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UBC Theses and Dissertations

Demodulator compensating networks Pritchard, John Robert Gilbertson 1959

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DEMODULATOR COMPENSATING NETWORKS by JOHN ROBERT GILBERTSON PRITCHARD B . A . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1957 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the s t a n d a r d s r e q u i r e d from c a n d i d a t e s f o r the degree of Ma s t e r of A p p l i e d S c i e n c e The U n i v e r s i t y of B r i t i s h Columbia A p r i l , 1959 ABSTRACT r T h i s t h e s i s d e a l s w i t h t h e study o f some demodulator l e a d - l a g n e t w o r k s . S p e c i f i c a l l y the problem has i n v o l v e d a n a l y s i s and d e s i g n , accompanied by e x p e r i m e n t a l v e r i f i c a t i o n o f a new approach t o the r e a l i z a t i o n o f p h a s e - l e a d and phase- l a g networks f o r a p p l i c a t i o n i n ac servomechanisms„ A n a l y s i s has been made of s e v e r a l c i r c u i t s , d i f f e r e n t i n p h y s i c a l l a y o u t but o p e r a t i n g on t h e same b a s i c p r i n c i p l e . By computing the parameters w h i c h d e s c r i b e t h e s t e p r e s p o n s e of the p a r t i c u l a r network, an e q u i v a l e n t t r a n s f e r f u n c t i o n i s obtained,, T h i s t r a n s f e r f u n c t i o n i s the d e s c r i b i n g f u n c t i o n f o r t h e l i m i t i n g case o f i n f i n i t e c a r r i e r - t o - s i g n a l - f r e q u e n c y r a t i o . E x p e r i m e n t a l work was done w i t h an e l e c t r o - m e c h a n i c a l network, c a p a b l e o f g e n e r a t i n g l o w - f r e q u e n c y s i n u s o i d a l - m o d u - l a t e d s i g n a l s . Phase and a m p l i t u d e c h a r a c t e r i s t i c s o f an ac l e a d network, c e n t r e d a t a f r e q u e n c y o f UOO c y c l e s per second, were o b t a i n e d . S i n c e o n l y i n t h e l i m i t ^ c can the network W s be r e p r e s e n t e d e x a c t l y by a d e s c r i b i n g f u n c t i o n , e x p e r i m e n t a l and a n a l y t i c r e s u l t s f o r t h e network were compared t o check the l i m i t i n g d e s c r i b i n g f u n c t i o n as a p r a c t i c a l r e p r e s e n t a t i o n . i i In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I fur ther agree that permission for extensive copying of t h i s t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by his representat ives . It i s understood, that copying or p u b l i c a t i o n of t h i s thes is for f i n a n c i a l gain s h a l l not be allowed without my wri t ten permission. Department of S£JLCJ^-(LAJI f H ^ L A t The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , Canada. T a b l e of Contents T i t l e Page i A b s t r a c t 11 L i s t o f I l l u s t r a t i o n s i v Acknowledgment v i 1.0 I n t r o d u c t i o n 1 2.0 Methods of A n a l y s i s 3 3.0 C i r c u i t 1 7 k.O C i r c u i t 2 11 5.0 C i r c u i t 3 16 6 .0 C i r c u i t k 22 7.0 C i r c u i t 5 27 8.0 Feedback Methods of M o d i f y i n g the C i r c u i t Parameters 31 , 8 . 1 V a r i a b l e Conduc t ion Angle 31 8.2 S e r v o - M i x i n g Network 33 9.0 D i s c u s s i o n o f the V a l i d i t y o f the E q u i v a l e n t C i r c u i t 39 10.0 Measurement of Phase and Gain C h a r a c t e r i s t i c s of C i r c u i t 1 k2 11.0 C o n c l u s i o n s k? Appendix I DC Lead Network Appendix II R e f e r e n c e s 52 i l l L i s t of I l l u s t r a t i o n s Fig„ 2 0 1 Lead Network Response to a Unit-Step Input h „ 2 Complete Equivalent C i r c u i t f o r an AC Lead Network h o 3 T y p i c a l Output Voltage 5 0h Approximating of Condenser Voltage 5 05 DC Lead Network 6 F i g . 3 „ 1 Voltage Wave Forms 7 o 2 Schematic f o r the D e f i n i t i o n of the Q u a n t i t i e s to be used i n the A n a l y s i s 7 . 3 C i r c u i t Operation 8 „H Output Voltage Waveform 8 „5 DC Lead Network 9 „6 Equivalent DC Lead Network 1 0 F i g . k.l Diprose's C i r c u i t 1 1 ,2 Transformer Equivalent C i r c u i t 1 1 C i r c u i t f o r Diode Non-Conducting 1 2 C i r c u i t f o r Diode Conducting 1 2 ,5 Equivalent C i r c u i t 1 2 ,6 Equivalent C i r c u i t f o r Diprose's Network 1 2 F i g . 5°1 Schematic of Lyons' Network 16 , 2 S i g n a l C i r c u i t of the Secondary f o r State 1 16 . 3 S i g n a l C i r c u i t of the Secondary f o r State 2 1 7 oh H a l f - C y c l e Equivalent C i r c u i t 17 05 Output Voltage Waveforms 1 8 06 DC Discharge C i r c u i t 1 8 07 AC S i g n a l C i r c u i t 1 8 „ 8 Equivalent C i r c u i t f o r Lyons' Network 2 0 F i g . 6 . 1 Weiss and L e v e n s t e i n ' s Network 2 2 0 2 Output Voltage Waveforms 2 2 .•3 H a l f - C y c l e Equivalent C i r c u i t 2 3 „*+ AC Charge C i r c u i t 2 3 .5 DC Discharge C i r c u i t 2h .6 Equivalent C i r c u i t 2 1 ? o 7 DC Equivalent Lead Network 2 6 F i g 0 7»1 Schematic of Lyons' Lag Network 2 7 , 2 Condenser Charge C i r c u i t 2 7 03 Condenser Discharge C i r c u i t 2 8 oh DC Equivalent C i r c u i t 3 0 F i g . 8 . 1 Voltage Output f o r a V a r i a b l e Conduction Angle 3 1 0 2 Conduction C i r c u i t 3 1 „ 3 DC Equivalent Lead Network 3 2 »*+ Servo-Mixing Network w i t h AC Input and DC Feedback 3 3 i v L i s t o f I l l u s t r a t i o n s cont'do F i g . 8o5 DC Feedback C i r c u i t ' 33 06 AC I n p u t C i r c u i t 3*+ „7 E q u i v a l e n t Network f o r the S e r v o - M i x i n g Network 36 08 Network 1 37 09 Network 2 37 F i g . 9.1 Schematic f o r C i r c u i t 1 39 F i g . 1 0 o l Schematic f o r D e t e r m i n a t i o n of M u l t i p l i e r P r o p o r t i o n a l i t y C o n s t a n t k2 .2 M u l t i p l i e r Waveforms U-2 „3 B l o c k Diagram o f Measurement System *+3 A M u l t i p l i e r Schematic f o r Gain-Phase D e t e r m i n a t i o n s Appendix I F i g . 1.1 DC Lead Network *+8 v ACKNOWLEDGMENT The a u t h o r e x p r e s s e s h i s a p p r e c i a t i o n f o r the generous a s s i s t a n c e and guidance g i v e n by Dr„ E. V, Bohn, h i s s u p e r v i s o r . The a u t h o r wishes t o acknowledge the numerous o c c a s i o n s on which Members, S t a f f and Graduate S t u d e n t s of the Department of E l e c t r i c a l E n g i n e e r i n g c o n t r i b u t e d t o the development and u n d e r s t a n d i n g of t h i s project„ The a u t h o r thanks t h e N a t i o n a l R e s e a r c h C o u n c i l f o r i t s a i d which eased h i s f i n a n c i a l burdens. Awards i n c l u d e a B u r s a r y f o r t h e S e s s i o n 1957-58 and a S t u d e n t s h i p f o r t h e S e s s i o n 1958-59. v i 1.0 I n t r o d u c t i o n In servomechanism design, compensation i s o f t e n necessary to meet system s p e c i f i c a t i o n s . In open-loop sys- tems, problems may a r i s e i n achieving a d e s i r e d t r a n s i e n t response. I n closed-loop systems, s t a b i l i t y margins may re q u i r e gain-phase compensation. The compensating networks commonly used are l e a d - l a g networks. In servomechanisms, i n f o r m a t i o n may be tr a n s m i t t e d by a dc voltage l e v e l or w i t h a modulated c a r r i e r s i g n a l . For dc lead networks, a