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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 in the Department of ELECTRICAL ENGINEERING We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF APPLIED SCIENCE Member of the Department of Mechanical Engineering Member of the Department of Elec t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA August, 1954 Abstract This i n v e s t i g a t i o n i s concerned with the analysis and synthesis of contactor servomechanisms. The techniques employed are based on Kochenburger*s quasi-linear representation of the contactor describing function for sinusoidal input signals to the contactor. The frequency-response method of analysis and synthesis, which has been found p r a c t i c a l for treating l i n e a r servomechanisms has been applied by Kochenburger to the contactor servomechanism and i s explained here. By t h i s method i t i s possible to deter-mine whether the system possesses absolute s t a b i l i t y . The root-locus method o f synthesis which has been applied to l i n e a r servomechanisms i s applied to the contactor servomech-anism. The root-locus describes the roots of the closed~loop system f o r a l l values of the control signal amplitude. The root-locus method i s valuable when considering the problem of r e l a t i v e s t a b i l i t y . For a simple contactor with no hysteresis e f f e c t , Kochenburger*s vector form of the contactor describing function can be used d i r e c t l y to obtain the root-locus. The contactor appears as a variable gain element f o r the various control signal amplitudes. The contactor has no e f f e c t on the open-loop roots but the variations i n the contactor gain cause the roots of the closed-loop to travel along the root-locus obtained from the open-loop roots of the system. The root-locus can also be obtained when the contactor possesses hysteresis. Kochenburger»s vector form i s modified to the Laplace transform form of the contactor describing function. This form of the describing function shows that not only are the positions of the roots varying for the closed-loop but also f o r the open-loop. A model was constructed to check some of the theory. The assumed o v e r - a l l open-loop transfer functions approximated the a c t u a l . Even for the assumptions made, the experimental work has v e r i f i e d q u a l i t a t i v e l y and to some degree qu 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 Table of Contents i page 1. Introduction 1 2. General comments on contactor servomecnanisms 8 3. Mathematical representation of l i n e a r components 12 4. The frequency-response of contactor servomechanisms 18 5. The quasi-linear representation of the contactor 23 describing function. 6. S t a b i l i t y c r i t e r i a f o r contactor servomechanisms 26 7. Hoot-locus method of synthesis 32 8. Analysis of the model by the frequency-response and root-locus methods a) Kahn's method for obtaining transient responses 37 b) Analysis of the model 38 c) Application of a simple phase-lead type 44 compensating network 9. Root-locus method when contactor has hysteresis 46 10 2 The experiment a) The c i r c u i t and i t s operation 51 b) C a l i b r a t i o n of the model 53 c) Tests and re s u l t s 56 11. Summary and conclusions 60 12. References 62 1 3 . Acknowledgements 63 i i L i s t of I l l u s t r a t i o n s Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. Table 1. Figure 6. Figure 7. Figure 8. Figure 9. Figure 10. Figure 11. Figure 12. Figure 13. Figure 14. Figure 15. Figure 16. Figure 17. Figure 18. Figure 19. Figure 20. Block diagram of single-loop servomechanism Controller Characteristics Block diagramof a contactor servomechanism Typical contactor c h a r a c t e r i s t i c s Representative phase-lead network Symbols and units adopted Relation of control signal and correction signal f o r a contactor with hysteresis page 2 2 8 9 13 following 17 19 Graph of amplitude of harmonic components following 20 of correction signal against amplitude of control signal Plot of the fundamental harmonic contactor following 24 describing function, G r - ^ Block diagram of l i n e a r single-loop servomechanism Frequency polar-locus p l o t Super-posed frequency and amplitude l o c i Examples of super-posed l o c i Roots i n the p-plane Root-locus of single-loop servomechanism Single-loop contactor servomechanism Graphs for Kahn's semi-grapnical method Block diagram representation of the model 26 27 28 30 33 34 37 following 37 39 Super-posed frequency amplitude l o c i for the representation of the uncompensated model 39 following 40 Graph of the control signal amplitudes which give points of equilibrium for the representation of. the uncompensated model Root-locus f o r the representation of the following 41 uncompensated model i i i L i s t o f i l l u s t r a t i o n s - c o n t ' d . F i g u r e 2 1 . F i g u r e 2 2 . F i g u r e 2 3 . F i g u r e 2 4 . F i g u r e 2 5 . F i g u r e 2.6. F i g u r e 2.7. F i g u r e 2 8 . F i g u r e 2 9 . F i g u r e 3 0 . F i g u r e 3 1 . F i g u r e 3 2 . F i g u r e 3 3 . T a b l e 2 . T a b l e 3 . P l o t s o f t r a n s i e n t r e s p o n s e s t o r e f e r -e n c e s t e p i n p u t s . F r e q u e n c y l o c u s f o r . t h e r e p r e s e n t a t i o n o f t h e c o m p e n s a t e d m o d e l G r a p h o f 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 s w h i c h g i v e p o i n t s o f e q u i l i b r i u m f o r t h e r e p r e s e n t a t i o n o f t h e c o m p e n s a t e d m o d e l R o o t - l o c u s f o r t h e r e p r e s e n t a t i o n o f t h e c o m p e n s a t e d m o d e l P l o t s c o m p a r i n g t r a n s i e n t r e s p o n s e s f o r t h e . u n c o m p e n s a t e d a n d c o m p e n s a t e d m o d e l P l o t s n e c e s s a r y t o o b t a i n t h e f u n d a m e n t a l h a r m o n i c 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 , gjYj_ O p e n - l o o p z e r o - p o l e c o n f i g u r a t i o n w i t h c o n t a c t o r h y s t e r e s i s R o o t - l o c i w i t h c o n t a c t o r h y s t e r e s i s C i r c u i t d i a g r a m o f t h e m o d e l S c h e m a t i c d i a g r a m o f t h e m o d e l C h a n g e i n r h e o s t a t v o l t a g e d r o p a g a i n s t t i m e B l o c k d i a g r a m o f t h e m o d e l T r a n s i e n t r e s p o n s e s t o a s t e p d i s t u r b a n c e 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 -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 p a g e f o l l o w i n g 43 f o l l o w i n g 44 f o l l o w i n g 4 5 f o l l o w i n g 4 5 f o l l o w i n g 4 5 f o l l o w i n g 48 f o l l o w i n g 48 f o l l o w i n g 49 f o l l o w i n g 51 f o l l o w i n g 51 55 f o l l o w i n g 55 f o l l o w i n g 57 f o l l o w i n g 58 A m p l i t u d e s a n d f r e q u e n c i e s o f s e l f - s u s t a i n - f o l l o w i n g 58 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 I* Introduction In the l a s t two decades considerable progress- has been made i n the science of servomechanisms and feedback control systems. Most of the l i t e r a t u r e on these automatic controls deal with continuous control and very l i t t l e deals with discon^ tinuous control. This i s very s i g n i f i c a n t when i t i s considered that Hany of the f i r s t control systems were of the discontinuous type. The development of any exact science requires that relationships be expressed mathematically. I t i s then under** standable that the continuous type of control was more amenable to theore t i c a l i n v e s t i g a t i o n because relationships could be expressed as continuous functions. The discontinuous controls on the other hand have not been conducive to mathematical investigations especially when considering synthesis techniques, hence the preponderance of theory on continuous controls over discontinuous. However, many discontinuous types of controls are used and possibly more would be used i f t h e i r performance were more c l e a r l y understood. Therefore the i n v e s t i g a t i o n of the analysis and especially the synthesis techniques of discontinuous control systems has d e f i n i t e p r a c t i c a l importance. In this paper, the type of discontinuous control system to be studied i s that referred to as on-off, r e l a y or contactor type servomechanism* Figure 1 i s a block diagram representing a simple feedback control system. + -M Error Control Signet/ Coritrol/ad ) £ ' £/a.r»<Lnts D Systam 0 Figure 1. Block diagram of single~loop servomechanism This f i g u r e represents a f a i r l y general single-loop servomechanism consisting of a reference input, control elements, controlled system and a d i r e c t feedback comparing the controlled variable with the reference input. For continuous control of the control elements the relationship of the correction signal, D, to the steady-state error E s , may be represented as i n figure 2a. ; [ (or«tttvn S/yna! 0 1 .Correction 5'<tn«l D  J JtloJ Jtiajy-state trt-or _a) Continuous SgrvoirxLcfian'S/n -$>) Contactor SzrVomzctictmsm Figure 2:. Controller 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 passing through the o r i g i n i l l u s t r a t i n g that not only i s the correction f o r the error continuous but also l i n e a r . This condition i s the one f o r which most of the theory has been developed. If the control elements of figure 1 are substituted by a contactor servo-mechanism the 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 figure ab. I t may be observed from t h i s graph that no correction takes place within a ce r t a i n zone* This zone i s known as the inactive or dead zone. When the error i s outside this zone the c o n t r o l l e r introduces a constant value of correction signalvto the controlled system. I t i s an a l l or nothing type of control and the correction signal i s a discontinuous function of the error. This type of control i s widely used because of i t s s i m p l i c i t y . I t frequently requires much le 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 cost. The c o n t r o l l e r could consist of an electromagnetic r e l a y , hence the term relay servomechanism, or some sort of contact-making device activated, for example, pneumatically, h y d r a u l i c a l l y or e l e c t r i c a l l y , hence the term contactor servomechanism. The manner of operation of the control suggests the other name pn-off servomechanism. For purposes of uniformity the term contactor servomechanism w i l l be adopted i n t h i s paper and used exclusively henceforth; The early investigators approached the analysis of con-tactor servomechanisms with two s i g n i f i c a n t points i n mind. F i r s t , mathematical relationships could be expressed f o r the several con-d i t i o n s of control that i s for p o s i t i v e , negative and zero correction. This means that although the control system i s non-linear when con-sidered over i t s entire 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 set up fo r each condition of i t s operation. The equation chosen would depend on the condition of operationi The boundary conditions for the controlled variable would have to be matched for 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 o s c i l -4 l a t i o n s . This f a c t i s seen to be reasonable by considering the manner by which the contactor servomechanism corrects f o r error. Once the error i s outside of the i n a c t i v e zone a step input of correction signal i s introduced to the controlled system and t h i s same value of the s i g n a l p e r s i s t s u n t i l the error again, comes into the i n a c t i v e zone. With s u f f i c i e n t l y high gain i n the forward transfer function of the open-loop system the controlled variable i s caused to overshoot i t s stable p o s i t i o n , that i s , tne error travels outside the inactive zone and correction takes place as before i n the opposite d i r e c t i o n * Because of the type of c o r r e c t i o n signal a tendency e x i s t s f o r the switching process of c o r r e c t i o n to p e r s i s t hence causing a generation of sustained o s c i l l a t i o n s of the controlled v a r i a b l e . The early investigators were armed with these two f a c t s . The two i n i t i a l papers written on the subject were one by Ivanoff 1 and the other by Hazen2} both, i n 1934. Both, inv e s t i g a t o r s approached