UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Demodulator compensating networks Pritchard, John Robert Gilbertson 1959

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1959_A7 P7 D3.pdf [ 2.73MB ]
Metadata
JSON: 1.0105075.json
JSON-LD: 1.0105075+ld.json
RDF/XML (Pretty): 1.0105075.xml
RDF/JSON: 1.0105075+rdf.json
Turtle: 1.0105075+rdf-turtle.txt
N-Triples: 1.0105075+rdf-ntriples.txt
Original Record: 1.0105075 +original-record.json
Full Text
1.0105075.txt
Citation
1.0105075.ris

Full Text

DEMODULATOR COMPENSATING NETWORKS by JOHN ROBERT GILBERTSON PRITCHARD B . A . S c , U n i v e r s i t y of B r i t i s h Columbia,  1957  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE i n t h e Department of Electrical  We a c c e p t t h i s  Engineering  t h e s i s as conforming  to the  standards 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 Master  of Applied Science  The U n i v e r s i t y o f B r i t i s h  April,  1959  Columbia  ABSTRACT r This  t h e s i s deals  lead-lag networks.  with  t h e s t u d y o f some d e m o d u l a t o r  Specifically  a n a l y s i s and d e s i g n ,  t h e problem has i n v o l v e d  a c c o m p a n i e d by e x p e r i m e n t a l  verification  o f a new a p p r o a c h t o t h e 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 n e t w o r k s f o r a p p l i c a t i o n i n a c servomechanisms„ Analysis  h a s b e e n made o f 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 physical layout  but operating  o n t h e same b a s i c p r i n c i p l e .  By  computing t h e parameters which d e s c r i b e  of  t h e p a r t i c u l a r network, an e q u i v a l e n t  obtained,,  This  for the l i m i t i n g  transfer function  the step  transfer function i s  i s the describing  case of i n f i n i t e  response  function  carrier-to-signal-frequency  ratio. E x p e r i m e n t a l w o r k was done w i t h a n network, capable of generating lated  signals.  be  sinusoidal-modu-  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, centred were o b t a i n e d .  low-frequency  electro-mechanical  a t a f r e q u e n c y o f UOO  Since only  represented exactly  i n the l i m i t  by a d e s c r i b i n g  ^ c W s  cycles per second, can the network  function,  experimental  and  a n a l y t i c r e s u l t s f o r t h e n e t w o r k were compared t o c h e c k  the  limiting describing  f u n c t i o n as a p r a c t i c a l  ii  representation.  In p r e s e n t i n g t h i s t h e s i s the requirements  f o r an advanced degree at the U n i v e r s i t y  o f B r i t i s h Columbia, it  i n p a r t i a l f u l f i l m e n t of  I agree t h a t the L i b r a r y s h a l l make  f r e e l y a v a i l a b l e f o r reference  and study.  agree that permission f o r extensive  I  further  copying of t h i s  thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s .  It  that copying or p u b l i c a t i o n of t h i s t h e s i s  i s understood, for f i n a n c i a l  g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n .  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.  Table of  Title  Contents  Page  i  Abstract List  of  11 Illustrations  iv  Acknowledgment  vi  1.0  Introduction  1  2.0  Methods o f  3  3.0  Circuit  1  7  k.O  Circuit  2  11  5.0  Circuit  3  16  6.0  Circuit  k  22  7.0  Circuit  5  27  8.0  F e e d b a c k Methods o f M o d i f y i n g  Analysis  the  Circuit  Parameters ,8.1 8.2  9.0 10.0  Variable  Conduction Angle  Servo-Mixing  D i s c u s s i o n of Measurement o f Circuit  11.0  31  Network  the V a l i d i t y  of  the E q u i v a l e n t  Phase and G a i n C h a r a c t e r i s t i c s  1  I  DC L e a d N e t w o r k  Appendix  II  References  33  C i r c u i t 39 of k2 k?  Conclusions  Appendix  31  52 ill  L i s t of I l l u s t r a t i o n s Fig„ 2  0  „2  1  o3 h 5 0  0  Fig. 3 „ 1 o2 .3 „H „5 „6 F i g . k.l ,2 ,5 ,6 Fig.  5°1 ,2 .3 oh 05 6 07 „8 0  Fig. 6 . 1 2 .•3 „*+ .5 .6 o7 0  Fig  0  7»1 ,2 03 oh  Fig. 8 . 1 2 „3 »*+ 0  Lead Network Response t o a U n i t - S t e p I n p u t 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 T y p i c a l Output V o l t a g e A p p r o x i m a t i n g of Condenser V o l t a g e DC Lead Network  h  V o l t a g e Wave Forms Schematic f o r the D e f i n i t i o n of t h e 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 C i r c u i t Operation Output V o l t a g e Waveform DC Lead Network E q u i v a l e n t DC Lead Network  7 7 8 8 9 1 0  Diprose's C i r c u i t Transformer Equivalent C i r c u i t C i r c u i t f o r Diode Non-Conducting C i r c u i t f o r Diode C o n d u c t i n g Equivalent C i r c u i t 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  1 1 1 1 1 1  h 5 5 6  1 1 2 2 2 2  Schematic of Lyons' Network 16 S i g n a l C i r c u i t of t h e Secondary f o r S t a t e 1 16 S i g n a l C i r c u i t of t h e Secondary f o r S t a t e 2 1 7 Half-Cycle Equivalent C i r c u i t 17 Output V o l t a g e Waveforms 1 8 DC D i s c h a r g e C i r c u i t 1 8 AC S i g n a l C i r c u i t 1 8 E q u i v a l e n t C i r c u i t f o r Lyons' Network 2 0 Weiss and L e v e n s t e i n ' s Network Output V o l t a g e Waveforms Half-Cycle Equivalent C i r c u i t AC Charge C i r c u i t DC D i s c h a r g e C i r c u i t Equivalent Circuit DC E q u i v a l e n t Lead Network  2 2 2 2 2 3 2 3 2h 2 ? 2 6  Schematic o f Lyons' Lag Network Condenser Charge C i r c u i t Condenser D i s c h a r g e C i r c u i t DC E q u i v a l e n t C i r c u i t  2 2 2 3  V o l t a g e Output f o r a V a r i a b l e Conduction Angle Conduction C i r c u i t DC E q u i v a l e n t Lead Network S e r v o - M i x i n g Network w i t h AC Input and DC Feedback iv  1  7 7 8 0  3 1 3 1 3 2 3 3  Fig.  8o5 6 „7 0  08 09 Fig.  9.1  Fig.  10ol .2 „3 A  L i s t o f I l l u s t r a t i o n s cont'do DC F e e d b a c k C i r c u i t ' AC I n p u t C i r c u i t 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 Network 1 Network 2 Schematic f o r C i r c u i t  1  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 Constant M u l t i p l i e r Waveforms B l o c k Diagram o f Measurement System M u l t i p l i e r Schematic f o r Gain-Phase Determinations  33 3*+ 36 37 37 39 k2 U-2 *+3  Appendix I Fig.  1.1  DC L e a d N e t w o r k  *+8  v  ACKNOWLEDGMENT  The  author expresses h i s a p p r e c i a t i o n f o r t h e generous  a s s i s t a n c e a n d g u i d a n c e g i v e n by Dr„ E. V, B o h n , h i s s u p e r v i s o r .  The  a u t h o r wishes t o acknowledge  t h e numerous o c c a s i o n s  on w h i c h Members, S t a f f a n d G r a d u a t e S t u d e n t s o f t h e D e p a r t m e n t 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 t h e development and  understanding of t h i s  The its  project„  author thanks t h e N a t i o n a l Research C o u n c i l f o r  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 a n d 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.  vi  1.0  Introduction I n servomechanism d e s i g n , compensation i s o f t e n  n e c e s s a r y t o meet system s p e c i f i c a t i o n s .  I n open-loop s y s -  tems, problems may a r i s e i n a c h i e v i n g a d e s i r e d response.  transient  I n c l o s e d - l o o p systems, s t a b i l i t y margins may  r e q u i r e gain-phase compensation.  The compensating networks  commonly used a r e l e a d - l a g n e t w o r k s . I n servomechanisms, i n f o r m a t i o n may be t r a n s m i t t e d by a dc v o l t a g e l e v e l o r w i t h a modulated c a r r i e r  signal.  