n a l y t i c r e s u l t s are w e l l known. De- s i g n s p e c i f i c a t i o n may i n c l u d e , from the point of view of a t r a n s i e n t response to a u n i t - s t e p i n p u t , the value of the output f o r t = 0, the value of the output f o r t = <><> , the decay time constant; o r , from s t a b i l i t y c o n s i d e r a t i o n s , the maximum amount of available phase s h i f t , and the frequency at which t h i s maximum occurs. Depending on the p a r t i c u l a r net- work, one has two or three Independent choices of design parameters. C o n v e n t i o n a l l y , compensation i n ac systems has been achieved by e i t h e r of two methods. The f i r s t of these i s the passive system, w i t h compensation being achieved by the R-L-C elements. The r e a l i z a t i o n of the ac network i s obtained by the usual low-pass-to-band-pass tran s f o r m a t i o n s . S i m i l a r c h a r a c t e r i s t i c s may be obtained by employing p a r a l l e l - T and bridge-T networks. These networks are tuned to the c a r r i e r frequency and hence these c h a r a c t e r i s t i c s are q u i t e s e n s i t i v e to any v a r i a t i o n i n t h i s c a r r i e r frequency. Gen- e r a l l y f o r these networks, the at t e n u a t i o n at the c a r r i e r f r e q u e n c y i s s e v e r e and the phase c h a r a c t e r i s t i c s a r e e f f e c - t i v e over a v e r y l i m i t e d b a n d w i dth. The second method o f a c h i e v i n g ac compensation r e q u i r e s a demodulator, a dc n e t - work, and a m o d u l a t o r . Here t h e envelope i s peak d e t e c t e d , the f i l t e r e d s i g n a l i s a p p l i e d t o the dc network and t h e n chopped t o r e i n t r o d u c e the c a r r i e r f r e q u e n c y . The m e r i t s o f a network o f t h i s type a r e the independence o f c a r r i e r f r e q - uency and the l a r g e amount o f phase s h i f t a v a i l a b l e . The main d i s a d v a n t a g e of t h i s network i s the presence of a f i l - t e r w h i c h l i m i t s the amount of a v a i l a b l e phase l e a d and the c i r c u i t c o m p l e x i t y i n v o l v e d . T h i s t h e s i s c o n c e r n s i t s e l f w i t h the d e t a i l e d s t u d y of some demodulator l e a d - l a g n e t w o r k s . These networks have a resemblance t o the demodulator-dc-network-modulator- type systems. However they o p e r a t e on a d i f f e r e n t p r i n c i p l e i n t h a t t h e d e m o d u l a t i o n o p e r a t i o n i s not c o m p l e t e l y c a r r i e d out and t h e r e i s no m o d u l a t o r . An a n a l y t i c t e c h n i q u e has been dev e l o p e d based on the s t u d y o f t h e t r a n s i e n t r e s p o n s e t o a s t e p i n p u t . Parameters d e s c r i b i n g t h e t r a n s i e n t r e s - i ponse have been found and t h e s e a r e used t o o b t a i n an e q u i - v a l e n t c i r c u i t w h i c h i s v a l i d o n l y f o r i n f i n i t e c a r r i e r - t o - s i g n a l - f r e q u e n c y r a t i o . P r e v i o u s a n a l y t i c t r e a t m e n t o f v a r i o u s demodulator networks has been s u p e r f i c i a l because o f the d i f f i c u l t y i n o b t a i n i n g e q u i v a l e n t c i r c u i t s . U s i n g l i m i - t i n g f u n c t i o n s , the a u t h o r has o b t a i n e d e q u i v a l e n t dc n e t w o r k s . E x p e r i m e n t a l work was d i r e c t e d towards d e t e r m i n i n g the ade- quacy of t h e s e n e t w o r k s . 2.0 Methods o f A n a l y s i s 3 F o r a n o n - l i n e a r system i t i s i m p o s s i b l e t o g i v e a g e n e r a l d e f i n i t i o n o f a t r a n s f e r f u n c t i o n . F o r c e r t a i n t y p e s of n e t w o r k s , i t i s c o n v e n i e n t t o d e f i n e an e q u i v a l e n t t r a n s f e r f u n c t i o n or d e s c r i b i n g f u n c t i o n o f the system. The d e s c r i - b i n g f u n c t i o n method uses the f u n d a m e n t a l component o f the output r e s p o n s e t o a s i n u s o i d a l i n p u t s i g n a l , t o d e f i n e a t r a n s f e r f u n c t i o n . As the d e s c r i b i n g f u n c t i o n p l a y s the r o l e o f the t r a n s f e r f u n c t i o n f o r the fundamental f r e q u e n c y , a s u i t a b l e e q u i v a l e n t l i n e a r c i r c u i t can be o b t a i n e d . F o r d e m o d u l a t o r - t y p e networks i t i s not p o s s i b l e t o o b t a i n a d e s c r i b i n g f u n c t i o n w h i c h i s v a l i d except i n t h e l i - m i t as the r a t i o ^c. y oo° The subsequent a n a l y s i s o f t h e s e networks r e l i e s upon the q u a s i - s t a t i o n a r y n a t u r e of t h e s i g - n a l s . That i s , the a m p l i t u d e of t h e s i g n a l s i s s l o w l y t i m e - v a r y i n g but may be assumed c o n s t a n t f o r a s h o r t p e r i o d o f time w i t h o u t a p p r e c i a b l e e r r o r . I n o r d e r t o e s t i m a t e t h e b e h a v i o r of a p a r t i c u l a r s i g n a l , q u a s i - s t a t i o n a r y F o u r i e r a n a l y s i s has been applied„ S i n c e the t h e o r y of t h e s e ac l e a d networks i s dev e l o p e d from a t r a n s i e n t r e s p o n s e v i e w - p o i n t , c o n s i d e r b r i e f l y the t r a n s i e n t r e s p o n s e o f a dc l e a d network t o u n i t - s t e p i n p u t . I f a u n i t - s t e p v o l t a g e i s ap- p l i e d a t the i n p u t , the o u t p u t w i l l s u d d e n l y r i s e t o an i n i - t i a l v a l u e , f o l l o w e d by a decay t o a s t e a d y s t a t e v a l u e as the condenser c h a r g e s . F o r an ac l e a d network, i f the i n p u t i s a s u d d e n l y a p p l i e d u n i t a m p l i t u d e s i n e wave, the envelope of the o u t p u t w i l l s uddenly r i s e t o an i n i t i a l v a l u e f o l l o w - ed by decay t o steady s t a t e v a l u e . T h i s i s i l l u s t r a t e d i n F i g . 2. Input Output a. DC Network b. AC Network F i g . 2.1 Lead Network Response t o a U n i t - S t e p I n put R e f e r r i n g t o Fig„ 2.1, the f o l l o w i n g parameters a r e d e f i n e d w h i c h c h a r a c t e r i z e the l i m i t i n g d e s c r i b i n g f u n c t i o n . These a r e : « - v i ( 0 ) oi A ~ P * 2 o l P G = I o ( 0 ) and 2.2 v l ( 0 ) T, the time c o n s t a n t a s s o c i a t e d w i t h the r e s p o n s e t o a s t e p i n p u t . v"^(t) r e p r e s e n t s the envelope o f the s i g - n a l v ( t ) . Then "a" and "T" may be used t o d e f i n e an e q u i - v a l e n t dc l e a d network (see Appendix I ) . The s c h e m a t i c may be w r i t t e n f o r the s i g n a l f u n d a m e n t a l as shown i n F i g . 2.2 v ^ ( t ) - > G > Y ( s )  1 1 a, T -> v Q ( t ) F i g . 2.2 Complete E q u i v a l e n t C i r c u i t f o r an AC Lead Network To account f o r h a l f or f u l l wave o p e r a t i o n and t h e p o s s i b l e use o f t r a n s f o r m e r s , the T r a n s f o r m a t i o n R a t i o "G" has been i n t r o d u c e d . 5 I n o r d e r t o d e t e r m i n e "a" and "T", c o n s i d e r F i g . 2 . 