the problem assuming a condition of sustained o s c i l l a t i o n s In the steady-state. Ivanoff, who was p r i m a r i l y interested i n temperature regulation assumed a symmetrical rectangular c o r r e c t i o n signal wave being introduced to the controlled elements, which, i n his case was a heat plant. He analysed the rectangular wave into i t s Fourier series and assumed a transference c h a r a c t e r i s t i c f o r his heat plant. He was able to demonstrate the r e l a t i o n s h i p of the steady-state amplitude and frequency to the i n a c t i v e zone width. By e m p l o y i n g the Fourier analysis of the rectangular wave he also showed that the predominant effect of the rectangular correction wave was due to i t s fundamental component, Hazen assumed a s i n u s o i d a l response f o r the controlled variable and then used d i f f e r e n t i a l equations d i r e c t l y . This method requires that d i f f e r e n t i a l equations be 5 solved for each, correction interval initiated by tlie contactor and that boundary conditions be matched for each switching operation* This method becomes very awkward even for relatively simple cases. Hazen, however, was able to demonstrate many of the peculiar characteristics of contactor servomeonanisms such as the effects due to the inactive-zone and back-lash with his direct differential equation approach. Methods of analysis based indirectly on the transient response have been developed. Weiss 3 and MacColl 4 applied the graphical phase-plane method, which had been used in non-linear systems, to the analysis of contactor servomecnanisms. The differ-ential equations are put in such a form that the solution i s obtained i n terms of the f i r s t derivative with respect to time of the con-trolled variable and the controlled variable. If the controlled variable be displacement then the f i r s t derivative with respect to time i s velocity. 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 velocity versus displacement i s known as the phase-plane; Tra-jectories i n the phase-plane are plotted i n the different regions and from the web of the trajectories 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 plot. Recently Fliigge-Lotz 5 has given extensive treatment of this method with specific application to guided missiles. Kahn6 has developed a 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 limitations in that they are satisfactory only for systems which can 6 be described by a second order d i f f e r e n t i a l equation. Kahn&has further developed a graphical method of p l o t t i n g the transient response for a r b i t r a r y disturbances* Prac-? t i c a l l y , the solution of only one d i f f e r e n t i a l equation i s necessary and t h i s i s the open*loop response of the contactor control signal to the step c o r r e c t i o n s i g n a l . This method w i l l be elaborated upon l a t e r as i t w i l l be used to obtain transient responses f o r c e r t a i n portions of the i n v e s t i g a t i o n to follow. The disadvantage of a l l the methods described i s that they are incapable of d i r e c t use f o r synthesis. This, of course, i s a serious disadvantage to the designer of the contactor servo-mechanisms. Kochenburger 7 has introduced a novel method of analysis which i s conducive to synthesis 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 . This approximation enables the contactor to be represented i n terms of a quasi-linear describing function. The chief advantage of t h i s approximate method, which sets i t apart from other methods, i s that i t allows l i n e a r techniques to be applied to non-linear mechanisms, so that other techniques already developed f o r l i n e a r control systems may also be used. The writer suggests i n this paper that the root-locus method developed by Evans® f o r l i n e a r oontrol system syn-thesis may be used as an added technique for the synthesis of con-tactor servomechanisms i n conjunction with the Kochenburger des-c r i b i n g function and i t s modifications. The object of this paper therefore i s to outline 7 Koehenburger's method of analysis and synthesis of contactor servomechanisms and to show how the root locus method of synthesis may be used, with the concept of the describing function of the contactor, to determine 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 constructed f o r the purpose of checking some of the theory developed. Unfortunately not a l l the theory developed could be checked experimentally because the hysteresis effect i n a contactor could not be reproduced i n the contactor of the model© The experimental re s u l t s obtained are based on the assumptions that p o s i t i v e and negative corrections are equal and a l l components of the servomechanism other than the contactor are l i n e a r . These assumptions are necessary but only approximately true. Therefore two types of error are to be expected i n the r e s u l t s ; the f i r s t source of error due to the approximate method of analysis used and the second due to the assumption that components other than the contactor are l i n e a r which was not exactly the case for the model. In the chapters to follow, Kochenburger'a frequency-response method and the root-locus method with the concept of the quasi-linear representation of the contactor describing function w i l l be explained. These two methods of analysis and synthesis, w i l l be applied,in. anal*» yzing the model f i r s t without a compensating network, then with a simple phase-lead compensating network. The mathematical analysis w i l l be followed by a description of the model, and a report on the experimental tests performed and the res u l t s obtained. 2» General Comments on Contactor Servomechanisms 8 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 •>rr<. <:tiar\ Control/iol El<unants Cont rolU<i 0 Figure 3* Block diagram of a contactor servomechanism The block diagram for the contactor servomechanism i s quite s i m i l a r i n form to the l i n e a r continuous type. The power amplifier which i s e s s e n t i a l to the control elements of the con-tinuous servomechanism i s replaced by the contactor means which acts as a discontinuous power am p l i f i e r . The compensating network i s very often included i n the control elements to aid the character-i s t i c s of the system. This compensating network i s whenever possible placed on the low power side of the contactor for purposes of economy. In fi g u r e 3 the output i s compared d i r e c t l y to the reference input. Direct feedback i s not e s s e n t i a l f o r contactor servomechanisms and more complicated transfer functions may be included i n the feedback loop. With a compensating network the error s i g n a l i s altered before being applied to the contactor. Without a compensating network the error i s , of course, the same as the control signal to the contactor. 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. In these representations symmetrical operation of the contactor i s assumed, which means that the negative and p o s i t i v e corrections are equal i n magnitude. It i s convenient to express the correction signal as having a unit dlmensionless amplitude so that the p o s i t i v e correction w i l l be plus one while the negative w i l l be minus one. A Correction •S/aoal D + i Corraction S/qnal * 0 y Control S/qitcf/ c 3 -I -H Cotltrvl 5/amxl c a) With mactiva zotf<z, Oy Figure 4; Typical contactor c h a r a c t e r i s t i c s b)With indctivd Zona. ,Ccl anjhyste^sti Zon<Lti Figure 4a represents the c h a r a c t e r i s t i c s of a contactor with i n a c t i v e zone -only, the width of the in a c t i v e zone being Da» No correction w i l l take place unless the control s i g n a l i s greater than Ca/2' and l e s s than -0^/2. The inactive zone width represents the range of permissible error within which no correction takes place and i t can be no larger than design s p e c i f i c a t i o n s . Making the inactive zone smaller than necessary complicates the problem of s t a b i l i t y . As the in a c t i v e zone width approaches zero the system w i l l at best maintain sustained o s c i l l a t i o n s about some equilibrium point. Figure 4b represents the c h a r a c t e r i s t i c of a contactor with i n a c t i v e and hysteresis zones. The physical meaning of 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 control signal required to cause correction i s greater than the control signal required to cease correction. This pheno^ menom may be observed, f o r example, i n electromagnetic relays where the c o i l current necessary to close a relay i s greater than that necessary to open the r e l a y . Of course the assumed c h a r a c t e r i s t i c of the contactor to account for hysteresis i s i d e a l i s t i c * Never-theless the e f f e c t of t h i s i d e a l i s t i c c h a r a c t e r i s t i c may be shown and i t can be reasonably assumed that an actual relay with hysteresis w i l l have much the same e f f e c t . I t w i l l be shown that hysteresis i n a contactor has an adverse effect on the s t a b i l i t y of a system. B a s i c a l l y the performance c r i t e r i a of contactor and l i n e a r continuous servomechanisms are the same. The control system attempts to maintain the controlled variable equal to some desired value of reference input. S t a t i c accuracy i n contactor servomechanisms, which i s the possible range of error under steady-state conditions, i s of prime importance. This accuracy i s determined by the i n a c t i v e zone. Dynamic accuracy i s the measure of error when the system i s responding to a disturbance. This accuracy may be improved by increasing the rate of response of correction or e f f e c t i v e l y as i n continuous servomechanims increasing the amplication of the con* t r o l l e r . This 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 introduces the problem of s t a b i l i t y ; For purposes of dynamic accuracy i t i s important that the frequencies, associated with the o s c i l l a t i o n s , be as high as possible* S t a b i l i t y requirements demand that 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 of the damping of these o s c i l l a t i o n s i s referred to as the 11 r e l a t i v e s t a b i l i t y of the system, A d i s t i n c t i o n must be made between s t a b i l i t y requirements for contactor and l i n e a r servcn-mechanisms. For contactor servomechanisms self-sustained o s c i l l a t i o n s of f i n i t e amplitude are quite possible and at times permissible 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 necessity increase and are destructive* With respect 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 contactor servomechanisms, the s t a b i l i t y of the system i s dependent on the type of disturbance introduced into the system, A p a r t i c u l a r system could, f o r example, be absolutely stable, that i s incapable of maintaining self-sustained o s c i l l a t i o n s , f o r a step input of disturbance while be i n a state of sustained o s c i l l a t i o n f o r a " v e l o c i t y " input of disturbance. This i s unlike l i n e a r servomechanisms i n which the frequency and damping of the transient o s c i l l a t i o n s are independent of the disturbance and dependent solely on the natural 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 transfer functions. For* tunately, however, i t i s r a r e l y necessary to design contactor servos mechanisms for continuously variable control of output* This g r e a t l y s i m p l i f i e s the analysis of contactor servomechanisms as well as the equipment necessary f o r t h e i r suitable operation. 