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The analysis and synthesis of contactor servomechanisms Paris, Armand Pierre 1954

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THE ANALYSIS AND SYNTHESIS OF CONTACTOR SERVOMECHANISMS  by ARMAND PIERRE PARIS  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of ELECTRICAL ENGINEERING  We accept this thesis as conforming  to the  standard required from candidates f o r the degree of MASTER OF APPLIED SCIENCE  Member of the Department of Mechanical Engineering  Member of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA August, 1954  Abstract T h i s i n v e s t i g a t i o n i s concerned with the a n a l y s i s and s y n t h e s i s of c o n t a c t o r servomechanisms.  The techniques  employed  are based on Kochenburger*s q u a s i - l i n e a r r e p r e s e n t a t i o n o f the c o n t a c t o r d e s c r i b i n g f u n c t i o n f o r s i n u s o i d a l input s i g n a l s t o the  contactor. The  frequency-response method of a n a l y s i s and s y n t h e s i s ,  which has been found p r a c t i c a l f o r t r e a t i n g l i n e a r servomechanisms has been a p p l i e d by Kochenburger to the c o n t a c t o r and i s e x p l a i n e d here.  servomechanism  By t h i s method i t i s p o s s i b l e to d e t e r -  mine whether t h e system possesses a b s o l u t e  stability.  The r o o t - l o c u s method o f s y n t h e s i s which has been a p p l i e d to  linear  anism.  servomechanisms i s a p p l i e d to t h e c o n t a c t o r servomech-  The r o o t - l o c u s d e s c r i b e s the r o o t s o f t h e c l o s e d ~ l o o p  system f o r a l l values o f t h e c o n t r o l s i g n a l amplitude.  The r o o t -  locus method i s v a l u a b l e when c o n s i d e r i n g the problem of r e l a t i v e stability. F o r a simple c o n t a c t o r with no h y s t e r e s i s e f f e c t , Kochenburger*s v e c t o r form of the c o n t a c t o r d e s c r i b i n g f u n c t i o n can be used d i r e c t l y  to o b t a i n t h e r o o t - l o c u s .  The c o n t a c t o r  appears as a v a r i a b l e gain element f o r the v a r i o u s c o n t r o l s i g n a l amplitudes. but  The c o n t a c t o r has no e f f e c t on the open-loop r o o t s  the v a r i a t i o n s i n the c o n t a c t o r gain cause the r o o t s o f the  c l o s e d - l o o p to t r a v e l along the r o o t - l o c u s obtained from the open-loop roots of t h e system. The r o o t - l o c u s can a l s o be obtained when t h e c o n t a c t o r possesses h y s t e r e s i s .  Kochenburger»s v e c t o r form i s m o d i f i e d  to the L a p l a c e transform  form of the c o n t a c t o r  describing function.  T h i s form of the d e s c r i b i n g f u n c t i o n shows t h a t n o t o n l y a r e the p o s i t i o n s of t h e roots v a r y i n g f o r the c l o s e d - l o o p the  but a l s o f o r  open-loop. A model was c o n s t r u c t e d  t o check some of the theory.  The assumed o v e r - a l l open-loop t r a n s f e r f u n c t i o n s the a c t u a l .  approximated  Even f o r the assumptions made, t h e e x p e r i m e n t a l  work has v e r i f i e d q u a l i t a t i v e l y and t o some degree q u a n t i t a t i v e l y the p r e d i c t i o n of the model performance.  i  i  Table of Contents  page 1.  Introduction  2.  General  3.  Mathematical r e p r e s e n t a t i o n of l i n e a r components  12  4.  The frequency-response of c o n t a c t o r servomechanisms  18  5.  The q u a s i - l i n e a r r e p r e s e n t a t i o n o f the c o n t a c t o r describing function.  23  6.  Stability  26  7.  Hoot-locus method of s y n t h e s i s  8.  A n a l y s i s of t h e model by the frequency-response and r o o t - l o c u s methods  9. 10 2  1  comments on c o n t a c t o r  servomecnanisms  c r i t e r i a f o r c o n t a c t o r servomechanisms  8  32  a)  Kahn's method f o r o b t a i n i n g t r a n s i e n t responses  37  b)  A n a l y s i s of the model  38  c)  A p p l i c a t i o n of a simple phase-lead compensating network  Root-locus  type  method when c o n t a c t o r has h y s t e r e s i s  44 46  The experiment a)  The c i r c u i t  and i t s o p e r a t i o n  51  b)  C a l i b r a t i o n o f t h e model  53  c)  Tests and r e s u l t s  56  11.  Summary and c o n c l u s i o n s  60  12.  References  62  13.  Acknowledgements  63  ii  L i s t of  Illustrations  page Figure  1.  Block  Figure  2.  Controller Characteristics  Figure  3.  Block  Figure  4.  Typical contactor  Figure  5.  Representative  Table  1.  Symbols and  Figure  6.  R e l a t i o n of c o n t r o l s i g n a l and c o r r e c t i o n s i g n a l f o r a contactor w i t h h y s t e r e s i s  19  Figure  7.  Graph of amplitude of harmonic components of c o r r e c t i o n s i g n a l a g a i n s t amplitude o f control signal  f o l l o w i n g 20  Figure  8.  P l o t of the fundamental harmonic c o n t a c t o r f o l l o w i n g 24 describing function, G r - ^  Figure  9.  Block diagram of l i n e a r s i n g l e - l o o p servomechanism  26  Figure  10.  Frequency p o l a r - l o c u s p l o t  27  Figure  11.  Super-posed frequency  Figure  12.  Examples of super-posed  Figure  13.  Roots i n the p-plane  33  Figure  14.  Root-locus  34  Figure  15.  Single-loop contactor  Figure  16.  Graphs f o r Kahn's s e m i - g r a p n i c a l method  Figure  17.  B l o c k diagram r e p r e s e n t a t i o n of the model  39  Figure  18.  Super-posed frequency amplitude l o c i f o r the r e p r e s e n t a t i o n of the uncompensated model  39  Figure  19.  following Graph of the c o n t r o l s i g n a l amplitudes which g i v e p o i n t s of e q u i l i b r i u m f o r the r e p r e s e n t a t i o n of. the uncompensated model  40  Figure  20.  Root-locus f o r the r e p r e s e n t a t i o n of the uncompensated model  diagram of s i n g l e - l o o p servomechanism  2 2  diagramof a c o n t a c t o r  servomechanism  8  characteristics  phase-lead  9  network  13  u n i t s adopted  and  following  amplitude  loci  17  28 30  loci  of s i n g l e - l o o p servomechanism  37  servomechanism following  37  f o l l o w i n g 41  i i i  List  of  illustrations -  cont'd.  page Figure  21.  P l o t s of t r a n s i e n t ence s t e p i n p u t s .  refer-  following  43  Figure  22.  Frequency locus f o r . t h e representation of the compensated model  following  44  Figure  23.  Graph of the c o n t r o l s i g n a l amplitudes following which g i v e p o i n t s of e q u i l i b r i u m f o r the r e p r e s e n t a t i o n o f the compensated model  45  Figure  24.  R o o t - l o c u s f o r the r e p r e s e n t a t i o n the compensated model  following  45  Figure  25.  P l o t s comparing transient responses for the. uncompensated and compensated m o d e l  following  45  Figure  2.6.  P l o t s n e c e s s a r y to harmonic contactor  o b t a i n the fundamental following d e s c r i b i n g f u n c t i o n , gjYj_  48  Figure  2.7.  Open-loop contactor  zero-pole configuration hysteresis  Figure  28.  Root-loci  with  Figure  29.  Circuit  Figure  30.  Schematic  Figure  31.  Change time  Figure  32.  Block  Figure  33.  Transient  responses  contactor  diagram of  the  diagram of  i n rheostat  responses  of  with  hysteresis  model  the  model  voltage  diagram o f the  to  drop  a  step  48  following  49  following  51  following  51  against  model to  following  disturbance  55 following  55  following  57  Table  2.  A m p l i t u d e s and f r e q u e n c i e s o f s e l f - s u s t a i a - f o l l o w i n g ed o s c i l l a t i o n s w i t h o u t c o m p e n s a t i o n  58  Table  3.  A m p l i t u d e s and f r e q u e n c i e s o f s e l f - s u s t a i n ed o s c i l l a t i o n s w i t h c o m p e n s a t i o n  58  following  I*  Introduction In the l a s t two  decades c o n s i d e r a b l e progress- has  been made i n the s c i e n c e of servomechanisms and feedback c o n t r o l systems.  Most o f the l i t e r a t u r e on these automatic  d e a l w i t h continuous tinuous c o n t r o l .  controls  c o n t r o l and very l i t t l e deals w i t h d i s c o n ^  T h i s i s very s i g n i f i c a n t when i t i s c o n s i d e r e d  that Hany o f the f i r s t  c o n t r o l systems were of the d i s c o n t i n u o u s  type. The development o f any exact s c i e n c e r e q u i r e s t h a t r e l a t i o n s h i p s be expressed m a t h e m a t i c a l l y . standable t h a t the continuous  I t i s then under**  type of c o n t r o l was  more amenable  to t h e o r e t i c a l i n v e s t i g a t i o n because r e l a t i o n s h i p s could be expressed  as continuous f u n c t i o n s .  The d i s c o n t i n u o u s c o n t r o l s  on the other hand have not been conducive  to mathematical  i n v e s t i g a t i o n s e s p e c i a l l y when c o n s i d e r i n g s y n t h e s i s techniques, hence the preponderance of theory on continuous discontinuous.  c o n t r o l s over  However, many d i s c o n t i n u o u s types of c o n t r o l s  are used and p o s s i b l y more would be used i f t h e i r performance were more c l e a r l y understood. a n a l y s i s and  Therefore the i n v e s t i g a t i o n o f  the  e s p e c i a l l y the s y n t h e s i s techniques of d i s c o n t i n u o u s  c o n t r o l systems has d e f i n i t e p r a c t i c a l importance.  In t h i s paper,  the type of d i s c o n t i n u o u s c o n t r o l system to be s t u d i e d i s t h a t r e f e r r e d to as o n - o f f , r e l a y o r c o n t a c t o r type servomechanism*  Figure feedback c o n t r o l  +  1 i s a b l o c k diagram r e p r e s e n t i n g  ) £ '  Figure  1.  Signet/ Control £/a.r»<Lnts D  Coritrol/ad  0  Systam  B l o c k diagram of s i n g l e ~ l o o p  servomechanism  This f i g u r e r e p r e s e n t s a f a i r l y g e n e r a l s i n g l e - l o o p c o n s i s t i n g of a r e f e r e n c e i n p u t , system and w i t h the  simple  system.  Error  -M  a  c o n t r o l elements,  a d i r e c t feedback comparing the  reference input.  servomechanism controlled  controlled  variable  For continuous c o n t r o l o f the  elements the r e l a t i o n s h i p of the c o r r e c t i o n s i g n a l , D, steady-state error E , s  may  be  1  0  .Correction D  Figure  the  2a. 5'<tn«l J  JtloJ  Jtiajy-state trt-or  _a) Continuous  to  r e p r e s e n t e d as i n f i g u r e  ; [ (or«tttvn S/yna!  control  -$>) Contactor  SgrvoirxLcfian'S/n  SzrVomzctictmsm  2:. C o n t r o l l e r c h a r a c t e r i s t i c s  This r e l a t i o n s h i p i s drawn as a s t r a i g h t l i n e p a s s i n g through the o r i g i n i l l u s t r a t i n g f o r the one  e r r o r continuous but  f o r which most of the  also l i n e a r .  theory has  c o n t r o l elements of f i g u r e 1 are mechanism the  t h a t not o n l y i s the  correction  This c o n d i t i o n  been developed.  substituted  If  is  the  the  by a c o n t a c t o r servo-  c o n t r o l l e r c h a r a c t e r i s t i c would be  as  represented  3  i n f i g u r e ab.  I t may be observed from t h i s graph that no  c o r r e c t i o n takes place w i t h i n a c e r t a i n zone* as the i n a c t i v e or dead zone. the c o n t r o l l e r introduces the c o n t r o l l e d system.  This zone i s known  When the e r r o r i s outside t h i s zone  a constant value of c o r r e c t i o n s i g n a l v t o  I t i s an a l l o r nothing type of c o n t r o l and  the c o r r e c t i o n s i g n a l i s a discontinuous f u n c t i o n of the e r r o r . This type of c o n t r o l i s widely used because o f i t s simplicity.  I t f r e q u e n t l y requires much l e s s equipment than con-  tinuous c o n t r o l l e r s , hence i t s many savings, i n weight and c o s t . The c o n t r o l l e r could c o n s i s t of an electromagnetic r e l a y , hence the term r e l a y servomechanism, o r some s o r t of contact-making device a c t i v a t e d , f o r example, pneumatically, h y d r a u l i c a l l y o r e l e c t r i c a l l y , hence the term contactor servomechanism.  The manner of operation  of the c o n t r o l suggests the other name pn-off servomechanism. For purposes of u n i f o r m i t y the term contactor  servomechanism w i l l be  adopted i n t h i s paper and used e x c l u s i v e l y henceforth; The e a r l y i n v e s t i g a t o r s approached the a n a l y s i s of cont a c t o r servomechanisms w i t h two s i g n i f i c a n t p o i n t s i n mind.  First,  mathematical r e l a t i o n s h i p s could be expressed f o r the s e v e r a l cond i t i o n s of c o n t r o l that i s f o r p o s i t i v e , negative and zero c o r r e c t i o n . This means that although the c o n t r o l system i s non-linear when considered over i t s e n t i r e range of operation i t may be considered l i n e a r and d i f f e r e n t i a l equations can be s e t up f o r each c o n d i t i o n of i t s operation. of operationi  The equation chosen would depend on the c o n d i t i o n  The boundary conditions f o r the c o n t r o l l e d v a r i a b l e  would have to be matched f o r each contactor switching  operation.  Secondly, experience had Indicated that contactor servomechanisms were o s c i l l a t o r y by nature and tended to maintain sustained  oscil-  4 lations.  This f a c t i s seen to be reasonable  by c o n s i d e r i n g the  manner by which the c o n t a c t o r servomechanism c o r r e c t s f o r e r r o r . Once the e r r o r i s o u t s i d e o f the i n a c t i v e zone a s t e p i n p u t o f c o r r e c t i o n s i g n a l i s i n t r o d u c e d to the c o n t r o l l e d system and  this  same value of the s i g n a l p e r s i s t s u n t i l the e r r o r again, comes i n t o the i n a c t i v e zone.  With s u f f i c i e n t l y h i g h g a i n i n the  forward  t r a n s f e r f u n c t i o n of the open-loop system the c o n t r o l l e d v a r i a b l e i s caused to overshoot  i t s s t a b l e p o s i t i o n , t h a t i s , tne e r r o r  t r a v e l s o u t s i d e the i n a c t i v e zone and b e f o r e i n the o p p o s i t e d i r e c t i o n *  c o r r e c t i o n takes p l a c e  as  Because of the type o f c o r r e c t i o n  s i g n a l a tendency e x i s t s f o r the s w i t c h i n g p r o c e s s of c o r r e c t i o n to p e r s i s t hence c a u s i n g a g e n e r a t i o n of s u s t a i n e d o s c i l l a t i o n s of the c o n t r o l l e d v a r i a b l e . The e a r l y i n v e s t i g a t o r s were armed w i t h these two The  two  i n i t i a l papers w r i t t e n on the s u b j e c t were one  and  the other by Hazen } both, i n 1934. 2  by  facts.  Ivanoff  Both, i n v e s t i g a t o r s approached  the problem assuming a c o n d i t i o n o f s u s t a i n e d o s c i l l a t i o n s In steady-state.  I v a n o f f , who  was  a heat p l a n t .  the  p r i m a r i l y i n t e r e s t e d i n temperature  r e g u l a t i o n assumed a symmetrical being i n t r o d u c e d  r e c t a n g u l a r c o r r e c t i o n s i g n a l wave  to the c o n t r o l l e d elements, which, i n h i s case  He analysed  1  was  the r e c t a n g u l a r wave i n t o i t s F o u r i e r  s e r i e s and assumed a t r a n s f e r e n c e c h a r a c t e r i s t i c f o r h i s heat p l a n t . He was  a b l e to demonstrate the r e l a t i o n s h i p of the  amplitude and frequency  to the i n a c t i v e zone w i d t h .  