Contribution to the synthesis of stabilizing transformers and RC circuits in electric machinery arrangements

Contribution to the synthesis of stabilizing transformers and

RC circuits in electric machinery arrangements

Citation for published version (APA):

Oen, T. T. (1961). Contribution to the synthesis of stabilizing transformers and RC circuits in electric machinery

arrangements. Nijhoff. https://doi.org/10.6100/IR30017

DOI:

10.6100/IR30017

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Published: 01/01/1961

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CONTRIBUTION TO THE SYNTHESIS OF

STABILIZING TRANSFORMERS AND

RC

CIRCUITS

IN ELECTRIC MACHINERY ARRANGEMENTS

PROEFSCHRIFT

"fER VERKRIJGING VAN DE GRAAD VAN DOCTOR 1N DE TECHNISCHE WETENSCHAP AAN DE

TECH-NISCliL HOGESCHOOL TE EIN DHOVEN, OF GEZAG

VAN DE WAARNEMEND RECTOR MAGN1F1CDS

DR. IR. W. L. H. SCHMID, HOOGLERAAR IN DE AFDELXN"G DER WERKTUXGI30UWKUNDE, VOOR

EEN COMMISSIE UIT DE SEN AAT TE VERDEDIGEN

OP J)TNSDAG 25 APRIL 1961, DES NAMIDDAGS TE 4 UUR

DOOR

TAN TIONG DEN 'I': L ~'n(T ,;:, o~n(,(: H N l SC l-:l. IN Gl';Nl E U Ii.

GrrBOnEN T'll ~:.;r1)1~A,MA1U

'S-GRAVENHACE

DIT l' H OEl':;ClIH 1)''f IS GOEDGEKE U RD DOOl< DE J'ROMOTOI{

The author is most grateful to Messrs. N. V. Eledrotechnische Industrie voorheen 'Willem Smit & Co., Slikkerveer, Holland, who enabled the achievement of this thesis; in particular, he is greatly indebted to Dr. Ir.]. Gilt ay for hiE.; criticiml and encouragement

TO MY PARENTS TO MY WJFR

CONTENTS

SUMMARY . • . . . • . . .

-Chapter 1

TRANSIENT PERFORMANCE OF A CONTROL SYSTEM

1.1. Introduction. . . . 1.2_ Overall inverse transfer function 1,3. Wyschnegradski diagram .

1.31. Parametric equations. 1.32. Interpretation. . , . I A. Pedormance criteria . . .

1.4L Analog computer mechanization,

1.42. Mathematical expressions for the integral of squared

5 7 10 10 12 16 16 error. . 17 1.5, Bode diagram . . _ . . . , , __ . . . . . 19 Chapter 2

CONTROL AMPLIFIEll WITH STABILIZING NETWORK

2.1_ IntJ.'Oduction. . . . , 21

2.2. Transfer function of the control amplifier 22

2.3. Transfer function of the control amplifier with the stabi· lizing network . , . , . . , . . _ . . . , . . . 25

2.4. R/:alizahility domain of th~ C\hbili~ing network. . . 33 2.4 I. Stabilizing tmmJorrncr. . . 34

2.411. M llttwrn<l.tical cxpressions for the tran:;for-il"H:r parameters . . . . . 34 2.412. H.(;alizability conditions. . . 35

2.413. Provisionallimit5 and S()IrW Himplifications. 39

2.414. Note on th,~ (ksign of the stabili~illg fidd

windin/S 40

2.42. Stahilizing

He

circuit 42

2.421. Math,~matical Holution for the RC <;ireuit parameters . . . 42 2.422. Healir,ability conditions. . . 43

2.423. P[,ovisionallimits . . . 45

2.424. Note on tfH~ !lcSigIl of the stabilil.ing fidd

winding 45

2.43. Comparison betwl~('lI sLtbilizing tran5-forn}(,r a.nd

He

circuit . . . 46 2.5. Determinatiull of Ihr~ parameter values 47

Chapter 3

STABILIZING -rHAN',POHMER DESICN

3.1. introduction. . . 49 3.2. n(~sign with complete fn"l'dom 50

3.21. Core-type transformers with one and with two rect-angular coils; shcll-type transforIl1cr wit.h

rectangu-lar coil. . . 50 3.211. F1HH.lamcntal equations. . 50 3.212. Sollltion . . . 53

3.72. Con:-lypc transformers with One and with two

circu-lar coils; shellMtYJH~ transformer with circ\lhJ.r coil. 57 J.23. N()llI(jg-rams and gn~phs 59 3.24 . .l)(:~ign prllecdufe . . . 61 3.25. Remarks on air g"p and dissipation 72

3.3. Design with constraint . . . . . 72

3.31. Dimension .atios of the iron Core specified 72

3.32. Punching of the transformer laminations spe6fied 76

3.33. Iron core specified. . . " 76

Chapter 4

CALCULATION EXAMPLES AN'D EXPERIMENTAT. R£sur.TS

4.1. Introduction. . . 79

4.2, Current regulating system. , . . . 79

4,21. Open-loop frequency-response function. 79

4.22. Determination of the parameter values. 8 I

4.23. Justification of approximate trH.nsfer function. 83

4.24. Design of the stabilizing transformer. 84

4.25. Experimental results. . 85

4.3. Voltage follow-up system . . 88

4.31. Overall transfer function 88

4.32. Wyschnegradski diagram . 89

4,33. Dynamic effectiveness . . 9[

4,34. Justification of approximate transfer function. 95

4.35. Experimental results. . . 95

REFERENCES. • . . . • . . . • . • 98

SUMMARY

In electric machinery anangements constituting control systems it often occurs that the power amplification is achieved in ;;evencl amplifying stages. The first amplifying stage or the first two stagef; are frequently realized by a rotating amplifier witho\lt feedback effects. If in this case the control :>ystem is not stabk or too oscillatory, it has then often been found to be useful to derive from the output voltage of the above-mentioned control amplifier an. mmf forming a negative feedback in series with tlH~ re~uJtant input mrnf of the control amplifier concerned with the aid of ,\ stabilizing t.ran5former or a simple stabiJ.i;,':ing RC circuit; the, steady-state perfOl:-mance of the control amplifier, however, is principally not affected, The present thesis if.; restricted to control amplifiers of the above-mentioned type and to thl' ;~foresaid 5ta-bilizing networks.

Chapter 1 presents some principles of the theory of lirl(,ar control systems which are applied in the calculati(m examples. It starts with the derivation of a theorem by which the overall inverse transfer function of a control system Can be obtained from a gener, ali;,':ed block diagram in whiCh the influcnce of the input signal

truv~b di,ectly along one single path to the output_ If this latter premisE' is not satisfied by a given control system as a whole or if there is no need for thc overall inverse transfer function to be determined, the theorem may be applied to only part of thfl sy5t!:lm. Then it is shown that the stability of,~ control system having tW() flee parameters can be investigated by me;mS of the Wyschnc-gradski diagram on the coordinntes of which arc plotted the free "ystem parameter~_ Regions of stability, if any. an~ found OIl this diagram_ In a region of stability th(' free parameters an, given such values that the effectiveness of the control "Yotem is satisfaetory_ For obtaining the relatively mO'$t favourablc adju;,;tment in a region

of stability, "orne suitable pcdonnallc(o criterion i5 n~(luin,~d. In particubr, till' integral of squan~d error critcrion i~ di~G\N;(~d m()1"e

c!os(-'\y, which criterion can I)(~ cVllluatcc\ eithe, by nl(~ans of an

(d(~clronic analog compuLer or by aritbmetic

Finally, tlw Bode diagram is bridly described. Herewith, ;1. ~uitablc "d. of values for the fn.'(·· system parameters m;.\y IJt~ found

by applying :tpart from ;\ stabiHty criterion SC.llTl(·' rksign criterion

on Utis frcquencY-f(':-;POIISc b;t~i".

