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Synthesis of exponentially-tapered distributed rc networks realizing dirving-point immittance functions Chinn, Henry Ronald 1966

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©  Henry Ronald Chinn 1967  THE SYNTHESIS OF EXPONENTIALLY-TAPERED DISTRIBUTED RC NETWORKS REALIZING DRIVING-POINT OMITTANCE  FUNCTIONS  by  HENRY RONALD CHINN F.R.M.T.C., R o y a l Melbourne T e c h n i c a l C o l l e g e , 1956, B.E.,  U n i v e r s i t y of New  M.E., U n i v e r s i t y o f New  South Wales,  I960,  South Wales, 1963.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY i n the Department of Electrical  We a c c e p t  Engineering  t h i s t h e s i s as c o n f o r m i n g t o t h e required  Research Supervisor  standard  .,  Members of the Committee ..  Head o f the Department Members of the Department of E l e c t r i c a l  Engineering  THE UNIVERSITY OF BRITISH COLUMBIA DECEMBER, 1966  In p r e s e n t i n g  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 o f the requirements  f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y 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 r e f e r e n c e and study,  I f u r t h e r agree t h a t p e r m i s s i o n f o r extensive  copying o f t h i s  t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s .  I t i s under stood t h a t copying  or p u b l i c a t i o n of" t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed w i t h o u t my w r i t t e n p e r m i s s i o n .  Department  of_El e_cJ^ri^^ ;  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date  ABSTRACT  In t h i s d i s s e r t a t i o n , f o r m a t i o n s are realization  nentially-tapered  a special  'the  equivalent  through the and  ideal  RC  networksc  r e a l i z a t i o n by  trans-  These  the  expo-  synthesis  uniform d i s t r i b u t e d  circuits  distributed  circuits  RC  f o r the  are  developed f o r the  network.  and  expo-  These d i f f e r  uniform d i s t r i b u t e d  presence of p o s i t i v e  RC  n e g a t i v e lumped  from network  elements  transformers. It  i s found t h a t  n a t e d from the  equivalent  tapered d i s t r i b u t e d a positive-real  the  lumped elements must be  c i r c u i t s developed f o r  c o n f o r m a l t r a n s f o r m a t i o n to s y n t h e s i s problem.  elimi-  exponentially-  RC. networks b e f o r e i t i s p o s s i b l e  problem i n t o a lumped LC  to  change the  of d r i v i n g - p o i n t  Hence, through  RC  pensate f o r the  s e c t i o n s are  by  a function  that  configurations presented. facilities  equivalent  s y n t h e s i s procedure can  n e c e s s a r y to approximate any function  presented.  Additional  used i n these r e a l i z a t i o n s  lumped elements i n the  B e f o r e the  reali-  immittance f u n c t i o n s i s d e v e l o p e d .  V a r i o u s cascade network r e a l i z a t i o n s are distributed  apply  synthesis  R i c h a r d s ' Theorem, a cascade s y n t h e s i s procedure f o r the zation  RC  case.  equivalent  nentially-tapered  immittance f u n c t i o n s by  distributed  p r o c e d u r e s i n c l u d e the  New  conformal  used to d e v e l o p s y n t h e s i s p r o c e d u r e s f o r  of d r i v i n g - p o i n t  networks as  positive-real  be  circuits.  applied,  specified driving-point i s r e a l i z a b l e by  A digital  one  ii  i t is  immittance  of the  computer w i t h  i s deemed n e c e s s a r y f o r t h i s purpose.  to com-  network  plotting  TABLE OF CONTENTS  Page LIST OP ILLUSTRATIONS •  •  •  ACKNOWLEDGEMENTS 1.  2.  v  »  INTRODUCTION  viii  .  1  1.1  Review  1.2  Proposed S y n t h e s i s P r o c e d u r e s  o f P r e v i o u s Work  1 4  THE CHARACTERISTICS OR UNIFORM AND EXPONENTIAL RC SECTIONS, AND A SUMMARY OF WTNDRUM'S SYNTHESIS 2.1  Introduction  2.2  The Two-Port  2.3  Cascaded U n i f o r m RC S e c t i o n s  2.4  Wyndrum s S y n t h e s i s Procedure  2.5  . Parameters o f t h e U n i f o r m RC  11 ,... c.  .. . o .  Realizahility  2.4.2  P o s i t i v e - R e a l Conformal T r a n s f o r m a t i o n  2.4.3  R i c h a r d s ' Theorem . . . .  17  2.4.4  S y n t h e s i s Procedure  18  The Two-Port Section  Conditions  14  2.4.1  ... 16  ^............... ,  Equivalent C i r c u i t s  Cascaded E x p o n e n t i a l RC S e c t i o n s  2.7  Conjecture 2.7.1  14  Parameters o f the E x p o n e n t i a l RC  2.6  2 .8  8  1  2.5.1  3.  «  19  „  21  ................  24  Cascaded Ex p o n e n t i a l RC S e c t i o n s Under the R e s t r i c t i o n o f the C o n j e c t u r e  2.7.2  P o s i t i v e - R e a l Conformal T r a n s f o r m a t i o n s  2.7.3  Realizahility  Discussion  23  Conditions  ,  .. 26 27 30  e  THE SYNTHESIS OF DRIVING-POINT IMMITTANCE FUNCTIONS .  .. 31  3 .1  Introduction  3.2  S y n t h e s i s Procedures f o r Cascaded S e c t i o n s w i t h S h o r t - C i r c u i t e d Stubs and Lumped Elements i n Shunt . ............ . 39 3.2.1  .......  24  R e a l i z a t i o n w i t h a Minimum Number of RC S e c t i o n s and Lumped Elements iii  31  42  3.2,1.1 3.2.2  3.2.2.1  5.  Practical  S y n t h e s i s Procedure  3.3  S y n t h e s i s P r o c e d u r e s f o r Cascaded S e c t i o n s w i t h O p e n - C i r c u i t e d Stubs and lumped Elements i n Series .  3.4  A S y n t h e s i s Procedure f o r a Cascade of S e c t i o n s i n P a r a l l e l Together w i t h Lumped Elements i n 3.4.1  4.  S y n t h e s i s Procedure  R e a l i z a t i o n which F a c i l i t a t e s Implementation  S y n t h e s i s Procedure  3.5  A S y n t h e s i s Procedure f o r a Cascade of S e c t i o n s i n S e r i e s Together w i t h Lumped Elements i n  3.7  Discussion  THE APPROXIMATION PROBLEM 4 »1  IihiijX*ociiicijioi-i  4.2  Functions  4*4  Disenssion  o*e•  0©*o»»»«#«»oe»»»*»»«»*»«o-#9«  of E x p o n e n t i a l P o l y n o m i a l s  ...  »®©«»®»»oe*o»«c««»*»»*»o®e»9«'»*»»ff#«  CONCLUSIONS .... a  9 « # *  iv  LIST OP ILLUSTRATIONS  Figure 1.1  Page S y n t h e s i s Procedure U s i n g P o s i t i v e - R e a l Conformal T r a n s f o r m a t i o n  1 ©2  TILL© XJXII J_ o^ni  1.3  The E x p o n e n t i a l RC S e c t i o n  2.1  E q u i v a l e n t C i r c u i t s o f the U n i f o r m RC S e c t i o n .... 12  2.2 2.3  Cascade of RC S e c t l An E q u i v a l e n t C i r c u i t Secti  2.4  An A l t e r n a t i v e E q u i v a l e n t C i r c u i t  3.1  Cascade of E x p o n e n t i a l RC S e c t i o n s w i t h Compensat i n g Lumped Elements Added i n S e r i e s w i t h the  3.2  Cascade of E x p o n e n t i a l RC S e c t i o n s w i t h Compensat i n g Lumped Elements Added i n Shunt w i t h the  3.,3  Equivalent C i r c u i t  o f an O p e n - C i r c u i t e d  3.4  Equivalent C i r c u i t  of a S h o r t - C i r c u i t e d Exponen-  3.5  Cascaded S e c t i o n s w i t h S h o r t - C i r c u i t e d Stubs and Lumped Elements i n Shunt .........................  3.6  E q u i v a l e n t C i r c u i t o f Cascaded S e c t i o n s w i t h S h o r t - C i r c u i t e d Stubs and Lumped Elements i n S]hl\\AlIl't'  R.0  3  S © c ~t n_ on  #a«o»»#oos>a<B(?««eo9»«  5  6  of the E x p o n e n t i a l RC of the Exponen-  Exponential  • « e » « » » * « * » « * * « 6 * » « e » » » « « 0 » 0 » » * 9 e e »  3.7  A l t e r n a t i v e C o n f i g u r a t i o n o f Cascaded S e c t i o n s w i t h Short C i r c u i t e d Stubs,and Lumped Elements i n  3.8  Cascaded S e c t i o n s w i t h Lumped Elements i n  Open-Circuited  V  Stubs and  37  3 8  Figure  Page  3.9  Equivalent C i r c u i t of Cascaded Sections w i t h Open-Circuited Stubs and Lumped Elements i n  3.10  A l t e r n a t i v e C o n f i g u r a t i o n of Cascaded Sections w i t h Open-Circuited Stubs and Lumped Elements i n  3»11  Cascade of Sections i n P a r a l l e l Together w i t h Lumped Elements i n Shunt . o o o o c » » . » . s < , » » . . s . 0  0  0  9  0  S  54  i  3.12  Equivalent C i r c u i t of Cascade of Sections i n P a r a l l e l Together w i t h Lumped Elements i n  3.13  A l t e r n a t i v e Configuration of Cascade of Sections i n P a r a l l e l Together w i t h Lumped Elements i n  3.14  Cascade of Sections i n Series Together w i t h Lumped Elements i n S e r i e s ..«»„.<>•<>»....«.»»».*.  61  E q u i v a l e n t C i r c u i t of Cascade of Sections i n Series Together w i t h Lumped Elements i n S e r i e s .  62  3.15 3.16  A l t e r n a t i v e C o n f i g u r a t i o n of Cascade of Sections i n S e r i e s Together w i t h Lumped Elements i n  3.17  Network R e a l i z a t i o n .„*.*»»  4.1  Magnitude-Frequency Curves f o r Synthesis Example  4.2  R e a l i z a t i o n of Z^(s) by a Uniform RC Network ...  4.3  R e a l i z a t i o n of 2 (s) by an Exponential RC Net-  4.4  R e a l i z a t i o n of 2^(s) by an Exponential RC  4.5  Magnitude-Frequency Curves f o r Synthesis Example o  2  o o o & o o  Q  o o o o o a o a a 9 # o $ s 'a o o 6 e & o o o a a o o o Q Q e o  O  *  *  4.6  R e a l i z a t i o n of ^ ( s ) by a Uniform RC Network ...  4.7  R e a l i z a t i o n of ( s ) by an Exponential RC Netvi Y  c  76  80 81  Figure 4„8  Page Realization 1M©  "fc'W  QH^k  o f ^ ( s ) by  an E x p o n e n t i a l  RO  s o o « o o - « f i o » o « » e « o o 9 » e e » t 5 < J o o o o » 9 « * « > t t o o * * » o »  vii  82  ACKNOWLEDGEMENTS  I wish, t o t h a n k my r e s e a r c h  s u p e r v i s o r , D r . A.D. Moore,  f o r h i s encouragement, g u i d a n c e , i n t e r e s t  and s u g g e s t i o n s  the c o u r s e o f t h e r e s e a r c h work and t h e w r i t i n g  of t h i s  during dis-  sertation.,  I a l s o w i s h t o t h a n k Dr. L. Young f o r h i s i n t e r e s t  and h e l p f u l  d i s c u s s i o n s on t h e p r a c t i c a l r e a l i z a t i o n o f t h i n -  f i l m networks. Thanks a r e due t o my c o l l e a g u e s and s u g g e s t i o n s ,  p a r t i c u l a r l y with regard  for helpful to d i g i t a l  discussions computer  programming. Thanks a r e a l s o due t o M i s s L. B l a i n e f o r t y p i n g this  dissertation. I gratefully  acknowledge t h e f i n a n c i a l  assistance  o f 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 o f Canada t h r o u g h i t s g r a n t A-68 f o r 1963-64 and i t s award o f a S t u d e n t s h i p  viii  f o r 1965-67.  1,  1.1  Review  INTRODUCTION  o f P r e v i o u s Work  I n t e g r a t e d c i r c u i t s and t h i n - f i l m n e t w o r k s a r e b e i n g i n c r e a s i n g l y used because of t h e i r s m a l l e r p h y s i c a l d i m e n s i o n s and g r e a t e r r e l i a b i l i t y , ,  The r e a l i z a t i o n o f p a s s i v e  distributed  parameter n e t w o r k s i n t h e f o r m of t h i n - f i l m s t r u c t u r e s i s c u r r e n t l y under i n v e s t i g a t i o n by c i r c u i t theorists„ Linear, passive, time-invariant, distributed-parameter n e t w o r k s , h e r e a f t e r r e f e r r e d t o s i m p l y as d i s t r i b u t e d - p a r a m e t e r n e t w o r k s , a r e c h a r a c t e r i z e d by i m m i t t a n c e f u n c t i o n s t h a t a r e not r a t i o n a l i n t h e complex f r e q u e n c y s.  Various  approaches  have been used t o r e a l i z e d i s t r i b u t e d - p a r a m e t e r n e t w o r k s t o s a t i s f y network s p e c i f i c a t i o n s " ' " .  These approaches c a n be  d i v i d e d i n t o two g r o u p s . 2  The f i r s t group o f methods i n c l u d e t h o s e ' a network a p p r o x i m a t i n g  3  i n which  t h e s p e c i f i e d r e s p o n s e i s s e l e c t e d on  t h e b a s i s o f r e s p o n s e c u r v e s o b t a i n e d by a n a l y s i s , as w e l l as t h o s e w h i c h a r e based upon a p p r o x i m a t i o n s , dominant p o l e s ^ .  s u c h as t h e use o f  Such methods a r e n e i t h e r as e l e g a n t n o r as  g e n e r a l o f a p p l i c a t i o n as t h o s e of t h e second group. The second group o f methods a r e s y n t h e s i s methods, because t h e d i s t r i b u t e d - p a r a m e t e r network i s e x a c t l y and s y s t e m a t i c a l l y r e a l i z e d a f t e r t h e s p e c i f i e d network f u n c t i o n i s a p p r o x i m a t e d by a s u i t a b l e network f u n c t i o n .  