3 , which shows a t y p i c a l v o l t a g e o u t p u t a p p e a r i n g a c r o s s a l o a d r e s i s t o r r Q . The t e r m i n a l s of the condenser a r e s y n c h r o n o u s l y r e v e r s e d by a d i o d e b r i d g e so t h a t the c u r r e n t e n t e r i n g C i s u n i d i r e c t i o n a l . Both V 0 and V c a r e t a k e n t o be q u a s i - s t a t i o n - v c +1 i — + v. C v. F i g . 2.3. T y p i c a l Output V o l t a g e a r y s i g n a l s . One may w r i t e t h e q u a s i - s t a t i o n a r y average c u r - rent i n t o the condenser a s , rTT f fi- ICt) = 1 J i d0 = l j | _ v c ( t)+sin0| d 0 = 1 TT o TT o( r 0 / n I f the change i n v c per cycle i s s m a l l , then c : (see F i g . 2.k below) "LUC + 2 r o r o 2M w i t h At = 2/T w c R e w r i t i n g t h u s : A vc = I A t c I t can be seen t h a t l i m A vc _ dv. w c v - 7 ° ° At d t 2.5 and l i m I = ~ v c + 2TT 2 .6 w c v _ ' "^o v c ( t ) A t ^ i F i g . 2.h A p p r o x i m a t i o n of Condenser V o l t a g e 2.3 The approximate waveform used i n the a n a l y s i s i s shown i n s o l i d l i n e s , w h i l e the a c t u a l waveform i s shown i n d o t t e d l i n e s . The p r e c e e d i n g e q u a t i o n s may be s o l v e d f o r v c g i v i n g an e x p o n e n t i a l f u n c t i o n w i t h time c o n s t a n t T. I n o r d e r t o o b t a i n t h e o t h e r two parameters a and G, c o n s i d e r a q u a s i - s t a t i o n a r y F o u r i e r a n a l y s i s o f the o u t p u t v o l t a g e . From F i g o 2.3, i t i s seen t h a t IT f 2 J T v j ( t ) = 1 / 2 v osin0 d0 = 1 J -v c+sin0Jsin0 d0 . jTJo n °\ I From the d e f i n i t i o n s o f " a " and "G", t h e s e parameters a r e e a s i l y o b t a i n e d . W i t h r e f e r e n c e t o F i g * 2.2, the complete e q u i v a l e n t c i r c u i t has been d e t e r m i n e d w i t h the a i d o f t h e l i m i t i n g d e s c r i b i n g f u n c t i o n . F o r d e s i g n p u r p o s e s , i t i s c o n v e n i e n t t o have an e q u i v a l e n t dc network r e s e m b l i n g as c l o s e l y as p o s s i b l e the a c t u a l c i r c u i t . C o n s i d e r F i g . 2.5. F o r t h i s c i r c u i t one G 1 o b t a i n s a time c o n s t a n t o f T' = R e C 9 w i t h 1 = 1 + 1_, and ^ R e r s r o r s ^ the f a c t o r a = r s + r o F i g . 2.5 DC Lead Network The p r o c e d u r e f o r d e t e r m i n i n g an e q u i v a l e n t c i r c u i t i s : ( i ) Equate l o a d r e s i s t o r s . ( i i ) Shunt the condenser w i t h a r e s i s t o r r s t o account f o r the a t t e n u a t i o n e f f e c t s . ( i i i ) W i t h r s i n c l u d e d compute an e q u i v a l e n t r e - s i s t a n c e R e t h r o u g h w h i c h C might d i s c h a r g e w i t h t ime c o n s t a n t T*. To f i n d C , equate T' w i t h the time c o n s t a n t o f the ac network. 3.0 C i r c u i t 1. The f i r s t d e m o d u l a t o r - l e a d network t o be d i s c u s s e d i s the 6-diode g a t e . A complete d e s c r i p t i o n may be found i n Appendix I I , R e f e r e n c e 1. B r i e f l y , w i t h r e f e r e n c e t o F i g . 3»2 the c o n t r o l v o l t a g e VR b l o c k s the d i o d e s D l and D2 so t h a t c u r r e n t i s passed t h r o u g h t h e network. The i n p u t v o l t a g e w i l l t h e n appear a t t h e o u t p u t . F o r a b e t t e r u n d e r s t a n d i n g o f the o p e r a t i o n , the i n p u t , o u t p u t and c o n t r o l waveforms a r e shown i n F i g . 3.1 Input vi Output v D 7 C o n t r o l VR F i g . 3«1 V o l t a g e Waveforms C o n s i d e r the c i r c u i t shown i n F i g . 3»2. The f o l - l o w i n g e q u a t i o n s , v a l i d o n l y d u r i n g the p e r i o d t h a t D l and D2 a r e b l o c k e d , may be w r i t t e n f o r F i g . 3«2: r — +V 1VR i ^ l r x —K) — T - H D l " _L I V„ i x = V - v o + 3Z sr. r i . ~ 3 i 2 = v o + v *2 = h+ A 3 3.1 3.2 3.3 D2 +VR S u b t r a c t i n g 3«1 from 3°2 y i e l d s -V i 2 - i x 2v F i g . 3-2 Schematic f o r t h e D e f i n i t i o n o f the Q u a n t i t i e s t o be used i n the A n a l y s i s . __o r l 3 .h c o n t ' d . S u b s t i t u t i n g 3 i n t o 3«3 g i v e s "3 A l s o U = i>|+ 2 V o 1-5 3 . 6 8 S u b s t i t u t i n g 3»6 i n t o 3 . 5 y i e l d s *3 = v ^ ( l + 2) = V Q I 0 r r-i 1 o 1 1 w i t h Then 1 R o vrt =-v. 1 + 2 3 . 7 3 . 8 3 . 9 The problem i s t o d e t e r m i n e v c as a f u n c t i o n o f time w i t h the c o n d i t i o n v c ( 0 ) = 0 and a u n i t s t e p a p p l i e d a t the i n p u t . C o n s i d e r the c i r c u i t shown i n F i g . 3 .3 v r + F i g . 3 . 3 C i r c u i t O p e r a t i o n - v c+v^ c o n d u c t i o n = 0 n o n - c o n d u c t i o n The o u t p u t v o l t a g e waveform v c a t any time t i s shown i n F i g . 3.h v. PTT F i g . 3 . ^ Output V o l t a g e Waveform. Then a c c o r d i n g t o s e c t i o n 2.0, one may w r i t e t h e q u a s i - s t a - t i o n a r y average c u r r e n t i n t o t h e condenser as i ( t ) =. 3.10 JL (2\d0 = 1_ f 2 7 Z V d0 = 1_ r ^ v c ( t ) + s i n 0 T d 0 = - v c ( t ) 4 i _ ZKJo 4 27CJ R 2^Ji R J 2R 7TR The d i f f e r e n t i a l e q u a t i o n f o r the condenser v o l t a g e i s d t C 2RC V 7TV The s o l u t i o n o f 3»H» w i t h the c o n d i t i o n v c ( t = 0 ) = 0 , i s v c = 2' l - e x p ( d , ) 3.12 I n o r d e r t o o b t a i n the parameters " a " and "G" mentioned i n s e c t i o n 2.0, c o n s i d e r the f u n d a m e n t a l of the o u t p u t v o l t a g e by a p p l y i n g F o u r i e r a n a l y s i s t o t h e q u a s i - s t a t i o n a r y s i g n a l r2TT rn v£(t) = 1J0 v osin0 d0 ^ 1J0 (-v c+sin0)sin0 d0 = -2V C + i 3 7T 77 JT . ? Then one o b t a i n s , v»(0)_ , e , , _ v ' ( 0 ) , . a = o = ir, = 5«3 and G = o = i = 1 THZZJ F% VTTol 1 2 The time c o n s t a n t o f the l e a d network i s T = 2RC, w h i c h i s apparent from 3«12. F o r d e s i g n p u r p o s e s , i t i s c o n v e n i e n t t o have as s i m i l a r a network c o n f i g u r a t i o n as p o s s i b l e t o t h e ac n e t - work. From the procedure o u t l i n e d i n s e c t i o n 2„0, the c i r - c u i t o f F i g . 3„5 may be used. F o r t h i s c i r c u i t the time C 1 c o n s t a n t i s T' = R e C w i t h 1=1+1 and the parameter R R e R R s L A / V W V S a i s R + R s . R s ^ —H F i g . 3.5 DC Lead Network E q u a t i n g p a r a m e t e r s , one o b t a i n s R =1-8 R+R s F or R s = If. 3 R E q u a t i n g time c o n s t a n t s i t i s found t h a t C 1 = C ft2 Then the f o l l o w i n g e q u i v a l e n t dc l e a d network may be drawn as shown i n F i g . 3 ° 6 Y(s) = 1 (1 + aTsN a 1 + T s ; a = 5.3 T = 2RC [-/WV~i M-.3R F i g . 3 . 