In Chapter 2 1h~ combination of t.he control amplifi(·r ;\ml the

~tabilizillg network is d\'alt with as a single unit. First thc multiple

illput to output tnl.ns(m· function of tb(~ control o.mplifier alolle is derived, tl](/ IIlagnctic coupling b(-ltw(!cn the varjoll~ control-field winding" beiIlg takl,n into account. Starting from Lhe obtained n,htionship the tr,\n,kr function of the control amplifier with i;h() stabili&ing wltwork is found.

Tlll~n a c()rlsilicration is givell for th(~ reali&abiiity domain of the

~hbilizing network al:;o in connection with the) requirement of ;1

lirnitcd clissipatiu)l. Taking into account th(~ condition that tl\(·~ nd-work pJ.ramctel"." ('!\eh of them being in a ccrtain rebti.op-,;ldp to I he transfer-fnndion parameter:>, must be physicldly realizable ,mel that the (lissipation in th,\ network under :stc,\.dy-sl8.Le condi-tions should be lwlow a permissible amount. limits are fOIllHI within which thc LI'clllSfel"-functioll parameters are to be chosen.

l~r()rn tlillse considcrati()rIs fO!' the stabilizing t.t<J.rlsfol'mer it follows that, fOI- obtainillg cither a minimum dissipation or :J minimum

~iz(·, ()( the transfOn1wr, a quantity of CO]lp(;r as largc as po~sibk

is to be used for the, sLl.bilizing field winding of the control ampli-fier. In the case of the stabili1.ing He circuit it I:; found that, in ord.er to obtain a minimllm size of the capacitor and a mini.mum dissipal.ion in the potllT\ti()m(~ter, if any, a )Illmber of turns for tlw stabilizing field winding of thc control mnplificr as large a~ p()s~iblc

i" rlu"iI'<lhle. In contrarli:;tinction to the silwl Lion with the ,;1:abilizing tr<Lll:;former, tlw win) section of thi, winding is subjed to :l lower

limit; a ~('dioll exceeding this low(~r limit implic·'s ,ITI inefficient US(~

of the copper material. From comparative con:;idcrationg it appuHs that tIl(: ~tahilizing transf0rl11er is to bt.· prdern'd to the stabilir.ing

He

circuit in many n·sjlccLs. It appeal's tmtltcr that, jf ,\ solution with it network o( ()ne type h,\~ been found, a "olutiorl consi.sting of a l\(.~twork of the otl)('1" type )JI'oviding th(·~ sarnt: control pn)lwftics with tJl(~ same stabilizing field winding rr\oly not always exist. This

chapter is concluded by a procedure for determining the values of the transfer-function parameters in such a manner that these values result in an acceptable dynamic effectiveness of the control system and, moreover, correspond to network parameters s;ltisfyi,ng the conditions for the realizability and the dissipation,

Chapter 3 contains the design of the stabili;>;ing transformer, the values of the resistances and the coefficients of self-inductance of the primary and secondary winding ,end the magnitude of the prE:-magnetizing current in the primary winding being given. On account of the premagnetizing current it is generally defdrable to provide the magnetic circuit with an air gap having a most favourable length, Two cases are considered here_

In the first ease there is a complete freedom in choosing the ratios between the dimensions of the iron core. By means of the theorem of Lagrange multipliers the ~;onditions are found for obtain-ing minimum costs, minimum weight or minimum volume of the transformer materiaL These conditions arc qualitatively the same for the three criteria. Due to the nonlinear magnetic properties of the tmnsfotmer iron, the ratios between the dimensions of the iron core depend upon several data which may be combined in one single parameter. Moreover, in the case of the minimum-cost criterion, these ratios naturally depend upon the relative price~ per unit vohlDle of copper winding and sheet·steel laminations; in the case of the mirlimum-weight criterion upon the rebtive ~pecific w'light:;. It has been found that one of these optimum ratios can be determined from two nomograms, the othe, ratios following directly from this one, The absolute magnitude of one of the di-mensions of the iron core is obtained from another nomogram; finally the design of the primary and secondary winding is obtained readily.

In the second case a constraint is given in the form of the prescription of the ratios between the dimensions of the iron core, of the punching for the transformer laminations or even of the whole iron Core. Assuming that the winding window should be completely filled in the first two subease5, the conditions for minimum material-volume are readily determined by the requirement that

a

5uitably variable geometric dim!:'l1sion of the transformer should attain a minimum after which the stabilizing transformer may be designed in a simple manner by U'sing graphs. Finn.lly, ebapter 4 is devoted to two calculation examples illl)s"

trati I'lg the fot'cg-oing and to the experiment::; for verifying the re!;ulh

tlJ(:~rc()f. The example'S compris(~ a current r(~gulating sy~te!11 n.nd a

voltage tollow-up syst~ml. In these :>y~h!fllS the first tW()

amplifying-!>tagcs have b(~(~rI realized by an ampliclyne, wIdth energized an c'xcit(.'r which, in turn, enc,~rgiz~,d a d-c generator. The arrangel11~mi:

of the {:urrcnt rq~\lluting system comprised a stabilizing trans-former c\csiglwd t() this encl. An flcccptablc dynamic dkctivcncss

of the voltage follow-up Sy~h'Hl was obtain(~(l hy a stabilizillg

He cirelli L It (lppl~an,d thd: a :;,ltisfactory awccmcnt exish l.a-iwccn the experimental results and those obtained with th(·~ aid of an

Chapter I

TRANSIENT PERFORMANCE OF A CONTlWL

SYSTEM

1.1_ INTRODUCTION

As is well-known, electric machinery arrangements often COll-,titute control systems. The control systems to be cOD.:>idered here can be defined as systems in which the outpul ,;ignal (controlbJ variable) or some function thereof is fed back continuously for comparison with the reference input (which is some function of the input signal) and in which the difference between these two quan-tities (error ~ignal) is used to control a source of power for mini-mizing the deviation of the output signal from its desired value. I'urtherrnore, according to the prnminent function of the control system a distinction may be made between follow-up and ,egI1-lating systems, which arc principally characterized respectively ;lS fol)()w,;:

In the follow-up systems the output signal sholl\d follow as accurately as possible the input signal which it'> cssentblly variable. In the regulating syotems the output signal should t(>m:lin as con~tant as possible in spite of certain disturbances in the system, the input signal being set at a fixed value. In accordance with the definition of the control system given above, at least one feedback loop must be present. FeedbRck may give rise to o:;cillahons of increasing or constant amplitude in any of th(" time-dependent variables, the first case being termed instability and the latter marginal stability; for our purposes both (~ases make the control system of no value. If a control system docs not produce spontaneously oscillations of increasing or constant amplitude, the system is said to be stable. However, not all stable system~ a,n~

in the output ,;ignaL The extent to which the control sy~tern is performing its function in this re!>pt~ct i~ dcsignated "s the df(~c'-ivl'­

nes~, Tfw dyn;lmic effectiven(~ss is considered to b<:', s,J.tisbctory

if some specifications an~ mct by the transient r!;'spoIlsC to a specific input or dist\lrbanc(~ signal.

The transieIlt response of the cont.I'oi system will be stndied here for a. sLldden chang" in t.h(; input signal or distmbance signal. In order to makt:> rna.thcmatical expressi(~ns easy, the step-hHl(:i:i()!l changp; are supposed to OCC\lr only once and to b(j

,n

small, that the system may be com,ickred to be a linear ,ysi:em in the restricted range of state,; ttn'()lIf;'h which the sy!:\hlm pa~se,~ in COIl,(~(pleTlCe of the di,turl.HUh~C.