121 3# Mathematical Representation of Linear Components In using Kochenburger rs frequency-response method or the root-locus method, with the contactor describing function, the assumption made i s that a l l elements other than the contactor are l i n e a r . These elements are represented mathematically i n terms of t h e i r transfer functions or ra t i o s of outputs to inputs. Referring to fig u r e 3 the transfer function G s(s) of the controlled system i s defined as G s ( s | = Qjs) (1) where 0(s) and D(s) are the Laplace transform of the controlled variable 0(t) and the correction s i g n a l D(t) respectively, subject to zero i n i t i a l conditions* I f , f o r example, the controlled system consisted of a servomotor, the transform function could a f t e r being factored have the following form G s ( s ) = Rs (2:) s ( T i s + 1) T i i s a time constant of the motor; R s i s the gain of the transfer function and i n t h i s case gives the slope of the response of the motor to a unit step Input a f t e r a s u f f i c i e n t l y long time. For t h i s reason R$ i s referred to as the runaway v e l o c i t y of the motor. A l l other l i n e a r components may be expressed i n a s i m i l a r manner. Consider the phase-lead compensating network of f i g u r e 5. R, in ohms I — — i Input vo/taji 0 Output vo/tayi Figure 5. Representative phase-lead network. The transfer function of t h i s network G c(s) i s 60 (a) = Rc(m2s-r- 1) Tgs+ 1 (3) where Rc = Rg , m2 - RjCi , T 2 - B3R3C1 Rl+^a R1+R2 Tg and mg are time constants of the transfer function. Transfer functions can be expressed as products the gain and the function o f s , g ( s ) , defined Gr(s) = Rg (s) (4) so f o r the transfer function given by equation (2) G s(s) = ( R s ) = R s g s ( s ) (5) TsTf][sTTT where gs(s) s 1 S ( T T S + 1) S i m i l a r l y for the transfer function of the phase-lead com-pensating network given i n equation (g) where G c(s) = Rc(m l S-i- 1) = H cg c(s) (T 2 .8+ 1) g c (s) s m i s -»• 1 (6) Since the transfer functions of the controlled system and the compensating network are l i n e a r they may be combined as the open-loop transfer function from contactor output or correction signal to the contactor input or control s i g n a l . For the system as represented i n figure 3, consisting of the controlled elements, a compensating network and the contactor, the^open-loop transfer function i s then G(s) - C(s) DTI7 iE = constant «0 (7) - G s(s)G c(s) (8) = R s S c g s ( s ) g c ( s ) = Rg(s) (9) Where R = RsRc , g(s) - gs(s)gc(s) , G s(s) and G c(s) are the transfer functions of the controlled system and compensating network respectively* The transfer functions Rg(s) as already described i n factored form w i l l be referred to as the Nyquist form of the transfer function. In t h i s form the complex variable i n a factor i s multiplied! by the time constant of the factor so that the numerator has factors of the form (ms+1) and the denominator has factors 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 plotted versus angular v e l o c i t y . However t h i s form of the transfer function i s not convenient for use with the root-locus method of synthesis. Consider the open-loop transfer function i n Nyquist form G(s) . Rg(s) = R (mis-H) (10) a l ' i l s ^ D l ' f a s + l ) The root-locus form of the transfer function given by equation 10 Is F(s) s Rmx (s+ 1 ) (11) ' mi TxT2 s(s + l )U^ 1 ) Tx T a and rewriting F(s) as the product of the gain and the function of s i n an analogous manner to G(s) - Eg(s) f o r the Nyquist form, f.(s) = A f ( s ) (12) for the root-locus form. Therefore f o r equation 11 A = Rmi (13a) "TlTa f ( s ) = (s + 1 ) V s(s+ 1 )(s+ 1 ) T± T 2 (13b) For the transfer function factored i n root-locus form the complex variable, s, i s m u l t i p l i e d by unity. The zeros and poles of F ( s ) , the root-locus form of the transfer function, may be obtained d i r e c t l y from the factored form of t h i s transfer function. Zeros and poles are those values of s which make the transfer function zero and i n f i n i t e respectively. For equation (13b) a zero occurs at - 1 , and poles at 0, - 1 , and - 1 mx T i T £ on the complex s-plane. I t w i l l be found most convenient to represent both the Nyquist and root-locus forms of the transfer function i n dimensionless time. This i s accomplished by substituting the complex variable s which has the dimensions seconds-1 by the dimensionless complex variable p where P = t b 8 (14) and t D i s the time base selected. The time base selected i s usually one of the time constants of the transfer function G(s). Transient solutions w i l l therefore be functions of dimensionless time 0 where i - t (15) ^ b t being the elapsed time i n seconds. To obtain the dimensionless-time Nyquist form of transfer functions, consider again the case of the open-loop transfer function G(a) given by equation 10. The time base selected w i l l be % r 1\ G(s) - R (mTS+1) = RT! (m-L^ s-•• 1) s(Tis-r l ) ( T 2 s + 1) T i Txs (T xs + 1 j ( T g T x S + 1) (16) For p s Tis S(p) r R T i U i P + l ) (17) m plp~+TTIyip +1) wAar* cf, = r p - and y 4 = A Separating the transfer function G(p) into a product of the gain and a function of p G(p) r Kg'(p) (18) f o r equation 17 K = . RTT_ (19a) and g ( p ) = (qiP+1) (19b) p l p T T J T y 2P + 1) S i m i l a r l y for the dimensionless-time root-locus form of the transfer function F ( p ) s B f ( p ) (20) and for the oase under consideration F ( p ) = RTid! ( p + 1 A = B f ( p ) (21) <U y 8 p ( p + l ) ( p + l ) 7Z where B r R^iqj. (22a) and f(p) - p + i _ (22:b) *1 P(P + l ) ( p ^ _ l _ ) y'a-For the transfer functions i n dimensionless-time form, the gain K of the Nyquist form and the gain B of the root-locus form have the same dimensions. This i s convenient for the study to follow because the transfer functions are frequently changed from one form to the other. Table 1 gives a synopsis of the adopted notation for the system properties i n t h e i r various forms and the signals throughout the system, along with t h e i r various dimensions. to follow page 17 Table 1 Symbols and Units Adopted Quantity Symbol Units Time base Elapsed time Complex variable of ™ Laplace Transform •H Elapsed time S3 P> <D Ej Complex variable of EH Laplace Transform tb t Ss/H-jW P=st t l -cr+-ju Seconds Seconds Seconds" 1 Dimensionless Dimensionless Controlled Variable c or Output ' H cJU Ref erence Input « ™3 Error td dra Control Signal S u o Correction Signal • H 'CO I B.I-0 C D Output-units Output-units Output-units Output-units Dimensionless Inactive zone Hysteresis zone Transfer Function Properties: a) Nyquist Form Of controlled elements Of compensating network. Oi 0) •H £ Over-all open-loop o Time constant i n £ numerator o) Time constant i n denominator >> CO V bj Root-locus form of Controlled elements Of compensating network Cd G s ( s ) = R s g a ( s ) r O ( s ) D(s) a c(s)=R cg c(s} =C(s) G ( s ) 5 G s ( s ) G c ( s ) m T E-slsJsAsfgtsJsOCs) D(s) g c ( s ) - A c f c ( s ) s c | s | Output-units Output-units Output-units Dimensionless Output-units Seconds Seconds Output-units Dimensionle s s to follow page 17 Table 1 (continued) Quantity Symbol Units Over-all open-loop F ( s ) = F s ( s } F c (s) Output-uni ts c) Dimensionless-time Nyquist form Overall open-loop G(p)=G a(p)G c (P) Output-units erties =Kg(p) erties K g(p) Output-units Dimensionless ;em Prop Time constant i n numerator Dimensionless • f 01 >> co Time constant i n denominator t b Dimensionless d) Dimensionless-time root-locus form Overall open-loop F(p)=F s(p)F c «Bf (p) B f ( p ) (P) Output-units Output-units Dimensionless o 18 4. The frequency-response of oontaotor servomechanisms. The frequency-response method of analysis and synthesis has been found valuable f o r l i n e a r servomechanisms^ This method, based on certain .approximations.can also be used f o r study of contactor servomechanisms. The o v e r - a l l open-loop transfer function G-(p) may be described i n terms of i t s steady-state response to sinusoidal inputs of various r e a l frequencies by substituting j u f o r p where u i s the dimensionless angular v e l o c i t y . Then G (ju) = Kg(ju) (23) g(ju) varies with the applied frequency only hence g(ju) i s a frequency variant portion of the system. The r a t i o of output to input of the contactor f o r sinusoidal inputs cannot be represented by a transfer function because of i t s non-linear 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 describing function of the contactor. It w i l l be found that the describing function i s dependent on the amplitude but independent of the frequency of the applied input s i g n a l . The contactor describing function i s then an amplitude variant portion of the system. Kochenburger*s frequency-response method b a s i c a l l y i s the combination of the frequency variant portion of the system and the amplitude variant portion of the system into one scheme subject 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 contactor with the c h a r a c t e r i s t i c s of fig u r e 4b 1 9 which has both an inactive and hysteresis zone. Let the control signal C be represented by C = Cmcosu<t> (24) The contactor w i l l i n i t i a t e a p o s i t i v e correction s i g n a l when C = Gd/2'+ C^/2 and cease correction when C = Cd/2 - Cfc/2. It w i l l i n i t i a t e a negative correction signal when C =-0^/2 - C^/2 and cease negative correction when C =-0^/2+0^/2. These instants of time are designated by the angles u<|) equals a - b, a + b,Tr +• a - b, and i r + a •*• b respectively. These r e l a t i o n s h i p s are shown i n figure 6a. a)Assontad sinuto/<Ja/ s/)apt of control j/jna/ ' 1 Correction Signal, D rvnJ<imc*taf componentf 0, . b) Rt3»ft<*"£ form of correction sicjnql and its fundamental component" Figure 6. Relation of control signal and correction signal for a contactor with hysteresis. The pulse width of the rectangular wave is given by the angle 2b as shown i n figure. 6b,. The fundamental harmonic component of the rectangular wave D]_ i s superimposed on the rectangular wave i n figure 6b and the angle, a, represents the phase lag of the fun-damental component of the correction signal behind the control signal, 20 I f the correction signal consisted s o l e l y of the fundamental harmonic component Di, then only s i n u s o i d a l l y varying signals would exi s t i n the system. The contactor would then appear, as a so-called quasi-linear transfer device i n that i t would operate as a l i n e a r amplifier f o r any given constant amplitude of control s i g n a l . The contactor would not operate as a t r u l y l i n e a r device because of i t s nonlinear r e l a t i o n s h i p between correction s i g n a l and control s i g n a l amplitude. Considering the contactor as such a quasi-l i n e a r device the frequency-response method may be used f o r any p a r t i c u l a r signal amplitude. This approximation of the control signal, which neglects the higher order harmonics of t h i s rectangular wave, i s essential to the concept of the quasi-linear describing function. 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 be considered. An analysis of the rectangular wave of the correction signal 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 are a function of the pulse-width which i n turn i s a function of the amplitude of the control signal and the i n a c t i v e and hysteresis 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 harmonics with respect to the fundamental for a contactor with i n a c t i v e zone only. The amplitudes are pl o t t e d against the r a t i o of the amplitude of the control signal and one-half the i n a c t i v e zone. The graph of the amplitude of the fundamental i s taken as being assymptotic to unity. From the graphs i t may be observed except for very small values of the abscissa that the amplitude of the fundamental i s greater than those amplitudes.of i t s harmonics. But the amplitudes of the harmonics can no way be considered as being n e g l i g i b l e as f a r as the correction signal only i s concerned. However^ the transfer functions of the controlled elements usually act to suppress the 21 higher harmonics. Consider f o r example the controlled element to be a servomotor whose transfer function i s of the form K$ P(y]P+ 1) The transfer function described i n terms of i t s steady-state response to sinusoidal inputs of r e a l frequencies i s obtained by substituting f o r p, ju, j3u, j5u for the fundamental, t h i r d harmonic and f i f t h harmonic respectively. I t i s obvious from making these substi-tutions that the higher harmonic components w i l l be greatly suppressed by the controlled element so that although the higher harmonic com-ponents appear i n the correction signal t h e i r effect on the output sign a l can be considered to be n e g l i g i b l e . If signals other than the fundamental are n e g l i g i b l e i n the output then the error and the control signal can be said to contain only the fundamental component. This i s the basis for con-sidering the contactor c h a r a c t e r i s t i c i n terms of a quasi-linear describing function gj>-j_ where the subscript 1 indicates that the describing function considers only the fundamental component of the correction s i g n a l . This describing function neglects the higher harmonics with the following j u s t i f i c a t i o n as set f o r t h by Kochenburger. 