6 E q u i v a l e n t DC Lead Network W.O C i r c u i t 2 . The f i r s t published reference t o demodulator-lead networks i s a b r i e f d e s c r i p t i o n given by Diprose (Appendix I I , Reference 2 ) . The network i s shown below. r 2 : l F i g . k„l Diprose's C i r c u i t . The assumptions f o r network ope r a t i o n are: the voltage l e v e l of the reference s i g n a l (REF) alone determines whether the diodes conduct or not; and, i d e a l transformers and i d e a l diodes are employed. I f the turns r a t i o s are conveniently chosen as i n d i c a t e d one may w r i t e the f o l l o w i n g ac equivalent c i r c u i t , F i g . h.2. c w i t h r i = 2 ——-j |————>s~-[ J-**—-—•o r 2 ~ 1 C F i g . k02 Transformer Equi v a l e n t C i r c u i t . For ease i n a n a l y s i s consider the unbalanced equivalent c i r - c u i t under va r i o u s operating c o n d i t i o n s . The network has two operating s t a t e s determined by whether the diodes are conduc- t i n g or non-conducting. Consider t h e c i r c u i t w i t h the diodes non-conducting and an a r b i t r a r y voltage v c on the condenser as shown i n F i g . k03>. F i g . <̂,3 C i r c u i t f o r Diodes N o n - c o n d u c t i n g . As, v c i s s l o w l y v a r y i n g , the o u t p u t p r i m a r y t e r m i n a l s a r e e s s e n t i a l l y a s h o r t c i r c u i t and hence the c u r r e n t f l o w i n g i s g i v e n by, *d = l i e »+3i R The i n d u c t a n c e o f the p r i m a r y o f the o u t p u t t r a n s f o r m e r i s assumed t o be so l a r g e t h a t t h i s c u r r e n t remains unchanged d u r i n g t h i s c o n d u c t i o n c y c l e o f the d i o d e s . The v o l t a g e - v c must now appear a c r o s s the o u t p u t . F i g . h.h shows the e q u i v a l e n t c i r c u i t when the d i - odes a r e c o n d u c t i n g . F i g . k0h C i r c u i t f o r Diodes C o n d u c t i n g . Then c l e a r l y , v. = i c R L + v c lt.2 Hence . / i c = e i ~ v c Diodes c o n d u c t i n g ^.3 R L Diodes n o n - c o n d u c t i n g Now i = i c + i d h.k Then as e x p l a i n e d i n s e c t i o n 2.0, one may w r i t e d v c = I k.5 d t C W i t h the averages t a k e n over t h e a p p r o p r i a t e r a n g e , i may be foun d . 2K 1 = 1 . I i d0 = 1 2nJ° 2n w i t h 0 = wt rK r2rc K d 0 + J i d d0 if.6 S u b s t i t u t i n g and i n t e g r a t i n g y i e l d s , i = -v c(_I- + I) + 1 2R L R rt*i Now l e t 1 = _ 1 _ + 1 R e 2R L R R e C **.7 >f.8 ^-9 and T The s o l u t i o n o f i f . 5, w i t h the i n i t i a l c o n d i t i o n t h a t v c - 0 f o r t=0, i s : v c = 1 R e 1-exp-t I u,.io TTRL 1 R7c / W r i t i n g t h e o u t p u t v o l t a g e v G as a q u a s i - s t a t i o n a r y F o u r i e r s e r i e s and s o l v i n g f o r the a m p l i t u d e o f the fundamental V Q , one o b t a i n s , as v Q = 2 e Q , v ' ( t ) = 1 - l ^ c ^.11 K A l e a d network may be d e s c r i b e d by parameters w h i c h may be o b t a i n e d from the r e s p o n s e t o a u n i t - s t e p i n p u t . These parameters a r e " a " , "T" and "G". U s i n g e q u a t i o n k„1 and e q u a t i o n i f . 2 one o b t a i n s : T = R eC a = v£(t=0) _ o(oo) 1-h R - " i ^2(,2RL+R) v o ( t = 0 ) = 1 = 1 v!(0) 1 C o n s i d e r the c i r c u i t shown i n F i g . k.5 Hi C 2R, L V W A V v W J R„ F i g . E q u i v a l e n t C i r c u i t . F o l l o w i n g t h e pr o c e d u r e s o u t l i n e d a t t h e end o f s e c t i o n 2.0, one o b t a i n s from ( i i ) , l-k_ R = 2 R L 7t 2(2R L+R) R S' + 2RL S o l v i n g f o r 1_ g i v e s : 1 = TC^-h + ? t 2 R< R S mr m From ( i i i ) , i t i s apparent 1 = 1 + 1 R« R S 2R L Hence e q u a t i n g time c o n s t a n t s , y i e l d s C_ = CJ_ R| R E C 1 = C7T2 The e q u i v a l e n t c i r c u i t i s shown i n F i g . k.6 ^R '^VWvVW—> 8 H 2RT 1 F i g . i+,6 E q u i v a l e n t C i r c u i t f o r D i p r o s e ' s Network. Then ' a = 1 1- kR 7f*(2RL+R) and T = 2 R L R 2R£+R w i t h Y ( s ) = 1 (1 + aTs) a (1 + Ts) 5.0 C i r c u i t 3. Continuing the study of ac lead networks one f i n d s i n Appendix I I , Reference 3 the f o l l o w i n g c i r c u i t : REF Output F i g . 5.1 Schematic of Lyons' Network The s i n u s o i d a l input i s i n synchronism w i t h the r e - ference voltage which c o n t r o l s the diode bri d g e . I t i s assum- ed that the voltage l e v e l of the s i g n a l i s s m a l l compared to reference v o l t a g e . In F i g . 5 . 1 , consider the s t a t e of network when D l and D2 are blocked. For an input s i g n a l r e l a t e d to the sec- ondary the f o l l o w i n g c i r c u i t a p p l i e s : D3 + } ~ ~ L C b a. b. F i g . 5.2 S i g n a l C i r c u i t of the Secondary f o r State 1. N e g l e c t i n g the source impedance, the c i r c u i t o f F i g . 5.2 (a) may be reduced t o F i g . 5.2 ( b ) . Next c o n s i d e r t h e s t a t e of network when D3 and D*+ are n o n - c o n d u c t i n g . Then the i n p u t s i g n a l r e l a t e d t o t h e secondary sees the f o l l o w i n g c i r c u i t . a. b. F i g . 5.3 S i g n a l C i r c u i t o f the Secondary f o r S t a t e 2. U s i n g s i m i l a r assumptions as b e f o r e t h i s c i r c u i t may be r e - duced as shown. Then f o r e i t h e r s t a t e o f the network t h e same c i r c u i t i s a p p l i c a b l e f o r the i n p u t s i g n a l . R e f e r r i n g c i r c u i t components t o t h e p r i m a r y , one o b t a i n s , w i t h the understanding t h a t the condenser must be s w i t c h e d e v e r y h a l f c y c l e , the c i r c u i t o f F i g . 5.*+: F i g . 5 .^ H a l f - C y c l e E q u i v a l e n t C i r c u i t . The o u t p u t r e s p o n s e t o a u n i t s t e p .input i s shown i n F i g . 5 . 5 . v Q ( t ) t= 0 F i g . 5.5 Output V o l t a g e Waveform, 18 I f t h e r e i s an a r b i t r a r y v o l t a g e v on the conden- s e r , as shown i n F i g . 5.6, w h i c h i s s l o w l y v a r y i n g , one may w r i t e : i d - 2. 5.1 F i g . 5.6 DC D i s c h a r g e C i r c u i t . The t r a n s f o r m e r t u r n s r a t i o i s c o n v e n i e n t l y chosen so t h a t a r a t i o o f 1:1 e x i s t s between the p r i m a r y and one- h a l f o f the s e c o n d a r y . Then the c i r c u i t of F i g . 5.7 may be used i n the a n a l y s i s o f ac s i g n a l s . W i t h r e f e r e n c e t o F i g . 5.*+ 1 2 r 3 = R 3 r 2 = R 2 C = C, F i g . 5.7 AC S i g n a l , C i r c u i t , L e t v x , v Q and v c r e f e r t o i n p u t , o u t p u t and condenser v o l - t a g e s , r e s p e c t i v e l y , as i n d i c a t e d . The f o l l o w i n g e q u a t i o n s can be w r i t t e n : v l = ^ c + v c + ( i c + ^ r 3 v l = r 2 i 2 + ( i c + *2> r 3 v o = ( i c + i 2 ) r 3 5.2 5.3 E l i m i n a t i n g i 2 from equations 5.2 and 5.3 gives v l " v c = M r l + r2> +J-1_<Y 1 - i c r 3 ) r 3 + " p 2 According to s e c t i o n 2 . 0 , i t i s p o s s i b l e to w r i t e the f o l l o w i n g d i f f e r e n t i a l equation, d v c = i 5.6 dt C where 1 = 1 / i d0 and i = i - i , . To compute 1, take a ft JO c d . time average of 5.1 and 5 .5, n o t i n g that v x = s i n 0 and v c i s s lowly v a r y i n g . Now def i n e the f o l l o w i n g equivalent r e - s i s t a n c e s , R i = r-. + r 2 r 3 r 2 + r 3 and Then 1 = 1 + 1 Ri rI i = - + 2 R £2- 1 + r T T R i 7r ( r 2 ̂  '"3 S u b s t i t u t i n g 5.9 i n t o 5.6 g i v e s , d v c = 1 dt RC -vc+ Z v2 JR. ?r ( r 2 + P 3 T R l 5.7 5.8 5.9 5.10 S o l v i n g t h i s d i f f e r e n t i a l equation g i v e s , w i t h c o n d i t i o n v c ( 0 ) | l - e x p j - t v„(t) = 2 r 2 R ' c TT ( r 0 + T^FR1 • : 5.11 2" • L 3 / " 1 " 1 ' To obtain the network parameters, consider the en- velope v 0 ( t ) of the output v Q ( t ) . With 5 .2 , 5.3 and 5.