In ,;ynthesizing control systtorns thc ovcmll tn\flsfer function of tllp syskrn or of p"rt th(~re()f (for instance in c(Hlverting block elia· grams) may be n~qllired. A theorem will bo given for obbining the rcciprocal (:xprcs,ioll of thi~ ovt'r,lll transfer function, the so-called overall inverse transfer 'f\111dioIl, from thc transfer functions of the

component parts,

It is of frequ(mt occurrence that .. n acceptable <:ff(~div("Il(:~ss of a system cannot be achieved withnllt special mea';llr(:~S. In this case we can consider to provkh~ the system with a sLabili:oing ndwork by which the ovc:ntll tn:tnsfcr function comprises a Hum b(,r of new

paramet(~rs, which in principle can be cho8en. with freedom and,

tfwrdoI'c, are designah~d as free panUJ1!.~u:r". If in general n is the number of these p'l.mmctcrs, then n:giow; of stability and instability can in. wmcral be indicated in the tt-dimem;ional space having coordinates on which thli parametcrs are thought to b~ plotted. In the literatun:> SlW(:i~il attention is given to the ca~(~ 1'1 '-' 2 and,

in addition, Uw cocfficients of the denominator p()lynomial of tll(~

overall transfer function are linear functions of the free paramet(~rs

[1{1-3]. TTl this case the configllro.tion in th~ two-dimensional space bears tlH' name Wyschnegradsky diagram. In view of tbe bet tho.t this diagram will h(~ applied to the investigation of the stability of the follow"up system descrilH\d in the calculation exampk;, thi,; subj(:ct wlll be treat<~d more closely,

After finding a region of stability the free parilTnders should b(~ giv(~n such values in this regioll, that the dfeclivcncss is satidactory, For comparing tlll': dynamic effectivencss with various o.cljustmento; in a region of stability, use m\lst be made of ~ome suitable mea:mre of errol'. As ~:t criterion for th" optimum tnLTlsient response to some

specific input it may then be "tated, that tbis measure of errol' shall be a minimum (performance criterion),

Finally, we will briefly discuss the Bode diagram which will be used in obtaining an acceptable effectiveness of the regulating system treated in the calculation examples. The application of a.

stability criterion and some dt,sign (~riterion on this

frequency-r!'sp()n,;~ ba,;i" may lead to a suitable set of valll~'; for the free system paramders.

1.2. OVERALL INVERSE TRANSFER FUN CrION

Fig. 1.1 shows the generalized block diagram of a control system. In this system the influence of the input sign,d R travels through the blocks having the transfer functions G./, (i = I, ... , n), briefly termed (;rhlocks, along one single path to the outpllt. Tl)(' ontput signal of block Gt is fed back through block H"j to summation point

c

L - - - 1 H , c , I - - - l

Fig. 1.1. Generalized block diagram with one single signal path froTT! inp"t to outl',)t.

j, (j = I, ... , i), the output signal Ej

*

of which operates as an

input signal for block Gj . The output signal of t.lw control system IS denoted by C.

The overall inverse transfer function RIC of the control ~y:;tem

I'an. be found as the sum of a number of terms. Each of them corre"ponds to one of the possible paths from the output to the input. On such a path ,J" G"block is traversed only ag<~iu~t and an H-block only with the direction of the signal. The term concerned comprises a number of factors, e,J"ch of which

corres-1\1: Note (hilt 1;~1c.~h time-dependent val'i;"J.ble:-. i':S dewJtcd by a latin mallu:-;rHL~ and Ih~

ponds to a (;- or H-function found on Ow givcll path. A fudor corrnsponclillg lo a c.:-functiou or

,IT!

II-function is eC[11al to tlllO

inv(~t's(; value' of the' function or to LhlO function itself n~speclivdy.

The proof of this theorem wili be shown h,'n,' by induction. With rdcrel\c(.~ to fig. 1.1 the foliowing n~lati()n" are readily found:

(1.1) £} ~ (;J-dlj-I .. ~ (;'#;,11:.';,

U -

I, .. " 11), (1.2)

.1, J where

and

Equation:> (l.2) ~Ir(~ (kvdopecl as follows:

(;3H~,1 F~ 1 ••• +G,,-lHn·-1,I.fn -1

+

(;,Jln .1 E,,--N.

-(:llil I (I I (;2112.2)Ed- ... --[--(;n-lffnl,2E,,--1+

1 G.,,1-l'll,~ h'/i--O (1_3)

(;" II~n"1+

·+·(1 +G.nfl"",.)Fn=O

Now con"ilkr lhe dctenninilili.

1)(le)..,.., I-I-G1HJ ,1 (;J[~,L' . ·Gk--··zH/>;--~,) ---C1 1-1-(,'2112,2- --Gk-~rrk--.2.2

o

-(;~

..

'(;k dh-"2,3

o

o

o

o

-Grc -2

o

(;" llh 1.,1 G k-1H k-l,~ C; f,;-1 HI,:_.1,3 (;/cHIf,1

(;,Jh.3

(;kH />;,:, 1--1-'(;,,-1.11,.-'1,/>;-1 (;/{}l k.k -1 ---Gk -- 1 1 I (;kJl k,l.;

(f ttl(' cofadol' of the clement of til" 11.11 row and t hi:~ ii,l> column of ])(lr) is designated by

f)Y;',

it is rcadily fOll11d thaI.

/,---1

nY'I; -"-'

DU-l). I

r (;,.

'I j

( 1.5) which rd;dioll abo holds for

i

= I, if we dciine ])(0) -,-- 1. In partlelllar, we have 11 .1

D\':;,=

I1G

j • (1.6) 'i. -1

s

_ (1.4)

By expanding DOC) by the la~t row we obtain further

D(k) = (1

+

G/cH/c,k)D(I,l)

+

Gk-lD)'~;H'

After expanding

D1!:'l,-1

by the last column

this

relation turnS into

/;-1

D(k) = (1

+

GkH k,k)D(i:-l)

+

G/;-I L;

G,Jh,JD\;·;.::_\),

i··l

which, by llsing (1.5), becomes

,. k

D(l,) = D(k--l)

+

:E

H",jD(j-I)

II

G,.

j." 1 ',: -i

(1.7) Considering (1.3) as linear equations in E 1, .• " R II, we get

D(I,,)

E =~H

" D(n) ,

hom which, making use of (1.1) and (1.6), the overall inv(~rsf.

transfer function is found:

N. D(n)

C I, (1.8)

II

(;j

'i·--1

Taking into 2.ccount (1.7) for k = n, this relation may be wr'itten a,

it D(n-I) DU-I)

_. =, - - -

+

~ H" j ...

-C n }'-1 '. /'-'1

II

Gt

II

Gi

j .. 1 'f. ~,

or, if Ck denotes the output :signal of a control system with tlw

blocks G 1, " ' , G /; and the aSf',o6ated H -blocks, a,

R . 1 R n R

, ... _- =

-.---.+.

Z

Hn,J-'-:

en

(rn Cn -) ; .. ·1 C,-.1 (1.9) use being made of the fact that (1.8) naturally abo a.pplies, if herein n is rephlced by k and if per definition Co eq\lld:s R.

Now assumc that the theorem applies to a number of G-blocks equal to n - 1 and less, then it is found from (1.9) that thc theorem applies to n G-blocks as will be se"n from fig. 1.1, if we verify the con"equences of the addition of block Gn and blocks H n,;' (f = I, .. " n) to a system having only n - 1 G-blocks. If 1'1 ""'" I, (1.9) turm; into the obviolls relation

R

1

- =

-G-l

"

+

Hl,l.

From llii" it <1ppear~ that the theorem is correcl for one G-bl()(;k <1]"[(1 lherefore, it "ppli!is to 2, 3, etc, G-blocks as well, i.!~. the theorem is gelHlrally valid.