1, The normal frequency spectrum of a rectangular wave involves progressively smaller amplitudes for increasing orders of the harmonic components. 2. Most servomotors (Kochenburger here considered a p a r t i c u l a r type of controlled element) serve as e f f e c t i v e low-pass f i l t e r s and minimize the importance of the higher-harmonic components. Kochenburger, by more exact analytic methods and by te s t , has compared res u l t s with those predicted by his approximate method and found the comparisons to be quite good. This frequency-response method when applicable i s apparently s a t i s f a c t o r y engineering approximation. 23 5. The quasi-linear representation of the contactor describing function.  In the preceding section the form of the control signal C was assumed to be G = Cmcosuc|> which implies the duration of the p o s i t i v e and negative correction signal pulses to be the same. This i s not the most general form of control signal but i t i s the case which w i l l be considered i n t h i s paper. For unequal p o s i t i v e and negative pulses of the correction signal the control signal would be of the form C ~ G 0 + Cm c o s u* w n © r e G 0 i s a constant, descriptive of the average value of the control s i g n a l . The choice of the form of the control signal presupposes a knowledge of the form of self-sustained o s c i l l a t i o n s of the control s i g n a l . For an assumed control s i g n a l with an average value of zero i t i s implied that the input ha.Se zero average rate of change of input and that a l l signals within the system have zero average value. For a control signal C = C^cosuQ the fundamental of the rectangular wave i s as shown i n figure 6b. A Fourier analysis of the rectangular corrective wave yi e l d s a fundamental component ^1 = Dimcos(u(t) + /D| ) (25) Here D i m = 4 sanb (2.6a) fol = -a (2.6b) where b * 1 (cos-^Ca-Cn) + cos~1(GCi+-Ch) (2;7a) ^ (2Cm F (2Cm ) and a = l ( c o s - 1 ( C d - C i 1 ) ~ cos" 1 (0^+0^) (27b) 2 (20 m ) (20m ) The contactor describing function may be expressed i n a manner sim i l a r to the l i n e a r transfer function as the r a t i o 24 of tne output to the input. The describing function gjji_ i s defined as the ratio of the Laplace transform of the correction signal and the control signal subject to zero i n i t i a l conditionsnor gD]_ « L[pi mooa(u»+ / D i ) l (28) L[GiaCOSU<|)] This expression w i l l be found useful when considering, the relat ive s tabi l i ty 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 -rectly 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 in vector form as obtained from the ratio of the output vector and the input vector. The describing function Gp^ in vector form is GDI = Dim /&1 * 4sinb/-a (2:9) — T T Cm This form of the describing function of the contactor is the one developed by Kochenburger. It is a function of inactive 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 ratios of inactive and hysteresis zones. The plot of the magnitude i l lus tra tes that for values of less than unity that the magnitude Is gero Cd/2 or no correction signal i s present because the control signal is within the: inactive zone. With an increase i n control signal the magnitude of the describing function increases then reaches a 2 5 maximum, and decreases u n t i l i n the l i m i t as the amplitude of the control signal approaches i n f i n i t y the magnitude of the describing function approaches zero. When hysteresis i s present the phase angle i s greatest f o r small values of control s i g n a l and t h i s phase angle decreases with an increase i n control s i g n a l . There i s zero phase-shift when the hysteresis effect i s non-existent. 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 single-loop l i n e a r servomechanism with unity feedback and an open-loop transfer function equal to Kg(p), as represented i n fi g u r e 9. I +^ £T — • — < i — 0 J Figure 9. Block diagram of l i n e a r single-loop servomechanism. The r a t i o of the output and input for sinusoidal e x c i t a t i o n i s given by O j j u j - Kg(iu) (30) I 1+ Kg(ju) The inverse response r a t i o jE(ju) may be expressed i n the 0 following form •1 (ju) » g- 1(ju)-VK (31) 0 K where g " 1 ( j u ) = 1 g( Ju) A s i m p l i f i e d version of the Nyquist c r i t e r i o n for s t a b i l i t y which applies f o r a single-loop system i n which d i r e c t feedback i s employed and which has a minimum-phase forward transfer function may be stated as follows: A system s a t i s f y i n g the con-d i t i o n s and having an inverse response r a t i o as i n equation 31 w i l l be stable i f , for a polar-locus plot of g~!(ju) drawn for values of u from -ooto*«o t the point - K i s not enclosed and such a system w i l l be unstable i f , for the polar-locus p l o t of g _ 1 ( j u ) drawn for values of u from - o o t o t o o , the point - K i s completely enclosed. Tmoq. aKiS 27 Figure 10, Frequency polar-locus p l o t Figure 10 shows a t y p i c a l frequency polar-locus p l o t . For a transfer function gain of K-j_, the system, according to the c r i t e r i o n i s stable. However f o r a gain of K 2 "the system i s unstable. Actually for 0< K<K D the system w i l l be stable and for K>Kb "the system w i l l be unstable. For the gain equal to Kb 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 s e l f -sustained o s c i l l a t i o n s . Consider now a contactor servomechanism with an o v e r - a l l open-loop transfer function equal to Kg(p). For a single constant control signal amplitude the contactor describing function w i l l determine the p a r t i c u l a r value of the gain and phase-shift due to the contactor. The describing function G T ^ , may be considered i n the same manner as a conventional transfer function. Therefore the equivalent gain of the system w i l l be KGr^. This value of the gain however i s not a constant and depends on the amplitude of the control s i g n a l . Applying the s i m p l i f i e d Nyquist c r i t e r i o n f o r s t a b i l i t y that has already been stated i t may be found that the system i s stable for some amplitudes of the control signal and unstable for others. The s i m p l i f i e d Nyquist c r i t e r i o n restated f o r the contactor servomechanism i s as follows: For an o v e r - a l l open-loop transfer function Kg(p) and a contactor describing 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 of g * 1 ( j u ) drawn f o r values of u from -o» to+o, the p o i n t - KG-T^, 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 of g - l ( j u ) drawn f o r values of u from -«o to+«°, the poin t - KGrr^  i s completely enclosed. I n order to 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 values of - KGD I must be considered. Therefore a second l o c u s i s drawn j o i n i n g a l l the - KG.DT. vectors f o r a l l p o s s i b l e ampli-tudes of the c o n t r o l s i g n a l . This locus i s a p p r o p r i a t e l y termed the amplitude locus and i s super-posed on the graph of the frequency locus p l o t t e d f o r the f u n c t i o n g ~ 1 ( j u ) . Conditions in <ix«tmpl<L plottaxi <Cc/ZK=.or Jfitassaction u~3.l>CmlQl2 = /.24-k Imoa. amy J8 F i g u r e 11. Super-posed frequency and amplitude l o c i Consider f i g u r e 11 which i s an example of the super-posed l o c i . Only a p o r t i o n of the frequency p o l a r - l o c u s p l o t f o r u p o s i t i v e i s p l o t t e d f o r the f u n c t i o n g ^ t j u ) r j u ( j u + 1) (32.) The amplitude 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 Ca/K .05 and C n /Ca = .2. The frequency and amplitude l o c i f o r u 2~ negative w i l l be the complex conjugate of the l o c i f o r u p o s i t i v e . For t h i s example i n figure 11 the frequency and amplitude l o c i intersect at 0^0^/2 = 1.24. The values of - KGj^ f o r 1.2< Oj]y/cC3:/2< 1.24 w i l l be completely enclosed by the frequency polar-locus p l o t . Therefore the system w i l l be unstable f o r 1.2i< C^/CCL/2< 1.24. The values of - KGr^ f o r values of Cm/ca/2> 1.24 are not enclosed by the frequency polar-locus p l o t . Therefore the system w i l l be stable f o r > 1 • 24. Keeping i n mind that Ca/2: represents one-half the i n a c t i v e zone width and Cm the amplitude of the sinusoidal control signal, for very small disturbances the amplitude of o s c i l l a t i o n w i l l increase because the system i s unstable f o r small amplitudes, up to Cny/Cc\/2'= 1.24. For amplitudes of o s c i l l a t i o n greater than this value the system i s stable. Therefore the condition where the two l o c i i n t ersect represents a point of equilibrium. This point of e q u i l i b -rium i s described as a point of convergent equilibrium because with reference to the amplitude at t h i s point higher amplitudes of o s c i l -l a t i o n decrease and lower amplitudes inorease. Therefore at t h i s point of convergent equilibrium o s c i l l a t i o n s w i l l tend to be main-tained. At the i n t e r s e c t i o n of the l o c i the angular v e l o c i t y u r 3.1. Therefore from the super-posed l o c i i t i s expected that the s e l f -sustained 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 angular v e l o c i t y of 3.1. Kochenburger has checked this p a r t i c u l a r example experimentally and found the agree-ments concerning amplitude and frequency within 6§ per cent and 3 per cent respectively. Figure 12 i l l u s t r a t e s several configurations of frequency and amplitude l o c i . From the c r i t e r i o n of s t a b i l i t y the system represented by figure 12a i s stable f o r a l l operating conditions and i s c a l l e d an absolutely stable system while that of figure 12b i s unstable for a l l operating conditions. /1 Tma< at/3 7 30 /1 Tma< Rial dXIS Figure 12. Examples of super-posed l o c i . The system represented by f i g u r e 12c has two points of equilibrium* Point P represents a convergent point of equilibrium already des-cribed. Point Q represents a divergent point of equilibrium so c a l l e d because for amplitudes s l i g h t l y less than that amplitude at point Q the system i s stable therefore effecting a decrease i n amplitude while for amplitudes s l i g h t l y greater the system i s unstable there-fore effecting an increase i n amplitude. This point of i n t e r s e c t i o n of the l o c i then does not correspond to a point of sustained o s c i l -l a t i o n but i s rather a boundary condition and one f o r which amplitudes w i l l tend to s h i f t away. For t h i s system represented by figure 12c for small disturbances the system w i l l o s c i l l a t e at the amplitude and frequency determined by point P . I f a s u f f i c i e n t l y large dis«-turbance were to cause an amplitude of o s c i l l a t i o n greater than that determined by point Q, then the amplitude would grow and the system would be unstable. The problem so f a r considered has been that of s t a b i l i t y only. I t i s very often not s u f f i c i e n t to consider just the problem 31 of s t a b i l i t y * Relative s t a b i l i t y or the amount of damping present f o r o s c i l l a t i o n s a f t e r a disturbance must very often be considered, Kochenburger approaches t h i s problem from his amplitude and frequency l o c i using the well known concept of the M c r i t e r i o n used i n l i n e a r servomechanismsi M i s defined as the j 0 ( j u ) | and i n l i n e a r servomeonanisms the maximum value of M, Mp, i s often chosen by r u l e of thumb. Linear systems are often considered to have s u f f i c i e n t damping i f Mp<1.3. Kochenburger suggests, from results obtained from< experimental tests, that an Mp of 2.0 near the cut-off point and of 1.3 f o r higher c o n t r o l - s i g n a l amplitudes, provides 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 for most applications. I t i s suggested i n the work to follow that the root-locus method used i n the synthesis of l i n e a r systems can be used to great advantage f o r contactor servomechanisms esp e c i a l l y with respect to.the r e l a t i v e s t a b i l i t y of a system. From the root-locus the s t a t i c servomechanism gain can be determined d i r e c t l y f o r a s p e c i f i e d amount of damping. 