*+? one may o b t a i n , v 0 | ' l + 1+1 h v W l + 1) V 5.12 L r l r 2 r 3 J r l r 2 r l In order to examine the output fundamental, apply Fourier ana- lysis to 5 .12. v',(t) = r ^ ( r l * r 2 ) r l r 2 + r l r 3 + r 2 r 3 \*t r l r 2 + r x r 3 + r 2 r 3 Substituting for v c with 5.11 yields, Vc.lt) = r ^ ( r x + r 2 ) - 8 i £ a R r l r 2 + r l r 3 + r 2 r 3 ( r 2 + r 3 ) 2 (Ri)2 X one has 1-exp - t RC* 5.13. Then according to the definitions of section 2 . 0 , G a G = - 3 r 2 r l r i r 2 + r i r 3 + r 2 r 3 ' ̂ T f ^ R S ^ T 2 r J , ( r l r 2 ) r l r 2 + r l r 3 + r 2 r 3 5.1^ 5.15 To determine an equivalent ci r c u i t by the procedure of section 2 . 0 , consider Fig. 5 . 8 . r ^ v v v v v w w - i r r s . 2 C - / W W W W V Fig. 5 .8 Equivalent Circuit for Lyons'Network, To account for the attenuation effects, the resistor r g is introduced into network configuration. To find r g , the f o l - lowing relations are obtained and solved: ?1 r 3 ( r l + r 2 ) 8 r 2 r 3 R - r^rx+ r g + r g ) r 1 r 2 + r 1 r 3 + r 2 r 3 K2 ( l y h ^ ( R ^ r l r 2 + r l V r 2 r 3 + r s C i 2 + r 3 > The above e q u a t i o n g i v e s r „ = 8 r-i 1 • #2 r l r 2 + r l r 3 + r 2 r 3 To e v a l u a t e the e q u i v a l e n t condenser compute the d i s c h a r g e r e s i s t a n c e R e o f F i g . 5.8 and then o b t a i n T e = R e G ' = RC. From t h i s p rocedure one o b t a i n s , C = C ( l + R_) r s F o r the c i r c u i t of F i g . 5.8 one has Y ( s ) = G 1 + aTs a 1 + Ts w i t h T = RC , f rom 5.13 1, f rom 5.1̂ a G, f rom 5.15 6.0 C i r c u i t h. F i g . 6.1 shows the c i r c u i t diagram of a demodulator lead network mentioned i n Appendix I I , Reference k. o— v i R- R- D l i t 3 Ri Ri R, ^ C F i g . 6.1 Weiss and Levenstein's Network, The f o l l o w i n g assumptions are made i n the a n a l y s i s : ( 1 ) i d e a l diodes are employed, ( 2 ) transformers are i d e a l , ( 3 ) the reference voltage controls' the diode bridge. This synchronously switches the t e r m i n a l s cd and ab so that the current i n t o the condenser i s u n i d i r e c t i o n a l . For a con- s t a n t amplitude sine input a s t a t e of e q u i l i b r i u m f s eventual- l y reached whereby the charge added during a c y c l e i s e x a c t l y balanced by discharge of the condenser through various r e s i s - t o r s . Hence the f o l l o w i n g waveforms, as shown i n F i g . 6.2, occur at the output f o r t h i s ac step i n p u t . C\j ft./ 'o t= 0 t= Oft F i g . 6.2 Output Voltage Waveforms. Then the following c i r c u i t i n F i g . 6 .3 may be con- sidered for analysis with the understanding that the conden- ser i s to be switched every half cycle. — I H — - A A A A A A A / W -2R3. R. F i g . 6.3 Half-Cycle Equivalent C i r c u i t , For analysis consider the c i r c u i t shown i n F i g . 6.M-, v. F i g . 6.h AC Charge C i r c u i t . The following equations may be written: v i = V ( i c + V R2 v i = ^ t + ( i c i 2 ) R 2 v~ 1=1 +1 R T 2R3 2RL 6.1 6.2 6.3 6A ( i c + i 2 ) R 2 Eliminating i 2 from 6.1 and 6.2 gives, v+ - v = i P R 2 + R 2 ( V i " ^ ^ ^ 1 c c d Rt + R2 : Consider F i g . 6.5 for the c i r c u i t describing the dc discharge. From the description of operation i t i s ap- parent that for a voltage v c on the condenser, discharge w i l l occur through the equivalent resistance R. Rif R: R 3 . 1 =_!.+ 1_ + 1_ R 2 R i j . 2 R 3 R 2 R, F i g . 6,5 DC Discharge C i r c u i t . As v c i s s l o w l y v a r y i n g , one has Ld c R~ 6.5 Now according to s e c t i o n 2.0, i t i s p o s s i b l e to w r i t e the f o l l o w i n g d i f f e r e n t i a l equation, d v c = i 6.6 dT" c r2fc K- where i = 1 / i.d0 , and i = i c + i d . To cqmpute i , take the time average of 6.*+ and 6.5. By s o l v i n g these equations f o r I c and I d , one o b t a i n s , I = -2V C + 2 R i 7T R 2+R t S u b s t i t u t i n g 6.7 i n t o 6.6 y i e l d s , -2v c + 2 f t ft Ri+17 dv~ -£ = 1_ 6 . 7 6.8 dt RC Sol v i n g t h i s d i f f e r e n t i a l equation g i v e s , w i t h the i n i t i a l c o n d i t i o n that v c ( 0 ) = 0 , v c ( t ) = l . S t Il-exp z21 71 Rt̂ R2- RC 6.9 I f v ^ ( t ) i s the fundamental of the output, then by applying F o u r i e r a n a l y s i s to the output w i t h the assumption that v c i s slowly v a r y i n g , one o b t a i n s , iv'(t) = 1 / v osin0 d0 = 1 / (-v„+sin0) sin0 d0 6.10 V ' ( t ) = 1 - K Hence v ^ ( t ) "^c 6.11 7T Using the d e f i n i t i o n s of a and G as given i n s e c t i o n 2.0, one f i n d s , 1 ,'. 6.12 a = XoiPJ = W R t + R 2 G = VQ(0)= 1 = 1 vjW I 6.13 In s e c t i o n 2.0, a procedure f o r determining an equivalent c i r c u i t has been o u t l i n e d . Consider the c i r c u i t of F i g . 6.6. C I — V W A / S M / V — 1 r _ R, F i g . 6.6 Equivalent C i r c u i t . To account f o r at t e n u a t i o n e f f e c t s and employing equation 6.12, one o b t a i n s , 1 - ifRt = R2 6.lh ^•2(R t +R 2) r s+R 2 S o l v i n g f o r 1_. g i v e s , 1 = 1K2 "+ |TC£ - 1. 1_ R 2 6.15 Associate r e w i t h the time constant T e of the equivalent c i r - c u i t , i . e . , T e = r e C . By i n s p e c t i o n one o b t a i n s , 1_ = !_ + !_ r e r s R2 6.16 S u b s t i t u t i n g 6.15 i n t o 6.16 g i v e s , " 1_ = ^ " 2 a + 1_\ r_ *T VR t R 2 ; 6.17 Equating time constants y i e l d s , w i t h the a i d of F i g . 6.6 and equation 6.9, C' = C R 2 r e S u b s t i t u t i n g 6.17 i n t o 6.18, one obtains C - o f 6.18 6.19 Hence i t i s apparent that an equivalent c i r c u i t i s given by F i g . 6.7. 1T C ^ R t 1X2 X J+ R ' R- F i g . 6.7 DC Equiv a l e n t Lead Network. Then w i t h Y(s) =• 1 1 + aTs a 1 + Ts a = 1 l 3 R t 7 r2(R 2+R t) T = RC 2 7.0 C i r c u i t 5. °ne f i n d s i n Appendix I I , Reference 3» "the network of F i g . 7.1. I t w i l l be shown that t h i s network i s an ac l a g network. F i g . 7.1 Schematic of Lyons' Lag Network. The operation of a s i m i l a r type network has been described i n s e c t i o n 5.0. Consider the s t a t e of the network when D.l and D2 are conducting. Then the charge c i r c u i t f o r the condenser i s shown i n F i g . 7 .2. v o F i g . 7.2 Condenser Charge C i r c u i t . The diode bridge synchronously reverses the terminals of the condenser so that the current e n t e r i n g i s u n i d i r e c t i o n a l . The f o l l o w i n g equations may be w r i t t e n : V J - V Q = i 7.1 r l v o ' v c = 1 c 2 r b 7.2 2 v o = ^ " i c 2 ^ r 2 7.3 The equations may be solved f o r i C 2 i n terms of v^ and v c g i v i n g , I c 2 = A v i - B v c 7. 1* w i t h A = 2 r 2 2 r x r 2 + ( rx+r 2)r b B = 2 ( r l + r 2 ^ 2r^T2 T̂r̂ +T̂ 7r̂ Next s o l v i n g equations 7 . 1 , 7 . 2 , and 7.3 f o r v Q i n terms of v^ and v c g i v e s , v o = A l v i + B l v c 7 . 5 w i t h Ax = r 2 r b r l r b + 2 r l r 2 + r 2 r b Bx = 2 r l r 2 r l r b + 2 r l r 2 + r 2 r b The discharge c i r c u i t of the condenser i s shown i n F i g . 7.3 where i t i s assumed that inductances are n e g l i g i b l e f o r the q u a s i - s t a t i o n a r y type discharge. + 1 f t , v ' c l £b 2 F i g . 7.3 Condenser Discharge C i r c u i t . Then i c l = 2 v c 7.6 r b To o b t a i n v c as a f u n c t i o n of time, one may w r i t e according to s e c t i o n 2 . 0 , d v c - i dt C where I i s the average net charging c u r r e n t . Then I = I c 2 - I c l 7.7 Now I c 2 may be computed from 7.