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Synthesis of elementary distributed amplifiers using an iterative method Walton, Norman 1955

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SYNTHESIS OP ELEMENTARY DISTRIBUTED AMPLIFIERS USING AN ITERATIVE METHOD by NORMAN WALTON B.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1954 A t h e s i s submitted i n p a r t i a l f u l f i l m e n t of the requirements f o r the degree of MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l E n g i n e e r i n g We accept t h i s t h e s i s as confirming to the standard r e q u i r e d from candidates f o r the degree of MASTER OF APPLIED SCIENCE Members of the Department of E l e c t r i c a l E n gineering THE UNIVERSITY OF BRITISH COLUMBIA December, 1955. 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 permission f o r extensive copying of 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 of my depart-ment or, i n h i s absence, by the U n i v e r s i t y L i b r a r i a n . I t i s understood that any 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 without my w r i t t e n p e r m i s s i o n . ABSTRACT This t h e s i s d e s c r i b e s the design of two p a r t i c u l a r types of conventional d i s t r i b u t e d a m p l i f i e r s and t r e a t s a proposed s p l i t - b a n d a m p l i f i e r . The method used f o r the c o n v e n t i o n a l designs i s an i t e r a t i v e s y n t h e s i s process developed at Stanford U n i v e r s i t y i n 1952. The o b j e c t i v e was to i n v e s t i g a t e the p o s s i b i l i t y of producing d i s t r i -buted a m p l i f i e r s of s u p e r i o r performance. Only a m p l i f i e r s with a f l a t amplitude response were considered since the c a l c u l a t i n g equipment a v a i l a b l e was inadequate f o r the computations i n v o l v e d i n producing a m p l i f i e r s with other types of response c h a r a c t e r i s t i c s . Three designs of one form of conventional d i s t r i b u t e d a m p l i f i e r were c a r r i e d out. These were a m p l i f i e r s with ladder networks f o r delay l i n e s and with both delay l i n e s i d e n t i c a l except f o r a p o s s i b l e d i f f e r e n c e i n impedance l e v e l . None of the three a m p l i f i e r s had t h e o r e t i c a l c h a r a c t e r i s t i c s which j u s t i f i e d an attempt to c o n s t r u c t them. The other conventional a m p l i f i e r was one employing l a d d e r - l i k e networks f o r the delay l i n e s with each of the l i n e s symmetrical about the mid-point of i t s l e n g t h and with both l i n e s i d e n t i c a l except f o r a p o s s i b l e d i f f e r e n c e i n impedance l e v e l . An attempt to design one of these a m p l i f i e r s produced new i n f o r m a t i o n beyond t h a t r e p o r t e d i n the o r i g i n a l work at Stanford U n i v e r s i t y . Furthermore, when using the i t e r a t i v e technique to design t h i s ampli-f i e r , there seemed to be reasonable doubt as to whether i i i or not the process was always convergent. No d e f i n i t e o p i n i o n on the matter was formulated since i t would have r e q u i r e d that the c a l c u l a t i o n s be continued through more c y c l e s of i t e r a t i o n than could reasonably be c a r r i e d out with the computing equipment at hand. F i n a l l y , a proposed s p l i t - b a n d a m p l i f i e r was i n v e s t i -gated and i t s t h e o r e t i c a l gain-bandwidth c h a r a c t e r i s t i c s were compared with those of a conventional d i s t r i b u t e d a m p l i f i e r . I t showed.a s l i g h t advantage, but t h i s was f a r outweighed by c e r t a i n s e r i o u s inherent disadvantages and the p r o j e c t was d i s c o n t i n u e d . The i n v e s t i g a t i o n has shown that i t i s i m p r a c t i c a l to c a r r y out the c a l c u l a t i o n s i n v o l v e d i n the i t e r a t i v e prodecure when using a hand c a l c u l a t o r . A l s o , some doubt as to the general convergence of the i t e r a t i v e synthesis process has been r a i s e d . TABLE OF CONTENTS Page Table of Contents i v . L i s t of I l l u s t r a t i o n s . v. Acknowledgement v i . CHAPTER I I n t r o d u c t i o n I I The Design of D i s t r i b u t e d A m p l i f i e r s f o r Maximally-Flat Amplitude Response 5 1. The General T r a n s f e r F u n c t i o n and the D e f i n i t i o n of Maximal F l a t n e s s . . . . . . . 5 2. The Max i m a l l y - F l a t D i s t r i b u t e d - A m p l i f i e r Stage . . 7 3. O u t l i n e of the I t e r a t i v e Procedure f o r Max i m a l l y - F l a t Response . . . . 10 I I I An A p p l i c a t i o n of the I t e r a t i v e Synthesis Process . 12 1. The Form of the D i s t r i b u t e d A m p l i f i e r to be Considered 12 2. I d e n t i c a l N a t u r a l Modes i n Both Networks . . 13 3. Intermediate Steps and F i n a l R e s u l t s of the Design . 15 4. Separation of L e f t - H a l f - P l a n e Roots and R e a l i z a t i o n of Component Values i n the Net-works 18 5. The Gain-Bandwidth F a c t o r f o r a S i n g l e -stage D i s t r i b u t e d A m p l i f i e r . 19 IV Design R e s u l t s f o r Two Non-Minimum Phase Ampli-f i e r s 22 V The Design of D i s t r i b u t e d A m p l i f i e r s Containing Symmetrical Networks 25 1. C o n s i d e r a t i o n f o r S i m p l i f y i n g the Design Procedure . 25 2. D a r l i n g t o n ' s Synthesis of Reactance Four-Poles 25 VI A S p l i t - B a n d D i s t r i b u t e d A m p l i f i e r 40 VII Conclusion . 47 APPENDIX I 49 BIBLIOGRAPHY „ . . Ro V LIST OF ILLUSTRATIONS F i g u r e Page 1. P e r c i v a l ' s Wave A m p l i f i e r , to f o l l o w . 1 2. A S i n g l e - s t a g e D i s t r i b u t e d A m p l i f i e r , to f o l l o w . . 7 3. A Two-Tube D i s t r i b u t e d - A m p l i f i e r Stage . . 12 4. I n i t i a l Component Values f o r a Two-Tube A m p l i f i e r . . 16 5. Network Element Values a f t e r One Cycle of I t e r a t i o n . 19 6. Two-Tube Stage with I d e n t i c a l N a t u r a l Modes, to f o l l o w . . 19 7. Two-Tube Stage with C o i n c i d e n t Real Modes, to f o l l o w 20 8. Improved Twc-Tube Stage with I d e n t i c a l Natural Modes, to f o l l o w . . . 22 9. A Three-Tube A m p l i f i e r with I d e n t i c a l Networks, to f o l l o w 23 10. A G r i d Network as a Reactive Four-Pole, to f o l l o w . 26 11. I n i t i a l Element Values f o r a Two-Tube A m p l i f i e r Containing Symmetrical Networks . . 33 12. A G r i d or P l a t e Network a f t e r One Cycle of I t e r a t i o n f o r the Choice, B 1 = - 1 - 2p 2 - 1.3075p 4. . . . . . 37 13. G r i d or P l a t e Network a f t e r One Cycle of I t e r a t i o n f o r the Choice, B 1 = - 1 - 2p 2 - 3.3075p4 . . . . . 37 14. Network Element Values at, Successive Stages of the I t e r a t i o n Procedure, to f o l l o w . . . . . 38 15a. Single-Stage D i s t r i b u t e d A m p l i f i e r , to f o l l o w . . . 