32 7. Root-locus method of synthesis. 8 The root-locus method was developed by Evans. This method uses the roots of the open-loop transfer function to f i n d the roots of the closed-loop transfer function. For the l i n e a r single-loop servomechanism represented i n figure $ the open-loop transfer function was Kg(p) i n Nyquist form, and this transfer function can be expressed as Bf(p) in root-locus form. The r a t i o of output to input i s given by the expression, 0 (p) = Bf (p) (33) 1 1+ Bf (p ) The problem of fin d i n g the roots of this closed-loop transfer functions appears i n the form of f i n d i n g those values of p which make the denominator of equation 33 equal to zero. This condition i s s a t i s f i e d i f the open-loop transfer function . Bf(p) = - 1 (34) or i f f ( p ) " 1 = - B (35) f ( p ) ~ ^ may be written i n vector form f ( p ) " 1 = N & 0 (36) For the value of the p to be a root of the closed-loop system N/£° - - B (37) Therefore N = B ' (38a) T ° - 18oU+2n) • ' (38b) where n i s any integer. To fi n d the root-locus the following procedure i s adopted. Consider the open-loop transfer function. 1 (39) Bf(p) ~ B L (P+ Jd (P+ i ) v l ?2 where the dimensionless time constants q.^ , Y^> y2'» a r e r e a^- numbers. The zeros and poles of the open-loop transfer function are plotted on the complex p-plane as i n f i g u r e 13. A zero i s located at -1_, and a pole at -°1 , and -1 . The two requirements q.1 yi 7Q imposed regarding the phase angle and the amplitude of the transfer function are considered separately. Figure 13* Roots i n the p-plane. F i r s t a locus i s obtained which makes the phase-shift angle of f (p)*"1 equal to 180°(1 + 2n). This i s done by choosing an exploratory p point and joining the positions of the zeros and poles to the p point to form vectors which represent a l l the complex factored terms involved i n the open-loop transfer function. The angle of each vector i s measured with respect to a l i n e p a r a l l e l to the p o s i t i v e r e a l axis. This operation has been performed i n figure 13, I f the exploratory p point i s on the root locus then the sum of the angles must equal 180°(H-2n) or from figure 13 - 8 2 - 6]+*! s 180°(1+ 2n) (40) This procedure i s carried out u n t i l the complete "180° phase locus'* or root-locus i s obtained on the complex p-plane. The next step i s to f i n d the amplitude of the inverse transfer function on the root-locus, this amplitude being equal to tae s t a t i c gain B of the open-loop transfer function. The s t a t i c gain B f o r any point on the root-locus f o r this example i s given by B = Pr+ 1 I n y a Pr + *1 * i .on where p r is/the root-locus and the factors are the absolute values of the vectors joining the poles and zeros to pr» Therefore, given the poles and zeros of the open-loop transfer function and the s t a t i c gain B, the roots of the closed** loop system can be obtained d i r e c t l y from the root-locus. Figure 14. Root locus of single-loop servomechanism. Figure 14 i l l u s t r a t e s the root-locus for a single-loop servo-mechanism with open-loop poles at 0, »I , and - i • The root formed y i 72 from the pole - i , travels i n an increasingly negative d i r e c t i o n y2 along the negative r e a l axis. The roots formed from the poles at 0 and - i approach one another, at f i r s t , then form a complex con-y i jugate p a i r with a further increase i n gain. From this root-locus i t may be seen that the troublesome roots are those which have formed into the complex conjugate p a i r . For a s u f f i c i e n t l y high gain they may appear i n the r i g h t half p-plane so that the complex conjugate 5 5 p a i r w i l l have p o s i t i v e r e a l parts r e s u l t i n g i n an unstable system. However the gain can be set so that the complex conjugate 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 large value of the negative r e a l part to ensure s u f f i c i e n t damping for the natural o s c i l l a t i o n s which would r e s u l t from a disturbance to the system. Consider a single-loop contactor servomechanism with poles of the o v e r a l l open-loop transfer function as those i n figure 14. The contactor i n t h i s discussion i s assumed to have no hysteresis zone. For zero hysteresis, the vector form of the contactor describing function has zero phase-shift f o r a l l amplitudes of the control s i g n a l . Therefore f o r a l l sinusoidal control signal amplitudes the value of the describing function shows the contactor to appear as a variable gain element, the gain being dependent on the amplitude of the control s i g n a l . I t i s important to observe that the contactor c h a r a c t e r i s t i c expressed i n terms of i t s describing function has no e f f e c t on the open-loop roots of the system. The open-loop zeros and poles are obtained from the l i n e a r portion of the c i r c u i t and remain fi x e d i n the p-plane. However, the closed loop roots are not f i x e d and are dependent on the e f f e c t i v e open-loop gain, BGj^, which i s a product of the s t a t i c l i n e a r gain and the contactor describing function. For a p a r t i c u l a r value of control signal amplitude the contactor gain i s given by the describing function and for a known value of the l i n e a r gain the closed-loop roots of the system may be obtained from the root-locus. Therefore as the amplitude of the control s i g n a l i s varied, the roots of the closed-loop system vary i n a manner determined by the root-locus. The root-locus i n conjunction with the describing function of the contactor can be used as a synthesis 36 technique f o r contactor servomechanisms and becomes quite e f f e c t i v e when considering the problem of r e l a t i v e s t a b i l i t y . This description on the use of the root-locus i n synthesizing contactor servomechanisms has been general. However i n the following section,which considers a p a r t i c u l a r example, the use and the significance of the root-locus method w i l l be treated i n s p e c i f i c d e t a i l . o 37 8. Analysis of the model by the frequency-response and root-locus methods*  a) Kahn's method for obtaining transient responses. The transient response of a control system to a disturbance i s of major s i g n i f i c a n c e . This statement i s true for both contactor and l i n e a r servomechanisms. 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 follow. Such responses are important when correlating r e l a t i v e s t a b i l i t y from the root-locus i n an analogous fashion to l i n e a r servomechanisms. The transient responses are a l l obtained by application of Satin 1 s semi-graphical method. The following i s a summary of this method. Consider the single-loop contactor servomechanism as represented i n figure 15. Figure 15. Single-loopj contactor servomechanism For a step reference input the transient response of the controlled variable could have the form as shown i n f i g u r e 16a and the correction signal would then be as represented i n figure 16b. controlled variable f a r the quantities expressed i n t h e i r Laplace transforms l i s 0 The relationship between the correction signal and the D ( s ) - Iff} - ^ ( s ) (42) to follow page 37 FtQuri G-raphs hr Kahn*5 saint-yrafAtca/ itidthod 38 where l£a) i s a term containing i n i t i a l conditions. For i n i t i a l conditions equal to zero (43) For D(t) as shown i n fi g u r e 16b D(s) r f [ l - e ~ t l S - e - t 2 S + e'^s + B'^S - e ^ 5 S - ...] (44) To obtain the transient response graphically i t i s necessary to solve one equation only, and that i s the inverse Laplace transform L - l [ G i s i ] = 0 R ( t ) (45) o The physical s i g n i f i c a n c e of this inverse transform i s that i t i s the output of the controlled variable which re s u l t s from a steady ap p l i c a t i o n of the correction signal +1 when the cont r o l l e d variable i s i n i t a l l y at r e s t . Therefore to obtain the complete transient response the "runaway responses"' can be added graphically since 0(t) s 0 R ( t ) - OR(t-ti) - OR(t-tg) + 0R(t-t3)+ ... t>0 t > t i t>t 2 t>t3 (46) and t h i s has been shown i n figure 16c. In a l l the transient responses which follow, zero i n i t i a l conditions have been assumed for s i m p l i c i t y . b) Analysis of the Model The contactor of the model contained n e g l i g i b l e hysteresis e f f e c t . Therefore this analysis i s based on zero hysteresis zone for the contactor. Further the o v e r - a l l open-loop transfer function of the model can be approximated by the transfer function G-(p) * K_ (47) PTP^TI ) D 39 Figure 17 i s a block diagram re p r e s e n t i n g t h i s system. For a s i n u s o i d a l input s i g n a l the 1 + Contactor mo.clns o f>(p*)z 0 Figure 17. Block diagram r e p r e s e n t a t i o n o f the model to input r a t i o of output i s gi v e n by A 0 ( j U ) - &Di G - U u ) (48) where GUu) = Kg(ju) s K JuUu+ I ) 2 The i n v e r s e frequency l o c u s g _ 1 ( j u ) = j u ( j u + l ) 2 i s p l o t t e d i n f i g u r e 18. The amplitude l o c u s -KGr^ w i l l t r a v e l along the negative r e a l a x i s since no p h a s e - s h i f t e x i s t s f o r G Q ^ when the contactor has no h y s t e r e s i s zone. For r a t i o s of Cm/ba/2 equal to u n i t y and i n f i n i t y , Gr^ equals zero. Therefore the amplitude locus has i t s o r i g i n at the o r i g i n of the complex plane f o r 0^0^/^ r 1.0 then proceeds along the negative r e a l a x i s to a minimum p o i n t then reverses and f i n a l l y as the c o n t r o l s i g n a l amplitude approaches i n f i n i t y i t again approaches the o r i g i n . From the 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 the system w i l l be s t a b l e f o r values of K G T J 1 < 2;.0 and w i l l be abs o l u t e l y 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 amplitudes, i f the maximum value of K G j ^ ^ K G T ^ m a x . ) , i s l e s s than 2.0. P o i n t s to follow of equilibrium occur f o r KGJJ^ =r 2.0. ForCKG^max.^ 2..0, two points of equilibrium, both for u = 1.0, occur on the super-posed l o c i . The point of equilibrium for the smaller control signal amplitude i s a divergent point of equilibrium while that for the larger control signal amplitude i s a convergent point of equilibrium. For example, the amplitude locus drawn i n figure 18 i s for Cfl/K r .25. From f i g u r e 8 thereforeCKG^max.) » 2.44 and KGr^ equals 2.0 2for Cm/cd/2 equal to 1.11 and 2;.33. Therefore the divergent point of equilibrium occurs f o r Cm/ca/2. = 1.11 and the convergent point of equilibrium f o r Cm/ca/2: r 2.33. A p l o t of the amplitudes of the control s i g n a l which give points of equilibrium 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 equilibrium are those of s p e c i a l interest because these points represent the possible self-sustained o s c i l -l a t i o n s of the system. It i s d i f f i c u l t to ascertain the meaning of the points of, divergent equilibrium for small values of the amplitude 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 of the frequency response method do not apply. From figure 7 for C m / C a < 1.1 the frequency spectrum of a rectangular wave does not involve pro-gressively smaller amplitudes for increasing orders of the harmonic components. However th i s has no effect on the analysis because of such small amplitudes involved. The graph of figure 19 shows the system to be absolutely stable f o r values of Ca/K > _ l . For Ca/K^, 1 the system i s capable of self-sustained o s c i l l a t i o n s and the amplitudes of the s e l f -sustained o s c i l l a t i o n s increase with a decrease ,in Ca/K. The 2~ frequency of these self-sustained o s c i l l a t i o n s , regardless of ampli-tude, remains constant at u = 1.0. 