*+ where v c i s slowly v a r y i n g and = sin0, hence I = 2A - Bv c - 2vc The d i f f e r e n t i a l equation becomes + 1 v. = 2A 7.8 dt ¥ ftC W l t h i = i ( B + 2_) 7 ' 9 r b Equation 7.8 may be solved w i t h the c o n d i t i o n v c ( 0 ) = 0 , g i v i n g v c ( t ) = 2AT 7 t c 1-exp (-t) T 7.10 To o b t a i n the envelope of the output apply F o u r i e r a n a l y s i s to the q u a s i - s t a t i o n a r y output s i g n a l i n 7.5. 1 v 0 ( t ) = 1 / (Aisin0 + B!V c)sin0 d0 2 nJo Hence, v^Ct) = A-.+ hB1 v c ( t ) 7.11 Then from the d e f i n i t i o n s of a and G one obtains 1 8 ABiT , _ I = fenfCJ 5 and G = A l . F o l l o w i n g the procedure o u t l i n e d i n s e c t i o n 2,0, the c i r c u i t shown i n F i g . 7 A i s obtained. la r 2 T F i g . DC Eq u i v a l e n t C i r c u i t . To account f o r the steady-state a t t e n u a t i o n of a dc step i n p u t , one f i n d s , w i t h the a i d of 7.11, that ^ = f f ^ I 7.12 w i t h R-L = r l r 2 r i + r 2 and 1 = 2(1+%) K r D To o b t a i n c ' i t i s necessary to compute the d i s - charge r e s i s t a n c e of F i g . 7.1+. 1 = 1 + * = 1 + B 6 2 As the time constant i s given by 7.9, the f o l l o w i n g r e l a t i o n holds C* = C , 1 , = CK ( l + r b + R l ) 7 * 1 3 R e ( B + f - ) 2 r s *7 8.0 Feedback Methods of Modifying the C i r c u i t Parameters 8.1 V a r i a b l e Conduction Angle By va r y i n g the conduction angle of the diode bridge i t i s p o s s i b l e to vary a, t and G. By s u i t a b l e arrangement of the c o n t r o l voltages of F i g . 3.2, the output voltage w i l l appear as shown i n F i g . 8.1, where the conduc- t i o n angle i s 7C- 2 e><. I ^ 0 v 2fr V ( t ) o / v \ 0( 7t-(X t = o F i g . 8.1 Voltage Output f o r a V a r i a b l e Conduction Angle. During conduction the c i r c u i t i s as shown i n F i g . 8.2, v ^ t ) < v Q ( t ) o- F i g . 8.2 Conduction C i r c u i t . Then by s e c t i o n 2.0, one may v/rite d v c = I dt C Now i t Is apparent that 1 = A J ( ~ v c + s ln0) d0 = _JL_ 27TR -be 27TR -vc(/T-20C) + 2cosOC 8.1 S u b s t i t u t i n g i n t o the d i f f e r e n t i a l equation and s o l v i n g y i e l d s , w i t h v c(0) =0, v c ( t ) = 2cos<X ( 1 _ e x p l-(^-2cX ) t ] \ 8.2 c ?T-2<X I1 P [ 2RC 7C J I Now applying F o u r i e r a n a l y s i s to the q u a s i - s t a t i o n a r y output, (ZK r7r-« v ' ( t ) =1 I v ( t ) sin0 d0 = _1 / (-V c+sin0)sin0d0 Tt o rrJ« ] = 1 K -2v ccos<x + (K-2<\) +sin2(X 8.3 The parameters a, T and G may be obtained from d e f i n i t i o n s i n s e c t i o n 2.0. From 8.2, the time constant i s T = 2 R C - Z L 7T-20< 8.if a = 8.5 The f a c t o r a i s obtained from 8 . 3 , w i t h ICL(0) = _ I _ v^£oT 1 -p (X 8 c o s 2 w l t n % = (7C-2« )(7t-2(X +sin2<X ) The transformation r a t i o i s , then, r _ v l ( 0 ) _ -, " TjTo) ~W W-2« +sln2<*> To o b t a i n an equivalen c i r c u i t f o r these parameters consider F i g . 8 . 3 , and the procedure o u t l i n e d i n s e c t i o n 2.0. 8.6 C I—*vwww—1 Ro F i g . 8.3 DC Equivalent Lead Network. Ac c o r d i n g l y , one may w r i t e , R + R< R a = 1- Hence (3 By computing the time constant of the c i r c u i t of F i g . 8 . 3 , and equating t h i s w i t h 8.U, one o b t a i n s , C = 2C 7t p(7r_2c< ) This network o f f e r s the p o s s i b i l i t y of v a r y i n g the maximum a v a i l a b l e phase s h i f t 0m and the frequency at which i t occurs by c o n t r o l l i n g the conduction angle. These parameters as a f u n c t i o n of "a" are given i n Appendix I by 1.1^ and 1.16 . 8.2 Servo Mixing Network In Appendix I I , Reference 3, a network i s shown ca- pable of employing an ac s i g n a l w i t h dc. feedback to o b t a i n , as d e s i r e d , e i t h e r a lead or l a g e f f e c t . The schematic i s given i n F i g , S.k. The diode bridge o p e r a t i o n has been des- c r i b e d i n s e c t i o n 7.0. O - A A / V W V W V Ji r l -o v Q ' output 1:2 1 (-AWvVvWv C r HI- F i g . 8.>+ Servo Mixing Network w i t h AC Input and DC Feedback. V t FEEDBACK Considering only the dc feedback v o l t a g e , the c i r - c u i t i n F i g . 8.5 i s analysed. i r-i c l "V/ - c l F i g . 8.5 DC Feedback C i r c u i t . The f o l l o w i n g equations may be w r i t t e n . V + V c = i £b Hence Hence 1 = 2 ( v + v ) v c = ( 1 c r i ) r i = i + 12 c l r 8.7 8.8 8.9 8.10 S u b s t i t u t i n g 8.9 i n t o 8.10 gives i c l =|I + ( f + I) v c r b r b r c 8.11 Consider only the ac i n p u t , the c i r c u i t , shown i n F i g . 8 .6, i s analysed. F i g . 8.6 AC Input C i r c u i t . 3M- The f o l l o w i n g equations may be w r i t t e n . v c = Ci - i c 2 ) r i = I i - Y o V o - V r » = i r b = r b (v< - v j 2 2 ! V w i - "O' E l i m i n a t i n g the currents and s o l v i n g f o r v Q gives _ v c v = 2rn + V 1 r b o 2 f i + r b x 1 2 r x + r D 8.12 8 .13 8.15 Then i C 2 may be obtained i n terms of v^ and v c from the above equations. 2 2 1 i C 2 = v i ( 2 r x + r b ^ - v c ( 2 r i + r b + 8.16 In order to obtain the net charging current i G of the condenser, one has, combining 8.16 and 8.11, i C = *c2 - - i d 8.17 = A v ± - Bv c - 2V 8.18 r b w i t h A = 2 8.19 2*1 + r b B = 2 ) - 8.20 r ^ r 2r^+rb Now i t i s p o s s i b l e to w r i t e , according to s e c t i o n 2.0 d v c _ l c To compute i c , consider 8.18. I t i s assumed that V and v c are slowly v a r y i n g and that Vj[ = s i n 0 and V i s a dc step v o l t a g e Hence one o b t a i n s , L = A 2„ r > v „ 2V 8.21 C ~ V i - BV C - ? - S u b s t i t u t i o n i n t o the d i f f e r e n t i a l equation g i v e s , dv c I K d t " + f V ° ~ T 8.22 w i t h 1 _ B 8.23 T ~ C and K = I I .Ŝ V, - 8.2»+. S o l v i n g f o r v c w i t h v c ( 0 ) = 0 gives v _ ( t ) = K ( l - exp - t ) 8.25 Now applying F o u r i e r a n a l y s i s to q u a s i - s t a t i o n a r y output g i v e s , i v i ( t ) = 1 lv 0(.t) s i n 0 d0 v'(t) = h r-,Avrt+ A r b V-o - i c _ i 8.26 With the aid of 8.2k and 8.26, one may write 8.26 as I 8 A 2 r l -+ Ar 7 r B r D T J - - E X P T ; 8.27 i 8.27 This network offers the possibility of obtaining opti- mum response with a controlling feedback voltage as the overall effect of the network may be either lead- or lag- compensation depending on the feedback. From 8.27 one sees that v^Ct) i s composed of two parts: a contribution from the ac signal input and a contribu- tion from the dc signal feedback. This suggests the equivalent circuit of Fig, 8.7 . V, Network 1 Network 2 Adder o Fig. 8.7 Equivalent Network for the Servo Mixing Network. To obtain Network 1 consider that part of v^(t) which contains Vj_. Following the procedure for determining equivalent networks in section 2,0, consider Fig. 8.8. 3 7 Fig, 8 . 8 Network 1 , This network must respond to a step input of amplitude v\ as V i 8 A 2r 1 ri - t ^ A r b - (1-exp —) + TC2B To account f o r a t t e n u a t i o n at " t " =<XJ, r g must s a t i s f y the f o l - lowing equation: r l + r b + r s f t 2 1 B 8 . 2 8 S o l v i n g t h i s equation g i v e s , = 8 l r s B d - ^ J ) Computing the time constant of Network 1 'and equating t h i s w i t h 8 . 2 3 gives C = C -g Next consider the c o n t r i b u t i o n of v ' ( t ) a s s o c i a t e d o w i t h the step feedback V . The l a g - l i k e response suggest the F i g . 