steady-state By  employing  the F o u r i e r a n a l y s i s of the r e c t a n g u l a r wave he a l s o showed t h a t the predominant e f f e c t of the r e c t a n g u l a r c o r r e c t i o n wave was to i t s fundamental component, f o r the c o n t r o l l e d v a r i a b l e and directly.  due  Hazen assumed a s i n u s o i d a l response t h e n used d i f f e r e n t i a l  equations  T h i s method r e q u i r e s t h a t d i f f e r e n t i a l equations  be  5  solved for each, correction i n t e r v a l i n i t i a t e d by tlie contactor and that boundary conditions be matched f o r each switching operation*  This method becomes very awkward even f o r r e l a t i v e l y  simple cases. the  Hazen, however, was able to demonstrate many of  peculiar c h a r a c t e r i s t i c s of contactor servomeonanisms such  as the effects due to the inactive-zone and back-lash with h i s d i r e c t d i f f e r e n t i a l equation approach. Methods of analysis based i n d i r e c t l y on the transient response have been developed.  Weiss  3  and M a c C o l l applied the 4  graphical phase-plane method, which had been used i n non-linear systems, to the analysis of contactor servomecnanisms.  The d i f f e r -  e n t i a l equations are put i n such a form that the s o l u t i o n i s obtained i n terms of the f i r s t derivative with respect to time of the cont r o l l e d variable and the controlled v a r i a b l e .  I f the controlled  variable be displacement then the f i r s t derivative with respect to time i s v e l o c i t y .  The plane formed by having the f i r s t derivative  of the controlled variable with respect to time as the ordinate and the  controlled variable as abscissa usually expressed i n terms of  v e l o c i t y versus displacement i s known as the phase-plane;  Tra-  jectories i n the phase-plane are plotted i n the d i f f e r e n t regions and from the web of the t r a j e c t o r i e s the s t a b i l i t y of the control system can be determined.  The transient response of the system can  also be obtained graphically from the phase-plane p l o t .  Recently  Fliigge-Lotz has given extensive treatment of t h i s method with 5  s p e c i f i c application to guided m i s s i l e s .  Kahn has developed a 6  method which Is quite analogous to the phase-plane.  His graphical  method i s based on plots of the f i r s t derivative of the controlled variable versus time.  Both these methods of analysis suffer serious  l i m i t a t i o n s i n that they are s a t i s f a c t o r y only f o r systems which can  6 be d e s c r i b e d by a second order d i f f e r e n t i a l  equation.  Kahn&has f u r t h e r developed a g r a p h i c a l method o f p l o t t i n g the t r a n s i e n t response f o r a r b i t r a r y d i s t u r b a n c e s * Prac-? tically, and  the s o l u t i o n o f o n l y one d i f f e r e n t i a l equation i s  necessary  t h i s i s the open*loop response o f the c o n t a c t o r c o n t r o l s i g n a l  to the step c o r r e c t i o n s i g n a l .  T h i s method w i l l be e l a b o r a t e d upon  l a t e r as i t w i l l be used to o b t a i n t r a n s i e n t responses f o r c e r t a i n p o r t i o n s of the i n v e s t i g a t i o n to f o l l o w . The  disadvantage of a l l the methods d e s c r i b e d i s that  they are i n c a p a b l e o f d i r e c t use  for synthesis.  T h i s , of  i s a s e r i o u s disadvantage to the designer of the c o n t a c t o r  course, servo-  mechanisms. Kochenburger  7  has i n t r o d u c e d a novel method of a n a l y s i s  which i s conducive to s y n t h e s i s based on the frequency-response of the system.  This method i s based on approximations which are  v a l i d f o r most systems encountered i n p r a c t i c e . enables  the c o n t a c t o r to be r e p r e s e n t e d  describing function.  The  T h i s approximation  i n terms o f a q u a s i - l i n e a r  c h i e f advantage o f t h i s approximate  method, which s e t s i t apart from o t h e r methods, i s that i t a l l o w s l i n e a r techniques other techniques a l s o be used.  to be a p p l i e d to n o n - l i n e a r mechanisms, so t h a t a l r e a d y developed f o r l i n e a r c o n t r o l systems  The  may  w r i t e r suggests i n t h i s paper t h a t the r o o t -  l o c u s method developed by Evans® f o r l i n e a r o o n t r o l system synt h e s i s may  be used as an added technique  f o r the s y n t h e s i s of con-  t a c t o r servomechanisms i n c o n j u n c t i o n with the Kochenburger desc r i b i n g f u n c t i o n and The  i t s modifications.  o b j e c t of t h i s paper t h e r e f o r e i s to o u t l i n e  7  Koehenburger's method o f a n a l y s i s and s y n t h e s i s of c o n t a c t o r servomechanisms and to show how the root l o c u s method of s y n t h e s i s may be used, w i t h the concept c o n t a c t o r , t o determine  of the d e s c r i b i n g f u n c t i o n o f the  the r e l a t i v e s t a b i l i t y  of a contactor  servomechani sm. A model was c o n s t r u c t e d f o r the purpose of checking some of  the theory developed.  c o u l d be checked  U n f o r t u n a t e l y not a l l the theory  developed  e x p e r i m e n t a l l y because the h y s t e r e s i s e f f e c t i n a  c o n t a c t o r c o u l d not be reproduced  i n the c o n t a c t o r of t h e model©  The experimental r e s u l t s obtained a r e based on the assumptions t h a t p o s i t i v e and n e g a t i v e c o r r e c t i o n s are equal and a l l components o f the servomechanism other than the c o n t a c t o r a r e l i n e a r . assumptions a r e necessary but o n l y approximately two  types o f e r r o r a r e to be expected  true.  These Therefore  i n the r e s u l t s ; the f i r s t  source o f e r r o r due t o the approximate method of a n a l y s i s used and the second due to the assumption  t h a t components o t h e r than the  c o n t a c t o r a r e l i n e a r which was not e x a c t l y the case f o r the model. In  the c h a p t e r s to f o l l o w , Kochenburger'a  method and the r o o t - l o c u s method with the concept  frequency-response  o f the q u a s i - l i n e a r  r e p r e s e n t a t i o n o f the c o n t a c t o r d e s c r i b i n g f u n c t i o n w i l l be e x p l a i n e d . These two methods of a n a l y s i s and synthesis, w i l l be a p p l i e d , i n . anal*» y z i n g the model f i r s t without a compensating network, then w i t h a simple phase-lead compensating network.  The mathematical  analysis  w i l l be f o l l o w e d by a d e s c r i p t i o n o f the model, and a r e p o r t on the experimental t e s t s performed  and the r e s u l t s o b t a i n e d .  8 2»  General  Comments on Contactor  Servomechanisms  Figure 3 represents a t y p i c a l single-loop contactor servomechanism. Control  EI<L/nertts  Hahrtna J Input -\~ f \ terror Compensation  Contactor means  £>  <:tiar\ •>rr<.  F i g u r e 3* Block diagram of a c o n t a c t o r  Control/iol El<unants  Cont rolU<i  0  servomechanism  The b l o c k diagram f o r the c o n t a c t o r servomechanism i s q u i t e s i m i l a r i n form to the l i n e a r continuous  type.  The power  a m p l i f i e r which i s e s s e n t i a l to the c o n t r o l elements of the cont i n u o u s servomechanism as a discontinuous  i s r e p l a c e d by the c o n t a c t o r means which a c t s  power a m p l i f i e r .  The compensating network i s  very o f t e n i n c l u d e d i n the c o n t r o l elements to a i d the c h a r a c t e r i s t i c s of the system.  T h i s compensating network i s whenever  p o s s i b l e p l a c e d on the low power s i d e of the c o n t a c t o r f o r purposes of economy. In f i g u r e 3 the output reference input. servomechanisms  i s compared d i r e c t l y to the  D i r e c t feedback i s not e s s e n t i a l f o r c o n t a c t o r and more complicated  i n c l u d e d i n the feedback l o o p .  t r a n s f e r f u n c t i o n s may  be  With a compensating network the  e r r o r s i g n a l i s a l t e r e d before being a p p l i e d to the c o n t a c t o r . Without a compensating network the e r r o r i s , of course,  the same  as the c o n t r o l s i g n a l to the c o n t a c t o r . Figure 4 represents  t y p i c a l c h a r a c t e r i s t i c s of contactor  9 means.  I n these r e p r e s e n t a t i o n s symmetrical  o p e r a t i o n o f the  c o n t a c t o r i s assumed, which means that the n e g a t i v e and p o s i t i v e c o r r e c t i o n s a r e equal i n magnitude.  I t i s convenient t o express  the c o r r e c t i o n s i g n a l as having a u n i t d l m e n s i o n l e s s amplitude so that the p o s i t i v e c o r r e c t i o n w i l l be p l u s one w h i l e the n e g a t i v e w i l l be minus one. Corraction S/qnal  A Correction •S/aoal  * 0  D  y  -H  +i  Control  c  S/qitcf/  Cotltrvl 5/amxl  3  c  -I  b)With indctivd Zona. ,Ccl anjhyste^sti  a) With mactiva zotf<z, Oy  Zon<L i  F i g u r e 4; T y p i c a l c o n t a c t o r c h a r a c t e r i s t i c s F i g u r e 4a r e p r e s e n t s the c h a r a c t e r i s t i c s o f a c o n t a c t o r w i t h i n a c t i v e zone -only, the width of t h e i n a c t i v e zone b e i n g Da» No c o r r e c t i o n w i l l take p l a c e u n l e s s the c o n t r o l s i g n a l i s g r e a t e r than Ca/2' and l e s s than -0^/2.  The i n a c t i v e zone width r e p r e s e n t s  the range o f p e r m i s s i b l e e r r o r w i t h i n which no c o r r e c t i o n  takes  p l a c e and i t can be no l a r g e r than design s p e c i f i c a t i o n s .  Making  the i n a c t i v e zone s m a l l e r than necessary c o m p l i c a t e s the problem of s t a b i l i t y .  As the i n a c t i v e zone width approaches zero the system  w i l l a t best m a i n t a i n s u s t a i n e d o s c i l l a t i o n s about some  equilibrium  point. F i g u r e 4b r e p r e s e n t s the c h a r a c t e r i s t i c o f a c o n t a c t o r w i t h i n a c t i v e and h y s t e r e s i s zones.  The p h y s i c a l meaning o f  t  10 hysteresis i n a contactor  as may  be observed from t h i s f i g u r e i s  that the c o n t r o l s i g n a l r e q u i r e d to cause c o r r e c t i o n i s g r e a t e r than the c o n t r o l s i g n a l r e q u i r e d to cease c o r r e c t i o n .  T h i s pheno^  menom may  r e l a y s where  be observed, f o r example, i n electromagnetic  the c o i l c u r r e n t n e c e s s a r y to c l o s e a r e l a y i s g r e a t e r necessary to open the r e l a y . o f the c o n t a c t o r  Of course the assumed c h a r a c t e r i s t i c  to account f o r h y s t e r e s i s i s i d e a l i s t i c *  t h e l e s s the e f f e c t of t h i s i d e a l i s t i c and  i t can  than that  c h a r a c t e r i s t i c may  Neverbe shown  be reasonably assumed t h a t an a c t u a l r e l a y w i t h h y s t e r e s i s  w i l l have much the same e f f e c t .  I t w i l l be shown that  i n a c o n t a c t o r has an adverse e f f e c t on  hysteresis  the s t a b i l i t y o f a system.  B a s i c a l l y the performance c r i t e r i a o f c o n t a c t o r l i n e a r continuous servomechanisms are the same.  The  and  c o n t r o l system  attempts to m a i n t a i n the c o n t r o l l e d v a r i a b l e equal to some d e s i r e d value of r e f e r e n c e  input.  S t a t i c accuracy i n c o n t a c t o r  servomechanisms, which i s  the p o s s i b l e range of e r r o r under s t e a d y - s t a t e prime importance. zone.  conditions, i s of  This a c c u r a c y i s determined by the i n a c t i v e  Dynamic accuracy i s the measure of e r r o r when the system i s  responding to a d i s t u r b a n c e .  This a c c u r a c y may  be improved  by  i n c r e a s i n g the r a t e of response o f c o r r e c t i o n or e f f e c t i v e l y as i n continuous servomechanims i n c r e a s i n g the a m p l i c a t i o n troller.  of the  con*  T h i s Improvement i n accuracy tends to cause o s c i l l a t i o n s  i n the dynamic response which i n t r o d u c e s  the problem of  stability;  For purposes of dynamic accuracy i t i s important t h a t the a s s o c i a t e d w i t h the o s c i l l a t i o n s , be as h i g h as p o s s i b l e *  frequencies, Stability  requirements demand t h a t o s c i l l a t i o n s be s u f f i c i e n t l y damped.  The  measure o f the damping of these o s c i l l a t i o n s i s r e f e r r e d to as  the  11 r e l a t i v e s t a b i l i t y of the system, between s t a b i l i t y requirements mechanisms.  A d i s t i n c t i o n must be made  f o r c o n t a c t o r and l i n e a r servcn-  For c o n t a c t o r servomechanisms s e l f - s u s t a i n e d  o s c i l l a t i o n s of f i n i t e amplitude  are q u i t e p o s s i b l e and a t  times  p e r m i s s i b l e while f o r l i n e a r servomechanims self**sustained o s c i l l a t i o n s of n e c e s s i t y i n c r e a s e and are d e s t r u c t i v e * With r e s p e c t to the problem of s t a b i l i t y i t i s i n t e r e s t i n g to note that f o r c o n t a c t o r servomechanisms, the s t a b i l i t y of  the  system i s dependent on the type of d i s t u r b a n c e i n t r o d u c e d i n t o system,  the  A p a r t i c u l a r system c o u l d , f o r example, be a b s o l u t e l y  s t a b l e , t h a t i s i n c a p a b l e of m a i n t a i n i n g  self-sustained oscillations,  f o r a step i n p u t of d i s t u r b a n c e w h i l e be i n a s t a t e o f s u s t a i n e d o s c i l l a t i o n f o r a " v e l o c i t y " i n p u t of d i s t u r b a n c e . l i n e a r servomechanisms i n which the frequency  and  This i s unlike damping o f the  t r a n s i e n t o s c i l l a t i o n s are independent o f the d i s t u r b a n c e  and  dependent s o l e l y on the n a t u r a l modes of o s c i l l a t i o n of the  servo*  mechanisms determined from t h e i r system t r a n s f e r f u n c t i o n s .  For*  t u n a t e l y , however, i t i s r a r e l y necessary  to d e s i g n c o n t a c t o r  mechanisms f o r c o n t i n u o u s l y v a r i a b l e c o n t r o l of output*  servos  This g r e a t l y  s i m p l i f i e s the a n a l y s i s of c o n t a c t o r servomechanisms as w e l l as the equipment necessary  f o r their suitable operation.  121 3#  Mathematical  R e p r e s e n t a t i o n of L i n e a r Components  I n using Kochenburger s frequency-response r  method or  the r o o t - l o c u s method, w i t h the c o n t a c t o r d e s c r i b i n g f u n c t i o n , assumption linear.  the  made i s t h a t a l l elements o t h e r than the c o n t a c t o r are  These elements are r e p r e s e n t e d mathematically i n terms o f  t h e i r t r a n s f e r f u n c t i o n s or r a t i o s of outputs to i n p u t s . R e f e r r i n g to f i g u r e 3 the t r a n s f e r f u n c t i o n G ( s ) s  of  the c o n t r o l l e d system i s d e f i n e d as G ( s | = Qjs)  (1)  s  where 0 ( s ) and D(s) a r e the L a p l a c e t r a n s f o r m of the c o n t r o l l e d v a r i a b l e 0 ( t ) and  the c o r r e c t i o n s i g n a l D ( t ) r e s p e c t i v e l y , s u b j e c t  to zero i n i t i a l c o n d i t i o n s * I f , f o r example, the c o n t r o l l e d system c o n s i s t e d of a servomotor,  the t r a n s f o r m f u n c t i o n could a f t e r being f a c t o r e d have  the f o l l o w i n g form G  s(s) =  (2:)  Rs s ( T i s + 1)  T i i s a time constant of the motor; R  s  i s the g a i n o f the t r a n s f e r  f u n c t i o n and i n t h i s case g i v e s the s l o p e of the response o f the motor to a u n i t step Input a f t e r a s u f f i c i e n t l y l o n g time.  