40 15b. S p l i t - B a n d D i s t r i b u t e d A m p l i f i e r , to f o l l o w . . . . 40 16. Comparison of a S p l i t - B a n d and a Single-Stage D i s t r i b u t e d A m p l i f i e r , to f o l l o w . . . . . 44 17. Comparison of S p l i t - B a n d and Cascaded D i s t r i b u t e d A m p l i f i e r , to f o l l o w . . . . . . . . 45 A l . A Cascaded D i s t r i b u t e d A m p l i f i e r . 49 A2. I d e n t i c a l Reactive Ladder Networks . . . . . . . . . 50 ACKNOWLEDGEMENT The m a t e r i a l r e p o r t e d i n t h i s paper i s a r e s u l t of work done under the sponsorship of the Defence Research Board, Department of N a t i o n a l Defence, Canada, through DRB Grant Number 5503-04. The author wishes to express h i s a p p r e c i a t i o n f o r the a s s i s t a n c e r e c e i v e d from Dr. A.D. Moore of the Department of E l e c t r i c a l E n g i n e e r i n g , The U n i v e r s i t y of B r i t i s h Columbia. The B r i t i s h Columbia E l e c t r i c Railway Company Limited Graduate S c h o l a r s h i p i n Engineering helped to make the author's post-graduate s t u d i e s p o s s i b l e . SYNTHESIS OP ELEMENTARY DISTRIBUTED AMPLIFIERS USING AN ITERATIVE METHOD 1. I INTRODUCTION The distributed amplifier i s a device based on a d i s -closure i n a B r i t i s h patent granted to W. S„ Percival^" i n 1935. It came into i t s own as a p r a c t i c a l amplifier i n 1948, following the publication of an analysis by Ginzton, Hewlett, 2 Jasberg and Noe. The reasons for the usefulness of the amplifier l i e i n g the natural limitations of the cascade amplifier. Elmore was shown that a conventional cascade amplifier with a given gain has a minimum rise-time i n response to a step-function at i t s input, regardless of the number of i d e n t i c a l stages of amplification used. The l i m i t a t i o n on rise-time i s a d i r e c t consequence of the fact that the over-all gain for such an amplifier i s a function of the product of the stage gains. On the other hand, i n a distributed amplifier, the over-all gain i s d i r e c t l y proportional to the sum of the tube transconductance so that the rise-time l i m i t a t i o n of W. S. Percival, "Thermionic Valve C i r c u i t s , " B r i t i s h Patent 460,562, July 24, 1935 - January 25, 1937. 2 E. L. Ginzton, W. R. Hewlett, J. H. Jasberg and J. 0. Noe, "Distributed Amplification," Proceedings of Institute of Radio Engineers r XXXVI, 956-9597 (1948). 3 W. C. Elmore, "The Transient Response of Damped Linear Networks with Particular Regard to Wideband Amplifiers," Journal of Applied Physics. XIX, 55-63, (1948). Figure 1. Percival's Wave Amplifier. 2. the cascaded amplifier i s not present. Thus i t i s possible to build a distributed amplifier, using conventional elements, which gives results not attainable with a cascade amplifier even with an i n f i n i t e number of elements. Figure 1 shows a block diagram of Percival's proposed wave amplifier. It i s assumed that the a r t i f i c i a l trans-mission lines are r e l a t i v e l y l o s s l e s s , that they contain the tube capacitances, that they are designed to have the same phase v e l o c i t y v^, and that they are properly terminated so as to avoid r e f l e c t i o n s . Thus, any input voltage wave ( E^ n) w i l l travel to the right on the grid l i n e with l i t t l e d i s -t o r t i o n . As E. reaches the f i r s t g r i d connection (T, ), i t xn J.g w i l l produce a similar forward wave (F^) propagating to the right and a backward wave (B^), propagating to the l e f t at the f i r s t plate connection ( T ^ ) i n the plate l i n e . B^ w i l l be absorbed i n the left-hand termination of the plate l i n e , while F^ w i l l t r a v e l to the right with a phase vel o c i t y v . E. w i l l t r a v e l to T n , producing two more waves F„ p i n 2g' * 6 2 and Bg at Tg^, and Fg i s i n phase with F^ a r r i v i n g at Tg^. F^ and Fg add and continue to propagate to the right at the vel o c i t y v . At T 0 , F 0 adds to F, and F 0 and so on. The p op 6 I d forward wave, F, grows i n t h i s manner as i t travels toward the load. Thus, the voltage amplification E„. ,/E. i s 7 out i n d i r e c t l y proportional to the sum of the tube transconductances. By extending the grid and plate networks and adding more tubes, increased gain can be obtained i n such an amplifier without any apparent s a c r i f i c e i n bandwidth or rise-time. Furthermore, a greater bandwidth or a shorter rise-time along with a 3. spec i f ied gain can be obtained i f one i s w i l l i n g to use more components. The increased need for wideband ampli f iers and the present in teres t i n ampl i f iers with prescribed gain, phase, or t r a n -sient-response charac ter i s t i c s have led to improvements i n the 4 design methods appl ied to d i s t r i b u t e d ampl i f i ers . Pederson 5 and Horton each advanced the state of the a r t by applying g network theory, and i n 1952 Moore devised an i t e r a t i v e synthe-s i s method for r e a l i z i n g cer ta in types of d i s t r ibuted ampli-f i e r s with prescribed amplitude response. It should be pointed out that the c i r c u i t s designed for wideband appl icat ions using th i s procedure are only a guide to those used i n the f i n a l construct ion. That i s , the presence of the d i s t r ibuted effects of inductance and capacitance at the frequencies employed must be considered. To use the i t e r a t i v e method, one must f i r s t choose the network, and then, for th i s network, i t i s possible to f ind the component values which w i l l give a close approximation to the prescribed response. Though developed for the synthe-s i s of d i s t r ibuted ampl i f i ers , the technique has been applied D. 0. Pederson, "The Analys is and Synthesis of D i s -t r ibuted ampl i f iers with Ladder Networks," TR-No. 34 (Contract N6onr 25107), Stanford Univers i ty (Electronics Research Laboratory), May, 1951. 5 r W. H. Horton, "A Further Theoret ical and Experimental Investigation of the P r i n c i p a l of Dis tr ibuted Ampl i f i ca t ion ," Ph.D. D i s ser ta t ion , Stanford Univers i ty , 1951. A. D. Moore, "Synthesis of Dis tr ibuted Ampli f iers for Prescribed Amplitude Response," TR-No, 53 (Contract N6onr 25107), Stanford Univers i ty (Electronics Research Laboratory) 1952. 4. 