These  synthesis  methods can be f u r t h e r d i v i d e d i n t o two subgroups. I n one of t h e s e subgroups, c o n f i g u r a t i o n s o f d i s t r i b u t  p a r a m e t e r n e t w o r k s w h i c h have i m m i t t a n c e r a t i o n a l i n t h e complex f r e q u e n c y  functions that are  s are found,  so t h a t t h e  u s u a l lumped-parameter s y n t h e s i s p r o c e d u r e s c a n be a p p l i e d . Heizer^ "'"^ and Hesselberth"'""'" have r e p o r t e d v a r i o u s c o n f i g u r a -  t i o n s o f t h i n - f i l m d i s t r i b u t e d RC n e t w o r k s t h a t w i l l g i v e r a t i o nal  immittance The  functions.. o t h e r subgroup o f s y n t h e s i s methods uses p o s i t i v e  r e a l (p-r) conformal  t r a n s f o r m a t i o n s t o change t h e d i s t r i b u t e d -  parameter n e t w o r k problems i n t o lumped-parameter n e t w o r k problemso  The u s u a l lumped-parameter s y n t h e s i s p r o c e d u r e s  can t h e n be a p p l i e d and t h e d i s t r i b u t e d - p a r a m e t e r n e t w o r k s realized. In  T h i s s y n t h e s i s a p p r o a c h i s d e p i c t e d by P i g . 1.1.  t h i s approach, t h e d i s t r i b u t e d - p a r a m e t e r network f u n c t i o n  must f i r s t be o b t a i n e d , i . e . , t h e s p e c i f i e d n e t w o r k f u n c t i o n must be a p p r o x i m a t e d by an i r r a t i o n a l f u n c t i o n t h a t the r e a l i z a b i l i t y c o n d i t i o n s . used t h i s t e c h n i q u e  W y n d r i m ^ " " ^ and O'Shea"^ have  i n t h e i r synthesis procedures.  used t h e p - r c o n f o r m a l  satisfies  t r a n s f o r m a t i o n w = tanh  a/SRC  Wyndrum  't  to  change t h e d i s t r i b u t e d RC network f u n c t i o n i n t o a lumped LC network f u n c t i o n .  He has proposed a cascade s y n t h e s i s p r o -  cedure t o r e a l i z e d r i v i n g - p o i n t (d-p) i m m i t t a n c e by u n i f o r m to  functions  ( u n t a p e r e d ) d i s t r i b u t e d RC ( h e r e a f t e r a b b r e v i a t e d  u n i f o r m RC  ) networks.  t r a n s f o r m a t i o n w = cosh  O'Shea used t h e p - r c o n f o r m a l  *JsRC't  t o .change t h e d i s t r i b u t e d  RC network f u n c t i o n i n t o a lumped R l network f u n c t i o n , and proposed P o s t e r - t y p e r e a l i z a t i o n s  ( i . e . , r e a l i z a t i o n s based  upon p a r t i a l - f r a c t i o n e x p a n s i o n s ) o f d-p i m m i t t a n c e  functions  CONFORMAL  Fig.  TRANSFORMATDN  1.1  - Synthesis Procedure Using Conformal Transformation.  INVERSE TRANSFORMATION  Positive-Real  4  by u n i f o r m RC networks*  However, p h y s i c a l r e a l i z a t i o n of  networks i s not as p r a c t i c a l as the cascaded o b t a i n e d by Wyndrum's s y n t h e s i s 1.2  Proposed S y n t h e s i s Wyndrum and  P i g . 1.2  u n i f o r m RC  these  networks  procedure.  Procedures  O'Shea used the u n i f o r m RC  s e c t i o n of  as the b a s i c network element i n t h e i r s y n t h e s i s  procedures, 13 I t has been r e p o r t e d RC networks may RC networks.  have u s e f u l p r o p e r t i e s not p o s s e s s e d  width  by  passive uniform  F o r ease of f a b r i c a t i o n , d i s t r i b u t e d p a s s i v e  structures consist of u n i f o r m  that tapered t h i n - f i l m  of c o n d u c t i v e , d i e l e c t r i c  composition  and  constant  and r e s i s t i v e  thickness.  The  that  films  effective  a t any p o i n t of a t a p e r e d t r a n s m i s s i o n l i n e w i l l  assumed to be t h a t of the r e s i s t i v e f i l m and  RC  be  modulation  of t h i s w i d t h w i t h r e s p e c t t o the a x i a l c o o r d i n a t e produces r e s i s t a n c e and Under these  c a p a c i t a n c e t a p e r f u n c t i o n s which are  reciprocal.  c o n s t r a i n t s , s o l u t i o n s of the d i f f e r e n t i a l  equations  of nonuniform RC networks are known f o r some t a p e r f u n c t i o n s such 17 17 as the l i n e a r and e x p o n e n t i a l t a p e r s . The e x p o n e n t i a l l y t a p e r e d d i s t r i b u t e d RC  ( e x p o n e n t i a l RC  ) s e c t i o n of F i g . 1»3?  with  the a d d i t i o n a l degree of freedom p r o v i d e d by i t s t a p e r parameter, may  have c h a r a c t e r i s t i c s not o b t a i n a b l e by a u n i f o r m RC s e c t i o n .  The  e x p o n e n t i a l RC  a s p e c i a l case.  s e c t i o n I n c l u d e s the u n i f o r m RC s e c t i o n as 17 18 Furthermore, H e l l s t r o m and Schwartz have  shown o t h e r d i s t r i b u t e d RC the e x p o n e n t i a l RC are concerned.  s t r u c t u r e s which are e q u i v a l e n t to  s e c t i o n i n s o f a r as t e r m i n a l c h a r a c t e r i s t i c s  These e q u i v a l e n t s t r u c t u r e s do not c o n s t r a i n the  5  H  RESISTIVE  FILM  DIELECTRIC FILM m  1, O  CONDUCTIVE FILM  AA/V  2  J  o  + 2  V  SYMBOLC REPRESENTATION  Fig.  1 . 2 - The U n i f o r m  RC  Section.  Fig.  I  5  [  1  J  1.3  -  The  Exponential  HC  Section.  7 r e s i s t a n c e and c a p a c i t a n c e f u n c t i o n s t o v a r y A simple  inversely.  e x t e n s i o n of Wyndrum's s y n t h e s i s  f o r the r e a l i z a t i o n networks t o the case  of d-p immittance  procedure  f u n c t i o n s by u n i f o r m RC  of e x p o n e n t i a l RC networks would  from the p - r c o n f o r m a l  t r a n s f o r m a t i o n w = tanh AJ k  i n s t e a d o f w = tanh ^ s R C '£ .  result  + sRC  t  However, t h i s i s inadequate f o r  the development o f a s y n t h e s i s procedure  u s i n g e x p o n e n t i a l RC  networks. In t h i s d i s s e r t a t i o n , s y n t h e s i s procedure immittance  i t w i l l be shown how Wyndrum's  can be extended t o the r e a l i z a t i o n  f u n c t i o n s by e x p o n e n t i a l RC networks.  s t r a i n t s must be s a t i s f i e d . the  taper constants  and  the s y n t h e s i s procedure  u n i f o r m RC s e c t i o n s .  of d-p  C e r t a i n con-  These c o n s t r a i n t s do n o t a p p l y when  of the e x p o n e n t i a l RC s e c t i o n s become z e r o reduces  t o Wyndrum's procedure f o r  2.  THE CHARACTERISTICS OE UNIFORM AND EXPONENTIAL RC  SECTIONS, AND A SUMMARY OE WTNDRUM'S SYNTHESIS PROCEDURE  2.1  Introduction The c h a r a c t e r i s t i c s  o f u n i f o r m and e x p o n e n t i a l RC  s e c t i o n s w i l l he d i s c u s s e d i n t h i s  chapter.  The t w o - p o r t p a r a m e t e r s o f t h e u n i f o r m RC s e c t i o n and — 12 13 of t h e e x p o n e n t i a l RC s e c t i o n have been d e r i v e d p r e v i o u s l y ' ' 19 20 '  b u t w i l l be p r e s e n t e d  inductance  here f o r completeness.  Series  and s h u n t conductance w i l l be assumed t o be  present  to make t h e a n a l y s i s more g e n e r a l . To show how i t can be extended t o t h e r e a l i z a t i o n o f d-p i m m i t t a n c e f u n c t i o n s by e x p o n e n t i a l RC n e t w o r k s ,  Wyndrum's  p r o c e d u r e f o r t h e r e a l i z a t i o n o f d-p i m m i t t a n c e f u n c t i o n s by u n i f o r m RC n e t w o r k s w i l l be 2.2  presented.  The' Two-Port P a r a m e t e r s o f t h e U n i f o r m RC S e c t i o n Consider  i n P i g . 1.2.  t h e u n i f o r m RC s e c t i o n of l e n g t h t  1 2 9  1 5  shown  The s e c t i o n i n t h i n - f i l m f o r m c o n s i s t s o f conduc-  t i v e , d i e l e c t r i c and r e s i s t i v e l a y e r s .  We have_  Z(x) = R + s L , Y ( x ) = 0 + sC, R = series resistance/unit length at x = 0 L = series inductance/unit length at x = 0, G k shunt c o n d u c t a n c e / u n i t x = 0,  length at  C = shunt c a p a c i t a n c e / u n i t  length at  0,  x =  x = d i s t a n c e a l o n g the s e c t i o n , and The  (2.1)  s = complex f r e q u e n c y . s e r i e s i n d u c t a n c e of t h i n - f i l m RC networks  i s negligibly  s m a l l but i s i n c l u d e d i n the a n a l y s i s f o r c o m p l e t e n e s s . i t i s assumed t h a t t h e d i e l e c t r i c f i l m i s l o s s y * shunt  Because conduc-  tance i s included. By c o n s i d e r i n g an e l e m e n t a l l e n g t h of l i n e , we d e r i v e the t e l e g r a p h e r ' s dV dx = ~  „I -  can  equations  \  T Z ( z ) l  (2.2)  = -T(x)V  w h i c h y i e l d t h e Sturm-Li ouv i l 1 e  d_lY  1 |Z dV _  ,2  =  Z dx dx  dx  equations  d ! l . 1 dT d j _ ,2 Y dx dx dx  =  ( 2 o 3 )  F o r t h e s p e c i a l case of the u n i f o r m (untapered) d7 dY where ~ = 0 and ~ = 0 , dx CLX  line,  t h e s e e q u a t i o n s become  2  M £ - (R+sL) (G+sC)y = 0 dx^ L± _ (G c)(R+sL)I = 0 dx^  .  + s  (2.4)  The s o l u t i o n s of E q u a t i o n s Y = A  y  e  + B_  e  are  (2.4)  10  -i-  .  I = A  i  -tfx e  Yx  -o  +  (2.5)  e  where A , B , A. and B. a r e f u n c t i o n s r e l a t e d t o the boundary conditions, and  =  Y  y / (R+sL)(G+sC) '  (2.6)  A the p r o p a g a t i o n f u n c t i o n of the u n i f o r m RC =  section.  But, from E q u a t i o n ( 2 . 2 ) , I = i.e.  A  =  i  R+sL  v  R+sl  +  (2.7)  v  B v  R+sL  -y  and  l  (2.8)  R+sl  The boundary c o n d i t i o n s a r e , w i t h r e f e r e n c e t o E i g . 1.2, V = V  1  and I = I- a t x = 0,  V = V  2  and I = - I  L  at x =  t.  Hence we o b t a i n _ R+sl  A  T,  R+sL  6  e  T h e r e f o r e , Y_ =  2  +  e**I + I l 2 x  +  2 = G+sC  c  s  c  h  (2.9)  L  coth  ]  V  l  X  J  ^  J  l  +  l  +  Thus, the o p e n - c i r c u i t impedance  C+sC  CO t h * ?  C^C G+sC  c  c  o  s  c  t  ^  h  h  ^  J  J  2  2 (2.10)  matrix i s c h ^ G+sC cs (2.11)  G+sC  CSC  G+sC  CO  11 and  the s h o r t - c i r c u i t admittance matrix i s  Y  R+sL  CO t  -Y  oh Yi R+sL cs  h ^  (2.12)  Y  -Y c s c h Yi  CO thYe  R+sL  R+sL  commonly" used e q u i v a l e n t c i r c u i t s ^ "21" , d e r i v e d  The  1  (2,11) and ( 2 . 1 2 ) , a r e shown i n  from the parameter matrices Pig. 2.1. 2.3  Cascaded U n i f o r m RC S e c t i o n s S i n c e Wyndrum's s y n t h e s i s p r o c e d u r e uses c a s c a d e d  u n i f o r m RC s e c t i o n s , t h e d-p i m m i t t a n c e f u n c t i o n s o f cascaded s e c t i o n s w i l l be d e r i v e d . t h e cascade o f RC s e c t i o n s shown i n P i g . 2.2,  Consider th For the i  s e c t i o n , t h e i n p u t impedance r-r 2 2 i •'• " i + l ' y  (2.13)  s ;  Z where (z2jJ^±* ( i 2 ^lil i ^ "^ (2 2 ^ i) , + ^ ° P ~ i i " k impedance th parameters o f t h e i s e c t i o n and Z^ -^( ) i s t h e i n p u t impedance Z  ( s )  =  (  z  } a n (  z  z  a  Z o o  r  e  e  A  en  c  rcu  s  +  th  of t h e (i+1)  U X i  sect ion. ( 2 . 1 1 ) , we t h e r e f o r e have,  S u b s t i t u t i n g from E q u a t i o n f o r a cascade o f u n i f o r m RC s e c t i o n s , (R^SIK)  Z (s) ±  Y  ±  +  Y GothY 2  c o t h y±e±  ±  +  ±  (G  1  +  B  z  ±  z. (s)  c )  + 1  1  G +sC ±  t a n h YJ  ±  * ~Y~  i z i + 1  (s) "77"  a-+sc.  1 + -^r-±  (s)  i + 1  Z  i  +  1 ^  tanhy.f.  (2.14)  12  Y G-hsC  ianh l  2  O  q  +  + r-r^rCSCh G-t-sC  V  1  YC  V  2  O  Fig.  2.1  -  Equivalent Uniform  RC  Circuits Section.  of  the  Zj (s)  a-V  th  Yj (s)  SECTION  Fig.  2.2  (i+1)  th  SECTION  - Cascade  o f RC  SECTION  Sections.  VJ4  14 = ^/(R  where Y  ±  + s L ) (& +sC ) ±  ±  '  ±  (2.15)  = the p r o p a g a t i o n f u n c t i o n of the i " ^ and R. , L. , G. , C. , t.  section,  are the parameters of the i ^ *  section.  1  th Similarly,  the i n p u t admittance a t the i  a e c t i o n of  an RC cascade i s  i  Y  (  s  )  =  zTTsT i  or  V " > - ^ l l h " (, ^l  T  22  where ( y ^ ) ^ ^ i 2 ^ i of the i ^ *  a n <  ^  s e c t i o n , and  1  ^22 ^ i  a  r  e  ^'  tie  "'  s i l o r  1  +  l  (  a)  } - c i r c u i  < 2  "'  parameters  :  ^ ( s ) i s the i n p u t admittance of the  (i+l)"kk s e c t i o n . T h e r e f o r e , f o r a cascade of u n i f o r m RC s e c t i o n s ,  from  E q u a t i o n (2.12), (G.+sC,) + V. c o t h K f T , ^ ( s ) T.(s) = - i ^ = }f c o t h Y.e 4- ( R s L 7 ) Y (s) 1  1  ±  1  ±  1  1  +  1  ± + 1  i+  R.+sL. R.+sL. 1  1  R.