F o r dc l e a d n e t w o r k s , a n a l y t i c r e s u l t s a r e 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 t h e p o i n t o f v i e w o f a t r a n s i e n t response t o a u n i t - s t e p i n p u t , t h e v a l u e o f t h e o u t p u t f o r t = 0, t h e v a l u e o f t h e o u t p u t f o r t = <><> , t h e decay time c o n s t a n t ; 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 , t h e maximum amount o f a v a i l a b l e phase s h i f t , and t h e f r e q u e n c y a t w h i c h t h i s maximum o c c u r s .  Depending on t h e p a r t i c u l a r n e t -  work, one has two o r t h r e e Independent c h o i c e s o f d e s i g n parameters.  C o n v e n t i o n a l l y , compensation i n ac systems has  been a c h i e v e d by e i t h e r o f two methods.  The f i r s t o f t h e s e  i s t h e p a s s i v e system, w i t h compensation b e i n g a c h i e v e d by the R-L-C elements.  The r e a l i z a t i o n o f t h e ac network i s  o b t a i n e d by t h e u s u a l low-pass-to-band-pass t r a n 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 o b t a i n e d by employing p a r a l l e l T and b r i d g e - T networks.  These networks a r e tuned t o t h e  c a r r i e r f r e q u e n c y and hence t h e s e c h a r a c t e r i s t i c s a r e q u i t e s e n s i t i v e t o any v a r i a t i o n i n t h i s c a r r i e r f r e q u e n c y . e r a l l y f o r these networks,  Gen-  the attenuation a t 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 t h e p h a s e  c h a r a c t e r i s t i c s are  t i v e over a v e r y l i m i t e d bandwidth.  The  effec-  second method o f  a c h i e v i n g a c c o m p e n s a t i o n r e q u i r e s a d e m o d u l a t o r , a dc n e t w o r k , and a m o d u l a t o r . the f i l t e r e d chopped  Here t h e e n v e l o p e i s peak d e t e c t e d ,  s i g n a l i s a p p l i e d t o t h e dc n e t w o r k and  to r e i n t r o d u c e the c a r r i e r  a network of t h i s u e n c y and  type are the independence  t h e l a r g e amount o f p h a s e  main disadvantage of t h i s network ter which l i m i t s circuit  frequency.  then  The m e r i t s o f  of c a r r i e r  shift available.  freqThe  i s the presence of a  t h e amount o f a v a i l a b l e p h a s e  f i l -  l e a d and  the  complexity involved. This t h e s i s concerns i t s e l f w i t h the  s t u d y o f some d e m o d u l a t o r have a r e s e m b l a n c e type systems.  to the  l e a d - l a g networks.  detailed These  networks  demodulator-dc-network-modulator-  However t h e y o p e r a t e on a d i f f e r e n t  principle  i n t h a t the demodulation 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 o u t 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  b e e n d e v e l o p e d b a s e d on t h e s t u d y o f t h e t r a n s i e n t to a step input.  response  Parameters d e s c r i b i n g the t r a n s i e n t r e s i  p o n s e h a v e b e e n f o u n d and t h e s e a r e u s e d t o o b t a i n an valent c i r c u i t which i s v a l i d signal-frequency r a t i o . various demodulator the d i f f i c u l t y  only f o r i n f i n i t e  Previous analytic  equi-  carrier-to-  treatment of  networks has been s u p e r f i c i a l because  i n obtaining equivalent c i r c u i t s .  Using  t i n g f u n c t i o n s , t h e a u t h o r h a s o b t a i n e d e q u i v a l e n t dc E x p e r i m e n t a l w o r k was  directed  quacy of t h e s e n e t w o r k s .  towards d e t e r m i n i n g the  of limi-  networks. ade-  2.0  Methods o f A n a l y s i s For  general  3  a non-linear  system i t i s impossible  d e f i n i t i o n of a transfer function.  of networks, i t i s c o n v e n i e n t t o d e f i n e f u n c t i o n or d e s c r i b i n g bing  to give  a  For c e r t a i n types  an e q u i v a l e n t  f u n c t i o n of the system.  transfer  The d e s c r i -  f u n c t i o n method u s e s t h e f u n d a m e n t a l component o f t h e  output response t o a s i n u s o i d a l input  s i g n a l , to define  transfer function.  function plays  As t h e d e s c r i b i n g  a  the r o l e  of the t r a n s f e r f u n c t i o n f o r the fundamental frequency, a suitable equivalent For obtain  linear  c a n be  a describing  f u n c t i o n which i s v a l i d  except i n the l i -  The s u b s e q u e n t a n a l y s i s o f t h e s e  n e t w o r k s r e l i e s upon t h e q u a s i - s t a t i o n a r y nals.  obtained.  demodulator-type networks i t i s not p o s s i b l e t o  a s t h e r a t i o ^ c . y oo°  mit  circuit  nature of the s i g -  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  varying  b u t may  be assumed c o n s t a n t f o r a s h o r t  period  time without appreciable  error.  behavior of a p a r t i c u l a r  signal, quasi-stationary  a n a l y s i s h a s b e e n applied„  briefly  network t o u n i t - s t e p  Fourier  S i n c e t h e t h e o r y o f t h e s e ac  I f a unit-step voltage  p l i e d at the input, the output w i l l tial the is  value,  followed  of the output w i l l ed  lead  i s ap-  s u d d e n l y r i s e t o an  ini-  by a d e c a y t o a s t e a d y s t a t e v a l u e a s  condenser charges. a suddenly applied  view-  t h e t r a n s i e n t r e s p o n s e o f a dc  input.  of  In order to estimate the  lead networks i s developed from a t r a n s i e n t response point, consider  time-  F o r an ac l e a d n e t w o r k , i f t h e i n p u t u n i t a m p l i t u d e s i n e wave, t h e e n v e l o p e  suddenly r i s e  t o an i n i t i a l  by d e c a y t o s t e a d y s t a t e v a l u e .  This  value  follow-  is illustrated i n  Fig.  2.  Input  Output  a. DC  Network  Fig.  b. AC  2.1  Lead Network Response Input  R e f e r r i n g t o Fig„ 2.1, defined  Network to a  Unit-Step  the f o l l o w i n g parameters are  which c h a r a c t e r i z e the l i m i t i n g  describing  function.  These a r e : « A  ~  v  i  (  0  oi  )  P  2ol  *  P  G = Io(0) vl(0)  and  2.2  T, t h e t i m e c o n s t a n t to a step nal  v(t).  input.  associated  v"^(t) r e p r e s e n t s  T h e n " a " and "T" may  w i t h the response  the envelope of the  be u s e d t o d e f i n e a n  v a l e n t dc l e a d n e t w o r k ( s e e A p p e n d i x I ) . The  sig-  equi-  schematic  may  be w r i t t e n f o r t h e s i g n a l f u n d a m e n t a l a s shown i n F i g . 2.2  v  ^(t)  -> Fig.  2.2  G  >  Y(s) a, T  Complete E q u i v a l e n t a n AC L e a d N e t w o r k  1  ->  Circuit for  1  v (t) Q  To a c c o u n t f o r h a l f o r f u l l wave o p e r a t i o n a n d t h e p o s s i b l e use of transformers, has  been  the Transformation  Ratio  "G"  introduced. In order  w h i c h shows a t y p i c a l v o l t a g e resistor r reversed  Q  .  output appearing  across  The t e r m i n a l s o f t h e c o n d e n s e r a r e  by a d i o d e b r i d g e  unidirectional.  2.3,  t o d e t e r m i n e " a " a n d "T", c o n s i d e r F i g .  Both V  synchronously  so t h a t t h e c u r r e n t e n t e r i n g C i s  and V  0  a load  c  are taken  t o be q u a s i - s t a t i o n c i—  v  +1  + v.  v. C  2.3.  Fig. ary  signals.  T y p i c a l Output  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 a v e r a g e  r e n t i n t o the condenser rTT  ICt)  Voltage  as, f  fi-  = 1 J i d0 = l j | _ v ( t ) + s i n 0 | TT o TT o ( r / c  d0  =  1  0  I f the change  with  in v  c  At  per c y c l e  n  "LUC + 2 o o r  2.3  r  i s s m a l l , then  : ( s e e F i g . 