7 to other ampl i f i er c i r c u i t s . The purpose of th i s invest igagt ion was to extend the work already done on d i s t r ibuted ampli f iers by using the i t e r a t i v e method, with the object of assessing by test ( the p r a c t i c a b i l i t y of the appl i ca t ion of network theory to this problem. It was hoped that pract icable ampl i f ier designs with contro l led frequency charac ter i s t i c s and with superior gain-bandwidth factors could be produced. The types of d i s t r i b u t e d ampl i f iers chosen for invest igat ion were non-minimum-phase ampl i f iers and conventional forms of ampli-f i e r s containing symmetrical networks. An exploratory study was also made for a sp l i t -band d i s t r ibuted ampl i f i er . Only designs for maximally-f lat amplitude response were considered since the ca l cu la t ing equipment avai lable (a hand ca lculator with 8 s i gn i f i cant f igures) was impraci tca l for carrying out those ca lcu lat ions necessary to produce ei ther l inear-phase or Chebyshev amplitude c h a r a c t e r i s t i c s . G. A. Caryotakis , "Iterat ive Methods i n Ampl i f i er Inter stage Synthesis," TR-No. 86 (Contract N6onr 25107), Stanford Univers i ty (Electronics Research Laboratory) May, 1955. 5. I I THE DESIGN OP DISTRIBUTED AMPLIFIERS FOR MAXIMALLY-FLAT AMPLITUDE RESPONSE 1. The General T r a n s f e r Function and the D e f i n i t i o n of Maximal  F l a t n e s s . The a m p l i f i c a t i o n f u n c t i o n of a d i s t r i b u t e d a m p l i f i e r made up of lumped l i n e a r elements may be w r i t t e n i n terms of the complex-frequency v a r i a b l e , p, as s / \ a^ + a^p + . . . . s o o . a^p b 0 + b x p + b r p r where the a's and b's are r e a l constant c o e f f i c i e n t s , and r>s. I t i s w e l l known t h a t the u s u a l m a x i m a l l y - f l a t (or B u t t e r -Q worth) low-pass t r a n s f e r f u n c t i o n i s one having a l l zeros of t r a n s m i s s i o n a t i n f i n i t y and a l l p o l e s of t r a n s m i s s i o n e q u a l l y spaced on a s e m i - c i r c l e i n the lext- n-^.. ~ J . »^ .e ^-plane centred about the o r i g i n . Such an arrangement i s not r e a l i z a b l e i n a d i s t r i b u t e d a m p l i f i e r without using c o i n c i d e n t pole-zero p a i r s , because s i n Equation 2:1 i s not normally zero. There-f o r e , i t i s necessary to r e d e f i n e the term " m a x i m a l l y - f l a t " f o r our purposes. To do t h i s , we d e f i n e a power-gain f u n c t i o n , . « « - o ( P ) o ( - , ) - - 2 - 2 > which has numerator and denominator polynomials of degree 2s and 2r r e s p e c t i v e l y . By d i f f e r e n t i a t i n g t h i s f u n c t i o n once with r e s p e c t to p, i t i s apparent t h a t the d e r i v e d f u n c t i o n w i l l have 2 ( r + s ) - l f i r s t - o r d e r zeros i n the f i n i t e r e g i o n of G.E. V a l l e y , J r . , and H. Wallman, "Vacuum Tube A m p l i f i e r s " . New York: McGraw-Hill Book Co., 1948. 6. the p-plane so t h a t the o r i g i n a l f u n c t i o n has 2 ( r + s ) - l f i r s t -order s a d d l e - p o i n t s i n the same r e g i o n . To produce a maximally-f l a t f u n c t i o n under our d e f i n i t i o n , the g r e a t e s t p o s s i b l e number of saddle p o i n t s must he made c o i n c i d e n t at the o r i g i n of the p-plane. How many of these 2 ( r + s ) - l saddle p o i n t s can be made to c o i n c i d e at the o r i g i n ? . This q u e s t i o n can e a s i l y be answered i f one remembers t h a t c o i n c i d e n c e of saddle-points a l s o i m p l i e s c o i n c i d e n c e of f u n c t i o n a l values a t those p o i n t s . For i n s t a n c e , a c l o s e d curve may be drawn around any saddle-point of order (2k-l) i n such a way t h a t the value of the f u n c t i o n passes through 2k complete c y c l e s i n c i r c l i n g the p o i n t once, i . e . , any value found on the curve i s repeated 2k times i n e n c i r l i n g the s i n g u l a r i t y . However, G(p)G(-p) at the o r i g i n i s a r e a l p o s i t i v e constant, G (0). I f the o r i g i n i s a s a d d l e - p o i n t of 2 order ( 2 k - l ) , t h e value G (0) can be approached through the same values from 2k d i f f e r e n t d i r e c t i o n s about the o r i g i n so that p=0 i s a 2 k - f o l d root of the equation N(p)N(-p)/D(p)D(-p) = 2 . • G (0). Since/no pol e s of G(p) appear at the o r i g i n f o r a low-2 pass f u n c t i o n , the equation becomes G (0) D(p)D(-p) - N(p)N(-p)=0, which can have at most 2_r c o i n c i d e n t r o o t s . Therefore, no more than ( 2 r - l ) d e r i v a t i v e s of the general squared-amplitude f u n c t i o n , G(p)G(-p), can be zero a t the o r i g i n . Because G(p)G(-p) i s an even f u n c t i o n i n p, only ( r - l ) of these d e r i v a t e s are zero by the choice of the c o e f f i c i e n t s of the f u n c t i o n . That i s , a l l odd d e r i v a t i v e s of any even f u n c t i o n must be zero* so t h a t only ( r - l ) degrees of freedom i n a p p r o x i -mation are used i n s p e c i f y i n g maximal f l a t n e s s at band cent r e . A f t e r u s i n g two more degrees of freedom to s p e c i f y G(0) and the bandwidth, only (r+l) of the o r i g i n a l (r+s+l) a v a i l a b l e parameters (the c o e f f i c i e n t s i n Equation 2:1) have been used. Therefore, a f t e r maximal f l a t n e s s , bandwidth, and mid-band gain have been s p e c i f i e d , a general low-pass t r a n s f e r f u n c t i o n has as many degrees of freedom a v a i l a b l e f o r f u r t h e r s p e c i f i c a t i o n s as there are f i n i t e zeros of t r a n s m i s s i o n . 2. The M a x i m a l l y - F l a t D i s t r i b u t e d - A m p l i f i e r Stage. I t i s now p o s s i b l e to c o n s i d e r the design problems i n v o l v e d i n s y n t h e s i z i n g d i s t r i b u t e d - a m p l i f i e r stages with m a x i m a l l y - f l a t amplitude response i f one accepts the v a l i d i t y of the f o l l O A v i n g g statement shown to be t r u e by Moore : f o r a d i s t r i b u t e d - a m p l i f i e r stage with networks i n the form of simple ladders, o p e r a t i n g between f i x e d generator and load r e s i s t a n c e s , with a s p e c i f i e d r e l a t i o n s h i p between tube transconductances, and with a r b i t r a r y r e v e r s e - t e r m i n a t i n g r e s i s t a n c e s , there are j u s t three more degrees of freedom i n approximation than n a t u r a l modes of o s c i l l a t i o n . The (r+3) degrees of freedom may be considered to be ( i ) the r c o - o r d i n a t e s i n the p - p l a n i d e f i n i n g the n a t u r a l modes of o s c i l l a t i o n (or p o i n t s of i n f i n i t e g a i n ) , ( i i ) the two a r b i t r a r y parameters expressing the r a t i o of t e r m i n a t i o n r e s i s t a n c e s to reverse terminations i n the networks, ( i i i ) the l e v e l of t r a n s -conductances i n the set of u n i l a t e r a l elements. Of course, any qne of these may be predetermined or l i m i t e d i n i t s v a r i a t i o n , or exchanged f o r some other parameter. Here, i t i s assumed t h a t two r e a c t i v e networks and n vacuum tubes are connected i n a known c o n f i g u r a t i o n with s p e c i f i e d generator and load r e s i s t a n c e s , with s p e c i f i e d r e l a t i o n s h i p s between the tube transconductances, and with a r b i t r a r y r e v e r s e -t e r m i n a t i o n r e s i s t a n c e s as shown i n F i g u r e 2. There are (r+3) degrees of freedom i n s y n t h e s i s , of which two are o b v i o u s l y the reverse t e r m i n a t i o n s . The remaining (r+l) are e x a c t l y the Figure 2. A Single-Stage Distributed Amplifier 8. r e q u i r e d number needed to o b t a i n maximal f l a t n e s s w i t h s p e c i -f i e d mid-band g a i n and bandwidth.