8 . 9 as an equivalent c i r c u i t f o r Network 2 . O-AAAAAAWV r l C" r T . F i g . 8 . 9 Network 2 . 38 This network must respond to a step input of amplitude V as V 8Arx _ t ^Bp^(l-exp fpr). To account f o r a t t e n u a t i o n at t , the f o l l o w i n g r e l a t i o n must h o l d : s 8Ar- Hence r s = r]_ i j R - t i - ^ - ) - 1 Computing the time constant of Network 2 and equating w i t h 8„23 g i v e s , C« = KC f r b + l f r c \2" 9,0 D i s c u s s i o n of the V a l i d i t y of the Eq u i v a l e n t C i r c u i t . Consider the network shown i n F i g . 9.1. A u n i t amp- l i t u d e sine^-wave input v^ i s a p p l i e d at t = 0. v ± '<\ > R F i g . 9.1 Schematic f o r C i r c u i t 1. The operation of the gate G i s such that the input to the con- denser i s a half-wave sine s i g n a l . During the non-conduction pe r i o d of the gate, the condenser input t e r m i n a l i s considered open c i r c u i t . During, the conduction p e r i o d of the gate, the s i g n a l i s assumed to be generated from a zero impedance source, During the conduction p e r i o d the f o l l o w i n g d i f f e r e n - t i a l equation i s v a l i d . v c + C R l£r = s i n wt 9.1 a t The s o l u t i o n of 9.1 i s v c = A n exp + P s i n wt + Qcos wt where T = RC, P = 1 0 _ 1 n "1 + (tuCR)^ ~ 1 +. (uJ T ) 2 ' Q = - »^CR. - = -">T 1 + (u/CR) 2 1 H<o T ) 2 ' and A n i s to be determined f o r each conduction i n t e r v a l . As v e ( 0 ) = 0, then A„ = U.CR • = u>.T" c o r + ( u ) C R ) 2 1 +( UJ T ) 2 For J£jb = 1, v c ( l ) = A Q Let a = e x p r - f t % KM T J 1 + exp(̂ r) During the i n t e r v a l = 1 to w t = 2, there i s no change i n v c . Then, matching boundary c o n d i t i o n s gives A 0 ( l + a ) = A 2 -A Q Hence A 2 = A Q (2+a) Now f o r "J t = 3, v c ( 3 ) = A 2a + A Q S i m i l a r matching may be done f o r other times, and the r e s u l t s a r e , r e l a t i n g to A 0, v c ( l ) = A Q (1+a) v c ( 3 ) = A 0 (l+2a+a 2) v c ( 5 ) = A Q (l+2a+2a2+a3) I n general f o r ^ t = (2n+l), v c ( 2 n+l) =AD i ^ J [ ( l - a n ++ Let x = K vol Then v c ( 2 n+l) = | ( 1 . a n + 1 ) [i - x 2 ( 0 . 0 1 7 9 9 ) + x^O.OOOW,. Due to the nature of the ope r a t i o n of the gate G, then v c ( 2 n+l) = v c (2n+2) From the a n a l y s i s of c i r c u i t 1 i n . s e c t i o n 3 one - t < 2RC' o b t a i n s , v e ( t ) = g I 1 - e x p ( ^ r ) To compare the two r e s u l t s set t = (2n+2)7~~ . Using the d e f i - n i t i o n s of a = exp(-^-T^) and T = RC, one obtains •e<*> = a (1 - a n + 1 ) In comparison one has, v e(2n+2) 0 , u, , . • ^ - ( j ) = 1-x 2(0.01799)+ x4-CO-.000^3*0 . f. c Then i t i s apparent that l i m v c ( 2 t l + 2 ) _ . o vcTtT That i s , the r e s u l t s from e i t h e r approach are i d e n t i c a l i n the l i m i t . Now from the d e f i n i t i o n of x, l i m = l i m as x = ft . x -5» - o W - * - o « u)T The l i m i t i n g case f o r u; , the c a r r i e r frequency, approaching i n f i n i t y , i s equivalent to x approaching zero. Since the e r r o r term enters q u a d r a t i c a l l y , the l i m i t i n g e q uivalent c i r c u i t w i l l be v a l i d f o r the usual range of o p e r a t i o n i n servo-systems, i . e , 10.0 Measurement o f Phase and Gain C h a r a c t e r i s t i c s o f C i r c u i t 1. The t h e o r y o f the c i r c u i t 1 shown i n F i g . 3.2 has been developed u s i n g F o u r i e r a n a l y s i s . P h y s i c a l r e s u l t s o f F o u r i e r a n a l y s i s may be o b t a i n e d by employing the time - a v e r a g e o u t p u t o f a m u l t i p l i e r . The system used i s shown i n F i g , 10.1. >+00 v l M V o v 2 Network F i g . 10.1 Schematic f o r D e t e r m i n a t i o n o f M u l t i p l i e r P r o p o r t i o n a l i t y C o n s t a n t F o r c a l i b r a t i o n t h e f o l l o w i n g waveforms were employed: v ^ ^fOO cps v o l t a g e o f l ^ . l U peak v o l t s v 2 *+00 cps v a r i a b l e - a m p l i t u d e square-wave. The a c t u a l waveforms a r e shown i n F i g . 10.2. a. I n p u t Waveforms b. Output Waveforms F i g , 10.2 M u l t i p l i e r Waveforms. The m u l t i p l i e r d e f l e c t i o n i s p r o p o r t i o n a l t o the dc l e v e l o f the o u t p u t . f* Output = 1 A A sin0 d0 = A 1 A 2 2KJo~1 2 ~ W ~ Now the m u l t i p l i e r ' s dc output V Q i s p r o p o r t i o n a l to the time average of the product of the two i n p u t s . Hence the propor- t i o n a l i t y constant Ic i s defined by V 0 = E AxA 2 Tt By s u i t a b l e measurements i t was found that =0.06^25 •(T-^-^r) "•*(•• 579) \ 2 A „ • -2A{ K t J f y j 2(l*f.l*f): The measurement of phase-gain c h a r a c t e r i s t i c s i s achieved by the system shown i n F i g . 10,3. A modulated c a r - r i e r i s generated by the synchro TI which i s d r i v e n by a v a r i - able speed motor at the r a t e <^s. The modulated s i g n a l i s ap- p l i e d to a second synchro T2 which i s used as a manual phase s h i f t e r to vary the phase of the envelope of the s i g n a l . This phase-shifted modulated s i g n a l i s a p p l i e d to the network. The output of the network i s a p p l i e d to one of the m u l t i p l i e r ' s i n p u t s . A known comparison s i g n a l generated by a r e s o l v e r i s a p p l i e d to the second m u l t i p l i e r i n p u t . The time average of the product of the inputs i s recorded by the galvonometer, F i g . 10.3. Block Diagram of Measurement System. For F i g . 10.3, the f o l l o w i n g terms are define d : TI Synchro Transformer M E l e c t r o n i c M u l t i p l i e r T2 Synchro Transformer G Galvonometer R Resolver w g modulation frequency Consider the f o l l o w i n g arrangements of m u l t i p l i e r inputs shown i n F i g . 10.h. Resolver v l V 0 1 M System v 2 G Galvonometer M E l e c t r o n i c M u l t i p l i e r | v l c = V-^cos w/gt s i n u)Qt l v l s = V ^ i n u J g t s i n u c t v 2 = V 2 s i n (u; gt+(X)sin t u c t F i g . 10.k M u l t i p l i e r Schematic f o r Gain-Phase Determinations In the d e f i n i t i o n of v 2 , CX i s the combined phase s h i f t i n t r o - duced by the network and the v a r i a b l e transformer, i . e . , 0( = 0 (network) +V( transformer) Let V o c = k v i c v 2 = k V^cos w s t s i n c o c t V 2 s i n ( u ; s t +o( )sino»ct = k V 1V 2(sinc< cos* 1 u; gt + cos6(cosu»st s i n u j g t J s i n ^ u i Q t With s u i t a b l e sum and d i f f e r e n c e angle formulae, one o b t a i n s , l i n < * i - J j j s i n ( u l e ~ "V* + 'sin(t*> c+u? s)t| d(u> et) V o c = * V g s i n - = k V,V 0sin« 1 . 1.2.7C = k ViVosinoc 1 2 27C 5 ' * Then V Q C i s made zero by choosing <X.= 0, i . e . , 0 = —Y Employing v-^, i t may be shown s i m i l a r l y V o S = k V-jVocosa With W •= o , one may solve f o r V 2, ob t a i n i n g For computation of the gain of the network, the peak output amplitude V"2 must be compared w i t h that peak s i g n a l V i which would pass through the system i n the absence of any com- pensating network. I f the input s i g n a l has a peak V, then as G = one has ][i = The gain of the system i s V Y = V 2 = 2 V 2 = 8 V O S V i V k vxv The experimental s t u d i e s involved measurement of the gain-phase c h a r a c t e r i s t i c s as a f u n c t i o n of s i g n a l frequency. The c a l c u l a t e d network parameters a and T provided the g a i n - phase c h a r a c t e r i s t i c s of the l i m i t i n g e q u i v a l e n t c i r c u i t . A comparative p l o t of r e s u l t s i s shown i n F i g . 