2.k  c  2M  below)  = 2/T w c  Rewriting thus:  A c t v  A  I t c a n be s e e n w  c  v  v - 7 ° °  At  dt  v (t) c  Fig.  2.5  = I c  that  A c _ dv.  lim  and  l i m I = ~ c + 2TT c v _ ' "^o v  2.6  w  At^i 2.h  cur-  A p p r o x i m a t i o n of Condenser  Voltage  5  The  a p p r o x i m a t e w a v e f o r m u s e d i n t h e a n a l y s i s i s shown i n  solid  l i n e s , w h i l e t h e a c t u a l w a v e f o r m i s shown i n d o t t e d  lines.  The  preceeding  an e x p o n e n t i a l  equations  may  be  solved for v  f u n c t i o n w i t h time constant  o b t a i n the other  two  p a r a m e t e r s a and  G,  T.  In order  consider  a  2.3,  From  i t i s seen t h a t  vj(t) = 1 /  2  IT  v s i n 0 d0 = 1 J n  J  F r o m t h e d e f i n i t i o n s o f " a " and e a s i l y obtained.  f  o  jT o  equivalent  to  quasi-  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 of the output v o l t a g e . Figo  giving  c  With reference  c i r c u i t has  2 J T  -v +sin0Jsin0 d0  .  c  °\  I  "G",  these  to Fig*  parameters  2.2,  the  are  complete  been d e t e r m i n e d w i t h the a i d of  the  limiting describing function. For e q u i v a l e n t dc  design purposes, i t i s convenient network resembling  actual circuit.  Consider G  as c l o s e l y as p o s s i b l e  F i g . 2.5.  1  For  s  F i g . 2.5 The  DC  this circuit  obtains  a time constant  T'  9  = R C e  ^ r  the f a c t o r  a = s r  the  of and  r  +  r  o  Lead Network  procedure f o r determining (i)  Equate l o a d  (ii)  Shunt the  an e q u i v a l e n t  With r  s  condenser w i t h a r e s i s t o r r  s  e  through which C  w i t h time constant  T*.  time constant  To  to  effects.  i n c l u d e d compute an e q u i v a l e n t  sistance R  w i t h the  circuit i s :  resistors.  account f o r the a t t e n u a t i o n (iii)  r  an  one  w i t h 1 = 1 + 1_, e s o R  ^  t o have  might find C,  o f t h e ac  re-  discharge equate  network.  T'  3.0  1.  Circuit  The f i r s t is  d e m o d u l a t o r - l e a d n e t w o r k t o be  the 6-diode gate.  A c o m p l e t e d e s c r i p t i o n may be f o u n d i n  A p p e n d i x I I , R e f e r e n c e 1.  B r i e f l y , with reference  t h e c o n t r o l v o l t a g e VR b l o c k s  then appear a t t h e output.  in Fig.  t o F i g . 3»2  t h e d i o d e s D l a n d D2 s o t h a t c u r  rent i s passed through the network.  operation,  discussed  The i n p u t v o l t a g e  For a better understanding  3.1  7  Output v  Control  D  VR  Fig.  3«1  Voltage  equations,  Waveforms  t h e c i r c u i t shown i n F i g . 3»2.  Consider  valid  only during  V  v  x  x  v  +  v  2  r  i. ~ 3  -V  F i g . 3-2 Schematic f o r the D e f i n i t i o n o f t h e Q u a n t i t i e s t o be u s e d i n the A n a l y s i s .  *2 =  The  h  +  A  3.1 3.2 3.3  3  S u b t r a c t i n g 3«1 f r o m 3°2 2v i -i 3 .h __o l 2  fol-  t h e 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 i = - o 1VR i ^ lr —K) —T-H Dl " _L I V„ i = o 3Z sr. +  D2 +VR  of the  t h e i n p u t , o u t p u t a n d c o n t r o l w a v e f o r m s a r e shown  I n p u t vi  lowing  will  x  r  cont'd.  yields  Substituting  U"3  3 i n t o = i>|+  3«3 2 V  8  gives  1-5  o  Also  3.6  Substituting  *3  3»6  = v ^ ( l + 2) ro r-i 1  R  yields VQI  =  0  1  3.7  1  + 2  1  1  with  i n t o 3.5  3.8  o  v =-v.  Then  3.9  rt  time w i t h at the  The  problem i s to determine v  the  condition  input.  v (0)  Consider the v  = 0 and  c  circuit  as  c  a function  a unit  step  shown i n F i g .  of  applied 3.3  r  + - v +v^ c  = 0  3.3  Fig. The Fig.  Circuit  conduction  non-conduction  Operation  output v o l t a g e waveform v  a t any  c  t i m e t i s shown i n  3.h PTT  v.  3.^  Fig.  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, tionary  average current  i(t)  =.JL ( \d0 ZKJo  i n t o the  = 1_  2  4  one  27CJ  f  2  7  Z  R  V  may  write  condenser d0  = 1_  2^Ji  the  quasi-sta3.10  as  r^v (t)+sin0T c  R  J  d 0 =  -  v  c  (  2R  t  )  4  i_  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 t h e c o n d e n s e r v o l t a g e i s  C  dt The  2RC  7TV  V  s o l u t i o n o f 3»H» w i t h t h e c o n d i t i o n v ( t = 0 ) = 0 , i s c  v  l-exp(d,)  = 2'  c  3.12  In order t o o b t a i n the parameters s e c t i o n 2.0,  " a " and  "G" m e n t i o n e d  c o n s i d e r the fundamental of the output 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  r2TT v£(t)  = 1J  7T  signal  rn  v s i n 0 d0 ^ 1J  0  in  o  77  obtains, v»(0)_ , a = o = ir, THZZJ F %  0  (-v +sin0)sin0 d0 = - 2 V JT c  C  + i . ?  3  Then one  The  =  e  ,  , _ v'(0) and G = o  5«3  VTTol  time c o n s t a n t of the l e a d network  similar work.  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 -  F r o m t h e p r o c e d u r e o u t l i n e d i n s e c t i o n 2„0, may C  be u s e d .  L  c o n s t a n t i s T'  A/VWV R  Fig.  3.5  s DC  S  1=1+1 R R  R  e  a  i s  ^  R  +  R s  = R C  cir-  with  e  and t h e  parameter  s  .  —H  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 R+R  s  the  For t h i s c i r c u i t the time  1  R  R  which i s  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  c u i t o f F i g . 3„5  or  i s T = 2RC,  3«12.  apparent from For  , . = i = 1 1 2  =1-8 F  s  = If. 3 R  Equating time constants i t i s found that C  1  = C ft  2  Then t h e f o l l o w i n g e q u i v a l e n t d c l e a d n e t w o r k may as shown i n F i g . 3 ° 6  Y ( s ) = 1 (1 + aTsN a 1 + Ts  ;  a =  5.3  T = 2RC [  - WV~ /  i  M-.3R Fig.  3.6  E q u i v a l e n t DC Lead Network  be d r a w n  W.O  Circuit 2 . The f i r s t p u b l i s h e d r e f e r e n c e t o d e m o d u l a t o r - l e a d  networks i s a b r i e f d e s c r i p t i o n g i v e n by D i p r o s e Reference 2 ) .  (Appendix I I ,  The network i s shown below. r :l 2  F i g . k„l  Diprose's  Circuit.  The assumptions f o r network o p e r a t i o n a r e : t h e v o l t a g e of t h e r e f e r e n c e s i g n a l  (REF) a l o n e d e t e r m i n e s whether t h e  d i o d e s conduct o r n o t ; a n d , i d e a l t r a n s f o r m e r s d i o d e s a r e employed.  level  and i d e a l  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 t h e f o l l o w i n g ac e q u i v a l e n t c i r c u i t , F i g . h.2. c  ——-j  |————>s~-[  w  i  t  h  r  J-**—-—•o  i  =  2  r2~ 1  C Fig. k2  Transformer E q u i v a l e n t  0  Circuit.  For ease i n a n a l y s i s c o n s i d e r t h e unbalanced e q u i v a l e n t c u i t under v a r i o u s o p e r a t i n g c o n d i t i o n s .  cir-  The network has two  o p e r a t i n g s t a t e s determined by whether t h e d i o d e s a r e conduct i n g or non-conducting.  Consider t h e c i r c u i t w i t h the diodes  n o n - c o n d u c t i n g and an a r b i t r a r y v o l t a g e v as shown i n F i g . k 3>. 0  c  on t h e condenser  Fig. As, v  c  ^<,3  C i r c u i t f o r Diodes  i s s l o w l y v a r y i n g , the output  essentially a short c i r c u i t  Non-conducting.  primary  terminals are  and hence t h e 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 R  »+ i 3  The i n d u c t a n c e o f t h e p r i m a r y  of the output  assumed t o be s o l a r g e t h a t t h i s  transformer i s  c u r r e n t remains unchanged  during t h i s conduction c y c l e of the diodes.  