-The i t e r a t i v e process produces m a x i m a l l y - f l a t stages i f one a r b i t r a r i l y decides on valuas f o r the reverse terminations and any other parameters corresponding to extra degrees of freedom (such as the resonant f r e q u e n c i e s of shunt branches w i t h i n the networks produced by mutual inductance between adjacent c o i l s ) . In c e r t a i n cases, i t w i l l be e i t h e r con-v e n i e n t or necessary to i n t r o d u c e c o n s t r a i n t s which reduce the f l a t n e s s of the amplitude response i n order to e l i m i n a t e c h o i c e s o c c u r r i n g i n the design or r e s t r i c t i o n s imposed by the network c o n f i g u r a t i o n . The t r a n s f e r f u n c t i o n f o r the a m p l i f i e r given i n F i g u r e 2 i s s ( p ,=% = ! $ - 2!3> where D(p) = D g ( p ) D p ( p ) 2 : 4 . That i s , D(p) i s the product of the polynomials r e p r e s e n t i n g the n a t u r a l modes of the g r i d and p l a t e networks, and N(p) i s the sum of polynomials, each of which represents the zeros of t r a n s m i s s i o n f o r one tube. The c o n t r i b u t i o n to N(p) f o r a s i n g l e tube i s the product of the transconductance of the tube, the numerator of the t r a n s f e r impedance between the g r i d con-n e c t i o n and the i n p u t t e r m i n a t i o n i n the g r i d l i n e , and the numerator of the t r a n s f e r impedance between the p l a t e connection and the output t e r m i n a t i o n i n the p l a t e l i n e . Now, i f we de f i n e H(p) = G~*(p), the r e c i p r o c a l of the squared-amplitude f u n c t i o n Moore, op. c i t . , p. 30. 9. becomes, Suppose t h a t c e r t a i n c o n s t r a i n t s have r e q u i r e d the use of (r-k) degrees of freedom, l e a v i n g ( k - l ) to s p e c i f y maximal f l a t -ness and the remaining two to set the gain l e v e l and bandwidth. The s p e c i f i c a t i o n of maximal f l a t n e s s of the amplitude f u n c t i o n by using ( k - l ) c o e f f i c i e n t s of G-(p) produces a saddle-point of order ( 2 k - l ) i n the f u n c t i o n H(p)H(-p) at the o r i g i n of the p-plane, or a 2 k - f o l d r o o t of H(p)H(-p) - H 2(0) = 0. From Equation 2:5, these must be the zeros of the polynomial D(p)D(-p) - l T ( 0 ) N ( p ) N ( - p ) . A l l remaining zeros must be con-t a i n e d i n a c o n s t r a i n t f u n c t i o n C(p)C(-p), an a l g e b r a i c p o l y -nomial of degree 2 ( r - k ) . C(p)C(-p) g i v e s the l o c a t i o n of 2 those zeros of H(p)H(-p) - H (0) which cannot be used to achieve maximal f l a t n e s s under r e s t r i c t i o n s imposed e i t h e r f o r s i m p l i c i -t y or due to some requirement of the network c o n f i g u r a t i o n i t s e l f . Upon i n t r o d u c i n g an undetermined constant m u l t i p l i e r , 2 J , a l l (r+l) degrees of freedom w i l l have been expended; the r e s u l t i n g expression i s D(p)D(-p) - H 2(0)JN(p)N(-p)j = J 2 ( - p 2 ) k C ( p ) C ( - p ) ..... 2:6. I f the half-power frequency, a) Q, i s chosen as the measure 2 of the stage bandwidth, H( J W q)H(-jw o)/H (0) = 2; t h e r e f o r e , from Equation 2:6, J 2 = H 5(0)|N(jw o)N(-ja) o)/u) o 2 kC(jw o)C(-ju) o)] .......... 2:7. Then, f o r the s p e c i f i e d mid-band g a i n H~"*(o), the h a l f -power bandwidth u>0> and the c o n s t r a i n f u n c t i o n C(p), the n a t u r a l modes may be found by s o l v i n g the equation D(p)D(-p)=H 2(0) I N(p)N(-p)+N(jco 0)N(-jco o). 10, k 2 =2~ %2 C(p)C(-p) \.2:8, C ( j u ^ C ( - j u o | / For the m a x i m a l l y - f l a t d i s t r i b u t e d - a m p l i f i e r stage, Equation 2:8 i s the b a s i c r e l a t i o n s h i p upon which the i t e r a t i v e design procedure i s founded. 3. O u t l i n e of the I t e r a t i v e Procedure f o r Maximal l y - F l a t Response. Assume t h a t the polynomials D (p) and D (p), c o n t a i n i n g 6 3? the approximate n a t u r a l modes of g r i d and p l a t e networks r e -s p e c t i v e l y , are known f o r an a m p l i f i e r stage with a gi v e n con-f i g u r a t i o n and s p e c i f i e d t e r m i n a t i n g r e s i s t a n c e s and tube t r a n s -conductances. In most cases, during the design procedure, i t w i l l be found convenient to normalize the design parameters by s e t t i n g 0)Q = 1, H(0) = -1, and one r e s i s t a n c e i n each network equal to u n i t y . The b a s i c steps i n u s i n g the i t e r a t i v e method to determine the networks which w i l l g ive a m a x i m a l l y - f l a t amplitude response ar e : ( i ) from D (p) and D (p), design an approximate p a i r of g p networks u s i n g a p p r o p r i a t e methods based on D a r l i n g t o n ' s s y n t h e s i s technique"^ (see chapter 3 of t h i s t h e s i s f o r some of the main p o i n t s i n t h i s procedure), ( i i ) from the networks, compute the polynomial N(p) g i v i n g the zeros of t r a n s m i s s i o n and f i n d N ( j w o ) N ( - j w Q ) , ( i i i ) by methods a p p r o p r i a t e to the p a r t i c u l a r case, determine C(p) ( t h i s w i l l be d i s c u s s e d i n d e t a i l when i t a r i s e s ) , S. D a r l i n g t o n , "Synthesis of Reactance Four-Pole" which Produce P r e s c r i b e d I n s e r t i o n Loss C h a r a c t e r i s t i c s . " J o u r n a l of Mathematics and Ph y s i c s , XXVIII, 257-353, (1939) 11. ( i v ) from Equation. 2:8, determine D(p)D(-p) to s a t i s f y the requirement f o r maximal f l a t n e s s and e x t r a c t the zeros l y i n g i n the l e f t h a l f - p l a n e (the n a t u r a l modes), choose appropriate sets of new n a t u r a l modes f o r the two networks, and form D (p) and D p(p)> (v) begin again w i t h ( i ) and repeat the c y c l e u n t i l no f u r t h e r v a r i a t i o n i s considered necessary. I t should be noted that i f N(p) were known the i t e r a t i o n could be s t a r t e d at ( i i ) . N(p) could be obtained from the element values of a given network c o n f i g u r a t i o n . 