41 The r a t i o Cg/K i s s i g n i f i c a n t . For this case, K, a represents the slope of the output response to a unit step correction signal a f t e r a s u f f i c i e n t l y long time. Therefore the greater the K the quicker the correction response. However, the s t a b i l i t y of the system i s not only dependent on t h i s gain factor but also on the i n a c t i v e zone, Cg. This i s a property peculiar to contactor servomechanisms. The graph of figure 19 although separating the system into an absolutely stable reg/rin for Gd/Kv 1 and an o s c i l l a t o r y region f o r Ca./K^ 1 gives no information about the r e l a t i v e s t a b i l i t y of the system i n the absolutely stable region. The conventional M-criterion method employed i n l i n e a r servomechanisms may be used as previously mentioned. For this example the root-locus method w i l l be employed to investigate 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 root-locus forms of the o v e r - a l l open-loop transfer function are the same* F(P) S JL £ G(P) = _K (49) P(p + D a P ( P + D 2 where B = K and g(p) = f(p) s 1 T P(P + 1) f ( p ) has three poles on the complex p-plane, one at the o r i g i n and a double pole at p » -1. These poles as well as the root-locus are plotted i n figure 2 0 . For sinusoidal signals i n the System the e f f e c t i v e open-loop gain .is B G j j j S i n c e K = B, then KGTJI s B G D l * Values of Ca/K are also p l o t t e d on the root-locus. 2 The value of KGDT corresponding to the value of Cd/ K o n ^ e root-locus 2 i s the maximum value of KGy^ f o r that value of Ca/K. For example, 2~* to fo l low page Jj-l 1 -— • —-  t i 't j j . • ; i ' • . i • - - ^ - l - t — ' — • { — , - , - ; - j - :• — i 1 ' : ! ' : ; 1 i [ : '1 ! I • ' u r i ••;-[•-•• j C/n\= A 1. ' i ; I i : 1 i 1 1 ; ' i i i ; | 4 1 1 ! J • i ; i j ] i i 1 j i i • — ! i \ i i j \ i i i I i / ! j 1 i • i i i 1 I -I i i i i 1/ 0 l | i i • ! / j i _ 1 1 i •• 1 • i ! f .-v. i 1 - 1 t 1 , / 1 /• • i ; 1 ! i * 1 A*5* J ! I j j i r<3f / t 1 I / J | ! t " -6 4 i 1 1 1 > 1 V 1 ••• f £ J ! 1 I • • j < cr >S / A 'S ti • i i 5~ i | > j j Oc / r ( • i . j 1 i - I 1 j i 1 " 7 V i 1 i ! 1 1 , • I \ i i 1 . i i 1 • 1 i f 1 ! . l l t i _ - -• • I • I - 1 • F/aure. 20. fact-lows for tho, refreientatfOn of thi. vtKdrtf>as>sctt<id /#dcJ«./ 42 the values KGQ 1 r . 3 7 6 and °d/ K = 1 . 6 9 correspond to the same point on the root-locus of fi g u r e 2 0 . Therefore f o r a r a t i o of Gd/K = 1 . 6 9 2 the maximum value of KGrj-^  i s equal to . 3 7 6 and t h i s point on the root-locus determines the minimum damping for o s c i l l a t i o n s i n the system. Similar information regarding s t a b i l i t y can be obtained from the root-locus as from the super-posed amplitude frequency l o c i . The complex conjugate poles cut the imaginary axis at u = 1.0 which frequency corresponds to the frequency of convergent equilibrium i n the frequency-response p l o t . The open-loop gain from the root-locus at u x 1.0 i s KGTTJT^ r 2.0 which also corresponds to the value already obtained. From the root-locus the se t t i n g of the gain can be set f o r the desired minimum damping. For example, i f the minimum damping r a t i o <r/u i s to be no less than 1//3* then from f i g u r e 2 0 , KGD- can be no greater than . 3 7 6 . The r a t i o of Cd/K i s therefore 2 1 . 6 9 . Given the inactive zone C^, which i s determined from the requirements f o r s t a t i c accuracy, the gain K can be found. The root-locus gives some information 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-quanti-t a t i v e l y * . The contactor describing function GTJ-^  i s independent of frequency and dependent only on the amplitude of the control s i g n a l . However, the root-locus implies that for a p a r t i c u l a r amplitude of control s i g n a l that only one frequency i s possible. This i s true because the root-locus determines the natural mode of o s c i l l a t i o n of the system yet the gain 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 root-locus determines from the describing function the amplitude of the control s i g n a l f o r that natural mode of o s c i l l a t i o n . In f i g u r e 21 are plotted transient responses of the con-t r o l l e d output to a step input of the reference. The response i n fig u r e 21a i s f o r Ca/K = .4. From the root-locus the maximum value 2~ of KG/jjj_ i s 1.59. The angular frequency and damping f a c t o r corres-ponding to thi s value of KGr^ on the root locus are u - .91 and T : -.04 respectively. From the transient response plotted the pre-dominant frequency i s u » .796. For an increase i n the value of the r a t i o Ca/K i t would 2~ be expected that the transient response f o r a s i m i l a r type of d i s -turbance would have a lower frequency and greater damping. For Ca/K = »5| the maximum value of KG^ i s 1.2:7 and the frequency and "a damping fac t o r on the root-locus corresponding to this value are u s .81 and<Ts -.09. From the p l o t i n figure 21b the predominant frequency i s u : .689 and i t i s seen that the damping i s slig;htly greater i n this case than i n the previous one. I t i s quite reasonable that the predominant frequency of the transient response i s approximately equal to that given f o r the condition of minimum damping on the root-locus. This i s true especially i n t h i s case because minimum damping also corresponds to the maximum possible value of the angular frequency. Further this minimum damping occurs for a maximum value of KGT^, f o r which Cm/Cd/2'm 1.4 which means that the control s i g n a l amplitude i s r e l a t i v e l y small and the system i s close to coming to r e s t . I t i s , of course, very d i f f i c u l t to come to any d e f i n i t e conclusions regarding the transient response from an inspection of the root-locus because the transient responses are dependent on the form of the external disturbance. 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 experience and practice a s a t i s f a c t o r y rule of thumb c r i t e r i o n could be developed. However, the root-locus with the contactor describing function does y i e l d a p r a c t i c a l technique to help the designer i n choosing the gain of the system for 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) Application of a simple phase-lead type compensating network. For the system just analyzed the r a t i o C^/K had to be 2~ r e l a t i v e l y large to ensure s t a b i l i t y . The application of a com-pensating network allows this r a t i o to be made smaller. This means fo r a p a r t i c u l a r value of ina c t i v e zone, ,that K can be made larg e r which r e s u l t s i n improved dynamic accuracy. The compensating network added to the basic model i s the type shown i n figure 5 which has a transfer function Therefore the o v e r - a l l open-loop transfer function i s now F(p) » B(p-r-.65) (51) P(P-t-l) s(P + 2,.8) Placing the above transfer function i n i t s Nyquist form G(p) =• K(1.588p + 1) (52.) P(P+ l)k(.357p + l ) 8 1 1 ( 1 K = .65 B r *2S5B (53) The frequency-response of g ~ 1 ( j u ) - j u ( j u + l ) 2 ' (,3573m-1) (54) U.588Ju-t-l) i s shown i n figur e 22, This p l o t cuts the r e a l axis at -3.69 f o r u = 2*0. Therefore the boundary condition for s t a b i l i t y i s KOD-L S 3.69. The system w i l l be absolutely stable f o r K G ^ < 3,69 which i s an improvement over the uncompensated case which was absolutely stable f o r KG D L< 2.0. Figure 23 i s a s i m i l a r graph to that i n figure 19 f o r the uncompensated case. It consists of a p l o t of the amplitudes of the control signal which give points of equilibrium versus the r a t i o Oa/K. The forms of the two graphs are a l i k e . In the region 2: of possible self-sustained o s c i l l a t i o n s , the amplitudes of o s c i l -l a t i o n increase with a decrease i n the r a t i o C^/K . The frequency of the possible self-sustained o s c i l l a t i o n s w i f l now be u = 2.0 i n comparison to u - 1.0 f o r the uncompensated oase. The root-locus i s shown i n figur e 24. The eff e c t of the compensating network i s to include a zero at p = -.63 and a pole at p - -2.8 i n addition to the pole at p - 0 and the double pole at p = -1. In an analogous manner to the uncompensated case, the values of KGJJ^ as well as the values of Ca/K corresponding to the 2 maximum value of K G ^ are plotted on the root-locus. The e f f e c t of the addition of the compensating network on the transient response i s demonstrated i n figur e 25. This f i g u r e gives the transient responses of both the uncompensated and com-pensated systems to equal step inputs of the reference f o r equal values 6f the r a t i o Ca/K equal to .40. The improved dynamic response 2; of the compensated case i s obvious. In fi g u r e 25a which i s the transient response with no compensation, the contactor w i l l have to undergo many switching operations before the output response f i n a l l y rests i n the inactive zone. In contrast i s the response of the compensated system shown i n figure 25b. In t h i s case merely two pulses of the correction signal are necessary to bring the system to re s t . to follow page ^ yur~£ 2J4. Koot'/ocu^ for th<l r<tpr<L)<int®C/on of fJfv. company (a4 rr*>rf.-to follow page *J-5 1 : . . ^ ... „ . t • ; - - | - 7 - | - : - T - ; - l - i •""! " i ' i f : ! > • 1 1 i i ! > \& : i . i i 1 i i : A A A A 1 - T ~ M j -:- — • i C i =i i \ 1 \ 1 / l \ /1 \ M ' 1 M l : i | 1! i , i 1 1 / \ f- / i \ /' I \ \ f 1 ! J 1 • i I 1 f | 1 "• r 1 • -i ! • 1 ' j ' 1 ! 1 | . i I C . I I f 1 • j ' i - I a \ —- — j M r i 1 "3 ! • i j . *N / i 1 | j i / j 1 i a J\ i ' I J ! ) > I .3 0 ; • ! -D i ! -i : j i ton hss fcirr)<L __ j • i I c L B * •a s (Lntja t.ion a. uric Off i i i ' 11 t 1 i i • -\ ' I i i i •-. 1 -i i >. J .1 • _ 7TS/C • f • 1 i •N i . 1 \C:, lit - . • > . ; / -c $- it s - •> • • -• 1 • o • - ' < t 2 0 0 T h) tat dtti en_. i 1 i _ U i *• - - - > i t i I 9, Root-locus method when contactor has hysteresis 46 For the contactor with no hysteresis zone the vector form of the contactor describing function G ^ has zero phase-shift fo r a l l amplitudes of the control s i g n a l . The contactor therefore appears^' as a variable gain element, the gain depending on the ampli-tude of the sinusoidal control s i g n a l . The variations i n the gain of the contactor hawpno eff e c t on the o v e r - a l l open-loop roots and affected only the closed-loop roots of the system. For the contactor with a hysteresis zone, phase-shift exists i n the vector form of the describing function G^. Therefore the argument that the contactor has no effect on the o v e r - a l l open-loop roots no longer applies. I t w i l l be shown that the contactor with hysteresis now not only affects the closed-loop roots of the system but also the open-loop roots of the system. This of course complicates the root locus considerably. Referring back to chapter 5 where the describing function of the contactor was represented as a r a t i o of the Laplace trans** forms of the correction signal and control s i g n a l , and given by equation 28 SDi s L [Dim cos(u<t> + /P].)] L [Gm cos u<t>] where from equations 2:6a and 26b respectively D l m z 4 s i n b "* IT A>1 = -a Expanding equation 28 s. L[DIHJICOS ud>) (cos a)4(sin u<t>) (sin a jj] L [ C m cos ua» ] (55) 47 and placing equation 55 into i t s Laplace transform i t can be expressed i n the form Dlia cos Cm ?.a f p T- u tan a 1 - h i ( 5 6 ) The describing function i n i t s Laplace transform introduces a pole at the o r i g i n and a zero at p = - u tan a. This open-loop zero i s not f i x e d on the p-plane and varies as a function of u, the angular frequency, and the phase-shift angle, a. Therefore the describing function of the contactor with hysteresis not only a f f e c t s the roots of the closed-loop system but also those of the open-loop system. For zero hysteresis, a s o for a l l amplitudes of the control s i gnal, cos a then equals unity and tan a equals ^ zero, therefore the describing function g j ^ degenerates to SDi * pirn (57) which fo r t h i s case i s exactly the same as the vector form of the describing function which i s to be expected f o r zero hysteresis. Considering the gain, D i m - cos a, of the describing Cm function; the r a t i o Dim i s equal to the absolute value of the Crn vector form of the describing function G T ^ . T i l e describing function can then be rewritten (58) e D i = I C D I I c o s a p + u t a n a j P In figure 8, the product of the l i n e a r gain, K, and I G T J J , KlGj^l , as well as the phase-shift angle -a have been plotted against the amplitude of the control signal f o r various constant r a t i o s of Ch/Cd« From these two graphs i s plotted a new graph, KlGj^l cos a against the amplitude. It must be remembered that the gain K i s that for the Nyquist form of the l i n e a r open-loop transfer function. The graph of K I G D J cos a and the phase-shift angle '-a against the control signal amplitude f o r the r a t i o Ch/Ca= 0£2: has been plotted i n figures 26a and 2:6b respectively. The method of obtaining the root-locus for a p a r t i c u l a r example w i l l now be considered. The l i n e a r open-loop transfer function chosen i s the same as that which was assumed to represent the model that i s G(p) = K (59) P(P + D 2 the root-locus form i s then B ( P ) - , B .D (60) plp+ 1 ) 2 ; where B s K The example i s worked out f o r the r a t i o of hysteresis zone to inactive zone r 0--12' The r a t i o of output to input i s given by . |(P) r SDT. B(p) (61) and l + SDiB(p) SDiB(p) tt B'Gpxl cos a(p + u tan a) (62) p 2 (p+ l ) 2 For a phase-shift angle, -a^,and fo r the exploratory p vector having an imaginary component u^ the pole and zero con-fi g u r a t i o n i s as shown i n figure 27. The four roots of the closed-loop system f o r the open-to follow page k& t ; _ _ i t + -i ! i . ! • ' i • • • i .