::1,11

«

S.2 Line-commutated Converter in Single-phase Bridge Connection

S. Static Converter as a Power Actuator for DC Drives

(rom controller

2

109

.

Y1 =]iarc sm Y

rl~LL~~

~I~I: Y I;====IJ

T­ !.

Il0ug,

__ .

~IIOug2

a 0:1

0: 2

0:1

0:2

~

-Y1 (7:) wt=7:

b Fig. 8.11. Principie of an electronic firing control circuito

(a) Block diagram; (b) Mode of operation

UD~

Uo =

",. a(," \ /";{t !.. '

_ '.

I

'"

;'

l,

'. .'. ;.... . •

ROiO

I

(,o .

I \

I \

I \

I \

I

'.

~

/

I

I

1~ 1\ I~\ "

I

;"

I

F

\ \

I

_"">L-_. .

\

rot = r

\ \ \ \

,/

Fig. 8.12. Single phase converter with l'esistive load

only begin to conduct if the line volt age exceeds the back voltage, IUAI > e, because otherwise the thyristors would be reversely biased; the current ceases to flow as soon as this condition is violatedj Fig. 8.13 shows the effect. A furtlu:r cornplication arises, if the DC circuit contains an inductance J~ II in ,t () !lecami(: th(' arlllatlLre indllctance with its

UI)

8,2 Line-commutated Converter in Single-phase Bridge Connection

8. Static Converter as a Power Actuator for DC Drives

110

8 ~

/

~

The solution, satisfying the initial condition iD(al) = O, and with the abbre­ viation LD/ R D = TD is

,

e

\

\ \ I

I I

iD(r)

\ \ \ \

I

,

I

rot = r

UA = --

1

\ VI + (wT D )2

RD

[sin(r - arctanwTD) -

I

e-(r-a,)!wTD

1"11 A 1

- .!!...- [1- e-(r-a')!'OTD] RD

energy can maintain the current iD during part of the interval where

< e. This is drawn in Fig. 8.14 for different values of the firing angle a

;tlld assuming e = E ~;I.ill neglected.

= const.. The internal impedance of the line voltage is

lO

/ I /1

J/

1

",

E

(8.6)

At f31 < aI + 7r the current iD becomes zero, iD(f31) = O, see Fig. 8.14, at which time the thyristors return to blocked condition; the current remains zero until r = aI + 7r, when the thyristors 2, 2' are fired and the sarne transient is repeated. Thus in steady-state the direct current is discontinuous, consisting of separate pulses. The mean of the current is a function of all pararneters, _ iD

'\.

sin(al - arctanwTD)]

/

1"ilJ:. 8.13. Single phase converter with back volt age and resistive load c;I.()II~d

111

1

= ;:

j{3'

iD(r) dr

= F[a,

UA, wTD , R D , E].

(8.7)

a,

I

~31a1

I

I ,I

III)

I' I

rl

I

I

I

/

--úA I' ...... __ ......

I

/

' I...1 ...... _

//

0.,+11"

A

UD =

iD(aI) = iD(al

:a1

2n rot = r

a2

,

rll

I /{ JI

,

1 /I

'

'U , \ HIII ,

/,/. 1


I'

~

/1 1

•

(I)};)

+ 7r)

= O;

(8.9)

thus it assumes the form UD = E

Discontinuous and continuous current

,b:"

(8.8)

This integral may be simplified with the help of Eq. (8.5) considering the periodic boundary condition

",..-'

115

J

uD(r) dr .

a,

I I

assume an initial zero current state, iD =diD/dt = O and conse­ = E , At time wt = r = aI, with the condition UA sin aI > E, 1,11<: I,hyrisl:ors 1, l' are fired so that the current iD begins to rise with finite :,Io(l(: and lIuickly exceeds the latching value iL. With the thyristors 1, l' ('olld udi II"; ;wd ;11, /I (1') 'Il A (r), the following diffcrential equation holds, {"

~ j

·1

(lll(~ntly '/Lo

r, I

When evaluating Eq. (8.5) the impedance (RD, LD) must be augmented by a smallline-side impedance as is shown lateI'. A similar expression holds for the mean voltage

-- U ......

I I

I

o

Lct

I

l I I I, I

I'

I '"

a

I"i!;. tl.14.

I

/

I

rot = r

~3+n

+ RDiD



(8.10)

where iD is given by Eq. (8.7). This load characteristic uD(iD) for a = const. and different values of the back volt age E is highly nonlinear as long as the current is discontinuous; this is due to the variable period of conduction, f31 - aI < 7r, as is seen in Fig. 8.14. When graduaJly reducing the firing angle, a < aI, with the same initial conditiolls as 1wfore, the amplitude and the duration of the current pulses illc\,(',as(: 1IIII.il :I, lilllitiug case occurs, a = a2, where one pair of thyristors in lir('d ;11, UI(' 111.111]('111. (,]1(' other pair ceases conducting, f32 = a2 + 7r; the ('1111"1'111. 1.11<'11 ollly t,i1lwlll':; 'I,('II~ OIlC(~ (:v('ry IIn,lr p(:riod.

8,2 Line-commut~ted Converter in Single-phase Bridge Connection

8. Static Converter as a Power Actuator for DC Drives

112

If the firing angle is still further reduced, a < a2, the current will not reached zero by the time the other thyristors are fired. This then con­ :;LiLntes the region of continuous conduction, where the current is transferred iII a commutation transient directly from the outgoing thyristors 2, 2' to the illCOllling thyristors 1, l' and the current is no longer discontinuous. The bordering case to continuous conduction follows with the periodic J,ollll(lary condition II ; tvC

iD(T) = _UA_ RD

113

1 -,,;c1=+==;(=w=;;:T

:===;)""'2 D

[sin(T - arctanwTD) -

e-(T- 0 3) / wTD

[1-

e-(T- 0 3) / wTD]

-!

sin(a3 ;- arctanwTD)]

+ i D(a3) e-(T - 0 3) / wTD,

(8.16)

where the initial value iD(a3) in steady-state is obtained with the help of Eq.

-lo(a2) = iD(a2 + 'Ir) = O

(8.11)

I'rolll Eq. (8.6), UA

) i + (wT

)2 (1 D

+ e-

wTD

7r /

)

sin(a2 - arctanwTD )

+ E (1 -

e- 7r / wTD )

= O. (8.12)

Sol ving for a2, the condition for continuous conduction can be written as

(t

D < a2 = arctan w TD - arcsin [ E.)1 + (w T )2 tanhC W'lrTD )

UA .

1. (8.13)

l<'oJ" (~ach value of the back voltage E a firing angle a2 exists, at which colltinuous conduction begins. With WTD » 1, i.e. assuming a I 1<'1 1'1 'c 1.1 y sllJool.h current, Eq. (8.13) yields the obvious result 1.11<" 1;1.1\1',<: 0["

"2 /fI!

!T

ú;\ cos a -> E

,

wllidl IIW
(8.14) UD must exceed the back voltage

ror

CIlIll.illllOUS load current to flow. > !\ II L!t( ' preceding equations are valid for E < O, that is, for rectifier as wdL as illv(~rter operation. !\ s wa.'; Illcntioned before, continuous current flow occurs if the firing angle is rl ~
'i n (It:,)

= iJ)(üa -+- 'Ir)

lu ' [IC(' \.1[(' Sollll.ipll IIl;ty

> O;

1)(' writl:clI as

(8.15)

(8.15). li the mean value of the current is mainly of interest as is the case with drives, a much simpler solution exists for continuous current in steady-state. By integrating Eq. (8.5) over a half period and inserting the condition (8.15) for periodicity, we find

J

0 3 + 7r

-iD(a3)

=

:;1

1 (UA sin T ­ E) dT RD

=

R1 (uD(a3) - E) , D

(8.17)

0 3

where the average voltage UD is given by Eq. (8.3) . Hence the converter acts as a linear controllable voltage source only if the current in the load branch is continuous. The mean current i D (a2), at which discontinuous operation sets in, Eq. (8.13), is an important design parameter for the possible selection of a smoothing choke, resulting in an adequate value of the normalised time con­ stant wTD . As is shown in Fig. 8.16 this current also depends on the back volt age E, its peak occurs around E ~ O, where the AC component contained in UD(T) assumes a maximum. With large drives the specification may call for the maximum discontinuous current to be less than 10% of nominal current, i D (a2) < O.lIDr, to limit the ripple current which produces additionallosses in the armature and may impair commutation. However, with medium and smaller drives the smoothing choke is frequently omitted for reasons of cost, so that only the armature inductance is left for filtering the current. This may result in discontinuous current of half nominal value, affecting not only the los ses but also the design of the current controller, as will be explained in Sect. 8.5. Another important effect neglected so far is the finite commutation in­ terval caused by the impedance in the AC circuit, which consists mainly of the leakage inductance and winding resistance of the transformerj also, line inductances may have been inserted for protective reasons to limit the rate of change of currents (including fault currents) through the thyristors. At medium and larger power rating, the line impedance is predominantly induc­ tive, R;1 ::<:: O.lwL;\, which for approximate calculations justifies neglecting the f('sisl.iv(~ cOlllponnnt, Tlt(, liuI' :-l ide illdllcLallce L;\, even tholl gh it, is small whcn compared with UI(' illllllri..'I.JlIT I ,,) 'ill UI(' 1011,<1 circllil., cnIlSI'J» \.II C liJlI ~ ('.IIIT( ~ 1l1. to !>( !('. O lll l: a

8.2 Line-commutated Converter in Single-phase Bridge Connection

8. Static Converter as a Power Actuator for DC Drives

11'1

sLat.e variable subject to continuous change onlYi hence iA(T) and the cur­ )"('Ilt.S in the thyristor branches can no longer exhibit the discontinuities seen iii Fig. 8.6. Instead, the rise and decay of the thyristor currents and, cor­ f"(~spondingly, the reversal of the alternating current iA(T) now assume the f"mlll of continuous commutating transients requiring a finite time, called the coJJlmutation interval Tc or overlapi naturally, this effect occurs only with nllltinuous load current because in discontinuous current mode the currents h"gill in each period with zero initial condition. For example it may be assumed that the thyristors 2, 2' of the circuit iII F'ig. 8.15 a are initially conducting and that at T = 0:, Le. with positive lill(, voltage UA, the thyristors 1, l' are fired. Since the currents iD and iA ;\.IC c()lltinuous on account of L D and LA, the thyristors 2, 2' remain also mlld IlcLing at first, so that the four thyristors represent temporarily a short circllit connection, UD = UB = O, decoupling the line current from the load circllit. Hence, from Kirchhoffs laws, it follows with wt = T, = iD ,

(8.18)

= iA

(8.19)

'Í'Fl - 'i T2 LU

LA .

,

diA b + R' A tA

(8.21)

TIl\" illiCial conditions are

(o)

cc.

10'/,::(11') -

- iA (o:)

= iD(O:)

'/';1 -,

"

I,

a+rc

wt =r

Lo I

i

'--~r

!e

i

TI

L A/ RA we find for o: ~ T ~ o:

+ Tc (8.22)

UA

= ~ + (wLAF

i;; i\w p(:ak valuc of the (steady state) short circuit current at the line termi­ lIalH. 1"1"0111 r'~qs. (8.18), (8.19) the thyristor currents are obtained, (r)

; ' l '~ ( I')

I : ':1 ')

f'in(n)

~ l i r.((I')

I

I 'I

1'--­

i

1\

--~,

\

I I I I DI I

I

II I I I I I

wt=r

....

I

wt = r

d Fig. 8.15. Commutation of a single phase converter

+ Tc) = O

(8.23)

after this instant, Eq. (8.5) is again valido The various volt ages and currents, when including the line impedance, are seen in Fig. 8.15 a,b, assuming continuous current iDo All the currents are now continuous functions oftimei in contrast, the volt ages UD(T) and UB(T) exhibit characteristic discontinuities during commutation, when all thyristors are conducting. As a consequence of the finite commutation interval, the line current ex­ periences a further shift in time resulting in an additional component of reactive power due to commutation, increasing the reactive power caused by the delayed firing control as discussed before, Fig. 8.7 Another effect of the line impedance is a loss in output voltage due to the temporary short circuit,

1tO(T) = 0,

.

1

'

I

/>,

O ai1

c

a

I

I I I I T2111

Ro

iT2(0:

wll(')"(:

;',/"1

Uo

I I

I I

I

.

iA(T) ~ iA sin(T - arctanwTA) -[iD(O:) + iA sin(o: - arctanwTA)] e-(T-a)/wTA

'tA

I

These results apply only as long as the four thyristors are conducting. The commutation terminates at time o: + Tc, when

O,

It.. , 01"':(' of" Lhe llIuch smaller inductance LA, the line current changes rapidly wlll'f"(':ls Llw load current may be assumed to be approximately constant dur­ 1111', III(' shorL commutation interval, iD(T) ~ iD(O:). Hence, with the abbre­

"I:d.ioJl

\

/1

ue

(8.20)

= UA ,

. + RD ZD =-E .

diD . dT

w Lo - -

io'/'I

t1Uo

---,-,,-

+ iT2

'iTi

115

a

~

T

~

o:

+ Tc

i

';'A(T)I ,

iI.

ill( r }l·

c;m;;(';; ii drop

01"

i.Il(~

JIlcan voltage by

(8.24)

8.2 Line-conunutated Converter in Single-phase Bridge Connection

8. Static Converter as a Power Actuator for DC Drives

II(i

_

"J+

1

L1uD ~

1r

TC

-L1uD (8.25)

UA dr ,

" Wllich may be estimated from the current change in the AC circuito With the mentioned before,

~;illl[llification

RA «WLA, \VI'

lilld from Eq. (8.20)

---

.d"lll)

~

1

-

1r

"J+

TC UA dr

~

1 -wLAiA(r)

I"+TC

1r

"

"

1

= -WLA [iA (a + rc) - iA(a)].

L1uD ~ wLAIDn iD = k iD . UDO .,fiUA IDn I Dn

1 kllce

n



Uoo

(8.28)

The normalised impedance factor k is mainly determined by the leakage reactance of the line transformerj a common value is k ~ 0.05 to 0.10. The commutation interval rc, which usually lasts only a few degrees, de­ pends also on the firing angle aj this is seen from a rough estimate of the integral in Eq. (8.25)

discontinuous current

"2

(8.27)

which indicates that the line-commutated converter acts like a controlled voltage source having an apparent internal resistance proportional to the reactance of the AC linej of course, this drop of the terminal volt age does not involve real power losses, as it is caused by commutation at the AC side of the converter. As a consequence, the load curves of the converter uD(i D) for 0'= const. are drooping in the continuous current region (Fig. 8.16), even though all ohmic resistances apart from that in the load branch have been neglected. ln contrast to the intermittent current range where the curves are strongly nonlinear, the characteristics in the continuous current region are linear, exhibiting only a slight droop. By normalising Eq. (8.27) with the no-Ioad volt age UDO and nominal current I Dn , we find

(8.26)

the voltage-time-area lost during commutation is related to the flux ('liallgc in the line inductance.

2 ­ -WLA iD, 1r

1r

Uo

~

117

~ 4.J.UD ~ -1

a= 0°

1r

J' .

"+T

c

UA sm r dr

~

M2

. a -v rc UA sm

(8.29)

1r

"

continuous current

which shows that for constant load current the overlap is minimal at a ~, when the AC volt age has its peak amplitude. A more accurate result is obtained by evaluating the integral, Eq. (8.25),

60° 0,5

1\\ ':

'

i

r/ "'it~.

900

a

":n

120°

150°

8.10. Clmractcristics of single phase converter with inductive load

rc

~

arccos cosa - .,fiwLA-] UA iD - a.

[

The knowledge of the overlap is of particular importance when the con­ verter operates as inverter, i.e. with large firing delay a, because then it must be assured that the commutation is completed and in addition the outgoing thyristors have recovered and are ready to block forward volt age when the line voltage changes signj otherwise an unintentional refiring may occur. Since the back voltage is negative during inverter operation, E < O, this would con­ nect in series two voltages (E, UA) having (for a half period) the sarne signj the result wOllld be a short circuit condition with a fast current rise which is only limit<~d by the smoothing inductance. This malfunction is called a

Ir (I. f;IIIIÍcÍI')lI,ly 1i~1'I',(' slIIooLhillP; illdllct.:I.Il(,p Ln is employed, s\lell tllnt, tltc , Ippll ' ('llt n' I.II. '('I," I.nilwcI ill i,,( I) lIItty IH' IU'I',I<'d,'d, i ,1 (n ,·1 r,,) ~ 'I:,,((\') :=::!

COllllJllIl.al.ioll hilllr<~.

I (/ ; I' ~q

ifi

(H . ~~n ) 11,1'11 .I' /I'ld,I . ! 1

VI

,II. ii I'," <1m!, pr"lll,d.l."ml ln !.II('

)1I , ~d (' 111"1"1'111.

(8.30)

III )"i,,;. H. I'1 l,lti H di~t1Ll'b<\l\c e is (kr-;cril)( ~ d iu more detail. The invert(~ r '(;:)0", wll<'1l nll I.n 1'!,I 'I'Il Ln illil.in,lIy aI. I.ltl~ r;;IJ(, IÍrl 111', il.ll/ ~l<- n,

1I !IH lIlIl<'d

118

8. Static Converter as a Power Actuator for DC Drives Commutating

voltage

,

Ua

\

I

I ){

c;

, -,

"

\

\

IW

~

mt = "t"

~

mt = "t"

iT1 lJ

' U

!

"' :

T2

I

I~ (

!

Back commutation to 2, 2'

I

c

·-

---'-rf------==',.l...r.oL.,----....

mt = "t"

Fi!;. 8.17. Commutation failure due to late firing of thyristors 1, l'

;ulditiollal firing delay ..:1a for the thyristors 1, l' is introduced. Due to the I"w voJl.agc 7J.A (r) in this interval the commutation proceeds quite slowly as ii. I.al\( ·.s I.illl!: to accumulate the necessary voltage-time-area. The commutat­ "11', 1,I'iI.lISi(·1I1. is cventually completed at a + rc, when the current through I.lty,i: ;I.ors 2, :l' liaS become zero. There is still some negative blocking volt age 11,/, ~, ' () arross the outgoing thyristors but it is small and the time until the v"II.II.I'.< · 'Ii /I chauges sign may be too short for the thyristors 2, 2' to recover 1.llI' il I'mwarcl blocking capability. ln other words, the time equivalent of the t· .~ I.illrLioll angle, = 7r - (a + rc) no longe r exceeds the recovery time tr which i ~, i IIcr<~;t.scd by the low reversed bias volt age across the outgoing thyristors. Tllc collscquence could be a spontaneous refiring of thyristors 2, 2' and IlI\ck-cOIlIIn\ltation, leading to the series connection of UA < O and E < O I.Ilro'lf~ h thyristors 2, 2' and a subsequent steep rise of the current. The next I'\'/,:' alar Wllunutation of the inverter is possible one period after the attempted fil'illf ~ (Ir thyristors 1, 1', where the controller must now assure a sufficient1y H.d vallC(·d timing of the firing pulses to successfully complete the commutation d, ~spil.<: I.hc irtcreased current. Otherwise a circuit breaker or fuse wo.u ld have 1.0 d,~;t.[' lhe fault. II. is n(~('lI that. th(~ danger of a commutation failure increases with the IlIiq·,I1il.lld(· (I]' lhe c1lL'n~lll and thc line-side ['('ad.ance, since both effeds pro­ I"" lí 1.1a(' (,Olllllllltatjoll illl.(~rv;d r,: . III ord(~r 1.0 flllly Ill.Ílise nlc. voHagc of lhe lllv,· rl.v r, II ::ldlitl'·1I1. I 11 11 1'. II i I.lldn of t.l1(~ (·xl.lll<'l.i,," 11111',1 0 ,lIlUy h(· a.'is lIl·(·d hy a ilq' III I' I." ",,"11'1>1 ")('1'. 11"w"v"I', iII vil·w "r 1.III ',"I,l i l.I,,,ml '·'''"(1I< ·xity 1111.1 LIli'

8.3 Line-commutated Converter in Three-phase Bridge Connection

119

remaining uncertainty in case of a sudden voltage dip of the AC line voltage (which would unexpectedly increase the overlap), extinction angle control is seldom used except with very high power installations, such as HVDC, where operation of the inverter at the maximum firing angle is important for power factor considerations and the cost of the control system is less sig­ nificant [F21,77]. The usual practice with drives is to limit the firing angle to "safe" values, for instance a < a max = 150 0 , and to prevent excessive currents by a fast acting controlloop.

8.3 Line-commutated Converter in Three-phase Bridge Connection With the exception of traction drives having a single phase supply the single phase circuit discussed in the preceding section is normally used only for low power applications, such as the supply of field windings. Beyond a few kW there are strong incentives for three-phase converter circuits. The bridge con­ nection shown in Fig. 8.18 is the circuit most commonly used with thyristors or power transistors; it requires two more thyristors but has a number of important advantages: • The three-phase line is symmetrically loaded in steady state. • The line currents have a lower harmonic contentj as a consequence there is less distortion of the line volt ages than with a single phase circuito • The sarne is true for the direct voltage uD(r) which contains ripple com­ ponents of higher frequencies and lower amplitudes, thereby permitting a reduction of the filter components and causing lower losses in the load. • The dynamic performance of three-phase converters is superior because thyristors are fired at shorter intervals; hence the reduced delay for control signals permits more rapid controI. ln contrast to the single-phase bridge circuit (Fig. 8.5) where the thyris­ tors had to be fired in pairs, the commutations in the three-phase circuit (Fig. 8.18) alternate between the upper and lower row of thyristors, so that in steady state six regularly spaced firing transients occur during each line period. Hence the three-phase bridge is a "six-pulse converter" , whereas the single phase bridge with two firing instants per period belongs to the class of "two-pulse converter" circuits. With Hg-arc valves, having a much higher volt age drop than thyristors, center tap circuits without series connection of valves had been in widespread use; they were the natural choice with multi-anode valves with a common Hg­ cathode. However, since the introduction of thyristors as individual switching dp'vjc< ~s witll sllla.ll physical dimensions and low voltage drop these aspects have h ecllllu' ohsold,· .

I 'lO

8.3 Line-commutated Converter in Three-phase Bridge Connection

8. Static Converter as a Power Actuator for DC Drives 5

io

iTtt (x-----,.

~r

t

~r

1II.J 1.

-L O

I

I S2 1

-LA i1

!

U1

II RO I

a

!e

rr 1 2

U2

!

5

S1

Uo

iT4

121

13

U3

,,

!

I

b

i

I

::

A!!T1 /1

Fig. 8.18. Three-phase line-commutated converter bridge

~I

i

O::

I

=

I

-----v T1

c

'!'ltc AC voltages in Fig. 8.18 are assumed to represent the line-to-neutral vI)IL:I./.o;(~s of a balanced three-phase system. The thyristors 1, 3, 5 connect the i\ C lillc-I,crminals in cyclic sequence to the upper DC bus SI, the valves 2,4, 1\ LI) !.II(' lower bus S2. In normal steady-state operation and with continuous d l", ·( 1. clllTcnt, each thyristor in the upper and lower row carries the current !'", I:!()"; a]so, the two thyristors connected to one AC line-terminal do not IIClIlllally collduct at the sarne time, as this constitutes a short circuit of the I li: I.I'rlllill a ls. Figures 8.19 and 8.20 show various volt ages and currents in steady-state ('()lIdil.i()llS with the firing angles aI = 45° and a2 = 135° , belonging to r(' r Lili(~r- and inverter-operation respectively. The line inductances LA are 11( 'I',II 'cLcd for the time being so that no overlap between subsequent thyristors oCCIlrs; thc constant direct current iD = ID is again assumed to be generated I,)' :l cl1rrent source. The conducting thyristors are indicated in each 120° illl.<'rval , always comprising one of the upper and one of the lower row; the Jirilll~ aHcrnates between these two groups; hence the output volt age uD(r) C ( 'lI sis1.s of (i0° sections of sequentialline-to-line voltages, for example UI2 = 'If, I '/I. ~ . As IOllg as thyristor 1 is conducting, thyristor 3 sees a negative bias v()lt. agc 1<>1' 'lt2 - 'U I < Oano can only be fircd arte!' 'U 2 - 'UI has become positive. 'I'hi:: "lla(.l1l'a!" ('irillf.!,' instant (r ~. n/O for Lltyrislor 1) serves as a ref­ (')'( ' 111'(' r()(· 1.11.1' lirill P; ;\.IIJ-\ II ~ ( ~ , which cau ag:l.ill vn.ry fr<1I1l (Y O (lllilxillll1111 (1111.1'11'1. v"It. II,I':c', ,I, ·,·t.lfi(·r) L" rr ( ~""I." (lI d llillllllll 1111 Lp 11 I. VOJt I I,I~( ' 1 illv<'rl,l'r ),

I·

: .

'

27r

wt

='t'

I

:

7r

, 2n

,~3

,

,I

i

d

I I



I

r::-+l

,a 11

3 :IJi~ II n

I.

2nl

wt =

T

Fig. 8.19. Voltages and currents of a 6- pulse converter at ai

= 45°

An appropriate safety m argin 7r - ama:>: is again required which must exceed the maximum overlap plus recovery time tr to prevent commutation failures. With the definitions for the line-to-neutral voltages uI(r)

= V2UA

sinr 1

U2(r) = V2U A sin (r - 2 7r) 3

7r

u 3(r) = V2UA sin (r - 43

)

,

j

and the assnmed simplifications the mean of the output voltage is

8. Static Converter as a Power Actuator for DC Drives

122

8.3 Line-commutated Converter in Three-phase Bridge Connection

dominates in the first, forward blocking voltage (UTl > O) in the second case. This indicates that rectifier operation is insensitive to overload, while at the inverter limit the danger of commutation failure always lurks in the background, because negative voltage is required across the outgoing thyristor for a sufficient time to let it recover its forward blocking capability. ln Figs. 8.19 d and 8.20 d the current ii in one of the AC terminaIs is drawn, the fundamental component of which is again lagging the pertinent line-to-neutral voltage Ui by the firing angle o: (neglecting overlap). A com­ parison with Fig. 8.6 d shows that the harmonics content of the line currents is reduced; whereas all odd harmonics of line frequency I were present in the single-phase circuit, there are additionally no odd harmonics of frequency 31 in the three-phase circuito Rence the lowest order harmonics of the line currents, assuming perfect symmetry of line volt ages and firing sequence, are the 5th and the 7th in the three-phase circuito 80 far, it had again been assumed that the commutation between the conducting thyristors occurs instantaneously; in reality, it requires finite time because of the presence of leakage inductances in the supply lines and the thyristor branches. As mentioned before, line inductances are often intro­ duced deliberately to limit the rise of the current during commutation; if no transformer for supplying the converter is used, this is mandatory. Again the line impedance is assumed to be purely inductive, RA « WLA.

3

,I

rr_-

3TC

-",- 4 Inverter

Uo(1:) L)

1/ _.-:

I

-1- - -7 -­

,

I

TI ,

wt = 1:

:TC

(;

,II.

r

SI

iT1

5~I

wt = 1:

TS

Uo

I" .r: · )·I.:lO. Voltages and currents of a 6- pulse converter at

0<1

= 135

0

,'l

3 II./i ~ -

/

'Ir

'Ir

i+ a

~+Q

=

3V6 - UA cos ('Ir)li+a r + 'Ir 6 f+a

io

'Ir sin(r+-)dr 6

3V6 =- UA coso:

'Ir

= UDO

i2

coso:.

!

U2

control curve follows a cos-function. When the converter is rl.ed to the 230/400 V three-phase line, the maximum mean output voll,;W;(~ ii-> (1)O 540 V. 'I'IU' olltput voltal~e 'uJ)(r) in Figs. 8.19 and 8.20 contains harmoncis, thc rn'<[I'H'lIfiC':-l (II' wltic.h aJ'(~ lIlultiplcs of G/, i.ç. :\00 Ilz rOl' a 50' R,z Iinc~. V.. II.:q',e' 1\11 <1 ('IIITe'ul. (Ir ii. I.hyriH(.or l U'C' pllll.l\'<1 111 Jo'i, ç. R.H) (' fOI' )·(·('I.ifi er ­ l\cr(' I.on, lhe

' U
1,1

=

I" i". K ~O

e'

1'111"

I lIve'r!.!'I' o\,,'rnU(III .

TI

T5

5

L, Iuo

,----,

j II

LA

11"v" 'I't1(' I tlnl l v "II, I\I ~( '

('11'1'1

O)

a

LA i2

iI

i3

U3!

UI! b

Uo

~.lO

1"8, ,

LA

(8.31) ( ' IIII I ...

fi

i

IT6

f+a

3V6 / (Ul-u2)dr=--UA

_

•

6

6 .. ~+a

io

io

iO

,/ Iii

123

U2!

LA

T2

iI

i3 6

u3!

UI! -

U2!

c

Fig. 8.21. Sequence of commutation in three-phase converter bridge

Th(' COllllIIlIl.al.ion dcscribed in Fig. 8.21 proceeds in three stages: lnitially the tltyriHl.ors ri aliei G lIlay carry the consta.nt current iD = lD (Fig. 8.21 a); nl. I.illll' I ;~ I " 1.I1'y f' isl.or I. is firec] 1I\lI,(Ii)lf~ to ti\(' Ritllation in Fig. 8.21 b,

I~!J

8. Static Converter as a Power Actuator for DC Drives

8.3 Line-commutated Converter in Three-phase Bridge Connection

125

wlwre two AC terminaIs are connected through thyristors 1 and 5; due to the ~:lIlall inductances LA a short circuit current rapidly builds up, transferring U J( ~ current ID from thyristor 5 to thyristor 1. As soon as iT5 has become 'I,n u , thyristor 5 blocks and the circuit finds itself in the position shown in

'f

"'i g. 8.21 c. The overlap during which three thyristors are simultaneously

('olt
+ iT5

= iT6 ~ ID ,

(8.32) ,

i

-:'fc:+­

d w L A dT (in - iT5) ~ UI - U3 .

(8.33)

.,._._._.~) ,

~

din

..;' ~._._.J. ,

,

1\ \ I

2wLA ~ ~ UI - U3

;;:.

= V6UA

sm(T

-7r/6),

\

'f

"

i..

(8.34)

,----­ -' .-. ­ . \ - iT2

1(':ldÍlIg with the initial condition /'f' I (r)

~

in(~

+ a) = O to

(7r)]

-v'6U - - [ cos a - cos r - 2wL A 6 A

,

'T5

, _____ J,

: o.