12 I I I AN APPLICATION OF THE ITERATIVE SYNTHESIS PROCESS 1. The Form of the D i s t r i b u t e d A m p l i f i e r to be Considered. i A d e s c r i p t i o n s i m i l a r to that i n the preceeding chapter can be found i n a report by Howard B. Demuth"''^. His report i l l u s -t r a t e s the usefulness of the method when high-speed d i g i t a l computers are employed f o r the computations. To give the reader a l i t t l e i n s i g h t i n t o the problems inv o l v e d i n c a r r y i n g out a design, the f o l l o w i n g s e c t i o n w i l l describe i n d e t a i l the design of a two-tube d i s t r i b u t e d - a m p l i -f i e r stage having i d e n t i c a l n a t u r a l modes i n the g r i d and p l a t e networks and the c o n f i g u r a t i o n shown i n Figure 3* Figure 3 . A Two-Tube D i s t r i b u t e d - A m p l i f i e r Stage. Howard B. Demuth, "An I n v e s t i g a t i o n of the I t e r a t i v e S y n t h e s i s o f D i s t r i b u t e d . A m p l i f i e r s " . TR-No. 77 (Contract N66nr 25107) Stanford U n i v e r s i t y ( E l e c t r o n i c s Research Labo-r a t o r y ) August, 1954. 13. This p a r t i c u l a r a m p l i f i e r , a s i d e from being r e l a t i v e l y easy to design by the i t e r a t i v e method, has the i n t e r e s t i n g p r o p e r t y of producing a non-minimum phase response, which to our knowledge has not been encountered p r e v i o u s l y i n d i s t r i -buted a m p l i f i e r s . Furthermore, i t was hoped t h a t the use of i n f i n i t e reverse t e r m i n a t i o n s would r a i s e the gain-bandwidth f a c t o r . I t was r e a l i z e d t h a t t h i s might be u n d e s i r a b l e i n some a p p l i c a t i o n s due to the l a r g e r e f l e c t i o n c o e f f i c i e n t i n t r o d u c e d . 2. I d e n t i c a l N a t u r a l Modes i n Both Networks. The d i s t r i b u t e d a m p l i f i e r chosen i s a s p e c i a l case i n which both networks must be a l i k e , except f o r a p o s s i b l e d i f f e r -ence i n impedance l e v e l . For such an a m p l i f i e r , D(p) i s a p e r f e c t square, and the coin c i d e n c e of the zeros of D(p) r e q u i r e s t h a t the degree^ r , of D(p) be an even number. There are then g s a d d l e - p o i n t s of H(p)H(-p) = ^ J P J ^ J ' P J i n the l e f t h a l f - p l a n e (the complex p-plane) s i n c e h a l f of the a v a i l a b l e degrees of freedom (the c o e f f i c i e n t s of p i n D(p)) have been used i n s p e c i f y i n g equivalence of the two networks. This has e l i m i n a t e d a l l choice i n the d i s t r i b u t i o n of n a t u r a l modes between networks. Although the r e s u l t i n g networks may not g i v e optimum response, the labour r e q u i r e d i n the design process i s g r e a t l y reduced. This i s apparent i n two ways, by e l i m i n a t i n g the choices which must otherwise be made, and by reducing the degree of the polynomial, which must be solved to f i n d the n a t u r a l modes. The assumptions t h a t H(0) = -1 and LOQ = 1, as suggested e a r l i e r , help g r e a t l y to s i m p l i f y the development of the 14. equations necessary f o r t h i s s p e c i a l case. A d d i t i o n a l i n f o r -mation i s t h a t C(p), the unknown c o n s t r a i n t - f u n c t i o n , i s of degree and N(p) i s the usual polynomial c o n t a i n i n g the approxi -mate zeros of t r a n s m i s s i o n . By d e f i n i n g c i j i c ( - j { = C o + ° 2 P 2 + C 4 p 4 + + C r - 2 P r " 2 + V * 3 S 1 ' and 2 D(p)D(-p) =(d 0 + d 2 p 2 + + d r _ 2 p r " 2 + d r p r ) 3:2, these q u a n t i t i e s may be s u b s t i t u t e d i n t o Equation 2:8 to give d o 2 + 2 d oV 2 + ( 2 d o d 4 + d 2 2 ) P 4 + ^ W ^ V * 6 + ( 2 d o d 8 + 2 d 2 d g > d 4 2 ) p 8 + .... + ( 2 d r d r _ 2 ) p 2 r - 2 + ( d r 2 ) p 2 r = N(p)N(-p) + ( - p 2 ) r / 2 ( c 0 + c 2 p 2 + .. + c r_ 2p r"" 2+c rp r)N ( j)N ( - j ) .. 3 In t h i s equation, the unknown c o e f f i c i e n t s , d Q , d 2 , d^, .... r d A* d n t can be found s u c c e s s i v e l y from the TT r e l a t i o n s h i p s r—4 T—C £ r-2 between the c o e f f i c i e n t s of powers of £ up to p on both s i d e s of the equation. However, the next equation, found by equating c o e f f i c i e n t s of p , i s l i n e a r i n two unknowns, c Q and d r . S i m i l a r l y , a l l e's up to c r _ 2 can be expressed l i n e a r l y i n terms of d , whereas the expression f o r c contains r r 2 the q u a n t i t y d . Thus, only the two unknown d and c remain, r r o The f i n a l equation necessary f o r s o l u t i o n i s found from the normalized c o n s t r a i n t - f u n c t i o n ; i . e . by choosing u>o = 1, the normalized c o n s t r a i n t - f u n c t i o n of Equation 3:1 giv e s c Q - c 2 + c 4 - c 6 + ... + ( - l ) r / 2 ~ 1 c r _ 2 + ( - l ) r ^ 2 c r = 1 ... 3:4. I f the known expressions f o r the e's are s u b s t i t u t e d i n 15. Equation 3:4, a s i n g l e q u a d r a t i c equation i n d y r e s u l t s , having the form 2 d *4" U.d ""§• V — 0 • •••©••. •••••o..«o.©o.....»ft»oo 3:5, r r where U and V are r e a l constants. In choosing the proper s o l u t i o n of the q u a d r a t i c , d r must be negative i f 77 i s odd and p o s i t i v e i f i s even. T h i s i s evident i f one considers that d r i s the c o e f f i c i e n t of the h i g h e s t power term of an even f u n c t i o n formed from the product D (p)l> (-p), and must neces-• g g s a r i l y be negative i f the degree of D (p) i s odd. S i m i l a r l y d must be p o s i t i v e i f the degree of D (p) i s even, r g The i t e r a t i v e process i s then c a r r i e d out as before, beginning with an approximate expression f o r N(p), s o l v i n g f o r the d - c o e f f i c i e n t s which determine the polynomial g i v i n g the n a t u r a l modes of each network, d e s i g n i n g the networks ( i d e n t i c a l , except f o r a p o s s i b l e change i n impedance l e v e l ) , and f i n d i n g a c l o s e r approximation f o r N(p). 3. Intermediate Steps and P i n a l R e s u l t s of the Design. Since i t i s p o s s i b l e to begin a design from the knowledge of e i t h e r N(p) or D(p), i t was thought t h a t a low-pass, constant-k, n - s e c t i o n f i l t e r f o r the g r i d and p l a t e networks of the d i s t r i b u t e d a m p l i f i e r would be a s u i t a b l e s t a r t i n g -p o i n t from which N(p) c o u l d be obtained. For these networks, both R^=l and W q=1, and the component values r e s u l t i n g from these assumptions are shown i n F i g u r e 4. From the i n f o r m a t i o n given i n F i g u r e 4, the expression f o r N(p) can be determined a t once by f i n d i n g s e p a r a t e l y the zeros of t r a n s m i s s i o n f o r each tube and adding the r e s u l t a n t 16. polynomials. E . a -j-out + 1 _ E. C / V in ( p i . O i 2 2 1 31 F i g u r e 4. I n i t i a l Component Values f o r a Two-Tube A m p l i f i e r * The zeros f o r the f i r s t tube occur i n the g r i d l i n e when, 2p + j = 0 or 2 p 2 + 1 = 0 3:6, and i n the p l a t e l i n e when 2p + i • 0 or 2 p 2 + 1 = 0 3:7. P Hence, the polynomial c o n t a i n i n g the zeros due to the f i r s t tube i s g m 1 ( 2 p 2 + l ) 2 3:8. For t h i s a m p l i f i e r , the second tube does not c o n t r i b u t e any f i n i t e zeros of t r a n s m i s s i o n , so the corresponding p o l y -nomial i s a constant, p r o p o r t i o n a l to gnig. Therefore, N(p) = -gm 1(2p 2+l) 2-gm 2 3:9, and s e t t i n g gm^ = gm2 = ^ giv e s 17. - N(p) = 2 p 4 + 2p 2 + 1 I f G(0) = H ^(O) = -1, as s p e c i f i e d e a r l i e r , then the constant terra i n D(p) w i l l be u n i t y . The squared-magnitude f u n c t i o n of N(p) i s N(p)N(-p) = 4 p 8 + 8 p 6 + 8 p 4 + 4 p 2 + 1, and N ( j ) N ( - j ) = 1. . The occurrence of i d e n t i c a l n a t u r a l modes i n g r i d and p l a t e networks makes D(p) = D ( p ) D ( p ) = D (p) 2 = D (p) 2 3:10. s P 8 P Therefore, using Equation 3:3 and knowing t h a t r = 6, the equation i n v o l v i n g the d - c o e f f i c i e n t s becomes d Q 2 + 2 d Q d 2 p 2 + ( 2 d Q d 4 + d 2 2 ) p 4 + ( 2 d 0 d 6 + 2 d 2 d 2 ) p 6 + ( 2 d 2 d 6 + d 4 2 ) p 8 + 2 d 4 d 6 p 1 0 + d 6 2 p 1 2 = 1 + 4p 2 + 8 p 4 + ( 8 - c Q ) p 6 + ( 4 - c 2 ) p 8 - c 4 p 1 0 - c 6 p 1 2 .. 