10 . 5 . For C i r c u i t 1, a i s f i x e d at 5.3 so that experimental s t u d i e s involved v a r i a t i o n of T. The values of T chosen were 7.25 msecs and *+.21 msecs. These correspond to frequencies of maximum phase s h i f t o f 9.6 cps and 16.7 cps r e s p e c t i v e l y . he Y ( s ) R l R 2 E q u i v a l e n t C i r c u i t T 1 R T Radians F i g . 10.5 11.0 C o n c l u s i o n s . By c o n s i d e r i n g the l i m i t i n g case as ^ c > ^ , a s i m p l i f i e d t h e o r y was d e v e l o p e d t o o b t a i n network parameters from the t r a n s i e n t response t o a s t e p i n p u t . W i t h the a i d o f the p a r a m e t e r s , l i m i t i n g e q u i v a l e n t c i r c u i t s have been ob- t a i n e d whose q u a l i t a t i v e u n d e r s t a n d i n g i s immediate. Study has been done t o d e t e r m i n e the range of p r a c t i c a l r e p r e s e n - t a t i o n of the l i m i t i n g e q u i v a l e n t c i r c u i t . A n a l y s i s p r e d i c t e d and experiment c o n f i r m e d t h a t the l i m i t i n g c i r c u i t was adequat f o r the u s u a l case i n s e r v o - s y s t e m s where ^ s < 1 I t has been shown t h a t b a s i c l e a d - o r lag-compen- s a t i o n may be a c h i e v e d w i t h s e v e r a l d i f f e r e n t systems. F u r t h e r s t u d y o f compensating networks has r e v e a l e d t h a t , i n s p e c i a l c a s e s , network parameters a r e c o n t r o l l a b l e by s u i t a b l e f e e dback. W i t h the development o f s u i t a b l e e q u i v a l e n t c i r c u i t s c a r r i e r - s y s t e m compensation u t i l i z i n g demodulator networks can be r e a l i z e d g i v i n g wide a p p l i c a t i o n . For n o n - l i n e a r systems where a d j u s t a b l e g ain-phase c h a r a c t e r i s t i c s might b e s t a c h i e v e s p e c i f i c a t i o n s , c i r c u i t s e mploying f e e d b a c k - c o n t r o l l e d p a r a - meters would appear t o i n t r o d u c e a new f i e l d of i n t e r e s t i n g p o s s i b i l i t i e s . Appendix I . DC Lead Network. Consider the network below. r. - v w w v v w • v w w v w v Fig..1.1 DC Lead Network, as w i t h The t r a n s f e r f u n c t i o n f o r t h i s network may be w r i t t e n Y(s) = G o 1 + Tjs T l T. 1 + T 2S ~ C^r2rh + r 3 rt+) r 2 + r 3 + Tk = c ( r 1 r 2 r i f + r - ^ r u ̂  r 2 r ^ ) r 1 ( r 2 + r ^ + T̂ .) + ( r ^ + r^)r,2 = r x ( r 2 + r 3 + ruJ r l ( r 2 + r 3 + V + ( r 3 + V r 2 1.1 1.2 1.3 Then i t i s apparent that 3 parameters w i l l be s u f f i c i e n t to sp e c i f y gain and phase c h a r a c t e r i s t i c s . Of s p e c i a l i n t e r e s t i n the network response to a u n i t - step input because of the p a r t i c u l a r development of the theory i n the body of the t h e s i s . Then v j (s) = G Q (1+T-^s) 1 , i s the i (l+T sT s Laplace Transform of the output voltage v£ . * 2 P / In the us u a l manner the transform may be i n v e r t e d to y i e l d v i ( t ) = G 0 l - ( l - T 1 ) e x p - t ~2 1 Then f o r t = 0 i n 1.5, one has, v j ( 0 ) = GQ_1 1 2 1.5 1.6 and for t =»*> in 1.5, v1(<x») - Gc 1.7 Combining 1.6 and 1.7 yields, vx(0) o V-t(e*>) To O * 1.8 For the network i t is often necessary to know the frequency of maximum phase shift and the amount of phase shift at the maximum. From the definition of "a" in section 2.0 one has a = £l 1.9 For real frequencies, 1.1 may be written YUu>) = |Y| expUe^) = Y exp(j0) exp(;J02) 1.10 where tan0^ = u> aT 2 and tan0 2 = OJ T 2 1.11 and 0 = 0 X - 0 2 1.12 The frequency of maximum phase shift is obtained by differen- tiating 1.12. M = = o = 5 £ i . ̂ 2 d t A j d u> d uu d0. 1.13 The derivatives of 1.13 may be obtained from I.11. These equa- tions yield the following result, aTgCos 2^ = T 2cos 20 2 The various cosines may be obtained from I.11, so that one has, or, I.Hf To obtain the amount of phase shift consider tan0 = tan($1-^2) Then f o r the maximum phase s h i f t , i t may be shown that tan0 = L±*=D 1.15 m \/a 2 An equivalent form of 1.15 i s s i n 0 m = a T } 1.16 m a + l From the t r a n s f e r f u n c t i o n i t i s apparent that there are three choices to be made i n order to s p e c i f y the phase and amplitude c h a r a c t e r i s t i c s . These choices may be made i n a v a r i e t y of ways depending upon circumstance. For example they could be chosen on the network's response to a u n i t - s t e p i n p u t . In t h i s case the parameters might be the output amplitude i n i - t i a l l y , the output amplitude f i n a l l y , and the time constant f o r the decay to the steady s t a t e . Another bas i s of choice a r i s e s i n s t a b i l i t y problems where concern might be the frequency of maximum phase s h i f t and the amount of phase s h i f t at tha t par- t i c u l a r frequency. I t should be noted that i n F i g . 1.1 that n f may be eli m i n a t e d but the c i r c u i t would r e t a i n i t s performance charac- t e r i s t i c s be appropriate m o d i f i c a t i o n of r 2 and r ^ . Then, e s s e n t i a l l y , there are four elements to be chosen. The concept of three independent choices has been made p r e v i o u s l y , w h i l e the e x t r a choice or degree of freedom determines the impedance l e v e l s . For some of the c i r c u i t s discussed i n the t h e s i s , F i g . 1.1 needs m o d i f i c a t i o n . The f i r s t case i s f o r r\+ = . Hence Y(s) = r l 1 + Cs ( r 2 + r 3 ) [r 1 + r 2 T ( l > C s ( r ^ r i r 2 H r 1 + r 2 ' As a l l the important r e l a t i o n s have been derived g e n e r a l l y , nothing s i g n i f i c a n t a r i s e s i n t h i s s i t u a t i o n . = oo and r ^ = 0. Then Of very p a r t i c u l a r i n t e r e s t i s the case of r ^ >. Y ( s ) = £ l ^ 1 + Csrg = 1 r 1 + r 2 1 + Cs r j r s - 2- r l + R 2 * Now i t i s apparent that G Q = 1 and T-̂  = aT 2 . Then the a parameter a may be checked or determined by measurement. I t i s apparent f o r t h i s case that the network r e q u i r e s only two parameters f o r d e f i n i t i o n of gain-phase c h a r a c t e r i s t i c s . Appendix I I . References. 1. Millman, J . , and Puckett, J.R., "Accurate Linear B i d i r e c t i o n a l Gate", Proc. IRE. V o l . if3, Ro. 1 January 1955, pp. 29-37. 2. Diprose, K.V., "Discussion of D. M o r r i s ' s 'A Theore t l c a l and Experimental Method of Modulation A n a l y s i s f o r the Design of AC Servo Systems'"., T u s t i n , A., ed., Automatic and Manual C o n t r o l . London, Butterworth!s: S c i e n t i f i c P u b l i c a t i o n s , 1952, p. 536. 3. Lyons, L.F., "Wide Band AC Rate Networks", Conven- t i o n Record IRE. 1955. pt. 10, p.. .173, *f. Weiss, G., and Levenstein, H., "A-C Servos", T r u x a l J.G., C o n t r o l Engineers' Handbook. Toronto, McGraw H i l l , 1958, p. 6-61.. 5. Bohn, E.V.., Lectures of E.E. 563, (Servomechanisms) U n i v e r s i t y of B r i t i s h Columbia, 1958-59.

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