1':lilJlillation of iT5 results in

'f

.

T3

- iT4

- iT6

Fig. 8.22. Commutation transients of three-phase converter

(8.35)

-ij-+a+TC

LlUD

:1.11<1

I

3

= :;

~(UI-U3)dr.

(8.39)

-ã"+a /',/" ( / ) ~ J J) i I :,

::0"11

:IS '/''/'I>

'Il '/'/'Io

(

Ij

-

in (r) .

(8.36)

Integrating Eq. (8.34) with the limits O ~ in ~ ID results in

i!) zero,

' I' ü+rc

)

=0,

(8.37)

i+;"'.+TC 1 2' (UI

- U3) dr = w L A I D ;

(8.40)

-ij-+a

ii", ('1I11111l11tatiOll is concluded and Eqs.(8.32), (8.33) are no longe r valido

I"Í/';IJI'(' 8.22 shows a sequence of commutations as obtained by digital :illllld:tI,ioll of the converter circuit [E6, M17] where for the sake of clarity /.lI(' ('OIIllll1ltation intervals have been enlarged. Because of the equal induc­ 1.;\,\1,,(, :; iii (:ach line terminal, the potential of the upper bus SI corresponds Co LI\(, IIJ(~i).Il of the two line-to-neutral volt ages affected by the short circuit ('olld ÍLÍolI. Hence, the output volt age during overlap is 1

'IIO(T)::::::: ;;(UI +U3) - Uz, whicli l'I~II:Wh n Ins,~ H '~ , 'l ; í 1.,;

i.1l

HI P r" ll \lJ\lIIIIrI. í "l

OllLpllt

(8 .38) vol1:;I,f2:(~ accor
the voltage drop caused by the commutation is again a linear function of the load current,

--

3

LlUD = -wLAID;

7r

a fictitious ohmic resistor in would have a similar effect. verter in Fig. 8.23 show a range. As was cxpbilH:tl in the

(8.41) series with the controllable volt age source uD(a) The load characteristics of the three-phase con­ corresponding droop in the continuous current

case of t:hp sillgJe-phase converter, a load CÍrcllit illdllctivc: illlpcdaw',· iII : :<'ri(~s with a back voltag( ~ ]i) Citll 1<-0<1 1.<1 di n..c"t!,'IIIII
consistill/,: "r lt

(illil,,'

8. Static Converter as a Power Actuator for DC Drives

126

Ilighly nonlinear; this is seen at the left of Fig. 8.23. With an appropriate clloice of the smoothing inductance LD the region of discontinuous current C
uoo discontinuous current

~:1~; : /: ­ 1

a=O°

\ \

~inuous current I

~

\

\

I

-

a= 15 0 io

,.'

90 0

r

/

"'1\ ( H.4:1. I,oa d

The dynamic behaviour of the six-pulse converter is similar t0 that of the two-pulse converter with the exception that now the steady-state firing intervals are 1r/3, instead of 1r, permitting faster access to the output volt age UD(T). The waiting interval is now between O and 1/6 f corresponding to 3.3 ms for a 50 Hz-supply. The firing control of a three-phase converter is simply an extension of the firing scheme for single phase converters (Fig. 8.11), employing several time-displaced saw-tooth functions to sample the output signal of the con­ troller. For example, Fig. 8.24 displays voltage transients UD(T), neglecting commutation, which follow a step change of the firing angle from rectifier- to inverter operation and back. When the firing is delayed, the current carrying thyristors simply remaín conducting until the line volt age has changed its sign, while in opposite direction an immediate response is possible. It may even occur, as seen in Fig. 8.24, that some thyristors are skipped because of a multi pie commutation with several- in principIe ali - thyristors temporarily conducting, with the ones connected to the highest and lowest line volt age potential eventually prevailing.

0,5

tI /

127

60 0

\ 1

8.3 Line-commutated Converter in Three-phase Bridge Connection

a

150

IOn

n+,g. --::c;;}.,\

:1 ":2,

multiple commutation

n

2n+ "6

wt=1'

0

curves of a 6- pulse converter

'I 'Ice l.clllporary short circuits of the AC terminaIs during commutation lIavc ()[ (".OUfse an immediate effect on the waveforms of the AC voltages, pos­ ,:ildy inlluencing neighbouring equipment. One of the terminal voltages UBl wil.h rc ~SP( ~ct to neutral, Fig. 8.18, is plotted in Fig. 8.22. ln a weak supply sys­ 1.('111 haviug a large internal impedance, the danger of inadvertent feedback to 1.11(' ("'Ollvcrter through the synchronising signals exists which may even cause ill.~ l.a.bility [A11]. Apart from the obvious, but not always practical, solution (Ir lill.c rillg t.he synchronising signals, there are various "supply-independant" lirilq( S(".hClll(~S wit.h internal oscillators that permit stable operation even with hil':hly di:;l,oJ"l.cd lill( ~ volt ages [H61,M16]. Three-phase firing circuits are abo nVl~ih"k iII 1.11< ' (onll ()r il)I.(~gral,cd circuits (chips). .

Fig. 8.24. Control transíents of a 6- pulse converter

A feature of the three-phase bridge connection is that two thyristors have to be fired simultaneously for start-up and in discontinuous current mode, one in each rOWj otherwise the current could not begin to flow. This is achieved either by employing wide pulses (LlT > 1r/3) so that they overlap or by generating for each thyristor two short firing pulses, 1r /3 apart. Normally the second pulse would be applied to a thyristor that is already conducting and be of no consequence. If the first solution is preferred, the firing pulse may be chopped into narrow pulses to redu ce the voltage-time-area and the size of the pulse transformers. The iupllt current ofthe converter, drawn in Figs. 8.19 d and 8.19 d with­ oul t;ü.illg COllllllllt.a.tioH into account, is the sarne as the current that would fl ow iII I.hl' prilll1\.ry willdiug of a Y /Y- <:ollll(~ded transformer supplyill g the convI'rl." I". Illlw" v"r, til.· w:l.vc{llnll nl" ti II' prill llu''y 1·.lIrJ"(~1I1, dlall f1:( ~ S ir (III( : 0(' I.h(~

128

8.3 Line-commutated Converter in Three-phase Bridge Connection

8. Static Converter as a Power Actuator for DC Drives

129

2ia

ia

io

J'O

i3

~'

n' ~

: 2nl

r

a1

iO

a1

ia

b

U

U

o

a2

a

a2

III

U

02

o

~ b

Fig. 8.26. 12-pulse-converter consisting of two 6-pulse converters in series- and paraliei connection

ip2

i p1

ip3

a Supp/y

F ig . 8.25. Three-phase converter supplied by Y /6- connected transformer. (n) C ircuitj (b) Waveform of primary current

I.r,wsJ"ormer windings are .6-connected as is explained in Fig. 8.25. Assume 1.";tI, Lhe constant load current I D is initially flowing through the thyristors ·1 a.lI
1' )1 .~alce of simplicity - equal numbers of turns, the following equations then Ilold for the primary currents Zp l

=

i3 -

iI

= 2iD ,

i p2

= iI

-

iz

= -iD

'lp3

= iz -

i3

= -iD

(8.42)

,

Pecrllutation of the six possible states yields the primary current waveform /1'1 (T) shown in Fig, 8.25 b which also contains the 5th and the 7th as the lowesL ord(~r harmonics. With the appropriate ehoj('(õ of tlw llllmber of turns !.II<' Y -
1·. IlLTI ~ lI!. wavI~f()l.'lll S .

1'\n ~qll'~ltI.l'y in I.h(~ Jla.~!. ,

o!.l.ailI;dti"

wi!.h ii, ('OllVI ' r!.l ' r

II" I.h .' 11. I.h ,y l'l n!.ol'lI " .lI d Lo

I... JlllÍ liI'd iI'

1)('

Lll
t'llllllIlOld ,l' 1I0W, (,III! oulput volLal1;t~

11IWillU () II(' I.h'yl' in l\ lI ' 1"' 1' l'I'H. llch WHH ill s llfl ir in ll!. (·(lIllll'(l.I'd i II

nn 1'1 I'r! . ~i (l 1\ 1I' 1 ~c1vlI, itl , HW ' R !lla'y !.lil'lI

1111/11·,, <1 III' IlO' ril ':: (,olllll'l '111'l'. l w ll vl dl d d 1," Y l l il l.ol'/i

(·<)llIpll'i.l'

three-phase circuits are connected in series, as seen in Fig. 8.26 a. Rather than taking precautions for equal voltage sharing between series connected blocking thyristors one now relies on the output voltages of the transformer to produce the correct voltages. AIso, if a Y / .6-connection with its inhe1'­ ent 30° phase shift is chosen for the secondaries there is an added benefit in the reduction of harmonics on the load-side, where only multiples of 12 f occur, as weU as at the line-side, where the harmonics are those of a 12-pulsc converter, with the 11 th and the 13 th being the lowest harmonics; there is a further improvement in dynamic response because the maximum waiting interval is now 30°. Another similar circuit is seen in Fig. 8.26 b where two six­ pulse converters supplied from a Y / .6-connected transformer are connected in paraUel, employing two separate chokes or an interphase transformer for summation of the currents. The principIe of achieving 12-pulse operation by se1'ies connection of complete converters is applied also on HVDC transmis­ sions with hundreds of MW, where many thyristors are still series connect(·~ tl in each converter branch. This brief introduction to the fundamentaIs of line-commutated converl~ ers seems adequate for dealing with most drive problems. A more detailcd analysis such as harmonics, circuitry 01' protective measures is given in tlw specialised Iiterature [50, 58J. It should have become clear that multiphas(~ static power converters are electrical actuators of superb dynamic perfor­ mance providing an excellent tool for controlling large output power (MW) with low powcr control signals (mW) in ve1'y short time (ms). This combjna~ tiOIl is \I11lllal.dl\~d hy ;),IlY othe1' type of power actllator. On the o the!' hawl iI. :;\Jollld 1)(' )lol.,'d I.Ilal. dlw 1.0 I.\w switchilll-!; adioll sL;t.tic

pow( ~ r COII\lI'I'I., ' I':; arc'

1:-)0

8. Static Converter as a Power Actuator for DC Drives

inherently nonlinear. Since they have no internal energy storage the output power is instantaneously transferred into the AC line causing harmonics and reactive power in the supply system; strong grids are therefore required to support the operation of large power converters. Even shorter response times are feasible with force commutated converter circuits, discussed in Chapo 11; while the power ratings available today are adequate for drives, they do not yet reach the leveIs needed in energy supply systems.

8.4 Line-commutated Converters with Reduced Reactive Power I II the preceding sections it was shown that line-commutated converters draw l"<'active power from the line; this is inherent in the principIe of control by ddayed firing which results in lagging current at the AC side of the converter. Siuce the phase shift of the fundamental component of the line current is dircctly related to the firing angle a, the situation is particularly sever e at n ~ ~, where the line current is, apart from the losses, purely reactive. This ('olJdition is unavoidable, if the converter is supplying the armature of a DC IlIotor because when the motor starts from standstill the induced volt age is illitially zero and the converter must be controlled dose to a = ~; hence
-

iO

2'

iD(T)

~

O,

UD(T)

~

O

restrict the operation to a single quadrant in the UD, iD-plane; with traction drives employing series wound motors this would not be objectionable.

iA

-

uA i02

DI

O2

i

":f'\ í\ I.

io

iOI

TI

i T2

TI

T

Ô

a

n a+n I

2

a

O I iOI

wt == T

.

I

I

I

I

. I

I

I .

i

02

I I

wt == T

-r=r::-=t-f----~.

1'

O

b Fif.;. 8 .28. flalf-controlled single-phase converter. (n) CirclIit, clial'.ralllj (h) Waveforms of volt.rtgps nao currents

'1'111' Hil.lllll.ioll 11I1I.'y I)(~ t.oh~rablc wil.h l:i t.r:onf~ s lIpply sy s t(~ IlI S huI. is Illlac­ ""pl.ld,I,' "II 1. J'lII'Ll lIlI driv" ll, wl)(')"(' t,1i, : r·(':t('(,iv(' ('li rn''lIt, iII c(lluhillaLioll wil.h

I I I I

---CTt-_:Tlj-t I

,

8.27. Single-phase converter with a shunting diode

~+2n wt==T

I

Uo

I"i~~.

131

the large inductance of the power supply comprising the catenary would cause heavy voltage fluctuations, in addition to the harmonic distortion. Mainly for the benefit of traction drives special circuits and control techniques have been developed with the aim of limiting the reactive power at reduced output volt­ age. Some of these circuits contain diodes as well as thyristors [D25, S57]. A simple though uneconomical solution is the use of a diode parallel to the load, as shown in Fig. 8.27 for a single-phase bridge circuit. The diode has the effect of cutting off the negative edges of the output voltage, essentially acting as bypass during the intervals when UD would otherwise be negative. Thus the trailing end of the line current (during interval1l' < T < 1I'+a in Fig. 8.6 d does not flow through the converter and the line as it is short circuited through the diode. The result is an offset of the fundamental component of the line current in leading direction and a correspondent improvement of the power factor. This has, of course, the effect that the circuit can no longer act as inverter; the conditions

a,1

UA

iII

8.4 Line-commutated Converters with Reduced Reactive Power

wt == T

8. Static Converter as a Power Actuator for DC Drives

1:\2

8.5 Control Loop Containing an Electronic Power Converter

Another circuit with similar properties is shown in Fig. 8.28 a, which I,dongs to a class of half-controlled converter circuits containing diodes in "lace of some of the thyristors. Here too, the line current cannot flow against I.I\( ~ line voltage and thus cause a temporary power flow from the DC to the I\C side, since the opposite diode would conduct and act as bypass. Figure K.2 8 b shows a simplified diagram of the volt ages and currents, neglecting IIw commutation intervals. Again the trailing end of the line current is cut­ ,,:r, tltereby advancing the fundamental component and improving the power f";wto!". Clearly, this circuit could be easily extended to form a half-controlled 1I1r< "'-phase bridge connection. Uo

iO 2U01O

a=O

8.5 ControI Loop Containing an EIectronic Power Converter

2 a=­ _____ _ _______1f. UOIO ­ '

iO "' i l~ '

1' . 2 ,. . Bonsl.·· and -buck connection of controlled and uncontrolled converter

i\ 1.,·('I'lIiqll(, ror improving the power factor at the input of the converter ,:: 1)111.11111 '" iII I"ig. 8. 29 a, showing a "boost and buck"-scheme of a three­ 1,1,11: :" I< '("t ili" r co llt.aining only diodes, and a three phase thyristor converter; 1,\, "IIIJll oy illg a transformer with Y / 6. secondary windings the harmonics :,il.lliI.l.ioll Ill ay oe improved as well. This connection makes use of the fact 1. 11:1.1, ;, CIlllverlcr operating at a = O or a = a max has a tolerable power factor wltil<- ojlcral.ioll at a ~ ~ should be avoided, if ever possible. The total mean IlIII.plll. voltage, assuming continuous current, covers the range II"

Ifi

+ Un 40 cos a m

1\X

:S

UD

:S UD

10

+ UD 20

.

(8.43)

H y .';( d('("I.illg thc voltagcs according to the condition I I , ) lO I (lo ~IJ (:OS Ü'''" X , ~C

in the centre, U D ~ UDlO, where the controlled converter, accounting for about half the total power, is fired at a ~ ~. At full output voltage, a = O, the circuit corresponds to Fig. 8.26 a, causing approximately 12-pulse line interactions. The principIe of boost-and-buck operation may be generalised by splitting the full voltage range ±Uo and assigning the bands of ±Uo/n to n series­ connected thyristor converters which are supplied from separate transformer secondaries. The control is then arranged on a sequential basis such that only one section is phase-controlled with O < a < a-max, while ali n - 1 other sections are operating either with a = O or with a = a max · 8equential phase control is of particular interest if the converter needs to be subdivided anyway for reasons ofthe high voltage required. This technique has also found application for rai! traction drives [K70].

-~--~-- - _ . _~---

(1

Uo

133

O,

1.1,.. ( ' Illllpld( ~ lir ~: 1. (jlladralll. of I,\w 'iTiJ, i o -plnw' is within reaeh. Tha.t. this cl l ('llil. 1i:1 :1 a 1)( ' 1.1.1' 1' l"( 'aC"i'.iv(' »OW ('[ hablill<"l' til /uI. a sirll~I(' c()n :v( ~ rt.( ~ r cin:lI;it r" II I1 WII II'IHII 1,11<" elido I.h a l. rOI" JIlillillllllll IUI" !II H.lriullI!1I voll.n'I':<', i.l' . "11, /1 N () IUe! " I' "I ' ,,, ,, ,, h,, 1. 1t ;i. ·, ' Id "lI lI "I ",,'td ,,' wi U, /',1" ,, 1 I'0 W"I ' l'ilC"i,or. II. iii ,,"I y

Line-commutated converters are ideal actuators for electrical drives; a main feature , apart from their simplicity, proven reliability and practically unlim­ ited output power is their excellent dynamic performance. AIso, electronic controllers can be directly connected to the firing circuit which consists of integrated circuitry and logic devi ces operating at signallevel. When a controlloop containing a converter is to be designed, the question arises, how the dynamic behaviour of the converter should be described by a mathematical model. The difficulties stern from the fact that the firing of a thyristor is a discrete process - the firing angle is not a continuous function of time - and that the steady-state and dynamic behaviour of the converter is highly nonlinear. Pulse- and phase-modulated systems of this or similar kinds have been frequently dealt with in the literature but no coherent theory exists which is simple enough for general application [G31, L3, P46, 824]. A detailed analysis of dynamic processes in line-commutated converters is rather involved; the transients are described by nonlinear difference equa­ tions which defy an analytical solution. Linearisation of the equations is of course possible but the validity is then restricted to the vicinity of the chosen operating point. However, as a first order approximation it is found that the converter dynamics can be entirely neglected as long as the control plant, fed by the converter, has a low-pass characteristic, the controller contains an illt.egrating term and the controlloop is to be well damped [F12, L35]; these couditiolls are usually met on drive applications. Near the stability limit this Silllplificatioll is of course no longer valid; there it is necessary to revert to Lll( ; ac(".lIril. t. ( ~ IlIodd ( ~ qllations . 1\11,,1.1[("1" :: illlpk ll\( )( ld tha t. lms serv( ~d wdl ill practice is based on the oh:;(' l" v; Ii,j"lI 1.11:11. ; 1 varia.hl!'. w;I,it.illl ~ 1)('I"i(1(1 "!:q)s('s III1t.il a s lIl a Il cha.ng(~ of t,he

1:J1

8. Static Converter as a Power Actuator for DC Drives

8.5 Control Loop Containing an Electronic Power Converter

illput signal Yl to the firing circuits (Fig. 8.11) shows an effect by shifting I.lte first firing pulse. As the waiting period depends on the instant at which I.he input signal is changed, varying between zero and 20/6 ms = 3.3 ms for i i si x-pulse converter operating on the 50 Hz supply, a mean equivalent delay I.ime Tr = 1.67 ms is postulated [52]. This heuristic mo dei produces useful n ~ slllt s , as long as well damped transients of the closed loop are the objective IIIl t étgain, as the stability limit is approached, considerable deviations from 1.1)(' predictions must be expected. The most effective method of neutralising the nonlinear and discontinuous 1l('!J
e Uoo

io Ref lon

---

II)

nuf

~

,

io

e!

--",

~=--

io

Fig. 8.31. Simplified block diagram of current control loop

the current sensor is installed on the AC or DC si de of the converter. Sensing the line current is simpler because current transformers with an output recti­ fier offer inexpensive isolation; a DC current sensor in the load circuit, which may be faster, more accurate and responding to the sign of iD, is likely to be more costly. The use oflow levei control signals, typically 10 V, which are electrically isolated from line potential is a necessary condition for safety rea­ sons. When designing the controller, the delay of the converter and the filter time constant, which are both small in comparison with the time constant T D , are lumped together to Eorm an equivalent lag Te ~ 4 ms. A proven design method for the current control loop was discussed in Sect. 7.2. lnstead of the PlD-controller that would be desirable with a slowly responding rotating generator, a Pl-controller is quite adequate here. ln fact, a PlD-controller, whose step response begins with an impulse, may cause inconsistent results in conjunction with the discrete firing circuit so that the additional lead term is usually omitted. Hence the open loop transfer function oE the current control scheme is

~Gc,i

T i s+l 1 GD TiS T e s+lTD s+l'

(8.44)

where Gc,i is the gain and Ti the integrating time constant oE the current controller. A typical time constant of the load circuit is TD = 30 ms. Because of Te « T D the transfer function may bc simplified in the region of the cross-over frequency, which determines the band width of the closed loop, FoL(s) ~ G c,iG D rn

1 rn

T i s+l

-~ Te S + 1 .

(8.45)

L I_ _ _ _ _ _ _ _

Fil{. 1-1.:10. COllverter with current control

'1'11 (' circni(. in Fi g. 8.30 is now transformed into the linearised block dia­ iu Fil';. R.:n , wh( ~ r< : the converter with its control circuit is approxirnatcd I,.v 11.11 (:qUiVldl 'lll. 111('11.11 dd<J,y, which for a fi-pul se umvcrter and 50 Ih-sllpply II 'l ~" r.~ 1.'1 111.' 1 I,'m ::IIl()(ll.hiug t.iI,' I·UlT('1I1. rl 'I'llbac/( sigual) TI:' 2 IIlIl IIlay III ' 111 11 11 11111',[ wil.h Il In x 1'111 11" 1·IlIIV,'rl."I' . 'l'lli:! 1111 111<' .1"1)"11.1 11 all'l ll "II whl'l,IIl 'I'

1 ~ 1 1\l1I

}Q.. lon

FoL(s)

Lo

135

This is in line with the "symmetrical optimum" mentioned in Sect. 7.2. The best choice of parameters is [KI8,37]

Ti wltl ~n '

= a2T~ ) (f.

GOL

= Gc,iGD = ~a TD Te

I I "2 J) f'ollowti

Jlai r (lI' l'I I\" lIV: t.lIII':';

II

frOI/l

I 01' f)

'

(8.46)

t.he Iksin'd dalllpillg ratio D of the complex 1/ ) 2 ir; fr<'qlll'lIl.1y ('hos(:11.

I :I(i

8. Static Converter as a Power Actuator for DC Drives

The design of the current controller leads to the closed loop transients sllOwn in Fig. 8.32. The curves were obtained by accurately simulating the opcration of the three-phase bridge circuit, including the leakage inductances (,1' Lhe transformer and the thyristor branches. The effects of the commuta­ Liotl are seen on both AC and DC voltages uBdT), UD(T). The transients illdllde step changes of the current reference as well as large synthetic load .I isl.lll'bances caused by inverting the back voltage e. Ikspite the severe simplifications made in designing the current controller, t.11I' I,l'
WVV\f1A

f\f\f\A 9 til \[Jll

I+-+l 10ms

I ~~

137

8.5 ControI Loop Containing an EIectronic Power Converter

The transients in Fig. 8.32 are better damped than would be expected on the basis of the assumed parameters (a = 2, D = 1/2). The most likely reason is the fact that the assumed mean delay represents a somewhat pessimistic approximation of the dynamic response of the converter; another cause is the sim plification 1 ----~

TD

S

+1 ~

1 --­ TD S

'

when applying the rules of the symmetrical optimum. The operating conditions assumed for Fig. 8.32 were so selected that con­ tinuous current prevails except for one brief interval, where the load current becomes discontinuous. As mentioned before, the current is in general not al­ ways continuous; for example, if the smoothing reactor is omitted, the region of discontinuous current may cover a large portion of the operating range. If a current controlloop, the controller of which has been designed on the basis of continuous current, is operated in the region of discontinuous current, the transients become very sluggish and the quality of control deteriorates. The reason for this is that the gain of the converter is greatly reduced when the current becomes discontinuous; in addition, the load circuit is no longer characterised by the relatively large time lag TD, as the current begins at zero in every firing interval, i.e. every 3.3 ms with a 6-pulse converter. Therefore the PI-controller, Eq. (8.44), is poorly tuned to the controlled planto If consistency of the dynamic response is important, which is the case for instance with servo drives on machine tools, the current controller should be readjusted during discontinuous current operation, i.e. use of an adaptive current controller becomes desirable [B90, 835, 848]. The fact that the volt age gain of the converter is reduced with discontin­ uous current is qualitatively explained with the help of Fig. 8,33 for a 2-pulse converter .

.~ , ________JI~t1~__~~____~____~~__~~~~~~\,::oé~.~ Discontinuous current I,'i,<, H.:.12. Computed transients of current controI Ioop

Wlicll studying the details of Fig. 8.32 more closely, differences become at time tI, t2, following the inversion of the back voltage e. The ,( ':J,;;O Il is t.hat at 1;1 the back voltage was reversed immediately afier a thyristor ",1..1 hCCll fired, while at t 2 the reversal took place slightly before a firing i'1I1 ~; (' was dll('. Rellte a waiting interval was unavoidable at tI, whereas at I:~ 1./11' ('(lI1v('r[,(~r could rcspoJld immcdiately. The differellce in overshoot of LI\(' \"IITCIII. is i\. COIIK<'ljuc nce o[ Lhe discOlLtil1l1 o u~ nud lloulillear n a.tllre of tlw (·/ 'II V.TI.pr. Tbt' l'onlr,rll l lNolllly JHI,r tially Sllc.c('s,qfll.l iII ('.OlllpcllsaLÍ.JII'; L!tis dl(~d" 1'·( ' n~II: ;( ' 1,1,11' d l ll!.u rlI HIJ( ·(' i i'; 1I.(' 1.i1l,~ ,lil'('d.I:,Y 1111. 1./11' plalll..

wt =-r

ai'l'ar(~llL

Pig. 8 .33. VoItage gain of converter for continuous and discontinuous current

If wil.h cO ll lillUOIIS t \lrrcnt (ú :::: ú:J) the Jiring angle is reduced by a small 1.1", p() :l il. iv( ~ V()ltlLg<~·· l.iTll(~ ;tn'lt, n ,.. "r < 7f, is illC1'ci\sed, whik tlw

11.11 Itlll IIt. /\(~,

1«'I',nl.i",' v"IL IIf'," U III<'

1\,1'('11" /I

•

T'

/I

I

11' ,

i: i I'('dll('('(\' Wit,h disc()J1I . iIJII()II :~

1:\H

R. Static Converter as a Power Actuator for DC Drives

9. Control of Converter-supplied DC Drives

iI! II

fi

1\ fi

fi

fifi II

f\ ~

10ms

vv

Fil!:. 8.34. Step response of current controlloop with adaptive current controller

CIlITCnt (o: = o:d the situation is differentj the interval, where UD(T) > O, is st.ill illcreasing when o: is reduced by Llo:, but so is the time, where UD(T) < O. IÍPJl c:e, the voltage gain dUD / do: is much smaller than in the case of continuous ClllTent. The block diagram of the current control loop for discontinuous current colTesponds to that shown in Fig. 8.31 but the normalised voltage gain of the (·()Jlvcrt.er is now considerably lowerj also, the lag element with time constant ' /'i> is llOW replaced by a lag having a time constant approximately equal to \.II<' liliul-\ interval T D ~ 3.3 ms, which is about l/lO of the former value. ( :I( 'arl y t.his calls for a different design of the controllerj for example, an illl.( 'I',I:\1 colltroller with the transfer function , I,' ( . . ) .', ' .

1 7."i S

with

Ti =

G'o (Te + T1)

,

(8.47)

,ILII :,('l.l'Ii s(,,1 hy a small time constant TI, i.e. high gain, could now be ex­ 1>1'1 1."" 1.,) pr()duce fast and well damped closed loop transients. III md
Static converters are ideal electronic actuators for DC drives because of their practically unlimited output power and excellent controllability. The speed of response is usually adequate to handle the electromechanical transients occurring in drives. Line-commutated, phase controlled converters or, as they are also called, converters with natural commutation, are the most frequent choice for industrial applications, where a three-phase supply is availablej this is due to the simplicity of the circuits requiring a minimum number of active and passive components. For vehicle drives, where no AC catenary or independent AC supply is available or where the reactive current and the harmonics caused by a line­ commutated converter would be unacceptable, it may be necessary to employ forced commutated converters having a more complex circuitry and involving higher lossesj a special situation exists also with small DC servo drives, where the response of a line commutated converter may be insuflicient to cope with the stringent dynamic demands and where a chopper converter supplied by a DC link and operated with a higher switching frequency is necessary.