3:11. Equating c o e f f i c i e n t s of powers of p_ on both sides of Equation 3:11 immediately g i v e s the val u e s : CLQ = 1» d 2 = 2 and d^ = 2. The expressions f o r the c o e f f i c i e n t s CQ to Cg i n terms of d c can be s u b s t i t u t e d i n t o the normalized 2 c o n s t r a i n t - f u n c t i o n to g i v e the q u a d r a t i c dg - 2dg - 1 = 0 . The proper choice between the r o o t s of t h i s equation makes d g = -0.4142; t h e r e f o r e , D (p)D (-p) = -0.4142p 6 + 2 p 4 + p 2 + 1. 18. 4. Separation of L e f t - H a l f - P l a n e Roots and R e a l i z a t i o n  of Component Values i n the Networks. I f the even polynomial f o r D (p)D (-p) i s of eighth 8 6 degree or lower, the l e f t - h a l f - p l a n e (LHP) zeros can be separated from those i n the r i g h t - h a l f - p l a n e (RHP) w i t h -12 out a c t u a l l y f i n d i n g the i n d i v i d u a l zeros of the polynomial , For even polynomials of degree higher than e i g h t , i t i s necessary to remove f a c t o r s u n t i l the remaining f a c t o r i s of e i g h t h degree or l e s s . T h i s s i m p l i f i c a t i o n i s p o s s i -b l e here. A f t e r having c a r r i e d out the computations, the expression f o r the n a t u r a l modes f o r the g r i d and p l a t e networks becomes D„(p) = 0.6436p 3 + 1.940p 2 + 1.371p + 1. 8 Because each of the networks has one i n f i n i t e t e r m i -n a t i o n , the d r i v i n g - p o i n t impedance i s ^ A where Z i l - V p B 3 : 1 2 » D g ( p ) = A + pB. R e a l i z a t i o n of g i v e s the element values shown i n F i g u r e 5. With t h e . i n f o r m a t i o n given i n F i g u r e 5, a new expression f o r N(p) can be obtained and by r e p e a t i n g the process out-l i n e d above the element values w i l l converge to the d e s i r e d accuracy. Moore, op. c i t . . p. 90. 13 D a r l i n g t o n , op. c i t . . p. 277. 19. Figure 5. Network Element Values after One Cycle of Iteration. For the a m p l i f i e r being designed, a f t e r three c y c l e s of i t e r a t i o n , the element values a r r i v e d at are those appearing i n Figure 6. These values v a r i e d not more than three percent from those found i n the previous c y c l e . The expression f o r the t r a n s f e r f u n c t i o n f o r t h i s a m p l i f i e r i s S(p) = .1.716p 4 + 1.852p 2 + 1 (0.6098p° + 1.852P* + 1.347p + l ) which gives the amplitude and phase response shown i n Figure 6. I t i s obvious from the diagram that t h i s a m p l i f i e r has a non-minimum phase response. 5. The Gain-Bandwidth Factor f o r a Single-Stage D i s t r i b u t e d  A m p l i f i e r . 14 A f i g u r e of merit suggested by Pederson can be used to compare the r e l a t i v e advantages of one single-stage Pederson, op. c i t . . p. 25 20. 15 d i s t r i b u t e d a m p l i f i e r with another. But, as Moore has i n d i c a t e d , t h i s f i g u r e of merit should not be used to compare cascaded a m p l i f i e r s s i n c e i t does not account f o r the g r i d -c i r c u i t impedance l e v e l which should be i n c l u d e d i n some way to g ive a s a t i s f a c t o r y c r i t e r i o n of performance. The f i g u r e of merit (GB r g^) i s given i n i t s general form i n Equation 3:13 and i s based on the assumption t h a t the tubes i n a stage have i d e n t i c a l c h a r a c t e r i s t i c s and that C the g r i d capacitance to p l a t e capacitance r a t i o i s ^ = 1.2. P Then (GB) , = (l/2-pciwer bandwidth)x(l+C /C )x(voltage g a i n with r © J . Q p u n i t transconductance)x(minimum p l a t e capacitance) ..... 3:13. A comparison of the a m p l i f i e r d i s c u s s e d above with an a m p l i f i e r design from Moore's work shows t h a t the f i g u r e of merit of the former, GB , = (0.96)(1+1.2)(2)(0.329) = 1.39, r e i i s c o n s i d e r a b l y lower than the f i g u r e of merit of the l a t t e r ; i . e . G B « i = (0.92)(1+1.2)(2)(0.533) = 2.16. The comparison i s v a l i d s i n c e Moore's a m p l i f i e r i s a l s o a s i n g l e stage a m p l i f i e r c o n t a i n i n g the same number of tubes. T h i s a m p l i f i e r and i t s response c h a r a c t e r i s t i c s are shown i n F i g u r e 7. The only r e s t r i c t i o n on t h i s design was t h a t the g r i d and p l a t e networks c o n t a i n i d e n t i c a l r e a l modes. An Moore, op. c i t . . p. 24. Moore, op. c i t . . p. 75. Figure 7. Two-Tube Stage w i t h Coincident Real Modes. i n s p e c t i o n of the p o s i t i o n of the zeros i n the t r a n s f e r f u n c t i o n of t h i s a m p l i f i e r , which i s G(p) = U 8 3 9 p 2 + 1.117p + 1 : , (p +2.546p +2.493p+l.22)(p +1.875p +1.277p+.8198) shows that the a m p l i f i e r has a minimum-phase c h a r a c t e r i s t i c . The only advantage our design has over Moore's i s that i t could he connected to a load having a f i n i t e impedance without the use of ex t r a coupling devices required f o r h i s a m p l i f i e r . 22. IV DESIGN RESULTS FOR TWO NON-MINIMUM PHASE AMPLIFIERS An improvement i n the gain-bandwidth f a c t o r f o r the design considered i n the previous chapter was made by p l a c i n g a peak-i n g c o i l i n both the g r i d and p l a t e l i n e s which r a i s e d the value of the shunt c a p a c i t a n c e s . The design of a two-tube a m p l i f i e r having networks of the c o n f i g u r a t i o n mentioned above was c a r r i e d out, and a f t e r three c y c l e s of i t e r a t i o n the p o l y -nomial expression f o r the n a t u r a l modes i n the g r i d and p l a t e networks was D (p) = 0.7599p 4 + 2.188p 3 + 3.150p 2 + 1.991p + 1. O The r e a l i z a t i o n of the network gave element values with-i n three percent of those r e a l i z e d i n the previous c y c l e of i t e r a t i o n . The polynomial f o r the zeros of t r a n s m i s s i o n produced by these element values was N(p) = 3.023p 4 + 2.459p 2 + 1. The a m p l i f i e r ' s element values and i t s phase and ampli-tude response are shown i n Fi g u r e 8. The r e l a t i v e f i g u r e of mer i t i s GB i = ( I . l l ) ( 2 . 2 ) ( 2 ) ( 0 . 