1.3. WYSCHI'II\(;HA()SJ{] DlA(;W\M

Let th(1 (l(morninator polynomial o£ the ovendl ham·der function of a control system b(·'

P(s) 2,; a,sl,

,. \I

(I. 10)

wlwtc s designates th(~ complex v,.-6abk of the J _aplacc trans-formalion and a" (i = 0, " ' , n) am linear functiorl~ o[ two free

parameters % and 7', which functions will be writt(:'n ,IS

(III )

It. i~ well known tll,!.l stability b a~~l1rcd if all tlHj roMs of the

(~ha!'acteri.stic (I([llation - whicb m;IY be obtained by l:q uating (1,10)

to );cro '. have negative 1"(1,,1 parts; root::; of tlli" kind will [)(' htidly l!:'nn(;d sbble roots,

The number of st.able root~ depend" upon the values of Hand 71.

This d('p("nckncc can be repnls(~nled in a diagram in which 'H and v

ani the coordinates lInd OIl which am indicated regiorb having" (li fkrcnt nUl1lb(:1" o[ stable ro()t.~. This rcpres(:'lltation is designatnl

as the WyschrH:gradski diagn(lTl.

1.31. Para11utri(' cljlwhons

In the Wy:;chncgracbki di<l.gram, region~ having a different number of ;;lahle roots an',eparated from each other by a boundary curv(' 011 which at lea;;1: one root of t.he characteri,tic equation hus

only an imaginary purl. So, thl' i.·cluations in u ,l.Tld V detl'lTTlilling

thi" boundary curve may be obtained by :;ubstituting iT! the c.haracteristic equation for s the purely imaginary variable

iw

(f lwing the imaginary unit):

P(jw) = 0

Fron1 this equation, it [(,!lows in th!: first place, that

or k

L.

a~i( - (ilZ)i ~ 0 i 0 (1.12) (1.13)

with

n

if n is ev!)n

if n is odd and secondly. that

or with I W ~ aBHl(- ( 2)' "..., 0 i-O n-2 if n is even l= 2 n - I if n is odd, 2

SubstitutioII of (LlI) in (1.13) and (Ll4) yields Ueu

+

Ve7J

+

We

=

0 and U aU

+

Vo7J

+

W 0 = 0 respectively, where " I Ue = ~ a u,2t(- (2_{)'i, U} Q = 0)

1:

UU,2i+l(- w2)1; ":--0 "j=() k ~ (1.14) (1.15) (l.I6)

V. =

1:

Uj),;lf(-w2)i, Va = W

L

av,2i+l(-0)2).; (1.17)

bO ,---0

/, I

We = b aw,2i(- (2)i, Wo"'-' (I)

1:

aW,2Hl(- ( 2)i_

~-I) '/;=0

Solving (LlS) and (L16) for u, and v we find for the boundary curve the parametric equation~

and where D", U = -D D" 7 J = -D ' D'Il; = VeWQ ~ VoWe,

Dv = UaW. - UeWo, D = U"V" - UoVe,

1

) (118) (U9) (1,20) (L2I)

Since in (1.18) u and 11 are ,~vell functions of (I), t.hc graph of (1.18) may be tnlccd after substituting successive n~al values for (I) horn

;.;ero to pJ ufi infinity.

We now cOllsiril'r a value of (.Ii for which two of

ttw

(ld(,rrninLl.llts

( 1 . 19), (I. 20) and (1. 21) va.1!i~ll. For such it value the third d"krm 1l1in<lnt also vanishes and in the U1I planc a so-called sinf.{ular line

(~OITi.:sjlm\(b lo it. Such line, also belong to tht: graph of (1.18)

and the equation thereof may

tw

obtained by sulisi:ituling the concerned valur, of OJ directly in (1.15) nr (1.16), It slr()l.Ild be noted

that the values of (() equal to ;.;uo and infinity always c()rn~sp()n(l

to singular lilll~s. the cqu<ltiolls ()f which an: found by eqmdilJg un and an. wri1 kn as functiolls (If 11 and 11 according to (1, II), n:sF'cl-iveiy t() zer(). Fig. 1,2 shows a graph of (1.18): curve a b(~illg lhe

IlllTlnsillg"ular v,rt and tire. lincs b, c ;1.11(1 d the sing\llar par'L

Fig. l.2. \Vy~chl'lq~,-adski dii.Lgl'nnl; curv~' a I'nl'fusontillg the !i.()Il-singu I;~r

pnrt (UpOll wliil·1i pUilll /1). til(' lin"s b, " al]([ II the sini;l\l<tr part. I .32. J ntcrprct(l!ion

An interprctnt.ioll of lhe ~raph of (1. 18) describ(~d irl lhe previous

~i.'ction will 1.)(' givt'll by means of the following criterion,

Tf J) a.sS\llm~s a positivll (or' ncg-ative) v,due for SOll1l~ val Ill'. of (JJ

- to which, thel"i~r()n:, correspomb a (,crtain point of the

non-~ingubr part of tlit: boundary CllrVl' --. then, facing the direction

right-) hand side of the said part of thiO curve in the proximity of the point considered has one stable root more than the right" (or left-) hand :side; the imaginary part of the [oot concerned is, at least approximately, equal to j-timcs the considered value of 0). It is supposed in this case that the '14 and v axes C(ln~titute a

(u, v) ~y<;t(cm (as shown in fig. 1.2 and not it (v, #) system). In the following a proof of thi,; criterion is given ill a m()n~ concise form tkm that given by K. Tashiro, which is referred to in [RJ]. Consider a small region in the u-v plane around a certain point A on the boundary curve corresponding to some value of w for which D d()(~~ not vani::;h. If in the region considered one moves from a point beyond the line to point A, a pair of conjugMe complex roots of t.he n roots of the characteristic equation at the firM mentioned point passes continuously into a p;]jr of opposite purely imaginary roots. Let further be considered the one of this pair of conjugHte complex rooh which at A becomes purely imaginary and equals i-tirn"~ tbe considered va.lue of 0>. At point A, the direct.ion in which an in-(:rt:'roental change ow of w from the Gonsidered valu8 wit.h the real part a of the root wno;idered equal to zero i:; positive and lIH~ di,ection in which an increJJl8ntai change 811 of a with constant w is positive define the positive directions of the axes of <t local coordinate syst8m (8('),80) or (iJo',8w) having A for its origin. In the same point A we can imagine a local coordinat8 system (8'11, ov) sirlce a (01!, 8u) system would correopond to a (v, t~) ~y5tem which

ha~ been exd11ded

*.

The aforesaid criterion can now be altl,rnativdy stated a~ follow5.

If 0# and OV constitute a (iJu, 8v) system <I.nd

n

is negative or positive, then [Jw and oa constitute a (ow, 00-) or it (00', fJwl

system r8~pe('.tively (see fig. 1.2).

However, it is known that with a (au., ilv) system the functional determinant o(U, v) (hi OU -ow 00 (1.22) - - - = _ . " . o(w, a) ow 017

is positive or negative depending6n whether iJw and oa (:Ol),;titute a (00), fJa) or a (ila, 8M) ,;ystem. Therefore, it is now only to be proved that the functional determinant and D have opposite signs for the considered value of w ,d point A. This m,~y be shown by ~ TJ,c ;ituatiol1 jn thC:! (,.;1.$C of ~I ((Jvj c:rt) :=:.y:=:.tC!m will bu iOU'lHl ca~ily hy ~W(l,yiq~ till.:

expressing in (1.22) the parti;!l derivatives of u and '/J with re~pf\d to (I) in those with re~pf,(;\: to IJ for the considered point A; the

recjuired relations can he found as follows.

Let a' -I- jill bu a zero of (1,10), then the polynomial P(s) can be

divi(ie(l by s - a' _ ..

iw.