For  this  reason R$ i s r e f e r r e d to as the runaway v e l o c i t y of the motor. A l l o t h e r l i n e a r components may manner.  Consider the phase-lead  be expressed i n a s i m i l a r  compensating network of f i g u r e  5.  R, in ohms I — — i  Input  vo/taji  Output vo/tayi  0  F i g u r e 5. R e p r e s e n t a t i v e phase-lead network. The t r a n s f e r f u n c t i o n o f t h i s network G ( s ) i s c  where Rc =  Rg  60(a)  = R (m s-r- 1) Tgs+ 1  ,  - RjCi ,  m  2  c  (3)  2  T  2  - B3R3C1  Rl+^a  R1+R2  Tg and mg a r e time constants o f the t r a n s f e r f u n c t i o n . T r a n s f e r f u n c t i o n s can be expressed  as products  the g a i n and the f u n c t i o n o f s , g ( s ) , d e f i n e d Gr(s) = R g ( s )  (4)  so f o r the t r a n s f e r f u n c t i o n g i v e n by equation (2) G (s) = ( s  R  ) = R g (s)  s  s  s  (5)  TsTf][sTTT where g s ( s ) s  1 S(TTS +  1)  S i m i l a r l y f o r t h e t r a n s f e r f u n c t i o n o f the phase-lead comp e n s a t i n g network g i v e n i n equation (g) G (s) c  R (m -i- 1) = H g ( s )  =  c  lS  c  ( T . 8 + 1)  c  2  where  g (s) s c  m  i  s  -»• 1  (6)  Since the t r a n s f e r f u n c t i o n s of the c o n t r o l l e d system and the compensating network are l i n e a r they may be combined as the open-loop t r a n s f e r f u n c t i o n from contactor output or c o r r e c t i o n s i g n a l to the contactor input or c o n t r o l s i g n a l .  For the system  as represented i n f i g u r e 3, c o n s i s t i n g of the c o n t r o l l e d elements, a compensating network and the contactor,  the^open-loop t r a n s f e r  f u n c t i o n i s then G(s)  -  C(s)  DTI7  iE = constant « 0  Where R  =  RsRc ,  -  G (s)G (s)  =  R S g (s)g (s)  =  Rg(s)  s  s  (7) (8)  c  c  s  c  (9)  g(s) - g s ( s ) g c ( s ) , G ( s ) and G ( s ) are s  c  the t r a n s f e r functions of the c o n t r o l l e d system and compensating network r e s p e c t i v e l y * The t r a n s f e r functions Rg(s) as already described i n f a c t o r e d form w i l l be r e f e r r e d to as the Nyquist form of the transfer function.  I n t h i s form the complex v a r i a b l e i n a  f a c t o r i s m u l t i p l i e d ! by the time constant of the f a c t o r so that the numerator has f a c t o r s of the form (ms+1) and the denominator has f a c t o r s of the form (Ts+ 1 ) .  The advantage of t h i s form i s  obvious f o r use with the modified form of the Nyquist c r i t e r i o n of s t a b i l i t y where attenuation  i n decibels and phase-margin are  p l o t t e d versus angular v e l o c i t y .  However t h i s form of the t r a n s f e r  f u n c t i o n i s not convenient f o r use w i t h the root-locus method of synthesis.  Consider the open-loop t r a n s f e r f u n c t i o n i n N y q u i s t G(s) . Rg(s) = R  The r o o t - l o c u s  (mis-H) al'ils^Dl'fas + l )  form  (10)  form of the t r a n s f e r f u n c t i o n g i v e n by  equation 10 I s F(s)  Rmx  s  (s+ 1 ) ' mi  TxT  (11)  s(s + l )U^1 Tx  2  T  ) a  and r e w r i t i n g F ( s ) as the product of the g a i n and of s i n an analogous  the r o o t - l o c u s  function  manner to G(s) - E g ( s ) f o r the N y q u i s t form,  f.(s) for  the  A f(s)  =  form.  (12)  T h e r e f o r e f o r equation  11  A = Rmi  (13a)  "TlTa f ( s ) = (s + 1 )  V  s(s+ 1  T± For  )(s+ 1 ) T  (13b)  2  the t r a n s f e r f u n c t i o n f a c t o r e d i n r o o t - l o c u s  the complex v a r i a b l e , s, i s m u l t i p l i e d by u n i t y . p o l e s o f F ( s ) , the r o o t - l o c u s may  The  form of the t r a n s f e r  zeros and  function,  be obtained d i r e c t l y from the f a c t o r e d form of t h i s  function.  form  transfer  Zeros and p o l e s are those values of s which make the  t r a n s f e r f u n c t i o n zero and i n f i n i t e r e s p e c t i v e l y .  For  (13b) a zero occurs at - 1  , and - 1  m  , and p o l e s at 0, - 1 Ti  x  equation  T£  on the complex s-plane. I t w i l l be found most convenient the N y q u i s t and r o o t - l o c u s  to r e p r e s e n t both  forms of the t r a n s f e r f u n c t i o n i n  dimensionless time.  This i s accomplished by s u b s t i t u t i n g the  complex v a r i a b l e s which has the dimensions seconds-1 by the dimensionless complex v a r i a b l e p where P = t 8  (14)  b  and t  D  i s the time base s e l e c t e d .  The time base  selected  i s u s u a l l y one of the time c o n s t a n t s of the t r a n s f e r G(s).  Transient  function  s o l u t i o n s w i l l t h e r e f o r e be f u n c t i o n s o f  dimensionless time 0 where  i - t  (15)  ^b  t being the e l a p s e d time i n seconds. To o b t a i n the dimensionless-time N y q u i s t form of transfer functions,  c o n s i d e r again the case of the open-loop  t r a n s f e r f u n c t i o n G(a) g i v e n by equation 10. s e l e c t e d w i l l be % G(s) - R  The time base  r 1\  (mTS+1) s(Tis-r l ) ( T s + 1)  =  2  RT! (m-L^s-•• 1) Ti  Txs ( T s + 1 j ( T g T x S + 1) x  (16) For p s Tis  S(p) r R T i U i P + l )  (17)  plp~+TTIyip +1)  m  wAar* cf, = r p - and y = 4  A  S e p a r a t i n g the t r a n s f e r f u n c t i o n G(p) i n t o a product of the g a i n and a f u n c t i o n o f p G(p)  r  Kg'(p)  (18)  for  equation 17 K . =  and g ( p )  (19a)  RTT_  =  (qiP+1)  p l p T T J T y P + 1)  (19b)  2  S i m i l a r l y f o r t h e dimensionless-time r o o t - l o c u s form of  the t r a n s f e r f u n c t i o n  F(p)  s  Bf(p)  (20)  and f o r the oase under c o n s i d e r a t i o n F(p)  = RTid! ( p + 1 A <U  = Bf(p)  (21)  y p(p+l)(p+l ) 8  7Z where  B r R^iqj.  (22a)  and f(p)  -  p+i_ *1  (22:b)  P(P + l ) ( p ^ _ l _ ) y'aFor t h e t r a n s f e r f u n c t i o n s i n dimensionless-time  form,  the g a i n K of the Nyquist form and the g a i n B of the r o o t - l o c u s form have the same dimensions. to  T h i s i s convenient f o r the study  f o l l o w because the t r a n s f e r f u n c t i o n s are f r e q u e n t l y changed  from one form to the o t h e r . Table 1 g i v e s a synopsis of the adopted  notation for  the system p r o p e r t i e s i n t h e i r v a r i o u s forms and the throughout  the system, a l o n g w i t h t h e i r v a r i o u s  signals  dimensions.  to f o l l o w page 17  Table 1  Symbols and U n i t s Adopted  ™ •H S3 P>  Units  Symbol  Quantity Time base Elapsed time Complex v a r i a b l e of L a p l a c e Transform E l a p s e d time  Seconds Seconds  tb t Ss/H-jW  Seconds" Dimensionless  P=st -cr+-ju  Dimensionless  1  <D  Ej EH  Complex v a r i a b l e of L a p l a c e Transform  Controlled Variable or Output ' cJU Ref erence Input « ™3 E r r o r t d dra C o n t r o l S i g n a l S u o Correction Signal c H  •H 'CO  tl  Output-units  I n a c t i v e zone H y s t e r e s i s zone Transfer Function Properties: a) N y q u i s t Form Of c o n t r o l l e d  elements  Output-units Output-units Output-units Dimensionless  I B.I-0 C D  Output-units Output-units  Cd  G (s) R g (s)rO(s) D(s) s  =  s  a  Output-units Dimensionless  Of compensating network.  a (s)=R g (s} C(s)  £  O v e r - a l l open-loop  G(s) G (s)G (s)  o £  Time constant i n numerator  m  Seconds  o)  Time constant i n denominator  T  Seconds  c  c  c  =  Oi 0)  •H  >>  CO  b j Root-locus V  5  s  Output-units  c  form  of C o n t r o l l e d elements  E-slsJsAsfgtsJsOCs) D(s)  Output-units  g (s)-Acf (s)sc|s|  Dimensionle s s  c  Of compensating network  c  to f o l l o w page 17 Table 1  (continued) Quantity  Over-all  open-loop  ;em Prop erties  c) Dimensionless-time Nyquist form O v e r a l l open-loop  •f  01  >> co  Units  Symbol F ( s ) F ( s } F (s)  Output-uni t s  G ( p ) = G ( p ) G (P)  Output-units  =  s  c  a  c  =Kg(p) K g(p)  Output-units Dimensionless  Time constant i n numerator  Dimensionless  Time constant i n denominator t  d) Dimensionless-time r o o t - l o c u s form O v e r a l l open-loop  o  Dimensionless  b  F ( p ) = F ( p ) F (P)  Output-units  «Bf (p) B f(p)  Output-units Dimensionless  s  c  18 4.  The frequency-response of oontaotor servomechanisms. The frequency-response method of a n a l y s i s and  has been found v a l u a b l e  f o r l i n e a r servomechanisms^  synthesis  This method,  based on c e r t a i n .approximations.can a l s o be used f o r study of contactor  servomechanisms. The o v e r - a l l open-loop t r a n s f e r f u n c t i o n G-(p) may  described  i n terms of i t s s t e a d y - s t a t e  i n p u t s of v a r i o u s r e a l f r e q u e n c i e s  response to s i n u s o i d a l  by s u b s t i t u t i n g j u f o r p where  u i s the d i m e n s i o n l e s s a n g u l a r v e l o c i t y . G (ju)  =  be  Then  Kg(ju)  (23)  g ( j u ) v a r i e s w i t h the a p p l i e d frequency only hence g ( j u ) i s a frequency v a r i a n t p o r t i o n of the  system.  The r a t i o o f output to i n p u t of the c o n t a c t o r s i n u s o i d a l i n p u t s cannot be r e p r e s e n t e d by a t r a n s f e r because  of i t s n o n - l i n e a r  I t w i l l be found that  of the a p p l i e d i n p u t s i g n a l .  function  the d e s c r i b i n g  i s dependent on the amplitude but independent The c o n t a c t o r  i s then an amplitude v a r i a n t p o r t i o n of the Kochenburger*s  function  c h a r a c t e r i s t i c s . Kochenburger has  expressed t h i s r a t i o by what i s c a l l e d the d e s c r i b i n g of the c o n t a c t o r .  for  function  of the frequency  describing  function  system.  frequency-response method b a s i c a l l y  i s the combination of the frequency v a r i a n t p o r t i o n of the and the amplitude v a r i a n t p o r t i o n o f the system i n t o one subject  system  scheme  to i n t e r p r e t a t i o n by the frequency-response method used  i n l i n e a r servomechanisms. Consider a c o n t a c t o r  with the c h a r a c t e r i s t i c s of f i g u r e 4b  19  which has both an i n a c t i v e and h y s t e r e s i s  zone.  L e t the control  s i g n a l C be r e p r e s e n t e d by C = The  C cosu<t>  (24)  m  c o n t a c t o r w i l l i n i t i a t e a p o s i t i v e c o r r e c t i o n s i g n a l when  C = G /2'+ C^/2 and cease c o r r e c t i o n when C = C /2 - Cfc/2. I t d  d  w i l l i n i t i a t e a negative c o r r e c t i o n s i g n a l when C =-0^/2 - C^/2 and  cease n e g a t i v e c o r r e c t i o n when C =-0^/2+0^/2.  instants  These  of time a r e designated by the angles u<|) equals a - b,  a + b,Tr +• a - b, and i r + a •*• b r e s p e c t i v e l y .  These r e l a t i o n s h i p s  are shown i n f i g u r e 6a.  a)Assontad sinuto/<Ja/ s/)apt of control j/jna/  ' Correction Signal, D 1  rvnJ<imc*taf component 0, . f  b) Rt3»ft<*"£ form of correction sicjnql and its fundamental component" Figure The  6.  R e l a t i o n of c o n t r o l s i g n a l and c o r r e c t i o n s i g n a l f o r a contactor w i t h h y s t e r e s i s .  p u l s e width o f the r e c t a n g u l a r  as shown i n figure. 6b,. rectangular  wave i s g i v e n by t h e angle 2b  The fundamental harmonic component of the  wave D]_ i s superimposed on the r e c t a n g u l a r  wave i n  f i g u r e 6b and t h e angle, a, r e p r e s e n t s t h e phase l a g o f t h e fundamental component o f t h e c o r r e c t i o n s i g n a l behind t h e c o n t r o l s i g n a l ,  20 I f the c o r r e c t i o n s i g n a l c o n s i s t e d s o l e l y of t h e fundamental  harmonic component D i , then o n l y s i n u s o i d a l l y v a r y i n g  s i g n a l s would e x i s t i n the system.  The c o n t a c t o r would then  appear,  as a s o - c a l l e d q u a s i - l i n e a r t r a n s f e r d e v i c e i n that i t would operate as a l i n e a r a m p l i f i e r f o r any g i v e n constant amplitude o f c o n t r o l signal.  The c o n t a c t o r would not operate as a t r u l y l i n e a r d e v i c e  because  of i t s n o n l i n e a r r e l a t i o n s h i p between c o r r e c t i o n s i g n a l and  c o n t r o l s i g n a l amplitude.  C o n s i d e r i n g the c o n t a c t o r as such a q u a s i -  l i n e a r d e v i c e the frequency-response method may p a r t i c u l a r s i g n a l amplitude.  be used f o r any  This approximation of the c o n t r o l s i g n a l ,  which n e g l e c t s the h i g h e r o r d e r harmonics  of t h i s r e c t a n g u l a r wave,  i s e s s e n t i a l to the concept of the q u a s i - l i n e a r d e s c r i b i n g The j u s t i f i c a t i o n f o r t h i s approximation w i l l now  function.  be c o n s i d e r e d .  An a n a l y s i s of the r e c t a n g u l a r wave of the c o r r e c t i o n s i g n a l w i l l show the r e l a t i v e importance of the harmonic components. The r e l a t i v e amplitudes of the harmonic components a r e a f u n c t i o n of the p u l s e - w i d t h which i n t u r n i s a f u n c t i o n of the amplitude o f the c o n t r o l s i g n a l and the i n a c t i v e and h y s t e r e s i s zones.  Figure 7  i l l u s t r a t e s the r e l a t i v e importance of the t h i r d and f i f t h with r e s p e c t to the fundamental only.  f o r a c o n t a c t o r w i t h i n a c t i v e zone  The amplitudes are p l o t t e d a g a i n s t the r a t i o of the amplitude  o f the c o n t r o l s i g n a l and o n e - h a l f the i n a c t i v e zone. the amplitude of the fundamental unity.  harmonics  From the graphs i t may  The graph of  i s taken as b e i n g assymptotic to  be observed except f o r v e r y s m a l l  values of the a b s c i s s a t h a t the amplitude of the fundamental i s g r e a t e r than those amplitudes.of i t s harmonics. of the harmonics  can no way  But the  amplitudes  be c o n s i d e r e d as being n e g l i g i b l e as  f a r as the c o r r e c t i o n s i g n a l o n l y i s concerned.  However^ the t r a n s f e r  f u n c t i o n s of the c o n t r o l l e d elements u s u a l l y a c t to suppress the  21 higher harmonics.  Consider f o r example the c o n t r o l l e d  element  to be a servomotor whose t r a n s f e r f u n c t i o n i s of the form  K$ P(y]P+ 1)  The t r a n s f e r f u n c t i o n d e s c r i b e d i n terms o f i t s s t e a d y - s t a t e response to s i n u s o i d a l i n p u t s of r e a l f r e q u e n c i e s i s o b t a i n e d by s u b s t i t u t i n g f o r p, j u , j 3 u , j5u f o r the fundamental, t h i r d harmonic and harmonic r e s p e c t i v e l y .  fifth  I t i s obvious from making these s u b s t i -  t u t i o n s that the h i g h e r harmonic components w i l l be g r e a t l y suppressed by the c o n t r o l l e d element so that a l t h o u g h the h i g h e r harmonic components appear i n the c o r r e c t i o n s i g n a l t h e i r e f f e c t on the output s i g n a l can be c o n s i d e r e d to be n e g l i g i b l e . I f s i g n a l s o t h e r than the fundamental are n e g l i g i b l e i n the output then the e r r o r and the c o n t r o l s i g n a l can be s a i d to c o n t a i n o n l y the fundamental component.  This i s the b a s i s f o r con-  s i d e r i n g the c o n t a c t o r c h a r a c t e r i s t i c i n terms of a q u a s i - l i n e a r d e s c r i b i n g f u n c t i o n gj>-j_ where the s u b s c r i p t 1 i n d i c a t e s that the d e s c r i b i n g f u n c t i o n c o n s i d e r s o n l y the fundamental component o f the correction  signal. This d e s c r i b i n g f u n c t i o n n e g l e c t s the h i g h e r harmonics  w i t h the f o l l o w i n g j u s t i f i c a t i o n as s e t f o r t h by 1,  Kochenburger.  