9.1 DC Drive with Line-commutated Converter When connecting a line-commutated converter, for example the three-phase bridge circuit in Fig. 8.18, to the armature of a DC machine (Fig. 7.2), it should be kept in mind that the converter can only operate in two quadrants of the uD/iD-plane as seen in Fig. 8.23. Because of UD = Ua , iD = ia it follows from Eqs. (5.6, 5.8) that, assuming constant sign of the flux
9.1 DC Drive with Line-commutated Converter

9. Control of Converter-supplied DC Drives

1'10

Braking of the drive in the 2nd quadrant by regeneration is not imme­ di a.Ldy feasible; of course, there is always the possibility of non-regenerative 1)J;tking with the help of external resistors. Two-quadrant drives of this type are suitable for unidirectional loads, Lhe t.orque of which contains a large component of friction, such as paper- or I'rillting machines, calenders, also for pumps or fans; in principIe, it would also IIC applicable to hoists without self-Iocking gears, but rever se torque is usually (I('('ded there too because of counter weights or for helping to accelerate in LlJ( ~ lowering direction. With the driving torque being proportional to the product of armature ('lIrrent ia and flux Pe there are two options for achieving reverse torque, i.e. f()lIr-qu adrant operation: reversing the armature current or the main fluxo 11()(.b lIlethods are in use, though with markedly different preference. 1

2

3

La

iaRef (11/ 1,)1

ia

(I)

~ ia

before the switching command is actually issued; the firing pulses must be inhibited until the commutation is completed and the outgoing thyristors are blocked. If the current transducer senses the load current at the converter side of the reversing switch (as is shown in Fig. 9.3) or with current transformers at the AC side, the current controlloop is unaffected by the switching state. However, the sign in the speed controlloop is reversed due to the fact that a given output current of the converter now produces reverse torque; to offset this effect, one more inversion is needed at the input of the speed controller, as is also indicated in Fig. 9.3. Another possibility for reversing the torque of a DC motor is to reverse the flux Pe while leaving the direction of the armature current unchanged. This can be done by reversing the field through contactors or a bidirectional converter (of considerably reduced power than for the armature). However, since considerable energy is stored in the magnetic field, flux reversal requires llluch more time than inverting the armature; even when applying 3 times Hominal field voltage, the reversing transient may take up to a second on large drives, during which time the armature current must be kept at zero; hence, a lI oticeable zero-torque interval occurs, which is detrimental for the response of I.he controi. A motor subjected to this duty must be fully laminated; also, the voltage induced by transformer action in the armature must be considered. For these reasons field reversing is rarely applied today. There is now a general preference for fully electronic reversal of the ar­ Ill ature current with the help of bidirectional or dual converters. The cost of I.ltcse schemes has been reduced as a result of advanced semiconductors and IlIanufacturing techniques. The most common circuit of this type is shown in I"i g. 9.4; it consists of two six-pulse bridge-converters C 1 and C2 , connected iii opposition. Clearly, to exclude short circuits between line terminals, only

II·IH . 11 .1. (:;lHc ad e control scheme of DC motor with two-quadrant converter úJ

'I'IJ(~

ó\.I'I11aturc can be reversed with the help of a reversing contactor or, as iII I''ig. 9.3, four thyristors, of which one diagonal pair is conducting. .'·;ill('(· I.lIyristors cannot be blocked electronically, armature reversal can take I'lan: ()uIy at zero armature current; hence, a switching command is only .. lr(·(·Live WIWIl the conditions

úJ

11.,'(

= O,

iD = O

~ i llllll(.atll·(Hlsly

(9.1)

fulfilled. The first signal indicatcs that the superimposed :lP('(',[ ("oul.rolleJ cnlls for a r<~v<: rsal of tOJ:quc while I.Iw scconcl confirms that LI)(' ( ' 111'''''111. COII (.r()1 I('r bas n :dllccd Lhe artlla(.IIJ'(' 1'\11' 1'('11(. (.0 'I.ero, p oss ibly after 1.11<' (·OIIV( ·I'(.I'l' hU H 111'( ' 11 opnal,iul,( (.(,lllpOJ'II.J'il y l1 [il l.l1 ill v<,J'I.('I'. SillC(~ I.lwn! lIlay 1)(' 1;<1 1111 ' 1111I ·PI·I.III II1.'y wil." !.II(' ~: ('('(llId ('(111<1 11, 1,, 11 I II I.I!, . rlU' l' 01' i II I... n II i 1.1.('11 I. 1'1111"111. , ii . 1/1 11111'(1,1. .1111. 1.,. \V .lil. Ii :.11<111 1111 111Vii i \''1 /1.1> 1.1", '·(lII V('II."1 1;1 .. ..rwd,

111'( '

1

=úJRef 1

Field weakening

----------\~

::l!oWII

'i. "

141

1 O

úJ

"­ -... úJ= úJRef2

Eftect of current limit

7

,./

Unidirectional I:urrent of (:unveI1er in ;> O

I" l l~' 1I,:l.

ia = ia max

Speed control

mn

2mn

úJ= úJRef3

()I' ''I 11I.i ll/ \ rltl I!',"

"r a

I.w" ··<[IIi1.dr;',1I1. dr.'l v"

m

142

9. Control of Converter-supplied DC Drives

9.1 DC Drive with Line-commutated Converter

Supp/y 123

a torque reversal, for instance ia Ref < O and the current controller responds by lowering the armature volt age UD = U a with the consequence that the armature current eventually reaches zero, ia = O. As soon as this condition holds, the firing pulses for the conducting converter CI are blocked by the command module, while those of the incoming converter C2 are activated. Hence the switching conditions are

úJRef úJ

-

~

;0 ;ORef

143

[ia Ref > O1\ ia = O] --+ CI On, [iaRef < O1\ ia = O] --+ C I Off,

I. I

;0 úJ

Fig. 9.3. Four quadrant operation of a DC drive by armature reversal

one of the converters can be allowed to conduct at any one time; hence only one thyristor of each pair produces conduction- and switching- losses so that the pair can be mounted on the sarne heat sink. However, having opposite polarity, they must be electrically insulated. At lower power ratings, com­ plete thyristor modules are available, having the necessary interconnections built-in. Supp/y oncommon heat s;nk

(ú

ia

úJ

Fig. 9.4. DC drive employing a three-phase dual converter without circulating current

Tlw cllrrent controller supplies
C2 0ff, C2 0n.

Additional features must be incorporated to enhance the safety of operation and to improve the control transients. For example, a waiting interval of 2 to 5 ms may be included before firing pulses are issued to the incoming converter, to be certain that the current in the outgoing converter is really zero. Activating the firing pulses of the incoming converter before alI opposing thyristors are safely blocked would cause a line-side short circuit current that cannot be suppressed by control but must be cleared by fuse links or by a breaker. On the other hand, the firing pulses for the outgoing converter must not be blocked as long as current is flowing because, should the converter operate as inverter, this could lead to a commutation failure, as explained in Fig. 8.17. Therefore, the decision to switch from one converter to the other is of considerable consequence; in particular, the state ia == O must be ascertained safely and with a high resolution. For instance, a converter having a nominal current of 1000 A may contain two arms where at the criticaI time a residual current of 100 mA is flowing; if now the opposite converter is turned on, a short circuit is still likely. Sometimes two current sensors are employed on large drives, one having a linear characteristic over the full current range, to be used for feedback control, while the other serves as a zero-current detector for the command module; its characteristic would have to be linear in the low current region only. AIso note that the currents through the snubber circuits may affect the current measurements, increasing the noise of the signal. Clearly, great emphasis must be given to the reliability of the sensors and control components if this circuit is to operate safely under practical conditions. lndeed, this has initially been a problem that has delayed the general use of this exceedingly simple converter circuito ln the meanwhile I,ite problems have been overcome and reversible converters with antiparallel I.hyristors are produced in large numbers in the form of very compact units covering a wide power range from a few kW to about 10 MW. For the higher ratings modular designs with air- or water-cooled heat sinks are available, SecL ll.4. Hcsides the control scheme shown in Fig. 9.4 there are a number of vari­ ;d.ioJls; for example, only one firing circuit might be used with the output Plll:;(':; heiJlI'; switdu:d from OIlC converter to tlte other. AIso, the drive is usu­ idly (·xl.(·lId(·d Lo illdlld(: lidd w('alwlliul': Il~; diHClI: ,:;"d ill S(:c!:. 7.3. Thc field

9. Control of Converter-supplied DC Drives

1~~

wíllding would then be supplied from a separate unidirectional converter of :;IIl;dler size. Thcre are other ways to further improve the performance of the control. '1'111: adaptive current controller has already been mentioned in Sect. 8.5; also, 1.11(' ('(lrrent controller may be given a suitable initial condition, to optimise 1.11<: cllrrent transient after the zero current interval, when the incoming con­ V ( ' rI .( ~ r is activated. The best initial condition depends on the existing current rd('n ~ IIc(' and the magnitude of the back volt age which is known from the lirillJ'; I,' ;\'11<1 whcre a completely continuous transition of armature current ,,11.1 I.l>r'llw is specified. This is achieved with another dual converter cir­ 1'1111., ::I",wlI ill Fig. 9.5 a, where both two-quadrant converters are conducting ::illllilI.:LII('()llsly. This calls for separate transformer secondaries as well as de­ mllplil'1': reactors because of the instantaneous volt age differences existing I)('Lw(:('1I Lhe two converters C 1 , C 2 • As seen from Fig. 9.5 a the armature CIII'I'(:IIL call assume either sign, '/." ,- 1. 1

--

(9.2)

l2

dq wlId iIlg on which of the currents prevails. The srna:llcr of the currents rep­

a cirenlatillg current which flows throur,h I.he cOllvcrtcrs but bypasses 1.111' lo;\.( I brallch,

n :;;('lIl.s

IIliIlUIJ~)

ii :.flolll.! [)(' kq>1. a:: I " ' ! I('/,I V"

I'H\V I "

(9.3)

1.,.

[' "IIII':!

1111,,111 a~; ()II

145

9.1 DC Drive with Line-commutated Converter

1'":i::iI,j',, 1." 1','<111 ('( ' !.II(' ;I:::;o('i:~t.(:d n.d.ivl: alld 1.1,,' ,>1,.111'1' 1i :11I,1, 1.1" , ,·I,' ·u,l n l.illl'. ('1111'1'1'111. ::hould h('

L(7RC

ii

ú>,

ú>1

Lo ia

\.:J

-

e

L(7RC

e

C I Rectifier

C I Rectifier

C2 1nverter

C2 1nverter

C I Inverter C2 Rectifier

C I Inverter C2 Rectifier

ia" mM

i2 -------'

a

b

Fig. 9.5. Reversible converter with circulating current supplying DC machine

continuous so that there are no unnecessary waiting intervals delaying possi­ ble control action, The operating states of the two converters are indicated in the four quad­ rants of the torque/speed- plane in Fig. 9.5 b; for mM > O, the armature­ and circulating currents flow through C 1 whereas the auxiliary converter C 2 carries only the circulating current; the inverse is true for mM < O. To keep Lhe circulating current at a sufficiently low leveI, such as 10% of nominal cur­ rent, the mean voltage of the auxiliary converter must c10sely track the mean voltage of the main converter which is determined by the armature of the Illotor. If this condition is maintained all the time, both converters are active ;wd ready to accept the motor current in a continuous transient, without any waiting interval. This two-variable control is effected with the help of the two lirillg circuits. The auxiliary converter must be controlled to satisfy the condition UDl

+ UD2

= Rc(i1

+ i 2)

~ O,

(9.4)

I . (~ . the sum of the mean volt ages should be approximately zero; however, Lili,; is not true for the instantaneous voltages UD1(T), UD2(T), because one III' Lhe converters operates as rectifier, the other as inverter. The two volt ages 1\.1111 th( ~ ir sum are plotted in Fig. 9.6 for three different conditions, neglecting ,·I)III1I1I1t.af:ion and internal voltage drops. Clearly at 01 = 02 = ~ the sum 01' !.lu' iJlsLanl.all<XHlS voltagcs assumes very large values eventhough the sum "I' I.he 1IH':t1l va.lues is :r.( ~ ro. To prevent I.hcsc large alternating voltages from I'II.IIJi illl ( I' )( ', 'HH iv,' (' II ITI'II L", Lhe c.Íl'clllal.il\l!; I:IIIT('IIL rcad.ors Dc' seen in Fig. 9.5 1 111(' II 'qu i n'.! , Si ""I> 1.111'1'1' i i; nlw ll,YH "III' whirh (,;I.rrit's 1.111'. slllall ('ollt.iIIIlOIl S

1,1(;

!lo Control of Converter-supplied DC Drives

II/II

...

a1 !: a2 !:

= 45 (X2 = 135 ~

;1 II I \ \ J

90 0

0

(XI

II II I I

9.1 DC Drive with Line-commutated Converter

I' I \ \.J

,i2Ref

,,

h I' I' j' 1 \ 1 \ 1 \ 1\ \ , ,I , 1 , 1 1 'I 1 1 ,1 ,1

: 'I '/ '~ 1

a2 =45

0

a1

,

~

,,

0

= 135

0

rot!:

147

T

,,

,,

,,

..

~

iaRef

v

I, 1 \ 1

b

ii

~

UDt W Ref

$~

--1M Lo

W

rot = T

I"ig. 9.6. Voltages in reversible converter with circulating current

-LC

~I

~

U02

Lc

a Fig. 9.7. Control of reversible converter with controIled circulating current

cir<:lIlating current, these reactors need not be entirely linear; in fact, by allowillg the oue carrying the armature current to saturate, their physical size i:; redllccd . For smoothing the armature current ia, a separate filter reactor /'/1 IlIay h(~ induded in the armature circuito Io'or oht.ailling the desired current distribution for the respective main­ .11101 ;LIIXili;l.I·y-collverters, given a certain load current ia, one might think li, :; 1. 01 ali OpCll loop control scheme, where both firing angles are controlled 1"11I1.I 'y ;I,cmrdillg to 0:2 ~ 7r-O:l, which approximately fulfills condition (9.4). IIl1wI 'vC"I" , UI!: result would be far from satisfactory since the characteristics rv( .'l ) (Ir I.h(' Lwo firing circuits are likely to be somewhat different; also the CIIITI·1I1. d( ~ p c l1dent voltage drops would have to be taken into account. Thus I.Iw circlllatillg current would either be too large causing unnecessary los ses iII 1,11<' I.mllsfor mer, converters and reactors, or it would become discontin­ IIOII S, pr<: venting fast response, should the need for a reversal of armature CIIIT('UI. arise. The c10sed loop control scheme shown in Fig. 9.7 a promises a. vcry silllplc a nd effective solution to this problem; it is proven in numerous I\.Jlplications over many years [K24]. 'I'h( ~ idea. behind this scheme is as follows, Fig. 9.7: Based on the current 1('1(~I(,IIc(~ 'ia Ref prescribed by the speed controller, separate current references ,i I 110'1, 'i~ Ilo!" a.re derived for each of the converters. This is done by an electronic I'lIlId,ioll [l;ml C.I'ator, the characteristics of which are shown in Fig. 9.7 b; they lIT :oi O dlOse ll I.haL t.lre' eOllditioIlS 'I, 1

u"r

·1. :;' 1I, \f

'I,,,

IIl i rl( i l 11 ,,1, 'I 'J 11 0' 1 )

JI " f ,

i"

II ,· !'

(' 0"111 .•

are fulfilled, with the circulating current reference assuring continuous current flow. Thus the task of controlling armature current and circulating current is assigned to two separate current controlloops as described in Sect . 8.5. The circuit shown in Fig. 9.7 a can also be extended to include field weakening beyond base speed. With this control method the drive can operate in all quadrants of the torquejspeed plane allowing smooth transitions between the quadrants. How­ ever, this is achieved with considerable additional cost and complexity (trans­ former, reactors). Some savings may be possible, for example with continuous rolling mills, when the converter for reverse torque is only used for occasional braking and can be designed for a lower current rating; mOre substantial economies are made by turning to the reversible converter without circulat­ ing current that Was discussed before. This circuit permits also the use of the full rectifier voltage (o: = O), whereas with the circuit in Fig. 9.5, the rectifier voltage must be limited to the maximum value that can be supported by the inverter operating at maximum firing angle O:max. With today's state of con­ trol the loss in dynamic performance caused by the discontinuity at current zero is very slight, rendering the circuit in Fig. 9.4 the best solution for most applications.

1·lfI.

H.:l

!).

De

Contrai ofConverter-supplied DC Drives

D rives with Force-commutated Converters

("Ol! verter circuits dealt with so far have in common that they are fed Imlll siJlgle or three-phase alternating voltages; this is a prerequisite for phase c,,"l.rol cmploying natural commutation. As soon as a valve, having forward I)ias voltage, is fired, one of the supply voltages assumes the task of com­ 1IIIII.ating the load current from the previously conducting to the newly fired

'1'11('

vd.l v,:.

III a Ilumber of potential applications AC supply volt ages are not avail­ :tI,I", so t.hat this simple method of commutation is not feasible. Because a ,', IIld IIcl.illg thyristor cannot be turned off by electronic control, special pro­ visi" II s i Il the power circui tare then required to ensure that the thyristor to 1)(' li red has, at least temporarily, forward bias voltage and that the outgoing Lllyrisl.or becomes nonconducting and is briefly exposed to reverse voltage as :1. COlldition for blocking. A multitude of circuits exists which perform this "I'orcer! commutation". ln rccent years thyristors have been developed that can also be switched 011' hy suitable control signals while conducting (Gate-Turn-Off thyristors, ( :TO); tltis is a major improvement, even though the current gain for turn­ 011' is very low, requiring a short high pulse of inverse control current. These ,kvic,:s ltave in the last 10 years been developed to the point, where they (":1.11 n:place normal thyristors with several kV and kA rating in high power d.""lical.iollS, such as traction drives. 'I'1i,' 1I10S!. common switching device at lower power permitting electronic 1.11111 011' is (.Iw bipolar transistor. Since it can only conduct, when it is ac­ 1.1 \1.'1 1., 'd I,y ;1. I)ase current, it may be switched on and off at will, as long as i.l1(' :: W iI.eI I i I'1~ lasses are not too high and the transistor is protected against IIVI'I v"ll.ag':s caused, for example, by an inductive load. Bipolar transistors iII I':I.::CII( Icd Imm (Darlington circuit) are now available for volt ages exceeding III"" V ;llId Clll'rents of several hundred A, so that converters for medium size drlv,':-': II!, Lo about 500 kW can been built, but the design ofthe base control (·II'('lIil.s n:ll1a.ins cumbersome. i\ Ilol.her arca where promising developments are under way are field ef­ 1",'1. I.ransist.ors (MOSFET) for low power applications. These transistors are "llaracl.('l'isrd by very short switching time and, consequently, high switch· il'l', [J('.qllPIlCY beyond the audible range; being inherently volt age controlled, Ul('y ['('<["in: practically no steady state gate current, similar to vacuum tubes. Ilowcv(:r, caused by the relatively large gate capacil.y tlwrc is a substantial dlil.l'f~i IIJ'; C1ll'rellt which must be supplied by the conl.rol device for switching; :rI:;", ti", Oll~ st.a.te resislance is larp;er than wil.h hipolar l.ransistors. Mm(: n:c('lll.ly Insllbt"r! Gate llipolar "'n UIHi: iI.on-1 (1(; 11'1') lmve bepll de­ V,'loi'"d ii:; Ilil';11 (l0W('!' dcvic('s, COlllhillill/', 1.\[(' li i'l',h ('1111'('111. carryill/~ capaciLy "II,I!'"I;I.r I.ril.lI:: i::!.or:; wiLh Lhe :;illlpl,' ("""!.r,,l "r li,'l" dl','cl. I.rall:;i.~l.o n, iI.S wdl n: : Idr.h ::w ll.lllill /l, ~" )(>",I. 1<: 11'1":: l U '" ,,1'I'lll '"I.I ,\' 1.111' III,!:; L ,,1'()lIli:~i!l/ ~ :.; wi t.c1li 11/'; d" v II" '1< r, ,, Idl,.h,,,' Ili 'Wi>)' I1,p"lil" l.l.í" ll l1 . l'I'Ji ,'ld ll /( ilrl.1) 1.111' h V, "i\ I' I ./ ~ i,," . i\

9.2 DC Drives with Force-commutated Converters

149

further possibility, but still at an early stage of development, are MOSFET­ controlled thyristors (MCT) but their potential cannot yet be assessed. Clearly, the work on new and improved high power electronic switching devices is continuing at a rapid pace and a state of consolidation is not in sight. This strongly affects the development of the circuitry and control meth­ ods for converter equipment. ln this section it is shown with the example of a sim pie force-commutated chopper circuits for DC drives, how the new devices may result in better dynamic performance, simplifications and, possibly, cost reductions.

L

L

~o~--~~"~--,

ia

ia

Uo

Ua

a

Load

Load

Uo

b

Fig. 9.8. Principie of single phase chopper circuits

(a) Step-down chopper (b) Step-up chopper

ln Fig. 9.8 a,b two basic circuits for DC/DC choppers are seen for gener­ ating from a constant direct bus volt age with low losses a controllable direct volt age for supplying a DC load. The electronic switches in conjunction with an inductive energy storage are operated with a controllable On/Off duty cycle to produce a desired mean voltage U a across the output terminais; the circuits could be used as single quadrant supply for the armature of a DC mo­ tor. The electronic switch is carrying only forward current, reverse blocking is not necessary with the diodes included. As an example that is mainly of historical interest, a typical converter cir­ cuit of the type of Fig. 9.8 a, employing forced commutation with an ordinary thyristor is depicted in Fig. 9.9. Its purpose is to convert the constant direct volt age Uo to a lower adjustable volt age U a < Uo for controlling the speed of a DC series wound motor. This problem arises with vehicles, supplied through a DC rail or catenary 01' from a storage battery. As the thyristor TI which corresponds to the switch in Fig. 9.8 a cannot be turned off by electronic con­ trol, a resonant circuit with the auxiliary thyristor T 2 is provided with the purpose of creating a zero intersection of the current iTl, thus facilitating the blocking of thyristor TI' When using a GTO which can also be switched off electronically, the same effect could be achieved more directly. T lw 1)(:/1)(: (,()Ilv(~rt.er operates a.s a chopper, which, by means of an i,ll'drillllC ::wil.rll, «III1\(~cl.S th(~ illdllctiv(~ load with the pa.rallel shunting diode /II 1", r! "dicillI.1' LI) 1.11(' cOII::/.:!.IIL ";lIpply voll.:"';i>. III' varyiug Lh(~ dllty cyrk, til('

l i .!)

'L

9.2 DC Drives wíth Force-commutated Converters

Control of Converter-supplied DC Drives

kK-----­ _-­

Tum-off Thyristor (GTO)

a

......,.;tVI

~

1- - - - - -T- - -. - - ,

Ro

. fJ

'Tf

Re

or

Uo

I I I I i Df L2 I ___ .J

--1--__

-"'" 1__ O2

ua

l;e Ue ~--zí:

J

Fig. 9 .9. DC/DC chopper supplying a series-wound motor

J

~t2

\----,

"

/'

ID f

/Cctificr

.

1,\ iTl

-,

I

I J,lttery

- ____ ....

........

I I I

Lo

ia

--- -

Thyristor with force tum-oft I

151

t

~t3

I

t tI F;ring 01 T,

\

\Y

V

j t,

r--ZL= /~;

\I

voltage U a is changed. The switching frequency is chosen sufficiently lIigh (> 100 Hz) so that a relatively smooth continuous current ia flows in the load circuit. The converter shown in Fig. 9.9 is of the single quadrant variety, wilic\) is adequate for a series motor but with additional components it could 1)(' ('X I.(~lldcd for two- or four-quadrant operation as well. The circuit should ollly \)(' collsidered as an example; there are many different versions having ::illlilar proper Lies. The operation of the DC/DC converter will be explained iii :: I.,'adY Klatc where ali variables are periodic functions with period T . The (' III v(':: iII I<' i!(. 9.10 have been obtained by a detailed simulation of the circuit 1 111 :1. d i/';iLd (,o ll1puter. '1'1 1<' IIlotl e of operation is briefly as follows : The main thyristor TI is I)('Ji odi c; dl y Jircd at time tI + vT, v = 0, 1, . . . connecting the supply volt age (I" 1.0 1.11, · i11dll ctive load circuit, ua(t) = Uo, at the sarne time blocking the :illlllllill! '; di( )(le DI which had carried the continuous load current i a. This (IIII.' ;,·S i.llC pos itively charged capacitor, ue(td > 0, to be discharged through 1.11<' J'( ~S()Ilitl1t. commutation circuit consisting of Le, L2' R2' D 2; after the first Ilall' wav(~ t.his transient is interrupted by D 2 as the current is about to change il.s ;;ig11 , II'
( .~ "'/1" . di

1 "1 •

( ~ Io ~i )

iT2

----""\

t t2 - T

----"'\ \

2

1 Hring 01 T

íV02 . . , !., ,~,I01, ----j v',

,_,___ -; r·,

Fig. 9,10. Símulation results for DC/DC converter circuit

The bias volt age across thyristor TI eventually becomes positive at time t3, when U e passes through zero; in the interval t3 - t2 the recovery of TI must have been completed. The rechargingofthe capacitor continues until Ue = Uo, when the freewheeling diode DI takes over the load current and T 2 reverts 1.0 blocking condition. The electronic switch then remains in this state, with Iloth thyristors blocked and Ue > O, until the next switching cycle begins. ln Fig. 9.10 the commutation interval has been enlarged for better visibility. Clearly it is criticaI for the functioning of the circuit that the alternating volt age of the commutating capacitor is of sufficient magnitude to ensure an adcquate hold-off interval t 3 - t2 when the main thyristor is reversely biased. I[ this time becomes too short because the charge on the capacitor was in­ ~i il{fic ient or there is excessive load current, TI will refire and a short circuit colldition develops which can only be cleared by a fuse link or a breaker. ln I,"is r< :sp ect a commutation failure of this circuit is more serious than with a l i ll<'~ <:O lllJ\lill.al(:d converter where the situ a tion could be resolved by advanc­ illf', LIli' 111',1, firillf'; illSL;Ult.. 'T'hi H d al1{';(~r 01' l()s ill ~ cOl1t rol over the converter

I r,2

9. ControI of Converter-supplied DC Drives

9.2 DC Drives with Force-commutated Converters

('xists with alI thyristor circuits employing forced commutationj it can only Iw avoided through careful design of the circuit and by having a fast cur­ l('[ll. control which prevents excessive load current before it approaches the ,'01 II II Illtation limito Âllother detail is seen in Fig, 9.10 at time t4 where the capacitor, having 1)(" '11 clIarged beyond Uo due to the effect of the source inductance Lo, returns ;;"",,' 01" its charge to the battery through diode D 2 j since it is desirable to h:IV(' I.Iw capacitor well charged for added commutation capability, this loss ,,j rh:I.II~(' SllOUld be prevented by using instead of D 2 another thyristor that i:: I i1'1'<1 I.o!-\cther with TI. '1'11(: ,~witching sequence is repeated at a frequency f between 50 Hz and IWlllaps 1 kHz, depending on the type and power rating of the thyristors. '1'1.(' control effect is achieved by pulse-width modulationj for example, if Lhyrist.or TI is fired at the frequency f with a fixed phase, the On-Off ratio is all.ned by advancing or retarding the firing instant t2 of thyristor T 2 . When 1I"I;I( ~clillg the commutating transients the volt age U a can be approximated I',l' a periodical square wave having constant amplitude Uo and variable pulse­ wi
V'a

t2 - tI

< -- = - T - < 0.9 Uo

ing branchesj nevertheless, digital simulation is an indispensible tool at the design stage. Naturally, after the converter design is completed, much simpler models can be used for designing the control system. An extensive specialised literature exists on De/De converters describing numerous details, particu­ larly with regard to commutation circuits [3,48,51).

lL iSf

W

ia

Load curve WRef

(9.6)

1.11 i," avoids the large power losses that would occur with a series resistor. UI" rOIlJ"se , ali components in the converter circuit involve some losses which hil v,' 1.0 h(~ \; Xalllilled very carefully at the design stagej still, the over-all dliri"I\(,l' uf ;t I.ypical De/De converter may be better than 95%. The limits (I ~ II) )" alie! (t2 - tI) -t T cannot be reached by continuous pulse-width (", 1111.1'" hl'c;I.IISI' some time is needed for the commutating transients. 1"'(1111 I.Ili,~ silllplified description it is apparent that the analysis and, con­ 1:'·'1'"'lIl.Iy, I.he d(;sign of a converter with forced commutation is much more , (1111 pi", I.h;\'1I t.hat of a line-commutated converter. The volt ages and currents, 1,,1 n:IIIIJllc I.hose directly affecting the thyristors, can only be determined by 1.('!lII)IIS Ilalle! calculations in the successive intervals, by measurement on a l:dlOlill.ory IIl1it or by simulating the complete circuit on a digital computer. VI'I,\' dJici(:II1. programs have been developed for this purpose which require "1I1 ,l' :1. t.opologi cal description of the circuitj from this information the differ­ I'lIl.ial ('qllatiollS are automatically generated. With given initial conditions, t.Il,' 1I11111nical illtegration proceeds until either a control input occurs (e.g. lil'illl'. 01" a Ll,yristor) or one ofthe components reaches a discontinuity, such as iI. di,,,[P hlockiug, at wlIich time new equations are cldillcd (or new parameters "III.I'I'C
L

GTO

j

Uo

Mmax

I

I

a

01

Ua

De chopper

I

I

j

153

CD

b mM

Fig. 9.11. (a) Simplified drive control scheme of drive with DC/DC converter, (b) steady-state characteristics

The De/De converter, like the line-commutated converter, is a power amplifier with fast response having limited overload capacityj hence it calls for dfective protection which is best provided by closed loop control. With vehicle e!rives which are often coupled to a large load inertia this is of particular illlportance because acceleration should often take place at a chosen current limito Therefore cascade control having an inner current loop is well suited ;l1so to De/De converters. As was shown in Sect. 8.5, very simple dynamic modeIs are quite adequate ror the design of the control system, provided the innermost loop contains ­ I,csicles the switching converter - a simple control plant with low-pass char­ :t,c.(",(,ristics. When neglecting the commutation transients and assuming con­ I.ÚIIIOIlS load current ia, the converter in Fig. 9.9 looks from the load side like ; \.11 illlpressed voltage source, where u a alternates between Uo and zero with variablc duty cycle. Hence the circuit can be represented by a mechanical :, wil.ch as seen in Fig. 9.11 a that may be operated by an On-Off-controller It:t.villg a lIarrow hysteresis loop of width 2L1. With the real converter, the ,', 1I11.rolkr woulcl issue firing pulses to the main- and auxiliary thyristors. TIl(' hlock diaf';ram of a series wouncl motor, Fig. 6.3, is quite nonlinear, 11111. wil.h ali ()II ·(){J"-coltt.wll(,r for (,11\, illlwr CIIJT(' lIt control loop no prob­ 1"111 1: ''',' "111'011111."(1"1. '1'11(' :;yHI.\'1I1 1',',11111.:1 iII a, "'"i1,;i ·Iillear illIWJ" dos('.d loop

11, ,1

~) .

( :() IILro] of Converter-supplied DC Drives

9.2 DC Drives with Force-commutated Converters

"0 I lu

' ', I~~~~~~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~ ., ,: , ~ l,r)

t

1 m,

'''~

: m)

.,

"l~~ I" i l-t, 11,12,

Silllul nted starting transient of drive with De/DC converter

wil.lt :r.ppro xilllal.c!y unity gain. ln the superimposed speed controlloop the 1I11'{ 'II IUl ic:d illLegra tion is added (neglecting friction), however with variable I '.I U II, 1 H' (, ; LlI S (~ Lhe torque of a series motor is determined by Peia :::::! i ~. The " '111 111, W()lIld Iw different speed transients depending on the operating point 01 !.III' IlIOI.UI'. '1'0 avoid this effect, it may be advantageous to add a nonlinear 1'r 1l H'l.i oll g( ~ l wrat or producing a torque signal, '/II' M '/}/.)