The quotient, abo a polynomial in s, is

(knoted by Q(,-). Then the following' identity holdii:

F(s) .. ,' (s _ .. fJ - iw)(!(s) _{( 1.23)}

It is evident l.h;it the coefficients in Q(5) <lr(l to be consid<~rwl

as functi()n~; of wand cr, MOI'eovcr, th(" coefficients in P(s)

depend l1pOIl OJ and cr since in principle a variation of the

para-meter,; '/./ and v invoiv(", a variation of OJ and a' too. However, the

codficlcnts au",!, U1',l, and aw,i,

U

= 0, ' , " n) are independent of (I)

and (Y. Partial differentiation of (1.23) with [hP(~ct to w yields

oPts) . . . 8Q(s)

--~.. -- - lQ(S) 1 (s - IJ - }(ll) . '-' •

(IW 8m

If $ i~ chosen to be e(lllal to iJ -I- jU), we obtain

J

(lP(s) } , .

t

.. ··· .. ·.

ow

s=a+jw = - JQ((Y -I- 1(1))·

Similarly, hy differentiating with re~pect to (j howev(cr, we find

From the h,t two relation:; it follows generally, that

(1.24) Applying (1.24) to the considered point A, ,It. which 11 = 0 and

OJ has a cerbin valu~, and iiubscquently equating the) real and

imagimlry parts on both sides of the equation obtained, we get

analogous to (1.12) through (1.17) the relations

iJ·u 811 8u OV U. ---

-I·

V.·--- = -, Uo-;- - Vo ()w

ow

()(Y f)'j (1.25) and (!U 8v Ua _.;._-

+

V Q dw 8m 8u Dv U~·-;-

+

V ,--(!cr e (j11 (1.26)

respectively. By solving (1.25) ,md (1.26) for

au/ow

and

au/ow,

we deduce

and

a2t UeVe

+

U"V o

ou

V~

+

V~ all

ow

= - - - " .. D - -

a;; -

---v

ocr

00)

(I;

+

u~ OU. U,V. ·1- UoV"

ov

D

~+.

D O(J

Substitution of these expre~sions in (t .22) for point

A

gives, after some rearrangements,

o(u,

v) o(m,a)

( u~--(J'u _ocr

+

Ve-,-(7)

)2

+

(au

Uo' .. -

+

Vo -.-iJu

)2

iJo (Jcr 00

-.n

(1.27)

Since in (1.27) the mlmerator of the right-hand member is always

positive, the functional determinant and

n

consequently kwc

opposite signs for the considered value of co at point A, Q.E.D. The situation in which One side of the curve ha:s one stable root more than the other side is indicated in the graph by providing the Curve on the side with the greater number of stable roots with a hatching. Since u and v are even functions of w, the curve is traced twice, viz. at <l variation of co from zero to plus infinity and from minus infinity to zero. However, the sign of D alternates at (J) = 0

,end (I) = 00, so that the same sidle of the curve is hatched once more (curve a in fig. 1.2); this double hatched side then has two stable (conjugate complex) roots more than the other side.

Each singular line is also provided with a single or a double hatching according to whether one value of IV is associated with the line (see the lines band c in fig. 1.2) or two (opposite) value" (~ee line d in fig. 1.2); the first cage applies only to the singular lines corre,ponding to OJ = 0 and (v ==; 00. On which side of the

singular line the hatching is to be made, is evident from the situation of the hatching of the non-singular part of the figure in the proximity of the point which is located on this non-5ingular part and at which OJ has the same value as on the singular line concerned (sec fig. 1.2).

Further points of intersection of the singular line., with each other need not be taken into account.

After the diagram has been provided with the aforesaid lHdching, starting with the number of t>table roots in }l certain region the numbers of stable roots in the other regions can be found without

any difficulty (in t11l~ figure the.~e numbers an~ indicated in circles).

TtJi:4 i:-; (:vi(lcllt from the comicliTatioll of a tnln:;ition from one region to ,ulo1 hCt' ero~sing a bOllTldary not provided with a hatching just

before thc' boundary, bul provided with a Hingle OT Ii double hatchint,;

just. a.ft.t:r tllC' boundary. This transition TCMlltS in an increase of the numtwr of ,table. roots by one or two n'spcdivcly, while the

nllHlilcr' decrcaf>cs similarly, when th(·~ t.ransition t,lkr-'S pbce in the:: opposite S(:ll.'(~' A n:gioll in which tll(: number of ,table root~ appear', to be equal to the: degree of tilt: characteriotic equation i~ a region of stability. It will be ckm jkll. such a n:gio!l may b(' lacking.

1.4, PEHF()!<MA!':CI': CIllTEI{IA

For oblaining the optimuIll tram;i(.~l1l. respons,' ill <1 region of

stabilit.y some performance critc:rion mllst be Icppliec1. In the,

lit.cmlure many p(~rf()!'mallee crit(':ria are kn()wn, e.g. I H4 .. ·6l-How(:v(::r, mo,l of lhem s(:','rn I'cally u~dlll only wh<:')l a ~uitable

computc!' is available. 'What cri.terion is to bc~ a.pplied depend~ nn

the spl:'cifi(~:di()llo of the control ~yslcm \Imler consideration. One of

t.1I" crileria, ck:oigmdul as the integral of squan:(l error critc:'rioll, Call be evaluate:(l by means of i:1.1I ,·kctronic analog comput(':T as well

as by ,~rithllldic. As we will apply this criterion in th,' calculation

cxarnph~, i I: will be cliscu:-;~(,d further. This criterion call be statnl

<\:,; follows.

For a step-function in put signal the squan: o[ the c\l'viatioll of '-he output ,-;ign:tl f[,om its final value, intc'gratccl over tilt) time from the st.(:p to infinity, "hall be a minimum

Irn

91-Pn)c(',:dillg in this way, large dcviati()Tl~ both in tl}(l positive un(l

in the ncgativ,~ :,wns8 will normally be avoided.

lAl. A -nalor; cmnputa meclumization

In an electronic an~\log computcrlH10. III all variables ()ccurring in the control system are n-~pn:scntcd, in gell(~ral, by d,:d.l"ic volta.W~s; how(·:vcr, the time remains the in<.itlpCndent V<lriablc.

Starting from the block diagram with lhe various tro.ns\(:'r [unction:> of the building blocks of a control Systl'nl, (1 diagram of

the compnt.m" sdup is developed in wtlich the compuh':r components are eO]Jm:dccl in acconhnce with the block di<Ignlnl. It may he

1I()1c(1 hcre that the: transfer functions of the huilding blocks should be realized in such a manner, that the adjustment of en.ch of th(~ frc(~ pammet<:rs to be studied is as dimcl as po",ihlc [R 12, 13]. Ttwn tlt(: quantitative a.djustment of the kn()wn paral1let(~rs may

be performed, so that subsequently the analog computer providet> the possibility of studying the free parameters.

for this purpose a step-function input signal r(t) is fed to the analog of the follow-up system after which the course of the output signal crt) may be visualized by meanS of a reco,ding device (see fig. 1.3). A" all. example, the integral of squared error criterion will be applied hcrc. The function crt) i~ added to an adju~table voltage (~(lU,Li t.o "'- c(oo), after which the sum operates as each of both inputs of a multiplier, the output of which is integrated over the time, Finally, the integrator output is recorded fb fl. function of

"" (J"'J/Qg of ·toliOW-lJp system

/ !cm-

c{wJj;dt

'I

Fig. ] ,3. Ana]og computer seU,p for the evalu'ltion <If th" int"fintl of oquare,] error criterion,

time, The fact that the integration cannot be (~,<rried along to infinity i~ not an obj,cc;tiol1, !t jg sufficient to carry out the integra-tion up to the moment upon which the tr'"f\;;ient. phenomena may be considered as practically died out; this moment can be dMived hom the visua1i2;ed course of the integrator output or of the output signaL The free p,rmmeters are adjusted in a region of ~tability so that an (absolute) minimum is found, Then the set of values of the free parameters is read from the computer.