+sl. 1 + - J j r ^ . T  2.4  Wyndrum's S y n t h e s i s Procedure  2.4.1  Realizability  +  i  (  s  )  tanh  v  . . YJ  (2.17)  ±  12-15  Conditions  A d-p immittance f u n c t i o n F ( s ) can he r e a l i z e d by a network  of u n i f o r m RC s e c t i o n s , under the p - r c o n f o r m a l t r a n s -  formation  22 w = tanh Yt  (2.18)  1 6  '  15 if  and  o n l y i f F (s) can be expressed  ( 8 ;  ~~  as  (e *-l)  ^ ( e ^ D . e ^ l )  2  i=i (2.19)  where K i s a r e a l p o s i t i v e c o n s t a n t , y i s d e f i n e d by E q u a t i o n ( 2 . 6 )  9  s i n c e the p - r c o n f o r m a l t r a n s f o r m a t i o n (2.18) a p p l i e s to a l l s e c t i o n s of the network, |N . | , | D . | <f 2 t o ensure t h a t the Immittance under the t r a n s f o r m a t i o n (2.18) i s an LC D ."> N. and  o n l y one  and D. may  coefficient  function  function,  i n the s e t of IT.  be z e r o t o ensure  t h a t the p o l e s and  zeros  J of the t r a n s f o r m e d immittance m = p or m .- p-1  function  t o s a t i s f y the degree  "of the transformed 1C immittance and g ( s ) depends upon the n a t u r e of F ( s ) an impedance f u n c t i o n  or an admittance  P  interlace  requirements function,  i»e  0 ?  whether i t i s  f u n c t i o n , and upon the  v a l u e s of the s e r i e s i n d u c t a n c e s and shunt conductances RC  9  of the  sections. In t h i s d i s s e r t a t i o n . , we  s p e c i a l c a s e s , namely I.  s h a l l be concerned w i t h f o u r  ?  I f F ( s ) i s an admittance i n d u c t a n c e i s not  f u n c t i o n and  series  negligible,  then g ( s ) = 1 + s t ;  (2.20)  16 II„  I f P ( s ) i s an admittance  f u n c t i o n and s e r i e s  inductance i s n e g l i g i b l e , then III,  g(s) = 1  (2.21)  I f F ( s ) i s an impedance f u n c t i o n and shunt conductance  i s not n e g l i g i b l e .  then g ( s ) = s + ~ where IV.  V  s  (2.22)  = 7 ^ = 77 G^ G  I f P ( s ) i s an impedance f u n c t i o n and shunt conductance  i s negligible,  then g ( s ) = s  (2.23)  The p o l y n o m i a l f a c t o r s t h a t appear i n the immittance f u n c t i o n P(s) of Equation polynomials i n t h i s 2.4.2  (2.19) a r e r e f e r r e d  t o as e x p o n e n t i a l  dissertation.  P o s i t i v e - R e a l Conformal  Transformation  Under the p - r conformal t r a n s f o r m a t i o n (2.18) and making use o f E q u a t i o n s t i o n s of E q u a t i o n s  (2o20)-(2.23)  ?  the d-p immittance  (2.14) and (2.17), when p r e m u l t i p l i e d by  "^yjp" t become  „  w +X. f . ~ (w)  where f.(w) and f . -, (w) a r e d-p immittance complex v a r i a b l e w and g ( s ) .  ?  and ^  f u n c t i o n s o f the  depends upon the n a t u r e of P ^ s )  Thus, we have f o u r s p e c i a l c a s e s , namely, I. I I .  func-  I f P.(s) i s an admittance  f u n c t i o n and  17  g(s)  i s g i v e n by E q u a t i o n (2*20) o r (2.21)  thenT.  = R. I.  ° 1  X  (2.25) I  I f I \ ( s ) i s an impedance f u n c t i o n and g ( s )  I I I , IV.  i s g i v e n by E q u a t i o n (2.22) o r (2.23),  then3 = 0 t ±  I t c a n be shown  ±  ~  (2.26)  ±  t h a t t h e f u n c t i o n f^(w) o f  E q u a t i o n (2.24) i s a lumped LO d-p i m m i t t a n c e f u n c t i o n o f t h e complex v a r i a b l e w,  trans•  Note t h a t E q u a t i o n (2,24) i n d i c a t e s a b i l i n e a r formation  between t h e i m m i t t a n c e f u n c t i o n s f^(w) and f ^ ^ ( w ) , +  so t h a t R i c h a r d s ' 2„4»3  Richards'  Theorem may be a p p l i e d . Theorem ^ 2  Richards' cascade s y n t h e s i s  9  2  4  Theorem i s u s e f u l i n t h e development o f a  p r o c e d u r e and may be s t a t e d as f o l l o w s :  I f f ( w ) i s a p - r f u n c t i o n , and i  a f . (w) f.  l+l  \  (w) = 8  x  w f . (a) -P /\  (2.27)  1  ^ a f ( a ) - w fA.v)  1  i  where a, (3 a r e r e a l p o s i t i v e c o n s t a n t s ,  then f ^ ( w ) i s a l s o + 1  a p-r function. Now c o n s i d e r E q u a t i o n (2*24).  t h e d-p i m m i t t a n c e f u n c t i o n f ( w ) o f i  We have  f . (1.) = h-  (2.28)  J i  1  and  f  i + 1  (w) =  f (w) - w ^ i / t . - w f . >w; 01 i ±  J1  f (w) - w f ( l ) i  f  i  ( l )  T TU ^) - ' VV f\  Tf, 7(1) T T ~ T S  =  T  ^  2 L  18 S i n c e f\ (vr) is  a p - r d-p  a p - r d-p  immittance  a = 1 and  8 = f.(oc) = ^k—  .  S i n c e f.(w)  3 i  f u n c t i o n , ^j_ i_(' )  is  w  is  f^(w)  i s an LO  a l s o an LC immittance  of ^ . ^ ( w ) ^  of the form  ^  also  s  (l-w ).  immittance  1  Erom E q u a t i o n (2.29), i t and denominator  function,  f u n c t i o n , by R i c h a r d s ' Theorem, w i t h  1  +  immittance  a v e  function.  i s apparent the common f a c t o r .  a  Thus, fj_ 3_( ) i w  g  +  numerator  This  factor  of lower rank than  so t h a t a cascade s y n t h e s i s procedure can be f o r m u l a t e d .  2,4.4 Given:  S y n t h e s i s Procedure as s p e c i f i e d by E q u a t i o n (2.18), and f-^(w), a immittance  f u n c t i o n which s a t i s f i e s the  d-p  realizahility  c o n d i t i o n s of S e c t i o n 2.4.1. Step 1:  Calculate f-^(l). Hence h- = f ( l ) 01  (2.30)  1  ±  where ^  i s d e f i n e d by E q u a t i o n (2.25) or (2.26).  The parameters  o f the f i r s t RC s e c t i o n can then be  determined through E q u a t i o n s (2.15), and Step 2:  (2.20)-(2.23),  Prom E q u a t i o n s (2.28)-(2.30), w i t h i = 1, and c a n c e l l i n g o the f a c t o r (l-w ) from the numerator and denominator, determine ^j_ 2_(' )> i . e . , w  +  f.(w) f. (w) = f ( l ) + 1  Step 3:  ±  Hence c a l c u l a t e and i  •= f  i + 1  - w f.(1) . * v  f  ( w )  (2.31)  fj_ ]_(l) +  (l)  (2.32)  Ji+1 where "f. , i s d e f i n e d by E q u a t i o n (2.25) or (2.26). l+l 0  The parameters  of the second RC s e c t i o n can then be  19 determined Step  A'  Repeat is  Steps  if  that  The  For  Therefore,  last  the  the  (2. 2 0 ) - ( 2 . 2 3 ) .  »».  » until  f-^w)  cascade  f u n c t i o n , and  is  open-  short-circuited  function. — E x p o n e n t i a l RO  of the  e x p o n e n t i a l RC  e x p o n e n t i a l RC  Z(x)  k  (R+sL)  Y(x)  k  (G+sC) e ~  taper  the  e  constant  Section  s e c t i o n of l e n g t h  19  '  £  2  k  2  z  k  (2.33)  x  of the  f o r the  s e c t i o n and  +  2  -  ¥•  k  -  may  s e c t i o n of E i g .  (R+sl)(G+sC)V =  take  the  be  positive  1.3)•  form  0  (C+sC)(R+sL)I = 0  , (2.34)  solutions V = A  V  e  I = A  e  ±  where A  V  . B  V  (k+/)x  ( ^ k +  . A.  and  k x/k  2  1  x  B.  i  , . -  JX-^Y  p  TT  V  + B. are  e™  ( k  ~^  (2.35!  ) x  functions related  to  the  boundary  conditions, Y  20  section,  Sturm^LiouVille equations  _ 2k g  2 ^4 dx  3,  s e c t i o n i n the  impedance  ( i t i s positive  £l dx  2,  and  1.3»  i s the  or negative  with  3 for i =  Two-Port Parameters  shown I n P i g .  where k  the  admittance  Consider  (2.15),  realized.  i f f-^(w) i s a n  f-^(w) i s a n  2.5  2 and  completely  Note circuited  from Equations  +  (R+sL)(G+sC)'  (2.36)  20 and (Y+k)  and Y-k)  are the propagation  n e n t i a l RO s e c t i o n .  Prom E q u a t i o n  k+Y  I =  -(k-Y)x  ,  R+sL  v  e  i . e . ,'  A.1 = - if—-C B R+sl v  and  B. = A l R+sl v  f u n c t i o n s o f t h e expo-  (2.2),  k-Y ~ R+sL  ^ v  -(k+Y)x e  (2.37)  '.2.38)  The boundary c o n d i t i o n s a r e , w i t h r e f e r e n c e t o R i g . 1.3,  1^ a t z = 0,  V = V-j^ and I = V =  t  and I = -1^ a t x = -(k+Y)£  so t h a t  A = R+sL v k+y B  = -  kg  1 ,1 + 1 2 - e  8  e  R+SL  ke  e  ^  ( k  l  X  ^  +  (2.39)  - e and  V  * G+sC  l =  c o i thV0  + a+sc  Y  ^  k£  + e V. = e  H  + e  c s c h  G+sC  k  X cschYc I G+sC 2ke  Y  G+sC  L  i  x  n  k G+sC  coth  [2 AO)  Thus, t h e o p e n - c i r c u i t impedance m a t r i x i s ^— ~ G+sC  0  t h ^  ^ c s c h Yl G+sC  + G+scy  n  n  H k-e  VL G+sC  C S C H  2~kl( Y  M e  k ' n n COth 0 v - -z r: \ G+sC G+sC (2.41)  21 and  t h e s h o r t - c i r c u i t admittance m a t r i x i s  R+sL,  (2.42) 2.5.1  Equivalent  Circuits  Equivalent  c i r c u i t s f o r the e x p o n e n t i a l  RC s e c t i o n  w i l l now be d e r i v e d . L e t the e q u i v a l e n t  circuit  c o n s i s t of a T - c o n f i g u r a t i o n  of impedances i n cascade w i t h an i d e a l t r a n s f o r m e r . the  o p e n - c i r c u i t impedance m a t r i x  of t h i s e q u i v a l e n t  to the o p e n - c i r c u i t impedance m a t r i x  i n the e q u i v a l e n t  circuit  i s then as shown i n P i g .  circuit  of the i d e a l  can be determined i n  terms o f the parameters o f the e x p o n e n t i a l circuit  equating  (2.41) f o r the e x p o n e n t i a l  RC s e c t i o n , t h e impedances and the t u r n s r a t i o transformer  By  RC s e c t i o n .  The  2.3.  S i m i l a r l y , by assuming a Jt - c o n f i g u r a t i o n of impedances i n cascade w i t h circuit  an i d e a l t r a n s f o r m e r  and u s i n g the s h o r t - c i r c u i t admittance m a t r i x  we o b t a i n the e q u i v a l e n t circuit  i n the e q u i v a l e n t  circuit  can a l s o be obtained  of F i g . 2.4.  (2.42),  The e q u i v a l e n t  by a p p l y i n g the T - J C t r a n s f o r m a t i o n  to the T - c o n f i g u r a t i o n of impedances i n the e q u i v a l e n t  circuit  of F i g . 2.3o These e q u i v a l e n t  circuits,  elements and i d e a l t r a n s f o r m e r s , uniform  except f o r the lumped  are s i m i l a r t o those of the  RC s e c t i o n (as shown i n F i g . 2.1).  The symmetric  G p—VWW-i  G*sC  fan/)  G r-VVVVWi  /2  0  r ^2  sC  O  /DE4Z.  Pig.  2.3  - An E q u i v a l e n t E x p o n e n t i a l RC  F i g . 2.4  Circuit  of  the  Section.  - An A l t e r n a t i v e E q u i v a l e n t of the E x p o n e n t i a l RC  Circuit  Section.  -O  23 p o r t i o n s o f t h e e q u i v a l e n t c i r c u i t s , namely the T-network o f F i g , 2 , 3 and t h e it-network o f F i g . 2 . 4 , do n o t r e p r e s e n t any p h y s i c a l l y r e a l i z a b l e u n i f o r m RC s e c t i o n because o f the n a t u r e o f the Y f u n c t i o n  The e q u i v a l e n t c i r c u i t s become those  a  o f the  uniform  RC s e c t i o n when the t a p e r c o n s t a n t k o f t h e e x p o n e n t i a l RC section i s zero. S y n t h e s i s p r o c e d u r e s f o r e x p o n e n t i a l RC n e t w o r k s w i l l be d e v e l o p e d w i t h the a i d o f these e q u i v a l e n t circuits» A s e a r c h o f t h e l i t e r a t u r e shows t h a t t h e e q u i v a l e n t c i r c u i t s f o r t h e e x p o n e n t i a l RC s e c t i o n have n o t been p u b l i s h e d previously. 2o6  Cascaded E x p o n e n t i a l RC S e c t i o n s  C o n s i d e r a cascade o f e x p o n e n t i a l RC s e c t i o n s w i t h t h e th i o n of F i g . 2 . 2 . I c o n f i g u r a t i o n o f F i g . 2 . 2 . F o r the i " s e c t i o n , f r o m E q u a t i o n s ( 2 . 1 3 ) and ( 2 . 4 1 ) , we o b t a i n 2k. I. (R.+sL.) e n  /  1  \  =—  V ) s  1  1  1  Y  kjk^  ±  1  1  i  i  (s)  1  1+1  1  i  +(R +sL )(G +sC ) i  + k.) Z  1  2k.e. - k.) e + (G +sC ) Z  (Y± c o t h YJ± where  + (Y, c o t h Y. I  1  i  7  i  (s) (2.43) (2.44)  i + 1  and k^, R^, L^, G^, C^, &^ are the parameters o f t h e i^* s e c t i o n . 1  S i m i l a r l y , from E q u a t i o n s -2k.  (G.+sC.) e  y  V  I  s  1  \  ;  1  -  1  1  ( 2 . 1 6 ) and ( 2 . 4 2 ) ,  + (Y.  cothY.t.  1  - k.) Y. , (s)  1  1  X  1+1.  -2k.e. (K  coth Y L ±  + k.) e  1  1  + (R +sL ) Y i  ±  i + 1  (s)  (2.45) F o l l o w i n g Wyndrum's s y n t h e s i s t e c h n i q u e RC n e t w o r k s , the d-p immittance may  f o r uniform  f u n c t i o n s ( 2 , 4 3 ) and ( 2 , 4 5 )  be p r e m u l t i p l i e d by f a c t o r s o f the form  1  and t h e p - r  24 conformal  transformation  (2,18) a p p l i e d t o the r e s u l t a n t f u n c -  t i o n s , where g ( s ) i s g i v e n by one of the E q u a t i o n s and  Y by E q u a t i o n  tance  (2.36).  (2.20)-(2.23)»  However, the transformed  d-p immit-  f u n c t i o n s cannot be i d e n t i f i e d w i t h lumped-parameter d-p  immittance f u n c t i o n s , so t h a t no s y n t h e s i s procedure u s i n g Wyndrum's t e c h n i q u e  can be developed.  Attempts a t f i n d i n g some  other p - r conformal  t r a n s f o r m a t i o n t o accomplish  the change t o  lumped-parameter d-p immittance f u n c t i o n s proved u n s u c c e s s f u l . Comparison of the d-p immittance f u n c t i o n s and  (2.45) w i t h the e q u i v a l e n t c i r c u i t s  and w i t h those  (2.43)  of F i g s . 