IUleI

f

(~) Zal

,

ia

~

O,

(9.7)

an innel' torque controlloop instead of a current loop; no partic­ Irlal' n.C(,lIJ'étcy of the function generator is requil'ed. The result is seen in the Irlor!, di agl'étlI1 Fi g, 9.11 a, where all nonlinearities are now contained in the illllc r loo p. Tl w pulse-width modulated waveform is l~c n cratcd by the closed 1. ( IIIJlI<' COIl \. rol loop wit!tout tlw need for a li o lC \.,~nlld dock; the frequency ,'lt[l III' itdjll sl.(~d uy t he dlOicc of t,he hys t;(~reHi:4 wid l. h. J\, ~aill, wi t h a simple ()II/( )Jr.I' o H l. rlllkl' , \.he I' wiLchi ltg' fl'< ~qlJ(~ II CY III' UI<' ('oll vn\'( :r is not cO)H'i t a nt I lI ll. . ' xlt ih it.l1 ~~ II IIU/ i llllllll iII t.lu ' ("'n t, l'I \ III' 1.11.. ' " (ll d,I'ol n LUI.(I' rlRI, huI. t.his , ' III IIII I' H 11,0 )1lt1'l 1l / Ul I " I ~H l U! Lh l' I 1.1. i II Í'lu 11 11,1 rl " ''1III '' " I ',~t 111 Il ldl i ,· i.l' IIUy hil':h, Ilow " V( 'I, wl" '1I /•• ·.. 11111,'11 11 /Iw lt.d IÍ 1'I '; r l'l ''1 I1 ' ' I I(' V 1/1 11 1" ','111.-.1 1." fLV" irl 1111."1 r"I ' C' IIC ' C' Lo rOl'lll

155

with signalling circuits laid along the railway tracks, a constant frequency pulse-width modulator is preferable. If the vehicle contains several indepen­ dently fed drive motors, the firing of the converters may be synchronised out of phase to increase the ripple frequency of the current in the supply line. Figure 9.12 shows a simulated starting transient with speed-dependent load torque; the operating point in Fig.9.11 b moves from 1 via 2 to 3. The inertia of the drive was chosen very small to clarify the details of the switching operation; with an actual traction drive, the acceleration phase may last many seconds while the switching takes place at several hundred Hz. When selecting the parameters of the controllers, sim pIe design methods are again applicable, for example as discussed in [38J. As mentioned in Sect. 8.5, On/ Off control can be treated more accurately as a nonlinear discrete control system which however would involve considerable complexity; in view of the good results with the approximate methods this is normally not justified. As indicated in Chapo 6 the use of series connected motors is restricted to only a few applications; for high dynamic performance drives, such as reversible servo drives, De motors with separate excitation or permanent magnets are normally preferred; they require converters with four quadrant capability in the ia/ua-plane. The bridge type converter shown in Fig. 9.13 a is connected to the ar­ mature circuit of a DC motor; it may be supplied with constant volt age UD from a DC bus or a battery. The converter contains four electronic switches where two in each half-bridge are drawn in the form of a transfer switch (at the sarne time excluding accidental short circuits of the DC bus) ; the diodes which can be part of the electronic switches allow an inductive load current to continue during the short protective intervals, when alI contacts are open (similar to the red-light-overlap on a signal crossing). By assigning logic symbols SI, S2 to the otherwise ideally assumed switches, the voltage equation of the load circuit is Za La -

dt

+ R a Za' + e =

(9.8)

Ua ,

where, depending on the switching state Ua

holds,

1 = 2(S1 - S2)

UD

and

, ZD

1

= 2(S1 -

,

S2) Za

(9.9)

I r,j;

9. Control oE Converter-supplied DC Drives

9.2 DC Drives with Force-commutated Converters

iO

157

SUpp/y

2

c

La

3 io

Ra

4-Quadrant­ converter

$2 -1

U

o

ej ­

ii

U

a

C

UD __

I)

~/_

-I

I/ua

, S = -1' II..., ......L .......L._

T

S",.S

I~: ~o_

o r-:-J-_ ­ U - - - U-­ a

=r:;....::::... / I

T

S/i;

Ir

1...--.....1_

Symmetrical modulation

- U 12. _____ 2

~

__ _

Cyclical modulation

inductive load,

TIl(' pIIIs('-width-rnodulation (PWM) of the converter at the frequency I j'1' (';tII fullow different switching strategies, as illustrated in Fig. 9.13 I, w; I.ii 1.1I(' oIII. P III. vol tage U a during a switching period:

f

• \ l"ipol;11' IlIodlda.tioIl (1111'
,':y III 11I1'l.l'i(' ; t! Illudul ation

W iI.l, ,," i.~ .l lwdulation pattern the switches are operated in diagonal pairs, S, :-,"2, so tltat the short circuit interval is ornitted and the output vo ll.;1.1 ';<. ;dl.el'llules between the values UD and -UD. During the unavoidable (I" I I. iII I'·ig. 9.13 neglected) protective intervals the diodes are carrying the 1";1.'/ ( ' lIl'reIlL

Illodulation I.lri s lIlodulat.iOIl scheme the two transfer switches are operated se­ IJI[(·III.;all 'y, so I.hat. th e output is alternatively short circuited at the upper 01' lowcr sllpply bus. H(~ llc(~ the output voltage U a assumes a ternary wavc­ 1'01 ' 111, '11." '/I.u, O, " · ltf).Wh(~ n~as witlt symnwtri ca.l switchillg oldy Llte mean II I' Li", olll.plll. voll.a!,,(· ('.<1.11 I)( ~ cOllLrul!cd iu st.c;l
• (:y d ie
l'I i lllll ll ll .l l1l'.

J

~ir-J···~,

FiJ.C. 11.13 . Four-quadrant DC/DC converter with (,, ) (:il'('uiL, (b ) Different modulation patterns

1;,1'

Ta

- - -"--r

~1

-~

Ulllpolar modulation

Uo

Iiii 1'1 11011 i( ' H III' 1.11(' Olll.plll. volta",..,

'11." ,

Fig. 9.14. DC Servo drive with voltage source DC/DC converter

A four-quadrant converter with lGBT-switches is depicted in Fig. 9.14 as frequently used for DC servo drivesj protective circuitry is again omitted. For simplicit.yan OnjOff device is drawn for current controI but a linear current controller with constant frequency PWM would normally be preferred. A diode rectifier followed by a smoot.hing filter, whose capacitor C ab­ sorbs also the modulation-induced ripple components of the link current iD serves as the supply of the DC link with constant voltage UD. For instance, when rapidly braking the drive, power released from the kinet.ic energy flows back into the DC-link causing negative current iD and, because of the uni­ directionaIline-side rectifiers, couId result in an overcharge of the capacitor; this is prevented by dissipat.ing the energy in a resistive ballast circuit that can also be pulse-width-modulated, depending on the link voltage. ln view of the losses this is only practical with srnall drives or when it happens only occasionally; otherwise a reversible line-side supply (an active front-end con­ verter) is preferable as will be shown in Fig. 9.18 and further discussed in Sect. 13.2. Some of the steady state waveforms in a converter like the one in Fig. !J.14 are indicated in Fig. 9.15, showing the output voltage U a alternating hdween UD and -UD and the alternating current components of iD, which lllllSt be absorbed by the capacitor. The control can be arranged as beforej il.U illller cnrrent loop controlling the converter via a pulse-width modulator is illlportant. for safe operation. The current controller in Fig. 9.14 is again drawl1 ;IS
I!, fi

'l. Control

r.

of Converter-supplied DC Drives 2f,.

0.1')

'.0)

/l1' ~/iYlfPH21f3 I~Ref

""II

9.2 DC Drives with Force-commutated Converters

159

Phase shift (Ual. ial )

Ua

n

Jc1 c1 c1 c1 c1 [L o o o o O":

1r1 c1 r1 r1 r1" ;

i

U U U U U

U;

I~ il; . tI.15. Waveforms of DC/DC converter with symmetrical modulation .

al'pJicaliollS this justifies the assumption that the current controlloop acts as colll.rollable current source having instantaneous response. Current limit i" ac:bil ~ vI~d by limiting the current reference produced by the superimposed :iJ >('('d ('()lIl.roller. The next higher levei of control could be a position control !\lO p :IS showlI in Fig. 15.9, where the response may be further improved by 1" "<1 J'o)ward siguais from a reference generator. '1 'r;I.II ::i:; l.m e<mverters have the important advantage that they can be li wil.d", 5 kHz, thus enlarging the control bandwidth as '''"11 " 11 ,,<1 1.0 lillc-commutated converters. With field effect transistors or I( : I\'1' 'i:, I.hl ' rl(~quency can even be increased beyond the audible threshold 1(; !t 11 '1" so t,hat the drive is no longer emitting objectionable acoustic noise. ' ! 'II(' II>II/' quadrant converter shown in Fig. 9.14 is also suitable for produc­ ill/', 1\( : iII Lhe form of pulse-with-modulated voltages and currents, provided 11.11 J'n ~ qll('lI('.y is sufficiently below the switching frequency of the converter; i.l ti fi i.~ SCI~II iII Fig. 9.16 with the example of approximately sinusoidal out­ P"l. 1'111'1'1 ' 111. 'i " in a passive (e = O) inductive load circuito With the assumed (),, / O!!' colllroller the modulation is symmetrical and the switching frequency \ll lIil~S !ln!. Lhis could be remedied with a PWM mod1l1ator and fixed clock r/" ' qlll ~ lll'Y, Ali illspection of the curves reveals that the <:onverter passes cycli­ ",,!l y I, h011 i', h
rlnnnnnm~ij~~~~nn ~;1r1(]nnnon~~~~~r~~~u~4\r1~ ~ ~ i ~ ~ m~~[~~nUD' -UU~ij~~~~~ io

n

• I

Fig. 9.16. Waveforms of symmetrically modulated four-quadrant converter with AC output

• Cylindrical motors with a slim rotor hence low inertia, but otherwise con­ ventional design and • Disk motors with axial magnetic field and radial armature winding, which is directly printed on a fiber reinforced disk; the brushes are sliding on the printed conductors of the commutator. These motors are charakterised by a very small armature time constant and low inertia; typical data are given in Fig. 14.1. Because of the progress with controlled AC drives, servo motors are today mostly synchronous motors with permanent magnet excitation, Sect. 14.1. When discussing phase controlled converters in Chapo 8, the problem of line-side interactions by harmonics and reactive current was mentioned. This applies also to uncontrolled rectifiers, such as in Fig. 9.14, particularly when a simple capacitive filter without smoothing choke is employed which causes line currents in the form of short spikes at the volt age peaks. With the growing use of power electronic equipment these effects are now receiving increased attention in technical standards and regulations. With force controlled converters the reactive line currents can be greatly reduced while the harmonics are shifted to a higher frequency range, where it is easier to suppress them by filters; at the sarne time the problem of reverse power flow can be solved. This is shown with the schematíc diagram in Fig. 9.17 of a single phase converter which is of interest, apart from low power drives, on AC traction drives with intermediate DC link converter (an extension to three phases is di scussed ill Sect. 13.2) . The circuit is essentially an inverted four-quadrant I~OIlV(~ rt;t: r iII ),'ig , 0.13, where the inductive impedance LN is now represented II,Y i. 1!<: ka!t:\I';t: 01' t.1!I~ Iilu:-sirle tra nsformer. The simplified equations are

11111

(:olltrol of Converter-supplied DC Drives

!) ,

9.2 DC Drives with Force-cornrnutated Converters

ir­ 'Nb

y

controller. When shifting the current reference by to with respect to the line voltage UN, a reactive component of the line current results. Caused by the single phase supply, periodic fiuctuations of the link voltage UD at twice line frequency cannot be avoided but are greatly diminished by inserting a resonant círcuit tuned to double line frequency, as is indicated in Fig. 9.18; this is the usual practice on traction drives. The load-side DC link current iDL acts as a disturbance to the voltage control, its effect may be reduced by feed-forward to the current reference.

!lN

'NV/

',':!

UN(t) i

ON

k

lN

~;

i it)

1'15

iit) = bN(t)

2

-1

- -- -....

-.-.

5hort circuit

Transfer

interval

Interval

51 =52 =-1

5 1 =1,52 =-1 A

UN(t + to)! U N

Jo' i ~ . 9.1 7. Principie of a single phase line-fed four-quadrant converter for supplying ;) 1)( :

lillk

iN

'/'N

L N -

dt

+ RN

,

ZN

=

UN -

UB ,

(9.10)

iNRef

iOL

;1,lId

,Á'I/,f) dI.

, ' = ~DN

,

Uo

(9.11)

-ZDL,

F ig. 9.18. Control of a single phase line-fed four-quadrant converter

wlwJ'\ >'i/>/, is the load-side link current; furthermore 11./1

.: (,<" 1 -.52 )

uD

und

iDN

=

~(51 -

u/ V

82 )

161

iN

U

(9.12) 400

f---­

UN

I" del ::, II pprw::illla tely sinusoidalline current i N can only be achieved if the

1)(: lildí, 11:'t l1I lhe line voltage is much lower than UD, i.e. near the zero in­ I.I 'I'fil >('l.ioIlS oe 'U'N; hence it is necessary, as shown in Fig. 9.17 b, to create ', t'rll i Ill.crv;ds of the voltage UB for storing in the line-side inductance energy I.Il il.1. i:, slli>sequ(~ ntly transferred in the form of a current pulse to the link I'II,p;u 'iI.OI, "'hus it is necessary to employ a unipolar or cyclical modulation, w I w J'\ ~ I.hc COllverter bridge is temporarily short circuitcd at a DC bus. '1 '11(> switchillg frcquency of the converter is best chosen as an integer 1IIIdl.ipl(~ II of the line frequency to avoid in st ead y st.at(~ sllhharmonics of the l i lll ~ 1'11/'1'<:111.; with a 50 Il7. :mpply and a clode fn :IlIIl'Jlcy ror ius ta.nce of 2 kH z, n til) swiLchillf,'; (:yd(~f> wou ld fit illto one lillC voll.iI,l':(> Jlcriod, rcsultillg in a r.d rl.v :n llool.lt w;l.vd<mll ()f Lhe ('.111'1'<:111:, II pll:-ls i lll,' co//Crol ;'i dlCIIll! i;{ :, !t UWIl iII I" i,,;. n,I'H, whl'l'tj II. volt,ag., cOJl t rol 1""11 I~ II' '1/,11 iH rHlpI'I'illll)(IH,'d OH ali iIIU('l' " UITI' fll ''' ('III 1'\11' '/:N ,- '1'1-1\' I'IIlTI 'IJI. ,, 'I,'rl' II/ 'I' ; N 11 ,, 1 i!:1 d\,ri VI'd 1"1'0111 t,h" l i "I' vll lt HI'.~ ' ffll ~11, I,nilli l,,'; Il. Hil\ w ,,,i d a l Ii/\I' " 11111"111. 11'(" " ' 111 "' " !-III' IlI ftl\ l.! I.I\/'" ,II' wld, I, ln dnl"I'lll hll'd I, y LI ... v.. II';"F,"

o

11\

i OL

1\ A A A A 1\ A A

200

~lk\IViA\IViA~Vlk ~ i -200

-400

"';g. tI.lU. Si lllllhL"d loa
1\

lillc-fecl fOllr-lJ"adrant converter

Ifi ;~

!I .

Control of Converter-supplied DC Drives

1\ ~iiIlIlIlated load reversal is plotted in Fig. 9.19, initiated by inverting the

which results in a phase reversal of the line current in steady ::Iil.l c, i. c. feeding power back to the line [13]. The resonant circuit shown in leig. !l. I 8 is included in the simulation model, thus reducing the harmonics w ut.<:nt of the link voltage. I"IIITI'tll. i ,lL ,

10. Symmetrical Three-Phase AC Machines

The asynchronous or induction motor is the most widely used electrical drive motor; its invention at the end of the last century has given a strong impetus to the transition from DC to AC in the field of generation, transmission and distribution of electrical energy. Its main advantage is the elimination of alI sliding contacts, resulting in an exceedingly simpie and rugged construction. Induction machines are built in a variety of designs with ratings from a few watts to many megawatts. Unfortunately, the speed of line- fed induction motors cannot be continu­ ously varied without additional equipment or without incurring heavy power losses. Even though the problems of efficiently controlling the speed of induc­ tion motors have been investigated for decades, ali solutions realisable until some years ago were unsatisfactory with regard to complexity, efliciency, dy­ namic performance or cost. It is only due to the progress of semiconductor technology in the last 30 years t hat suitable static frequency converters can now be built at acceptable cost, making the induction machine a versatile adjustable speed drive for many applications. The substitution of the mechanical commutator, by a power electronic, solid state converter reduces the AC machine solely to the task of electrome­ chanical energy conversion, resulting in much higher power density per vol­ ume or mass. This is exemplified by two main line traction motors , the out­ lines of which are shown in Fig. 10.1, together with some characteristic data. The motor at left is a proven design, operated for many years in Intercity locomotives; it is a single phase series-wound commutator motor as men­ tioned in Chapo 6, voltage controlIed by transformer tap changer. At the right is a modem AC induction motor, fed and controlIed by a GTO volt­ age source converter. Clearly, the new motor is superior in every respect; it is also suitable for high dynamic performance torque control, thus alIowing ful! utilisation of the adhesive frictional forces between wheel and rail. As a result, a locomotive containing four of the new motors can produce a similar traction force as the former much heavier locomotive with six single phase motors, the torque of which is pulsating at double the supply frequency. Thc tbcory of the induction motor under dynamic conditions is somewhat iuvo!V<'d IlJ :citlls(: of t.he rotating rn af!,lId.ic lidrls , the spa tial relationships of whiclt dl'Jl<'lId ou :: 1" '1:<1 nlld loadj i!. .. 11.11 Iw In':Ü' :I[ h c l'(~ only in silllplifif~d

111 ,1

1II.

1', >I'" \, () II tlIe other hand, the equivalent circuits, derived for steady state "1)('r:ll.ioll with sinusoidal volt ages and currents, are inadequate when dealing wiUl I.rallsients or when the motor is supplied from a switching converter. ln I.h(' ['ollowing, the steady state condition will be treated as a special case of Uw dyuamic case. The mathematical model to be used is tailored to the needs of controlled d ri ves. It incorporates most of the qualitative features of an actual motor lilll. would not, of course, be accurate enough for the purpose of designing !.II(' Illaehine.

1.-:1 -

---,.p.

b)

S"ri e:~-wound

Lnll kW, 7730 Nm

Conto rating

,d, 1~):lIlJllin - l H~dll NIIl

IlilUI

III;"

:I ~ , ~ ,II h 1" 1~~ II 1'1 '. ' >I ;:

920mm

Three-phase AC ­ induction motor

tator motor

1428 kW, 9155 Nm at 1490 win- 1

Torque (5 min) Max, speed Mass Inertia

11600 Nm 4200 min- 1

a

2660 kg

0 s (ex,t)

22 kg m 2

,

F I,( . 10 . 1. ( ;lIlllparison of traction motors (Siemens). (II ) S ' "I·.I,· I'I"\.~,, o;eries-wound AC commutator motor, ( I, ) 'l'1>I ' I"'-pll;c~e AC induction motor

a.

I

ex

-TI

b ~s(ex-4TI/3)

S -TI

zs:

~s(ex)

'S:6~SV

~s(ex-2TI/3)

zs:

3

TI7KZ

c

10. 1 Mathematical Model of a General AC Machine I L i:: ...sslIllwd that the stator S of the machine is a hollow iron cylinder with ,.; rl'lda, (TOSS sectioIl, containing a concentric rotor R $0 that a llarrow airgap "I' I·{)W;(.alll. radial length h exists between the SIlI()oth cyJillctrical surfaces to wlll<,h sYlIlIlIl'I.ricaJ (.!trpc- plta:;c winctings of ll('I ': lil~ihk JI(~ight, arp assumed to III' :d.J.;u:lwd; t.Iwir sJliLl.ial illllperetlll'llS c1ixtrillllli,,":: IIlay h(~ thOllght o[ as IlI'illl·. prodll('('d hy slIiLallly plal'cd I.l,ill sLI'1\!,,\:J "I' ax ia l ('Illldlldol's or COll­ ""II'Liv,' :d,,·..I,::. Illlth 1I( ·nl.mls "I' UII' :iI,i1,r· "()IIIl, ,,, \.,·d wi lldilll (:-l .U:(· isola!.(·", l.\w

" I' 1,1, (" fOI." .. willdÍlIj': :U',· ,·,iI.ll<'l· "",,111" ' 1.,,01 I.,! ::1 i l> l' i "I'':: 1)1' :;b,,1'1. ('ir­ t"Ilíl.,·d illl',"1 " II II ,v. N :ó • Nu 11.1 '1' 1.1... 11111111""' ;0 "I 1,,11 Iti !...!1 1.111' 1111 1111 '1 11''' """:«' (.I ·,I·I
165

winding.The permeability of the fully laminated stator and rotor iron is as­ sumed to be infinite; saturation, iron losses, end-windings and slot-effects are ignored. The following discussion applies to two-pole motors; with multi-pole machines the synchronous speed is reduced correspondingly. This effect is exploited with pole-changing windings, however at some loss in utilisation of the motor.

§f,omm

11200 mm

,I) , 8 70 mm

!:1I11I1l11I

10.1 Mathematical Model of a General AC Machine

Symmetrical Three-Phase AC Machines

.... '/;71 '

ti

- TI

.. a

.........."',\

o



...

,

Fil!;. 10.2. Symmetrical AC machine

CrIlSS ~cctioll wilh two-layer airgap windings,

(I.) {lJ)wrapp"d slat.or windings,

(t') 1\"'1)('1'1'1.1"'(\1-1 di:·ll.rihIlLioll or xLa.l.nr willdillgs,

(Il)

(01) 'I'rllVl'II ; "I',

II.IIIJlI'II'I.III'IIH W:\VI '

ntll~(,d h.v

t.Il1" ·'·' I''':''''''

I'.II ..

r<:"L~

I{jG

10. Symmetrical Three-Phase AC Machines

10.1 Mathematical Model oC a General AC Machine

The variables are defined in Fig. 10.2 a. a is the angular coordinate in the stator wíth reference to the axis of stator winding 1 which is indicated by its central turn. The centres of the identical windings 2 and 3 are positioned 0 at a = -y = 120 and a = 2 -y = 240 0 respectively. Corresponding definitions hold for the rotor windings, where f3 is the angular coordinate, again with rderence to the centre of winding 1. é(t) is the angle of rotation of the rotor, Il)(~asured in the stator frame of coordinates, w(t) = dél dt is the instantaneous ;tIlgular velocity of the rotor. The magnetic field in the airgap of the machine has radial direction be­ LtllSC of the smooth parallel stator and rotor surfaces and the postulated irdillite permeability of the iron; since end-effects are neglected, we are deal­ illg with a two-dimensional magnetic field problem. The three stator currents isl(t), is 2(t), iS3(t) may exhibit any waveforms; I'()r: r(~asons of symmetry all currents are retained throughout the calculations, CV(~1l though one of the currents is redundant because, due to the isolated Ileutral, 'is1(t)

=O

+ is2(t) + iS3(t)

where is(t) = isl(t)

+ is 2(t) eh + iss(t) ej2 "(

(10.5)

is a time-dependent current vector in the complex plane perpendicular to the motor axis and is(t)

= is1(t) + is 2(t) e-h + iss(t) e- j2 "(

(10.6)

is the corresponding conjugate complex vector. The ampereturns 8s(a, t) are of course real, being a measurable physical quantity depending on a and t.

e jy

(10.1)

is:ft) !\/isit)e

is valícl at any instant, Le. the currents are balanced; this applies also to Ll _ COllllccted motors.

+ is 2(t) '!9(a - -y) + iss(t) '!9(a -

2í'r 2 -y)] ,-y= 3'.

J2y

2

N 8s (ex,!)

'S2(t) e fY

Wit,h these simplifications and definitions, the radial ampereturns wave 01' t.11( ~ stator windings consists of three spatially sinusoidal, stationary terms, w li ich a]'(~ modulated by the currents (-)" (II'. I) ' N S [is1(t) '!9(a)

167

s

Mt)e a

-ja

b

Fig. 10.3. Complex current vector

(10.2) !\\mrdillg to Fig. 10.2, 8s(a, t) corresponds to the stator ampereturns en­ <"1",,: ('<1 I,'y ii. ra.dial magnetic field line crossing the motor at angle a; because "I 1.1)(' ;1,SSIIllICd high permeability of the iron these ampere turns appear as III;q ~ "d()ltIotive forces at the two airgap crossings. The nondimensional spatial I'rrlll'l,i()1I - 1 ::; '!9(a) ::; 1 in Fig. 10.2 c characterises the ampereturns distri­ 1'IIt.io/l ()f one phase of the stator winding, it could be written as a Fourier ::ni(!:,; . Wh cn lhe windings are fed with alternating currents, each term in Eq. ( 10.:J.) o~cillat.es as a standing wave, fixed in space; their superposition results iII ;t, travdling wave 8s(a, t), as seen in Fig. 10.2 d. The spatial harmonics rllH.y he dÍluillished by reduced pitch and skew of the windings, resulting in /1 ( II') ., cos cr. I ty illl,roducillg rornplex llotation, :I~ ( ( ,:in

('OS IY

IU ld

'l (.

:lU)

,

etc ,

(10.3)

The construction of the current vector is(t) is shown in Fig. 10.3 for assumed values of iS l > O, is 2, iss < O. Magnitude and angle each vary with time according to

The current vector determines the instantaneous magnitude and angular po­ sition of the peak of the sinusoidally distributed ampereturns wave produced by the three spatially displaced stator windings. By combining Eqs. (1004, 10.7) the ampereturns rnay be expressed as a travelling wave, the peak of which follows the angle a = ((t) of the current vector, 8s(a , t) I

r t1w

. I)

,I,

N :.'

1':1 (1 ),'

I"

I

/7,(/ ) " I"

1

I

I

f

~"

I

1

( 10.,1)

cos(((t) - a) .

(10.8) three phase and velocity

d ( / di, i II I,) I( ~ ai q~ap of the mo(;or . Siw(' 1.111' CC>lllp)(' X v('d()r~; ddillt'd iu )';(p :. (10.;, In.S) rJ('scribe t,he spat.i:tl dl ll l.r'il'lll,i ,," (Ir II 1I ,1I 11 ~ III'I , iI' lidd ill II, 1'""1<' Iwrp"lIdi('lrlll.r I,,, 1.1". 11101,01' Il.li i : l ,

(v I () ' i (II

= Nsis(t)

st a t.or currents are sinusoidal and form a syrnmetrical LII(! alllpprct.lIfl1s wave revolves with constant magnitude

s'ysL(~1I1 I

J't·lI.r,(,H III·. ir'l ~ . 1';'1. (I().:~) 1>('('OJll(~S

(10.7)

is(t) = is(t) ej(t) .

10.1 Mathematical Model of a General AC Machine

lU. SyJlllll(~Lrical Three-Phase AC Machines

1()8

they are also called space-vectors or space-phasors [N7, 873, 31]. However, they are not to be confused with the constant complex phasors for describing steady state sinusoidal alternating quantities. The sarne reasoning as before holds for the ampereturns wave produced IJy the moving three-phase rotor windings,

0 R (!3, t)

= NR [iRl(t)

cosf3

+ i R2 (t)

cos(f3 - ,)

+ iR3(t)

B( 0.)

cos(f3 - 2,)] . (10.9)

Incremental number of turns

N 2

.2. cos

WILcn defining a rotor current vector,

in(t) = iRl (t)

+ iR2(t) eh + i R3 (t) ei 2, = iR(t) ei ç(t) ,

169

Â. LLA..

(10.10) Fig. 10.4. Distributed winding

fR(t) = iRl (t)

+ iR2 (t) e-h + iR3 (t) e-i 2,

= iR(t) e-H(t) ,

(10.11)

(.]I(~

ampereturns-wave, excited by the rotor currents and moving with the ro(.or, assumes the form en(f3, t) = ~ N R [iR(t) e- i {3

+ i~(t) ei {3]

,

i

~s, ~ ~

(10.12)

real;

(t)

Ns

i Le; drcct. ou the stator is obtained by substituting

li -- n

() II

(10.13)

- é )

(II', (, I.) -

t NR [iR(t) e-i (a-c) + i~(t) ei (a-c)]

.

(10.14)

TI,,· ",: :"ll.;ut!, waw is a superposition of the stator- and rotor-ampere turns

H(
= 0s(0, t) + 0 R (0,

é,

t) .

= 2h 110 [(1 + 0"5)0 5 (0, t) + 0R(0, é, t)],

'l/J51(t)

= ~~ ~ (1+0"5)110

(10.15)

2

[e j À+ e-j À] /

/ ~~

+ N5.~~lr 110 /

[i5(t) e-ia

+ is(t) ei a] dodÀ

À-~

~

À+~

[ejÀ +e- iÀ ] /

_~

[iR(t) e- j

(a-c)

+i~(t)ei(a-c)]

dodÀ.

À-~

(10.18) (10.16)

il.CCOllnts for magnetic leakage; 110 is the perrncability constant. A lIloddlillg of leakage effects is not possibk with lhe simplifications

wli('I(~ (rs

""':til(~d

(10.17)

.x+~

1!. 2

2

rOJ'(:e becomes effective at the two airgap crossings, causing a local dClIsit.y at the stator side of the airgap,

/I,.;(n, s, t)

I r Bs(a, c, t) da] d>',

°

1.01l10!.i vC

1

,L '"'>.l2:

where I is the effective axiallength and r the radius of the rotor. The integra­ tion over is a consequence of the non-uniform field in the airgap while the integration ove r À is required by the nonuniform distribution of the winding. Inserting Eqs. (10.4, 10.14, 10.16) results in

,';illC" I.IL<~ pcrllleability of the iron is assumed infinite, the combined magne­ flIIX

i.

in the interval < À < Thus the ftux linkage in stator winding 1 is obtained by a double integration,

The evaluation of the integral is greatly simplified by the complex notation
Jllad('.

.

(1

WIL< ~ il' calcll];.d,illg llllx

lillkagcs, the sp;l.t.i;t! di::I,ribIILioll 01' thc conductors 1111 \1; (. IIC (·oll c. id(·J'('d ;t:; ill
N~ I r

+ 0"5) ~ 71' 110 N ,r.; NlIlr Hh

II.

-

71'

Ito

1

= "3 L 5 ,

=

:.li llli'I(' 1'<'11 1111. 'lfl Olol.II,ill('
1

M

:J .