1.42, Mathematical expressions tor the integral of squared error Let the overall transfer function of a control system be

m C "j,::(j

L

b,s' (1.28) R n

:E

als·' ,,-0

in which rn does not exceed n, a condition which ·is easily satisfied in practice. Then, if the input signal R is a unit-step function at the time origin

t

= 0, the output signal C may be written as

m

:E

hiS' C = _~.~.:

_.,

0 ... __ , s

1:

ajst i-() (1.29)

Assuming that t!tf~ sy~Lem is Rtablc, the Meady-state condition is expressed by

Du

c(=)C'""-.

ao (1.30)

The Laplace transform of the deviation of the outpuL signal from

its final va)\l<' has, hking into account (1.29) and (1.30), the following form: with '11.-1

l:

b~si 8{c(t) - c(=)} = ~.--, 2,; al.s! ·1 () , /;0 b.,: "" /lIH - ---aHl, (i = 0, '-', n - 1) ao bt-t·l =

°

if i

>

m -_. 1. The integral of squared error

j ""

f

{c(I) - .:;(oo)}2dt

()

(1,31)

(1,32)

(':<ITI b(~ evuhl'ltt~,1 ill tlefrrlS of the coefficients ai,

U -

0, ... , n)

:111(1 I);.

U::::7

0, .. '. n - I) with the aid of n-row determinant"

[H.14] ~lk follows: where (- 1)" 1 II' I =....!....---'----H ' 2an an-l Cl,,-3 an a n-2 J-l--,,- 0 an 1 0 a'fl 0 0

...

(Hurwit;c' determinant) with

( 1.33) al-p, a2 n Cl3-n a4-n ao

while H' is obtained from H by rephcing thr' clements of the

where i

B

j =

.:E (-

I)k b;~"b~ 1,-0 with

b;

= 0 if

i <

0 and if

i

>

n - I.

For numerical c'l.lcuiations of (1.32) with n c::;; 6, the following expression can also be applied successfully [R 15J :

1 = __ • - a ds

+

lim _ s b , - b"

I

f

g2 g2 ,,2 ,,2

4nj

f

ef 0 $'->00 21/ Q

(L34) r

Herein te (or g.) and to (or go) designate the sum of the terms of the polynomial in the denominator

f

(or numerator g) of (1.31) with even and odd exponents of s respectively, while

r

denote'S a closed path of integration lying in the s plane, which enelo!;!::, all zeros of

I

Q but none of

h

In (1.34) the contour integral may be

found casily by applying Cauchy's residue theorem, while the evaluation of the limiting expression does not give risc to any difficulty since the degree of the numerator does not ex!;eed that of the denominator ao; Can. be :seen from (1.31). Various methods for the calculation of the integral of squared ertor are known in the lite-mture, e.g. [RI6-l8J.

1.5. BODE DIAGRAM

The block diagram of the regulating systems considered here is frequently capable of being converted into that shown in fig. 1.4.

In order that the output signal C should be as independent as p08sible of the disturbance signal R' at a constant input signal R,

R'

~

c

Fig. 1.4. Converted block diagram of 11 regulating sygtCIn. the variable E' - which i'S the sum of C' and R' - must rcm~l.in as constant as possible_ In other words, C' should follow the variation,; of R' in an opposite seIJ.~e. Consequently, when studying the transient performance, this regulating system may be considered as a follQw-up system with R' as the input and C' «:; the output signal while R i::; considered to be zero.

The tramient p,,'rforrnancc of this follow-up system can be

analyzed by means of characteristics in which the magnitude and

tiJ(~ jlha;;<,~ angie of the open-loop fn~(l\lency-n\~l)(l1"l"\ function, which nlay be obtained from the open-loop transfer function

(;lG~fl by rcpla6ug s by a purely imaginary arg\lment i/o ((') d('not('s

the angular fre(!1H~ncy), anI plotted against the frequency. The

trans;(~nt bclUJ.vi()ur of the system is c01Upit'tely dd('rrnirwd by

these curves

ll{

19l1f logarithmic SC,lieS an: used for the magIli tude

and the frl~quency, tlles(~ ()1Hves together arc designated as the

B()(1c diagram. Usually, B:rigg-lull loga.rithms arc llsed; the magnitude Hnd tl1" phase angle an:' given in decibels and degrees respectively. In the Ijtemtun~ mally s1.:lbility and design criteria on this fre-'lucncy-n:sprHlS(1 basi~ an: known, e.g. lEI, 20-22]. A Vt'ry simple Btability alld d(~~igrl criterion can be given for those ~y~tem~ which ,In.' s(abk l.lp to some value of the loop gain and nn~tahi<1 for hrger y,\lucs or which anI shbl(~ [()l' all vallics of the loop gain. Two

Qll:l1ltities related to the Bode diagram may ttHm ~w used as criteria in the synthesi", Om, of thcse quantities is designated as th(~ ph as,,'

Tnargin and is commonly defined a, ttl(, sum of 180 deg and the

phase angle in deg for th(~ fmqlH:llcy at which the magnitude "quais unity. The othn q\lantity, (ksignated as the gain margin, is cit/TimId ~\S thl' mimI'; vahu' in dB of the magnitude when the fn\([\](erlcy is such th,\t tlH~ phase angle is equal to minus 180 dcg. Til,) st.ahility crit(\rion can IIOW he stated as follows.

A control ,;ystem is stable if it hit.'; i\ po.-.;it.iv() phasc margin; oth(TWisc it L~ unstable IT~I, 231- For the design criteria, it is rccommcnckd that the phae;c and gain margin should I)(~ at

I(~,M (:irca 30 dcg and circa 8 dB respedively [R22, 24-27J.

These rules may n~s\llt ill a e;d of values for the fn:,- p,\r'I.m,"ters

,\fh'r whi(:h. if [c.quirec1, the effectivell(css of tht: cUlltrol ~yste1TI

may be verified, for it ~ho\lld be I'c'l.lized that the:.' p)la~(' allil the gain margi n ,tiOIl(: of U)llrsC do not provick ~\lfficicnt ba~iK for "yntIH:'sis.

Chapter 2

CONTROL AMPLIFIER WITH STABILIZING NETWORK

2.1. INTRoDucnoN

In electric machinery arrangement:> con"tituting control systems

a l.ow levd of the error power is usually required for obtaining ,~n

acceptable effectiveness and small control device~. However, a hi.gh p()wer amplification ratio may be implied. which lead'S t.o the usc of several amplifying ;;tage:;, e'l.ch of which involves a time lag. In cont>equence of this, such a system will often be ull:;tabl() or too oscillatory.

In the aforesaid arrangeIT1,~nb, the first: amplifying stage or the first two stages are frequently realized by a rotating amplifier without feedbilck effects such as the conventional d-c generator, the amplidyne, the magnicon. the two-stage magnavolt, the mpidyne and the cascade arrangement of two convent.ional d-c generators. If in this situation the control system is uMtable or if it is too oscillatory, it hao; often proved to b(· Ilseful i.o provide the syt>tem with a stabilizing network formed by a transformer or

,1

simple

He

circuit connected a, a feedback cirwit around the

above-m(~TItioned control amplificr which has in general several control.

field windings on the first stage lR28-31J, Tue stabilizing network produce", an rnmf from the output voltage of the control amplifier; owing to the derivative character of the network this mmf occurs only at variations of the output voltage which, in turn, are cfltlsed by variations of the res\lltant input mmf of the first stage. This varying input mmf is counteracted by the mmf produced by the stabilizing network, SO that the output voltage cannot follow the variations of the input mmf as rapidly as in the situation without stabilizing network. Therefore. the function of this network is to damp to a greater or smaller extent the oscillations in the transient

performance of the control sy;tl,m; the steady-stat,(, pl,rf()nnallc(~ of the control amplifier, however, is not affected in an essential Wily.