2.3 and 2.4?  of the u n i f o r m RC s e c t i o n , l e a d s t o the f o l l o w i n g  conjecture. 2„7  Conjecture I t i s c o n j e c t u r e d t h a t a cascade s y n t h e s i s procedure  using Richards' tance and  Theorem  '  f o r the r e a l i z a t i o n o f d-p Immit-  f u n c t i o n s by e x p o n e n t i a l RC networks can be developed i f  o n l y i f the lumped elements i n the e q u i v a l e n t c i r c u i t s of  the e x p o n e n t i a l RC s e c t i o n s can be e l i m i n a t e d . elements correspond parameter m a t r i c e s  The lumped  t o t h e r a t i o n a l terms t h a t appear i n the (2.41) and (2.42).  In the f o l l o w i n g , and i n Chapter 3, the s u f f i c i e n c y part  of t h i s c o n j e c t u r e w i l l be demonstrated by d e v e l o p i n g  s p e c i f i c s y n t h e s i s procedures.  No proof  of the n e c e s s a r y  p a r t of the c o n j e c t u r e has been found. 2.7.1  Cascaded E x p o n e n t i a l RC S e c t i o n s Under the R e s t r i c t i o n of the C o n j e c t u r e Let us r e c o n s i d e r the cascade of e x p o n e n t i a l RC  25  s e c t i o n s o f Fig» 2.2, the lumped elements  I f i t i s assumed p o s s i b l e t o e l i m i n a t e  o f t h e e q u i v a l e n t c i r c u i t o f F i g , 2.3,  namely t h e c a p a c i t o r s and r e s i s t o r s , f o r each e x p o n e n t i a l RC -t-M  s e c t i o n , then, f o r the I " s e c t i o n ,  (  z  l l i  = O^tcT  }  *iA A  coth  k ( z  ( z  12 i = }  22 i = }  ( z  6  21 1  ) i  =  =  s  risrr 1  1  Vi W~Ac-  1  W7Ttc-  1  e  V >  e  1  c  2k.c. 1  1  I E I e  1  o  t  h  c  V  s  c  ^  h  i  i  , „  t a n h YA  -  ,  + (G.+sC.) Z. , ( s ) / tf, 1  1  + (G.+sC.) Z . ( s )  1  + 1  t r " f  1  tanh  Y l / ^. ± ±  (2.46)  I t w i l l he shown l a t e r t h a t a p - r c o n f o r m a l t r a n s f o r m a t i o n c a n he a p p l i e d t o t h e d-p impedance Z ^ ( s ) and a cascade s y n t h e s i s p r o c e d u r e d e v e l o p e d w i t h t h e a i d o f R i c h a r d s ' Theorem. S i m i l a r l y , i f i t i s assumed p o s s i b l e t o e l i m i n a t e the lumped elements  o f t h e e q u i v a l e n t c i r c u i t o f F i g . 2.4,  namely t h e i n d u c t o r s and r e s i s t o r s ? f o r each e x p o n e n t i a l RC th s e c t i o n of t h e cascade, then f o r the i section, -2kA. , Y. e t a n h Y I + (R.+sL.) T (s) /Y, V R.+sL. -2k. t, ^ ~~ ~ e + (R.+sL.) Y. _(s) t a n h Y.t. /Y. n  1  s  1  =  :  1  1  1  1  x  1  l+l  1  1 '  (2.47)  A p - r c o n f o r m a l t r a n s f o r m a t i o n and R i c h a r d s '  Theorem  can a l s o be a p p l i e d t o t h i s d-p a d m i t t a n c e f u n c t i o n t o d e v e l o p  1  26  a cascade s y n t h e s i s The  procedure,  above d i s c u s s i o n has  ments i n the e q u i v a l e n t can be e l i m i n a t e d .  c i r c u i t s of the e x p o n e n t i a l RC  sections  Some n e t w o r k c o n f i g u r a t i o n s t h a t w i l l accom-  plish this elimination The  assumed t h a t the lumped e l e -  w i l l be p r e s e n t e d  i n Chapter  p-r conformal t r a n s f o r m a t i o n s  t h a t may  3. be  applied,  under the r e s t r i c t i o n of the conjectures, t o change the p r o b l e m i n t o a lumped-parameter s y n t h e s i s p r o b l e m w i l l be  discussed  next o 22  2.7.2  P o s i t i v e - R e a l Conformal Transformations I f the e x p o n e n t i a l RC d-p  i n the previous form  immittance f u n c t i o n s  s e c t i o n are p r e m u l t i p l i e d by f a c t o r s of  ^ , where g ( s ) depends upon the n a t u r e of the  f u n c t i o n and  upon the v a l u e s  ( 2 . 2 3 ) i n S e c t i o n 2.4°l)j>  s e c t i o n s (see E q u a t i o n s  the  immittance  of the s e r i e s i n d u c t a n c e s  shunt c o n d u c t a n c e s of the RC  derived  and (2.20)-  and  YI k y e ±  ±  = y k + ( R + s L ) (G+sC) ' I  (2.48)  2  and  i f the p - r c o n f o r m a l  transformation  w = t a n h Ye  (2.49)  i s a p p l i e d , t h e n t h e r e s u l t a n t d-p be r a t i o n a l i n the new ( 2 . 4 6 ) and  immittance f u n c t i o n s  complex v a r i a b l e w.  will  Thus, E q u a t i o n s  ( 2 . 4 7 ) become, r e s p e c t i v e l y , Z  I  (  W  Yi  )  1  w  Z  J  1  e  1  /' O i + z"j+1( w ) T  t  2k i  ( S )  2kl  w e«  \p  1  1  1  i + 1  1  /\t  ±  + *  1 + J  z i  +  i ^ )  (2.50,  and y.(w)  ^  i  ! :  YI  Y. ( s ) w  Jl _  e  -2k.2 1  .  1 1 "/ 3  . (w)  i  -2k.t,  1 1  .+ y  /3±  +  (2.51)  i i i — w  where ^\ i s given hy Equation  y  i  +  i^)  (2.25)  or ( 2 . 2 6 ) .  These d-p  immittance f u n c t i o n s are lumped LC f u n c t i o n s of the complex v a r i a b l e w. Because the transformation of the exponential RC d-p immittance f u n c t i o n i n t o a lumped LC d-p immittance f u n c t i o n i s done a t the s t a r t of the synthesis procedure ( E i g . l . l ) ,  Yt i s common t o a l l s e c t i o n s of the network. Instead of the p-r conformal transformation the a l t e r n a t i v e p-r conformal w = coth might be used.  (2.49)?  transformation  Yt  The.resultant transformed  (2.52)  d-p immittance f u n c t i o n s  considered would then be dual t o those obtained using the t r a n s formation ( 2 o 4 9 ) .  Except f o r the change i n correspondence w i t h  lumped immittances i n the w-domain, the use of the  transforma-  t i o n ( 2 . 5 2 ) would not g r e a t l y a f f e c t the synthesis procedure, so t h i s a l t e r n a t i v e w i l l not be pursued f u r t h e r . 2.7.3  R e a l i z a h i l i t y Conditions Because of the synthesis approach of R i g . 1 . 1 and the  p r o p e r t i e s of the p-r conformal transformation of Section 2 . 7 * 2 , the necessary and s u f f i c i e n t c o n d i t i o n s f o r a d-p immittance F(s) t o be r e a l i z a b l e by a network of exponential RC s e c t i o n s can be derived by c o n s i d e r a t i o n of the r e a l i z a h i l i t y c o n d i t i o n s  28 f o r the c o r r e s p o n d i n g lumped LC d-p immittance  f u n c t i o n f (w) ;  namely, t h a t f(w) i s r e a l i z a b l e i f and o n l y i f i t can be e x p r e s s e d as -  , 2  P  (W"  7t  ( H  + n.  +  V  3=1 where K  i s a real positive  constant,  m = p or m = p-1, n., d. a r e r e a l and p o s i t i v e , and  (2.54)  d.>n.o  S u b s t i t u t i n g from E q u a t i o n (2.49) and p r e m u l t i p l y i n g by the factor  Y? / \  Sis;  , where g ( s ) i s g i v e n by E q u a t i o n s  (2.20)-(2.23),  we o b t a i n  (2.55) TV where  E  A "  (n.+l)  ' j = l _ l _ m TT (d,+i)  3=1  n.-l  (2.56)  D  .  29 Therefore,  from E q u a t i o n s  (2.54) and (2.57), we  have  2+11.  and  (2.59:  f o r n . t o be r e a l and p o s i t i v e J  111 1 < 2  (2.60)  J  Similarly,  |D.|<2  (2.61)  A l s o , s i n c e d.>n.. therefore, and  D.>N.  (2.62)  o n l y one c o e f f i c i e n t  i n the s e t of N, and D. may  be z e r o .  Thus, we have the r e a l i z a h i l i t y c o n d i t i o n s : A d-p immittance f u n c t i o n E ( s ) under the p - r c o n f o r mal t r a n s f o r m a t i o n w = tanh Yi,  can be r e a l i z e d by a network  of e x p o n e n t i a l B.C s e c t i o n s ( p r o v i d e d t h a t the lumped elements i n the e q u i v a l e n t c i r c u i t s of these s e c t i o n s can be if  and o n l y i f E ( s ) can be expressed  as  * ( e ^ F  (  s  )  > ^ ( P  K  2  ^ 1 )  2  (  -  P  eliminated)  )  +  1  +  N . e  2  ^  +  l )  A  j=l (2.63) where K i s a r e a l p o s i t i v e  constant,,  i s g i v e n by E q u a t i o n  (2.48) ,  g(s) i s g i v e n by one of E q u a t i o n s m = p or m = p-1, IN  .  L  I  D ,  D . > N .. .  I <  2,  (2.20)-(2.23) ,  30 and 28 a  o n l y one c o e f f i c i e n t  i n the s e t o f N. and D. may be z e r o . j J  Discussion I t has been shown t h a t , i f i t i s p o s s i b l e t o e l i m i n a t e  the lumped elements from the e q u i v a l e n t  circuits  t i a l RC sections,, t h e d-p immittance f u n c t i o n s  o f t h e exponen-  of e x p o n e n t i a l  RC networks can be t r a n s f o r m e d i n t o lumped-parameter d-p immittance f u n c t i o n s o Equations  The t r a n s f o r m e d immittance f u n c t i o n s (see  (2,50) and (2,51). ) are b i l i n e a r t r a n s f o r m a t i o n s  the i n p u t immittances and the t e r m i n a t i n g Richards'  i m m i t t a n c e s , and  Theorem can t h e r e f o r e be a p p l i e d i n a cascade  procedure t h a t w i l l exponential  between  synthesis  extend Wyndrum's procedure t o t h e case o f  RC n e t w o r k s  B  Some network c o n f i g u r a t i o n s of e x p o n e n t i a l  RC s e c t i o n s  and  lumped elements t h a t w i l l meet t h e r e q u i r e d c o n s t r a i n t s ,  and  the s p e c i f i c  developed, w i l l  s y n t h e s i s procedures t h a t can be c o n s e q u e n t l y be p r e s e n t e d  i n the next  chapter.  3.  3.1  THE SYNTHESIS OE DRIVING-POINT IMMITTANCE FUNCTIONS  Introduction The  c o n j e c t u r e d i s c u s s e d i n Chapter 2 r e q u i r e s the  e l i m i n a t i o n o f t h e lumped e l e m e n t s f r o m t h e e q u i v a l e n t  circuit  of each o f t h e e x p o n e n t i a l RC s e c t i o n s b e f o r e a s y n t h e s i s cedure u s i n g Wyndrum's t e c h n i q u e  pro-  can be f o r m u l a t e d =. These  lumped e l e m e n t s p r o d u c e f i n i t e i m m i t t a n c e p o l e s , i f t h e s e r i e s inductances  o r t h e shunt c o n d u c t a n c e s o f t h e RC s e c t i o n s a r e  n o t n e g l i g i b l e , and i t i s n e c e s s a r y p o l e s be n e g a t i v e  t h a t the r e s i d u e s a t these  or zero.  Provided  that the taper constants  of the e x p o n e n t i a l  RC s e c t i o n s a r e a l l o f t h e same s i g n , a s i m p l e  cascade o f s e c -  t i o n s t o g e t h e r w i t h c o m p e n s a t i n g lumped elements might s a t i s f y the c o n s t r a i n t s i m p l i e d by t h e c o n j e c t u r e . dance f u n c t i o n o f t h e form g i v e n by E q u a t i o n  Thus, a d-p impe(2.63) c a n be  r e a l i z e d by a n e t w o r k w i t h t h e c o n f i g u r a t i o n of P i g . 3.1? i f the t a p e r c o n s t a n t s  o f t h e RC s e c t i o n s are- a l l n e g a t i v e , t h e  last section i s open-circuited —  f  and t h e r e s u l t a n t r e s i d u e o f 1 A °-  adjacent  RC s e c t i o n s a t t h e p o l e , s = - =- where Z' = , *-s i i s n e g a t i v e o r z e r o . Compensating elements can t h e n be added i n s e r i e s w i t h t h e s e c t i o n s t o produce a z e r o r e s i d u e a t t h e pole, s = - ~ , S i m i l a r l y , a d-p a d m i t t a n c e f u n c t i o n can be r e a l i z e d by a n e t w o r k w i t h t h e c o n f i g u r a t i o n o f F i g . 3°2, i f t h e t a p e r constants  o f t h e RC s e c t i o n s a r e a l l p o s i t i v e , the end s e c t i o n  i s s h o r t - c i r c u i t e d , and t h e r e s u l t a n t r e s i d u e o f a d j a c e n t  where T  RC s e c t i o n s at the p o l e , s = or zero.  P  = ^ P  , i s negative 1  Compensating lumped elements can then be added i n  shunt w i t h the s e c t i o n s to produce a zero residue at the p o l e ,  However, the above cascade network c o n f i g u r a t i o n s w i l l only r e a l i z e a r e s t r i c t e d c l a s s of d-p immittance funct i o n s , namely, those which w i l l ensure that the r e s u l t a n t r e s i due at the pole produced by the lumped elements i n the  equiva-  l e n t c i r c u i t s of adjacent RC s e c t i o n s i s negative or zero. The p o s i t i v e lumped elements at one port of the equivalent c i r c u i t of an exponential RC s e c t i o n can be i n e f f e c t i v e by s u i t a b l y t e r m i n a t i n g the s e c t i o n .  rendered  That i s , i f  the equivalent c i r c u i t of P i g , 2.3 i s used, the s e c t i o n should be o p e n - c i r c u i t e d as i n P i g , 3»3» but i f the equivalent c i r c u i t of P i g . 2 , 4 i s used, the s e c t i o n should be s h o r t - c i r c u i t e d as i n F i g . 3.4»  The negative lumped elements at the input ports  of these terminated s e c t i o n s provide negative residues at the poles produced by these elements.  Therefore, the o p e n - c i r c u i t e d  s e c t i o n can be used as a s e r i e s stub i n s t e a d of, or together with  ?  