,

(10.19) (10.20)

'l/JSl (t)

= %Ls [is(t) + is(t)] + %M

[iR(t) e j

171

10.1 Mathematical Model of a General AC Machine

10. Symmetrical Three-Phase AC Machines

170

ê + iR(t) e-j ê]

(10.21)

'file fiux linkage in the stator winding 1 is a physical integral quantity and Illust, of course, be real. The flux in the other stator windings is computed lilcewise, with the exception that the integration over À is shifted to , ± ~ and 2, ± ~ respectively, reflecting the position of the windings. This results

'l/JRl (t)

=

t LR [iR(t) + iR(t)] + t M

[is(t) e-j ê + is(t) e

j

ê] ,

(10.28)

which has a similar form as Eq. (10.21). However, as this fillX is defined in rotor coordinates, the stator current vector is now turned backwards relative to the position of the rotor. 2

(1

+ O'R) N;

lr

h

7r

/.Lo =

t LR

(10.29)

111

'l/JS2(t) = %Ls [is(t) e-h

defines the self inductance of a rotor winding. Likewise the flllX linkages of the other rotor windings are obtained,

+ is(t) eh]

+%M [iR(t)ej(ê-')') + iR(t)e- j (ê-')')] 'tPS3(t) = %Ls [is(t) e- j2 ')'

,

t

(10.22)

'l/JR2(t) = LR [iR(t) e-h

+tM[is (t)e-j(ê+')')+is(t)ei(ê+')')] ,

+ is(t) e j2 ')']

+% 111 (iR(t) ej (ê-2')') + iR(t) e-j (ê-2')')]

(10.23)

'l/JR3(t) = %LR [iR(t) e-

+t M

Tlw symmetry of these equations gives rise to the definition of a complex

v('dor of fiux linkages

(10.24)

=

+ M iR(t) e j

(10.25)

ê(t) .

','his IIIIX: vector describes the magnitude and angular position ofthe peak of 1.1((' :-; illllsoidal flllX density in the airgap ofthe machine. The exponential term aIJ;wllcd L() in (t) indicates that the rotor current vector must be rotated by

()r IIlcchanical rotation

COllversion of the stator current vector into rotor coordinates by Eq. (10.13) yields

("~:; ((J, c, t)

= t Ns

[is(t) e-j (.B+ê)

+ is(t) e

j

(.B+ ê )]

(10.26)

wilich produccs a fiux density on the rotor surface, corresponding to Eq. (IO,IIi)

/I}{(!1, c, t)

r' li

= 21h /.Lo [(1 + O'R)8R(!3, t) + 8 s (fJ , c, t)]

(10.27)

ilf'.ajll si.a.llds ror (",he tmavoidahle nmglldic 1r·;i1q~( ' . "d' ('I~raLioll aWlIlld t, h(~ ( ~ iI'Clllllf(~rclI(,(' (lI' ti", rul,ol', II.gll.iu prcsnpposing a

lil,l"ly d i ~ tri"IlI.('(ll'()tor WilldiJlI!', kad s to 1111 ( ' ~ pl' (' f1:il o ll fo r til(' JlIlX lillkitl~(~ iII

ro t"r

will"il" '. I ,

(10.32)

•

10.5. iR1

iS1

before its effect can be superimposed upon

~:;UII(' w;ty.

ê

The magnetic linkages described by Eqs. (10.21)-(10.23) are now used to derive the voltage equations for the stator and rotor circuits, depicted in Fig.

Rs

01' L1w stator current vector is(t). '1'11(' IIllx linkage of the moving rotor windings is computed in exactly the

or

(10.31 )

+ 'l/J R2 (t) eh + 'l/J R3 (t) ej 2')' L R iR(t) + M is(t) e-j

I.hal.

IIl!'illI~;

+ is(t)e j k+ 2 ')')]

[is(t)e- j (ê+2')')

!fR (t) = 'l/JRl (t)

I.cJ'lIlS

11.1'1,;1('

j2

+ iR(t) e ')']

')'

rotor flux .

allowing Eqs. (10.21)-(10.23) to be combined and disposing of the conjugate

(.11"

j2

(10.30)

These expressions are again simplified by forming a complex vector for the

!fs(t) = 'l/JSl(t) + 'l/JS2(t) eh + 'l/JS3(t) e j2 ')' ,

'~:.'Jt) = Ls is(t)

+ iR(t) eh]

l R, Rs

usd

u~

iS2

'l'SI

'l'R)

'l'S2

'l'R2

iS3 'l'S3

I

I

'l'R3

iR2 iR3

RR RR

I uR)

RR ["R' UR3

US3

_________________

~

L ________________ _

Fig. 10.5. Magnetic linkages and voltages

The tine-to-neutral voltages in the stator circuit are

. d'l/JSl RSZ81 + ----;[t = USl(t) U:.; 'i.:; ',~ U ; ';" . 1

d'ljJS2 _ 'tL'i2(t) , (li. dlf' :;;1 ti,

1I ;::I(t.) ,

(10.33)

ln

lO. Symmetrical Three-Phase AC Machines

10.1 Mathematical Model of a General AC Machine

wlwre Rs is the stator resistance per phase and USl, US2, US3 are volt ages of a.,iJitrary waveforms. These equations may again be combined by introducing ("/II\ljJlex vectors. By formal definition of a voltage vector, eventhough there is 110 immediate physical interpretation as with the currents,

"Ils (t) = USl(t) + US2(t) eh + US3(t) e j2 'Y, ;tllcI

(10.34)

with Eqs. (10.5,10.10, 10.25), this results in

. d'IjJs . dis d .. "S'/..5 + =RSIs+Ls dt +Mdt(1ReJé)=:J!s(t).

dt

1';lIlploying the rules of differentiation, this equation assumes the form lls is

dis

di

·

+ L S dt + M dtR eJ + j é

.

w M iR eJ é

= :J!s (t) ,

(10.36)

w!tere w = dê/ dt is the angular velo city of the rotor. The two terms containing t!t(, rotor current may be interpreted as voltages due to mutual induction and r/ lI.ation, respectively. l1y inserting Eqs. (10.21-10.23) into Eq. (10.33) and considering Eq. (10.1) whic:h is due to the isolated neutral, it is found that the line-to-neutral volt­ ; tf~('s aI' the symmetrical machine are also balanced at any instant,

'/ISI (t)

+ US2(t) + US3(t)

=O .

With the help of Eqs. (10.10, 10.32) they are again combined to form a vectorial equation,

. RR IR

(10,38)

(10,39)

clashc:d lines connecting the neutraIs in Fig, 10,5 are redundant in reality. 11/:111:(' t!te voltage equations are

/.Iw

"

fluiu/. 11'1l11l ~ 1

+ -(l1/Jra ,- = uw(t) , ri/. 111/' /,' .1 ,(/

d. _ . dt (1s e J é)

= :J!R (t)

,

(10.41)

+ UR2(t) eh +uR3(t)e j2 'Y

(10.42)

B RS ((3,

é,

N:~o

t) =

[i s (t)e- j (!3+é)

+ is(t) e jUHé )]

.

(10.43)

The magnetic field produced by the rotor currents themselves does not gen­ erate any net tangential forces with the rotor currents since there is no reluc­ Lance torque with the uniform air-gap; this could be shown by carrying out t!te caIculations with B R instead of BRS. The current distribution aR((3, t) along the circumference of the rotor is ;t!so, with the assumptions made, sinusoidal. According to Fig. 10.6, it is cldined as the spatial derivative of the rotor ampereturns,

- ~ 8GR((3, t) __ . NR [. -j{3 _ .• j{3] aR ((3 , t) - 2 8(r (3) - J 4 r lR e lR e .

(10.44)

T!te tangential force df acting on an axial strip of width r d(3 of the rotor sur­ ,'aC(~ is the product of radial flux density and axial rotor current distribution d!

= -B RS ((3,

é,

t) aR(!3, t) 1r d(3 ,

(10.45)

w!tidl Ilpon integration along the circumference yields the electric torque in clin:ctioll of rotation,

dt

({c/I 11.'1.

di

may be impressed by an external voltage source. ln case of a cage rotor, the voltages are zero and the rotor circuit is internalIy short circuited. The vector differential equations (10.35, 10.41) describing the electromag­ netic interactions of the symmetrical AC motor in steady state and transient condition are now to be supplemented by equations for the electrical torque and the mechanical transients. On a real motor the tangential forces are act­ ing at the sides of the slots, where the magnetic field enters the iron. Since there are no slots in our simplified model with airgap windings, the torque is computed with the help of the tangential Lorentz- forces exerted on the ax­ ial current carrying conductors orthogonally crossed by the radial magnetic field. The component of the flux density on the rotor surface which is caused by the stator currents folIows from Eq. (10.27)

'l'1c(' S
Ic./I'/./li

.

di = RR IR + L R di'R + M

(10.37)

IIIOst motors the neutral point of the stator winding is not accessible, ;:0 J.ltaJ. oltly the terminal voltages US12 = USl - US2 etc. are available for /'XI,CTII:d IIS(\ while the line- to neutral volt ages are dependent quantities. '1'11 i:: COIT( :sponds to an actual situation, where the motor may in fact be L1 /·/)/III('CI.(·d . With the terminal volt ages U12 = USl - US2 etc. the voltage VI'/'LIlI' iiI Eq. (10 ,34) can also be written as

i lll (I) + iR2(t) + iR3(t) == O ;

d'IjJ R

where alI variables are defined in rotor coordinates. The vector of the rotor voltages,

()IJ

(/ :: (/) = 1t12(t) - U23(t) e-h,

+

:J!R(t) = URl(t) (10.35)

173

!.II(' ·UII~(i.) '1/ /n

,

(I) .

271'

(10.110) 'III'M{I.)

l'/IU /,'

,/:lZ ( Uu s (f'i , It , o

/.)

11.11((1, /.) d(1 .

( 10.46)

10. Symmetrical Three-Phase AC Machines

J 74

10.2 Induction Motor with Sinusoidal Volt ages

1nserting Eqs. (10.43, 10.44) results in 2".

mM(t) = -

M. /

6 'Ir J

[is e- j (t3+ ê )

+ i*

ej (t3+ ê )] [iR e-j 13 - iR e j 13] df3 ,

· R Sls

. j ê) + L s didts + L O dtd (1R e = 'Y<s (t ) ,

(10.50)

· R R1R

' + L R didtR + Lo dtd (ls e- j ê)

(10.51)

-s

o

J

(10.47)

=

'Y
(t ) ,

~~ = mM(t) -mdt) = í Lo 1m [is (iR ej ê)*]

dê

where M is the coefficient of mutual inductivity defined in Eq. (10.20). When integrating over the full circumference of the rotor, the exponential terms containing f3 canceI. Hence we find mM(t) = -M 3 'Ir

/2". is iR e-j 2 ê

fSiR e J

-

o

j

.

ó

df3

= ~3 MIm [is(t) (iR ej Ó)*] (10.48)

the imaginary part of the bracket is equivalent to a vector-product, being proportional to the product of the two current vectors and the sine of the angular displacement X, as seen in Fig. 10.6 b.

dr

aR(f3,t)

dt

- mL (ê, w, t),

(10.52) (10.53)

= w.

Since the first two equations can be split into real and imaginary parts, this represents a set of 6 scalar nonlinear differential equations. They are valid for any waveforms of voltages, currents and at varying load torque and speed. The physical currents, for example in the stator windings, are easily de­ rived from the vectorial representation. With Eq. (10.1), the current vector, defined in Eq. (10.5), is

~ is1(t) + j

is(t) =

V;

[is 2 (t) - is 3 (t)]

(10.54)

hence (10.55)

is1(t) = íRe[is(t)] ,

BRs (j3,t)

175

and correspondingly, is 2 (t) =

is

í Re [is e-h] , etc.

(10.56)

Likewise the currents in the rotor winding are obtainedi of course, in case of a squirrel cage rotor, the notion of phase currents loses its physical meaning,

~ e ic

a

b

Fig. 10.6. Rotor current distribution and torque

When assuming that the stator and rotor windings have an equal number of turns, N~ = N s, and introducing the usual leakage factors,

Ls = (1 the

+ O"s) Lo,

M;;-:Lo

compld<' matlwntatical model of Lhe sYUIJllI'Lrjeal

with til(' llllllJwd illNI.ia J t.lw

L R =(1+O"R)L o ,

1Id.

10;1.<1

1.01"qlll'

t.JWIl aSSlllll(~K

:d. I.h(~ I'ollplillf':

01'

1.11(' followilll': !.II(' 1I1C/!.IH,

(10.49)

I~ellera.l fOl'lll,

AC machine mI, being

wil.h

lwen though the simplified theory remains applicable. The Eqs. (10.50-10.53) are the starting point for ali subsequent discus­ sions of transient and steady state phenomena. They describe the electrome­ chanical interactions of a symmetrical AC machine and can be adapted to various constraints, imposed by the type of machine used, as long as the basic assllmptions on which the model was based, remain valido This mainly refers 1.0 ra.dial symmetry and constant airgap, excluding saliency of the stator or rotor. Table 10.1 contains the rotor circuits of different nonsalient AC machines IIs('d in controlled drives that may be represented by the mo dei equations. 'l'hc details of the constraints are discussed in later chapters. 'file model equations contain, of course, also the special case of steady :: !.al.e opcration, when the AC motor is fed with sinusoidal impressed volt ages 1J)' a s'yulluetrica.l three-phase supply and is loaded with constant torque. To f01"l1l a link to tbis commonly used presentation and to gain a better IIlId('l"st.alldilll ~ of \.III: 1llOdel <'quntiolls, t,his cas!! will bc discussed first. With 1111

0,

H

( 'I ~I \"

IIIO!.Ol"

IO!.OI' 1' .' : Ii: I!."I' :: III!!}

or a wC/ulld rot.ol'

Itn,v,' 1""'11 ,1.11<11 '01 .

1'1 'S II 11.:-1 ,

w!tI'J"(' ('xLern a) ~ymmdrical

,'(1;

'II . ~-; y "'lfIetrical

Three-Phase AC Machines

10.2 Induction Motor in Steady State

'1'11 101. · 10.1. Mathematical models for nonsalient rotors

,,' di'I'"I" '''1. AC machines used in controlled electrical drives

.

iR

IJI R1 iR1

= '23/F

iRI

US2(t) = USl

i'RI

RR

US3(t)

IJI R2

.

(t - :J

-/2 [U e j 2 -s

.,

+lR

= USl

t-,)

(Wl

+ U·

-s

e-j

(WI

,= ""3 21l'

t-,)]

(10.58)

(t - ~~)

-/2 [U e j 2 -s

3 /

177

t-2,)

(WI

+ U·

-s

e-j

(Wl

t-2 ,)]

(10.59)

lR='2 F

IJI R3 iR3

a Synchronous motor with permanent magnets on rotor surta ce

~ !dR

b Synchronous motor with permanent magnets and damper winding

~IUF

~

c

The voltage phasor U S is a complex constant, having a magnitude equal to the RMS line-to-neutral voltage and an arbitrary phase angle TI which may be set at will because it is defined by the chosen instant for time zero. When the complex voltage vector according to Eq. (10.34) is formed, a very simple result is obtained,

=const.

iRI

~

= O

!dR=

Induction motor with wound ar cage rotor

­

URI

i R2

UF iR3

d Synchronous motor

wíth fie/d and damper windings

e Doub/y-fed inductíon motor

. . () = USl + US2 e J , + US3 e J 2,

Jfs t

3V2

.

= -2- -s U e J Wl

t .

,

(10.60)

the conjugate complex part vanishes, leaving a vector with constant mag­ nitude and constant angular velocity describing a circular path around the origino The stator and rotor currents form also symmetrical three-phase systems in steady state. With

with impressed rotor vo/tages

is 1 (t)

i (t) = 3 -/2 I e j -s 2 -s

IO.:l . l Stator Current, Current Locus

II. :i'y lIlllletrical three-phase system of sinusoidal voltages having the angular

'll.S J

(t) =

-/2 uS

COS(Wl

t

+ Td

= Re

[-/2 Us

= V2 [u ejwlt+U*e-jWlt] 2 -8 - s , w IIC:re rro.; = (Js

(;j T I

TI

ej

Wl

i

t]

t

(10.62)

,

RI

(t) =

-/2 2

[I-R e j

(WI

-w)

t+ r

-R

e-j (WI-W)

t]

,

etc.

(10.63)

n" (10.57)

is t.hc constant volLagc I'IJ:tJ;or; correspomlinG exprcssions

ii .. !" for I'lItuws ~ 1IJl
WI

where the current phasor Is is still to be determined. A similar definition ('xists for the rotor currents oscillating with slip frequency W2 = Wl - W

may be defined as follows, using complex notation,

ej

(10.61)

we find

10.2 l.lHluction Motor with Sinusoidal Symmetrical V olta/,!;es in Steady State

I'r(:<jIH:JlC:y Wl

= -/2 [I e jw1t + -s r e- jw1t ] , etc. 2-S

. . _ 3 -/2 I ej(WI-W)t. L/I(t) 2 -R

(10.64)

TI\(: rotor Cllrrcnt in stator coordinates, i.e. as seen by a stationary observer, III with ~ - wl. lWC;lUSC of constant sp(:ed, i /{(I)

, ,, I tI {l}

., ..

J/. '0

I/{

i (",I

t ..

(IO.(;[i)

I '/ H

lO

''-;'y"11I<'l.rical

Three-Phase AC Machines

10.2 Induction Motor in Steady State

II(~II(,( ' Ilw rotor ampereturns wave, moving with slip frequency across the rotor slIfface, rotates in synchronism with the stator current wave; this is a

prccondition for constant electrical torque. By introducing these expressions into Eqs. (10.50, 10.51) and omitting the redundant terms, the differential equations are converted with UR = O to algebraic equations for the phasors,

+ j Wl as LoH s + j W} Lo (Is + IR) = U s (RR + j w2 aR LoH R + j W2 Lo (Is + IR) = O.

(Rs

N orlllalising

(10.66)

(10.67)

Wl

n'snlts in (

~R + j Wl aR LO)

+j

IR

Wl Lo (Is

+ IR) = O.

(10.68)

I~:qu ations (10.66, 10.68) lead immediately to the familiar single phase equiv­ alcllL circuit seen in Fig. 10.7 a; its validity is restricted to steady state, asslllning sinusoidal symmetrical currents and constant load torque.

as Lo

/Is

aR Lo

lR

aLs

RS ~.

J.s 1/.,'

LO

Is

1.1/1

Y.s

S

(1-aJ Ls ,

=.1m

..!!....L 1-a s

1-/S

1 RR --2­ (1+aRJ S

lmR

Ns =N'R

;/

(1+aRJ1R

is RR

I

b 1

--lR 1+as i

II

is

y.s

Lsl

O(1+asJ SRR 2

1ms

·SWII7L R

1+J RR Zs=Rs+jw}Ls l+js'1: R

where

Sp

W}

RR a LR

1 wlaTR' with

F iH· 10.7. Silll\k pI IlL~(' 1.''1"ivaklll. c.ircllit,j "I' i,,
'"I

'"I jl!l

Il llLl,,,'

,,,"1 1'0(.,,,

:: I,at" .. Ili
1'0(,01'

111.1, ' ,

LR TR = RR '

(10.71)

which is called pull-out slip because a motor with zero stator resistance pro­ duces maximum torque at S = Sp, as will be shown later. The pull-out slip of normal medium size motors is Sp < 0.25. The steady state equivalent circuit in Fig. 10.7 a contains three synthetic inductances which, because of 1m = Is+IR' are not representing independent energy storages; hence it must be possible to redraw the circuit diagram with only two inductances. One possibility is shown in Fig. 10.7 b, where all the lea.kage is concentrated at the stator side. There is a different fictitious rotor resistance leading to a different current in the right hand side branch of the circuit but the stator impedances are identical. The new synthetic rnagnetising current ImR represents the rotor flux, as will be shown in a later chapter. A third equivalent circuit is seen in Fig. 10.7 c, where all the leakage is referred to the rotor side and where Ims stands for the stator flux. This arrangement can also be used for designing controls. The stator current is now computed with the added simplification Rs = O which is of little significance with larger machines operating with low slip at line frequency. Thus the normalised expression for the stator current phasor becomes I -s

(h) wil.h 1"111 (1 11 1'" ;,,<111<'1. 111,11'" (. -) wi!.11 1(l1I 10 I/ J.4' IlIdllc (, 11 1" '1\

=

. 1

c

(10.70)

+ as)(l + aR)

is the totalleakage factor of the motor, which has an important influence on the characteristics of the machine; it is chosen by suitable design parameters such as the shape of the slots and the length of the airgap. Normal values of a range between 0.05 for low leakage machines to about 0.20. The expression in Eq. (10.69) gives rise to the definition of yet another important parameter

1+as

(II) w;l.h

(10.69)

'

1

s = W2 = W} -w Wl

By inspecting the equivalent circuit in Fig. 10.7 a, the stator impedance per phase is derived. It may be written in the form

a = 1 - (1

the second equation with the slip frequency

179

=

S

U 1 + J Ü s. -s ,s j Wl Ls 1 + J sp

(10.72)

whj rJt is n liucar cowplex function umpping the real S/Sp-axis onto the Is­ pIHII( ~. A('('ordilll~, to t.Iw mies or confoJ'lllal Illapping, the result is a circular lorwl, 1.111' ('irei" di:"',1 i l,tIl , ...lS(l ('1I.lI('d ( h rmllll l l.'H (lI' I!('ylaud's circle . This is ""lIlirll"'.! 111'y LI", 1,')It>\v ll,,~ I.r; ~ II:lr(lrlll l l, I , i ,,,,

180

10. SymmetricaI Three-Phase AC Machines

Is = Iso [ 1 + O' 2 O'

10.2 Induction Motor in Steady State

1 - O' 1 - ~ ~l 2 O' 1 + J sp

_

= Iso [ 1+0' _1-0' e-j2arctans/ sp], 20' 20'

(10.73)

where

representing (for Rs = O) the no-load magnetising current as mentioned before. Minimum phase shift cp = arg(U 5,15), i.e. maximum power factor is reached when the current phasor forms a tangent to the circle. The optimal power factor then follows from the geometrical relations of Fig. 10.8, COS(()

rn

Iso = j

Il. s Wl

(10.74)

Ls

is the ideal no load current valid for S = O, i.e. when the rotor moves syn­ chronously with the revolving stator field. When no speed difference exists, no rotor current will be induced and there is no torque; hence the no-load current is a purely reactive magnetising current. This is in agreement with the equivalent circuits in Fig. 10.7 for S = O.

(10.75)

'

7r

cp = - -

2

1 S S - - arctanSp Sp

+ arctan -

(10.76)

O'

by differentiation , d(trSp) = O; the result is

VU.

=

(10.77)

p

Rence, not only the geometry of the current locus, but also the maximum power factor, the pull-out slip and the normalised slip at the nominal oper­ ating point are all determined by the leakage factor 0'. The magnitude of the normalised stator current,

.I.

!,Is

S 1 . / " Sp

,

~

1- cr e - j2 are ta{ Sp

/

cr JSD

Is Iso

(ínductíve)

-1

Fig. 10.8. Locus of stator current with stator resistance neglected, circIe diagram for an induction motor

is an even function of slip, as seen in Fig. 10.9 a. The normal operating region comprises only the central portion of the curve. At larger values of slip, for instance when starting the motor at S = 1, the current rises sharply so that the simplifications employed become questionable; this is particlllarly true for the iron saturation that is not negligible at high currents. The stator current at nominal slip, Sn/Sp = y'ci, is

IS n

Iso The complex locus Is (S/ Sp) is depicted in Fig. 10.8 with the voltage phasor assumed imaginary, U 5 = jUs, in order to place th e circle in the customary orientation. The curve covers the fllll rall~C "00 < S/Sp < 00; for S > O the motor runs below synchrolli~IIJ , drawillg acl.ive power, while for S < O iI, op( ~ ral.(~s as a gcnerat,ol' alld Sl1ppli('/ol 1\cl.iv( ~ pow(~ r to t1w lilln. Reactiv( ~ pownl' ror lllil(f,lwti satioll is il.lwaY ~1 11(.,..11'11 whi('h ,i!l n dmra('.I.(~ri!;tic of illdllt'l.ioll 1I 1I\f IIi Il ('1i Jlil.vil lj ( 110 illl."l'IlII.J v,,JI.II II.'·

til. S

II 1.1".

11 11111'("( '

whi('h ir, pn ·r;(·IJI. iII

1'1 111"' 111. HIi!! " " " '!1 il. rl II d llillllllll vILIII"

(10.78)

VI + (ir

Reactive current

cr 2 cr

, 1+ JSD

V1+(~ff;r

1

- Jso 2 cr

JSD

HY III·h,.IIl1 ll ll il Il lhe lllll('" .

1-0' 1 + O'

= -_.

this is usually defined as the rated or nominal operating point. The pertinent value of Sn/ Sp is obtained from the phase angle, Eq. (10.72),

~n Active current

(motoring)

Js

181

1

= y'ci

(10. 7!l)

,

while the asymptotic value of the current is 1

Is Iso

fim 5-400

(lO.BO)

O'

Tllis ull(krlillCS I.he importance of lhe feakage factor O' as a design 'I'lw III<\ Xillllllll ;tJl(.l ruilLimulU val!i(':; "f til" ac:t.ive current pO\V('r, ;1,S;; lIl1lilll'. Ii..: ('OIlSt.., ;tJ'(' 1'1'11..-111'11 H,1. S (lI' 1,111' .. ird,·. I" LIli' III' XI. ró( ·(·I.j"lI íl. 1:1 !.h"wlI I.h:d.

wl"'I"

1.111' 1."1

ii""

11 : ./1 111111' 11 iI ,1I

P('It!( vrdll'·:.

pararnd( ~ r. anel

acl.i v,~

1''''1'' i.". in tlw v( ~ rl.i( : (·n

LlI<':'w a.re

al ~ ()

1.11(' poilll.:.1

10.2 Induction Motor in Steady State

10. Symmetrical Three-Phase AC Machines

182

3 1 - (J U~ mM = - - - - - - -

~

2

ISO ~~--------------I1 - ---------------;~

--',

1 ,---

//;­

RS=O (J

=0,1

ISO I

I

-2

-1

a

Sp

S

2

-~ ~

mM = mpo

Sp

3 1-

úJ,mM\

1-----

mn mpo

I

p



.... _ _

I

: Motor

s

úJ 1 _ U úJo - Uso

=1

I I

I

-1

S

2

Sp

Generator

------------' -1

Fig. 10.9. Stator current and torque as a function of slip (Rs

= O)

10.2.2 Steady State Torque, Efliciency

An expression for the steady state torque is obtained by inserting the vec­ tors for sinusoidal symmetrical currents, Eqs. (10.62, 10.65), into the general formula for the torque, Eq. (10.48), mM = 3 Lo 1m (Is

IR) ,

(10.81)

yielding a constant value, as expected. From Fig. 10.7 a the following relation between stator- and rotor-current is derived, -j w} Lo I I -R= RR I S+jw} L R -s,

(10.82)

which leads to mM

'

:\ r ' lI

Uslw} Usolwo )

2

2

SISp + SpIS '

(J

U~o

SISp + SpIS '

(10.84)

(10.85)

(10.86)

mo=-------

mpo

b

2

=m p - - -

where

mM

-2

(

Sp

,

I

2

SISp + SpIS

where mp represents the peak torque, available at the pull-out slip S = ±Sp. The function of torque vs, normalised slip is plotted in Fig. 10.9 b. When the induction motor is part of a variable speed drive, the stator voltage Us and frequency w} may differ from their nominal values Uso, Wo shown on the nameplate; hence the following extension of Eq. (10.84) applies

/

~

w? Ls

(J

183

j

l ,~; TIll [ ali I S

wl

I,o

.i W I

Wil.li 1.111' 111'11' "I 1-:'1 , (I (J ,'tH) I,liil :

[,1/

(' 11 11

I.

1" ,

)"',111 11 111',""

(10.8:3) L..

2

(J

w5 Ls

is the maximum torque (for Rs = O) at nominal volt age and frequency. This indicates that the torque rises with the square of the flux; the reason is of course that the rotor current, being caused by magnetic induction, is also depending on fluxo The torque vs. slip curve seen in Fig. 10.9 b may be qualitatively inter­ preted as follows: When the rotor speed deviates from the angular velo city w} of the stator field, currents are induced in the rotor winding producing a torque with the tendency of reducing the speed difference. Below synchronism (S > O) the torque acts in the direction of the rotating field, i.e. the machine operates be­ tween standstill and synchronous speed as motor; above synchronous speed, the torque is opposed to the direction of rotation, hence the machine acts as generator, feeding electrical power to the line. Steady state supersynchronous speed can thus only occur if the machine is mechanically driven . Because of the reactive power problem, induction generators are rarely used except in small unattended hydro or wind power stations, but temporary regeneration is a normal feature of controlled AC drives. The occurrence of a peak torque can be explained by a simplified model of the rotor circuit as shown in Fig. 10.10. The rotor may be represented by a R-L-circuit that is supplied from a voltage source, the amplitude U2 of which is proportional to frequency W2. The current phasor

U20 ~ Wo I 2 = R 2 + j W2 L 2

U 1

-20- - jwOL2 1-j ~ W2 L2

(10,87)

as a function of frequency W2 is described by the semicircle shown in Fig. 10.10 b , While the magnitude h rises asymptotically with W2, the active componenl. H,(' / '.1

(}~:o ii.!"

I J'.~

1 II ~ 101 ..

1, ..1

ti/ :l l i ~

/I "

(IO,RR)

184

10.2 Induction Motor in Steady State

10. Symmetrica! Three-Phase AC Machines

J2

U20

L2

Re

'2

W 2 W

a

b

o

j WOL2

~

I I I I

Motor

0<S<1

C

(10.89)

'

wi Ih , ~xt:ernal resistors in the rotor circuit then offers a possibility of modifying lhe torqllc/speed characteristics by changing the pull-out slip Sp through the rcsist.ance. The resulting family of curves is depicted in Fig. 10.ll in the

;\,n :lls toIllcd orientation. ln combination with the dashed curves for inverse

of the field by interchanging two stator t erminais a fairly complete of the four quadrants in the torque/speed plane is achieved, albeit 1.\1(' sliape of SOlllC eurV($ is unfavourable. Lowcrilll-!; the st.at.or voltages, for iIl S t,"'Il(,( ~ I.hwll f.': 1t a tap cltanl'-(illg trans[oollcr, l"(~dIlCCI ; Lhe peak torqllc by the ::(\,,<1.1'" 01' l,b <' voll.a!';,' \'(~dIlCl.i()lI, f\lrt.ll<~r ('olllpr('::: :illl'; 1.11(' (",IlJ'V(~S itt bori7.Olll.al

H\ l tI,1i 1'

01' (·Olll.íIlIIOIl ! : 11 ( 1(" ' "

'·"lI l. lt" I.y

rol.o!'