The following considerations will be based upon a cOlltrol amplifier and a slabili,,;ing ndwork of Uw type~ mentioned above. Taking into account the magnetic coupling between each pelir of !.II(, cOlltrol.field windings, the transfer function (multiple input to output opemlional relationship) of the control \lmplifit'r wit.h

ttw

stabilizing network will he determined. lhc is mack of thc t.rar\sf(~r

function of the control amplifi('1' alone, which function will first. bt' derived. Then the realizability domain of the sLl.bilizing network will be investigated; this is of interest in connection with the dnh'rminatiOll of the values for the transfer-function parameters

in obta.inillg an accq)t.able tran~ient perforll1anct~ of t.he control system under consideration.

[n thc\ di'riv«ti(ll1~ th,' control amplifier is S\lppo~(~d to b,' of thc'

two-~tagc amplifier type. The conditions for the case of the

singk-"tage arnplifi('r ,I.TU readily found from tfH-l oht.'lim·.d n'su1ts by

:';1.1 bstituting zero and unity respectively for thc time constant

and tlw voltage amplification ratio of the s(c~cond stage of the two-shW' ,1mplifi,I:'r.

2.2. THANSFEH FUNCTION OF THE CONT1WL AMPLlFIEH

In deriving Ulto transfer funct.ion of the control amplifier the starting point is fonm~d by the configuration in fig. 2,1 ignoring

1.rv'I"

\~1~!

,

,.J

Fig, 2.1. ,Diagram of (I. two-stage control ampli.fkr, the dott('cl l\n('o

indicating the feedback stabilizing circuit.

for the time being the circuit shown dotted. This figure shows it two-stage amplifier having on the first stage a number of Control,

fjcc'ld windings magnetically coupled with "n.ch other. Thc ith

control-fidel winding is connected through a passive, linear network Ni to a voltage sourcc,

U

= 1, " ' , m),

For the first stage the ~ymbob employed are designated as follows'" :

t:.c< = emf of the voltage source connected to the primm:-y t.ermi-nals of network N."

E1 = rotational emf,

I", '-= current in the itl' control-field winding,

Zt(s) = resistance of the il.l) control-fidel winding iflcreascd by the impedance seen looking from the output terminab back into the network Ni , when the voltage ,;ource c!)rlflected

to the primary terminals is replaced by its internal impedance,

F/(s) = ratio of the open·circuit voltage of network N, to the voltage JEw

L,,,

= coefficient of self-inductance of the p.lo control-field

winding,

We< = number of turns per pole IMir for the itl! control-field winding,

1) = main flux per pole cro!;~ing the air gap, u/. = Hopkinson leakage factor,

p

= number of pole pair:;,

a --'- number of p,lirs of parallel paths in the d-c armature winding,

z

=

total number of armature conductors,

n = constant speed in revolutions per unit of time of the rotor, 11 = permeance of the magnetic circuit per pole prllr;

while for the second :;tage:

£2 = rotational emf,

If = field current,

Rr = resistance of the field circuit,

Lf = coefficient of self-inductance of the field circuit,

'T! = LfIR" time constant of the field circuit, kf = rotational emf per unit of field cu.rent.

Hereafter, nonJinearities and iron losse, will not be laken into con::;ideration. It will be assumed that the magnetic coupling between the various control-field windings with respect to each other is perfect and that all these windings have the same Hopkinson leakage factor (defined as the ratio of the total main flux to the

:$ In thi~ chapte!1 without further description G:ach time-dependent varLablc will be

r~pI'~:!.Cr"l.ted by ~t~ LJ,pl~c~. tr\l.n!3fQ1'\1~; while all q1J::Lntiti~~ 'l!i~O in thi~ th~!=;i!i are :!='uPPQ::;(d to be cAprc~~c~' h~ I,.mjt~ o( the r'~ti0Il.~ll:'t_e(1 GiOtai $y~t:ern, \Inl~$g othf!l'wise s,trlterl.

main flux crossing the ceir gap). It will further be assumed tl't<lt till: brush(~s are I()cah-~d in t.he n(~\ltnd :1-01W" while the effects of the coils of till; !lrrnab.lte winding undcrgoing eornrnuLl.t.ion an, sup· posl,d to be negligibk. Furthermore, in the dcrivations all initial (;()wlitl()fls arc CqUH tcd i.n :1-CI"O in acco[da flC('~ wit.h tlw dc·finihon of the concept of a transfer function.

By HlCarh of LlH: IIH:o[CJ\"I of Tlusvenill w(~ find for theil,lo control· field circuit

Fi(S)h", ~ Zi(S)]",

+-

spw,p,,1>,

whill' to t he magnetic circuit the following rda tion applies:

(nQ) = ;1

2: W,),,'

i I

(2.1)

(2.2) In orcier to eliminate the control·field current:; 1,.'"

U

~~. 1, .. " m) from (2, I) and (2.2), ,'qu,ltion (2.1) is fir..,t multipli('d by zo,,,IXc(,,)

after which both members of the obtaincd equation arc totalized for all m ('ontrol·fidd windings. Thus

n\l)

dilnin:d:i()Tl result: is readily [ound:

U';(' I)ping rn:vln of Utl: tdaliml<;hip

pw,;A

=

L".

(2.3)

FLlrther, we: have for the rotational emf of the first stage the relation

Elimination of Ii) from

with

P

E 1 ~::: nzrJ). ({ (2.3) and (2.4) rc:-mlts in 'm l~ Fi(S)

1:

.---.5'----_ E /. l Z/,(s) '" pnz,1w", he. = ... . -{{(j/l (2.4) (2.5)

In Lhis relatioll k,., repreStlnts the rotational emf of the first stage

p<:'r unit curren.t in the 1;('" contwl-fieJd circuit a"> will ;IIlP",]l" from

For the second stage of the amplifier we have KjK) E , . = -lTl which • I

+

fiTf llf K r = -. Rj (2_6) (2.7) designates the volt'lge amplification ratio of E~ to £1 under steady-state eonditions_

Fig. 2.2. G~'nemlizc~l blocl, dillgrarn ,,[ th" C()Iltrol amplifier. Finally. elimination of El from (2_5) ,tnd (2.6) provides the relationship between E~ and E_e,. (i ---,--c I, . , '. m):

'~ kJ'i(S) ]{t

:B --;---

E", _ i ' - I ZitS) 1::.2 =

(I

+

s

£

,L'2. __ ) (1

+

STr) .:"-·1 Z.;(s) (2.8)

On the basis of (2.8) the generalized block diagram n:pn,sentation of Lhe control amplifi('f may be shown in fig, 2.2,

2,3. THAN5F£1{ r'-UNCTlON OF THE CONTROL AMPLIFIER WITn THE

STABILIZING NETWOH_K

We now consider the case in which apart from the m control-fidd winding::; the control amplifier has a stabilizing field winding

to which the output voltage is fed back through <l stabilizing network

N"

as indicated by the dotted circuit in fig_ 2_1 _ The resistance, the coefficient of :self-inductance and the time constant of the stabilizing field winding will be denoted by R8 , L8 and TS

resp(·ctively. H"H'alh:f, the impcdance Z itS) = £', ,;orresponding to ndwork N", (i ~ 1, .. " m) is ",;;urn!;(l tel be practically ohmic. Furthermore, th(:' inlF:·d"r1(.~t· ill consequence of the codficitmt of scJf-indllcb.ncc of Lhe amplifier wiudi.ng bdwecIl the output tc:nninals i" iplOfed since its Val\H~ ;;; VMy much lcs~ than the otl\(~r

impcdances of intcn:"t:; moreover, it should be bornt, ill mind that with til(' \lmplificr type here considered t.hl) la~1. amplifying stage

is provickd with a compensating winding neutralizing exactly tlw

IJ.rmature reaction. l'irmlly, it should be noted that in ekcLric machinery arrangr:mc:nts t.he: control amplifier is ]tmdul by an impedance which, ill general, will be much higher than thtl resistance of the mnplifi/r wimlillg between the output t("rminals, so that the

sai.d IOled cloes 1101 affeet the 'no-low.!' ennditioll phenomena

(kj(Trnil1(~d by the rc\ation~ to be ckr'ivcd.