{.--.i~::-* • . -:—*::-.-, ••• ! 1 1 . i : i . i ' i ; , • i. • . . . : : . | L : _ . • •• : ' I i ; . • 1 .:_ : M -1 • t ; f '• i ' • ' > i i A J M !:; .; i . i - i • : I i i ; : . : •: 'f. \i:, Of')-- - :: 4 . . 1 -•' i , . . • . . i . . . - 1 f r -1 • ^ *-t • . t . . . .P _•_.. .. 'A. A-:.:-, ';;[:; •jv ;:i; : . ....;. f . . i • • • i : • . : ! ' • •::; ..j ~PU ---Ish) - 4 - ; co/7£"o':c tor = q.n't of \ — i - r f H - J • i / J ^ 1 J . . . : t . _ ; . com por u • • • • j -. — - - -' • : : ! i . . • ° r y p t : i • • • i - • • - — : - -—rlT- — i: •: ' ' J . : — r ': HI''-:: .:• i y [ / -a - H-/ . 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" ; • i i i 1; i • l . . ; • :. i' • • ' i i ~ ' ' - j : • f - T i :'T • l ! ; ! !' •' 11 ] jHl: 1 1 1 ; i t i : ; !!;!• i;!L tfii: HI; i-iiji •r ;n •iM 1—J , : : : . j : ' 1H : : : - . : ^. . : ' ' : i ' : rii!'- ii . : ' ; :::: '. i'.! fit Hi j t i , !j'j i i , ; - . , : i i \\\r Ha -i M : l l : i - i i . i ..}—i . .j i.. .,. i . i: ;.;;j|ii: ;r;r iiii - :i :i' ii11 | ! J::; ; .'.'.I r::. ;;;; ill' i i i if! i i i . i i r ! rf. - 14,1 1!] iii; I ill •!:;! :i;j i l l :; 1 i t : "-;<',-' ' i 1 • • • 1 " -f 1 ' 1 1 • t-i} i ; ; ; ; ! ; 7 . . _, T T r-l • ' -J i f - -i : . : ;:i: ! 11 m i . TiriTT ! : ! ': " * j ! '• i 1 • • iiiili!! iii? i.!-H : <d:: ; j ! i .: : ; :. ;;;!-;;•! ; ; ; ; ; ; ; ; ••:: :; \ . . I , • i ::;: ;fiiiiiii| f/jvrd 27. 0/»w -loop tdro-p^lt conf/juration tilth contactor bystateS'S 49 loop zero-pole configuration of figure 27 can be located approx-imately by inspection. Two of these closed-loop roots l i e on the negative r e a l axis; one of these l i e s between the zero and the o r i g i n while the other l i e s somewhere to the l e f t of the pole at -1.0. The other two roots, form a complex conjugate p a i r . Root-loci for constant values of the angle, a, can be constructed. These constant phase-shift r o o t - l o c i w i l l f o r u£ o have two roots on the negative r e a l axis:, as already described, one between the zero and the o r i g i n while the other w i l l be to the l e f t of the pole at -1.0. The complex conjugate p a i r w i l l break from the r e a l axis at the point where the complex conjugate roots f o r the root-locus of f(p) = 1 break the r e a l a x i s . This i s so p(p+ D 2 because f o r u = o, the zero and pole introduced by the describing function of the contactor are superimposed at the o r i g i n and cancel out the e f f e c t of one another. Therefore, the complex conjugate roots of a l l the constant phase-shift r o o t - l o c i w i l l converge at the same point on the r e a l axis. The troublesome roots which determine the natural 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 . Only these roots are traced on the r o o t - l o c i of f i g u r e 2.7, Further, for convenience, only the half 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 traced. The other h a l f , for u negative, can be located by inspection because of the symmetry of these roots with respect to the r e a l a x i s . Referring to figure 28 the r o o t - l o c i have been obtained for several constant values of the phase-shift angle. The e f f e c t i v e gains which equal B I G D J cos a are plotted on the constant to follow page phase-shift angle r o o t - l o c i and l i n e s of constant gain are constructed. Since i n t h i s case K = B these l i n e s are constant K I G Q ^ I C O S a l i n e s . In the general case where K ^ B , the values of B I G / D T J c o s a would have to he mul t i p l i e d by the appropriate factor r e l a t i n g K and B to obtain the constant K l G ^ I cos a l i n e s . Root-loci have been drawn fo r constant phase-shift angles of the contactor describing function. Constant gain l i n e s have also been super-posed on these r o o t - l o c i . The next step i s to obtain the actual root-locus f o r various values of the control signal amplitude. The procedure i s as follows: Referring to fig u r e 26, assume a r a t i o Ca/K =» 1.0. For an amplitude r a t i o Cm/Ca/2: r 1.2, a = 25° and ~~2 KlGj^lcos a = .44. From a knowledge of the phase-shift angle and the gain the root for t h i s condition can be obtained on f i g u r e 28. This procedure i s repeated f o r increasing values of the control signal amplitude and hence a root-locus for a l l amplitudes of the control signal for a constant r a t i o C^yK t 1.0 i s obtained. This: 2 procedure i s then repeated for other r a t i o s of Ca/K. A family of ~2 r o o t - l o c i for several values of the constant r a t i o Ca/K has been plotted i n f i g u r e 28. The r o o t - l o c i plotted i n figure 2,7 show that contactor hysteresis has an adverse e f f e c t on the system. This adverse e f f e c t due to hysteresis can of course be demonatrated also by the auper-posed frequency-amplitude l o c i and the Nyquist c r i t e r i o n . However the advantage of the root-locus method i s that the values of the frequencies and damping factors can be obtained d i r e c t l y f o r a l l values of control s i g n a l amplitudes. 51 10. The experiment. a) The c i r c u i t and i t s operation. The c i r c u i t diagram of the model tested appears i n figu r e 29 and a convenient schematic diagram of the model i n figure 30. With reference to these two figures, the voltage drop, V 0 , across the resistance R]_ of the potentiometer, which may or may not include the compensating network, i s energized by the voltage drop across the motor-driven rheostat. A second v o l t -age drop, V R, across R]_, i s produced by the potentiometer across the balancing voltage source. The voltages VQ and are opposite i n p o l a r i t y . The difference of these two voltages, VR - Vo, i s applied across the c r i t i c a l l y damped c i r c u i t galvanometer. I f the l i g h t reflected by the mirror of the c i r c u i t galvanometer and in 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 thyratrons are driven beyond g r i d cut-off and are non-conducting. Except f o r a narrow s t r i p on the face, each phototube i s completely masked by tape. The l i g h t spot shining on the phototubes i s rectangular i n shape. The edges of this spot p a r a l l e l to the axis of galvanometer rot a t i o n , are p a r a l l e l to the light-accepting s l i t s of the phototubes. 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 c i r c u i t galvanometer the l i g h t w i l l move off one of the phototubes causing the thyratron con-t r o l l e d by th i s phototube to 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 opposite d i r e c t i o n the l i g h t spot w i l l move off the other phototube causing i t s thyratron to con-duct. For 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 -HOv — - H Thyratron Photo multiplier Tube 93/-A J ,/2meqD. each y ^~ ' 1 \ )' Cr-ittcqlly damped* Record ma &al^onomet<tr frjure 1% Crtvtt dKtgrem of the model Motor X , y Q-dihanootethr' IOK+ Cref/v. ®C, Defection ontacto r /Mean % Armcttbre, ',urr&n"C •gOP"tTflg| 7 y x ^ - S 39.7 X r-AAAAAA-, LyvWSAA-j Com pen s <*ti na 4-1.2 j^-f f a . O 5 NO (xalvanOmetsjr\ Deflection, $ t2. \ 5 ± F/gure 30. Schew'ot/c Ptaqram of the Model on both phototubes and both thyratrons are non-conducting. This range determines the inactive zone of the contactor. The d i r e c t i o n of motor rotation depends on which thjir-atron i s conducting. Both the thyratron p l a t e supply voltage and the voltage applied across the f i e l d of the motor are al t e r n a t i n g . Consider that one of the thyratrons i s conducting. This thyratron conducts over a h a l f - c y c l e of the p l a t e supply alternating voltage I f the other thyratron conducts, i t w i l l do so over the other half cycle. However the current from either thyratron which i s the current through the armature of the motor i s u n i d i r e c t i o n a l reg-ardless over which half- c y c l e thyratron conduction takes place. But the d i r e c t i o n of the f i e l d current i s reversed over each h a l f -cycle. Therefore the direction of motor ro t a t i o n depends on which thyratron i s conducting. In summary, control takes place In the following manner. For a s u f f i c i e n t l y large difference of the voltages, VR - VQ, the c i r c u i t galvanometer w i l l be deflected, so that l i g h t w i l l move off one of the phototubes. The thyratron controlled by this photo tube w i l l conduct. The motor w i l l run and turn the shaft of the rheostat i n such a d i r e c t i o n as to correct for the unbalance. If the unbalance i s i n the reverse p o l a r i t y then the other thyratron w i l l conduct. The motor w i l l run i n the opposite d i r e c t i o n and turn the shaft of the rheostat i n the opposite d i r e c t i o n i n order to correct for the unbalance. From the c i r c u i t diagram i t i s seen that there are two galvanometers in the system, one referred to as the c i r c u i t gal-vanometer while the other is referred to as the recording galvan-53 ometer. A l l the calculations are based on the control signal which i s the input signal to the contactor and which f o r the model i s the deflection of the c i r c u i t galvanometer, ©Q]_. Since i t was im-possible to "break i n t o " the c i r c u i t to observe the d e f l e c t i o n of the c i r c u i t galvanometer, the recording galvanometer was introd-uced i n such a manner that the def l e c t i o n of the recording galvan-ometer, ©C2> would be i d e n t i c a l to the d e f l e c t i o n of 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 recording galvanometers. Neither galvan-ometer possessed electromagnetic damping on open c i r c u i t . The periods of the galvanometers on open c i r c u i t were 6.46 and 6.45 seconds f o r the c i r c u i t and recording 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 period of 6.46 seconds. S u f f i c i e n t resistance was added to each galvanometer so that both galvanometers were c r i t -i c a l l y damped. For c r i t i c a l damping, the time constant of each galvanometer i s therefore 6.46/2T = 1.029 seconds. The sensit-i v i t i e s were found to be 370 radians/volt and 477 radians/volt for the c i r c u i t and recording galvanometers respectively. The transfer functions of the galvanometers neglecting a i r damping and inductance in the galvanometers for the condition of c r i t i c a l damping galvanometer are therefore: for the c i r c u i t galvanometer: G f fi (s) = 5 7 0 radians/volt (63a) S -(Tf025s + 1) and for the recording galvanomter G g 2 (s) - 477 radians/volt (63b) (1,029s + l ) 2 54 The inactive zone was determined in the following manner. A voltage just s u f f i c i e n t to f i r e one of the thyratrons 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 voltage just s u f f i c i e n t to f i r e the other thyratron. The difference of the two voltages, which gives a measure of the inactive zone, was obtained. This procedure was repeated several times and the average value of t h i s voltage was found to be 5.9 micro-volts. Since the output of the system i s considered to be the d e f l e c t i o n of the galvanometer, the inactive zone, C a, i n terms of galvanometer de-f l e c t i o n i s Cd - 5.9 x IO" 6 x 370 = 2.18 x 10~ 3 radians. The transfer'function from the motor armature current to the voltage drop across the rheostat driven