The v o l t a g e - v  c  must now a p p e a r a c r o s s t h e o u t p u t . Fig.  h.h  shows t h e e q u i v a l e n t c i r c u i t  when t h e d i -  odes a r e c o n d u c t i n g .  Fig. kh 0  Then  C i r c u i t f o r Diodes  clearly, v. = i R + v c  Hence  L  . / ic = i ~ c e  v  R  lt.2  c  Diodes  i = i  c  + i  d  conducting  ^.3  L  Diodes Now  Conducting.  non-conducting  h.k  Then a s e x p l a i n e d c dt  i n s e c t i o n 2.0, one may w r i t e  = I C  d v  k.5  With the averages taken over t h e a p p r o p r i a t e  rK  2K  found. 1=1.  I i d0 =  1  2n ° J  w i t h 0 = wt  r2rc  K  2n  r a n g e , i may be  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 (_I2R  +  c  L  I) + 1 R rt*i  **.7  Now l e t 1 = _ 1 _ + 1 R 2R R  >f.8  and  ^-9  e  The  T  R  L  e  C  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  condition that v -0 f o r c  t=0, i s : v  c  =  1  R  e  I  1-exp-t  TTRL 1  u,.io  R7c /  W r i t i n g the output voltage  v  G  as a q u a s i - s t a t i o n a r y  Fourier  s e r i e s and s o l v i n g f o r t h e a m p l i t u d e o f t h e fundamental V Q , one  o b t a i n s , as v = 2 e , v'(t) = 1-l^c Q  Q  ^.11  K  A l e a d n e t w o r k may be d e s c r i b e d may be o b t a i n e d  from the response t o a u n i t - s t e p  These p a r a m e t e r s a r e " a " , and  equation  by parameters w h i c h  "T" a n d "G".  i f . 2 one o b t a i n s :  T = R C e  a = v£(t=0) _ o(oo)  1-h R  -"i  ^2(,2R +R) L  v (t=0) o  v  !(0)  = 1 = 1 1  input.  Using equation  k„1  Consider  the c i r c u i t  Hi  shown i n F i g . k.5  C  2R, L  V W A V v W J  R„ Fig.  Equivalent  Circuit.  F o l l o w i n g t h e p r 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 f r o m ( i i ) ,  l-k_  R = L 7 t ( 2 R + R ) R ' + 2RL 2  R  2  L  S  S o l v i n g f o r 1_ g i v e s : 1 = TC^-h + ? t R mr m R<  2  S  From ( i i i ) ,  i t i s apparent  1 =1 +1 R«  R  Hence e q u a t i n g C_ R| C  1  L  time constants, CJ_ R  =  2R  S  yields  E  = C7T  2  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 2R  '^VWvVW—>  8 Fig.  i+,6  H  T  1  Equivalent Circuit  f o r Diprose's  Network.  Then '  a =  1- kR  1  7f*(2R +R) L  and  T =  2  R  L  R  2R£+R  with  Y ( s ) = 1 (1 + aTs) a (1 + Ts)  5.0  Circuit  3.  C o n t i n u i n g the s t u d y o f ac l e a d networks one f i n d s i n Appendix I I , R e f e r e n c e 3 t h e f o l l o w i n g c i r c u i t :  REF  Output  Fig.  5.1  Schematic o f Lyons' Network  The s i n u s o i d a l i n p u t i s i n s y n c h r o n i s m w i t h the r e f e r e n c e v o l t a g e which c o n t r o l s the d i o d e b r i d g e .  I t i s assum-  ed t h a t the v o l t a g e l e v e l o f the s i g n a l i s s m a l l compared t o reference voltage. In F i g . 5.1, and D2 a r e b l o c k e d .  c o n s i d e r the s t a t e o f network when D l  For an i n p u t s i g n a l r e l a t e d t o the s e c -  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  a.  Fig.  5.2  + }  ~~L  C  b  b. S i g n a l C i r c u i t of the Secondary f o r S t a t e  1.  Neglecting  the source  impedance, t h e c i r c u i t  o f F i g . 5.2 ( a )  may be r e d u c e d t o F i g . 5.2 ( b ) . Next c o n s i d e r are non-conducting.  t h e s t a t e o f n e t w o r k when D3 a n d D*+  Then t h e 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. Fig.  Using  b.  5.3  S i g n a l C i r c u i t of the Secondary f o r S t a t e 2.  s i m i l a r assumptions as b e f o r e  d u c e d a s shown. same c i r c u i t circuit  c i r c u i t may be r e -  Then f o r e i t h e r s t a t e o f t h e n e t w o r k t h e  i s a p p l i c a b l e f o r the input s i g n a l .  components t o t h e p r i m a r y ,  understanding  this  one o b t a i n s , w i t h t h e  t h a t t h e c o n d e n s e r must be s w i t c h e d  cycle, the c i r c u i t  Fig.  5.^  Referring  every  half  o f F i g . 5.*+:  Half-Cycle Equivalent  Circuit.  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 . i n p u t i s shown in Fig.  5.5.  18  v (t) Q  t=  0 Fig.  s e r , as  5.5  O u t p u t V o l t a g e Waveform,  I f t h e r e i s an  arbitrary voltage v  shown i n F i g . 5.6,  which i s slowly  on  the  varying,  condenone  may  write: id -  Fig. The  5.1  2.  5.6  DC  transformer turns r a t i o i s conveniently  so t h a t  a r a t i o o f 1:1  h a l f of  the  secondary.  used i n the  Discharge C i r c u i t .  analysis  e x i s t s between the Then the  o f ac  chosen  p r i m a r y and  circuit  one-  o f F i g . 5.7  may  be  signals. With reference to F i g . 1  r  Fig. Let vx,  v  5.7  Q  and  AC  Signal,Circuit,  v  refer to input,  c  tages, r e s p e c t i v e l y , can  be  written:  as  indicated.  3  r  2 =R  o u t p u t and The  C  3  2  =  R  =  C,  2  condenser v o l -  following  equations  5.*+  v  v  v  l  ^ c  =  l = 2 2 r  o  =  ( i  Eliminating i v  i  2  c  +  +  +  i  c  v  +  c  ( i  2  ( i  )  r  c  ^  +  *2>  +  r  r  5.2  3  5.3  3  3  from e q u a t i o n s 5.2 and 5.3  l " c = v  M l r  +  r  2> J - 1 _ < Y 3 " 2 +  where 1 = 1  +  c 3 r  )  p  to section 2.0, i t i s possible to write  the f o l l o w i n g d i f f e r e n t i a l d v  i  1  r  According  gives  equation,  5.6  c = i dt C / i d0 and i = i - i , . c d  ft JO  To compute 1, .  take a  time average o f 5.1  and 5 . 5 ,  i s slowly varying.  Now d e f i n e t h e f o l l o w i n g e q u i v a l e n t r e -  noting that v  x  = s i n 0 and v  c  sistances, R  i = r-. + 2 3 r  r +r 2  Ri  Then  i = -  r  R  3  5.8  1 = 1 + 1  and  5.7  r  I  + 2 £21 rTTRi 7r ( r + ^ '"3  S u b s t i t u t i n g 5.9 i n t o 5.6 = 1 RC  c dt d v  5.9  2  gives,  -v + Z 2 JR. ?r ( r + P T R l v  c  Solving t h i s d i f f e r e n t i a l equation v„(t) = 2 2 TT ( r +  gives, with condition v (0) c  |l-expj-t ' • : 2" • 3 " 1 " '  c  r  R  0  T^FR L /  5.10  3  2  1  1  5.11  To o b t a i n t h e network p a r a m e t e r s , c o n s i d e r v e l o p e v ( t ) o f t h e output v ( t ) . 0  Q  With 5.2,  t h e en-  5.3 and 5.*+? one  may o b t a i n , v | ' l + 1+1 h v W l + 1) V L l 2 3J l 2 l 0  r  r  r  r  r  r  5.12  In order to examine the output fundamental, apply Fourier anal y s i s to  5.12. v',(t) =  ^  r  r  Substituting  for v  (  l  r  l 2 r  * +  r  )  l 3  r  r  +  with 5.11  c  Vc.lt) = r ^ ( r + l 2 r  +  r  r  2 3  \*t l 2 r  r  r r3 +  +  r  r r3  x  2  yields,  r )  x  r  2  -  2  l 3 r  +  r  2 3  8  (r  r  R  i£a 2  + r )2 3  1-exp  -t RC*  (Ri)2  5.13.  2.0,  Then according to the d e f i n i t i o n s of section one  X  has G a  r  ir2  G = J, r  r  ( r  l 2 r  +  ir3 r r  r  2  l 2 r  +  r  3  '  +  3  r  2  r  l  ^Tf^RS^T  5.1^ 2  5.15  )  l 3 r  +  r  2 3 r  To determine an equivalent c i r c u i t by the procedure of section 2 . 0 , consider F i g .  5.8. r^vvvvvww-i  rs r  . 2 - WWWWV  C  /  F i g . 5.8  Equivalent C i r c u i t for Lyons'Network,  To account for the attenuation e f f e c t s , the r e s i s t o r r introduced into network configuration. lowing r e l a t i o n s are obtained and  g  To f i n d r , the  solved:  g  is fol-  ?1  r  3  (  r  r r 1  l 2  +  +  2  r  r r 1  8  )  3  +  r r 2  2  K  3  r  2 3 R  ( l y h ^( R ^  The above e q u a t i o n g i v e s  r^rx+rg+rg)  -  r  r„  = 8  r  l  r  2  +  r  l V  r-i  r  2  resistance R From t h i s  the e q u i v a l e n t of F i g .  