8 9 ) = 4.35. r e i Although there i s no a m p l i f i e r of the same c o n f i g u r a t i o n with which to compare these r e s u l t s , the la r g e i n c r e a s e i n gain-bandwidth f a c t o r (about three times) over the f i r s t d e s i gn considered seemed to i n d i c a t e t h a t i t would be worth-while to design a three-tube a m p l i f i e r of the same type. This was c a r r i e d out, but only to a p o i n t where i t was ap-parent t h a t no a p p r e c i a b l e i n c r e a s e i n gain-bandwidth f a c t o r Figure 8. Improved Two-Tube Stage w i t h I d e n t i c a l Natural Modes• would be achieved* After completing the calculations for three cycles of iteration, the element values did not vary more than a few percent and the expression for the natural modes at the end of the third cycle became D (p) = 0.712p6 + 2.955p5 + 6.129p4 O + 6.982p3 + 6.311p2 + 2.715p + 1. Using D (p) to realize the network, the element values in 8 Figure 9. were obtained and these produced the expression for the zeros of transmission N(p) = 6.589p8 + 16.763p6 + 16.29p4 + ,5.655p2 + 1. Hence,the transfer function for the amplifier became r/ x 6.589p8+16• 76p6+16.29p4+5.6 55p2+l  Glp)=r S c — c c — z r — C — Q — TP» (0.7124p°+2.955p°+6.129p>6.982p°+6.3 l i p +2.715p+i)' and i t s relative figure of merit was approximately GB , = (1.08)(2.2)(3)(0.665) = 4.74. The small increase in gain-bandwidth factor obtained by the addition of an extra tube, and the rapidly increasing phase slope produced by adding the extra elements for these tubes, indicated that further investigation on this particu-lar type of amplifier would be unwarranted. Furthermore, to design a distributed amplifier of any configuration containing three or more tubes requires weeks of computation when using a desk calculator, even assuming familiarity with the design procedure. Figure 9 . A Three-Tube A m p l i f i e r with I d e n t i c a l Networks. Another drawback to the p r a c t i c a b i l i t y of such an a m p l i f i e r i s the f a c t t h a t i f two i d e n t i c a l a m p l i f i e r s are cascaded, n a t u r a l modes are produced on the r e a l -frequency a x i s thereby completely d e s t r o y i n g the maximally f l a t amplitude response. The proof of t h i s statement i s g i v e n i n Appendix I. 25. V THE DESIGN OP DISTRIBUTED AMPLIFIERS CONTAINING SYMMETRICAL NETWORKS 1. C o n s i d e r a t i o n s f o r S i m p l i f y i n g the Design Procedure. The time consumed i n f a c t o r i z i n g the polynomials en-countered i n the designs considered thus f a r i n d i c a t e d t h a t , with the type of equipment a v a i l a b l e f o r c a r r y i n g out the c a l c u l a t i o n s , any other d i s t r i b u t e d a m p l i f i e r to be designed should have as simple a s t r u c t u r e as p o s s i b l e so as to reduce the degree of the polynomials to be f a c t o r e d . One of the simplest s t r u c t u r e s f o r d i s t r i b u t e d a m p l i f i e r s i s one which has i d e n t i c a l n a t u r a l modes i n the g r i d and p l a t e networks, with each network symmetrical about i t s mid-point. The development of a design procedure f o r a m p l i f i e r s 17 of such a c o n f i g u r a t i o n was c a r r i e d out by Moore • The main p o i n t s are being reproduced here to give a b e t t e r i n -s i g h t i n t o the problem and to c l a r i f y a t l e a s t one p o i n t i n the procedure which caused some co n f u s i o n . C e r t a i n p a r t s of D a r l i n g t o n ' s work on the s y n t h e s i s of reactance f o u r - p o l e s are necessary f o r the development of the procedure; t h e r e f o r e , those p a r t s are presented here f o r the sake of completeness. 2. D a r l i n g t o n ' s Synthesis of Reactance Four-Poles. 18 A method de v i s e d by D a r l i n g t o n f o r the s y n t h e s i s of reactance f o u r - p o l e s can be used to design reactance Moore, o p . c i t . . p. 53. D a r l i n g t o n , op. c i t . . p. 260. networks which produce a p r e s c r i b e d i n s e r t i o n power r a t i o , 2a 6 , when placed between a generator of i n t e r n a l r e s i s t a n c e R^, and a load r e s i s t a n c e Rg. I n s e r t i o n power r a t i o can be d e f i n e d i n terms of the l o a d v o l t a g e E^Q present before i n s e r t i o n of the network, and Eg, the load voltage a f t e r the network i s i n p l a c e , by the expression °2a - <VV8 5 ! l-The q u a n t i t y a then becomes the a t t e n u a t i o n produced by the reactance network, i n nepers. I t was shown by D a r l i n g t o n t h a t an exact r e a l i z a t i o n u s i n g reactance net-works i s p o s s i b l e f o r any r a t i o n a l power r a t i o i f , f o r a l l r e a l to, 2a 4 R 1 B 2 E^A S —LJL- 5:2. (Rj+R,,) To demonstrate the method, l e t us c o n s i d e r one of the g r i d networks, suqh as t h a t shown i n F i g u r e 10, as a r e -actance^quadripole o p e r a t i n g between p r e s c r i b e d t e r m i n a t i o n s , R^ and R^. In the diagram, the equivalent-T r e p r e s e n t a t i o n of coupled c o i l s i s used. Although the c o n f i g u r a t i o n of t h i s g r i d network i s known, the element values have not been determined. How-ever, i t w i l l be assumed t h a t the n a t u r a l modes are known i n the form of a polynomial D g which has i t s zeros at the p o l e s of t r a n s m i s s i o n between the r e s i s t i v e t e r m i n a t i o n s . I f the network were a simple low-pass ladder, the s e r i e s branches would c o n t a i n inductance only and the shunt Figure 10. A G r i d Network as a Reactance Four - P o l e 27. branches would c o n t a i n capacitance only. In that case, a l l of the zeros of t r a n s m i s s i o n would be at p = 0 0 , and the numerator of the t r a n s m i s s i o n f u n c t i o n would be a constant. However, one must consider more general networks i n which there may be mutual c o u p l i n g between adjacent c o i l s ( D a r l i n g -ton's Type-C s e c t i o n s ) , so t h a t i t i s p o s s i b l e t h a t zeros of t r a n s m i s s i o n can occur symmetrically about the o r i g i n on e i t h e r a x i s , producing an even polynomial P (p) i n the numerator of the t r a n s m i s s i o n f u n c t i o n . I f i t i s assumed t h a t the constant terms i n the p o l y -nomials are equal, the complex i n s e r t i o n v o l t a g e r a t i o may be w r i t t e n E20 . V£ 5:3. E 2 « PglpT Therefore, D ( p ) D ( - P ) c2« = J t . .H, 5:4, and t h i s expression s a t i s f i e s equation 5:1 on the w-axis, but i s g e n e r a l i z e d to apply to the whole p-plane. I f A(p) and B(p) are d e f i n e d as even f u n c t i o n s , the polynomial D may be r e w r i t t e n as A + pB, so t h a t S e2a A 2 - p 2 B 2 5 l 5 . P 2 g Let us suppose that the input impedance of the t e r m i -nated network f a c i n g the generator r e s i s t a n c e R^ i s d e f i n e d as z-^l* S i m i l a r l y , l e t the impedance seen by Rg be d e f i n e d as Z g g . At any t e r m i n a l p a i r , such as t h a t where R^ i s 2 8 . connected, a complex r e f l e c t i o n c o e f f i c i e n t may be defined, R, - z,, r = _± i i . 