by the motor was ob-tained i n the following manner. It was assumed, viien a thyratron conducted and therefore the armature of the motor carried current, that the motor immediately attained a certain speed and maintained this speed constant as long as i t was running. This assumption was observed to be approximately true although some time would have to elapse before the motor came up to speed. This period i s very small compared witn the time constant of the galvanometer and hence as an approximation this period can be neglected. There-fore with this assumption the change i n the voltage drop across the rheostat caused by the motor r o t a t i o n would be proportional to t i me. The voltage difference between two settings of the rheostat arm p o s i t i o n was obtained together with the time required for the rheostat arm to travel the distance between these s e t t i n g s . From this, information and the assumption of constant motor speed, 55 the transfer function of t h i s p o r t i o n of the system can be defivad. For example, when the voltage output of the variable voltage supply, Vg, was equal to 445 v o l t s , the difference in voltage drop between two settings of the rheostat arm p o s i t i o n equalled 1.57 v o l t s / For the case where the rheostat arm required 8 seconds to traverse the distance between these two settings, the change i n the voltage drop across the rheostat appears as shown i n fi g u r e 31. / 5 7 " tirrta. In S tconds Figure 31 - Change i n rheostat voltage drop against time. Assuming, fo r convenience, the input to be a dimensionless unit step function, the transfer function, G^ U), of t h i s p ortion o f the system i s •1 G m(s) = 1 R m g m (s) * 1.57 ( 64) "s s 8^ Gm(s) = Rmg m( s) s -1965/s vo l t s (65) where R m = .1965 volts/second and g m ( s ) = 1/s The value of the gain, Rm, was changed by changing either the value of the supply voltage, V s, or the speed of the motor, or both. The system i s depicted by a block diagram i n figure 38. For the condition that the d e f l e c t i o n of the c i r c u i t galvanometer and the recording galvanometer be the same, 370 Rp]_ must equal 477 R p 2. R p i i s equal to 1^/23,100 and R p 2 i s equal to R2/22,400 for the uncompensated case. Rpj_ i s equal to R^/51,200 and R p 2 ; i s equal to R2/51,720 for the compensated case. R]_ and R 2 are expressed cr to follow page 55 partest/jjiucr} V •4. •MS T y \ ^ 56 i n ohms. The r e l a t i v e values of R]_ and R g were always such that the condition f o r equal d e f l e c t i o n of the galvanometers was s a t i s -f i e d . Both of the values of these resistances were always small compared with resistance of the galvanometer c i r c u i t s to ensure neg l i g i b l e e f f e c t on the positions of the poles of the galvanometer transfer functions* from figure 32, the o v e r - a l l open-loop transfer functions, G(s), for the uncompensated and compensated cases respectively are G(s) = R (66a) s(1.029s + l ) 2 G(s) = R (1.635s •» 1) (66b) s(1.029s + 1J« (,367s + 1) Where R = 370 R m R p l. The over-all open-loop transfer functions i n dimensionless-time form f o r the time base t ^ = 1.029 seconds for the uncompensated and compensated cases respectively are G ( P ) = K (67a) p(p + D 2 and G(p) = K(1.558p ( 6 7 b ) p(p + l ) 2 (.357p + 1) where p = 1.029s and K = R(1.02$) output-units, which for this case equals radians; The above o v e r - a l l open-loop transfer functions are i d e n t i c a l with those of the two systems analyzed i n chapter 8. c) Tests and Results. Transient responses for both the uncompensated and com-pensated cases were obtained f o r equal step input disturbances. The disturbance introduced appears as a step input of the c i r c u i t 57 galvanometer displacement and was obtained In the following manner. The contactor was set so that i t rested just on the edge of the inactive zone. The balancing voltage, V^, was then changed i n a l l cases by an amount equal to 13.5 micro-volts corresponding to a galvanometer d e f l e c t i o n of 5 x 10~ 3 radians. The f i e l d switch of the motor was l e f t open so that correction could not take place. A f t e r the c i r c u i t galvanometer had reached i t s stable position the switch i n the f i e l d c i r c u i t of the motor was closed to allow the system to correct f o r the de f l e c t i o n s . This appears to the system as a step input of 5 x ICP^ radians to the control s i g n a l . The transient response was observed on the re-cording galvanometer. Graphs of transient responses f o r the model uncompensated and compensated are drawn i n figure 33. Galvanometer d e f l e c t i o n i s plotted against dimensionless time 0, where 0 = t/1.029. From the graphs i t may be observed that the dynamic accuracy f o r comparable values of C^/2K i s improved by the compensating network. The e f f e c t of the compensating network on the frequency of the transient response i s evident from the graphs. For a r a t i o Cd/2K = .588 f o r the Compensated system the predom-inant frequency i s nearly twice as great as the predominant frequency f o r the uncompensated system with a r a t i o G^/2K = .583. From an inspection of the r o o t - l o c i f o r the two cases, t h i s increase i n frequency for the compensated case i s tobe expected. •> Values of amplitudes and frequencies of self-sustained o s c i l l a t i o n s were also obtained. Table 2 i s a tabulation of the amplitudes and frequencies of these o s c i l l a t i o n s f o r several to follow page 57 11: 1: -: I'1:i i 1.11. i1ii i Al1 LA ,..; i L i l l i 11 li i t rill iii 11 fijur& J j . 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 frequencies for the assumed o v e r - a l l open-loop transfer function are also tabulated. These values were obtained from the graph i n figu r e 19. Table 3 represents s i m i l a r results f o r the model with the compensating network. The predicted amplitude and frequencies f o r the assumed o v e r - a l l open-loop transfer function were obtained from the graph i n f i g u r e 23. Referring to the r e s u l t s i n table 2 for the uncompensated case. The predicted frequency i s greater than the observed frequencies while the predicted amplitudes are less than the obser-ved, f o r a l l eases. The higher predicted frequency i s probably due to the assumed transfer function of the mot or-rheostat which was approximated as R m/s. A more r e a l i s t i c transfer function for the motor-rheostat would be R m/s (T ms + 1) with T m, of course, being very large; The ef f e c t of T m i s to introduce another open-loop pole on the complex plane. The root-locus of the closed-loop system would therefore cut the imaginary axis at a value of u < 1 . 0 . Therefore the frequency of the self-sustained o s c i l l a t -ions, which i s determined by the point where the complex-conjugate roots cross the imaginary axis, would be l e s s than u = 1 . 0 . For such a case the amplitudes of the self-sustained o s c i l l a t i o n s would be greater for equal r a t i o s of Ca/2K than for the case where the crossing of the root-locus occurred at u = 1 . 0 . There is yet greater discrepancy i n the results f o r the compensated system as shown i n table 3. The system f o r the assumed ov e r - a l l open-loop transfer function i s not r e a l l y the system that has been compensated. Yet the predicted values of to follow page 58 Table 2. Amplitudes and frequencies of self-sustained oscillations without compensation. Predicted Observed C d/2K Cm/Cd/2 u C^d/ 2 u .354 A 0 1.0 1.83 .842 .266 2.2 1.0 2.44 .85 .213 2.8 1.0 3.87 .846 .177 3.4 1.0 4.58 .854 .152 3.9 1.0 5.57 .858 .118 5.01 1.0 7.15 .847 .106 5.8 1.0 8.06 .852 .080 7.9 1.0 10.4 .849 .073 9.0 1.0 12.5 .849 Table 3. Amplitudes and frequencies of self-sustained oscillations with compensation. Predicted Observed C d/2K Cm/cd/2 u C r / V 2 u .1397 2.2 2.0 6.6 1.27 .12 2.7 2.0 8.04 1.28 .105 3.2 2.0 9.9 1.28 .093 3.5 2.0 10.48 1.27 .076 4.4 2.0 12.6 1.29 .0698 4.8 2.0 13.6 1.25 .0599 5.7 2.0 16.2 1.25 & - This corresponds to an absolutely stable - state. 59 amplitude and frequency are based on adding a compensating network to the assumed o v e r - a l l open-loop transfer function. Therefore i t i s reasonable to expect that the re s u l t s obtained for the compensated case would not check as c l o s e l y as those f o r the uncompensated case. However, important conclusions can be drawn from these res u l t s . The frequencies of the self-sustained o s c i l l a t i o n s remain approximately constant regardless o f the amplitudes. The amplitudes of the self-sustained o s c i l l a t i o n s do increase with a decrease i n the r a t i o Gd/2K. The addition of the compensating network increases .the frequency of self-sustained o s c i l l a t i o n s . For the assumed o v e r - a l l open-loop transfer functions the increase predicted i s from u = 1.0 f o r the uncompensated case to u s 2 f o r the compensated. The observed increase of frequency i n the model i s from u & .85 for the uncompensated case to u « 1.27 f o r the compensated case. 11. Summary and conclusions. The quasi-linear representation of the contactor des-cribing, function 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 analysis and synthesis of contactor servomechanisms. Kochenburger * s method of analysis and synthesis based on the frequency-response has been explained and applied. The root-locus method of synthesis, which has been ap-p l i e d to l i n e a r servomechanisms, has been developed f o r the con-tactor servomechanism. This method appears to be valuable when considering the problem of r e l a t i v e s t a b i l i t y . For a simple contactor with no hysteresis e f f e c t , Koch-enburger' s vector form of the contactor describing function was used d i r e c t l y to obtain the root-locus. The contactor appeared as a variable gain element for the various control s i g n a l ampli-tudes. The contactor had no ef f e c t on the open-loop roots but the variations i n the contactor gain caused the roots of the closed-loop to t r a v e l along the root-locus obtained from the open-loop roots of the system. Root-loci were also obtained when the contactor poss-essed hysteresis. Kochenburger*s vector form was modified to the Laplace transform form of the contactor describing function. This form of, the describing function showed that not only were the positions of the roots varying f o r the closed-loop but also for the open-loop. The model constructed to check some of the theory was described. The o v e r - a l l open-loop transfer functions assumed f o r the mathematical analysis were only approximations of the actual. However, even for the assumption made, the experimental work v e r i f i e d q u a l i t a t i v e l y and to some degree quanti t a t i v e l y the prediction of the model performance. The methods of analysis and synthesis explained and developed, although being approximate, appear to be s a t i s f a c t o r y as engineering approximations. 62, 12. References. 1. Ivanoff, A., "Theoretical Foundations of the Automatic Hegulation of Temperature." Journal 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." Journal of the Franklin I n s t i t u t e , v o l . 218, no.3 (September, 1934), pp. 279-330. 3. Weiss, H. K., "Analysis of Relay Servomechanisms." Journal 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 Control, Princeton, N.J., Princeton University Press, 1953. 6. Kahn, D. A., "An Analysis 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 for 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, part 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. Vol. 69, p a r t i , 1950. pp.66-69, 13. Acknowledgements. The author wishes to express h i s indebtedness to his thesis director, Mr. W. A. Wolfe. He would l i k e to thank him for his guidance, patience and encouragement shown during the course of this research. The author would also l i k e to thank those members of the e l e c t r i c a l engineering department who from time to time assisted him se graciously. Acknowledgement i s also made to the B r i t i s h Columbia E l e c t r i c Company Limited who assisted the author by means of a scholarship and research grant. F i n a l l y , he would l i k e to thank his s i s t e r , Marcelle, and Mr. N. C. Frelone who typed out the thesis report. A. P. P a r i s . 

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