The normal frequency spectrum of a r e c t a n g u l a r wave i n v o l v e s  p r o g r e s s i v e l y s m a l l e r amplitudes f o r i n c r e a s i n g o r d e r s of the harmonic 2.  components.  Most servomotors (Kochenburger here considered a p a r t i c u l a r  type o f c o n t r o l l e d element) serve as e f f e c t i v e low-pass  filters  and minimize the importance of the higher-harmonic components. Kochenburger, by more exact a n a l y t i c methods and by  test,  has compared r e s u l t s w i t h those p r e d i c t e d by h i s approximate method and found the comparisons to be q u i t e good.  T h i s frequency-response  method when a p p l i c a b l e i s a p p a r e n t l y engineering  approximation.  satisfactory  23 5.  The q u a s i - l i n e a r r e p r e s e n t a t i o n of the describing function.  contactor  In the p r e c e d i n g s e c t i o n the form of the c o n t r o l s i g n a l C was assumed to be G = C cosuc|> which i m p l i e s m  the  of the p o s i t i v e and n e g a t i v e c o r r e c t i o n s i g n a l p u l s e s same.  This i s not the most g e n e r a l  to be the  form o f c o n t r o l s i g n a l but i t  i s the case which w i l l be considered p o s i t i v e and n e g a t i v e p u l s e s  duration  i n t h i s paper.  For unequal  o f the c o r r e c t i o n s i g n a l the c o n t r o l  s i g n a l would be of the form C ~ G  Cm  + 0  c o s u  *  w n  ©re  G  0  i s a constant,  d e s c r i p t i v e of the average value of the c o n t r o l s i g n a l .  The  choice  of t h e form of the c o n t r o l s i g n a l presupposes a knowledge o f the form of s e l f - s u s t a i n e d o s c i l l a t i o n s of the c o n t r o l s i g n a l .  For an  assumed c o n t r o l s i g n a l w i t h an average v a l u e o f zero i t i s i m p l i e d that the input ha.Se zero average r a t e of change o f input and a l l s i g n a l s w i t h i n the system have zero average For rectangular  value.  a c o n t r o l s i g n a l C = C^cosuQ the fundamental of the  wave i s as shown i n f i g u r e 6b.  the r e c t a n g u l a r  A Fourier analysis of  c o r r e c t i v e wave y i e l d s a fundamental component  ^1 = Di cos(u(t) + /D| )  (25)  Di  (2.6a)  m  Here  m  that  = 4 sanb  fol = - a  (2.6b)  where  b * 1 (cos-^Ca-Cn) ^ (2Cm F  and  a = l ( c o s - ( C - C i ) ~ c o s " (0^+0^) (20 ) (20m )  +  cos~ (G i+-Ch) (2Cm ) 1  1  C  1  d  1  (2;7a)  (27b)  2  m  The contactor  describing  f u n c t i o n may  be expressed  i n a manner s i m i l a r to the l i n e a r t r a n s f e r f u n c t i o n as the r a t i o  24 of tne output to the i n p u t .  The describing function gjji_ i s  defined as the r a t i o of the Laplace transform of the c o r r e c t i o n signal and the control signal subject to zero i n i t i a l conditionsnor gD]_ « L [ p i o o a ( u » + / D i ) l L[GiaCOSU<|)]  (28)  m  This expression w i l l be found useful when considering, the r e l a t i v e s t a b i l i t y of a system using the root-locus method for a contactor with both inactive and hysteresis zones.  For the time being  equation 28 w i l l be ignored because i t does not contribute d i r e c t l y to the frequency-response method.  The describing function  i s found to be more useful for the frequency-response method when i t is written i n vector form as obtained from the r a t i o of the output vector and the input vector.  The describing function Gp^  i n vector form i s GDI = Dim /&1 —  * 4sinb/-a T Cm  T  (2:9)  This form of the describing function of the contactor i s the one developed by Kochenburger.  It i s a function of i n a c t i v e zone,  hysteresis zone and control signal amplitude and independent of the frequency. In figure 8 i s plotted the graph of the magnitude and phase angle, of the. vector form of the contactor describing function, against the amplitude of the control signal for various r a t i o s of inactive and hysteresis zones. that for values of  The plot of the magnitude i l l u s t r a t e s  l e s s than unity that the magnitude Is gero Cd/2  or no correction signal i s present because the control signal i s within the: inactive zone.  With an increase i n control signal the  magnitude of the describing function increases then reaches a  25  maximum, and decreases u n t i l i n the l i m i t  as t h e amplitude  o f the  c o n t r o l s i g n a l approaches i n f i n i t y the magnitude o f the d e s c r i b i n g f u n c t i o n approaches z e r o .  When h y s t e r e s i s i s p r e s e n t  the phase  angle i s g r e a t e s t f o r s m a l l values of c o n t r o l s i g n a l and t h i s phase angle decreases  w i t h an i n c r e a s e i n c o n t r o l s i g n a l .  There i s zero  p h a s e - s h i f t when the h y s t e r e s i s e f f e c t i s n o n - e x i s t e n t .  86  6.  S t a b i l i t y C r i t e r i a f o r Contactor Servomeonanisms. Consider a l i n e a r s i n g l e - l o o p l i n e a r  w i t h u n i t y feedback  and an open-loop  servomechanism  t r a n s f e r f u n c t i o n equal t o  Kg(p), as r e p r e s e n t e d i n f i g u r e 9. I  0  £T  +^  J — • — < i —  F i g u r e 9. Block diagram o f l i n e a r s i n g l e - l o o p servomechanism. The r a t i o of the output and i n p u t f o r s i n u s o i d a l e x c i t a t i o n i s g i v e n by O j j u j - Kg(iu) I 1+ K g ( j u )  (30)  The i n v e r s e response r a t i o jE(ju) may be expressed i n the 0 f o l l o w i n g form •1 ( j u ) » g - ( j u ) - V K 0 K 1  where  g" (ju) 1  =  (31)  1 g( Ju)  A s i m p l i f i e d v e r s i o n of the Nyquist c r i t e r i o n f o r s t a b i l i t y which a p p l i e s f o r a s i n g l e - l o o p system i n which d i r e c t feedback i s employed and which has a minimum-phase forward t r a n s f e r f u n c t i o n may be s t a t e d as f o l l o w s :  A system s a t i s f y i n g the con-  d i t i o n s and having an i n v e r s e response r a t i o as i n equation 31 w i l l be s t a b l e i f , f o r a p o l a r - l o c u s p l o t o f g ~ ! ( j u ) drawn f o r values of u from -ooto*«o  t  the p o i n t - K i s not enclosed and such  a system w i l l be unstable i f , f o r the p o l a r - l o c u s p l o t o f g drawn f o r values of u from - o o t o t o o , enclosed.  _ 1  (ju)  the p o i n t - K i s completely  27 Tmoq.  aKiS  F i g u r e 10, Frequency p o l a r - l o c u s p l o t F i g u r e 10 shows a t y p i c a l frequency p o l a r - l o c u s p l o t . F o r a t r a n s f e r f u n c t i o n g a i n of K-j_, the system, a c c o r d i n g criterion i s stable. unstable.  However f o r a g a i n o f K  A c t u a l l y f o r 0< K < K  D  the  "the system i s  2  the system w i l l be s t a b l e and  f o r K>Kb "the system w i l l be u n s t a b l e .  For the g a i n equal to  the system, at l e a s t t h e o r e t i c a l l y , w i l l be capable of sustained  to  Kb  self-  oscillations. Consider now  a contactor  servomechanism w i t h an o v e r - a l l  open-loop t r a n s f e r f u n c t i o n equal to K g ( p ) . c o n t r o l s i g n a l amplitude the c o n t a c t o r  For a s i n g l e  constant  describing function w i l l  determine the p a r t i c u l a r value of the g a i n and p h a s e - s h i f t due the c o n t a c t o r .  d e s c r i b i n g f u n c t i o n G T ^ , may  The  the same manner as a c o n v e n t i o n a l the e q u i v a l e n t  control signal. stability  transfer function.  g a i n of the system w i l l be KGr^.  g a i n however i s not a constant Applying  t h a t has  already  be considered  and  to in  Therefore  This value  of  the  depends on the amplitude of  the  the s i m p l i f i e d Nyquist c r i t e r i o n f o r been s t a t e d i t may  be found that  the  system i s s t a b l e f o r some amplitudes of the c o n t r o l s i g n a l and unstable  for others.  the c o n t a c t o r loop  The  s i m p l i f i e d Nyquist c r i t e r i o n r e s t a t e d f o r  servomechanism i s as f o l l o w s :  t r a n s f e r f u n c t i o n Kg(p)  and  a contactor  For an o v e r - a l l opendescribing function G J J ^ ,  28 the system w i l l he s t a b l e i f , f o r a p o l a r - l o c u s p l o t o f g * ( j u ) 1  drawn f o r v a l u e s o f u f r o m -o»  to+o, the p o i n t - KG-T^, i s not  e n c l o s e d and such a system w i l l be u n s t a b l e i f , f o r t h e p o l a r - l o c u s p l o t o f g - l ( j u ) drawn f o r v a l u e s of u from -«o to+«°, t h e p o i n t - KGrr^ i s c o m p l e t e l y  enclosed.  I n o r d e r t o o b t a i n the complete p i c t u r e o f s t a b i l i t y a l l p o s s i b l e v a l u e s o f - K G D I must be c o n s i d e r e d .  Therefore  a second  l o c u s i s drawn j o i n i n g a l l t h e - KG.DT. v e c t o r s f o r a l l p o s s i b l e a m p l i tudes of t h e c o n t r o l s i g n a l . the a m p l i t u d e  T h i s l o c u s i s a p p r o p r i a t e l y termed  l o c u s and i s super-posed on t h e graph of t h e f r e q u e n c y  locus p l o t t e d f o r the f u n c t i o n g ~ ( j u ) . 1  Conditions  in <ix«tmpl<L plottaxi  k Imoa. amy  <Cc/ZK=.or Jfitassaction u~3.l>C lQl2 = /.24-  J8  m  F i g u r e 11. Super-posed frequency  and a m p l i t u d e  loci  C o n s i d e r f i g u r e 11 which i s an example of t h e super-posed loci.  Only a p o r t i o n of t h e f r e q u e n c y  polar-locus plot f o r u positive  i s p l o t t e d f o r the function g ^ t j u ) r j u ( j u + 1) The a m p l i t u d e Ca/K  (32.)  locus i s obtained from f i g u r e 8 f o r c o n d i t i o n s  .05 and C / C a = .2. n  The f r e q u e n c y  2~ n e g a t i v e w i l l be t h e complex c o n j u g a t e  and a m p l i t u d e  loci foru  of t h e l o c i f o r u p o s i t i v e .  F o r t h i s example i n f i g u r e 11 the frequency loci  i n t e r s e c t a t 0^0^/2 = 1.24.  1.2<  Ojy/c 3:/2< 1.24 ]  locus p l o t .  enclosed by the frequency p o l a r -  Therefore the system w i l l be u n s t a b l e f o r  1.2i< C^/CCL/2< 1.24.  f o r values of Cm/ca/2>  The v a l u e s of - KGr^  are not enclosed by the frequency p o l a r - l o c u s p l o t .  Keeping  Therefore  i n mind t h a t Ca/2: r e p r e s e n t s o n e - h a l f the  zone width and Cm the amplitude  of the  sinusoidal control  f o r very s m a l l d i s t u r b a n c e s the amplitude  F o r amplitudes  the system i s s t a b l e .  rium i s d e s c r i b e d as a p o i n t o f convergent  up  to  This p o i n t o f  value  loci equilib-  e q u i l i b r i u m because w i t h  at t h i s p o i n t h i g h e r amplitudes  and lower amplitudes  p o i n t o f convergent  signal,  of o s c i l l a t i o n g r e a t e r than t h i s  i n t e r s e c t represents a p o i n t of e q u i l i b r i u m .  l a t i o n decrease  inactive  Therefore the c o n d i t i o n where the two  r e f e r e n c e to the amplitude  the  of o s c i l l a t i o n w i l l i n c r e a s e  because the system i s u n s t a b l e f o r small amplitudes, Cny/Cc\/2'= 1.24.  1.24  > 1 • 24.  system w i l l be s t a b l e f o r  tained.  amplitude  The v a l u e s of - KGj^ f o r  w i l l be completely  C  and  inorease.  of  Therefore a t  oscilthis  e q u i l i b r i u m o s c i l l a t i o n s w i l l tend to be main-  At the i n t e r s e c t i o n of the l o c i  Therefore from the super-posed  loci  the angular v e l o c i t y u r  i t i s expected  t h a t the  self-  s u s t a i n e d o s c i l l a t i o n s w i l l have an amplitude  r a t i o Cm/ca/2: st 1.24  and a dimensionless  Kochenburger  angular v e l o c i t y of 3.1.  checked t h i s p a r t i c u l a r example e x p e r i m e n t a l l y and ments c o n c e r n i n g amplitude 3 p e r cent  found  has  the  and frequency w i t h i n 6§ p e r cent  agreeand  respectively. F i g u r e 12 i l l u s t r a t e s s e v e r a l c o n f i g u r a t i o n s o f  frequency and  amplitude  loci.  From the c r i t e r i o n of  the system r e p r e s e n t e d by f i g u r e 12a  stability  i s stable f o r a l l operating  c o n d i t i o n s and i s c a l l e d an a b s o l u t e l y s t a b l e system while t h a t of f i g u r e 12b  i s unstable for a l l operating conditions.  3.1.  /1 Tma<  at/3 7  30  /1 Tma<  Rial  F i g u r e 12. Examples o f super-posed The  system represented  Point P represents cribed.  dXIS  loci.  by f i g u r e 12c has two p o i n t s o f e q u i l i b r i u m *  a convergent p o i n t o f e q u i l i b r i u m a l r e a d y  des-  P o i n t Q r e p r e s e n t s a d i v e r g e n t p o i n t o f e q u i l i b r i u m so c a l l e d  because f o r amplitudes s l i g h t l y l e s s than t h a t amplitude a t p o i n t Q the system i s s t a b l e t h e r e f o r e e f f e c t i n g a decrease i n amplitude while f o r amplitudes s l i g h t l y g r e a t e r the system i s u n s t a b l e f o r e e f f e c t i n g an i n c r e a s e i n amplitude.  there-  This p o i n t o f i n t e r s e c t i o n  of the l o c i then does not correspond to a p o i n t o f s u s t a i n e d o s c i l l a t i o n but i s r a t h e r a boundary c o n d i t i o n and one f o r which amplitudes w i l l tend to s h i f t away. for small disturbances  For t h i s system r e p r e s e n t e d  by f i g u r e 12c  the system w i l l o s c i l l a t e a t the amplitude  and frequency determined by p o i n t P .  I f a s u f f i c i e n t l y l a r g e dis«-  turbance were to cause an amplitude o f o s c i l l a t i o n g r e a t e r than t h a t determined by p o i n t Q, then the amplitude would grow and t h e system would be u n s t a b l e . The problem so f a r considered only.  I t i s very o f t e n not s u f f i c i e n t  has been that o f s t a b i l i t y to c o n s i d e r j u s t the problem  31 of s t a b i l i t y * present  R e l a t i v e s t a b i l i t y o r the amount o f damping  f o r o s c i l l a t i o n s a f t e r a d i s t u r b a n c e must very o f t e n  be  considered, Kochenburger approaches t h i s problem from h i s amplitude and  frequency l o c i u s i n g the w e l l known concept of the M  used i n l i n e a r servomechanismsi and  M i s d e f i n e d as the j 0 ( j u ) |  i n l i n e a r servomeonanisms the maximum value of M, Mp,  chosen by r u l e of thumb.  from< experimental  the c u t - o f f p o i n t and of 1.3 provides  i s often  L i n e a r systems are o f t e n c o n s i d e r e d  have s u f f i c i e n t damping i f Mp<1.3. r e s u l t s obtained  criterion  Kochenburger suggests, t e s t s , that an Mp  of 2.0  to  from near  f o r h i g h e r c o n t r o l - s i g n a l amplitudes,  a s a t i s f a c t o r y degree of r e l a t i v e s t a b i l i t y f o r most  applications. I t i s suggested i n the work to f o l l o w t h a t the  root-  l o c u s method used i n the s y n t h e s i s of l i n e a r systems can be used to great advantage f o r c o n t a c t o r servomechanisms e s p e c i a l l y w i t h r e s p e c t to.the r e l a t i v e s t a b i l i t y of a system. l o c u s the s t a t i c  From the r o o t -  servomechanism g a i n can be determined  f o r a s p e c i f i e d amount of damping.  directly  32 7.  Root-locus method o f s y n t h e s i s . The r o o t - l o c u s method was developed by Evans.  8  This  method uses the r o o t s o f the open-loop t r a n s f e r f u n c t i o n to f i n d the r o o t s o f t h e c l o s e d - l o o p For the l i n e a r  transfer function.  s i n g l e - l o o p servomechanism  represented  i n f i g u r e $ the open-loop t r a n s f e r f u n c t i o n was Kg(p) i n Nyquist form, and t h i s t r a n s f e r f u n c t i o n can be expressed as B f ( p ) i n r o o t locus  form.  