/ / /

/

time duty, for instance when starting the motor or with crane hoist appli­ cations, are these los ses tolerablej sim pie control schemes of this type are in widespread use. The efficiency of the simplified symmetrically fed induction machine in steady state operation may be assessed as follows: Because of the neglected iron and stator copper losses the active power Ps supplied to the stator windings is passed on to the rotor, where it is transformed into mechanical power and heat losses. ln the steady state equivalent circuit (Fig. 10.7 a) alI this power is dissipated in the synthetic resistance RR/S, P

R

= 31 2 RR

(10.90)

RS'

ln reality the resistance in each rotor phase is only RR, hence the difference in power must correspond to mechanical power that cannot be modelled directly by a static circuit, Fig. 10.12,

Pm

[1 ]

l-S · = 31R2 RR S -1 = 31R2 RR -S-

(10.91)

This yields an estimate for the efficiency in the motoring region, O < S < 1, 1'", 'II",

}"( 'Sí HI.O!'K (lI'

vnl i'.HI'." t' b UI!'
-2

Fig, 10.11. Torquejspeed curves of a wound rotor induction motor

I" '(' I. LOII,

dI/tiNI ,",(,1,

S>1

/

phase sequence

"í

IlIn i ll

Reverse braking / / ----

with reverse

rol.aÜOll

l 'q l\I"III'i', til"

-1

Operation

wvnag<~

'1'11('

mM

mpO

--/-------

11I;\,xiUIIIIll torque but only influences the pull-out slip. A wound rotor motor

I',V

/

(:~-/S~=0,2

whidl is a measure for the overload capacity of the motor. The pull-out I"rqllv , Eq, (10.86), shows that the rotor resistance has no effect on the

rot.or

-0,5 ~/

-1

I"

1 + (7 2 fo

Generator

.R2

passes through a peak value; the curve, plotted in Fig. 10.10 c, has a similar shape as the torque characteristic in Fig. 10.9 b. This points to the reason for the decreasing torque of the induction mo­ tor ato higher slip frequency: Due to the rotor leakage, the rotor current is increasingly lagging, causing the rotor current wave to be shifted and eventu­ ally become orthogonal to the stator flux wave. Hence the torque is reduced whereas the stator and rotor currents continue to increase asymptotically. TIl view of the stability of the drive and the rising losses, steady state opnat.ion of the motor makes only sense below pull-out slip. WhCll declaring the point, where the power factor is optimal, as the nom­ ill:!.1 op,'rat.ing point, Sn/Sp = fo, the ratio of the peak to nominal torque

-

, RS= O

USO

w2 L 2

F i g. 10.10. Simplified mode! of rotor circuito (a) Circuitj (h) Locus of current phasorj ( c) Real current component

_

o

S
U20

'1I1' JI

W

W1

2

U20~O

'111'1.0

!:!L= Us

W

185

H.:;

{'II

W

(10.92)

I - S = -< l , WI

w..I1 11f1 /'", "1" "'11.11111 : I :~ ;~ 1 ~' · "(·l' n, I,,, I' ! .'í

()

186

10. Symmetrical Three-Phase AC Machines

PR _ _1_ =

rJg = Pm - 1 - S

Wl

w

10.2 Induction Motor in Steady State

<1

(10.93)

which means that the efficiency even of an idealised motor ü; lowered in proportion to speed, whether the reduction in speed is caused by additional rotor resistors or by reduced stator voltage. A similar situation exists in the generation region, where the losses rise also with the magnitude of the slip. With a real motor, having primary copper losses as well as iron and friction losses, the efficiency will drop below this idealised case.

iR

(JRLa

PR ~

Inputpower

RR

RR

S

~

Pv

Real

Model Rea1

for a = 0.1,

no = 750 l/min

--s;

fo

= 0.22

0.20

0.32

0.30

fs!

fo

= 0.22

0.30

0.32

0.40

cos 'Pn

l-a rra = 0,90

0.90

0.82

0.84

mpo mn

1~ 2 a

2.30

1.82

2.0

Sn

Pm ~ Mech. power

The explanation is, of course, that the induction motor functions like a mechanical clutch, where the product of speed difference and torque causes an unavoidable power loss. The definition of efficiency in the braking region at reverse speed, S > I, does not make sense, because electrical as well as mechanical power is supplied to the rotor and converted to heat. With high slip control schemes for short duty drives the main problem is the dissipation of the rotor heat loss. ln this respect a wound rotor motor with external rotor resistors is preferable, however at the cost of moving contacts. These results give ample evidence that an efficient adjustable speed AC drive can only be realised when either the slip power is recovered by some means or the stator frequency Wl is changed to track the mechanical speed, thus restricting the rotor frequency to small values. 10.2.3 Comparison with Practical Motor Designs

The simplified theory of a symmetrical induction motor in steady state points to the totalleakage factor (j as the most inflllcntial panl.111 ctcr of the machine. As was mentioned before, (j depends on thc :-;11,,1'(: or til!: Klots and increases with lhe airgap. Tlw seal,' or til ,: círc nlar curwlll lonl~; iH d('I ,"l'Jllill"" lIy Uw i(!.:a,] Ilo.. load ('111'1'('111 , l .,." wh i(' h lI hw "",)/,11,11-1 1.0 a la r/ :" ,' x!.,' .. t "II 1,ltI' II i 1'1','a." , 1",,1' 11I1'c!uwi(';d 1" ' /l i,I II " ',

Model for a = 0.05, no = 3000 l/min

)

Fig. 10.12. Power balance in the rotor

'1 1'1,(1

The scale of the normalised slip/torque curve, Fig. 10.9 b, is fixed by the pull-out slip Sp and the peak torque m p ; in the interest of good efficiency in steady state, Sp should be as small as possible. The peak torque is also affected by the leakage factor. ln Table 10.2 some of the characteristic parameters obtained with the simplified model are compared to those of standard motors, as taken from manufacturers lists on medium size (> 100 kW) 50 Hz motors, for a two pole­ and an eight pole-machine. The comparison indicates a fairly close agreement, at least from the standpoint of the user. Table 10.2. Comparison of idealised model with pararneters of standard motors

Rotor lasses

,, ,, 1 RR(S-l)

,'r'

1,111' IlI r j ',!I'I' LI 1/1, 1'/ ',"1' wil,11

IlJlrll. l l",)"

/I lrll" Í, WII

187

1',,11' 1II;I('I,illl': ; (Ir "qllit.l

= 2.35

10.2.4 Starting of the Induction Motor

When selecting suitable values for efficiency, power factor and overload capac­ ity at the nominal operating point, the starting performance of the motor is unsatisfactory; this is apparent from Fig. 10.13, where characteristic variables of a motor with a leakage factor (j = 0.05 are plotted against normalised slip. The conditions for acceptable performance at the nominal operating POill(. are:

a) The normalised slip at the nominal operating point should be about SnlSp ~ ..j(i to achieve a good power factor and adequate overload ca­ pacity_ b)The need for good efficiency rJn < 1 - Sn calls for a minimal value of nominal and, hence, pull-out slip. With motors above 100 kW, a nomill;d slip Sn ~ 0.02 is realisable; the pull-out slip would then be about SI' ~ O,lO. This llw:tns that an indllctioll motor with a short circuited, consta.lli f':" :iist,nIlC<' rotor wOllld llitvo to Ht.ari fnllll fi lalldstill (8 = 1) at SI Sp ~ 10 whirll c'o[' n 'R!JOlld N !.o n ll op e ra.l.ill ~: POilll, ",I, 1.111' I"i/',.

I O.H. ' 1'1((' il l,lIrl.ill/ \ 11I'1'f'mfll ll llc'l' ;11, S

rll':hl, 01' !.h, ~ circula,!' I , "!a n rll l'I,(' , j!1c·t1

filJ'

"y

dia,/~nulI ,

10.2 Induction Motor in Steady State

10. Symmetrical Three-Phase AC Machines

188

IS mM I 'mn

with Bp

COS <{>

Sn

Is

R==O

3

Snc

", ,

mn

2

-==-'0= 0224 Sp '

,,

l-O

COS <(>n =~

, '........ ........

== 0,9

mM

. . . . ---_'!!.n_ Nominal operating point

, 0,5

~

~ ~ 2.35, mn

0.66,

signifying a decisive improvement over the case with a short circuited ro­ tor winding. With larger motors the starting resistor may have the form of a tank of water with enhanced conductivity into which the electrodes connected to the sliprings are progressively submerged. b) A more elegant method requiring no moving contacts is the utilisation of eddy currents in the rotor bars leading to a concentration of the current in the outer portions of the conductors at higher rotor frequency (skin effect), thus increasing the effective rotor resistance at larger slip. As a consequence, the starting torque is improved with an accompanying reduc­ tion of current, while in normal operation at low slip the desirable features of the motor with short circuited rotor are maintained.

0= 0,05

~

coscp

Sn

I Sr

COS <{>n

= 1.0 would show the following starting performance at B = 1

- I ~ 3.2,

IS

4

2

3

4

5

Yci

189

S Sp

R2

Double cage

Fig. 10.13. Characteristic curves of induction motor with constant rotor resistance

-1 I

Is Sn

~

1

-lU

~

4.5 ,

cos cp

~

0.095 ,

~ ~ 0.47, mn

would be very poor, since low torque is combined with a high predominantly reactive current. When the motor is fed from a grid with a relatively large internal impedance, there could be an appreciable volt age drop, reducing the current linearly but the torque with the square of the voltage. This may even lead to a situation where the motor does not start under loadj the protective devices would then have to disconnect it from the line to prevent damage due to overheating. The conditions are particularly severe in small grids such as on board ships, where the generators are not by orders of magnitude larger than the largest motorsj this also calls for fast volt age control of the generators. Two options are available to improve the starting performance: a) Use of a motor with a wound instead of a cage rotor which permits the rotor resistance and, hence, the pull-out slip Bp to be temporarily increased at will, creating optimal conditions for starting as well as aI; operational speed. The rotor resistance may be shorted in steps by COllt.a.ctors controlled in a fixed time sequence or depending on stator CHr)'(!Il(. 01' spced (sec also Sect. 3.1.4). The larger part ofthe power losses dmil'l'. st.il.l't. o<:(',llrs ill Lhe startiug resistor, i.e. outside the motor, so UlilJ 1.11<'1'1' ;1,1\' 110 addit.iollal coolillg problC:lll~;, Wll(~u t.\l(~ operat.iIlI~ s()('(' d 11 11:1 (" 'V II 1'(· n.cll('d, i,ll(' slip-rin l(S call 1>c s!lol'l,,'d illt.l'I'l lII. lly wit.h ii ~ (I('('i ll. l 1111 '1' 111 11 11 11 111 wl li ll' t.l1(,'IH'II :d ll': i 11.1'11 lil'l,('" 1.(, !-tv" i ,1 LI,, ·, 1I1111< 1I'(' WII II' y f l'lt- Ll oll (lI ,,1 W(' 1'.I .

fi. 1""1., ,,· wl l. "

L2

(10.94)

/I

o,() r"

II I.I t l' l. I "I "

a

b

R I "R2

LI "L 2 Fig. 10.14. Cross section of eddy current rotors and equivalent circuit

There are various designs which tend to emphasise the effect of eddy currents in the rotor bars. Qne solution employing two cage windings is shown in Fig. 10.14 a. The starting cage (1), being of higher resistivity, is placed in open slots immediately below the rotor surface and exhibits low magnetic leakage; as a consequence this winding produces a high pull-out slip BpI. Because of the low leakage this winding dominates at high rotor frequency, i.e. at low speed. The operational rotor winding (2), in contrast, is characterised by a larger cross section of the conductors, possibly employing also material of higher conductivitYj the leakage reactance is increased because the bars are now embedded in the rotor iron. Clearly, this winding becomes mainly effective at low slip fre(]IlPIlcy, Figure 10.14 b shows the equivalent circuit with the two rotor Willdilli\S COllIH',c[,cd in parallcl. 1\';<:;1.11:';1' 01' II" n~, DI ~ L~ 1.111' I.wo willdiJlgs ar<~ cOlllplemclLtary. '1'11(' ('tlnl·1I1. 1",'11:: ,,(' !.II(' 1IIi1,dlilll: iH, "r ,' 111 II':!" , II<' 1011/':('1' ;1. ('irdc, 'I'ypic
1'111

I,. /

,'. 11

III.

111

..

,"iY"lIlldrical Three-Phase AC Machines

10.3 Induction Motor with Impressed Voltages

191

M

11111

~

,/

'Sn

?

mM

---------

mn

Should the motor be fed with sinusoidal symmetrical three-phase voltages of constant frequency and loaded with constant torque, the transients will eventually decay and the steady state condition discussed in Seet. 10.2 will emerge, where alI quantities in the stator are periodic funetions with the angular frequency Wl' A better insight into the dynamic behaviour of the machine is obtained by choosing a moving system of coordinates, where these steady state oseillations disappear. A suitable choice of coordinate system, independent of the waveform of the voltages, is that defined by the vector of stator voltages, Eqs. (10.34), (10.38),

'Y,S(t) = USl(t) + US2(t) eh + US3(t) ej2 --y

Jo'i(.;. 10.15.

(10.95)

1 S

Sr

= UI2(t) - U23(t) e-h = Us(t) ej

Characteristic curves or induction motor with eddy current rotor

À(t) ;

its instantaneous angular velocity ('llrves of an induction motor with double cage rotor are seen in Fig. 10.15; ;\ wmparison with Fig. 10.13 indicates the improvements achieved. Another d(~si~1I of a rotor winding with strong eddy-current effect is a cage with narrow I)ar,~ extending deep into the rotor iron; this is also seen in Fig. 10.14 a. WIt(~n line-starting larger motors it is sometimes necessary to reduce the ílllllSIt ClllTCnt by lowering the stator voltages or by using only part of the ,·;L:!i.(I/, willdillg, thus increasing the leakage factor. ln both cases the starting (,(lrqll(' is fllrLllCr reduced which limits this method to starts with light load. ' I 'II\' ,;i"'lll('sl. possibility for reducing the voltage during the start is, of course, " y 1/', 1'I '( '(lIIIJ{:ction of the stator windings. It entails a reduction of voltage 1\11<1 "111'1('11(, hy 1/../3 and of input power and torque by 1/3.

d>'

corresponds to the angular frequency of the stator voltages. Should the mo­ tor be supplied with sinusoidal symmetrical three-phase voltages of constant frequency and amplitude, Eq. (10.60) again becomes yalid

'Y,S (t) = 3 y'2 US ej

SlIpplied by Impressed Voltages of Arbitrary Waveforms Arl.,'!' I.his discussion of the steady state with sinusoidal symmetrical three­ voltages let us now return to the more general model of the symmetrical i11<1 Ilcl.iol1 motor in dynamic condition, when torque and speed are varying. Wc assume that the stator is supplied by impressed volt ages having, in prill('iplc:, variable frequency and arbitrary waveforlll , as long as Eq. (10.37) is rlllfillNI. lu fact, due to the isolated neutral point o[ the motor, even a viobl.ioll of thi::; eondition would be of 110 COllS(:qll('l1n:; iI. would simply mean I.ha!' I.he llel1tral of t:he WilldillgHaSSlIm(','; a dilf(:n:lIt pol.( 'lllial aoS the neutral "r 1.11(' slIpply vol(,ap;<'s '/1',' :1, '/},:;2, '1/.,'::1' AlHO, til(' H'HIlI!,!! IlIlly III' g'(:n('raliscd to rI. lill/,/,Iy h;wjl'l ~ ('()II~I.;l.Id, :tlld (:'111111 illl.('!'ll ld illlj"'t!HIII'\'H iII (';LeI. pha.;.;('; 1.I":y ('",ii.! I... ill<'llI,[(',1 iII U.., ,, 1.11.1."1' illlJII'd lu""' 1I (1,' ,'1 , ,, :: 1,(,) "r 1.111' 111"1.,,1',

(Wl t+Tl)

2

(10.97)

'

resulting in

us(t) '(I.;, I Hd Ilction Motor

(10.96)

Wl(t) = dt

3y'2

= -2- Us,

>.(t) = Wl t + TI ,

(10.98)

where U s is the RMS line-to-neutral voltage. All electrical variables are now defined in this coordinate system; i.e. they are transformed as they appear to a moving observer positioned at angle >.(t); by making use of Eqs. (10,7, 10.10) we define

= is(t) ej (-À) = lis(t)

is(t) e-j À

phasc

(10.99)

;l.Ild

iR(t) ej ê e- jÀ

= iR(t) ej (e+ê-À) = li R(t)

(10.100)

as the current vectors in moving coordinates. With sinusoidal symmetrical

I.hree-phase stator voltages of fixed frequency and constant load torque, these qll<1l1ti (,ies are constant in steady state, When multiplying Eq. (10.50) by e- j À w(~ JiJld ,

/(, " 1" . 1'

",

.1

\

Ih -;

I { ",. . ' .II.

t:

"À

J

1/'(1

d( 1./ "1'.1 , III 1

',\

)1 ' J

'Il.

i s r-· >--

(10,101 )

192

10. Symmetrical Three-Phase AC Machines

10.3 Induction Motor with Impressed Voltages

which is transformed by substituting

(10.102)

Developing the expression dI' · dis -jÀ iS -e =--

dt

dt

+.JW1 1.Zs -

(10.103)

correspondingly,

(10.104)

and inserting Eqs. (10.103), (10.104) into Eq. (10.101), the stator voltage equation in the moving reference frame results,

(10.105)

~ (L R 1iR + Lo 1is) + j (W1 -

w) (L R 1iR

+ Lo 1is)

mM (t) = ~ Lo 1m [is (iR e j ê)*] = ~ Lo 1m [lis ei R )*]

Lo mM (t) = -2 1m 3 (72 L s L R

mM(t) = =

These equations may be simplified by substituting the flux vectors, defined in Eqs. (10.25), (10.32) which are also transformed into moving coordinates, ./.

-j À

j 'f..R = ./. !:f.R e

1./.

= Ls I 'is + LI' o iR, = LR li-R

(ê-À)

(10.107)

+ L o_s li .

(10.108)

Hence the current vectors are

1·z s_- -Ls 1- (1./. -s 1· _ L1 (1./. -

'I-'

(7

ZR- - -

-

(7

'I-'

-R

R

1./.) + (7R -R 1 1./. ) 1+

- -1-

1

---

'I-'

(10.109)

'I-'

(10.110)

(7s -s

Inserting Eqs. (10.107-10.110) into Eqs. (10.105, 10.106) results in d 1./.

T' s -d-1:...s t

T/~ W 1",\" ,

+ [1 + J. W1 T S'J

111.4) lU-H -I-

rI

I· j

() 'l-'s - - 1- o 1 4) I = 1" l+(7R - I. . .<; Us t ,

1./.

(WI - w) TRJ ''1/'/1

(10.114)

(f ; ,'

-s

1 ) (1 -R 7jJ - -1- 1 7jJ ) J . 1+

- -1- 7jJ 1 + (7 R -R

*

(7 S

-s

(10.111)

~

Lo

3 (72 Ls L R

~ 13

(7

(7

1m [17jJ 17jJ*

~ 1m Lo

-R

-S

+

(1

1

17jJ 17jJ* J

+ (7s)(l + (7R) -R

(10.116)

[17jJ 17jJ*] .

-s

-S

-R

This is inserted into Eqs. (10.52, 10.53) for the mechanical motion. To arrive at dimensionless real equations, the transformed flux vectors are nOw normalised and splít in real and imaginary parts, representing the components that are parallel (x) and orthogonal (y) to the voltage vector. The flux linkages in steady state operation with sinusoidal voltages and nominal frequency, as indicated on the nameplate, and neglected stator resistance serve as flux references, 3 .12 Uso [xs(t) 17jJ -s (t) = -2- Wo

+ j ys(t)] ,

(10.117)

3 .12 17jJ 2 Uso Wo [XR(t) -R (t) =

+ j YR(t)] .

(10.111'\)

Tlle magnitude of the voltage vector in Eq. (10.98) is also referred to Illagnitude at nominal operating conditions,

3.12 Us(t) 'Ils(t) = -2- Uso Uso .

I,IH '

(10.] I!))

II. is llutccl that no restrictions with regard to waveform or frequency of t.hc

J

I .1

[(1 7jJ

(10.115)

= O.

(10.106)

'f..s = 'f..s e

(10.113)

are transient time constants. The expression for the electrical torque, Eq. 10.48, can be written in the form

The rotor equation Eq.(10.51) is transformed accordingly, with Y.R(t) == O,

1./.

LR

Since only the mixed terms of the product contribute to the imaginary part, this reduces to

d

+ dt (Ls 1is + Lo 1iR ) + i W1 (Ls 1is + Lo 1iR ) = us(t) .

RR 1i R +

(7

the torque is independent of the coordinate system because the inverse trans­ formations with e jÀ are cancelling. Hence, using Eqs. (10.109, 10.110) we have

. 'ê) d 1i . 1. d(lR e) -dte -jÀ = ---=!}, dt + J W1 iR,

Rs 1is

Ls

(7

T~ = Rs ' T~= RR

l d(. _j>..) di s _ jÀ .d>'. -jÀ d is --=-zse =-e -J-zse dt dt dt dt -

193

'1/' "

0.,

(10.112)

voH'lgcs lmve been imposed so far. WiI.h 1.11('1:(' ddillitiollS thr V()Jl.al~(' (~q1\atiolls (10.111, 10.112) and thc e.' ·

:i/,;II,()I'

pr":ir:i"ll

rOl'

LIli'

I,")'qll('

1-;'1' (IO.llli)

: 1.':: :11111<'

1.11(~ r()llowillf~ fonn

I !11

10. Symmetrical Three-Phase AC Machines

1'8 ~ (xs + jys)+ [1 + jWl T;J

) = Wo T -U Us + J'YR s 50 ' I

+ (TR

~ (XR + jYR)+ [1 + j

(Wl - w)TRJ (XR 1

= 2 (1 + (Ts)mpo (Ys XR -

eo

(10.120)

, dxs T s - - = -Xs dt

+ jYR)

.

+ - - - YR, 1 + (TR

XSYR) ,

(10.122)

, dXR 1 T R - - = ---Xs - XR dt 1 + (T 5

1 W2

SpR Wo

Pull-out slip at nominal frequency and Rs = O

RR 1 SpR = Wo (T L R = Wo T R '

(10.123)

(10.124)

T

_ Jwo mpo

(10.125)

m -

instantaneous normalised frequencies

"t.at.or frequency

Wl Wo lIlecltanical rotational r['(~qll eIlcy (speed) W

Wo I'U lor

frcq U(~llcy (,);\ (,)0

WI

W (.III!

(IO.1 2Ci)

1

(10.127)

(10.128)

(10.129)

, dYR 1 1 W2 T R - - = ---Ys - - - -XR -YR, dt 1 + (TS SpR Wo

(10.130)

d

C~o) = 2 (1 + (Ts)(Ys XR .

eo d (eco) Rs 1 Sps = Wo(T Ls = woT; ,

1 Us(t)

+ -- -YR,

Tm

M(~chanical time constant

:1.11<1 lhe

1

+ ---XR + - - 1 + (TR Sps Uso

, dys 1 Wl Ts = - - - Xs - Ys dt Sps Wo

wltcre mpo is the nominal peak torque for Rs = O, as defined in Eq. (10.86). With the following abbreviations for

Pull-out slip of rotor-fed motor, nominal frequency and RR = O

1 Wl -Ys Sps Wo

+-

(10.121)

+ (TS

195

six real differential equations emerge for the direct and quadrature flux com­ ponents as well as motor speed and positionj is an arbitrary angle of reference.

+ J YS) = O ,

- -1-- (xs

mM

+ jys)

(xs

1 ( XR - -1--

TR

10.3 Induction Motor with Impressed Voltages

W

mL

Xs YR) - (w, c, t) , mpo

(10.131)

(10.132)

Wo dt Wo These equations may be represented in graphical form by the block dia­ gram in Fig. 10.16, where the integrators with feedback are redrawn to form unity gain lag elements. Due to the rotational symmetry of the machine the angles e(t), À(t) do not enter the right hand side of the model equations Eqs. (10.127-10.131) with the exception of load torque mL that may depend on the mechanical angle. The block diagram describes the dynamic behaviour of the symmetrical motor, fed by impressed stator volt ages independent of waveformj the mag­ nitude us(t) of the volt age vector is given by Eq. (10.119), its instantaneous angular frequency wdt) by Eq. (10.96). As special case with Us =const., Wl = dÀ/dt = const., the transient condition of the motor operating on the symmetrical three-phase supply results, where U5 is the RMS value of the line- to- neutral volt ages. Because of the transformatíon into the voltage based coordinate system moving with stator frequency, the steady state values of the flux components and speed, assuming constant load torque, are constant. The steady state values are derived by setting the derivatives in Eqs. (10.127-10.131) equal to zero; the resulting system of algebraic equations is equivalent to the results obtained in Sect. 10.2. Of course, under dynamic conditíons the Hux components are no longer CO!l:;tant beeause there may be transient flux waves which are linked to the Hlator 01' rotor windings and, hence, move with the angular velo city Wl or L,)~ «.I I é.) rdat.ive to lhe c()ordjnat.(~ s'ys t.
111I

II). :~ yllllll( , Lrical Three-Phase AC Machines

l1., {I)

10.3 lnduction Motor with Impressed Voltages

OJdt)

I

.!:!.§...

USO

197

mL mpo

Uso

Oh

úJ

úJo

úJo

XSYR

I--=,

mpo

.!!!.b...

Fig. 10.17. Simplified block diagram of symmetrical induction motor

mpO

the fiux components, for instance in the statori by eliminating Ys we find mM mpo

, 2

Ts

d

Xs

'

1

dxs

Wl

2 + 2 T s dt + [ ( Wo ) 2] di2 1 + S;s

xS

=O,

(10.133)

with the eigenvalues I+-t---ó+ - - - - I

YSXR

81,2

.!:!!.... OJo

OJ2

Wõ "' i ,~. I O. LO . lllock diagram of symmetrical induction motor

r.."

h.v illq .r<'.'i.'ied volt ages of arbitrary wave form

.J1CIII.S OL'Lppropriate frequencies; as a result there are torque and speed iill.ioll S.

= Wo ( -Sps ± J. Wl) Wo

(10.134)

hence the pertinent transient is a damped oscillation with the angular fre­ quency Wl, belonging to a fiux wave that is linked to the stator winding and thus remains fixed in space. The damping is achieved by the dynamic short circuit through the volt age source at the stator terminals. Likewise the lower feedback circuit in Fig. 10.16 is found to be described by a differential equation with the eigenvalues

t. 111111< Vi l.!

WII( 'JI ill.'i pccting the general structure of Fig. 10.16 it is recognised that iI. (': UI I>(! I"(!
illvolved , which is due to the fact that the currents in the rotor winding are "I:lglld.ica lly induced rather than impressed through a galvanic contact; also, 1.11(' all[.';Jc between the stator- and rotor-ampereturn waves depends on the op(!lal.illg condition, whereas it is essentially fixed by the field- and armature­ i i , ( ~S iII th e case of a compensated DC machine. The presence of AC quantities ;t./ :;o 1.( ~11(1s to blur the view, even though this has been alleviated by the H'( I illa Le transformation. W h iII! 110
83,4

=

Wo

(

-SpR

± J. W2) Wo

(10.135)

The associated physical effect is a flux wave linked to the rotor winding which moves with slip frequency relative to the coordinate axis. Considering now the special case Wl = W2 = O, when the mechanically locked motor is fed with direct voltage, and taking the dashed cross-couplings iI/to account, two new non-interacting feedback loops are formed, where each (~lHbraces stator and rotor. The differential equations for transients in the :,:-axis may be combined to ,

,d?-xs

TsTR

-2-

dt

'

,dxs

+ (Ts +TR ) -

dt

+axs = O .

(10.136)

(·~igenvalues are negative real, belonging to aperiodic transientsi a cor­ IT:'; POlldiug equation is valid for the quadrature axis of the machine. Clearly I.II1'S(: ('qllal.iolls eorrespond to those of two-winding transformers, describing I.I/\' d l'(,()l/l'bll" ilW \( ~ l.ic illl.(~rad: ions in the two axes ofthe machinei under the ,dnl/lIlI't! 1,('1:1, ('llIlIlil.i"IIH ;1.11.1 ill vil:w or I.lw voIt.a.ge-based coordinate system,

!10th

19i;

10. Symmetrical Three-Phase AC Machines

10.3 Induction Motor with Impressed Voltages

5 p = 0.10,

'm (!o)

(01

(00

~

jO,5

(02 '=

'

(00

0,05 '=

const

0,15

-0,1

Re (!o)

Fig. 10.18. Root locus of induction machine for at different stator frequencies and speeds

W2

= 0.08,

= 0.05 =

the :/lux in the quadrature axis must be zero, Ys = YR = O, hence there is no torque. This is in agreement with the practical experience of a symmetrical induction motor with DC excitation at standstill. When analysing the block diagrarn in Fig. 10.16 more closely, i.e. for Wl, W2 f:. O, the mutual interactions of the two damped oscillators, coupled dircctly and through the speed, must be taken into account. This leads to a complicated resonant system of 6th order, the response of which is diflicult 1.0 assess. However, assuming Wl =const. and W2 ~ const., i.e. large inertia, l.hcre are considerable simplifications because the multiplications with Wl, W2
= 5 pR

= 5p ,

(J"S

= (J"R = (J"o,

T;

= T~ = To ,

(10.137)

tbe characteristic equation pertaining to Eqs. (10.127-10.130) is

[(To H 1)' - (1- u) + ;; (:;)']

+

1-

(J"

52 p

(Wl

+ W2)2 W

2

o

[(To H 1)' - (1- u) + ;; (::)']

_ O -.