e  procedure C  = C(l  5.8  one  the c i r c u i t Y(s)  with  s  of F i g . = G a  1+ 1+  T = RC, from 1, f r o m a  5.1^  G,  5.15  from  obtains,  5.8 aTs Ts 5.13  one has  r  s  C  2  i  +  l 2 r  r  e  3>  r  •  compute t h e  and t h e n o b t a i n T  + R_) r  For  condenser  +  1 #2  To e v a l u a t e  3  r  +  r  l 3 r  +  r  discharge  = R G' e  = RC.  2 3 r  6.0  Circuit  h.  F i g . 6.1 shows the c i r c u i t diagram of a demodulator l e a d network mentioned i n Appendix I I , R e f e r e n c e  k.  Dl R-  Ri  i t R-  ^  C  Ri  3  o— v  R,  i F i g . 6.1  Weiss and L e v e n s t e i n ' 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 d i o d e s a r e employed, ( 2 ) t r a n s f o r m e r s a r e ( 3 ) the r e f e r e n c e v o l t a g e c o n t r o l s ' the d i o d e b r i d g e . synchronously  ideal, This  s w i t c h e s the t e r m i n a l s cd and ab so t h a t the  c u r r e n t i n t o the condenser i s  unidirectional.  For a con-  s t a n t a m p l i t u d e s i n e i n p u t a s t a t e of e q u i l i b r i u m f s e v e n t u a l l y reached whereby the charge added d u r i n g a c y c l e i s e x a c t l y b a l a n c e d by d i s c h a r g e of the condenser t h r o u g h v a r i o u s r e s i s tors.  Hence the f o l l o w i n g waveforms, as shown i n F i g . 6.2,  occur a t the output f o r t h i s ac s t e p i n p u t .  'o  C\j t= 0 F i g . 6.2  ft./ t= Oft Output V o l t a g e Waveforms.  Then the f o l l o w i n g c i r c u i t  i n F i g . 6 . 3 may  be  s i d e r e d f o r a n a l y s i s w i t h the understanding t h a t the ser i s to be  con-  conden-  switched every h a l f c y c l e .  —IH— -AAAAAAA/W-  2R. 3  R.  Fig.  6.3  Half-Cycle  For a n a l y s i s consider  the  Equivalent  circuit  Circuit,  shown i n F i g .  6.M-,  1=1  R  T  2R3  +1  2RL  v.  6.h  Fig. The  AC Charge  f o l l o w i n g equations may  V  v  i  =  v  i  = ^ t  v~  (i  Eliminating  i  2  c  c  +  ( i  c  i  R  2  = i R P d  R  c  + i )  c  be  V  +  from 6.1  v+ - v 1  ( i  2  6.1  2  )  R  6.2  2  6.3 6.2  + 2 i Rt + R2 R  (  Consider F i g . 6.5 dc d i s c h a r g e .  written:  2  and 2  Circuit.  V  gives,  6A  "^^^ :  f o r the c i r c u i t  describing  From the d e s c r i p t i o n of o p e r a t i o n  parent that f o r a v o l t a g e  v  c  occur through the e q u i v a l e n t  i t is  the ap-  on the condenser, d i s c h a r g e w i l l resistance  R.  1 =_!.+  R  Rif R:  R  F i g . 6,5 As v  3.  2R  ij.  1_ 2 R  3  + 1_ R  2  R,  DC D i s c h a r g e  Circuit.  i s s l o w l y v a r y i n g , one has  c  L  6.5  c R~  d  Now a c c o r d i n g  t o s e c t i o n 2.0, i t i s p o s s i b l e t o  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,  c = i dT" c  6.6  d v  r2fc K  where i = 1 / i.d0  , and i = i  c  + i  the time average o f 6.*+ and 6.5. for I  and I , one o b t a i n s , I = -2V + 2 R i 7T R + R  c  d  .  To cqmpute i , t a k e  By s o l v i n g t h e s e e q u a t i o n s  d  6.7  C  2  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 , dv~£ =- 1_ - 2 v + 2 f t dt RC  6.8  c  ft Ri+17  Solving 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  condition that v ( 0 ) = 0 , c  v ( t ) = l . S t Il-exp z21 RC 71 R ^R c  t  6.9  2  I f v ^ ( t ) i s t h e fundamental o f t h e o u t p u t , then by applying F o u r i e r a n a l y s i s t o the output w i t h the assumption that v  c  i s s l o w l y v a r y i n g , one o b t a i n s ,  iv'(t) = 1 / v s i n 0 d0 = 1 / (-v„+sin0) sin0 d0  6.10  Hence  6.11  o  Vv '^((t t ) ) = 1 - " ^Kc 7T  t h e d e f i n i t i o n s o f a and G as g i v e n i n  Using  s e c t i o n 2.0, one f i n d s ,  a = XoiPJ =  W  1 R  ,'. 6.12 t R2 +  G = VQ(0)= 1 = 1  vjW  6.13  I  I n s e c t i o n 2.0, a procedure f o r d e t e r m i n i n g e q u i v a l e n t c i r c u i t has been o u t l i n e d .  Consider  an  the c i r c u i t  of F i g . 6.6. C  I — V W A / S M / V —  R,  1  r_  F i g . 6.6  Equivalent  Circuit.  To account f o r a t 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 ^•2(R R ) r +R t+  2  S o l v i n g f o r 1_. g i v e s , 1  s  6.lh 2  = 1K "+ |TC£ - 1. 1_ R 2  6.15  2  Associate r  e  w i t h t h e time c o n s t a n t T  cuit, i.e., T  of the equivalent  = r C . By i n s p e c t i o n one o b t a i n s ,  e  e  1_ = !_ + !_ e s 2  r  e  r  R  6.16  cir-  S u b s t i t u t i n g 6.15  i n t o 6.16  " 1_ = ^ " a  + 1_\  2  r_  *T R  gives,  R  V  t  6.17  ;  2  E q u a t i n g time c o n s t a n t s y i e l d s , w i t h t h e a i d o f F i g . 6.6 equation  and  6.9, C' = C 2r  R  6.18 e  S u b s t i t u t i n g 6.17 C  i n t o 6.18,  one o b t a i n s  of  -  6.19  Hence i t i s apparent t h a t an e q u i v a l e n t c i r c u i t i s g i v e n by Fig.  6.7.  1T  C  ^ t 1X2 R  X  J+  F i g . 6.7 Then  Y ( s ) =•  with  a =  1 1 + aTs 1 + Ts  a  l3Rt 2(R +R )  T = RC 2  R '  DC E q u i v a l e n t Lead Network.  1  7r  R-  2  t  7.0  Circuit  5.  °ne f i n d s i n Appendix I I , R e f e r e n c e 3» "the network of F i g . 7.1.  I t w i l l be shown t h a t t h i s network i s an ac l a g  network.  F i g . 7.1  Schematic of Lyons' Lag Network.  The o p e r a t i o n of a s i m i l a r t y p e network has been d e s c r i b e d i n s e c t i o n 5.0. C o n s i d e r the s t a t e o f the network when D.l and D2 a r e 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 d i o d e b r i d g e s y n c h r o n o u s l y r e v e r s e s the t e r m i n a l s of the condenser so t h a t the c u r r e n t 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 e q u a t i o n s may V J - V Q  v  i  7.1  l  r  v  =  be w r i t t e n :  o' c v  =  1  c2 b  7.2  r  2  o = ^  "  i  c2^  r  7.3  2  be s o l v e d f o r i 2 i n terms of v^ and  The e q u a t i o n s may  C  v  c  giving, Ic2  with  =  A  i  v  -  c  B v  7. * 1  2 2  A =  r  2 r r + (rx+r 2 )r b B = 2( l 2^ x  2  r  + r  2r^T2^Tr^+T^7r^ Next s o l v i n g e q u a t i o n s 7 . 1 , v ^ and v  and 7.3  for v  Q  i n terms of  gives,  c  v  with  7.2,  o  =  Ax =  A  l i v  2 b l b r  2  r  7.5  l c  B  v  r  r r  Bx =  +  r  +  2  l 2  r  r  +  r  2 b r  l 2 r  l b r  +  2  The d i s c h a r g e c i r c u i t  r  l 2 r  +  r  2 b r  o f the condenser  i s shown i n F i g . 7.3  where i t i s assumed t h a t i n d u c t a n c e s a r e 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 d i s c h a r g e .  + v  Fig.  7.3  1  ft,  ' cl  Condenser D i s c h a r g e  £b 2  Circuit.  Then i  c  l  =2vc r  7.6  b  To o b t a i n v according  as a f u n c t i o n o f t i m e , one may w r i t e  c  to s e c t i o n 2.0, c dt  - i C  d v  where I i s t h e average n e t c h a r g i n g I = I  Then Now I and  c  2  - I  c 2  7.7  c l  may be computed from 7.*+ where v  2A - B v  varying  h  i  ¥  i(B  =  7.8  ftC  2_)  +  r  Equation  becomes  + 1 v . = 2A  dt t  i s slowly  - 2vc  c  The d i f f e r e n t i a l e q u a t i o n  l  c  = sin0, hence  I =  W  current.  7  '  9  b  7.8 may be s o l v e d w i t h the c o n d i t i o n v ( 0 ) = 0 , c  v (t) c  = 2AT 1-exp (-t) 7t T  giving  7.10  c  To o b t a i n t h e envelope of t h e o u t p u t a p p l y  Fourier  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 output s i g n a l i n 7.5. 1 v (t) 0  =  1  2  Hence,  / (Aisin0 + B ! V ) s i n 0 d0 no c  J  v ^ C t ) = A-.+ hB  1  v (t)  7.11  c  Then from t h e d e f i n i t i o n s o f a and G one o b t a i n s 1  I  8 ABiT =  ,_  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 Fig.  DC E q u i v a l e n t  Circuit.  To account f o r the s t e a d y - s t a t e a t t e n u a t i o n o f a dc s t e p i n p u t , one f i n d s , w i t h t h e a i d o f 7.11, t h a t  ^ ff^I =  with and  R-L =  7.12  l 2 ri+r2 1 = 2(1+%) K r r  r  D  To o b t a i n c ' i t i s n e c e s s a r y t o compute t h e d i s charge r e s i s t a n c e o f F i g . 