The r a t i o 0  of output to input  (p) =  1  i s given by t h e expression,  Bf (p)  (33)  1+ Bf (p )  The problem of f i n d i n g the r o o t s o f t h i s c l o s e d - l o o p functions  transfer  appears i n the form of f i n d i n g those v a l u e s o f p which  make the denominator of e q u a t i o n 33 equal to zero. is satisfied  This  condition  i f the open-loop t r a n s f e r f u n c t i o n . Bf(p) =  or i f f ( p ) "  =  1  - 1  (34)  - B  (35)  f ( p ) ~ ^ may be w r i t t e n i n v e c t o r f(p)"  = N&  1  form (36)  0  For the value o f the p to be a root N/£° Therefore  of t h e c l o s e d - l o o p  -  - B  N  =  B  T°  -  18oU+2n)  system  (37) '  (38a)  •  '  (38b)  where n i s any i n t e g e r . To f i n d  the r o o t - l o c u s the f o l l o w i n g procedure i s adopted.  Consider the open-loop t r a n s f e r f u n c t i o n . Bf(p)  ~  1  B L (P+  Jd v  l  (39) (P+ i )  ?2  where the dimensionless The  time constants  q.^, Y^>  y '»  a  r  e  rea  2  ^-  numbers.  zeros and p o l e s of the open-loop t r a n s f e r f u n c t i o n  are p l o t t e d on the complex p-plane as i n f i g u r e 13.  A zero i s  l o c a t e d at -1_, and a p o l e at -°1 , and -1 . The two requirements q.1 yi 7Q imposed r e g a r d i n g the phase angle and the amplitude of the t r a n s f e r f u n c t i o n are considered  F i g u r e 13*  separately.  Roots i n the p - p l a n e .  F i r s t a l o c u s i s obtained angle o f f (p)*"  1  equal  which makes the  to 180°(1 + 2n).  e x p l o r a t o r y p p o i n t and  phase-shift  This i s done by  j o i n i n g the p o s i t i o n s of the  to the p p o i n t to form v e c t o r s which r e p r e s e n t  choosing  zeros and  the p o s i t i v e r e a l a x i s . f i g u r e 13, sum  poles  a l l the complex  f a c t o r e d terms i n v o l v e d i n the open-loop t r a n s f e r f u n c t i o n . angle of each v e c t o r i s measured w i t h r e s p e c t  an  The  to a l i n e p a r a l l e l  to  T h i s o p e r a t i o n has been performed i n  I f the e x p l o r a t o r y p p o i n t i s on the r o o t l o c u s then the  of the angles must equal 1 8 0 ° ( H - 2 n ) o r from f i g u r e 13 -8  2  -  6]+*!  s 180°(1+ 2n)  (40)  T h i s procedure i s c a r r i e d out u n t i l the complete "180° locus'* or r o o t - l o c u s i s obtained  phase  on the complex p - p l a n e .  The next step i s to f i n d the amplitude o f the  inverse  t r a n s f e r f u n c t i o n on the r o o t - l o c u s , t h i s amplitude being  equal  to tae s t a t i c g a i n B of the open-loop t r a n s f e r f u n c t i o n . s t a t i c g a i n B f o r any p o i n t on given  The  the r o o t - l o c u s f o r t h i s example i s  by B =  I  n  Pr+  1  P r + *1 *i  ya  .on where p  r  i s / t h e r o o t - l o c u s and  the f a c t o r s a r e the a b s o l u t e  of the v e c t o r s j o i n i n g the p o l e s and Therefore, t r a n s f e r f u n c t i o n and  values  zeros to pr»  g i v e n the p o l e s and  zeros of the open-loop  the s t a t i c g a i n B, the r o o t s of the closed**  loop system can be obtained  d i r e c t l y from the r o o t - l o c u s .  F i g u r e 14. Root l o c u s o f s i n g l e - l o o p servomechanism. F i g u r e 14 i l l u s t r a t e s the r o o t - l o c u s f o r a s i n g l e - l o o p servomechanism with open-loop p o l e s at 0, »I , and - i • The r o o t formed yi 72 from the p o l e - i , t r a v e l s i n an i n c r e a s i n g l y n e g a t i v e d i r e c t i o n y2 along the n e g a t i v e r e a l a x i s .  The  r o o t s formed from the p o l e s a t  0 and - i approach one another, at f i r s t , yi  then form a complex  jugate p a i r w i t h a f u r t h e r i n c r e a s e i n g a i n . i t may  From t h i s r o o t - l o c u s  be seen that the troublesome r o o t s are those which have formed  i n t o the complex conjugate may  con-  pair.  For a s u f f i c i e n t l y h i g h g a i n  appear i n the r i g h t h a l f p-plane so t h a t the complex  they  conjugate  55  p a i r w i l l have p o s i t i v e r e a l p a r t s r e s u l t i n g i n an u n s t a b l e However t h e g a i n can be s e t so t h a t the complex conjugate  system.  roots  appear i n the l e f t h a l f p-plane with a s u f f i c i e n t l y l a r g e value  of  the n e g a t i v e r e a l p a r t to ensure s u f f i c i e n t damping f o r the n a t u r a l o s c i l l a t i o n s which would r e s u l t from a d i s t u r b a n c e  to the system.  Consider a s i n g l e - l o o p c o n t a c t o r servomechanism w i t h p o l e s of the o v e r a l l open-loop t r a n s f e r f u n c t i o n as those f i g u r e 14.  The  in  c o n t a c t o r i n t h i s d i s c u s s i o n i s assumed to have  no h y s t e r e s i s zone.  For zero h y s t e r e s i s , the v e c t o r form of  the c o n t a c t o r d e s c r i b i n g f u n c t i o n has amplitudes of the c o n t r o l s i g n a l .  zero p h a s e - s h i f t f o r a l l  Therefore f o r a l l s i n u s o i d a l  c o n t r o l s i g n a l amplitudes the value of the d e s c r i b i n g f u n c t i o n shows the c o n t a c t o r to appear as a v a r i a b l e g a i n element, the g a i n being dependent on the amplitude of the c o n t r o l s i g n a l . i s important  to observe that the c o n t a c t o r c h a r a c t e r i s t i c  It  expressed  i n terms of i t s d e s c r i b i n g f u n c t i o n has no e f f e c t on the open-loop r o o t s of the system.  The open-loop zeros and p o l e s are  obtained  from the l i n e a r p o r t i o n of the c i r c u i t and remain f i x e d i n the p-plane.  However, the c l o s e d l o o p r o o t s are not f i x e d and  dependent on the e f f e c t i v e open-loop g a i n , BGj^,  are  which i s a  product  of the s t a t i c l i n e a r g a i n and the c o n t a c t o r d e s c r i b i n g f u n c t i o n . For a p a r t i c u l a r value of c o n t r o l s i g n a l amplitude the g a i n i s g i v e n by the d e s c r i b i n g f u n c t i o n and  f o r a known v a l u e  the l i n e a r g a i n the c l o s e d - l o o p r o o t s of the system may from the r o o t - l o c u s .  Therefore  contactor  be  of  obtained  as the amplitude o f the c o n t r o l s i g n a l  i s v a r i e d , the r o o t s of the c l o s e d - l o o p system vary i n a manner determined by the r o o t - l o c u s .  The  root-locus i n conjunction  with  the d e s c r i b i n g f u n c t i o n o f the c o n t a c t o r can be used as a s y n t h e s i s  36 technique f o r c o n t a c t o r servomechanisms and becomes q u i t e when c o n s i d e r i n g t h e problem of r e l a t i v e  effective  stability.  T h i s d e s c r i p t i o n on the use of the r o o t - l o c u s i n synthesizing contactor  servomechanisms has been g e n e r a l .  i n the f o l l o w i n g s e c t i o n , w h i c h  However  c o n s i d e r s a p a r t i c u l a r example, the  use and the s i g n i f i c a n c e of the r o o t - l o c u s method w i l l be t r e a t e d in specific  detail.  o  37 8.  Analysis  of the model by the frequency-response r o o t - l o c u s methods*  a)  Kahn's method f o r o b t a i n i n g The  transient  responses.  t r a n s i e n t response of a c o n t r o l system to a  d i s t u r b a n c e i s o f major s i g n i f i c a n c e . for  and  both c o n t a c t o r  and  This statement i s  l i n e a r servomechanisms.  true  Transient  responses  w i l l be examined i n the i n v e s t i g a t i o n to f o l l o w .  Such responses  are important when c o r r e l a t i n g r e l a t i v e s t a b i l i t y  from the  l o c u s i n an analogous f a s h i o n to l i n e a r servomechanisms.  rootThe  t r a n s i e n t responses are a l l o b t a i n e d by a p p l i c a t i o n of Satin s semi1  g r a p h i c a l method.  The  f o l l o w i n g i s a summary of t h i s method.  C o n s i d e r the s i n g l e - l o o p represented i n f i g u r e  contactor  servomechanism as  15.  0  F i g u r e 15.  Single-loopj c o n t a c t o r  For a step r e f e r e n c e  input  servomechanism  the t r a n s i e n t response of the  v a r i a b l e c o u l d have the form as shown i n f i g u r e 16a s i g n a l would then be as r e p r e s e n t e d i n f i g u r e The  and  controlled the  16b.  r e l a t i o n s h i p between the c o r r e c t i o n s i g n a l and  c o n t r o l l e d v a r i a b l e f a r the  the  q u a n t i t i e s expressed i n t h e i r L a p l a c e  transforms l i s D(s)  correction  - Iff} - ^  (s)  (42)  to f o l l o w page  FtQuri  G-raphs  hr  Kahn*5  saint-yrafAtca/  itidthod  37  38 where l£a) i s a term c o n t a i n i n g conditions  initial  conditions.  For i n i t i a l  equal to zero (43)  For D ( t ) as shown i n f i g u r e 16b D(s) r f  [l - e~  - e-  t l S  t 2 S  +  e'^  + B'^  S  s  - e ^  5  S  - ...]  (44) To o b t a i n  the t r a n s i e n t  response g r a p h i c a l l y i t i s n e c e s s a r y to  s o l v e one equation o n l y , and that i s the i n v e r s e L a p l a c e t r a n s f o r m  L  - l  [Gisi]  =  0 (t)  (45)  R  o  The  p h y s i c a l s i g n i f i c a n c e of this inverse  t r a n s f o r m i s that i t i s  the output of the c o n t r o l l e d v a r i a b l e which r e s u l t s from a steady a p p l i c a t i o n of the c o r r e c t i o n s i g n a l +1 when the c o n t r o l l e d is  i n i t a l l y at r e s t .  Therefore to o b t a i n  the complete  transient  response the "runaway responses"' can be added g r a p h i c a l l y 0(t)  and  s  0 (t) t>0 R  2  responses which f o l l o w , simplicity.  b)  Analysis The  effect. for  since  - O R ( t - t i ) - OR(t-tg) + 0 R ( t - t 3 ) + ... t>ti t>t t>t3 (46)  t h i s has been shown i n f i g u r e 16c.  for  variable  zero i n i t i a l  In a l l the t r a n s i e n t  conditions  have been assumed  of the Model c o n t a c t o r of the model contained n e g l i g i b l e  Therefore this a n a l y s i s  the c o n t a c t o r .  hysteresis  i s based on zero h y s t e r e s i s  F u r t h e r t h e o v e r - a l l open-loop t r a n s f e r  of the model can be approximated by the t r a n s f e r G-(p)  * K_ PTP^TI )  function (47)  D  zone function  39  F i g u r e 17 i s a b l o c k d i a g r a m r e p r e s e n t i n g t h i s system.  For a  sinusoidal input s i g n a l the  1+  Contactor  mo.clns  o  0 f>(p*)  z  F i g u r e 17. B l o c k diagram r e p r e s e n t a t i o n o f t h e model to input  r a t i o o f output i s g i v e n by A  0(j ) U  -  &Di  u  GUu) = Kg(ju)  where  (48)  G-U )  s  K JuUu+ I )  2  The i n v e r s e f r e q u e n c y l o c u s g ( j u ) = j u ( j u + l ) _ 1  figure  2  i s plotted i n  18. The a m p l i t u d e l o c u s - K G r ^ w i l l t r a v e l a l o n g t h e n e g a t i v e  r e a l a x i s s i n c e no p h a s e - s h i f t e x i s t s f o r G Q ^ when the c o n t a c t o r has no h y s t e r e s i s zone. infinity,  F o r r a t i o s o f Cm/ba/2 e q u a l t o u n i t y and  Gr^ e q u a l s z e r o .  T h e r e f o r e t h e a m p l i t u d e l o c u s has i t s  o r i g i n a t t h e o r i g i n o f the complex p l a n e f o r 0^0^/^ r 1.0  then  p r o c e e d s along t h e n e g a t i v e r e a l a x i s t o a minimum p o i n t then r e v e r s e s and f i n a l l y as t h e c o n t r o l s i g n a l a m p l i t u d e  approaches  i n f i n i t y i t a g a i n approaches t h e o r i g i n . From t h e s i m p l i f i e d v e r s i o n of the N y q u i s t the system w i l l be s t a b l e f o r v a l u e s o f K G T J  1  <  criterion  2;.0 and w i l l be  absolutely s t a b l e , that i s s t a b l e f o r a l l c o n t r o l s i g n a l i f t h e maximum v a l u e o f  KGj^^KGT^max.),  i s l e s s than 2.0.  amplitudes, Points  to follow  of e q u i l i b r i u m occur f o r points  of e q u i l i b r i u m ,  loci.  The  point  KGJJ^ =r  2.0.  ForCKG^max.^ 2..0,  both f o r u = 1.0,  occur on  of e q u i l i b r i u m f o r the s m a l l e r  amplitude i s a d i v e r g e n t  the  two  super-posed  control signal  p o i n t o f e q u i l i b r i u m w h i l e that f o r  l a r g e r c o n t r o l s i g n a l amplitude i s a convergent p o i n t of For  the  equilibrium.  example, the amplitude l o c u s drawn i n f i g u r e 18 i s f o r Cfl/K r  From f i g u r e 8 thereforeCKG^max.) » 2.44 Cm/cd/2 equal to 1.11  and  2;.33.  A p l o t of the points  KGr^  equals 2 . 0 f o r 2  Therefore the divergent  e q u i l i b r i u m o c c u r s f o r Cm/ca/2. = 1.11 e q u i l i b r i u m f o r Cm/ca/2: r  and  and  .25.  point  of  the convergent p o i n t  of  2.33.  amplitudes of the  c o n t r o l s i g n a l which g i v e  of e q u i l i b r i u m versus the r a t i o Ca/K  i s shown i n f i g u r e  19,  2  The  points  of convergent e q u i l i b r i u m are  because these p o i n t s  those of s p e c i a l i n t e r e s t  r e p r e s e n t the p o s s i b l e  self-sustained  oscil-  l a t i o n s of the system.  It i s d i f f i c u l t  the p o i n t s  e q u i l i b r i u m f o r s m a l l v a l u e s of the  of, d i v e r g e n t  to a s c e r t a i n the meaning o f  r a t i o because Kochenburger's j u s t i f i c a t i o n s f o r the use frequency response method do  not  apply.  the f r e q u e n c y spectrum of a r e c t a n g u l a r g r e s s i v e l y smaller components. such small  amplitudes  no  m  wave does not  e f f e c t on the  a  <  1.1  i n v o l v e proharmonic  a n a l y s i s because of  involved.  graph o f f i g u r e 19  s t a b l e f o r values of C a / K _ l >  shows the system to be .  of s e l f - s u s t a i n e d o s c i l l a t i o n s and sustained  the  From f i g u r e 7 f o r C / C  amplitudes f o r i n c r e a s i n g o r d e r s of the  However t h i s has  The  of  amplitude  o s c i l l a t i o n s increase  absolutely  For Ca/K^, 1 the system i s capable the amplitudes of the  w i t h a decrease ,in Ca/K. 2~  frequency of these s e l f - s u s t a i n e d o s c i l l a t i o n s , r e g a r d l e s s tude, remains constant at u =  1.0.  selfThe of  ampli-  41 The r a t i o Cg/K  i s significant.  For t h i s case,  K,  a  represents  the s l o p e of the output  response to a u n i t step  c o r r e c t i o n s i g n a l a f t e r a s u f f i c i e n t l y l o n g time.  Therefore  g r e a t e r the K the q u i c k e r the c o r r e c t i o n response.  the  However, the  s t a b i l i t y of the system i s not o n l y dependent on t h i s g a i n f a c t o r but a l s o on the i n a c t i v e zone, C g .  T h i s i s a p r o p e r t y p e c u l i a r to  c o n t a c t o r servomechanisms. The graph of f i g u r e 19 although s e p a r a t i n g the system i n t o an a b s o l u t e l y s t a b l e reg/rin f o r Gd/Kv  1 and an  oscillatory  r e g i o n f o r Ca./K^ 1 g i v e s no i n f o r m a t i o n about the r e l a t i v e of the system i n the a b s o l u t e l y s t a b l e r e g i o n .  The  conventional  M - c r i t e r i o n method employed i n l i n e a r servomechanisms may as p r e v i o u s l y mentioned.  For  stability  be  used  t h i s example the r o o t - l o c u s method  w i l l be employed to i n v e s t i g a t e the r e l a t i v e s t a b i l i t y of the system. For  t h i s case both the Nyquist  and  r o o t - l o c u s forms o f  the o v e r - a l l open-loop t r a n s f e r f u n c t i o n are the same* F(P) S  JL  £  P(p + D  where and  a  B =  G(P) = _K P(P+D  (49) 2  K  g(p) = f ( p ) s  1 P(P  T +  1)  f ( p ) has t h r e e p o l e s on the complex p-plane, the o r i g i n and  a double p o l e a t p » -1.  r o o t - l o c u s are p l o t t e d i n f i g u r e 2 0 .  