(10.138) Tlle

Wo

= 0.05

= consto

(10.139)

consto

W2

5 ps

W2

The result is plotted in Fig. 10.18 in the form of a root locus in the complex s - plane, omitting the conjugate branch of the locus. The shape of the curve indicates that there is little interaction between the two oscillators in Fig. 10.16 for Wl » W2, approximately leading to the "decoupled" eigenvalues mentioned before; on the other hand, the interactions are strong at low val­ ues of the stator frequency, Wl ~ W2, resulting in slowly decaying transients. Figure 10.19 depicts the decay of a :/lux component at different stator fre­ quencies, assuming the sarne initial conditions; the transient torque is also shown. There is a remarkable effect of stator frequency and speed on the dynamic behaviour of the nonlinear planto Therefore the induction machine, when operated in a wide speed range, represents a very difficult control plant, which calls for special control procedures; some will be discussed in Chapo 12.

j1

1

(J"

199

solnl.iOI\S or I.hi~ (~q1l'ati()n, i.e. th~~ ('igcllvalllf's or (.h(~ systf:tn, havc heen

cOlllplll. ed 1l1l111 \'rir:dly 1(lr di/fcJ"(~nl. valtll's or :JI.III.()r rn~qll( ~ l1C:y WI, a.:;S lllllill~ !.II( ' fiol[\l wl l'IJ'. 1,III'hu lC'!..'r:·:

tlxs

Wo

= 0,05 = const tlmM

-

w1

m~i Fig. 10.19. Flux and torque transients for at different stator frequencies and speeds

W2

= 0.05

= consto

As a qualitative test for the validity of the mathematical model of the induction machine, which was derived with considerable simplifications, a line start at nominal volt age and frequency has been computed by numerical intcgration of Eqs. (10.127-10.131); the load torque is initially zero but, aftcr stcady state speed has been reached, half the peak torque is applied as load. Tlte results are plotted in Fig. 10.20, assuming a low value of inertia so that I.!te start is completed in a few periods of line voltage; the other parameter:; are the sarne as before. Besides the flux components in direct and quadrature direction, the stator Cllrrents are of main interest. The associated vector is produced by inverne 1.1':tIlSrolllliltion with the help of Eq. (10.109)

200

10. Symmetrical Three-Phase AC Machines

10.4 Induction Motor with Unsymmetrical Line Voltages in Steady State

201

xs{t} I I I I I I

~ H ~.

II

t' 11111i •

"'o

a

IIIIIIIIII'~(\.~----------t-------

'Idlll" ~ II I o ,

w

~ "'o

I I

!~~'Ys(t) 151

W

"I

I

I

I

I

I

I

I

Wo

I; \ "

/! /)

\ mM

1m \ po ~

(.

r·\ ../·.. . ·------­

,/

"

.!!L(t) /

'

n

r./ úJo .-.-1. ,_,_._,_ './ ..... - . 1'­ mM

~.-

I

I I

mpo

Fig. 10.21. (a) Simplified block diagram of induction machine, (b) calculated transients

----­

/'-

mL mpo

I

Sta/1 at no load

b

Load forque applíed

1

m L ="2 mpo Fi,,, . 10.20. Simulated starting transient of induction motOr wil.lt I
= li-s e j wd =

iv (t,)

,.

:~

12

~ .- -

2

Uso Wo (7 Ls --

[(

_1_ [1'IjJ (t) _ _1_ 1'IjJ ] (7 L s -s 1 + (7 R - R xs -

_1 1 + (7R

XR)

+j

ejw1t

(YS _1 +_ 1 YR)]

ejw1 t .

(7R

(10.140) Wil,h

r~ q.

'i:>i (l)

(10.5 5) follows

= 1Re [is(t)]

J :~ (/ '~'II (Ai l ) ri

/, ::

f(

:/:,'; -

1 - Xli ) CO:-) W I/' -1 I {r ll

eIS

1 (T

li

lI n

) Sill Ü)lt'l, c

(10 , 111)

This is also plotted in Fig. 10.20; it shows the characteristic large inrush current which, as the motor approaches synchronism, settles down to the small sinusoidal no-load current until the load torque is applied, The currents in the other stator phases are similar. There are strong alternating components of torque which also affect the speed, even though with reduced amplitude because of the mechanical inertia. If the inertia is predominantly on the load side, the shaft and the mechanical coupling may be severely stressed during the start. The speed, as a function of time, follows a similar curve as obtained in Sect. 3,2 when the electrical transients were totally ignored, Fig. 3.12; this explains also the absence of the overshoot in speed which is present in Fig. 10.20, As an intermediate step between the quasi-steady-state model and the more complete model of the induction motor, the heuristic block diagram in Fig. 10.21 a may be helpful. It differs from the purely mechanical model by the inclusion of a lag T~, following the steady-state torque-slip-function. For small values of slip, where the torque function is linear, this clearly re­ sembles the block diagram of a De machine. A comparison of the starting transient computed with this model shows good qualitative agreement with Lhe transiellts obtained by integrating the Eqs. (10.127-10.131). Of course, Lh e os(:illa(,iom; Hl1perimposed on speed and torque cannot be represented but I.I)( ~ lJl( :all viI,I'WH ! ~ i'( : wdl luo
202

10. Symmetrical Three-Phase AC Machines

10.4 Induction Motor with Unsymmetrical Line Voltages

10.4 Induction Motor with Unsymmetrical Line

Voltages in Steady State

U5 + --

3"1 [U -51

+ eh -52 U + e_53 -h U ] ,

10.4.1 Symmetrical Components

U5 - --

3"1 [u -51

+ e -h -52 U + e_53 h U ] ,

For intermittent duty drives, such as cranes or hoists, where simplicity of hardware is more important than high efficiency, induction motors with un­ symmetrical voltage supply may be used. By way of example, a phasor di­ agram of sinusoidal but unsymmetricaJ line-to-neutral volt ages is shown in Fig. 10.22 a. Due to the isolated neutral of the symmetrical stator winding, the sum of the instantaneous phase voltages is still zero, Eq. (10.37); hence the phasor diagram forms a elosed triangle,

fl.o

e -fYl}se -jy l}s+

b

fl5-

u5 ++

U 52 = e-h U 5 +

U 5-

!)S­

c

+ fl.o ,

211" + eh U5 _ + fl.o , 1=3'

=t

U 12

[U 12

(10.14G)

eh U 23 ]

-

+ U 23 + U 31 = O

u 23 = (10.143)

ehfl5++e-hU5_+fl.o,

where U 5+ is the positive-, U5- the negative- and fl.o the zero-sequence volt age component per phase. The symmetrical components fl5+' U 5 .- form two symmetrical three-phase systems oE oppositc phase-sequencc. Solvilll( Eq. (10.14.1) yi(dds

(10.147)

is valid by definition. General1y the use of line-to-line volt ages does not ex­ elude the appearance of an arbitrary zero sequence component fl.o but, as seen before, it vanishes with a symmetrical load. If the motor is fed from a line possessing noticeable internal impedance, the symmetrical components of the terminal volt ages depend also on the load, i.e. the operating state of the motor. Should the voltages U 12' U 23' ~1 form a symmetrical three-phase system, e-j,,!

U 12

(10.148)

,

the result is

-

U53 =

(10.145)

Two of the line-to-line volt ages are sufficient to determine the positive ,Uld negative sequence components, because

It is known, that an unsymmetrical three-phase system may be de com­ posed into "symmetrical components" defined by

=

+ e-h)]

and correspondingly

Fig. 10.22. Decomposition of unsymmetrical balanced three-phase system

into symmetrical components

U 51

v

=0

-- 3"1 [U -12 - e -hu] -23

~s+À

~S2

!L23

!L12

The sarne holds naturally if the stator windings are Ll-connected.

!)S1

O.

1[U' -51- v- -U- 52 -e-h (fl52 - u 53) + u 52(1 + eh -' '--v---'

(10.142)

ejY!)s+

(10.144)

The expression for the positive sequence component is similar to that for the time dependent volt age vector, Eq. (10.34). Since the volt ages between the motor terminals and the isolated neutral are not directly available (the motor may be Ll-connected), it is appropriate to introduce the line-to-line voltages instead. By expanding Eq. (10.144) a we find, analogous to Eq. (10,38),

U 5+ =

fl 51 + U52 + U53 = O .

a

1 [U 51 + U 52 + u53] =

=

203

U -5+ -

1 e- j7r / 6 -12 U J3

-U - -51'

U-O - 5- -

.

(10.H!))

'l'hc case of a symmetricaJ induction motor fed by unsymmetrical snpply vol (.agci-i is fully containecl in the thcory developed in Scct. 10.1 bllt til<: sllb-· ~Cq'I('Ilt dis(".I\~;:;i()Jl oE som(~ cX
204

10.4 Induction Motor with Unsymmetrical Line Voltages

10. Symmetrical Three-Phase AC Machines

u (t) = 3 v2 [U U' e-i Wl tj -s 2 -s+ ei Wl t + -5-

(10.150)

,

a result, that can be interpreted as the superposition of two constant vectors rotating with equal angular velocity in opposite directioni hence the vectorial sum 1!cs (t) follows a periodic elliptical path, as seen in Fig. 10.23.

3..J2 U

2-

-s+

e jW1 t

components of Eq. (10.150). As long as iron saturation is ignored and the speed may be assumed constant due to a sufficiently large inertia, there is no interaction between the two symmetrical three-phase systems so that linear relations exist between volt age and current components,

Is+

= ys+ U s+,

Is-

3{2 -jw1 t 2- Us_e ­

w

= S and

Wl +W

ldS(t)

1

J

---­

Y

(10.151)

í (t)e jo = 3v2[I e- jw1t j , 2 -R+ eiw1t+I* -R-

(10.152)

-lI

while the rotor current vector in rotor coordinates is, with de:/dt COllst. ,

+ 1* e-j (wd- w ) tj -R-

=W

"(J I./tl. i 111 ': wi!."

I'c'I '(II, 'lId l lll \ 1111 K d('II: d !.y WIL VI'1i iII

1+

1

j

Wl

Ls

.1

2-S

JuS;: 1+j 2 S

S

= 3 Lo 1m [15+ DH + Is- IR-l = mM+ + mM-

mM

mM+

mM­

mpú

mpo

mpü

-=--+-­

5p

(10.153)

I l
(10.156)

.

These two components of the mean torque may be derived by the sarne reasoning as shown in Sect. 10.2, because each is produced by a symmetrical three-phase voltage system. ln accordance with Eq. (10.85) we find

~ .

l+jl~ 17 sp . s Ls 1 + Jsp

2

'I'hc spced w i::; counted in the direction of th<: positive sequClIce field. Rence 1.1)(', frcqllcllcy 01' tlte positive and negativ(' ;' ('
(10.155)

When substituting Eqs. (10.151, 10.152) into the torque equation (10.48), four terms arise, two of which are alternating torque components having the frequency 2 Wl i contributions to the mean torque are only produced by interactions between stator and rotor currents of the sarne sequencei hence the mean torque, counted in positive sequence direction, is mM

w"cre Is t, Is- are the symmetrical components of the stator currents which ;1.1"<: ddined in accordance with Eq. (10.143). Likewise the rotor current vector iII sLal.or coordinates may be written as

t

i

p

ln steady state, assuming constant speed, similar definitions hold for the vector of the stator currents,

v2 [I eiw1t+I* e- iw1t j i.(t)=3 2 -s+ -s, -s

Wl

= S-

Fig. 10.23. Volt age vector of unsymmetrical balanced supply

(WI- W )

=2_ S

the admittances, derived from Eq. (10.72) with Rs = O, are

Y s+ = .

,.Ln (t) -- 3 v2 [I j 2 -R+ e

(10.154)

i

Wl

Wl

/"" -

= Ys- U5-

where Ys+, Ys- are the positive and negative sequence stator admittances per phase. Because of the effective values of slip for positive and negative sequence components, respectively, Wl -

///

205

+ spS

Us+ Wo [

U50

Wl ]

2

2 2-S

5p

+~ 2-S

Us- wo] 2 [ Uso Wl

,

(10.157)

where mpo is again the peak torque, Eq. (10.86), at nominal voltage and Sp the pull-out slip for Rs = O, Eq. (10.71). Eqnation (10.157) may be interpreted as the net torque produced by two identical mol.ors coupled to the same shaft whidl are supplied by syrnrnetrical tltn'('-plt":;I' voll.;!.J'," sysl.('llls I, US (lf (~Il'lid fl't ~ljlICm(:y hllt oppo:-;ite pltasc :;('
"s

\, iI'. I II '1\

206

10. Symmetrical Three-Phase AC Machines

ls+

!

1s-

10.4 Induction Motor with Unsymmetrical Line Volt ages

!

-h) _. 1 _ - 1( h U U -s+ - -s­ 3 e - e U 23 - J v'3 U 23 - U AM

w

Us+!

L-+----J

I )

IUS-!

,

207

(10.159)

where U AM is the line-to-neutral volt age of the disconnected line-phase. When combining the two equations, we find L-+----J

U -s+ - y

YsU Zs+ U + y s- -AM - Z s+ + Z s­ -AM, s+ (10.160)

Us_ =

Ys+ Ys+ + Ys- U AM =

Fig. 10.24. I nduction motor with unsymmetrical supply volt ages, equivalent scheme

Eq. (10.157) clearly demonstrates the basic disadvantage of alI control schemes employing unsymmetrical supply; while the torque is reduced, the power losses are increased by the negative sequence voltage and current com­ ponents. By changing the rotor resistance (Sp) and with different circuit connec­ Lions, resulting in speed-dependent positive and negative sequence voltages, a variety of torque/speed-characteristics can be obtained. This is discussed with the help of three examples. 10.4.2 Single-phase Induction Motor

zsZs+

+ Zs-

U AM

;

Zs = l/Ys is the impedance of a stator phase in symmetrical three-phase operation. The sarne volt age magnitudes would result if the stator windings of the two coupled motors in Fig. 10.24 were connected in series to the symmet­ rical three- phase volt ages. According to Fig. 10.26 the positive and negative sequence admittances can be taken from the normalised circular diagram (Fig. 10.8), where Is/Iso = Ys/Yso was plotted for Rs = O as a function of slip. This indicates that a large difference exists between Ys+ = Ys(S) and Ys- = Ys(2 - S) for small values ofpull-out slip, i.e. for a motor with a cage rotor. jlm(~) Yo

AI

Ys(S)

1 1 1

!..S1 =0

----L­ --,-­

1 1 1 1 ;'

Ys +

y;;-=y;

------1

1.

a

R

Ys .(-) Yo

Starting capacitor

~

Fig. 10.26. Circular diagram showing positive and negative sequence admittance

;'

:l.C 'I Ih3= I}BC Fi!!:. 10.25. Symmetrical induction motor with single phase supply

Ily discolluec:ting one stator terminal of a Rylllluctrical induction motor, siuglc phase circ:uit in Fig. 10.25 rct;ul ts, A, II, n an' t:he terminaIs of the H.Y1l111I('!.ric:\.! lill(~ voltages. The cowll.,rnint, iUlpO!-lcd !Jy t.h(~ opcn circlIit is 1,11(,

1 "; 1

) '~"l l r :-: II V" j

lJ,,;

=H;

rUlld,II "! <'(111<1 11, ;<111 \; dl<1 1N11 rrnfll 1·:'1 J1 . ( 1.11 . 1<[1 ;) , ( IU. l·ll'i),

(10.1 ;)H)

If the motor operates near the positive synchronous speed (S ~ O), IYs+1 « IYs-1 holds, resulting in U s+ ~ U AM, U s- ~ O. This means that the negative sequence field is almost completely suppressed by the rotor wind­ ing due to the large relative speed 2 - S ~ 2; hence the motor operates with llearly symmetrical volt ages despite the fact that one terminal is not con­ nected. ln the vicinity of the negative synchronous speed the conditions are illverted. Th('! st.('ady-st.a.te torque of the single phase motor is obtained by substi­ LilLillg Eqs. (10. HiO) illlo Eq. (10.157)

208

10.4 Induction Motor with Unsymmetrical Line Voltages

10. Symmetrical Three-Phase AC Machines

I I Ys. Ys++Ys .

turned manually in either direction, the corresponding sequence prevails and the motor continues to accelerate at light load up to the operating speed. To make the motor self-starting, either some asymmetry has to be built into it, as is done with shaded-pole motors, where a short circuited turn is built into the stator winding, or a starting capacitor must be connected to the third or an auxiliary winding, as is shown in Fig. 10.25; in both cases the direction of rotation is fixed. The value of the starting torque depends on the design of the auxiliary winding and the capacitor. Besides the starting problems, the drawbacks of single-phase operation lie in the residual pulsating torque as well as reduced utilisation and efficiency of the motor which are always below that of a symmetrically fed motor. It is for these reasons that single-phase induction motors are only used at low power ratings, for example in household appliances.

I

I

Ys+ Ys ++ Ys.

2· Sp 2 S

Sp

Fig. 10.27. Symmetrical components of single-phase motor 11

10.4.3 Single-phase Electric Brake for AC Crane-Drives

.Q!.

úJ,

I

I

" mM+

I

, mpO

mM

I

mM· I

I

I

mpO

mpo I

I

I

I

_

m

I

I

----,---+

' ..... .....

209

­

mpO

Q

I

I

-1

3

Fig. 10.28. Torquejspeed characteristic of single-phase motor !l23 =

mM mpo

Fig. 10.29. Single phase supply of AC motor for braking duty

(10.161)

=

2

2

YS­ +Ys YS+ [ ~+s 1 sp --s

2

1

2-S s;-

+~ 2-S

IYs+Y:+yJ]

UAM Wo [

Uso

A simpIe intermittent-duty AC crane-drive required to operate as elec t.ri ... brake in the lowering direction may have the asymmetrical circuit COllllec~ tion shown in Fig. 10.29, where two stator terminals of a symmetrical wOIIJld rotor motor with large rotor resistor are short-circuited while single-phaH(' voltage is applied between the short-circuit connection and the remaillillf~ sta tor terminal [G23]. This is one of several possible connections having s ilJli lar properties. The analysis of the circuit is easy with the help of symmetricil.1

1

2

wd .

Since the torque rises with the square of the voltage, the negative sequence torque is even further reduced than the voltage. Some of the f1.Jnctions con­ tained in Eq. (10.161) are sketched in Fig. 10.27; by combining these curves the torque/speed characteristic of the singlcvlt;1.se motor is obtained, drllwll in the usual oricntation, Fig. 10.28. 1'11(' ClIJ V(~ is s'yl1llllctrica.l with rm;ped to the orif~ill hc("all s(: aI. S = 1., ys, = yC ; holdli alld I,\l(~ o[l[losiJl[!; t.orqU(· ("()[lIpOIl(·IIf. ~ 1

('11 1111<>1

O

i.I, ) 1I

U lInponcnts. J.llt,rodll<'ill{~ t.1l!:

('1 1.11<", ,1. 1I.'lIcl; lhe IlItlI.Ol" t!. ·V••I" p H IjO .1 ... 1, I.ol"tliw ;ü !; I.;lIId'!il.iU a li.! I, w l t.lulllf, (' xI,"l'lI n l ll Nl ii tl l,IIIU'O. Ir 1\ /1 11 11 111 ItIllt.OI :d, 1;l.:l.Ild nt.i\ll 1M

(/ I ::

/I

.·111'

cOllstraints ()f t.hc circlIle /I '.1 :1

.- {)

illl,.. I'~q t! (III H !,) , (10 I Iii) [, " :11111.: 1 iII • _.#'

(I().I(;~)

10. Symmetrical Three-Phase AC Machines

4 10

1

_

= Us- = "3 UAB

Us+

-

1

-v'3 UAM ,

10.4 Induction Motor with Unsymmetrical Line Voltages

(10.163)

illdicating that the symmetrical voltage components are of equal magnitude, illdependent of speed. Hence, the two machines in the equivalent circuit, Fig. 10.24, would have to be supplied by constant and equal three-phase voltages with two terminaIs of one motor interchangedj this renders the superposition (,f the two opposing torques particularly simple.

úJ

,,

, úJo 1

" I

,

m­M+ ',m po

",

\

m M. \ fn po \

\

\

3

\

------"-\---

\

\ 1.

\ \ \

,, ,

3

,,

Finally, Fig. 10.30 also contains the torquejspeed curve with symmetrical voltages but inverted sequence and short-circuited rotor resistance (c). This operation allows regeneration, however at higher than synchronous speed in lowering direction. It could be used as the "Full speed, Lowering" set-point of a contactor-controlled AC crane drive.

10.4.4 Unsymmetrical Starting Circuit for Induction Motor Smooth acceleration during start-up is important in some applications, such as textile machines, where there is a danger of breaking fibre threadsj an asymmetrical circuit like the one shown in Fig. 10.31 can provide a simpIe solution for this problem. An external impedance ZI is inserted between a line phase and one of the stator terminaIs of a symmetrical cage- or wound-rotor induction motor; the circuit is one of several similar connections. Assuming the impedance ZI to be constant, all volt ages and currents remain sinusoidal in steady statej the following equations hold: ZI

\

\

_ 1.

(b) Braking with De exeitation

\

I Sl + U 12

m

mpo

211

U 23

= U AB = (1 -

= U BC = (e-h -

e-h) U AM ,

eh) U AM .

(10.164) (10.165)

(a) Single phase braking A

,,

-----~

-1

':'~--\;)S;m~etrieal voltages reversed sequenee

"'l IJ:. JO.30. Torquejspeed curves of AC motor in unsymmetrical connection

'l'he torque curves are depicted in Fig. 10.30 assuming Sp = 2.0 caused by ;\. large external rotor resistor. Due to the reduced voltages the peak torques i UT only of the nominal peak torque, hence the motor is again only partially IIl.iliscd. Since the resulting torquejspeed curves lie in quadrants 2 and 4 the drive can only operate as an electric brake, regeneration is not possible with I.I.is scheme. An advantage of this connection, when used as a lowering brake in crane applications, is the fact that the resulting torquejspeed curve (a) passes I.I. rough the origin; unintentional speed reversal (raising) at light load does 1101. occur as would be possible with symmetrical supply and the rotor resis­ I.allcc sLill furl'.hcr illcreased. SillJilar hrakill ~: dmfacleris(,ics aJ"(~ liI/I.il.ill,'d with H sllIaller ro(,or J"(~SiH1.1~1I1·" 11'y r,'('djllf~ I,II!' til,nl.ol" windiJlI{ wi l.It ,11,'("<'1, (' lI.rT '·.fJl. (h). TIl(' lIlol.or 1''''' 11

k

"1' ( ' 1' 111." ' 1 rl' i 1111

\'dd ,v

1' 111 ('(' 111.

I,r ll.h'.

!J23 = !Jae Fig. 10.31. Unsymmetrical starting circuit

The s tat.or current can be expressed with symmetrical components, Eq. ( I O. 1~H), ;;, I

' :i I

;;, I

1\',.,' I 1/ ,'.,' I I Y·; (l ·.' .I ;

(10 . I (iii)

~!

10. Symmetrical Three-Phase AC Machines

I ~~

10.4 Induction Motor with Unsymmetrical Line Volt ages

likcwise the line-to-line voltages obtained from Eqs. (10.143) are U 12 = U 51

-

U 52 = (1 - e-h) U 5+

+ (1

U U - -53 U -- (e -h - eh) -5+ -52

U --23

maps the real Rl-axis into a circular arc in the complex plane, extending from RI = O to RI -+ 00. When the rotor is running, O < 8 < 1, the phasor diagram changes with the tendency of reducing the volt age R1I s1 , thus achieving improved symmetry. This corresponds to the situation found with the single-phase motor.

- eh) U 5- ,

+ (eh - e -h) -5U .

811bstituting these equations into Eqs. (10.164, 10.165) and solving for U 5+' lT. s _ results in

3 + Zl Ys U -5+ - 3 + ZI (Ys+ + Ys-) -AM U

-

u

=

-5-

(10.167)

,

-Zl Ys+ U. 3 + Zl (Ys + + Ys-) -AM

+ Ys - us-l

= -3 u 5-

(10.169)

is a measure of the asymmetry of the stator voltages. With these results the phasor diagram of the motor can be drawn for any chosen operating point. Tlte torquejspeed curves may be computed from Eq. (10.157); their shape dq)ends on the impedance Zl and the motor parameters, but they lie inside i.Ill~ :IIea which is bounded by the curves for Zl = O and Zl -+ 00. This is qll:dil.atively indicated in Fig. 10.32 for a low-power induction motor having I'ni rly large pull-out slip and assuming a resistive impedance, Zl = RI. Of ('1111 rsp, this and all similar arrangements are associated with additionallosses iII t.hc external components as well as the motor itself and are only suitable ror short duty operation. The function of the circuit shown in Fig. 10.31 is further illustrated by I I iS!:\lssing the phasor diagram of the stator volt ages in locked rotor condition. AI, 8 = 1 we have Y s + = Ys- = Ys(l). Hence it follows from Eqs. (10.167­ 10.169) with Zl = RI =

R I .

.L -SI

3 RI Ys(l)

3 + 2 RI Ys(l)

as a limit.ing case .'

hm (l{IL"; I)

U1

wil.11

H'XI

1.11<'

(RI -+

O :S RI

< 00

(10.170)

;

this includes again the single phase motor

= 'l U AM , 3

(Ir thc f;l.al.or vO! l.nW·!4 ai, ,<.,', IO ,:I:\) 1':'1 , ( 10 , I'/(l) 1'1'/11' 1'111 '111,:: li,

1l11 1l.':oJ' d i a l'/H III

:. 11 1. 1' 11,11',111.1 111 1 '

00)

U -AM ,

( 10'11 '\ '

úJ úJo

(10.168)

'l'ltis contains the limiting cases of the symmetrically fed motor (Zl = O) aud the single phase induction motor (Zl -+ 00) discussed in the previous paragraph. ln general, the positive and negative sequence voltages are com­ plicated functions of the motor admittances which in turn depend on speed and pull-out slip. The volt age across the series impedance Zl I S1 = Zl [Ys+ U 5+

213

I dq(' 'II ('I' tll. i 111':

mM mpQ Fig. 10.32. Torquejspeed curves of unsymmetrical starting circuit

for large pull-out slip

A {\--Rj=O

ZI=R I

S=1

\AB AR

3,C

!J Be

R I =oo

2,8

Fig. 10.33. Phasor diagram of asymmetrical circuit at standstill

The shaded area AR of the triangle (Fig. 10.33), formed by the terminal voltages fl 12 , U 23' ~l at standstill (8 = 1), can be expressed as

AR = ~ 1m [fl23 U;ll ;

(10.171)

lilll' llI'

o

1111,(1

1'1I11.-1.i'lll wil i eh

whjeh hy

ii '"

(10.172)

illsI'l'l.illl'; Eqs. (10.145, 10.14ô) ma,y be written as I,;

III}

1

(!?, I, "

(10,'173)

;, I '1

I () .

SymmetricaI Three-Phase AC Machines

A comparison with Eq. (10.157) shows that AR is a measure of the torque at standstill [28J; it vanishes for Rl -t 00 and assumes its maximum value at /lI = O, i.e. with symmetrical supply. The circuit shown in Fig. 10.31 can be extended by additional controllable i III pedances to allow continuous speed control, even four-quadrant operation hlll. llot regeneration. This has been realised in the past with saturable reac­ lo!s, the impedance of which could be varied by direct current in specially c , >llllccted control windings. Because of the high losses these drives were re­ ~;I.!ictcd to short duty operation; also the control was rather slow due to the si 11J';gish response of the large reactors, making this type of equipment obsolete I.<>day [L29J. A similar control effect, however with greatly reduced size and ("o s I. of the components as well as much more rapid response can be achieved hy illscrting antiparallel thyristors into the stator supply leads of the motor; l.!t is causes considerable nonlinear distortion of the motor voltages, resulting iII yet increased losses but it may be acceptable in short duty applications.

11. Power Supplies for Adjustable Speed AC Drives

An important result of the preceding chapter is that the application of AC motors in continuous duty adjustable speed drives calls for static converters of adequate power, generating three-phase volt ages of variable amplitude and frequency. This is necessary to maintain at all speeds a low rotor frequency, which is a precondition for acceptable overall efficiency of the drive. Con­ verters of this type are available today employing thyristors, gate turn-off thyristors (GTO), or switched power transistors (MOSFET and IGBT), but the complexity and cost of the converter equipment and control still exceeds that of line-commutated converters of a similar rating. While part of the cost for the converter can be recovered by the reduced cost of the AC motor as compared to a DC machine of comparable rating, the overall cost of the AC drive may still be somewhat higher than that of a standard DC drive. Of course, AC drives exhibit a number of important advantages, most promi­ nent in the case of an induction motor with cage rotor, that will inftuence the decision in their favour: • Power rating and speed of an AC motor is not limited by a mechanical commutator, • Reduced axiallength, volume and inertia of an AC motor, as seen in Fig. 10.1, • Rugged design, reduced maintenance and service requirements, • Drive is applicable in explosive and contaminated environment. The last features are of particular interest with traction and servo drives where many driven axes are incorporated into the machine tool or robot. Of course, there are also applications at high power and speed, where DC drives could not be built and only AC drives can provide a solution. Since the cost proLlems will diminish as the converter- and AC drive- technology matures, it is likely that DC drives will eventually be completely superseded by the more advanced AC drives . The basic lay-out of an adjustable speed AC drive is shown in Fig. 11 .1; iI. collsists of an AC motor, a static converter, which generates a variable voU.age, variable frequency AC system and the associated control equipment. 'I 'II< : i 11(".)"( ':1:-:«:<1 col1Ipkx:it.y of the c:onverter is callsed by the fact that inver­ I."I'Hwil.h fOI'( ·, 'd ("() li 11 11 II Latiol1 may n:q\lÍn ' :ul
11. Power Supplies for Adjustable Speed AC Drives

}I(i

11. Power Supplies for Adjustable Speed AC Drives

217

'J'1I1,l c 11.1. Adjustable speed AC drives, synopsis

° W';3

e-

~ ç1 ;.::::

~-:.

~iSal~ 0"",0.0.

m

~

°

u ;,.,. ". 0

~~

~ro~

~~]~~~~~

0.2:.-::; .... 0.2: (fJ d ~>, ..<:1013].<::

0.,<:; '" • ..<:11.()~ç1 .~~ 0.8 :::r::~.....:l "d

8..<:1 13 2'~ 13

u

s....