7. +. 1  1 = 1 +  *  + B  2  6  As the time c o n s t a n t holds  = 1  C* = C  i s g i v e n by 7.9, t h e f o l l o w i n g r e l a t i o n  ,1 R  e(  B +  ,  f-)  = CK ( l + b  + Rl)  r  2r  s  *7  7  *  1 3  8.0  Feedback Methods o f M o d i f y i n g t h e C i r c u i t 8.1  Parameters  V a r i a b l e C o n d u c t i o n Angle  By v a r y i n g t h e c o n d u c t i o n a n g l e o f t h e d i o d e b r i d g e i t i s p o s s i b l e t o v a r y a , t and G.  By s u i t a b l e  arrangement o f t h e c o n t r o l v o l t a g e s o f F i g . 3.2, t h e output v o l t a g e w i l l appear as shown i n F i g . 8.1, where the conduct i o n a n g l e i s 7C- 2 e><. I  ^  0  v  V (t)  2fr  v  o  \  /  7t-(X  0(  t = o F i g . 8.1  V o l t a g e Output f o r a V a r i a b l e Conduction Angle.  D u r i n g c o n d u c t i o n t h e c i r c u i t i s as shown i n F i g . 8.2,  v^t)  < v (t) Q  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 / r i t e d v  dt  c  = I C  Now i t I s apparent 1  =  that  A J (~ c 27TR -be v  +  sl  -v (/T-20C) + 2cosOC 8.1  n 0 ) d0 = _JL_ 27TR  c  S u b s t i t u t i n g i n t o t h e d i f f e r e n t i a l e q u a t i o n and s o l v i n g with v (0)  =0,  c  v ( t ) = 2cos<X ( _ l - ( ^ - 2 c X ) ] \ 1  c  c  ?T-2<X  I  1  exp  t  P  [ 2RC 7C  J I  8.2  yields,  Now 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 o u t p u t , (ZK 7r-« v ' ( t ) =1 I v ( t ) sin0 d0 = _1 / (-V +sin0)sin0d0 r  c  Tt  = 1  K  o  rr « J  - 2 v c o s < x + (K-2<\)  +sin2(X  c  ]  8.3  The parameters a, T and G may be o b t a i n e d from d e f i nitions  i n s e c t i o n 2.0.  From 8.2, the time c o n s t a n t i s  2RC-ZL  8.if  (0) = _ I _ a = ICL v^£oT 1-p  8.5  T  =  7T-20< The f a c t o r a i s o b t a i n e d from 8 . 3 , w i t h  w l t  (X  n  %  8cos  2  = (7C-2« )(7t-2(X +sin2<X )  The t r a n s f o r m a t i o n r a t i o i s , t h e n , r  _ v l ( 0 ) _ -, " TjTo)  ~W  W- « 2  +sln2  <*>  To o b t a i n an e q u i v a l e n c i r c u i t f o r these  8.6  parameters  c o n s i d e r F i g . 8 . 3 , and t h e p r o c e d u r e o u t l i n e d i n s e c t i o n 2.0. C I—*vwww—  1  Ro  F i g . 8.3  DC E q u i v a l e n t Lead Network.  A c c o r d i n g l y , one may w r i t e , R + R< R  a  = 1-  (3  Hence  By computing the time c o n s t a n t and  equating C  of the c i r c u i t o f F i g . 8 . 3 ,  t h i s w i t h 8.U, one o b t a i n s , = 2C 7t p(7r_2c< )  T h i s network o f f e r s t h e p o s s i b i l i t y o f v a r y i n g t h e maximum a v a i l a b l e phase s h i f t 0m and t h e f r e q u e n c y o c c u r s by c o n t r o l l i n g t h e c o n d u c t i o n a n g l e .  a t which i t  These parameters  as a f u n c t i o n o f " a " a r e g i v e n i n Appendix I by 1.1^ and 1.16 .  8.2  Servo M i x i n g Network  I n Appendix I I , R e f e r e n c e 3, a network i s shown cap a b l e o f employing an ac s i g n a l w i t h dc. feedback t o o b t a i n , as d e s i r e d , e i t h e r a l e a d o r l a g e f f e c t . g i v e n i n F i g , S.k.  The schematic i s  The diode b r i d g e 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  v  Q  -o ' output  1:2  1  (-AWvVvWv  C F i g . 8.>+  r  HI-  V t FEEDBACK  Servo M i x i n g Network w i t h AC Input and DC Feedback.  C o n s i d e r i n g o n l y t h e dc feedback v o l t a g e , t h e c i r c u i t i n F i g . 8.5 i s a n a l y s e d .  i r-i "V/  F i g . 8.5  cl -cl  DC Feedback C i r c u i t .  3MThe f o l l o w i n g e q u a t i o n s may be w r i t t e n . V + V Hence  1=2  v  Hence  = i £b  c  c  i  (  =  (  1  = i  cl  v  cr  +  i  8.7  )  8.8  )  v  8.9  r  8.10  12 r  +  S u b s t i t u t i n g 8.9 i n t o 8.10 g i v e s i  c  =|I b  l  r  +  ( f b  I) v  +  r  r  8.11  c  c  C o n s i d e r o n l y t h e 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  Circuit.  The f o l l o w i n g e q u a t i o n s may be w r i t t e n . v  c  = Ci - i  8.12  )r  c 2  8.13  i = Ii-Yo V o  -  Vr»  = i b2  = b  r  r  2 ! V(v< i  -- v"O'j  w  E l i m i n a t i n g t h e c u r r e n t s and s o l v i n g f o r v v  _ = c o 2fi + r v  2rn + 1 2r V  1  r  x  b  x  gives 8.15  b  + r  Q  D  Then i 2 may be o b t a i n e d i n terms o f v^ and v C  c  from t h e above  equations. i 2  =  C  i 2r  v  (  x  2 + r ^ -v ( c  b  2 r i  2 + r  1 + b  8.16  In order t o obtain the net charging current i  G  of the  condenser, one h a s , combining 8.16 and 8.11, i  = *c2 - - i d  C  = Av  - Bv  ±  8.17 8.18  - 2V b  c  r  with  A  2  = 2*1  = 2  B  +  8.19 r  b r 2r^+rb  r^  )  8.20  -  Now i t i s p o s s i b l e t o w r i t e , a c c o r d i n g t o s e c t i o n 2.0 d v  c  _  l  c  c o n s i d e r 8.18.  To compute i , c  I t i s assumed t h a t V and v  s l o w l y v a r y i n g and t h a t Vj[ =  s i n 0 and V i s a dc s t e p  c  are  voltage  Hence one o b t a i n s , L  C  =  A  > „ ~2„V i - rBV - 2VC  ?  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 dv dt"  I K f ° ~ T  c  +  8.21  v  V  gives, 8.22  with  1 _ B T ~ C  8.23  and  K = I I .S^V, -  8.2»+.  Solving f o r v  c  v_(t)  with v ( 0 ) = 0 gives c  = K ( l - exp - t )  8.25  Now 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 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 (.t) s i n 0 d0 0  v'(t) = h r-,Av +_ b o i  V-i  A r  rt  c  With the aid of 8.2k and 8.26, I8 A  2 r  8.26  one may write 8.26  l  -+  Ar  as 8.27  i 7rBr  T J D  --  E X  P  T  8.27  ;  This network o f f e r s the p o s s i b i l i t y of obtaining o p t i mum  response with a controlling feedback voltage as the o v e r a l l  effect of the network may  be either lead- or l a g - 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 contribution from the dc signal feedback. c i r c u i t of F i g , 8.7 V,  This suggests the equivalent  .  Network 1 Adder  o  Network 2 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  networks i n section 2,0,  consider F i g . 8.8.  equivalent  37  8.8  Fig,  Network 1 ,  T h i s network must respond t o a s t e p i n p u t of a m p l i t u d e 8A r 1 2  V  i  TC B  -  ri - —) t^+ (1-exp  A r  v\  as  b  2  To account f o r a t t e n u a t i o n a t " t " =<XJ, r lowing  g  must s a t i s f y the  fol-  equation:  8.28 r  l  +  b  r  +  r  ft  s  2  1  B  Solving t h i s equation gives, = r  l  8  s  Bd-^J)  Computing the time c o n s t a n t of Network 1 'and t h i s with 8.23 C  equating  gives = C -g  Next c o n s i d e r t h e 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 s t e p f e e d b a c k V . F i g . 8.9  The  l a g - l i k e r e s p o n s e suggest the  as an e q u i v a l e n t c i r c u i t f o r Network  O-AAAAAAWV r  l C"  Fig. 8.9  r  T. Network  2.  2.  38  T h i s network must respond t o a s t e p i n p u t o f a m p l i t u d e V as V 8Arx  _  t  ^ B p ^ ( l - e x p fpr).  To account f o r a t t e n u a t i o n a t t  , the  f o l l o w i n g r e l a t i o n must h o l d : 8Ars Hence r  s  = r]_  i -1  jR-ti-^-)  Computing t h e time c o n s t a n t o f Network 2 and e q u a t i n g w i t h 8„23 gives, C« = KC f b l f c \ " r  r  +  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 E q u i v a l e n t C i r c u i t . C o n s i d e r t h e network shown i n F i g . 9.1.  A u n i t amp-  l i t u d e sine^-wave i n p u t v ^ i s a p p l i e d a t t = 0.  v  '<\  ±  F i g . 9.1  > R  Schematic  f o r C i r c u i t 1.  