the compensating lumped elements i n the network c o n f i -  g u r a t i o n of F i g . 3 , 1 ,  while the s h o r t - c i r c u i t e d s e c t i o n can  be used as a shunt stub instead of  f  or together w i t h , the compen-  s a t i n g lumped elements i n the network c o n f i g u r a t i o n of F i g , 3 » 2 , These network c o n f i g u r a t i o n s , together w i t h the r e l e v a n t synthe' s i s procedures, w i l l be discussed next.  . COMPENSA TING  i-1)  th  ELEMENTS  SECTION  SECTION  F i g . 3.1  Cascade of Exponential RC Sections w i t h Compensating Lumped Elements Added i n Series w i t h the Sections.  (ii-1)  th  SECTION  SECTION  F i g . 3.2  - Cascade o f E x p o n e n t i a l RO S e c t i o n s w i t h Compensating Lumped E l e m e n t s Added i n Shunt w i t h t h e S e c t i o n s .  G  r-VWn  -4  O csch  y<f  s  sC  O  Pig.  3.3  - Equivalent Circuit Open-Circuited Section.  of  An  Exponential  RC  GfsC  -Xcsch R+sL  tt O  Y^ coth re R-hsL  O  F i g . 3.4 - Equivalent C i r c u i t of A S h o r t - C i r c u i t e d Exponential ..RC Section.  Shunt.  39 3.2  S y n t h e s i s P r o c e d u r e s f o r Cascaded  S e c t i o n s •with S h o r t -  C i r c u i t e d Stubs and Lumped E l e m e n t s i n Shunt C o n s i d e r t h e network c o n f i g u r a t i o n of cascaded  sections  w i t h s h o r t - c i r c u i t e d s t u b s and lumped elements i n shunt ( P i g , 3»5) and i t s e q u i v a l e n t c i r c u i t  ( P i g . 3.6). The l a s t s e c t i o n of t h e  cascade i s t e r m i n a t e d i n a s h o r t  circuit.  To e l i m i n a t e t h e lumped e l e m e n t s , we r e q u i r e at the input  junction,  _  l l I  _  I  .  (R +sl )S 1  1  !  1  I  l 11  II  , « t)  (R +sL )£ 1  1  "t"  UI  !!!  R-L + s L  1  _ 0  :  1  —  A  e  ^  2  k  ( : L: ) c R  + S  i+lA+l  (<  + 1 + S  L:  + 1  , )  (3  1  k  junction, i  t  (<  ^  i  +  +  1  +  S  1  „  L:  i_ +  1) J ° -- /  \  w  x  and a t t h e s e c o n d , and each s u c c e e d i n g , cascade k  that,  +  1  )^  +  1  (3»2)  in  .. + s l . l+l l+l  R.  where i = l ,  2, 3» <• • • , n-1,  and where t h e number o f cascaded s e c t i o n s i s n, in  To s a t i s f y in  E q u a t i o n s (3»l) and (3*2) w i t h n o n - n e g a t i v e R ^ and L^, i t i s necessary that k! <0 I  (3.3)  \>0  But Yt and k? a r e common t o a l l s e c t i o n s of t h e network a s i n g l e p - r c o n f o r m a l t r a n s f o r m a t i o n i s used. 1  1  1  1  because  Therefore, l e t >  '  40  •where k i s p o s i t i v e . Then E q u a t i o n s (3.1) and (3,2) become, r e s p e c t i v e l y , i  1  g(s)  +  , = o R, T  f e  1  2  r  " : e :- +  g(s)  R  1  1  f  I. £ . . R. A. l+l l + l l+l l+l  where g ( s ) i s g i v e n by E q u a t i o n ( 2 . 2 0 ) i  :: 1. A __1 p  n  in  1_  1  L.  i+lj (3»5)  o r ( 2 , 2 1 ) , and  l (2.44), -2k.i.  e' I  i\  1  I  I  R.+sL. 1  1  (2.45) and (3,4), we have . ,  ,  1  TT TT COth V R.+sL.  ^  1  ±  1  (s) /  v:  T  -2k.I. e " ~ + (R +sL ) T 1  ,  t a n h O.K. + (R.+sL. ) Y i i i+1 i  +  0  R.  l  From E q u a t i o n s  Y.(s)  R  L.  R.  R. i  + -fc— > =  n  n  ±  ± + 1  (s)  ft.  tanh l  I  ' l  ±  1  1  ?(s)  + Rl[  tanh  R j  i^i 1 + R Z e ±  2M  ±  e  2k e  Y  tanhVe  Y  i i i  (s) g(s) i + 1  (s)  •  g(s)  /H  +  R  (3.6)  where g ( s ) i s g i v e n by E q u a t i o n (2.20) o r (2.21), and h e r e i ~ 1 ^ 2 • j5 •  »9  9  f lie  Under t h e p - r c o n f o r m a l t r a n s f o r m a t i o n (2.49), we o b t a i n  2kg i R  .  e  E  .  1 1  I  +  RX: i i  y  2H  i  +  y  i  U  1 + 3  )  >) «  +  w  (3.7)  i, „'coth  A  41 where y  U)  i + 1  Yt  T  1 + 1  (e)  T h i s i s an LC d-p immittance f u n c t i o n i n the w-domain, because w  7j_+-|_(s) = OO f o r a s h o r t - c i r c u i t t e r m i n a t i o n , so t h a t y i + 2.^ ^ =  0 0  and  1  y ± (w) =  1  +  _x  X x_  X  (3.8)  w  which i s an i n d u c t i v e admittance i n the w-domain.  (3.7),  Prom E q u a t i o n  (1) = -±-r  y,  " R.g .  1  n  (3.9)  + -+-iT R.g. X X  and y ± (w) - w y ± ( l ) - (l/w - w) /  2H y  i1 ++ xl  ( w )  = '„* R.g.  y±(l)  xx  e  R  x i  [yi  -  1  /  R  ±  j [  ^ ( w ) - l / ( R ^ w)_ ( 1 )  ±  - w y (w)  2kg e  Rt  - w Qy.(l) y (w) - l / ( R ^ wf]  i i ] e  w  ±  (3.10) Because the s h o r t - c i r c u i t e d shunt stub has an admittance  1 R.Z.  1 , i t r e a l i z e s p a r t of the p o l e of y (w) a t w = 0 . i  w  I f the  x x i s c o m p l e t e l y removed, then y ( w ) w i l l have a zero a t pole i+1  w=0 and t h i s tion.  cannot be r e a l i z e d by the network under c o n s i d e r a -  Therefore,  the parameters  stub must be chosen such  that  of the s h o r t - c i r c u i t e d shunt  -1— ii  ti  / w y±(w) \  w=0  (3.11:  R.g . T h i s requirement may c o n f l i c t w i t h x xthe c o n s t r a i n t imposed by the conjecture  ( g i v e n by E q u a t i o n s ( 3 - 5 ) ) ,  i n which case the  s y n t h e s i s procedure breaks down. If Inequality  (3.11) I s s a t i s f i e d , then  i s a p-r f u n c t i o n because y^(w) and  y±(w)  xx  42 ii I I — a r e p - r . Hence, by R i c h a r d ' s Theorem (see E q u a t i o n 2.27)?  Vi  y - , i ( w ) , as g i v e n by E q u a t i o n  (3.10). i s a p . r . f u n c t i o n w i t h  n  a=l and 8 = — — - , a r e a l p o s i t i v e  constant.  V i A l s o , yj_ ]_("w) i s a r e a l i z a b l e  LO d-p  +  f u n c t i o n because y^(w) i s an LC d-p admittance the f a c t o r minator  (l-w ) can be c a n c e l l e d 1  i+1  ii  R 1  -e  X  1  ,I  1  +  (1) -  1  Rj_&j_r  (3*12), so t h a t d i f f e r e n t  II  .  II  R. A . , l+l i + l  ii  a  R  combinations  = 0  i+l  of the v a l u e s  n  ^  will  satisfy  networks may be r e a l i z e d  function.  Only two of these  and lumped  f o r the  possible  a minimum number  elements,  the r e a l i z a t i o n which f a c i l i t a t e s implementation  Equations  t  the r e a l i z a t i o n which r e q u i r e s of RC s e c t i o n s  (2)  1  in  II  w i l l be c o n s i d e r e d h e r e , namely  (l)  +  (5.12)  a r e many p o s s i b l e  same d-p immittance  = 0  in  R.  1  X  i t  realizations  Z  1  + 7A  R.e.  of the parameters  +  n  n  i  2ke  -r-r  gls.  There  y (w).  3  kg  Since  from the numerator and deno-  +  gls,  3.2.1  function.  of y ( " w ) ? 7± ±^^ °^ lower rank t h a t E q u a t i o n s (3»5) may be r e w r i t t e n as 1  and  admittance  i n thin-film  practical  form.  R e a l i z a t i o n w i t h a Minimum Rumber of RC S e c t i o n s and Lumped Elements Erom E q u a t i o n s  (3=9) and (3.12), i f — —  = 0, then  43 t  !  V i  2k£ if e, , >  Also,  (3.13)  It - ti  A  R  7,(1)  1  —  y. . ( 1 ) , w i t h -  R  it  P  0,  it  I+I i+1 1  t h e n —; R  and  ;—  i+l^i+l  -  — — = kt R. . i+I  hut,  if  then  2ke i tf ^  R i. ri  i  y i  +  (  1  )  2k£ e  t  - ^i+i  i  (3.14)  ( 1 )  R.e. i i  y  i + 1  d)»  i  with ,„ - °» i+i £  -n—-n— i+A+1  _2kr  1  1 + 1  2  R. e.  R  and  i  — i — = y . ,-,  —  i+l i+l  1  (l)  Synthesis  Step 1:  ±  + 1  + 1  (D  + V-r R.£.  (3.15)  i i  procedure i s t h e r e f o r e  as  follows  as s p e c i f i e d by E q u a t i o n s (2.48), (2.49) and  the r e a l i z a b i l i t y c o n d i t i o n s o f S e c t i o n 2,7-3  where r  (w H) JI  ^ • ^ > —l ^ — y  Q  y i  Erom E q u a t i o n (3.13), ~T  1 p^e and  J1  (3.4) and y-j_(w), a d-p a d m i t t a n c e f u n c t i o n w h i c h  satisfies and  2kf  Procedure  kt and ye. and  2  1  The s y n t h e s i s  Given;  1  R.1+1 A . 1+1  c  3.2.1.1  i  d)  y (i) x  1"  1_ t  R.1  (3.16)  44 (2.49) and  Hence, by E q u a t i o n s of  the f i r s t  circuited  m a i n c a s c a d e RC  shunt  ments a r e n o t Step  2:  s t u b c a n be  y. , (w) = 1  +  section  the and  determined.  parameters i t s shortLumped  ele-  required.  Erom E q u a t i o n ( 3 . 1 0 )  i.e.,  (3-4),  1  determine  2kg  y (w)  , , R.e.  "  y^^"")*  - w y (l) -  ±  y.(1)  w)/R £  (l/w  ±  i  - w y.  i  (w)  i i  (3.17!  w h e r e i = 1, from  remembering to c a n c e l  the numerator  T  — I  Now|w y  i + 1  must be  and n  |  |  U)J  the  w = = 0  ^i+l +  ( 1 )  -  1  Y£,  k£ and 2  Step  3:  y ( w ) , must be n  k  factor  (3.is:  R.e.  I f the i n e q u a l i t y  t h e n t h e p r o c e d u r e b r e a k s down and new  )  _2k^ 1  satisfied.  (l-w  denominator  V  >2  the  i s not  a fresh  satisfied,  start,  with  made.  e  e  I f y.  1  1  (l)4 i—~t  then, from Equations  —  1  +  (3.14),  R.e.  1  ^ i i+ i1 ^  W i + 1  1 = 0 R. 1. -, i+l i+l 1  and — T1 n — : = k£ R  2ke i  y  i  (D  3.19.  .R.e. i+l Lii i 2k£ i f y. T ( 1 ) > — J — t h e n , from E q u a t i o n s (3.15) R. t, n  but  1 + 1  1  +  1  l  l  4S  1  1  R. J .  2  2ke  y  1 + 1  (D  e  +  n  I I  l+l i+I  1  ii  1 y 2  II  R. £ . _ 1+1 1+1 and 1 = 0 i  2te  (1) 1  +  -  ^ T - T  R.0.  1  l  n  l  (3»2o;  II  R  i+1  Hence t h e parameters o f t h e second main cascade RC s e c t i o n and i t s s h o r t - c i r c u i t e d shunt s t u b , o r i t s lumped elements i n s h u n t , c a n be d e t e r m i n e d . Step 4:  Repeat S t e p s 2 and 3 f o r i = 2, 3, ..., n-2 u n t i l K y (w) o f t h e form — remains, where K i s a r e a l •^n' w ' n p o s i t i v e constant.  Step 5:  or  y (' ) w  c  a  n  D e  n  realized  (i)  i n e x a c t l y t h e same way as g i v e n by Step 3?  (ii)  by u s i n g E q u a t i o n s (3.19) i f y ( l ) ^ .  e  2kl  R I n-l n+l 1 = 0 so c  2k£  but i f y ( 1 ) > ^ T n-l^n-l  , then l e t  R  t h a t -Ti—rr = y n, ( l ) and R I  i  i  R £ n n  r  _1  R ii i n or  "2ke  ke  + R  y (D n  3»21)  n-l^n-l  ( i i i ) by u s i n g E q u a t i o n s (3.21) whatever t h e v a l u e of y ( 1 ) . The l a s t main cascade s e c t i o n i s t e r m i n a t e d i n a short c i r c u i t .  Note t h a t the s y n t h e s i s procedure  can be used what-  e v e r the v a l u e s of the s e r i e s i n d u c t a n c e s and shunt t a n c e s of the RO (Equations  3.2.2  sections.  (2.20) and  Yt  ( E q u a t i o n (2.48)) and  Prom E q u a t i o n s  (3-9) and  Implementation  (3-12), i f  = 0, R  -4-T R  A  (3*22)  V l  2  1  I  !  It  —  R. , I. , i+I i + l  R  then  ,  R_L  R^  =  d±  = i±  =  l  r  '  R^  A  = £ u  .L  ±  ±  l  )  C -,  1+1  (3 = 23)  1  ,  i = 1, 2, 3»  ,  from E q u a t i o n s  .. ., n (3.22) and  (3.23)J  ^ ±  ±  T h e r e f o r e , the main cascade  and  (  = L . ,  k^ = -k? = k  shunt  i+l  A  R  L. = and  1!  R.  1+1  2  i+1  If  y  = ^ T... T T - = He *  -kr~ in  then  l  = -4-n- =  1  Let  g(s)  (2.21)) must be m o d i f i e d a c c o r d i n g l y .  R e a l i z a t i o n which F a c i l i t a t e s P r a c t i c a l  then  conduc-  , from E q u a t i o n ( 3 . 4 ) .  (3-24)  s e c t i o n and i t s s h o r t - c i r c u i t e d  stub form a continuous s e c t i o n  as shown i n P i g .  can be c o n s i d e r e d as a c e n t r e - t a p p e d s e c t i o n .  3.7,  Practical  r e a l i z a t i o n i n t h i n - f i l m form i s then g r e a t l y f a c i l i t a t e d fewer  e v a p o r a t i o n mask shapes have to be made.  From E q u a t i o n  (3.10), we  have  because  48 y (D ±  y  i+l  (  w  )  2 k  g  7M  dA  - v y (i) -  ±  ±  =  y (l) ±  - w) y ( i ) / 2 ±  - v  y (w) ±  (3.25) The s y n t h e s i s p r o c e d u r e i s s i m i l a r t o t h a t o f t h e p r e v i o u s s e c t i o n and i s as f o l l o w s : 3.2.2.1 Given:  Synthesis Procedure kl  and Yl,  and  as s p e c i f i e d hy E q u a t i o n s (2.48),  (3.4),  and y-^(w) , a d-p a d m i t t a n c e f u n c t i o n w h i c h the  (2,49)  satisfies  r e a l i z a h i l i t y c o n d i t i o n s of S e c t i o n 2.7.3 y^D y ( w ) w=02  w  and where  >  1  Step 1: From E q u a t i o n s (3.22) and  (3.24),  y (D  1 CT  x  1*1 I  and —  R  '3.26]  = 0 1  Hence, hy E q u a t i o n s (2,48), (3.4) and (3.23), the parameters of the f i r s t main cascade s e c t i o n and i t s s h o r t - c i r c u i t e d shunt s t u b , i . e . , the f i r s t c o n t i n u o u s s e c t i o n , can be d e t e r m i n e d .  Lumped elements a r e n o t  required. Step 2: From E q u a t i o n (3.25), d e t e r m i n e y - ( w ) , i . e . , n  y  y. (w) = + 1  i  (  2  l  ) e  2kg  J  ±  M  ~ * i y  y  i  j l  " (^-w) ¥ d)/Z ( l ) - w y. (w) ' (  l  )  ±  (3.27) where i = 1, remembering t o c a n c e l the (l-w ) f a c t o r from the numerator and  denominator.  