S

~-ro~

1'1

UJ

tJ'"'

bOo;':::: Q) .bOl.() O"d

~2-

r;;.~~....:l"d

o o

o o

bO

ContraI

bO

Fig. 11.1. General scheme of AC motor control

(fJ

•

I

~ t) J; Q)

t-' .~

~

H

.... Q)

;>

13

'>

§~o

.... "" Q).3 ...c:: > o

Q)

-§,o~-5-§,gbO§S

~~$~~2-~9

. <:1

o.

;:l

~

­

'"

~

. <:1

UI,

.­

°

.... J; "

O)

..

<.J

I e- ~§"'~ U

..

o;>~..oo"",o. ~::g(j o..~ VJ

..<:10 -+J

;:l

g 00 ~~] ~ ç' o.:: ~ ~

~

S

H

I

.., S

ç1::;~~S

~t;O~o

..... - t-< ....,

U

UI;...·~;>

U

::Jt::,CJ ~O)

'tt1" I

8

"

.... oç1 tE

o

,"

.

1- ,

~

,.)

u '

....

Q) ~

n' -,'

;~

:~

i).1

r

I"

. <:1

O)



lJl

2:

..

O)

~"d

~Ç'o

(I)

0;>0

g~ 13 m t) ;:l

S..,;.a .... 0Q)

O)

~ ~ 0...>'< bO.2 13 13 00 ~oC~oo 0.0 or-lQ)~;>~ .-<

~,... . .- -u° e-<

.....:l~;>

. ~ ~

~~o

"d

....

'"J,

S ~

'1'1

I:

°

I­

>=l

d (l) ç1

'u

~

(,,1

VI

O)

R

0.:

° °

....,

ç1

0

ç1 O) V>

o r.l,+'

.~ 's-s ~ w.;,l

p'i. 0',

,. , Ir;

I~

-

~

~ .... ro

te

~ § t)

...,+-> u o

E S

~

f1

. 1-

S~ .... '"

'-'

Uo '

8.~ al ~·C

2: o. o."d 8 "'dO) 3.-<..<:1 >,~ ;.a V bO "d ç1 Q)

-:B ro

L -_ _ _

constant

Commutator / machine-side converter

I

DC - link

I

I

1­ -----+-1 ...._ _ _ _---I.

~



bO

AC - motor •mM' W

> UI' úJ

0

1

úJ

1

I variable

I

.... o

Õ Õ dO) .... I

bO

I

ç1 m "d

o U

'..sª

"a

úJo

o~

.... o

>=l

Line-sde converter

° lJl

2:

/

~.--_.-.-._-----_.----

~~"--::,",,.2~

.. ~

DC - machine

Supply

..

O)

~''' ~. ~ §' ~....

V"

~§~

"d

(l)

~

:1

110 ""'" ~

:.a2:~ .2~~

~ .&~ ~ ~ 2:

"foiS .~

32<..<:10 .-< ......bO"d .'"d - '-"' . <:1

,íl, '--,

§

° ~oo_

0...--::.

o-----.:;..~

o.iS ô.-::; 8 ~ o. '"

• I

....

~

o

~ ~ ~

U . ~/ ~ . ~I oI-' ;:. . II, ', I'~' Ü~ UOr1 I " ,

O)

....

Q)

of the inner torque controlloop which is much more complex than for a DC motor. The reason is that an AC motor with its simple mechanical design represents an involved nonlinear multivariable control plant.The motor must be fed with alternating currents of variable amplitude, frequency and phase; in addition, the rotor currents cannot be measured with ordinary cage motors. This is in contrast to a DC machine, which has a very simple decoupled control structure but a more complex mechanical designo The mechanical part of the control scheme is, apart from the smaller inertia of AC motors, the sarne for DC- and AC-drives. Fig. 11.2 indicates that the mechanical commutator of a DC machine acts like an electromechanical converter which feeds the armature conduc­ tors with alternating current of a frequency proportional to speed, Fig. 5.2 . On an AC drive, the functions of current conversion and torque generation are mechanically separate, thus creating new options for electromechanical interfaces .

'.;:Z~

ç1 ;:j

U "' O

.~ ~ I~

"d

o....

.. >") ° .3

0)....,

4-<

H

-"d

o

~o § 8 O ~

o

-,

~-------------- ______ I

.... .2o ~v ç1bO d ç1 .­

Fig. 11.2. Controlled AC-drive with DC-link converter

>=lm"d

~ .-d .§

0-

IS'

~ ..<:l

j;)"

§ ,?:l t

Table 11.1 presents an overview of controlled AC-drives, showing five ba­ fonns of power electronic converters for changing from the fixed line- to v;l.rial'](', 1I10Lor" frcqm:ucy. FOlll' of til(: circuits contain a DC-link for decou­ 1'1 i "I', 1.11(' I.W() :;id4'1 : 411' I.!t\: CO \lVerLCT , w 114'1'( ';\:; I, h( ~ cydo-collverter performs this

:;ic

I ],'~ .~ II )

--'

"114'1 ii. Li 4111 iII

; 1.

:1 ;111'..100 C'O\lVC'rnillll

:il,;lI'.4' f\ II 01' 1.114'

4'nllvc'rl.4'l':-1 are

;t.dapl, ;~hk to

11.1 Pulse width modulated Voltage Source Transistor Converter

11. Power Supplies for Adjustable Speed AC Drives

218

four-quadrant operation, they could also be used with variable speed genera­ t.ors, for instance in wind energy plants. ln the vertical column of Table 11.1 live basic types of AC-machines are listed that may be used for controlled drives; with the exception of the reluctance motor, which is characterised by salient poles, they can alI be treated with the mathematical model derived iii Sect. 10.1. ln principIe, most of the 25 possible converter-machine combinations could 1)(' realised but only those, where comments are entered, are presently applied i II practice; since each type exhibits particular advantages, it is unlikely that I.h(' lield will eventually narrow to just one or two major circuits as has been t . It(~ case with DC-drives. ln fact, new combinations may be added as new ~;lViLcbing devices and converter circuits appear on the scene. An example of Ih(' oIlgoing development are resonant-línk converters, presently restricted to Imv power applications such as switched mode power supplies; cyclo convert­ crs with forced commutation (matrix converters) are also a possible future opLioll. Only a few typical converter topologies that show promise to stay \ViII be discussed in this chapter.

Pulse width modulated (PWM) Voltage Source Transistor Converter (IGBT) 11. 1

TI\(~

rOIlf-quadrant DC/DC converter shown in Fig. 9.13 a can be extended by \,wo more arms, as seen in Fig. 11.3, to form a three-phase converter fo r 1."(' slIpply of AC-motors in the lower and medium power range, from ::11111.11 hiV;h d'yl1amic performance servo drives with speed and position control ( . I () I, W) Lo lIloSt auxiliary drives in industry, ranging up to several hundred li 'vV, II\. Itil;hcr power this two-Ievel circuit is also extended to multi-leveI ('oll/il',1I r:l.t.iOllS. The converter is suitable for supplying induction as well as Ily lI('h \( )1l01lS motors. W iLh Cite voltage source converter shown, the terminal voItages of the mo­ \'()f' ( '; \.11 assume the three possible values UD, O and -UD.When the motor is r('I'.( ·II('l' at.illg, the mean link current I D is reversed, calling for a two-quadrant :;lIl'pl'y of the DC link. As long as the power levei is low or when regenera­ Lioll OCCIll'S only occasionally, for instance during dynamic braking of a servo drive', Lhe DC link is normalIy supplied by diode rectifiers and the reverse I" ,wcr is clissipated in a balIast resistor, Fig. 9.14, as indicated also in Fig. I 1.:\ wiLh dashed lines. The pulse-width modulated ballast circuit then be­ I'.IIIS dischargc the link capacitor as soon as the link volt age rises above a pr('s('\. value. Of course, any other source of direct voltagc could also serve 1;)1 I'( '( 'dillg t.hp converter, such as a battery 011 an electrically driven vehick, wl\('re r(~I~: ('II( ~ rntion is il1lJ)ortant t.o sav(~ (m('rr~.Y. Wi\.h Il ipolnr t.mllsi st.!ll'K, Lhe :;wit.dlÍlI ~~ fr('qlwlI('.y i:-; lilllil.(~dto n r{ ~ W 1\1-/(" IlIll. \\'il.ll low 1"'111'('1' 11I:;III:t.I,(,·cI C a L{' 1111'01 111' ' 1'1' 1I. 1I11 íl~I . " rH (leIIT), H:-i :; ho wlI iII aclclillf~

L()

219

io

-', : : RB "

Uo

I

, I I' ,

.....

I~TB I I

iSI

iS2

iS3

iSI

Fig. 11.3. Switched IGBT- converter for three-phase AC motor (simplified circuit)

Fig. 11,3, or Field Effect Transistors (MOSFET), having a switching time in the order of a microsecond, the mean switching frequency may be above 16 kHz, i. e. beyond the audible threshold, so that no objectionable acoustic noise is emitted from the drive. An additional advantage is that the converter then exhibits a large bandwidth for controI. If the link volt age UD is of sufficient magnitude, fast current control loops can be designed which keep the stator currents in line with the alternating reference values; this effectively produces near ideal current sources for the stator windings of the motor, thus eliminating the effects or the stator volt age equation Eq. (10.50) on the dynamics of the drive. As a consequence, considerable simplification of the drive control is achieved because the stator voltage interactions are then dealt with by the current controlIers. These could either employ pulse-width modulators operating at constant frequency or simple On/Off comparators having a narrow hysteresis band, also indicated in Fig. 11.3. The resulting current waveform with a single-phase load and On/Off-control (symmetrical modulation) was shown in Fig. 9.16; it has the advantage of fast response but exhibits variable switching frequency which may be undesirable in view of interference problems. When extending the principIe to three-phase loads, as in Fig. 11.3, the hys­ t.eresis band is helpful to avoid interactions between the controUers because, in vicw of the isolated neutral of the machine winding, the stator currents are balanced and one of the controUers is redundant, it is only retained for sylllmetry. To produce a prcfcrred switching freqllency, a triangular clock signal may Iw added aI. LIt ~ ~\l\lJl\lillg poillt or t.Jj(' cmrclIL cnntrollers [829]. Another pos­ ::ihilit.y j:: 1.1)( , >1 YII('hlOllj'l,aLwJI (lI' UI(' (, ()IILIOII('\"~:, i.t:. :dl()wiJli~ a t,rallsj(.joll to

'2'20

11. Power Supplies for Adjustable Speed AC Drives iRef (I)

11.1 Pulse width modulated Volt age Source Transistor Converter

*j

-20~

'

7~;ef """---'

b

;f

No ,,,d

20

200 7

-20,

MM Load '

~;ef

'" F

)~,

I

I-

Stator current references i

A 200

221

, is

5ms

-I

~cc,n' ~

-200~

"

i à 200

F i(.!;. 11.4. Current waveform produced by a single-phase transistor converter with illdnclive load impedance and synchronised On-Off controller

only at equidistant sampling instants t = v To defined by a fixed clock Cn:qlwncy to = l/To [B45] . While Fig. 9.16 was derived with a simple free l'llllllillg On/Off-controller, the effect of synchronization is sketched in Fig. I 1,'1 , again for a single-phase converter with inductive load; the amplitude (lI' LIli: ripple superimposed on the load current is now increased, because the ::w; I.l'hillg op erations are delayed until the next sampling instants, thus cre­ :,",,;II/,; waiting intervals ranging between zero and To. If synchronised On/Off­ {'O lIl.r()III'I'~ a.re employed, the hysteresis band Ll can be omitted because there ,:' ""W :\. IlIillimum time interval To between subsequent switching operations. W hl 'll a very low ripple content of the currents is specified, as on machine I.,"I! CI",tI drives, the use of a pulse-width modulator in combination with i' 1""'; 1,1' I"llrrent controUer is preferable to On/Off current control. This is d l 'IIIIH, ~ I.], ; lt e d in Fig. 11.5 where measured current waveforms obtained with III<' :: ;1.111 " three-phase transistor converter are compared. Fig, 11.5 a depicts III<' ('IIIT"Jlt produced by a synchronised controUer having a sampling period 'I II '22'1,s, whereas the curves in Fig. 11.5 b show the result of linear current 1;0111,1',,1 with a pulse-width modulator operating at 5 kHz [H20].The audible 1IIIi :;I' is !llorc pronounced with PWM control unless a higher clock frequency i:: d()~I: JI, but the current waveform is much more acceptable. The reason for 1,111' illcn-:ased ripple current is the quantisation of the voltage-time area with 1.1)(' sYll chronised On/ Off - control. 1'lIbl:-width modulators are now available in a variety of integrated cir­ " II i!.:;, wltich gn'atly silllplifies the design of PWM COllverters, Thcre is also thc pnHs iliíliCy 01' so[l.wan:-based mo(lulatioll HHill/ '; fasl , si!!:J1 al proccssol'S which "II; ' I' III11illlil.('« 1I1'xihili I.y hy colllhilliJlI.r I'W M wi I.h H0l'hisLÍc,ÜI :d, I'udl as

-20

-200

()(TlIf

b) Currents with linear contrai, PWM at 5 kHz

Fig. 11.5. Comparison of current waveforms with different types of controllersi (a) Current waveform with synchronised On/Off-control, To = 22J.Ls (h) Current waveform with linear current controller and PWM ,

operating at 5 kHz

predictive or time-optimal current control; this is of interest with high power converters switching at a lower frequency [H67, H71], The assumption of virtual current sources for the stator windings is, of course, only valid as long as the ceiling volt age of the converter is not reached, This is demonstrated in Fig. 11.6 for a free running On/ Off - controUer when the frequen cy of the current reference is gradually increased. Due to the inductive load impedance, the necessary fundamental component of the volt­ age rises with frequency according to UI = h JR2 + (wIL)2, It is seen that, beginning with Fig, 11,6 b, there are signs of saturation as indicated by pro­ longed intervals where no switching occurs, which is due to insufficient supply voltage, Clearly, this results in a rising control error between the sinusoidal current reference and the piecewise exponential feedback signal and must be taken into account when designing a drive control system, A very effective method of pulse-width-modulation that is particularly slIitcu for fast switching converters is called vectorial PWM because it repre­ s(' IILs a n attempt to reproduce in a given time interval vTo < t :=:; (v + 1) To a v()l ta~t: v(~ctor 'U s IId (t) demanded by the current controUers [P29], For this iI. is illJp"rl.alll. 1,(1 rellle m!>!:r t.lt al. tll(: COll verter circnit of Fig_ 11.3, which in

·.! 22

;/

11. Power Supplies for Adjustable Speed AC Drives

.m.~; iReI (t), i(t)

/)

223

11.1 Pulse width modulated Voltage Source Transistor Converter

protective interval which lasts only a few rnicroseconds when transistors are used, can be assigned to a finite switching time in the converter model. When writing the volt age vector 1f.s(t) in terms of line- to-line voltages (10,38) which can assume the values UD, O and -UD , 1f.s(t)

" 2 = USI + US2 eJ"Y + US3 eJ "'( = U12

-

U23

-'

e

(11.1)

J"'(

u(t)

OOllPl~i

six distinct voltage vectors and two zero vectors result, as seen in Fig. 11.7 b, Clearly, when going from one comer of the hexagon to the next, only one leg of the converter needs to change its state,

1l.2

"Zero vectors' 2x

S1

Uo

S2

,,

S3

U

~ -SRef

'.

-1

(

\ .' U l -I

()ff

I I I I

I (.'

i(t)

u(t)

U23

"51 [

a

"'2"52 [

uS3

[

b

Available voltage vectors

Fig. 11.1. Switching model of converter in Fig. 11.3 and generated output voltage

, .,

"

vectors

,.,' iRe/t)

S;LI.llration effects of single-phase switched transistor converter with ( .,, / ( Hr cml't:1I1. control supplying a passive R-L load impedance 10' 1/\.

I I. H.

I",!" . I I. 7 a is represented by a switching model, must be so operated that 11t0l11' 0(' I.h(~ Icgs is short- circuiting the link voltage UD and that each motor 1.('J'llIill;d osition. Tlw switching :;t, at(~ (8 1, ,'!h, Sa) of the complete "4IIIVl'I'l.t'l' i ii !.Il(ili dcscribed hy a. tlm'c hit. biuary word h:wiug eight diffcrcnt V: dlWi l , ill(~ llIdilll': (I, I) I) aud ( I, - I, I) , (·"III,tI Y, I'I'O v( ~dor sl,at.(·s, wll(~rc t.11<' 1/101.01 1.1 'IIII ·ill ll.h 111'( ' li hol't. (~ir(,IIi1.('d "" . 1.111' 111'111'1 "I' I"wlil' 'ne IIml. Tht:

It is assumed that the converter is switched at a basic clock frequency = l/To, which may either be fixed or correspond to a multiple of the variable fundamental stator frequency, and that the current controllers have determined a command value 1f.SRef, valid in the next interval To; 1f.SRef is assumed to lie inside the first sector, as shown in Fig. 11. 7 b. Since this commanded voltage vector is normally not coinciding with one of the available,voltage vectors, it is to be composed by a switching sequence comprising the adjacent vectors 1f.l = 1f.S(Sl' 8 2 , 8 3 ) = 1f.s(l, -1, -1) = UD and 1f.2 = 1f.s(1, 1, - 1) = UD ej1T / 3 , while filling up the rest of the time interval with zero-vectors. The sub-intervals tI and t2 for the two adjacent vectors are to be computed from the following equivalence

lo

1f.s Ref

) h t2 U = (USa + J'USb Ref = 1f.l To + 1f.2 To = D

( tl To

+e

j

1T

/ 3 t2 ) To (11.2)

III

addil.ioll /, I I 1.1. 1- 2to

To holds, where to is the zero vector interval.

224

11. Power Supplies for Adjustable Speed AC Drives

t

Samplíng of current

!.- T. •

O

1

.. - - ­.-'

-

t 1 i+I

I......

Samplíng instants

-.I

T.

11.2 Voltage Source PWM Thyristor Converter

cussed later. The switching sequence pertaining to the example in Fig. 11. 7 b is plotted in Fig. 11.8, depicting the potentials of the motor terminaIs. Eq. (11.2) describes an idealised situation, where the protective intervals and the inherent delays of the switching devices are neglected. For the actual design of modulators, these effects must be taken into account, particularly the differences between tum - On and tum - Off times, which can cause con­ siderable distortion of the converter characteristics at low output volt age and frequency. Hardware- as well as software- based methods have been proposed for solving this problem [K35,S81,WlO].

of current

controllerTO

O. ,:",:---.

I

I

.

to I- : -!to I-! I ~ t2 ~ I

I

I

I/S1 ~S2

-1 . -

_.

I

I

I

I

I

.... _.

.-=.-:-1.._.J

º1~

~

1

1!:21

t

I

\A

._.-.

0_' _

º I ~21 ~ Iº1

1

Voltage vectors

11.2 Volt age Source PWM Thyristor Converter

Convener switching period 2T

o

I."ig. 11.8. Vectorial PWM with symmetrical switching sequence

Solving for tI, t 2 results in tI

1

USaRe!

USbRe!

To -

u;;- - v'3-U;;­

t2 To =

v'3-U;;­

2

~ ln = 1 _

USbRe!

USaRe! _

(11.3)

~

USbRe!

v'3

UD

'/:1

UD

> O. -

WlwlI ()lIlittillg zerO vectors, to = O, the resulting equivalent voltage vec­ 1."1 l'II
'//': I,,"f

= (1lI - 1l2)

tI

To

+ 112

;

(11.4)

= O and tI = To confirm this resulto Hence, by inclusion of dcsired voltage vector inside the hexagon defined by the six switching v('c/.ors lIlay be realised . Of course, the adjoining switching vectors must 1)(' choscn according to the sectorial position of 1ls ReC(t). If the switching rn'(/lI(~llcy io is high enough, fluctuations of the link volt age may be taken ill/.n accoullt by substituting a recent measurement of UD(t) for UD in the 1l1al.i.llg equations (11.3). T() relllove the ambiguity with respect to which of the two adjoining Hwil.chillg sUItes should be applied first, it is advantagcous to repeat tbe H11,11 1(' switchillg cyck with rcverse sequencc 50 that a syllundrÍCal switchilll.'; p;d.1.c1'll of kll/'. !.h 2'/'0 is ('.n~at(~d. Salllplillg l.ll(~ 1I10!.O!' cllrrcIlI.!-i, ill tll<~ C<~Jlt.(:r "r 1.11(' :i!im!. rircllil. illl.( 'rvlIlcl l.(I dilllilli::!iI'M1.111' n fl,llIpJill,~ lIoi:·;(" ;u, willlw di:.; !.II<' lilllitói for tI

l(J ally

I))()I!

225

The use of transistors (MOSFET, Bipolar) is presently restricted to lower and medium power applications, with lGBT's up to about 1 MW. With thyristors and GTO's, the rating of semiconductor converters extends to even higher power. A proven circuit, resembling the transistor converter in Fig. 11.3 is the thyristor converter shown in Fig. 11.9. Because of the constant link volt age and the low AC impedance of the DC link it is also called a volt age source converter. ln view of the higher power and the fact that the motor may have to operate for longer periods of time in the generating region, the line-side converter is usually an active front-end rectifier, able to feed power back to the line by reverse direct current ID. When employing gate tum off (GTO) thyristors, which are available in kV, kA- ratings, the converter circuit corresponds directly to the one in Fig. 11.3, with the transistors replaced by GTO's but with normal thyristors that cannot be blocked by gate signals, additional components are needed for implementing forced commutation of the machine-side converter; this is so because the stator windings of an induction motor do not contain current­ independent volt age sources as in the case of a line-connected transformer or a synchronous machine with an internal induced voltage. Hence, natural commutation of the machine-side converter feeding an induction motor is not feasible. This is understood when remembering that a converter with delayed firing calls for lagging reactive power which a normal induction machine is unable to supply. The converter depicted in Fig. 11.9 is an example of a variety of different circuitsj it is considerably simplified, showing only the main components. Instead of a pair of transistors as in Fig. 11.3, each phase contains a force­ commutated thyristor switch, comprising two main thyristors T, T', with ant.iparallel diodes D, D', two auxiliary thyristors AT, AT', a commutation capacitor C and an air cored inductor L. The resonant circuits in Fig. 11.9 have the ti\.sk of creating zero currentjnegative voltage intervals for the main Lby risl.o/'s '/',1" hy lirillg allxiliary thyrislors Al', AT'; briefly, a commutating JlI'()('I': ic: l.iI k" !i pla(', ' ;I: i f'()llows:

226

11. Power Supplies for Adjustable Speed AC Drives

Assume that the main thyristor T initially carries the load current iSl and that the commutating capacitor C is positively charged, Ue > Oj then firing of the auxiliary thyristor AT causes T to be blocked so that the load current flows temporarily through AT and C thus recharging the capacitor. At the sarne time a resonant circuit is formed consisting of AT, C, L and the diode Dj when about a half period of the oscillation has elapsed and the charging current tends to reverse its sign, AT and D are blocked, which leaves the capacitor with opposite voltage, ready for the next commutating transient when TI has to be extinguished. The load current is now fiowing through DI and the motor terminal 1 is connected to the lower DC bus. If the current pulse through the resonant circuit is of sufficient magnitude as compared to the load current, the commutation is almost independent of the load so that the voltage UlN can be approximated by a rectangular wave ±UD/2 with short but continuous commutating transients. The load current is l , Supply

----.--Generator \ Uo

Üo

Motor

-------r-----.. 0 Machine-side converter

iO

o

uo Inverter Rectifier Line-side converter

AT T AT' T'

lU 7N

Pulse width modulator Fig. 11.9. Thyristor converter with constant DC-link voltage (Mc Murray circuit)

hving co utinuous due to the motor leakage reactance, can in principie flow in citJwr dimction , even though alI four thyristors may be blocked temporarily, hCC
:wd

P "W l ' l

lI"w .

11.2 Voltage Source PWM Thyristor Converter

227

It is realised that the initial transient, when iS l is commutated from T to AT, is of course also continuous because smalI inductances, not shown in Fig. 11.9, limit the rate of change of the thyristor currents. Likewise there are paralIel RC snubber circuits connected to each thyristor to limit the rate of change of the voltages. A detailed analysis of converter circuits, taking alI these effects into account, is best performed by digital simulatioll'j this is particularly true for transient operation, when the initial conditions, for instance the charge on the capacitor when AT is fired, are changing. As with alI converter circuits, the function of the converter is eventualIy endangered when the required recovery time of the thyristors cannot be maintained, for example due to excessive load current or insuilicient initial charge on the commutating capacitor. Rence the accurate analysis of the commutating transients is of major concem to the circuit designer. Special types of thyris­ tors with short recovery time « 20f,Ls) are usualIy specified for pulse-width modulated converters, which limits the power rating to some MW. AIso, the occurrence of unacceptable load currents must be prevented by fast control action. The firing instants for the main- and auxiliary thyristors are determined by volt age and/or current controlIers. ln view of the limited switching fre­ quency, which is usualIy below 1 kRz for thyristor converters of higher power, the load currents can no longer be considered close to sinusoidal as was pos­ sible with transistor converters operating at a higher switching frequency. Rence PWM- thyristor converters of the type shown in Fig. 11.9 are nat­ uralIy suited to act as controlIable voltage sources, presenting to the load pulse-width modulated rectangular voltages, the fundamental components of which are prescribed by volt age command signals. Because of the large superimposed harmonics, caused by the chosen switching strategy, closed loop control of the fundamental currents at variable frequency is normalIy not practicalj it was different with transistor converters, where the switch­ ing frequency is higher and the currents are suiliciently filtered by the load impedances. Therefore, open loop voltage control through a pulse-width mod­ ulator is the standard solution, while the current loops are normally closed on the next higher levei of drive control, as will be shown in Chapo 12, The pulse-width modulators may again be of a variety of designs. One frequently used scheme with converters of higher power (Fig. 11.10) employs a triangular sampling signal [J8, M42, S29], the frequency to of which is a multiple n of the stator fundamental frequency ft to avoid subharmonics; n is reduced in stages as the stator frequency and volt age rise, to limit the switching frequency and to fulIy utilise the voltage capability ofthe converter. A drawback of this otherwise simple scheme is the appearance of large har­ monics ai the frequencies to ± 2ft . The load currents are smoothed by the leakage illductance of the motor which should not be chosen too smalI with this tyP(' 01' COllv(·rl.(\r. Ali tripkll harlllollic; a.re climinated from the cur­ l'('uLH iII :; 1., ':' 01.1' :: l.:t.I.,. dup 1.0 I.lw i :;ola.t., :d IlPIlLral
228

11. Power Supplies for Adjustable Speed AC Drives

are still noticeable additionallosses as well as sometimes objectionable noise cmitted by the converter and the motor. A typical current waveform with a PWM-converter and this type of "synchronous modulation" is shown in Fig. 11.12 b.

11.2 Voltage Source PWM Thyristor Converter Ü l1

n

2

UO

m=1 m=2 Numberof reversals per quarter-period m=4

Ar

v

t

I UI Ref

r;-;::::::-,

T

----.ó---.i

229

!1T> Tc

r

I' ,_o I

!

AT

u UI [Jo

m= 12

2

a

I

.

I

U

,,, ()II "

I I

I,

-1:1

I I

, I

I,

(1:) 1:2

1:

3

1:4

Voltage for m = 6

b

"

"\

1° Ttl

IN

1:

5

1:6

*

Uo -2

n 2 n 2

"wt=1: 1

W

1

t = 1:

Fig. 11.11. Optimised pulse-width modulation with precalculated switching patterns

/ "1

Triangular clock signal v(t) I"j,~. 11.10. Pulse-width modulator with a triangular sampling signal

' I'his unsatisfactory situation has given rise to the development of a mul­ f.if.II
J

allow the commutation to be completed. AIso, the losses in the converter caused by each commutation transient should be considered, which means that there is an upper limit for the switching frequency. At the maximum fundamental frequency, which corresponds to full output voltage of the con­ verter when feeding an AC motor, the optimal volt age waveform is a square wave with one reversal per half periodj this is included in the switching pat­ tern shown in Fig. 11.11. When realising a suitable modulator it is of course desirable to keep the required volume of memory smallj this can be achieved by storing only one quarter cycle O ~ T ~ ~ of the binary pattern for one phase and obtaining t.lte reruaining information by transposing and inverting this pattern. On the oLheI' hand , consi
(Ir 2 '0

lO )" ! HI.f'\IH pt '" qll adr' lllIl., t.hiH l ·t·llII ll.~ iII a IllclIlory

(Ir

11i

·.no

11. Power Supplies for Adjustable Speed AC Drives

11.2 Voltage Source PWM Thyristor Converter

II I! yt.( ~ which could easily be accommodated by one microelectronic chip of 11':1<1 only memory (ROM). At the very low end of the frequency range some (ll.bel' means of modulation must be employed because even 4 000 steps of :1.lIg111ar resolution per period will eventually prove to be insufficient. ln order to avoid continued fluctuations of the volt age amplitude, which would disturb the switching sequences and render the notion of "harmonics" 1IJ(~allingless, the superimposed control should contain a hysteresis band in 1.11(' amplitude channel; fine and rapid speed control are still possible through i.l1(~ frequency input of the modulator but the voltage would be changed more ~ :Imvly and temporarily in somewhat coarser steps. Some results of the current w:Lvdorms and spectra obtained with different modulators are shown in Fig. 1 1.12. The Curves were recorded on a 22 kw drive supplied by a voltage source I.hyristor converter [P44].

The load-independent commutation, resulting in low impedance motor volt ages makes the PWM converter an excellent choice for drives with high dynamic performance at power ratings where high frequency switching is not feasible; an example are high power traction drives with GTO-converters which require good dynamic performance to damp oscillations in the drive train and prevent wheel slippage during starting and braking.

1~L~ ~

+jQL

ULI

iLl

-I 11111 III 11111 IRlm IL;

U

~ 11111 Wulli±illu~ III11 ~U~

IN

231

machine-side PWM converter

•

t

Uo

iSI Une-side PWM converter

~II~I~I , I __~I~L\~~~M~o___

iSlv

17

29

;/

__ I

1

fifI

b

13, 17

29

43

~, fifI

I>'il-(_ 11.12_ Current waveforms and spectra achieved with different methods of I'lIls(· , widl.h modulation. (a) Optimised switching patternj (") SI! hharmonic modulation with triangular sampling signal

fi particular feature of the pulse-width modulator with optimised switch­

pattcrn is the fact that the pattern can easily be changed for different :1 pplical:iolls, placing more emphasis for instance on losses, torque pulsations