The o p e r a t i o n o f t h e gate G i s such t h a t t h e i n p u t t o t h e condenser  i s a half-wave  sine signal.  During the non-conduction  p e r i o d o f t h e g a t e , t h e condenser i n p u t t e r m i n a l i s c o n s i d e r e d open c i r c u i t . signal  During, the c o n d u c t i o n p e r i o d o f t h e g a t e , t h e  i s assumed t o be generated from a z e r o impedance source, During the conduction p e r i o d the f o l l o w i n g  differen-  t i a l equation i s v a l i d . v  c  +  C  l£r = s i n wt  R  9.1  at  The s o l u t i o n o f 9.1 i s v where  c  = A  exp  n  T = RC,  + P s i n wt + Qcos wt  P =  1 _ 1 "1 + ( t u C R ) ^ ~ 1 +. (uJ T ) ' Q = - »^CR. - = -">T 1 + (u/CR) 1 H<o T ) ' 0  n  2  2  and A As  n  i s t o be determined  2  f o r each c o n d u c t i o n  v ( 0 ) = 0, then A„ = U.CR c o  •  e  r  For  J£jb = 1, v ( l ) = A  Let  a = exp -ft %  c  r  K  M  T  J  Q  +  (  u  )  C  R  )  2  1 exp(^r) +  interval.  = u>. " 1 +( UJ T ) T  2  = 1  During the i n t e r v a l v . c  w t = 2, t h e r e i s no change i n  to  Then, m a t c h i n g boundary c o n d i t i o n s g i v e s A (l+a) = A 0  Hence  A  = A  2  Q  2  -A  Q  (2+a)  Now f o r "J t = 3 , v ( 3 ) = A a + A c  2  Q  S i m i l a r matching may be done f o r o t h e r t i m e s , and t h e r e s u l t s are,  relating to A , 0  (1+a)  v (l) = A  Q  v (3) = A  0  (l+2a+a )  v (5) = A  Q  (l+2a+2a +a3)  c  c  c  2  2  In general f o r ^ t = v ( 2 n + l ) =A c  Let  x =  (2n+l), i ^ J [ ( l- a + n +  D  K  vol  Then  v (2n+l) = | c  ( 1  .  n + 1 a  ) [i- 2( .01799) + x^O.OOOW,. x  0  Due t o t h e n a t u r e o f t h e o p e r a t i o n o f t h e gate G, then v (2n+l) = v c  c  (2n+2)  From t h e a n a l y s i s o f c i r c u i t 1 i n . s e c t i o n 3 one -t < v ( t ) = g I 1-exp(^ ) 2RC' To compare t h e two r e s u l t s s e t t = (2n+2)7~~ . U s i n g t h e d e f i obtains,  e  r  n i t i o n s o f a = exp(-^-T^) and T = RC, one o b t a i n s •e<*> = a ( 1 - a  n + 1  )  I n comparison one h a s , v (2n+2) ^-(j) = c e  , 1-x (0.01799)+ 0  2  u, , . • x -CO-.000^3*0 . . 4  f  Then i t i s apparent lim  v  c  that  ( 2 t l  +  o v TtT  2 )  _ .  c  That i s , the r e s u l t s from e i t h e r approach a r e i d e n t i c a l i n t h e limit.  Now  from the d e f i n i t i o n o f x,  lim x -5»-  = o  lim W-*-o«  as x =  ft . u)T  The l i m i t i n g case f o r u; , the c a r r i e r f r e q u e n c y , a p p r o a c h i n g infinity,  i s e q u i v a l e n t to x approaching z e r o .  S i n c e the e r r o r  term e n t e r s q u a d r a t i c a l l y , 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  will  be v a l i d f o r t h e u s u a l range o f o p e r a t i o n i n s e r v o - s y s t e m s , i . e ,  10.0  Measurement o f Phase and G a i n 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  theory o f the c i r c u i t  been developed u s i n g F o u r i e r Fourier  analysis  1 shown i n F i g . 3.2 h a s  analysis.  Physical  may be o b t a i n e d b y e m p l o y i n g  output o f a m u l t i p l i e r .  The s y s t e m u s e d v  >+00  the time-average  i s shown i n F i g , 1 0 . 1 .  l V  v  results of  o  M 2  Network Fig.  For  The  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 Constant  calibration the following  waveforms were employed:  v^  ^fOO c p s v o l t a g e o f l ^ . l U p e a k v o l t s  v  *+00 c p s v a r i a b l e - a m p l i t u d e  2  square-wave.  a c t u a l w a v e f o r m s a r e shown i n F i g . 1 0 . 2 .  a. I n p u t Waveforms  b. O u t p u t  Waveforms  F i g , 10.2 M u l t i p l i e r W a v e f o r m s . 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 the  output.  t o t h e dc l e v e l o f  f*  Output = 1  A A sin0 d0 =  2K o~1  Now  1 2 A  ~ W ~  2  J  A  t h e m u l t i p l i e r ' s dc o u t p u t V  Q  i s p r o p o r t i o n a l t o t h e time  average of t h e p r o d u c t o f t h e two i n p u t s . t i o n a l i t y c o n s t a n t Ic i s d e f i n e d V  0  = E AxA  Hence t h e p r o p o r -  by  2  Tt  By s u i t a b l e measurements i t was found t h a t • ( T - ^ - ^ r ) "•* •t•J f 579) K( y j A\ „ 2  •  -2A{  2(l*f.l*f):  =0.06^25  The measurement o f p h a s e - g a i n c h a r a c t e r i s t i c s i s a c h i e v e d by t h e system shown i n F i g . 1 0 , 3 .  A modulated  car-  r i e r i s g e n e r a t e d by the s y n c h r o TI w h i c h i s d r i v e n by a v a r i a b l e speed motor a t t h e r a t e  <^ . s  The modulated s i g n a l i s ap-  p l i e d t o a second synchro T2 w h i c h i s used as a manual phase s h i f t e r t o v a r y t h e phase of t h e envelope o f t h e s i g n a l .  This  p h a s e - s h i f t e d modulated s i g n a l i s a p p l i e d t o t h e network.  The  o u t p u t of t h e network i s a p p l i e d t o one o f t h e m u l t i p l i e r ' s inputs.  A known comparison s i g n a l g e n e r a t e d by a r e s o l v e r i s  a p p l i e d t o t h e second m u l t i p l i e r i n p u t .  The time average o f  the p r o d u c t o f the i n p u t s i s r e c o r d e d by t h e g a l v o n o m e t e r ,  F i g . 10.3. B l o c k Diagram o f Measurement System.  For F i g . 10.3, t h e f o l l o w i n g terms a r e d e f i n e d : TI  Synchro Transformer  M  Electronic Multiplier  T2  Synchro Transformer  G  Galvonometer  R  Resolver  w  modulation  g  frequency  C o n s i d e r t h e f o l l o w i n g arrangements o f m u l t i p l i e r i n p u t s shown i n F i g . 10.h. l  v  Resolver  V 0  M v  System  1  2  G  Galvonometer  M  Electronic Multiplier  |v  l c  = V-^cos w/gt s i n u) t  lv  l s  = V^inuJgt  v F i g . 10.k  Q  sinu t c  = V s i n (u; t+(X)sin t u t  2  2  g  M u l t i p l i e r Schematic Determinations  c  f o r Gain-Phase  I n t h e d e f i n i t i o n o f v , CX i s t h e combined phase s h i f t 2  intro-  duced by t h e network and t h e v a r i a b l e t r a n s f o r m e r , i . e . , 0( Let  V  = 0 (network) +V( t r a n s f o r m e r )  oc  =  k  v  ic 2 v  = k V^cos w  s  t s i n c o t V s i n ( u ; t +o( )sino» t c  2  s  c  = k V V ( s i n c < cos* u ; t + cos6(cosu» t s i n u j t J s i n ^ u i t 1  1  2  g  s  Q  g  W i t h s u i t a b l e sum and d i f f e r e n c e a n g l e f o r m u l a e , one o b t a i n s , V  oc = * Vgsin-* i - J j j l i n <  s i n ( u l  e ~ "V*  +  'sin(t*> +u? )t| d(u> t) c  = k V,V sin« 1 . 1.2.7C = k ViVosinoc 27C 5 ' * 0  1  Then V  Q C  2  i s made z e r o by c h o o s i n g <X.= 0, i . e . , 0 = —Y  s  e  Employing v-^, i t may be shown s i m i l a r l y V With  o S  = k V-jVocosa  W •= o , one may s o l v e f o r V , o b t a i n i n g 2  For c o m p u t a t i o n o f t h e g a i n o f t h e network, t h e peak o u t p u t a m p l i t u d e V" must be compared w i t h t h a t peak s i g n a l V i 2  w h i c h would pass t h r o u g h t h e system i n t h e absence o f any comp e n s a t i n g network. G =  I f t h e i n p u t s i g n a l has a peak V, then as  one has ][i =  The g a i n o f t h e system i s  V Y  = V V  2  i  =  2 V  V  2  = 8V  k  O S  vv x  The e x p e r i m e n t a l s t u d i e s i n v o l v e d measurement o f t h e 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 o f s i g n a l f r e q u e n c y . The c a l c u l a t e d network parameters a and T p r o v i d e d t h e g a i n phase c h a r a c t e r i s t i c s o f t h e 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 . comparative p l o t o f r e s u l t s i s shown i n F i g . 1 0 . 5 .  