49 Now,  W  y  i+l U  |w=0/ > 2^ — J  must be s a t i s f i e d .  =  (3.28)  0  I f the i n e q u a l i t y i s not  satisfied,  then the procedure breaks down and a f r e s h s t a r t must be made w i t h new Step 3:  Yt,  k£ and y  From E q u a t i o n (3.23),  R  i i+A+i  y =  1 + 1  (D  2  and -#r— = ke R. , i+l  — ^  Hence, the parameters  Repeat is  (3*29)  of the second continuous  and the lumped elements Step 4:  (w).  i n shunt can be  Steps 2 and 3 f o r i = 2, 3,  completely r e a l i z e d .  The l a s t  section  determined.  ..., u n t i l y-^(w) section i s short-  c i r c u i t terminated. As i n the p r e v i o u s case, the s y n t h e s i s procedure be used whatever conductances (Equations 3.3  the v a l u e s of the s e r i e s i n d u c t a n c e s and  of the RC s e c t i o n s .  (2.20) and  Yt  shunt  ( E q u a t i o n (2.48)) and  g(s)  (2.21)) must be m o d i f i e d a c c o r d i n g l y .  S y n t h e s i s Procedures f o r Cascaded Circuited  can  S e c t i o n s w i t h Open-  Stubs and Lumped Elements  i n Series  C o n s i d e r the network c o n f i g u r a t i o n of cascaded  RC  s e c t i o n s w i t h o p e n - c i r c u i t e d stubs and lumped elements i n s e r i e s as shown i n F i g . 3=8  and i t s e q u i v a l e n t c i r c u i t  shown  i n F i g . 3.9. T h i s c o n f i g u r a t i o n i s " d u a l " to t h a t of S e c t i o n Thus, the e q u a t i o n s , s y n t h e s i s procedures and remarks of  3.2.  50  S e c t i o n 3° 2 s i m i l a r l y a p p l y e x c e p t t h a t Z replaces  Y,  z replaces  y,  (G+sC) r e p l a c e s  (R+sl)  f  open-circuited replaces series replaces C. r e p l a c e s 1  |kj  short-circuited,  shunt,  R.,  *  1'  r e p l a c e s k except i n Equation  (3=4),  where k i s n e g a t i v e , (2.22)  and g ( s ) i s g i v e n by E q u a t i o n The  (2.23).  " d u a l " n e t w o r k c o n f i g u r a t i o n of F i g . 3-7  shown i n P i g . 3.4  or  is  3"10.  A S y n t h e s i s P r o c e d u r e f o r a Cascade of S e c t i o n s i n P a r a l l e l T o g e t h e r w i t h lumped Elements i n Shunt I f the RC  s e c t i o n s used as s h o r t - c i r c u i t e d s t u b s i n  the n e t w o r k c o n f i g u r a t i o n of S e c t i o n 3 . 2  are i n s t e a d p a r a l l e l e d  a t b o t h p o r t s w i t h the, r e s p e c t i v e p o r t s of the main cascade RC  s e c t i o n s , we  w h i c h has set  o b t a i n the network c o n f i g u r a t i o n of F i g . 3 » H »  the e q u i v a l e n t c i r c u i t shown i n F i g . 3»12»  of s e c t i o n s i n p a r a l l e l a r e t e r m i n a t e d  The  i n a short c i r c u i t .  I n o r d e r t o e l i m i n a t e the lumped e l e m e n t s , we that 1  ke  kt  i  i  2kl  R.2. 1 1  = 0  •2kt +  tt  1!  R.L 1 1  +  last  1  R. A. , i+l i+l  R. A. , i+l i+l  require  Fig.  3.11  - Cascade of Sections i n P a r a l l e l Together •with Lumped Elements i n Shunt  VJl  56 where k!#! = -k'.'t'.' = -k£ 1 1 i 1  (3.51) ' >  vJ  J  k i s positive, g(s)  i s given by Equation  A i L  T  ^  i =  1,  l  E.g.  i i  l  1  and  I  A  1  R.g.  l  II  —  R.  —  l  l tti — R. l  (2.21),  (5.52)  »...  —  I I  II  t!  i  i  1  R. 1. i i  l  1 1  in  L.  or  2, 5,  1  Let  R.  =  u  L.  (2.20)  A  =  1  (5.55)  h«i  the f i r s t of Equations (5.50) w i l l always be s a t i s f i e d  -  the second becomes 2k£ s i n h 2k£  1  gis;  V i  Erom Equations  ( )  Y  s  1  ^  (2..47) ,  + 1— Rl X  = 0  (5.54)  (2.48) and ( 5 . 5 1 ) , we obtain  2 cosh2k£ tanh*? + Rj  2  =  T  1  ^sTRTTT  2 cosh 2ke +  1  R L ±  () s  2 ^taxing sinh Yf s i n h  +  1 + 1  2  ££BL  Y.  + 1  (S)  tanh Yt (3.35)  where g(s) i s given by Equation (2.20) or (2.21), and here 1=1,  2, 3,  ..•,  n.  Under the p-r conformal transformation ( 2 . 4 9 ) , we have y.(w) =  Y.  2  =  R~T~  (s)  2 w cosh 2k£' + R ^  y  ( w ) - 2 sinh k£(l/w-w) 2  i + 1  2 cosh 2ke + R e ±  i  y  i + 1  (w) w (5.56)  ~  where y . ( w ) £ ^ +1  . (s)  T  + 1  This i s an LC d-p admittance f u n c t i o n i n the w-domain, because Y  i + 1  ( s ) =00 f o r a s h o r t - c i r c u i t t e r m i n a t i o n , i . e . , y  ( ) w  i + 1  =00,  and y  i  (  w  )  (3.37:  = CT7 w 11  which i s an i n d u c t i v e admittance i n the w-domain, Prom Equation (3.36), 2 i = H.e y  (  1  .3=38:  )  cosh 2kg [y (w) - w J UY\ - y ( l ) s i n h k g (l/w-> y.(1) - w y (w) 2  ±  y  i+l  ( w )  = i y  (  l  )  ±  ±  , / \ cosh2kg-1 i ^ ^ cosh2kg-l / -. \ i " 2cosh 2kg i 2cosh 2 k e i _ y ( l ) cosh 2k0 _ cosh2ke-l , . cosh2kg - l ^ l ' - ' i 2cosh2ke - ' -w y- (w)- 2cosh 2kg w (3-39) The term cos h2ke-i i r e a l i z e s part of the pole of y^(w) at 2 cosh2ke w=0. I f the pole i s completely removed, then y. ,(w) w i l l have y  r  y  ( l )  W  y  (  l  j  _  y  ( 1 )  ±  1  ( 1 ) U j  y  y  (  1  y  u  n  )  1  a zero at w=0  "T\J_  and t h i s cannot be r e a l i z e d by the network  c o n s i d e r a t i o n . Therefore, i t i s necessary that 2kg - 1 / N wy (w) w=0^^ cosh.. 2 cosh 2ke V y  ±  under  (3.40)  This requirement may c o n f l i c t w i t h the requirements of the conjecture, as given by Equation (3-34), i n which case the synthesis procedure breaks down. f i e d , then  I f the i n e q u a l i t y i s s a t i s -  cosh 2k£ - 1  y, U)  _2 cDstr 2k2  w  v f \ A cosh 2k£ - 1 i ^ ' because y {w) and cosh 2k£ w— y  ±  (  1  i s a p-r f u n c t i o n  > p _ r  2  H e un c e  "  Richards' Theorem (see Equation (2.27)), - j _ i . ( ) y  w  +  f u n c t i o n , w i t h a = 1 and (3 = y ^ ( l ) constant,  1 S  >  b  a  y P- » r  cosh 2kg, a r e a l p o s i t i v e  58 Also, y because  ( ^ ) i s a r e a l i z a b l e LC d-p admittance  i + 1  y (w)  function  i s an LC d-p admittance f u n c t i o n .  ±  S i n c e the  factor  (l-w ) can be c a n c e l l e d from the numerator  t o r of  yj_ ( 0' >  +1  y  i+i( ) w  i s  o f  ^  = 0  R. l  1  The s y n t h e s i s  Given:  be r e w r i t t e n as  -kg s i n h 2kg y . ( l ) +  gls  3.4.1  lower rank than y ( w ) . ±  E q u a t i o n (3.34) may 1  and denomina-  procedure can now  be  (3.41)  stated.  S y n t h e s i s Procedure kg and and  , as s p e c i f i e d by E q u a t i o n s (2.48), (2.49)  (3.4),  and y-^(w) , a d-p admittance f u n c t i o n which the r e a l i z a h i l i t y c o n d i t i o n s  satisfies  of S e c t i o n 2.7.3  and  where / T] I ^ cosh 2kg - 1 _ _ l U I w Q > 2 cosh 2kg —  y  y  U  =  Step 1:  l  /, \ (  l  )  From E q u a t i o n s (3-33), (3-38) and  (3-41), ^j-  and ^rr, = kg s i n h 2kg y. ( l ) R  —  (3.42)  l  Hence, by E q u a t i o n s (2.48), (3-31) and parameters  of the f i r s t  (3.33), the  s e t of s e c t i o n s i n p a r a l l e l  and the lumped elements i n shunt can be Step 2:  =  determined.  From E q u a t i o n (3-39), determine yj_ ]_(w)> +  cosh 2kg [ y ( w ) - w y ( l ] ] - y ( l ) i  y  i + 1  U) = y (i) ±  i  ±  sinh kg(l-w /w) 2  • y ( i ) - w y.(w) ±  (3.43)  2  59  where 1 = 1 ,  remembering t o c a n c e l the ( l - w ) f a c t o r  f r o m t h e numerator and denominator. Bo,,  [.  ^ ( ^ l ^ T c o s l f ^  must be s a t i s f i e d .  (3.44)  1  I f the i n e q u a l i t y i s n o t s a t i s f i e d ,  t h e n t h e p r o c e d u r e b r e a k s down and a f r e s h s t a r t  Ye, u  new  and y-|_(w) must be made.  with  A smaller value  of k£ s h o u l d be used. Step 3:  Prom E q u a t i o n s (3-33), (3.38) and (3-41), i  _ yi-fi  R. ,e. . i+i i + i  ( 1 )  2  and - k — = ke s i n h 2k£ y . ^ U ) R. , i+I  (3-45)  x + ±  Hence t h e parameters of the second s e t of s e c t i o n s i n p a r a l l e l and t h e lumped elements i n shunt can be determined. Step 4:  Repeat Steps 2 and 3 f o r i = 2, 3, ..., u n t i l y-^(w) i s completely r e a l i z e d . short-circuit  The l a s t s e t  of s e c t i o n s i s  terminated.  The above s y n t h e s i s p r o c e d u r e may be used f o r any v a l u e of the s e r i e s i n d u c t a n c e and shunt conductance o f the RC s e c t i o n s . or  Yl o f E q u a t i o n (2.48) and g ( s ) o f E q u a t i o n (2.20)  (2.21) must be m o d i f i e d a c c o r d i n g l y . I f t h e s e t of RC s e c t i o n s i n p a r a l l e l have e q u a l  l e n g t h s , i . e . , E q u a t i o n s (3.24) a p p l y , t h e n the network f i g u r a t i o n becomes t h a t shown i n F i g . 3-13.  con-  F i g . 3.13  - A l t e r n a t i v e C o n f i g u r a t i o n o f Cascade of S e c t i o n s i n P a r a l l e l Together w i t h Lumped E l e m e n t s i n Shunt.  o  F i g . 3-14 - Cascade of Sections i n Series Together w i t h Lumped Elements i n S e r i e s . OA  H  «-  \_IDEALj , J  r  |~ IDEAL "I J 1  ! ± c-r." GftsCj  s Cj  I L  L  II -wv  1  L  r  , a  />7 />7 K  sL  I  I  II  Y"  J  G"+sC" i  i  2  7  I  I  G;  s\  II  I  J  "/  /  \\~y  sCf  P i g . 3.15 - Equivalent C i r c u i t of Cascade of Sections i n S e r i e s Together w i t h Lumped Elements i n Series.  ro  P i g . 3.16  - A l t e r n a t i v e C o n f i g u r a t i o n of Cascade of S e c t i o n s i n S e r i e s Together w i t h Lumped Elements i n S e r i e s .  64 3.5  A Synthesis  Procedure f o r a Cascade of S e c t i o n s  i n Series  Together w i t h humped Elements i n S e r i e s , the network c o n f i g u r a t i o n of F i g . 3.14  Consider its  cascade of RC  sections i n s e r i e s together with  elements i n s e r i e s .  The  equivalent  T h i s network i s the S e c t i o n 3.4.  Therefore,  i s shown i n  Z replaces  Y,  z replaces  y,  " d u a l " of the c o n f i g u r a t i o n of  the e q u a t i o n s ,  remarks of S e c t i o n 3.4  (G+sC) r e p l a c e s  series replaces C^ r e p l a c e s  synthesis  s i m i l a r l y apply  except  that  short-circuited,  shunt,  R^,  |k| r e p l a c e s k except i n E q u a t i o n where  k is  negative,  and  g(s)  i s g i v e n by E q u a t i o n  The  procedure  (R+sl),  open-circuited replaces  (2.22)  (3.31)>  or  (2.23).  network c o n f i g u r a t i o n " d u a l " to t h a t of F i g ,  i s shown i n F i g . 3.6  lumped  3.15.  Fig.  and  circuit  3.13  3.16.  Example An  synthesis w i l l use  example w i l l now  procedures.  be worked out to i l l u s t r a t e  I t w i l l be assumed t h a t the  the network c o n f i g u r a t i o n of S e c t i o n 3 „ 2  lumped elements r e q u i r e d .  the  realization and  s y n t h e s i s procedure which m i n i m i z e s - t h e number of RC and  with  the sections  65 Let  l  Y  i  the d-p admittance  S  -iTiT  J  f u n c t i o n Y-^(s) be g i v e n by  (e ^ i)(e 2  2  y  +  e  -l)(e  W  0.67e ^ l) 2  +  +  (3.46) where g ( s ) i s g i v e n by E q u a t i o n (2.20) or (2.21), so t h a t under the p - r c o n f o r m a l t r a n s f o r m a t i o n (2.49), we  have  gig.; y  i  (  w  = n  )  Y  ^ Wow,  [  w y  l  ( w  i.e.  2  s  )  ^  10  =  x  (  2  t  (3.47)  9  + 4)  w (w  []|w 0  y (i)  and  +  i  = 4 =  2  '  2 5  0  -—^—  = Jg =  i—  -ii  2.0 y-i  lyi ^|*=0> (  (!)  2 "  Therefore, ^ ( s ) s a t i s f i e s  (  the r e a l i z a h i l i t y  3  '  4  8  )  c o n d i t i o n s of  S e c t i o n 2.7.3. A l s o , l e t kg = 0.2, Erom E q u a t i o n s  ^  J  n  ii  i_ in  2 k g  = 1.4918  (3.49)  (3-16)  1  1  i.e., e  y^ l (i)  =  2.0  2.0  = 0  (3.50)  = 2.9836  (3.5l)  2kg and,  R g 1  Erom E q u a t i o n  1  (3.17),  66 y (w)  = 2.9836  2  "wy (w)  i.e.  y (l)  1  0  )  (3.52)  = 0.7459  2  2ke - *r  y (D 2  and  w  = 0.4262  •w=0  2  + w(w + 7.0)  (  " w y (w) 9  = -1.1188  2kg w=0/^ 2 y 2 ( l ) -  (3.53)  ^ T - T  V l Thus, I n e q u a l i t y (3.18) i s s a t i s f i e d . 2kg S i n c e y ( l ) < ^ — , we have, from E q u a t i o n s 9  (3.19),  V i 1  y ( l ) = 0.7459  i i  1 II  V  2  = 0  it 2  1_  R and  From E q u a t i o n  i.e.  = 1.1128  (3.54)  (3.17), = 0.3709 <  v  w y^(w)  •w=0  y (l)  t 4  >  0  )  T  = 1.4836 = 1.8546  3  1 2  = 0.4475  2  y U) 3  y (i)  V l  2ke 2  2 k e  2  2  e  R g  e  = kg  in  2kg  y,(l) - V r = 0.3709 ^ R g 2  2  (3.55)  67 and  w  1 j=0/* 2 w=0/  y (w) 3  2kC  y (D  -  5  (3.56)  ^ T T  T h e r e f o r e , I n e q u a l i t y (3.18) i s s a t i s f i e d . Since y ^ ( l ) > — — -  R e 2  , "we have, from E q u a t i o n s  (3-20),  2  2k£  i y (l) 3  V 3  = 1.4837  + ^T-T  _  R  2^2_ 2k£  1t!  y (D  11  3  -  = 0.3709 R„ 2 c2  5 5  C  1_  = 0  i  R.3 and  2k£ T T T = 2.2134  Prom E q u a t i o n  (3.17),  y (w)  = 6.6402  4  i.e.  (3.57)  w y^(w)  w=0  y (l) 4  (3.58) = 6.6402 = 6.6402  2k£  1 2  = 2.2134  •3*3 2k? and  f  y  4 5|w 0>2 U  =  y  4 4  ( l )  ~ ^„ i u•3e"3  (3.59)  T h e r e f o r e , I n e q u a l i t y (3.18) i s s a t i s f i e d . U s i n g the r e a l i z a t i o n cedure have  ( i i i ) of Step 5 i n the s y n t h e s i s p r o -  of S e c t i o n 3.2.1.1, i . e . , u s i n g E q u a t i o n s  ( 3 . 2 1 ) , xe  Fig.  3.17  - Network  Realization.  03  69 1  0  1  y ( l ) = 6.6402 4  1 R  in  4  Hence, depending upon the f u n c t i o n Yt and the f r e quency and magnitude s c a l i n g used, the parameters o f the exponential  RC s e c t i o n s and the lumped elements r e q u i r e d i n  the r e a l i z a t i o n can be determined through E q u a t i o n s (3.4), the 3.7  (3-50),  (3.54),  (3.57) and (3.60).  (2.48),  F i g . 3.17 shows  c o n f i g u r a t i o n o f the network r e a l i z a t i o n . Discussion The  s y n t h e s i s procedures t h a t have been developed  f o r some cascade network c o n f i g u r a t i o n s of e x p o n e n t i a l  RC  sections and lumped elements demonstrate the s u f f i c i e n c y p a r t o f the  conjecture  discussed  i n Chapter 2.  The c o n j e c t u r e  requires  the removal of the r a t i o n a l terms i n the immittance parameters of the RC s e c t i o n s . tions discussed (or s e r i e s ) , use  T h i s has been achieved  i n the c o n f i g u r a -  by the use o f e x t r a RC s e c t i o n s i n p a r a l l e l  or as s t u b s , w i t h  of lumped elements.  the main RC s e c t i o n s , and the  The s y n t h e s i s procedures, however,  break down when the e x t r a RC s e c t i o n s to be r e a l i z e d a r e such t h a t they r e q u i r e p o l e s a t the o r i g i n  ( i n the w-domain) w i t h  r e s i d u e s l a r g e r than a c t u a l l y e x i s t .  In these cases, new  immittance f u n c t i o n s must be s p e c i f i e d and the s y n t h e s i s  70  procedures  repeated. The  s y n t h e s i s procedures  and the network r e a l i z a -  t i o n s r e d u c e t o Wyndrum's procedures when the t a p e r c o n s t a n t s zero.  of the e x p o n e n t i a l RC s e c t i o n s a r e  Then the c o n s t r a i n t s o f the c o n j e c t u r e no l o n g e r The  ferable  f o r u n i f o r m RC networks  apply.  r e a l i z a t i o n s o f S e c t i o n s 3»2 and 3»4 a r e p r e -  t o those  o f S e c t i o n s 3,3 and 3.5 f o r t h i n - f i l m  net-  works because of the common c o n n e c t i o n between a l l RC s e c t i o n s and lumped  elements. The  s y n t h e s i s procedures  are a p p l i c a b l e t o o t h e r  d i s t r i b u t e d - p a r a m e t e r systems d e s c r i b e d by s i m i l a r c h a r a c t e r i z i n g e q u a t i o n s , and may be extended t o the s y n t h e s i s of o t h e r nonu n i f o r m d i s t r i b u t e d RC networks by r e p l a c i n g each of the expon e n t i a l RC sections w i t h an e q u i v a l e n t s e c t i o n through the equivalences  of Schwartz  18  or H e l l s t r o m  17  4.  4.1  THE  APPROXIMATION PROBLEM  Introduction To r e a l i z e a d-p  immittance  f u n c t i o n by a network  of e x p o n e n t i a l RC s e c t i o n s , the i m m i t t a n c e  f u n o t i o n must be  a p p r o x i m a t e d by a f u n c t i o n of e x p o n e n t i a l p o l y n o m i a l s f o r m g i v e n by E q u a t i o n  (2.63).  One  of the  of the s y n t h e s i s  proce-  dures of C h a p t e r 3 can t h e n be used t o d e t e r m i n e the parameter v a l u e s of the e x p o n e n t i a l RC s e c t i o n s and lumped e l e m e n t s . 4.2  Functions  of E x p o n e n t i a l  Polynomials  As shown i n S e c t i o n 2 . 7 . 3 > a d-p  immittance  E ( s ) can be r e a l i z e d by a network of e x p o n e n t i a l RC ( p r o v i d e d the r a t i o n a l terms i n the i m m i t t a n c e each RC s e c t i o n can be e l i m i n a t e d ) i f and expressed  function  sections  parameters of  o n l y i f i t can  be  as  F(s) = K  ^  e  ^ ( e ^ 3=1  +  L . e  2  ^  +  i ;  J  (4.1)  where K i s a r e a l p o s i t i v e  constant,  i s g i v e n by E q u a t i o n  (2.48),  g ( s ) i s g i v e n by E q u a t i o n s m = p or m =  and  p-1,  ID.1/2,  IN.I,  D  (2.20)-(2.23),  ->N.,  o n l y one  c o e f f i c i e n t i n the s e t of N. and D.  may  be  zero.  72 F o r r e a l f r e q u e n c i e s , s = jw. A s p e c i f i e d d-p immittance a magnitude-frequency  c h a r a c t e r i s t i c curve, or a  curve, o r both, must be approximated g i v e n by E q u a t i o n network.  phase-frequency  by a f u n c t i o n of the form  (4.1) t o be r e a l i z a b l e by an e x p o n e n t i a l RC  The c l o s e n e s s o f f i t ,  approximation,  f u n c t i o n i n the form of  i . e . , the c r i t e r i o n o f " b e s t "  i s a t the d i s c r e t i o n of the c i r c u i t  and depends upon the problem  designer  a t hand.  13 Wyndrum  has p l o t t e d s e t s of n o r m a l i z e d magnitude-  f r e q u e n c y curves of the v a r i o u s e x p o n e n t i a l p o l y n o m i a l t h a t appear i n E q u a t i o n  factors  (4«l) f o r u n i f o r m RC networks w i t h  n e g l i g i b l e s e r i e s i n d u c t a n c e and shunt  conductance.  e x p o n e n t i a l RC networks, t h i s would i n v o l v e  For  plotting  s e t s of n o r m a l i z e d curves of the v a r i o u s e x p o n e n t i a l p o l y n o m i a l f a c t o r s f o r many v a l u e s of the n o r m a l i z e d t a p e r c o n s t a n t . Such curves would be r a t h e r t e d i o u s t o use i n any d e s i g n problem.  T h e r e f o r e , the u s e of a d i g i t a l computer w i t h  f a c i l i t i e s would make . i t e a s i e r t o approximate d-p  immittance  plotting  any s p e c i f i e d  function.  The approximated transformed by E q u a t i o n f(w) =  F  (  d-p immittance  f u n c t i o n F ( s ) i s then  (2.49) as f o l l o w s : s  )  p , 7\(w +n.) K_j=l L w m 2  =  7^(w +d.) d=i 2  J  ( . 4  2 )  73  P where K  = K  2  (  m  2  -  p  )  m  L  7T(2-D. 2 + N.  n 5. ==  and  d  r  f  2 + D. _ '1  =  (4.3)  p  j The t r a n s f o r m e d d-p i m m i t t a n c e f u n c t i o n f ( w ) can t h e n be r e a l i z e d  by one o f t h e s y n t h e s i s p r o c e d u r e s of C h a p t e r 3 .  I f f ( w ) i s an a d m i t t a n c e f u n c t i o n , t h e p r o c e d u r e o f S e c t i o n 3 . 2 or 3 - 4 may be u s e d , and i f f(w) i s an impedance f u n c t i o n , t h a t of S e c t i o n 3 . 3 o r 3 . 5 may be used. 4.3  Examples Two examples w i l l be d e s c r i b e d t o i l l u s t r a t e t h e  proposed s y n t h e s i s procedures. A FORTRAN IV program has been w r i t t e n f o r an IBM 7040 d i g i t a l computer  t o h a n d l e t h e problem of r e a l i z i n g d-p i m m i t -  t a n c e f u n c t i o n s by e x p o n e n t i a l RC n e t w o r k s . s i s t s o f a m a i n l i n e and f o u r s u b r o u t i n e s .  The program  con-  P r i n t e d output of  the magnitude- and p h a s e - f r e q u e n c y c h a r a c t e r i s t i c s ,  and t h e  network r e a l i z a t i o n , t o g e t h e r w i t h magnitude- and p h a s e - f r e q u e n c y p l o t s can be o b t a i n e d w i t h t h e program.  The s e r i e s i n d u c t a n c e  and shunt conductance o f t h e RC s e c t i o n s a r e assumed t o be z e r o , and t h e network r e a l i z a t i o n uses t h e c o n f i g u r a t i o n of S e c t i o n 3 . 4 or 3 - 5 , or of Section 3 . 2 or 3 . 3 -  74 4-3.1  Example The  No.  1  d-p i m p e d a n c e  function  t o be r e a l i z e d  i n this  example  i s g i v e n by t h e n o r m a l i z e d magnitude-frequency  cation,  shown i n E i g . 4.1, w i t h -6 d b / o c t a v e  specifi-  segments and  b r e a k f r e q u e n c i e s o f 0.065 a n d 2.0 r a d . / s e c . f o r t h e f r e q u e n c y range the  o f 0.01 t o 1 0 . 0 r a d . / s e c .  difference  half,  i s that  B  0.0501  =  Wyndrum's  Z,(s)  for  (s) =  Z  =  RC =  approximated  (4.4)  e  + ) -l) 1  2 y e  (  e  M ' ^ (e ^+e 1  4  + 2 y e  1  (4.5)  )  +l)  o f 0.7071  accounts  frequency n o r m a l i z a t i o n used. approximation  forrealization  by an expo-  is  0.68  c  and  c a n be  ^ s R C 'l, RC = g = 1 a n d t h e f a c t o r  RC n e t w o r k  Yt  function  approximation i s  s(e  A possible  where  .  <fcjgg  = 0.7071 * M  the d i f f e r e n t  nential  J  impedance  13  w h e r e ¥lk  J  o f one-  function  Z ( ) a  b y Wyndrum  specified  by a r a t i o n a l  except f o r  i n f r e q u e n c y n o r m a l i z a t i o n by a f a c t o r used  The  This function,  ^ * s(e {  ) -l) + 1  2 y e  (*  (4.6) (e +1.6e +l) m  m  /y/k^+eRc' £  g=  1  k t = -0.09 Another possible  e x p o n e n t i a l RC n e t w o r k i s  approximation  for realization  by an  7  C (j  =  1.414  C  }  7 _ 3.72 c?  =  37.284  -O (s)  O  o  P i g . 4.2 - R e a l i z a t i o n of Z ( s ) by a Uniform RC Network ( a f t e r Wyndrum). b  - ^ L  =  -rr^-r  ITT-15.993 c 2 "2 • W W  2.941  ri  AAAAr =  3  =30.260  3  - V W \  •o  490.963  90.289 2  A A / W =2.547  17  C  AAA/v 7 _ 75.553 2 3?  -ww7  o  '30.260  P i g . 4.3 - R e a l i z a t i o n of Z (s) by an Exponential TUJ Ne twork.  -\A/V\> Ijr,-2.941 11 C  V W \ -J-pr-U.9S2 22  L  Z  kfi'-kfi-k^-  4.4  - Realization RC  V W \  Network.  O  -1^ = 58.002 33  L  Z  =0.U  o f Z,(s) by an E x p o n e n t i a l  It , , , ,  ?  c  :  «  (  Yg = Jk +sRC  where  2  e  ^  l .  t  f  t  - l . »  e  « . l  ,,,,  '£  RC = I = 1 and  kg =  -0.14  To o b t a i n the a p p r o x i m a t i o n s ( 4 . 6 )  and ( 4 . 7 ) ,  the  v a l u e o f kg i s f i r s t chosen, f o l l o w e d hy a d j u s t m e n t o f the c o e f f i c i e n t s o f the e x p o n e n t i a l p o l y n o m i a l s and of the m u l t i p l i e r c o n s t a n t f o r the b e s t f i t , i n t h e l e a s t square s e n s e , t o the magnitude-frequency  specification.  The m a g n i t u d e - f r e q u e n c y c h a r a c t e r i s t i c s o f Z , Z  Q  and Z ^ a r e shown i n R i g . 4 . 1 ,  of Z ^ ,  Z  and 4 . 4 ,  q  and the network  Z^,  realizations  and Z ^ by RC networks are g i v e n by P i g s . 4 . 2 ,  4.3,  respectively. Hence, d e p e n d i n g upon the magnitude and f r e q u e n c y  s c a l i n g used, the parameters of the a c t u a l networks of RC s e c t i o n s and lumped elements can be d e t e r m i n e d . Note t h a t the m a g n i t u d e - f r e q u e n c y  characteristics  become a s y m p t o t i c a t low and h i g h f r e q u e n c i e s .  For a l l  n e t w o r k s , w i t h n e g l i g i b l e s e r i e s i n d u c t a n c e and shunt  RC conduc-  tance and w i t h o p e n - c i r c u i t t e r m i n a t i o n s , the s l o p e s of the l o w - and h i g h - f r e q u e n c y asymptotes a r e - 6 db/octave and - 3 db/octave, r e s p e c t i v e l y . 4.3.2  Example No. 2 The d-p impedance f u n c t i o n t o be r e a l i z e d here i s  g i v e n by the n o r m a l i z e d m a g n i t u d e - f r e q u e n c y  specification,  79  shown i n P i g . 4.5, w i t h a -3 db/octave segment and a break frequency of 0.4 rad./sec. f o r the frequency range of 0.01 t o 10.0 rad./sec.  For t h i s c l a s s of d-p impedance f u n c t i o n s , the  short-circuited  RC network c o n f i g u r a t i o n of Section 3.4 must be  used to r e a l i z e the f i n i t e low-frequency impedance. The s p e c i f i e d d-p impedance f u n c t i o n , f o r r e a l i z a t i o n by a uniform RC network, can be approximated by  V  a >  ^f ' 1?r 4y +0  i.6o  =  (e ^-l)  b  7 \ where  <••<»  (e ^+l)  2  ye k  +1)  Y  2  jikVe  RC = I = 1 k I = 0.  and  A p o s s i b l e approximation f o r r e a l i z a t i o n by an expon e n t i a l RC network  is  v  7" \ where Yc {s ' J v  Y (s) c  1.59  =  %  y  +  (e ^-l)  1  )  (e +l)  2  2U  (4.9)  Yl k V k + s R c ' e 2  RC = g = 1 and  kg =  0.10  Another p o s s i b l e approximation f o r r e a l i z a t i o n by an exponential RC network Y  ( \  n s  A  Y ( s ) = 1.54 d  is  ^7-7—r  ^(e (  E  4 y g  and  kg = 0.20  0.5e ^ l) 2  +  +  ^ _ J 2M '  ^ y^+sRc'? RC = t = 1  where  {e  +l)  ,.  (' > 4  10  _  16 12-  0.01  i  i  I I I  .02  .04  .06  0.1  i  <  .2  .4  FREQUENCY  i  I I I  .6  1  2  i  4  I I I  6  10  (rad./sec.)  