one  These p o l e s as w e l l as  s  B G  Dl*  Values o f Ca/K  the  For s i n u s o i d a l s i g n a l s i n the  System the e f f e c t i v e open-loop g a i n .is B G j j j S i n c e K = B, KGTJI  at  then  are a l s o p l o t t e d on the r o o t - l o c u s .  2 The v a l u e o f KGDT c o r r e s p o n d ing i s the maximum v a l u e of KGy^  to the v a l u e o f C d / 2 f o r t h a t v a l u e o f Ca/K. 2~*  K  o  n  ^ For  e  root-locus example,  f o l l o w p a g e Jj-l  to  — - • — — - t  1  i j  '  - - ^ -  i  t  C  j. l  •  ;  - t — ' — • { — ,  •  , -  ;-  1' : ! ' : ;  /n\=  J  1.  A  '  . i  [  •  j-  :•  1 i  : '1  —  i  I  ;  i  i  ] i  j i  j  i 1  \  i  j 1  i  i  ••;-[•-•• '  ;  —  i /  i  i  -  j  i •!  i i  |  I  I  i  i  i \ i  !  1  i  j  i  1 !  4  !  i ;  1  ;  1  I  i  i •  r  1  •  i  i i  i  ! I • 'u  :  1 i  • i  i  j  -  i '  /  1/ 0  i  !  •• 1 • i  1 t 1  -  |  l _  i  1  1  f .-v.  /• •  i  ;  / 1  ,  1  1 ! i  *  i  5  jj  f t  1  r<3  /  I -  J |  /  6  !  t  i  4  "  1 1  1  >  1  V  1 •••f  J  £ ! 1 I  ! I  J  1 A* *  •  • j <  cr  /A  >S  /  Oc  "r  • i  ti  'S  i  7(  .j  V  i  \  •  I  1  5~ i  • i  i  1  |  - I  > j  j  j  i  i  i  1  1  ! 1  1  1  i  i  .  ,  1  1  i  1  •  i f  !.  l l t  i _ -  -  • •  I  • I  F/aure. 20.  fact-lows  1 • for tho, refreientatfOn of  thi. vtKdrtf>as>sctt<id  /#dcJ«./  42  the v a l u e s K G Q r . 3 7 6 and ° d / = 1 . 6 9 correspond to the same p o i n t K  1  on the r o o t - l o c u s o f f i g u r e 2 0 .  T h e r e f o r e f o r a r a t i o o f Gd/K = 1 . 6 9 2  the maximum value of KGrj-^ i s equal to . 3 7 6 and t h i s p o i n t on the r o o t - l o c u s determines the minimum damping f o r o s c i l l a t i o n s i n the system. S i m i l a r i n f o r m a t i o n r e g a r d i n g s t a b i l i t y can be o b t a i n e d from the r o o t - l o c u s as from the super-posed amplitude frequency  loci.  The complex conjugate p o l e s c u t t h e imaginary a x i s a t u = 1 . 0 which frequency corresponds to the frequency o f convergent t h e frequency-response p l o t . at u x  1.0  i s KGTTJT^ r  2.0  The open-loop  equilibrium i n  g a i n from the r o o t - l o c u s  which a l s o corresponds to the value a l r e a d y  obtained. From the r o o t - l o c u s the s e t t i n g o f the g a i n can be s e t f o r the d e s i r e d minimum damping.  F o r example, i f the minimum  damping r a t i o <r/u i s to be no l e s s than 1 / / 3 * then from f i g u r e 2 0 , KGD-  can be no g r e a t e r than . 3 7 6 .  The r a t i o of Cd/K i s t h e r e f o r e  2 1.69.  Given the i n a c t i v e zone C^, which i s determined from the  requirements f o r s t a t i c accuracy, the g a i n K can be found. The r o o t - l o c u s g i v e s some i n f o r m a t i o n about  transient  responses at l e a s t q u a l i t a t i v e l y and i n many cases "semi-quantitatively*. The c o n t a c t o r d e s c r i b i n g f u n c t i o n GTJ-^ i s independent o f frequency and dependent o n l y on the amplitude o f t h e c o n t r o l  signal.  However, the r o o t - l o c u s i m p l i e s that f o r a p a r t i c u l a r amplitude o f c o n t r o l s i g n a l that o n l y one frequency i s p o s s i b l e . because  This i s true  the r o o t - l o c u s determines the n a t u r a l mode o f o s c i l l a t i o n  of the system yet the g a i n for-a, p a r t i c u l a r o s c i l l a t o r y  condition  on the r o o t - l o c u s determines  from the d e s c r i b i n g f u n c t i o n the  amplitude o f the c o n t r o l s i g n a l f o r that n a t u r a l mode of o s c i l l a t i o n . I n f i g u r e 21 a r e p l o t t e d t r a n s i e n t responses o f the cont r o l l e d output to a step i n p u t of the r e f e r e n c e . f i g u r e 21a i s f o r Ca/K = .4.  The response i n  From the r o o t - l o c u s the maximum v a l u e  2~  o f KG/jjj_ i s 1.59.  The angular frequency and damping f a c t o r c o r r e s -  ponding t o t h i s v a l u e o f KGr^ on the r o o t l o c u s a r e u - .91 and T:  -.04 r e s p e c t i v e l y .  From the t r a n s i e n t response p l o t t e d the p r e -  dominant frequency i s u » .796. For an i n c r e a s e i n the value of the r a t i o Ca/K i t would 2~ be expected t h a t the t r a n s i e n t response f o r a s i m i l a r type of d i s turbance would have a lower frequency and g r e a t e r damping.  For  Ca/K = »5| the maximum value o f K G ^ i s 1.2:7 and the frequency and  "a damping f a c t o r on the r o o t - l o c u s corresponding t o t h i s v a l u e a r e u s .81 and<Ts -.09. frequency i s u  :  From the p l o t i n f i g u r e 21b the predominant  .689 and i t i s seen that the damping i s s l i g ; h t l y  g r e a t e r i n t h i s case than i n the p r e v i o u s one. I t i s q u i t e r e a s o n a b l e that the predominant  frequency  of the t r a n s i e n t response i s approximately equal to t h a t g i v e n f o r the c o n d i t i o n of minimum damping on the r o o t - l o c u s .  This i s t r u e  e s p e c i a l l y i n t h i s case because minimum damping a l s o corresponds to the maximum p o s s i b l e v a l u e of the angular frequency.  Further  t h i s minimum damping o c c u r s f o r a maximum value o f KGT^, f o r which Cm/Cd/2'm 1.4 which means that the c o n t r o l s i g n a l amplitude i s r e l a t i v e l y s m a l l and the system i s c l o s e to coming to r e s t . I t i s , o f course, very d i f f i c u l t  to come to any d e f i n i t e  c o n c l u s i o n s r e g a r d i n g the t r a n s i e n t response from an i n s p e c t i o n o f the r o o t - l o c u s because  the t r a n s i e n t responses a r e dependent on  the form of the e x t e r n a l d i s t u r b a n c e .  Possibly with s u f f i c i e n t  Fiavra,  Zl. Plots  of  transient  r<tsponsa.s  to  r«fs.ra.ncq.  stap  inputs  e x p e r i e n c e and p r a c t i c e a s a t i s f a c t o r y r u l e of thumb c r i t e r i o n could be developed.  However, the r o o t - l o c u s  with the c o n t a c t o r  describing  f u n c t i o n does y i e l d a p r a c t i c a l technique t o h e l p the d e s i g n e r i n choosing the g a i n of the system f o r s a t i s f a c t o r y r e l a t i v e s t a b i l i t y , c)  A p p l i c a t i o n o f a simple phase-lead type compensating network. For  the system j u s t analyzed the r a t i o C^/K had to be 2~  r e l a t i v e l y l a r g e to ensure s t a b i l i t y .  The a p p l i c a t i o n o f a com-  p e n s a t i n g network a l l o w s t h i s r a t i o to be made s m a l l e r . f o r a p a r t i c u l a r v a l u e o f i n a c t i v e zone,  This  means  ,that K can be made l a r g e r  which r e s u l t s i n improved dynamic a c c u r a c y . The  compensating network added to the b a s i c model i s the  type shown i n f i g u r e 5 which has a t r a n s f e r  function  Therefore the o v e r - a l l open-loop t r a n s f e r f u n c t i o n F(p)  » B(p-r-.65) P ( P - t - l ) ( P + 2,.8)  i s now (51)  s  P l a c i n g the above t r a n s f e r f u n c t i o n i n i t s Nyquist form G(p)  =• K(1.588p + 1) P(P+ l ) k ( . 3 5 7 p + l )  (52.)  K = . 6 5 r *2S5B  8 1 1 ( 1  (53)  B  The  frequency-response o f g ~ ( j u ) - j u ( j u + l ) ' (,3573m-1) U.588Ju-t-l) 1  i s shown i n f i g u r e 22, u  =  2*0.  2  T h i s p l o t cuts the r e a l a x i s a t -3.69 f o r  Therefore the boundary c o n d i t i o n  KOD-L S 3.69.  (54)  for stability i s  The system w i l l be a b s o l u t e l y s t a b l e f o r K G ^ < 3,69 which i s an improvement over the uncompensated case which was a b s o l u t e l y  stable  f o r K G < 2.0. D L  F i g u r e 23 i s a s i m i l a r graph t o t h a t i n f i g u r e 19 f o r the uncompensated case.  I t c o n s i s t s of a p l o t of the amplitudes  o f t h e c o n t r o l s i g n a l which g i v e p o i n t s o f e q u i l i b r i u m v e r s u s the r a t i o Oa/K. 2:  The forms of the two graphs a r e a l i k e .  I n the r e g i o n  of p o s s i b l e s e l f - s u s t a i n e d o s c i l l a t i o n s , the amplitudes o f o s c i l l a t i o n i n c r e a s e w i t h a decrease i n the r a t i o C^/K .  The frequency  of t h e p o s s i b l e s e l f - s u s t a i n e d o s c i l l a t i o n s w i f l now be u  =  2.0 i n  comparison t o u - 1.0 f o r the uncompensated oase. The r o o t - l o c u s i s shown i n f i g u r e 24. compensating  network i s to i n c l u d e a zero a t p  =  The e f f e c t o f the -.63 and a p o l e  at p - -2.8 i n a d d i t i o n to the p o l e a t p - 0 and t h e double p o l e at p  =  -1.  I n an analogous manner to the uncompensated case, the  v a l u e s of KGJJ^ as w e l l as t h e v a l u e s o f Ca/K c o r r e s p o n d i n g t o the 2 maximum v a l u e o f K G ^ a r e p l o t t e d on the r o o t - l o c u s . The e f f e c t of the a d d i t i o n o f t h e compensating network on t h e t r a n s i e n t response i s demonstrated i n f i g u r e 25.  This f i g u r e  g i v e s the t r a n s i e n t responses o f both the uncompensated and compensated systems to equal s t e p i n p u t s o f t h e r e f e r e n c e f o r e q u a l values 6f the r a t i o Ca/K equal to .40. 2; of the compensated  case i s o b v i o u s .  The improved dynamic response  I n f i g u r e 25a which i s the  t r a n s i e n t response w i t h no compensation,  the c o n t a c t o r w i l l have to  undergo many s w i t c h i n g o p e r a t i o n s b e f o r e the output response f i n a l l y r e s t s i n the i n a c t i v e zone. compensated  I n c o n t r a s t i s the response o f the  system shown i n f i g u r e 25b.  I n t h i s case merely two  p u l s e s o f the c o r r e c t i o n s i g n a l are n e c e s s a r y to b r i n g the system t o rest.  to f o l l o w page  ^  yur~£ 2J4. Koot'/ocu^ for th<l r<tpr<L)<int®C/on of fJfv. company (a4 r  to follow page *J-5 .  .  ^  ...  1  „ .  •;--|-7-|-:-T- -l;  t  •  1  1  i  i  !  >  A  \&  C i =i1!  : i  i  \  i i i  1  \1  ,  A  \  1  / /  f-  /  1  .  1  i  i  •""! "  i  i  i  l\  \ \  I  /'  :  j  '  /1  1f  f 1  ' j'  i  | 1 1  1  • i I  • 1  |  !  1  . '  — •  |  !  I  "• r • 1  i  .  i I  -  I  j  —- —  i 1 i i  /  1 |  j  J\  a  >  i  /  i  I  i '  )  I  >  .3 0 ;  •  i c  L B *  ton  •a s (Lntja t.ion  a.  uric Off  •-  \  '  1  D  i ! i  '  i  1  j 1  i !  j  :  I  i i  I  i  -  i  t1 i  !  J  __ j •  hss fcirr)<L i  1  i i  i  -  •-.  _ 7TS/C  •  .1  !  MM l :  \ J1  \  . *N  j  C  1 •  f  - T ~ M j -:-  A  I  a M r "3 ! •  i  :  !  f  '  1  A i \  i  :  •  i  1  •N  i  >.  i J  •f  .  1 - .  \C:, lit  • >  . ;  c  -  •  •>  / it  $-  -  s •  -  • • 1  •  o  t  <  i  tatdttien_.  U i _  --  t  T  1  i -  0  0  2 i  h) *•  -'  > i  I  46 9,  Root-locus method when c o n t a c t o r has h y s t e r e s i s F o r the c o n t a c t o r w i t h no h y s t e r e s i s zone the v e c t o r  form o f the c o n t a c t o r d e s c r i b i n g f u n c t i o n G ^ f o r a l l amplitudes o f the c o n t r o l s i g n a l .  has zero p h a s e - s h i f t  The c o n t a c t o r  therefore  appears^' as a v a r i a b l e g a i n element, the g a i n depending on t h e amplitude o f the s i n u s o i d a l c o n t r o l s i g n a l .  The v a r i a t i o n s i n the g a i n  of the c o n t a c t o r hawpno e f f e c t on the o v e r - a l l open-loop r o o t s and a f f e c t e d o n l y the c l o s e d - l o o p r o o t s o f the system. For the c o n t a c t o r with a h y s t e r e s i s zone, p h a s e - s h i f t e x i s t s i n t h e v e c t o r form o f the d e s c r i b i n g f u n c t i o n G ^ .  Therefore  the argument that the c o n t a c t o r has no e f f e c t on the o v e r - a l l openl o o p r o o t s no l o n g e r a p p l i e s .  I t w i l l be shown t h a t the c o n t a c t o r  with h y s t e r e s i s now not o n l y a f f e c t s the c l o s e d - l o o p r o o t s o f t h e system but a l s o the open-loop r o o t s o f t h e system. complicates  the root l o c u s  T h i s o f course  considerably.  R e f e r r i n g back to chapter o f the c o n t a c t o r was r e p r e s e n t e d  5 where the d e s c r i b i n g f u n c t i o n  as a r a t i o o f t h e L a p l a c e trans**  forms o f the c o r r e c t i o n s i g n a l and c o n t r o l s i g n a l , and g i v e n by equation 28 SDi s L L where from equations D  [Dim cos(u<t> + /P].)] [Gm cos u<t>]  2:6a and 26b r e s p e c t i v e l y  lm z 4 sin b "* IT  A>1 = - a Expanding equation 28 s. L [ D I H J I C O S ud>) (cos L [ C cos ua» ]  a)4(sin  u<t>) ( s i n a jj]  m  (55)  47 and p l a c i n g equation 55 i n t o i t s L a p l a c e transform i t can  be  expressed i n the form Dlia cos?.a f p Cm - h i The a p o l e at the  T-  u  tan a  1 ( 5 6 )  d e s c r i b i n g function i n i t s Laplace transform o r i g i n and  a zero at p  zero i s not f i x e d on the p-plane and a n g u l a r frequency, and  - u t a n a.  =  open-loop  v a r i e s as a f u n c t i o n of u,  the p h a s e - s h i f t  d e s c r i b i n g f u n c t i o n o f the c o n t a c t o r the r o o t s o f the c l o s e d - l o o p  This  introduces  angle, a.  Therefore  w i t h h y s t e r e s i s not  system but  the  the  only a f f e c t s  a l s o those of the  open-loop  system. F o r z e r o h y s t e r e s i s , a s o f o r a l l amplitudes of c o n t r o l s i g n a l , cos a then equals u n i t y and t a n a equals ^ therefore  the zero,  the d e s c r i b i n g f u n c t i o n g j ^ degenerates to  SDi * pirn  (57)  which f o r t h i s case i s e x a c t l y the same as the v e c t o r  form of  the  d e s c r i b i n g f u n c t i o n which i s to be expected f o r zero h y s t e r e s i s . Considering  the g a i n , D i - cos a, of the m  f u n c t i o n ; the r a t i o Dim  i s equal  Cm  describing  t o the absolute  v a l u e of  the  Crn  vector can  form of the d e s c r i b i n g f u n c t i o n G T ^ .  then be  T i l e  describing  function  rewritten e  Di  =  ICDII  c  o  s  a  p  +  u  t  a  n  a  j  (58)  P In f i g u r e 8,  the product of the l i n e a r g a i n , K,  as w e l l as the p h a s e - s h i f t amplitude of the  and  IGTJJ  , KlGj^l ,  angle - a have been p l o t t e d a g a i n s t  control s i g n a l f o r various  constant r a t i o s of  the  Ch/Cd«  From these two  graphs i s p l o t t e d a new  a g a i n s t the amplitude. that f o r the Nyquist The graph of K  form o f the l i n e a r open-loop t r a n s f e r f u n c t i o n . the p h a s e - s h i f t angle '-a a g a i n s t  c o n t r o l s i g n a l amplitude f o r the r a t i o Ch/Ca= 0£2: has i n f i g u r e s 26a  and  a  I t must be remembered t h a t the g a i n K i s  cos a and  IGDJ  graph, K l G j ^ l cos  the  been p l o t t e d  2:6b r e s p e c t i v e l y .  The method of o b t a i n i n g the r o o t - l o c u s f o r a p a r t i c u l a r example w i l l now  be c o n s i d e r e d .  The  l i n e a r open-loop t r a n s f e r  f u n c t i o n chosen i s the same as t h a t which was  assumed to  represent  the model t h a t i s G(p) =  K P(P +  (59) D  2  the r o o t - l o c u s form i s then B  (P) -  where The  B s  , . plp+ 1)  (60)  B  D 2 ;  K  example i s worked out f o r the r a t i o of h y s t e r e s i s zone to  i n a c t i v e zone  r 0--12'  The r a t i o of output to i n p u t i s g i v e n .  |(P)  r  and  B(p)  SDT.  l  +  by (61)  SDiB(p)  SDiB(p) tt B'Gpxl cos a(p + u p  2  (p+  l)  tan a)  For a p h a s e - s h i f t angle, -a^,and f o r the p v e c t o r having  an imaginary component  f i g u r a t i o n i s as shown i n f i g u r e The  (62)  2  u^  exploratory  the p o l e and  zero con-  27.  