A  For C i r c u i t  1, a i s f i x e d a t 5.3 so t h a t e x p e r i m e n t a l s t u d i e s i n v o l v e d v a r i a t i o n o f T. *+.21 msecs.  The v a l u e s o f T chosen were 7.25 msecs and  These c o r r e s p o n d t o f r e q u e n c i e s o f 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 T  1 Equivalent  R  Circuit  T Fig.  10.5  Radians  11.0  Conclusions. By c o n s i d e r i n g t h e l i m i t i n g  simplified  t h e o r y was  case as  ^c  ^  >  developed to o b t a i n network parameters  from the t r a n s i e n t response t o a step i n p u t . the parameters, l i m i t i n g  With the a i d of  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 i m m e d i a t e . h a s b e e n done t o d e t e r m i n e t h e r a n g e o f p r a c t i c a l t a t i o n of the l i m i t i n g  , a  equivalent c i r c u i t .  and e x p e r i m e n t c o n f i r m e d t h a t t h e l i m i t i n g  Study  represen-  Analysis  predicted  c i r c u i t was  f o r the u s u a l c a s e i n s e r v o - s y s t e m s where ^ s <  adequat  1  I t h a s b e e n shown t h a t b a s i c l e a d - o r l a g - c o m p e n 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  that, i n  s p e c i a l c a s e s , n e t w o r k p a r a m e t e r s a r e c o n t r o l l a b l e by  suitable  feedback. W i t h the development  of suitable equivalent  circuits  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  be r e a l i z e d g i v i n g w i d e a p p l i c a t i o n .  For n o n - l i n e a r  can  systems  where a d j u s t a b l e g a i n - p h a s e 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 specifications, circuits  employing feedback-controlled  m e t e r s w o u l d a p p e a r t o i n t r o d u c e a new possibilities.  field  of  para-  interesting  Appendix I .  DC Lead Network.  C o n s i d e r t h e network below. r. -vwwvvw •vwwvwv DC Lead Network,  Fig..1.1  The t r a n s f e r as  Y(s)  function  f o r t h i s network may be w r i t t e n  = o 1 + Tjs 1 + T S G  1.1  2  with  Tl T.  ~ ^2h C  r  r  r  2  +  +  r  3  = (r r ri  r  +  2  r (r 1  r  l  ( r  2  1.2  k  + r-^ru^ r  f  2  r ^ )  + r ^ + T^.) + ( r ^ +  2  = rx(r  r  T  c  1  3 t+)  2  + r +  r  3  3  +  1.3  r^)r,  2  + ruJ V  +  ( r  3  +  V  2  r  Then i t i s a p p a r e n t t h a t 3 parameters w i l l be s u f f i c i e n t t o specify  g a i n 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 t h e network r e s p o n s e t o a u n i t -  s t e p i n p u t because o f t h e p a r t i c u l a r development o f t h e t h e o r y i n t h e body o f t h e t h e s i s .  Then v j ( s )  = G  (1+T-^s) 1 , i s t h e  Q  ( l + T sT  i  L a p l a c e T r a n s f o r m o f t h e o u t p u t v o l t a g e v£ . * I n t h e u s u a l manner t h e t r a n s f o r m may be i n v e r t e d  vi (t) = G  0  l-(l-T )exp-t  Then f o r t = 0  i n 1.5,  s  P /  to y i e l d 1.5  1  ~2  2  1  one h a s , v j ( 0 ) = G _1 Q  1  2  1.6  1.7  v (<x») - G  and for t =»*> i n 1.5,  1  c  Combining 1.6 and 1.7 y i e l d s ,  vo(0) x  V-t(e*>) O  For  1.8  To  *  the network i t i s often necessary to know the  frequency of maximum phase s h i f t and the amount of phase s h i f t at the maximum.  From the d e f i n i t i o n  of "a" i n section 2.0 one  has a = £l For  1.9  r e a l frequencies, 1.1 may be written YUu>) = |Y| expUe^) = exp(;J0 )  Y  exp(j0)  1.10  2  where  tan0^ = u> a T  and  0 = 0  X  - 0  and t a n 0 = OJ T  2  2  1.11  2  1.12  2  The frequency of maximum phase s h i f t i s obtained by d i f f e r e n t i a t i n g 1.12.  M = = o = 5 £ i . d0. ^ 2 dtAj  d u>  d  1.13  uu  The derivatives of 1.13 may be obtained from I.11.  These equa-  tions y i e l d the following r e s u l t , aTgCos ^ = T cos 0 2  2  2  2  The various cosines may be obtained from I.11,  or,  so that one has,  I.Hf  To obtain the amount of phase s h i f t consider tan0 = tan($1-^2)  Then f o r t h e maximum phase s h i f t , i t may be shown t h a t tan0 = m  L±*=D \/a 2  1.15  An e q u i v a l e n t form of 1.15 i s sin0 = T } m a + l  1.16  a  m  From t h e t r a n s f e r f u n c t i o n i t i s apparent t h a t t h e r e are  t h r e e c h o i c e s t o be made i n o r d e r t o s p e c i f y t h e phase and  amplitude c h a r a c t e r i s t i c s .  These c h o i c e s may be made i n a  v a r i e t y o f ways depending upon c i r c u m s t a n c e .  F o r example t h e y  c o u l d be chosen on t h e network's response t o a u n i t - s t e p i n p u t . I n t h i s case t h e parameters might be t h e o u t p u t a m p l i t u d e i n i t i a l l y , t h e o u t p u t a m p l i t u d e f i n a l l y , and t h e t i m e c o n s t a n t f o r the  decay t o t h e s t e a d y s t a t e .  Another b a s i s o f c h o i c e a r i s e s  i n s t a b i l i t y problems where c o n c e r n might be t h e f r e q u e n c y o f maximum phase s h i f t and t h e amount o f phase s h i f t a t t h a t p a r t i c u l a r frequency. I t s h o u l d be noted t h a t i n F i g . 1.1 t h a t n  f  may be  e l i m i n a t e d b u t t h e c i r c u i t would r e t a i n i t s performance c h a r a c t e r i s t i c s be a p p r o p r i a t e m o d i f i c a t i o n o f r  2  and r ^ .  e s s e n t i a l l y , t h e r e a r e f o u r elements t o be chosen.  Then, The concept  of t h r e e independent c h o i c e s has been made p r e v i o u s l y , w h i l e the  e x t r a c h o i c e or degree o f freedom d e t e r m i n e s t h e impedance  levels. For Fig.  some o f t h e c i r c u i t s d i s c u s s e d i n t h e t h e s i s ,  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  l 1 + Cs ( r + r )  Y(s) =  r  2  [r r T ( l > 1 +  3  Cs r^  2  (  r H  r i  r  1  2  +  r ' 2  As a l l the i m p o r t a n t r e l a t i o n s have been d e r i v e d  generally,  nothing s i g n i f i c a n t arises i n t h i s s i t u a t i o n . Of v e r y p a r t i c u l a r i n t e r e s t i s t h e case o f r ^ == and r ^ = 0. >.  oo  Then  Y(s)=£l^ r r 1  +  2  1 + Csrg 1 Cs r j r s r  Now i t i s apparent t h a t G  =  Q  = 1 a  l  +  R  1  -  +  2-  2  and T-^ = a T  * 2  .  Then t h e  parameter a may be checked or d e t e r m i n e d by measurement.  It  i s apparent f o r t h i s case t h a t t h e network r e q u i r e s o n l y two parameters f o r d e f i n i t i o n o f gain-phase c h a r a c t e r i s t i c s .  Appendix I I .  References.  1.  M i l l m a n , J . , and P u c k e t t , J.R., " A c c u r a t e L i n e a r B i d i r e c t i o n a l Gate", P r o c . IRE. V o l . if3, Ro. January 1955, pp. 29-37.  2.  D i p r o s e , K.V., " D i s c u s s i o n of D. M o r r i s ' s 'A Theore t l c a l and E x p e r i m e n t a l Method o f M o d u l a t i o n A n a l y s i s f o r the D e s i g n of AC Servo Systems'"., T u s t i n , A., ed., Automatic and Manual C o n t r o l . London, B u t t e r w o r t h ! 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", Convent i o n Record IRE. 1955. p t . 10, p.. .173,  *f.  W e i s s , G., and L e v e n s t e i n , H., "A-C S e r v o s " , T r u x a l J.G., C o n t r o l E n g i n e e r s ' Handbook. T o r o n t o , McGraw H i l l , 1958, p. 6-61..  5.  Bohn, E.V.., L e c t u r e s o f 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.  1  

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 4 0
China 2 1
Brazil 1 0
City Views Downloads
Ashburn 4 0
Shenzhen 1 1
Unknown 1 6
Beijing 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0105075/manifest

Comment

Related Items