F i g . 4-5 - Magnitude-Frequency Curves f o r Synthesis Example No. 2. oo o  81  = 1600 11  R  o  A  C  /  2  V  2.667  Re  2  W  Y (s) h  o-  Fig.  4.5  - Realization U n i f o r m RC  R  of " ^ ( s ) by a Network.  0.794  i i e  A  A  /  W  A A A A P J  = 0.032  Y (s) c  -—Tf—rr = 0-754  51  22  R  L -0.70  Fig.  4.7  - Realization  o f Y ( s ) by an  E x p o n e n t i a l RC  c  Network,  C  X  82  —L— = 0 772 1 1  K  -I—  R P 2 2  €  K  A A A A -  =  1.307  e  A A A A r  -WW—I  = 0.127 R"P"  7 77  0.772  o-  7 7  *T7  K  Fig.  4.8  - Realization an  -0.20  %  of Y ( s )  Exponential  d  RC  by  Network.  =  1.307  83 The same procedure as that used i n Example No. 1 was followed t o obtain the above approximations.  I t was found that  the same c o e f f i c i e n t values of exponential polynomials  could be  used f o r a l l the approximations. The magnitude-frequency c h a r a c t e r i s t i c s of these approximations are shown i n E i g . 4.5, and t h e i r network r e a l i z a t i o n s are given by E i g s . 4.6-4.8. The parameters of the a c t u a l networks of RC sections and lumped elements can then be determined by t a k i n g i n t o account the frequency and magnitude n o r m a l i z a t i o n s . 4.4  Discussion The network r e a l i z a t i o n s of Sections 3.4 and 3»5 have  been found to be more u s e f u l i n r e a l i z i n g d-p immittance funct i o n s than those of Sections 3.2 and 3.3 because the r e s t r i c t i o n imposed by the conjecture of Chapter 2, i . e . , I n e q u a l i t y (3.44), can be s a t i s f i e d by making the taper constant of the RC sections s u f f i c i e n t l y s m a l l , whereas I n e q u a l i t y (3.11) i s independent of the taper  constant. In the above examples, i t i s seen that the exponential  RC network r e a l i z a t i o n s r e q u i r e a d d i t i o n a l RC sections and lumped elements when compared w i t h the uniform RC network realizations.  I t i s p o s s i b l e to reduce the number of these  a d d i t i o n a l RC sections and lumped elements by s u i t a b l y comb i n i n g the simple cascade r e a l i z a t i o n of Section 3.1, the cascade-and-stub r e a l i z a t i o n of Section 3.2 or 3.3, and the r e a l i z a t i o n of Section 3.4 or 3-5. avoided  Lumped elements can be  by using stub sections instead with the c o n f i g u r a t i o n  o f S e c t i o n 3.4 The  or 3.5. e x p o n e n t i a l RC network i s capable  a s p e c i f i e d d-p immittance u n i f o r m RC network.  of  approximating  f u n c t i o n i n the same manner as the  In f a c t , a number o f d i f f e r e n t  RC networks w i t h v a r i o u s v a l u e s of t a p e r c o n s t a n t t h a t w i l l y i e l d s i m i l a r d-p immittance i l l u s t r a t e d i n the above examples.  functions.  exponential  can be o b t a i n e d T h i s has been  Preliminary results  indicate  t h a t the e x p o n e n t i a l RC network r e a l i z a t i o n s tend t o occupy l e s s a r e a than the u n i f o r m RC network. the r a t i o  In cases i n v e s t i g a t e d so f a r ,  of a r e a needed f o r e x p o n e n t i a l RC network r e a l i z a t i o n s  to a r e a needed f o r u n i f o r m RC network r e a l i z a t i o n s i s a p p r o x i m a t e l y 0.87 and  and 0.85  0.99 and 0.97  f o r the two a l t e r n a t i v e s i n Example No.  1,  i n Example No. 2.  Even i f t h e lumped elements a r e n o t c o n s i d e r e d , the o p e n - c i r c u i t e d and s h o r t - c i r c u i t e d E i g s . 3-3 and 3-4,  r e s p e c t i v e l y , do not form a d u a l s e t of  lumped c a p a c i t a n c e and i n d u c t a n c e different  e x p o n e n t i a l RC s e c t i o n s of  i n the w-domain because  p r e m u l t i p l y i n g f a c t o r s a r e used p r i o r t o t r a n s f o r m a t i o n  of the d-p immittances  i n t o the w-domain.  Poster s y n t h e s i s procedures,  Thus, Cauer and  i n the w-domain, cannot be a p p l i e d  t o the s y n t h e s i s o f e x p o n e n t i a l RC networks. L i m i t e d use of the proposed s y n t h e s i s procedures  seems  to i n d i c a t e t h a t the r e s t r i c t i o n s imposed by the c o n j e c t u r e of Chapter  2, through I n e q u a l i t i e s (3.11) and ( 3 . 4 4 ) , are always  satisfied and  i n any r e a l i z a t i o n , p r o v i d e d t h a t the i n e q u a l i t i e s  the r e a l i z a h i l i t y c o n d i t i o n s are s a t i s f i e d a t the s t a r t of  the r e a l i z a t i o n .  No proof of t h i s has been  found.  5.  CONCLUSIONS  The s y n t h e s i s p r o c e d u r e proposed by Wyndrum f o r t h e r e a l i z a t i o n o f d r i v i n g - p o i n t i m m i t t a n c e f u n c t i o n s by u n i f o r m d i s t r i b u t e d RC n e t w o r k s has been extended t o t h e r e a l i z a t i o n by exponentially-tapered  d i s t r i b u t e d RC n e t w o r k s .  The s y n t h e s i s t e c h n i q u e  i n v o l v e s the  approximation  t o a s p e c i f i e d d r i v i n g - p o i n t i m m i t t a n c e f u n c t i o n by a f u n c t i o n of e x p o n e n t i a l p o l y n o m i a l s .  The f u n c t i o n o f e x p o n e n t i a l  poly-  n o m i a l s must s a t i s f y t h e r e a l i z a b i l i t y c o n d i t i o n s f o r e x p o n e n t i a l l y tapered  d i s t r i b u t e d RC n e t w o r k s and the c o n d i t i o n  I n e q u a l i t y (3.11) formation  or ( 3 . 4 4 ) .  g i v e n by  A p o s i t i v e - r e a l conformal t r a n s -  i s t h e n used t o change t h e s y n t h e s i s problem i n t o a  lumped-parameter p r o b l e m so t h a t w e l l - k n o w n cascade s y n t h e s i s p r o c e d u r e s can be a p p l i e d .  The network r e a l i z a t i o n s a r e i n t h e  f o r m of cascades o f e x p o n e n t i a l l y - t a p e r e d  distributed  RC  s e c t i o n s w i t h e x t r a d i s t r i b u t e d RC s e c t i o n s and lumped elements added t o s a t i s f y t h e c o n s t r a i n t s imposed by t h e c o n j e c t u r e o f S e c t i o n 2.7. The e x p o n e n t i a l l y - t a p e r e d  d i s t r i b u t e d RC network can  r e a l i z e d r i v i n g - p o i n t i m m i t t a n c e f u n c t i o n s s i m i l a r t o those o f uniform  d i s t r i b u t e d RC n e t w o r k s , w i t h t h e advantages t h a t  n a t i v e r e a l i z a t i o n s can be o b t a i n e d constants  by u s i n g d i f f e r e n t  and t h a t l e s s a r e a i s o c c u p i e d ,  alter-  taper  and the d i s a d v a n t a g e  t h a t more s e c t i o n s a r e r e q u i r e d as a consequence o f t h e c o n j e c ture . I t has been shown t h a t t h e r e m o v a l o f the lumped e l e -  86  ments from the equivalent c i r c u i t s of exponentially-tapered d i s t r i b u t e d RC networks i s a s u f f i c i e n t c o n d i t i o n f o r the development of a synthesis procedure f o r the r e a l i z a t i o n of d r i v i n g - p o i n t immittance f u n c t i o n s . c o n d i t i o n has not been proven:  That t h i s i s a necessary  i t remains to be i n v e s t i g a t e d  further. A d r i v i n g - p o i n t immittance f u n c t i o n that s a t i s f i e s , , at the s t a r t of the synthesis procedure, the r e a l i z a h i l i t y condit i o n s f o r exponentially-tapered d i s t r i b u t e d RC networks and condition  the  given by I n e q u a l i t y (3.11) or (3.44) i s not guaran-  teed to be r e a l i z a b l e .  The p o s s i b i l i t y of i n c l u d i n g these  i n e q u a l i t i e s i n the r e a l i z a h i l i t y conditions merits f u r t h e r investigation. I t has been assumed that the e f f e c t i v e width at any point of a tapered transmission l i n e i s that of the r e s i s t i v e f i l m and that modulation of t h i s width w i t h respect to the a x i a l coordinate produces r e s i s t a n c e and capacitance taper functions which are r e c i p r o c a l . Experimental  v e r i f i c a t i o n of the above  and of the proposed d i s t r i b u t e d RC network r e a l i z a t i o n s of d r i v i n g - p o i n t immittance f u n c t i o n s remains to be done. No c o n s i d e r a t i o n has been given i n t h i s d i s s e r t a t i o n to the use of purely numerical techniques  to determine a d i s t r i -  buted RC network w i t h a d r i v i n g - p o i n t immittance f u n c t i o n that w i l l minimize some s p e c i f i c e r r o r c r i t e r i o n w i t h respect to a specified driving-point function. on the closed-form  Emphasis has been placed  s o l u t i o n to the problem.  REFERENCES  1.  M u l l i g a n , J.H., "The Role of Network Theory i n S o l i d - S t a t e E l e c t r o n i c s - Accomplishments and Future Challenges", IEEE Trans, on C i r c u i t Theory, V o l . CT-10, pp. 323-332, September, 1963.  2.  Castro, P.S., and Happ, W.W., " D i s t r i b u t e d Parameter C i r c u i t s and Microsystem E l e c t r o n i c s " , Proc, NEC, V o l . 16, pp. 4 4 8 - 4 6 O , I960.  3.  Happ, W.W., Castro, P.S., and F u l l e r , W.D., "Synthesis of S o l i d State D i s t r i b u t e d Parameter Functions", IRE I n t e r n a t i o n a l Convention Record, V o l , 10, Part 6, pp. 262-278, 1962.  4.  K e l l y , J„J., and Ghansi, M.S., "On the Dominant Poles of the D i s t r i b u t e d RC Networks", Technical Report 400-92, Dept. of E l e c t r i c a l Engineering, New York U n i v e r s i t y , March, 1964-  5.  H e i z e r , K,W., " D i s t r i b u t e d RC Networks w i t h R a t i o n a l Transf e r Functions", D o c t o r a l D i s s e r t a t i o n , U n i v e r s i t y of I l l i n o i s , 1962.  6.  H e i z e r , K.W., " D i s t r i b u t e d RC Networks w i t h R a t i o n a l Transf e r Functions", IRE Trans, on C i r c u i t Theory, V o l . CT-9, pp. 356-362, December, 1962.  7.  H e i z e r , K.W. , " R a t i o n a l Parameters w i t h D i s t r i b u t e d Networks", correspondence, IEEE Trans, on C i r c u i t Theory, V o l . CT-10, pp. 531-532, December, 1963.  8.  Barker, D.O., "Synthesis of A c t i v e F i l t e r s Employing Thin F i l m D i s t r i b u t e d Parameter Networks", IEEE I n t e r n a t i o n a l Convention Record, V o l . 13, Part 7, pp. 119-126, 1965.  9.  Woo, B.B., and Hove, R.G., "Synthesis of R a t i o n a l Transfer Functions w i t h Thin-Film D i s t r i b u t e d Parameter RC A c t i v e Networks", Proc. NEC, V o l . 21, pp. 241-246, 1965.  10.  Pu, Y., and Fu, J.S., "n-Port Rectangular-Shaped D i s t r i buted RC Networks", correspondence, IEEE Trans, on C i r c u i t Theory, V o l . CT-13, pp. 222-225, June, 1966.  11.  Hesselberth, CA., "Synthesis of Some D i s t r i b u t e d RC Networks", Report R-164, Coordinated Science Laboratory, U n i v e r s i t y of I l l i n o i s , August,- 1963.  12.  Wyndrum, R.W., "The Synthesis of D i s t r i b u t e d RC Networks", Technical Report 400-72, Dept. of E l e c t r i c a l Engineering, New York' U n i v e r s i t y D e c e m b e r , 1962.  88 13.  Wyndrum, R.W., "The Exact S y n t h e s i s of D i s t r i b u t e d RC works", T e c h n i c a l Report 400-76. Dept. of E l e c t r i c a l E n g i n e e r i n g , TTew York U n i v e r s i t y , . May, 196'3.  14.  Wyndrum, R.W., "The Exact R e a l i z a t i o n of D i s t r i b u t e d RC D r i v i n g P o i n t TFunctions", Wescon Convention T e c h n i c a l Papers, V o l . 8, P a r t 2, Paper 18.1, 1964-  15.  Wyndrum, R.W., "The R e a l i z a t i o n of Monomorphic T h i n E i l m D i s t r i b u t e d RC Networks", IEEE I n t e r n a t i o n a l Convention Record. V o l . 13, P a r t 10, pp. 90-95, 1965.  16.  O'Shea, R.P., " S y n t h e s i s U s i n g D i s t r i b u t e d RC Networks", IEEE I n t e r n a t i o n a l Convention Record. V o l . 13, P a r t 7, pp. 18-29, 1965. A l s o IEEE Trans, on C i r c u i t Theory. V o l . CT-12, pp. 546-554, December, 1965.  17.  H e l l s t r o m , M.J. " E q u i v a l e n t D i s t r i b u t e d RC Networks or T r a n s m i s s i o n L i n e s " , IRE Trans, on C i r c u i t Theory, V o l . CT-9, pp. 247-251, September, 1962.  18.  Schwartz, R„F., " T r a n s f o r m a t i o n s i n the A n a l y s i s of NonU n i f o r m T r a n s m i s s i o n L i n e s " , J . F r a n k l i n I n s t . , V o l . 278, pp. 163-172, September, 1964.  19.  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