COl)sidNing tll(~ st.abilizing fielel winding U~ on (m -I- 1)\"h control-fic·]tl winding with

wc' fill(l f['Om (2.8) , 1181'", I 1.(8) , - Aj .~--.. - . -L~

I

K_f

L

K",F,;{s)E", 7 " ' - + I ( s ) , : I 1£2 - ... -- .--- ---.. . ... ---.. ---.. --.--... ( 1 -I-STr. I· ,

sL." .... )

(1

+

STj) Lm+)('~) Tnt his n:latioi\ is (2.9)

/'., the rotational emf of UI(~ firRt stage per unit current in the sta.bili7.ing i'i("ld winding,

R,,, the quotient of k", and t.he toLal resistance of tlJ(~ith

control-fi(~ld circuit and

TIJ tll..:: sum of the timc: GOl"l:;tants of the m (ontTOI-iield circuits.

Fir~t, Ow UISC of the stabiliy~ing nn(:work N s repre:;enting a

stabilizing if,iIlsformer as illust:mt~d in fig. 2.3 will be di';(;11SSed, P(:rkd. rn;.lgnetie coupling bdwnen the primary and t.ltt'- secondary winding is assumed,

For the Rtabilizing transformer the following ;';YUlbob arc used: I1 ~ primary CllrItnt,

HI = resistance of tilt) primary winding,

/:.2 -- rcsist;l.ncc~ o[ the secondary winding,

L l = codfieient of self"inductance of the prjll1fl.TY winding, L~ ~ eodfici(::ni of odf-inductance of thll t'.(:condary winding,

M

=

coefficient of mutual inductance between the primary and the secondary wiTlding,

N = ratio of the secondary to tlllo primary turns.

rr .... ' · · ' - - - ,

L I

_, _ _ _ _ _ _ _ ..J

j<'jg:. 2.3. Dhgmm of tile SlabilLzing transformer.

The voltage ratio FlIl,+l(s) rdn.\.(:s to the no-load condition of tlw secondary side of the transformer, the secondary currcnt being consequently zero. Taking this into account we find easily

SlIM Fm+1{.) =

---y;-

(2.10) and £2 = Rdl

+

STIVI (2.11) with T j ' = LdNl" (2.12) whl'r~o RI , = Rl

+

R", (2.13)

if R" is the retjistance of the amplifier winding between the output terminals.

With perfcct magr1dic coupling between the primary and the secondary side of the stabili~ing transformer we have

M = (L1L~)'

and

Under these conditions rdalion (2.10), taking into m:colllli: (2.11),

tUniS i)).to

(2.15) SllbscqllCIli.ly, we find in a ~impje manm,r

. l?2'(1 -/-81') £", 1.[ (s) = -... -.-1

+

81'1' (2.16) with (2.17) N~' c- R2

+

R.. (2.18)

Substitution 0[ (2.15) and (2.16) in (2.9) yield"., aftf·r ;-,()me

rc,\rranw,nwIlh, when' :md willi !( I( 1 / ST) ~ K", F,(s)E", 'i I ( ... --... --.. -... ---.---.... : -r VTI' ) '-'-"--_" ... .

+-

T"

+

1'.f

+

l'

+

-r,'

+

AN· 1'8 s

+

.. I .. (-reT! 1 TOT·/-TjT+TfT,,' +1','TdsLITf(TcT-I-T,,'?"!')s:)

f., 7;,;1·"·':: .,'--" ....•..• , ](2' (2.19) (2.20) (2,21) (2.22) i{. is the vol taW: llrnplificn.tioll ratio of the rotational emf of til"

first stag(' to llil) vnltn.p;e acros:,\ the stabilizing field windinr; producing it Ilnder steady-state conditions.

Now w(: C()lhiiclcr the case in wliich the stabilizing ndwfwk N,\'

ill fig. 2.1 is rcpn.,sented by it simple RC cirC\lit Il.S indicated in

fig, 2,4. TIl(, syrnbo[;c; ill this figure an) (ksignatcd as follows:

C -- capacitance of th" Gl.pa.citoi',

R" .~ Tco,istall(~" of t:he series resistor, if

,I

ny,

I;

=

rativ of thfl potentiometer resis.Ll nce, to which thfl RC circuit is connected, to the total potentiometer resistance,

Negl(~cting Ra with respect to Rp we lind

F,n+l(S) = ~ and where with I

+

~T Zm+l(s}

=---c" ,

s T = RC R

=

1'(1 - f)Rp

+-

Rv

-I-

R,,-r - - - - - - j

Fi:;r. 2.4. Diagrmn 01 the stabilizing RC circuit.

(2_23)

(2.24)

(2_25) (2,26)

Taking into account (2_23) :'tnd (2,24), relation (2.9), after _,ome rearrangements, turns into

'"

Kf(1

+

S'T)

L:

Ko,Fi(s)Eol E₂= - - - " -- ."

---_!:::_"---1

1-

h

+

Tf

+

7'

+

K~7's')s

+

(2.27)

+

(-reT!

+

'reT

+

'1'/'1'

+

7'$7'$,}s2

+

Tf(To'T

+

'T.Tds3 whcre

(2.28) The relations (2.19) and (2.27) are of the "ame form and m:>y be

wriLll::n as

Kj(l ..

+-

87) :i: Ko, _Fj(s)Hc, F2 = _ _ _ _ _ _ .i •... _~ __ ••. _._._. _ _ _ ..

~ I -I-ST"

-+-

S~1"~

-+-

S31": '

(2.29)

in which

To<, T1t

and T;~ are defined in the c,\~e of tho stabilizing tra nsformcr by

i~i

and in tile case of lhe slabilizing RC circuit by T,x.

.=

TC .! .. Tf

-+-

T

-+-

j{E1""", T7t ~ TC7[

-+-

TeT ! TfT .! .. T"T,,', '1';l.

=,

'1'f(TlJT -I- T.<7'o')' (2.30) (2.31) (2.32) (2.33) (2.34) (2.35)

N ow the orckro of magnitude of the qualltiti(:~s '1'~, TiJ fInd Ty will

bl' compared with each other. All of the.~e (Illantili(o~ have the dimension of time and will be considen~d (I!; real positive numbers. 'fllis is allowed ~incl', a:> appear:; from tiJl: defining relationships, T~, T~ and

.<

un.' always ,',\<\1 ;11)(1 positive (an exception is tlH~ situat.ion with the single-stage amplifier where:' T:,: and h(~nC(l T1' nrc hero since T[ equals <lcro).

Tl}(~ fUlIcl.ioll of lll(~ s1:abili;.;ing network is to (\<\mp to a gmatiT

or smalkr extent o:,;cillatiol"l:; in tlJ(~ tJ"allsicnl performance of the cont.rol ,y';lcm concerned. This implies that one of the reciprocal values of the poles of transfer function (2.29) should be larg!'~

\·n()\lgh. ~il1c(' "T" is the sum of the said reciprocal valu"s, "T" slIould

Iw sllrrici(,rJt:ly large. Now the operation of the ~bbilizing network woul(l b(: mosl dkclivc if the other two n~ciprocal valuo;:; were zero, since then the denominator polynomial corresponds to only one single lagging dement. However, as :;tated bdorc:, 1"71 and T;~ are n.lwilys greater than zero.

The question might be put wbetlIer it. is possible for TS and '1'1'

to aSSUIlW ~rn<l II v<ll1l(~, compared to the value of T/X. To examine

lhi~, we consider the expressions for "T", '1'~f and

-r;',.

In order to hav(·.' a hrgf' vallI(' for T" wilh respect to the values of T{J and Ty, alleast

one: term hao to be found in the expre:;~ion for' T{>" which can be mad(·'