f o u r r o o t s of the c l o s e d - l o o p  system f o r the open-  to f o l l o w page k& t  i  t  ;  !  + -  i. •  __i : M  -1 ii  .:_ : I  ...::.|L:_. •  'i;,  •:  •::;  •  \i:,  Of')--  ~PU ---Ish)  ..j  - 4 -  ;  com  —  •° y  - - -  t  i . . .  - ::4 . . 1 - -  -1  ;• •  • • i  1  fr  ;  .P  of  J  .. . | .  H  irr~~-  —  i :.|  r  —  —-  .... | . .  —  -  —.  •::  .:  • " i  —  r — r  -i  :;::]Ti  1  ;  -..i  •:  ;' : ;'':  -  :'  :;::  : :;  :  ;:;  i : ' ' :  .... . '•'•<  .... :!:t  .  ::::  ::::  ; [ i; * '; ,(.. TT*T  * i • • -'  jHl: fit Hi iii  ;:i:  111; ti:;  . ::  jti,  ,  :.: 4 :!:_.; "i • '  tfii: HI; i-iiji  ii,;-., :ii  Ha  1!] iii;I ill •!:;!:i;j iiiili!! iii? \\\r  * j !  '• i • • 1  ! : ! ': "  27. 0/»w -loop  ; :  :: JL-  v.-i-  ;;  ::;:  '  I  tdro-p^lt  -i  M  ll:i  —  .:.:(.• •:. I..  4  —  p—  -  :--[--  '•  t  1  .i :.'„:. •.. ^  .... 1 • - -  i  .  ;  : i  ;  ;  it:  ;1  i.!-H  > i  i:/  .y_ __j  '.:-.;i:  _ at p  =bli_..  1  : j.; ; :  ii :.  :  ;!'.:  :  :  T"  ;  ;!::  .  1 ... .,.  :  ::: :  .  4  •  •  :  • >:..  iH.  ;. :: . r  : I : :•  . .,  .  ..!. >  i ::: 1 '..  P7  '. '.". \. ' . i . -*  .  • . : . : :.: ! " ;  ! • . .;  iii:.i i' i;  • :!.:: r *  • :. i' • • • f - T i 'T :  :::.j: :- i i . i ' ..}—i 1H  ill :  :  .:  : •—Tfi •  I  =L  j  •': •':'  - j : 1—J ,  •r;n •iM  :  ' t  ' i i ~ ''  !!;!• i;!L  !j'j  t..  T —  .  r  1: '  ;;;  t -  • l  ill' if! iii.iir! rf. - 14,1 ! 11 m i . TiriTT  f/jvrd  •  • t  i  ::! i ; : : . : :;  r r r f TH:  !' •' 11 ]  —  Pov !»/cj.  :  .j:  :::*  ; t•;  •iii  '. i'.!  .....  ....  ...  : :: j-: Ii; ... :.•:[ • .• ji :. :; ,  ::.:'::;:  -tr | ,  ::•.:  1; i  :•  '.:!: :  ::.. : : : :  :;t;- ; • ; : : .:::  "'i  i:... I:-' i • • ..::  •• i  :  ;•:: : ; : :  r  •l!; !  :i::j!: .  --  .  1  . ..1 . .  :  : : j: •  • :::: t  •  ::: ) ; :  : :  !  j  -rirHf '  J  :::: ' : . :  /  _ —— ,t  •i  /.L  t  dt f-  i  - H-  T ; —  : : ..|;  mOI, :;::;• •j  .: . 1 :, 1  ; ' ~"  ....  •• ' ' ' ' '  :  ITTT '-  • i- ;  ;  ::: •  :  ,  . : : i.  0  t •  :  • •;:: • I • • ; i • I:'' •  i  . ....;. f .  1  y  -  ~—Tri-  < -1  •a  I;::  ^  UT  n  ::•.[ . . :  •  ~ * •" r / ; ; . i ; : ; ; . : : j / •.  •  -  ;:::  ': HI''-::  ; J.J  *  .... ........  '-' i\  TT^TT  y  1  . i :i.  ^  J  i: •: '' J . : — r  . .TT....  • -• •  !'.':. :  -—  /  i  —rlT- — -a  (:.-: :  i  •jv :i;  [ /  \  i •  ;  :•.:!'•  .:• i  •-—:--  ;.: . /  —  i . i -  t  \  i  ••• ! 1  : ;  J...: ._;. •  — 4 -  -:—*::-.-,  •  •• • i  •• • i- •  • • i; ; ; ; -i  M  i i  . i  _•_.. ..  : ,; :  —nrrrr  >  'A. A-:.:-  ^ *-t •  ::-.\ . . .  ' • '  . !:.; *  • • • i  .t . . .  .r-  ;  .{.--.i~::-  ,';;[:;  i : : : i  :ii  '• i  i  |  .::  f  -  1  —i-rfH-  :  •• •ii.ill :..i±.LL .. i  i  l • ' :  i .•1  ' I  •  p  '•::!  : _ : _ ' .  ...  ;  AJ  :  co/7£"o':c tor =  por  r  •. ! •t •  _:..  i  •' i , . . • . . i...  q.n't • • • • j -.  u  pi  '  t ••  'f.  1 •' t  •  •  ;:.:  :::.(  !  i .  :::  | !  -.:  . .j i.. .,.  J::;  ;  1' "-;<','' i 1 • • • 1 "  -f  conf/juration  ^. . : ' ' : i ' :  .'.'.I  11  • t-i}  rii!'- ii ;;;;  : <d:: ;;;!-;;•! ; j ! i .: : ; :.  tilth  contactor  !  ;  7  iiii- i i' ii  ;r;r  :  r::. ;;;;  i  . : ' ; ::::  i ;.;;j|ii:  i.  :  11  . . _, T Tr-l • ' - i : . : J  if--  \ . . I, •  ;;;;;;;  :  ; ••::  :; i  bystateS'S  ::;:  iiiiiii|  ;f  49 l o o p zero-pole c o n f i g u r a t i o n of f i g u r e 27 can be l o c a t e d approxi m a t e l y by i n s p e c t i o n .  Two  of these c l o s e d - l o o p r o o t s l i e on  n e g a t i v e r e a l a x i s ; one of these l i e s between the zero and  the  the  o r i g i n while the o t h e r l i e s somewhere to the l e f t o f the p o l e at -1.0.  The other two  roots, form a complex conjugate  pair.  R o o t - l o c i f o r constant values of the angle, a, can constructed.  These constant p h a s e - s h i f t r o o t - l o c i w i l l f o r  be u£ o  have two r o o t s on the n e g a t i v e r e a l axis:, as a l r e a d y d e s c r i b e d ,  one  between the zero and the o r i g i n while the o t h e r w i l l be to the  left  of  the p o l e a t -1.0.  The  complex conjugate p a i r w i l l break from  the r e a l a x i s a t the p o i n t where the complex conjugate the r o o t - l o c u s of f ( p ) =  1 p(p+  break the r e a l a x i s . D  roots f o r T h i s i s so  2  because f o r u = o, the zero and p o l e i n t r o d u c e d by the d e s c r i b i n g f u n c t i o n o f the c o n t a c t o r are superimposed a t the o r i g i n and out the e f f e c t of one another.  Therefore, the complex  cancel  conjugate  r o o t s of a l l the constant p h a s e - s h i f t r o o t - l o c i w i l l converge at the same p o i n t on the r e a l  axis.  The troublesome r o o t s which determine t h e n a t u r a l mode of  o s c i l l a t i o n of the system are those which form the complex  conjugate p a i r . f i g u r e 2.7,  Only  these r o o t s are t r a c e d on the r o o t - l o c i of  F u r t h e r , f o r convenience,  o n l y the h a l f of the complex  conjugate p a i r r o o t - l o c i f o r u p o s i t i v e i s t r a c e d . for  The other h a l f ,  u n e g a t i v e , can be l o c a t e d by i n s p e c t i o n because o f the  symmetry of these roots with r e s p e c t to the r e a l  axis.  R e f e r r i n g to f i g u r e 28 the r o o t - l o c i have been o b t a i n e d for  s e v e r a l constant values o f the p h a s e - s h i f t a n g l e .  gains which equal  BIGDJ  cos a  are p l o t t e d on the  The  effective  constant  to follow page  p h a s e - s h i f t angle r o o t - l o c i and l i n e s of constant g a i n a r e constructed. KIGQ^I  of  a  COS c  BIG/DTJ  S i n c e i n t h i s case K  o  lines.  s  =  B these l i n e s a r e constant  In the g e n e r a l case where  K ^  the v a l u e s  B ,  would have to he m u l t i p l i e d by the a p p r o p r i a t e  a  f a c t o r r e l a t i n g K and B to o b t a i n the constant  KlG^I  cos a  lines.  R o o t - l o c i have been drawn f o r constant p h a s e - s h i f t a n g l e s of the c o n t a c t o r d e s c r i b i n g f u n c t i o n . been super-posed  on these r o o t - l o c i .  Constant  g a i n l i n e s have a l s o  The next step i s to o b t a i n the  a c t u a l r o o t - l o c u s f o r v a r i o u s v a l u e s of the c o n t r o l s i g n a l The procedure Ca/K =» 1.0. ~~2 KlGj^lcos  i s as f o l l o w s : F o r an amplitude  a  =  .44.  amplitude.  R e f e r r i n g to f i g u r e 26, assume a r a t i o r a t i o Cm/Ca/2: r 1.2,  a = 25°  and  From a knowledge of the p h a s e - s h i f t angle  and  the g a i n the r o o t f o r t h i s c o n d i t i o n can be obtained on f i g u r e This procedure  i s repeated f o r i n c r e a s i n g v a l u e s of the  s i g n a l amplitude  28.  control  and hence a r o o t - l o c u s f o r a l l amplitudes  of the  c o n t r o l s i g n a l f o r a constant r a t i o C^yK t 1.0 i s o b t a i n e d . This: 2 procedure i s then repeated f o r o t h e r r a t i o s of Ca/K. A family of ~2 r o o t - l o c i f o r s e v e r a l values of t h e constant r a t i o Ca/K has been plotted i n figure  28.  The r o o t - l o c i p l o t t e d i n f i g u r e 2,7 show that c o n t a c t o r h y s t e r e s i s has an adverse e f f e c t on the system. due  T h i s adverse  to h y s t e r e s i s can of course be demonatrated a l s o by  auper-posed frequency-amplitude  l o c i and t h e Nyquist  effect  the  criterion.  However the advantage o f the r o o t - l o c u s method i s t h a t the v a l u e s of the f r e q u e n c i e s and damping f a c t o r s can be o b t a i n e d f o r a l l v a l u e s of c o n t r o l s i g n a l  amplitudes.  directly  51 10.  The experiment.  a)  The c i r c u i t and i t s o p e r a t i o n . The c i r c u i t diagram o f the model t e s t e d appears i n f i g u r e  29 and a convenient schematic diagram o f the model i n f i g u r e  30.  With r e f e r e n c e to these two f i g u r e s , the v o l t a g e drop, V , a c r o s s the r e s i s t a n c e R]_ o f the p o t e n t i o m e t e r , which may 0  may  or  not i n c l u d e the compensating network, i s energized by the  v o l t a g e drop a c r o s s the motor-driven r h e o s t a t . age drop, V , R  across R]_, i s produced by the p o t e n t i o m e t e r a c r o s s The v o l t a g e s VQ and  the b a l a n c i n g v o l t a g e s o u r c e . in polarity.  A second v o l t -  are o p p o s i t e  The d i f f e r e n c e of these two v o l t a g e s , VR -  a p p l i e d a c r o s s the c r i t i c a l l y  damped c i r c u i t  Vo, i s  galvanometer.  I f the l i g h t r e f l e c t e d by the m i r r o r of the c i r c u i t galvanometer and i n turn r e f l e c t e d by the l i g h t - s p l i t t i n g  mirrors  f a l l s on the phototubes, the t h y r a t r o n s a r e d r i v e n beyond g r i d c u t - o f f and are non-conducting.  Except f o r a narrow s t r i p on the  f a c e , each phototube i s c o m p l e t e l y masked by tape.  The  light  spot s h i n i n g on the phototubes i s r e c t a n g u l a r i n shape.  The  edges of t h i s spot p a r a l l e l to the a x i s of galvanometer r o t a t i o n , are p a r a l l e l to the l i g h t - a c c e p t i n g s l i t s of the phototubes. For a s u f f i c i e n t l y l a r g e d e f l e c t i o n o f the c i r c u i t galvanometer the l i g h t w i l l move o f f one o f t h e phototubes c a u s i n g the t h y r a t r o n cont r o l l e d by t h i s phototube t o conduct.  For a s u f f i c i e n t l y  large  d e f l e c t i o n of the galvanometer i n the o p p o s i t e d i r e c t i o n the l i g h t spot w i l l move o f f t h e o t h e r phototube c a u s i n g i t s t h y r a t r o n to conduct. F o r a c e r t a i n range of galvanometer d e f l e c t i o n the l i g h t  remains  tio follow page 51 —-H  -HOv  Thyratron  Photo multiplier Tube 93/-A  J  ,/2meqD. each  y  \  frjure  )'  1% Crtvtt dKtgrem  ^~  '  1  Cr-ittcqlly damped* Recordma &al^onomet<tr  of  the model  X, y  Motor  Cref/v.  Q-dihanootethr' Defection  ®C, Contacto  r  /Mean %  Armcttbre, ',urr&n"C •gOP"tTflg|  7  y x^-S a.  IOK+  39.7 X  O  f  r-AAAAAA-, LyvWSAA-j  Compen s <*ti na  5  4-1.2 j^-f NO  \  (xalvanOmetsjr\  Deflection,  F/gure 30.  Schew'ot/c  Ptaqram  $  t2.  of  the  Model  5  ±  on both phototubes and both t h y r a t r o n s  a r e non-conducting.  This  range determines t h e i n a c t i v e zone of the c o n t a c t o r . The d i r e c t i o n of motor r o t a t i o n depends on which t h j i r atron i s conducting.  Both the t h y r a t r o n p l a t e s u p p l y v o l t a g e and  the v o l t a g e a p p l i e d a c r o s s t h e f i e l d of t h e motor are a l t e r n a t i n g . Consider  t h a t one of t h e t h y r a t r o n s i s c o n d u c t i n g .  conducts over a h a l f - c y c l e of the p l a t e supply  This  thyratron  a l t e r n a t i n g voltage  I f t h e other t h y r a t r o n conducts, i t w i l l do so over t h e other h a l f cycle.  However the c u r r e n t from e i t h e r t h y r a t r o n which i s the  c u r r e n t through the armature of the motor i s u n i d i r e c t i o n a l r e g a r d l e s s over which h a l f - c y c l e t h y r a t r o n conduction  takes p l a c e .  But  over each h a l f -  the d i r e c t i o n of the f i e l d c u r r e n t i s reversed  cycle.  Therefore  t h e d i r e c t i o n of motor r o t a t i o n depends on which  t h y r a t r o n i s conducting. In summary,  c o n t r o l takes p l a c e I n the f o l l o w i n g manner.  F o r a s u f f i c i e n t l y l a r g e d i f f e r e n c e o f t h e v o l t a g e s , VR - VQ, the circuit  galvanometer w i l l be d e f l e c t e d , so t h a t l i g h t w i l l move  o f f one of t h e phototubes. tube w i l l conduct. rheostat  The t h y r a t r o n c o n t r o l l e d by t h i s photo  The motor w i l l r u n and t u r n the s h a f t of t h e  i n such a d i r e c t i o n as to c o r r e c t f o r the unbalance.  the unbalance i s i n the r e v e r s e p o l a r i t y then the other w i l l conduct.  If  thyratron  The motor w i l l r u n i n the opposite d i r e c t i o n and  turn the s h a f t o f the r h e o s t a t  i n the o p p o s i t e  d i r e c t i o n i n order  to c o r r e c t f o r the unbalance. From the c i r c u i t  diagram i t i s seen that there are two  galvanometers i n the system, one r e f e r r e d to as the c i r c u i t vanometer while  the other  i s r e f e r r e d to as t h e r e c o r d i n g  galgalvan-  53 ometer.  A l l the c a l c u l a t i o n s a r e based on the c o n t r o l s i g n a l which  i s the input s i g n a l t o the c o n t a c t o r and which f o r the model i s the d e f l e c t i o n of the c i r c u i t  galvanometer, ©Q]_.  p o s s i b l e to "break i n t o " the c i r c u i t of the c i r c u i t  S i n c e i t was  im-  to observe the d e f l e c t i o n  galvanometer, the r e c o r d i n g galvanometer was i n t r o d -  uced i n such a manner t h a t the d e f l e c t i o n of the r e c o r d i n g  galvan-  ometer, ©C2> would be i d e n t i c a l t o the d e f l e c t i o n o f the c i r c u i t galvanomet er. b)  C a l i b r a t i o n of the model. B a l l i s t i c galvanometers of the D'Arsonval type were used  for  both the c i r c u i t  and r e c o r d i n g galvanometers.  N e i t h e r galvan-  ometer possessed electromagnetic damping on open c i r c u i t .  The  p e r i o d s of the galvanometers on open c i r c u i t were 6.46  6.45  seconds f o r the c i r c u i t  and  and r e c o r d i n g galvanometers r e s p e c t i v e l y .  For convenience i t was assumed that both galvanometers had an open c i r c u i t p e r i o d of 6.46  seconds.  Sufficient resistance  was  added to each galvanometer so t h a t both galvanometers were i c a l l y damped.  crit-  For c r i t i c a l damping, the time constant o f each  galvanometer i s t h e r e f o r e 6.46/2T = 1.029  seconds.  The s e n s i t -  i v i t i e s were found to be 370 r a d i a n s / v o l t and 477 r a d i a n s / v o l t for  the c i r c u i t and r e c o r d i n g galvanometers r e s p e c t i v e l y .  The  t r a n s f e r f u n c t i o n s of t h e galvanometers n e g l e c t i n g a i r damping and inductance i n the galvanometers f o r the c o n d i t i o n of c r i t i c a l damping galvanometer are t h e r e f o r e : for  the c i r c u i t  galvanometer: G i ff S  (s)  =  5  7  radians/volt  0  -(T 025s + 1)  (63a)  f  and f o r the r e c o r d i n g G  g 2  galvanomter (s)  -  477 (1,029s + l )  radians/volt 2  (63b)  54 The i n a c t i v e zone was A voltage just  determined i n the f o l l o w i n g manner.  s u f f i c i e n t to f i r e one of the t h y r a t r o n s was  applied  across the c r i t i c a l l y damped c i r c u i t galvanometer, then a v o l t a g e just s u f f i c i e n t the two  to f i r e t h e other  thyratron.  The d i f f e r e n c e o f  v o l t a g e s , which g i v e s a measure o f the i n a c t i v e zone,  obtained.  T h i s procedure was  value o f t h i s v o l t a g e was  repeated  found to be  s e v e r a l times and 5.9  micro-volts.  the average Since  output o f the system i s considered to be the d e f l e c t i o n o f galvanometer, the i n a c t i v e zone, C , a  was  the  the  i n terms of galvanometer de-  flection is Cd - 5.9 The to  x IO"  6  x 370  = 2.18  x 10~  radians.  3  t r a n s f e r ' f u n c t i o n from the motor armature c u r r e n t  the v o l t a g e drop a c r o s s the r h e o s t a t d r i v e n by the motor was  t a i n e d i n the f o l l o w i n g manner. conducted and  I t was  assumed, viien a t h y r a t r o n  t h e r e f o r e the armature of the motor c a r r i e d c u r r e n t ,  that the motor immediately a t t a i n e d a c e r t a i n speed and t h i s speed constant was  ob-  as l o n g as i t was  observed to be approximately  running.  t r u e although  T h i s assumption some time would  have to elapse before the motor came up t o speed. very s m a l l compared w i t n  the time constant  maintained  This p e r i o d i s  o f the galvanometer  and hence as an approximation t h i s p e r i o d can be n e g l e c t e d . f o r e with t h i s assumption the change i n the v o l t a g e drop  There-  across  the r h e o s t a t caused by the motor r o t a t i o n would be p r o p o r t i o n a l to  t i me. The  v o l t a g e d i f f e r e n c e between two  r h e o s t a t arm p o s i t i o n was for  o b t a i n e d together  the r h e o s t a t arm t o t r a v e l  s e t t i n g s of  the  w i t h the time r e q u i r e d  the d i s t a n c e between these s e t t i n g s .  From this, i n f o r m a t i o n and the assumption of constant  motor speed,  55 the t r a n s f e r f u n c t i o n of t h i s p o r t i o n of the system can be d e f i v a d . For example, when the v o l t a g e output o f the v a r i a b l e v o l t a g e Vg, was two  supply,  equal to 445 v o l t s , the d i f f e r e n c e i n v o l t a g e drop between  s e t t i n g s o f the r h e o s t a t arm  p o s i t i o n e q u a l l e d 1.57  volts/  For  the case where the r h e o s t a t arm r e q u i r e d 8 seconds to t r a v e r s e the d i s t a n c e between these two  s e t t i n g s , the change i n the  voltage  drop a c r o s s the r h e o s t a t appears as shown i n f i g u r e 31.  /57  "  tirrta.  In S  tconds  F i g u r e 31 - Change i n r h e o s t a t v o l t a g e drop a g a i n s t time. Assuming, f o r convenience, the i n p u t t o be a dimensionless step f u n c t i o n , the t r a n s f e r f u n c t i o n , G^U),  unit  of t h i s p o r t i o n o f  the system i s •1 G ( s ) "s  =  m  Gm(s) where  R  and  g (s)  The value  m  m  =  =  1 R g s m  Rmg ( ) s  m  .1965 =  * 1.57  ( 64)  8^ s  -1965/s v o l t s  (65)  volts/second  1/s  of the g a i n , R , m  of the supply v o l t a g e , V , s  The  (s)  m  was  changed by changing e i t h e r the v a l u e  or the speed of the motor, or both.  system i s d e p i c t e d by a block diagram i n f i g u r e 38.  For the c o n d i t i o n that the d e f l e c t i o n of the c i r c u i t galvanometer and  the r e c o r d i n g galvanometer be the same, 370  477 R . p2  R i i s equal p  to 1^/23,100 and R  f o r the uncompensated case. equal  p2  equal  i s equal to R /22,400 2  Rpj_ i s equal t o R^/51,200 and R  to R /51,720 f o r the compensated case. 2  Rp]_ must  R]_ and R  2  are  p2;  is  expressed  55  t o f o l l o w page  cr V  •4. partest/jjiucr}  •MS  T  y  \ ^  56 i n ohms.  The r e l a t i v e v a l u e s o f R]_ and R  g  were always such t h a t  the c o n d i t i o n f o r equal d e f l e c t i o n o f the galvanometers was fied.  Both of the v a l u e s of t h e s e r e s i s t a n c e s were always  satissmall  compared w i t h r e s i s t a n c e of the galvanometer c i r c u i t s to ensure n e g l i g i b l e e f f e c t on the p o s i t i o n s of t h e p o l e s o f the galvanometer transfer  functions* from f i g u r e 32, the o v e r - a l l open-loop t r a n s f e r f u n c t i o n s ,  G ( s ) , f o r the uncompensated G(s) =  R s(1.029s + l )  G(s) = Where R =  and compensated  (66a) 2  R (1.635s •» 1) s(1.029s + 1J« (,367s +  370 R  m  cases r e s p e c t i v e l y a r e  (66b) 1)  R . pl  The o v e r - a l l open-loop t r a n s f e r f u n c t i o n s i n d i m e n s i o n l e s s time form f o r the time base t ^ = 1.029 and compensated G  and  seconds f o r the  uncompensated  cases r e s p e c t i v e l y are  (P)  G(p)  K p(p +  =  =  (67a) D  2  K(1.558p p(p + l )  where  p  = 1.029s  and  K  = R(1.02$)  2  (  6 7 b  )  (.357p + 1)  o u t p u t - u n i t s , which f o r t h i s case equals r a d i a n s ;  The above o v e r - a l l open-loop t r a n s f e r f u n c t i o n s are i d e n t i c a l with those of the two systems a n a l y z e d i n c h a p t e r 8. c)  Tests and  Results.  T r a n s i e n t responses f o r both the uncompensated  and com-  pensated cases were o b t a i n e d f o r equal step i n p u t d i s t u r b a n c e s . The d i s t u r b a n c e i n t r o d u c e d appears as a step input of the c i r c u i t  57 galvanometer  displacement and was o b t a i n e d I n the f o l l o w i n g manner.  The c o n t a c t o r was i n a c t i v e zone.  s e t so t h a t i t r e s t e d j u s t on t h e edge o f the  The b a l a n c i n g v o l t a g e , V^, was then changed i n  a l l cases by an amount equal t o 13.5 m i c r o - v o l t s c o r r e s p o n d i n g to a galvanometer  d e f l e c t i o n of 5 x 1 0 ~  switch o f the motor was l e f t take p l a c e .  3  radians.  The  field  open so that c o r r e c t i o n could not  A f t e r the c i r c u i t  galvanometer had reached i t s  s t a b l e p o s i t i o n the s w i t c h i n t h e f i e l d c i r c u i t o f the motor c l o s e d t o a l l o w the system to c o r r e c t f o r the d e f l e c t i o n s . appears to t h e system as a step i n p u t of 5 x ICP^ control signal. cording  was This  r a d i a n s to the  The t r a n s i e n t response was observed on t h e r e -  galvanometer. Graphs o f t r a n s i e n t responses f o r the model uncompensated  and compensated are drawn i n f i g u r e 33.  Galvanometer  i s p l o t t e d a g a i n s t d i m e n s i o n l e s s time 0,  where 0 = t/1.029.  From the graphs i t may accuracy f o r comparable compensating network.  deflection  be observed t h a t the dynamic  v a l u e s o f C^/2K  i s improved  by the  The e f f e c t o f the compensating  network on  the  frequency o f the t r a n s i e n t response i s e v i d e n t from t h e graphs.  For  a r a t i o C /2K = .588 f o r the Compensated system the predomd  inant frequency i s n e a r l y twice as g r e a t as the  predominant  frequency f o r the uncompensated system w i t h a r a t i o G^/2K  =  .583.  From an i n s p e c t i o n o f the r o o t - l o c i f o r the two cases, t h i s i n c r e a s e i n frequency f o r the compensated case i s tobe expected.  •>  Values of amplitudes and f r e q u e n c i e s o f s e l f - s u s t a i n e d o s c i l l a t i o n s were a l s o o b t a i n e d .  Table 2 i s a t a b u l a t i o n o f the  amplitudes and f r e q u e n c i e s of these o s c i l l a t i o n s f o r s e v e r a l  to f o l l o w page  57  11: 1: -: I' :i i 1.11. i1ii i Al1 LA ,..; i L i l l i 11 li i t rill iii11 1  fijur&  Jj.  TransliinC  r<zspc}t,<L<> ta a st<ip  d/sturkiant  58  r a t i o s of C^/aK  f o r the uncompensated model.  The  predicted  amplitudes and f r e q u e n c i e s f o r the assumed o v e r - a l l t r a n s f e r f u n c t i o n are a l s o t a b u l a t e d . from the graph i n f i g u r e  open-loop  These v a l u e s were obtained  19.  T a b l e 3 r e p r e s e n t s s i m i l a r r e s u l t s f o r the model w i t h the compensating  network.  The p r e d i c t e d amplitude and f r e q u e n c i e s f o r  the assumed o v e r - a l l open-loop t r a n s f e r f u n c t i o n were o b t a i n e d 23.  from the graph i n f i g u r e  R e f e r r i n g to the r e s u l t s i n t a b l e 2 f o r the uncompensated case.  The p r e d i c t e d frequency i s g r e a t e r than the  observed  f r e q u e n c i e s w h i l e the p r e d i c t e d amplitudes are l e s s than the observed, f o r a l l e a s e s .  The h i g h e r p r e d i c t e d frequency i s p r o b a b l y  due to the assumed t r a n s f e r f u n c t i o n of the mot o r - r h e o s t a t which approximated  as R / s . m  A more r e a l i s t i c  motor-rheostat would be R /s  (T s +  m  very l a r g e ;  The e f f e c t of T  p o l e on the complex p l a n e .  m  m  was  t r a n s f e r f u n c t i o n f o r the  1) w i t h T ,  o f course, b e i n g  m  i s to i n t r o d u c e another  open-loop  The r o o t - l o c u s of the c l o s e d - l o o p  system would t h e r e f o r e cut the imaginary a x i s at a v a l u e of u<1.0.  T h e r e f o r e the frequency o f the s e l f - s u s t a i n e d  i o n s , which i s determined by the p o i n t where the r o o t s c r o s s t h e imaginary a x i s , would be l e s s  oscillat-  complex-conjugate  than u  =  1.0.  For  such a case the amplitudes of the s e l f - s u s t a i n e d o s c i l l a t i o n s would be g r e a t e r f o r e q u a l r a t i o s of Ca/2K than f o r the case where the c r o s s i n g o f the r o o t - l o c u s o c c u r r e d a t u = There  i s y e t g r e a t e r d i s c r e p a n c y i n the r e s u l t s f o r the  compensated system as shown i n t a b l e 3. o v e r - a l l open-loop that  has  1.0.  The system f o r the assumed  transfer function i s not r e a l l y  been compensated.  the system  Yet the p r e d i c t e d v a l u e s of  to follow page 58  Table 2.  Amplitudes and frequencies of self-sustained o s c i l l a t i o n s without compensation. Predicted  C /2K d  .354 .266 .213 .177 .152 .118 .106 .080 .073  Table 3.  Cm/Cd/2  A  0 2.2 2.8 3.4 3.9 5.01 5.8 7.9 9.0  Observed u  C^d/  1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0  1.83 2.44 3.87 4.58 5.57 7.15 8.06 10.4 12.5  d  .1397 .12 .105 .093 .076 .0698 .0599  &  -  .842 .85 .846 .854 .858 .847 .852 .849 .849  Amplitudes and frequencies of self-sustained o s c i l l a t i o n s with compensation. Predicted  C /2K  u  2  Observed  Cm/c /2  u  Cr/V  2.2 2.7 3.2 3.5 4.4 4.8 5.7  2.0 2.0 2.0 2.0 2.0 2.0 2.0  6.6 8.04 9.9 10.48 12.6 13.6 16.2  d  2  u 1.27 1.28 1.28 1.27 1.29 1.25 1.25  This corresponds to an absolutely stable - state.  59  amplitude and frequency are based on adding a compensating to the assumed o v e r - a l l open-loop  transfer function.  network  Therefore i t  i s reasonable to expect that the r e s u l t s o b t a i n e d f o r t h e compensated case would not check as c l o s e l y as those f o r the uncompensated case. However, important c o n c l u s i o n s can be drawn f r o m these results.  The f r e q u e n c i e s of the s e l f - s u s t a i n e d  oscillations  remain approximately constant r e g a r d l e s s o f the amplitudes.  The  amplitudes o f t h e s e l f - s u s t a i n e d o s c i l l a t i o n s do i n c r e a s e with a decrease i n the r a t i o G /2K.  The a d d i t i o n of the  network i n c r e a s e s .the frequency  of s e l f - s u s t a i n e d  d  For the assumed o v e r - a l l open-loop p r e d i c t e d i s from u = 1.0 the compensated.  compensating  oscillations.  t r a n s f e r f u n c t i o n s the i n c r e a s e  f o r t h e uncompensated case to u s 2 f o r  The observed i n c r e a s e o f frequency i n the model  i s from u & .85 f o r the uncompensated case to u « 1.27 compensated case.  f o r the  11.  Summary and c o n c l u s i o n s . The q u a s i - l i n e a r r e p r e s e n t a t i o n o f the c o n t a c t o r des-  c r i b i n g , f u n c t i o n enables e s s e n t i a l l y l i n e a r methods to be used i n the a n a l y s i s and s y n t h e s i s of c o n t a c t o r  servomechanisms.  Kochenburger * s method o f a n a l y s i s and s y n t h e s i s based on the frequency-response has been explained and a p p l i e d . The r o o t - l o c u s method o f s y n t h e s i s , which has been app l i e d t o l i n e a r servomechanisms, t a c t o r servomechanism.  has been developed f o r t h e con-  This method appears t o be v a l u a b l e when  c o n s i d e r i n g the problem of r e l a t i v e  stability.  For a simple c o n t a c t o r w i t h no h y s t e r e s i s e f f e c t ,  Koch-  enburger' s vector form o f the c o n t a c t o r d e s c r i b i n g f u n c t i o n was used d i r e c t l y t o o b t a i n the r o o t - l o c u s .  The c o n t a c t o r  appeared  as a v a r i a b l e g a i n element f o r t h e v a r i o u s c o n t r o l s i g n a l amplitudes.  The c o n t a c t o r had no e f f e c t on the open-loop r o o t s but the  v a r i a t i o n s i n t h e c o n t a c t o r g a i n caused the r o o t s o f t h e c l o s e d loop to t r a v e l along t h e r o o t - l o c u s obtained from t h e open-loop r o o t s o f the system. R o o t - l o c i were a l s o o b t a i n e d when the c o n t a c t o r possessed h y s t e r e s i s . L a p l a c e transform  Kochenburger*s v e c t o r form was modified  to the  form of the c o n t a c t o r d e s c r i b i n g f u n c t i o n .  T h i s form of, t h e d e s c r i b i n g f u n c t i o n showed t h a t not o n l y were the p o s i t i o n s of t h e roots v a r y i n g f o r the c l o s e d - l o o p but a l s o f o r the open-loop. The model c o n s t r u c t e d to check some o f the theory was described.  The o v e r - a l l open-loop t r a n s f e r f u n c t i o n s assumed f o r  the mathematical a n a l y s i s were o n l y approximations o f the a c t u a l . However, even f o r the assumption made, t h e experimental work v e r i f i e d q u a l i t a t i v e l y and t o some degree q u a n t i t a t i v e l y the p r e d i c t i o n o f the model performance. The methods o f a n a l y s i s and s y n t h e s i s e x p l a i n e d and developed, although b e i n g approximate, appear to be s a t i s f a c t o r y as e n g i n e e r i n g approximations.  62, 12.  References.  1.  Ivanoff, A., " T h e o r e t i c a l Foundations of the Automatic Hegulation of Temperature." J o u r n a l of the I n s t i t u t e of Fuel,(February, 1934) pp. 117-130.  2.  Hazen, H. L., "Theory of Servomechanisms." J o u r n a l of the F r a n k l i n I n s t i t u t e , v o l . 218, no.3 (September, 1934), pp. 279-330.  3.  Weiss, H. K., " A n a l y s i s of Relay Servomechanisms." J o u r n a l of the Aeronautical Sciences, vol.13 (July 1946 ) pp. 364-373.  4.  MacColl, L. A., Fundamental Theory of Servomechanisms, New York, N.Y., D. Van Nostrand Co., 1945  5.  Flugge-Lotz, I . , Discontinuous Automatic C o n t r o l , P r i n c e t o n , N.J., Princeton U n i v e r s i t y Press, 1953.  6.  Kahn, D. A., "An A n a l y s i s of Relay Servomechanisms? Transactions of the American I n s t i t u t e of E l e c t r i c a l Engineers, Vol.68, part I I , 1949 pp. 1079-1088.  7.  Kochenburger, R.J., "A Frequency Response Method f o r Analyzing and Synthesizing Contactor Servomechanisms." Transactions of the American I n s t i t u t e of E l e c t r i c a l Engineers, Vol.69, p a r t I , 1950. pp. 270-284  8.  Evans, W. R., "Control System Synthesis by Root Locus Method." Transactions of the American I n s t i t u t e of E l e c t r i c a l Engineers. V o l . 69, p a r t i , 1950. pp.66-69,  13.  Acknowledgements. The author wishes to express h i s indebtedness to h i s  t h e s i s d i r e c t o r , Mr. W. A. Wolfe. his of  He would l i k e to thank him f o r  guidance, p a t i e n c e and encouragement shown d u r i n g the course this research. The author would a l s o l i k e to thank those members o f  the  e l e c t r i c a l e n g i n e e r i n g department  who from time t o time  a s s i s t e d him se g r a c i o u s l y . Acknowledgement i s a l s o made to the B r i t i s h Electric  Columbia  Company L i m i t e d who a s s i s t e d t h e author by means of a  s c h o l a r s h i p and r e s e a r c h g r a n t . F i n a l l y , he would l i k e to thank h i s s i s t e r , and Mr. N. C. F r e l o n e who typed out the t h e s i s r e p o r t .  A. P. P a r i s .  Marcelle,  

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