5:6, A l R x + z x l ' so t h a t -11 _ _ i 5:7. R 2 i + r 1 The poles of the r e f l e c t i o n c o e f f i c i e n t are the n a t u r a l modes of the system, because R^ + = 0 at a n a t u r a l mode. S u b s t i t u t i n g the polynomial A + pB, which s p e c i f i e s the n a t u r a l modes, the expression f o r the complex r e f l e c t i o n c o e f f i c i e n t becomes r = A ' + p B ' _ 5 s 8 Ll A + pB where A' and B 1 are even polynomials i n p, and A 1 + pB 1 has zeros where R^ = z n * Using a well-known r e l a t i o n s h i p from i n s e r t i o n - l o s s theory = r (p)r ( - P ) = i - - 2 a 5 : 9 > i t can be r e w r i t t e n as, A' 2 - p 2 B ' 2 = A 2 - p 2 B 2 - 4 R l R s 9 . P 2 ( p ) 5:10. (Rj+RgT 8 2 2 2 The even polynomial, A' - p B 1 , contains zeros i n quadrantal symmetry i n the p-plane. Before proceeding i t i s necessary to s p l i t the polynomial i n t o two f a c t o r s , 19 H.W. Bode, "A General Theory of E l e c t r i c Wave F i l t e r s , " J o u r n a l of Mathematics and P h y s i c s . X I I I , 275-362 (1934). 29. A 1 + pB 1 and A 1 - pB 1. This can be done i n a number of d i f f e r e n t ways. In the f i n a l r e s u l t , i t w i l l be found that h a l f of these choices determine networks which are the duals of the othe r s . Because we know the network c o n f i g u r a t i o n , these choices can be r e j e c t e d immediately. Again, h a l f of the o r i g i n a l choices r e s u l t i n the r e v e r s a l of the networks, end f o r end, p o s s i b l y with a change i n impedance l e v e l . I f the problem i s analyzed i n d e t a i l , i t w i l l be found that there are two networks i n the s p e c i f i e d c o n f i g u r a t i o n f o r 2 2 2 every p o s s i b l e way of d i v i d i n g the zeros of A' - p B' i n t o two s e t s , m a i n t a i n i n g symmetry about the r e a l a x i s . A f t e r a choice has been made f o r A 1 and B', the s u b s t i -t u t i o n of Equation 5:8 i n t o Equation 5:7 produces an ex-p r e s s i o n f o r the d r i v i n g - p o i n t impedance, z _ R (A-A») + p(B-B') 5:1!. Z l l ~ ttl (A+A») + p(B+B') A knowledge of the d r i v i n g - p o i n t impedance f a c i n g the generator r e s i s t a n c e i s s u f f i c i e n t to permit the r e a l i z a t i o n of the network. One c o u l d a l s o r e a l i z e the network from the o p e n - c i r c u i t and s h o r t - c i r c u i t impedance expressions developed by D a r l i n g t o n and given here i n Equation 5:12. o A-A' „ n p(B-B' ) z01 = B l p(B+B<) Z s l = R l A+A' e o o • e • • 5*12 z02 " R2 p W I 2 s 2 = R2 S i m i l a r expressions can be found f o r the impedance f a c i n g the l o a d . In summary, i f a polynomial A + pB i s known, having zeros which are the n a t u r a l modes or p o i n t s of i n f i n i t e g a i n , 30. together with a polynomial P (p) g i v i n g the point* of i n f i n i t e S l o s s , both polynomials having the same constant term, a non-unique reactance s t r u c t u r e terminated i n the given r e s i s t a n c e s R^ and Rg can be found by the f o l l o w i n g procedure: ( i ) check t h a t c o n d i t i o n 5:2 i s f u l f i l l e d , 2 2 2 ( i i ) form the even polynomial A' - p B' , using 5:10, 2 2 2 ( i i i ) f a c t o r A' — p B' and choose A' and B', ( i v ) form Equation 5:11 to design the network. 3. I d e n t i c a l N a t u r a l Modes and Symmetrical Networks. The design problem becomes simpler i f the g r i d and p l a t e networks have the same n a t u r a l modes and each network i s symmetrical about the mid-point of i t s l e n g t h . This i m p l i e s t h a t the two terminations are n e c e s s a r i l y equal, so that two of the a r b i t r a r y parameters have been removed. Other degrees of freedom are a l s o l o s t i n s p e c i f y i n g symmetry so that the f u n c t i o n cannot be as f l a t near p = 0 but the so-l u t i o n should be more t r a c t a b l e . From D a r l i n g t o n ' s work, the c o n d i t i o n f o r symmetry i n a r e a c t i v e network i s t h a t the numerator of the r e f l e c t i o n c o e f f i c i e n t should be an odd polynomial i n p, i . e . A' = 0. Consequently, a network i s symmetrical i f B' i s even and i f The c o n d i t i o n f o r i d e n t i c a l n a t u r a l modes i n both net-works i s = P 2 - (A 2 - p 2 B 2 ) 5:13. D(p)D(-p) = (A 2 - p 2 B 2 ) 2 5:14. 31. Then i f r, i s the t o t a l number of modes of o s c i l l a t i o n f o r the stage, D(p)D(-p) must be a 2 r t h degree polynomial. Moreover, _r must be an even number due to the requirement of i d e n t i c a l g r i d and p l a t e networks, but r/2 must be odd f o r summetry w i t h i n each network. P $p), the square of the p o l y -8 nomial c o n t a i n i n g the zeros of t r a n s m i s s i o n between the terminations of one of the i n d i v i d u a l networks, must be of 2 2 lower degree than r. Therefore, from Equation 5:13, p B' must be of the r t h degree, so t h a t B 1 i s an even polynomial of degree (r/2 - l ) . Let P g 2 = e 0 + e 2 p 2 + e 4 p 4 + e s - 2 p S ~ 2 + e s p S 5 : 1 5 > where s_ i s l e s s than r , and l e t B* = b Q + b 2 p 2 + b 4 p 4 + ... + b r / 2 - 3 p r / 2 " 3 + b r / 2 - l p r / 2 ' " 1 - - 5 ! l 6 ' I t w i l l be assumed, as before, t h a t n o r m a l i z a t i o n has been introduced, so t h a t H(0) = -1 and co^  = 1. Using the same n o t a t i o n as i n the s e c t i o n d e a l i n g with i d e n t i c a l n a t u r -a l modes i n both networks, the normalized even c o n s t r a i n t -f u n c t i o n can be represented by • >0 + °**2 * = 2 ( ^ ) P 2 ( r - k ) i f only ( k - l ) degrees of freedom of the maximum number po s s i b l e , ( r - l ) , are l e f t to s p e c i f y maximal f l a t n e s s a f t e r the r e s t r i c t i o n s have been imposed. By combining Equations (2:8), (5:13), (5:14), (5:15), (5:16), and (5:17), the c o n d i t i o n f o r maximal f l a t n e s s , under the r e s t r i c t i o n s imposed, becomes 32. e 0 2 + 2 e 0 ( V b 0 2 ) p 2 + [f e2- b0 2 ) + 2 e 0 ( e 4 - 2 b 0 b 2 ^ 4 + [ 2 e 0 ( e 6 - 2 b 0 b 4 - b 2 2 ) + 2 ( e 2 - b Q 2 ) ( e ^ b ^ f j p 6 . . . + b r / 2 - 3 b 3 r / 2 - l p 2 " ? + ^ - l ^ = N ( p ) N ( - p ) + ( - p 2 ) k | ^ 0 + c 2 p 2 + . . + c 2 ( r _ k ) p 2 ( r - k ] N ( j ) N ( - j ) . . . 5:18. A s o l u t i o n can be obtained f o r a l l b's, i n the same manner as d i s c u s s e d p r e v i o u s l y , i f 2k = r/2 + 1. Then a l l the c o e f f i c i e n t s up to b 2k-4 c a n b e found s u c c e s s i v e l y by equating c o e f f i c i e n t s . A l l c f s can be expressed i n terms of b ryf 2 ^(=^2k 2^' a n d b y s u b s t i t u t i n g them i n t o the normal-i z e d c o n s t r a i n t - f u n c t i o n , C 0 " C2 + C 4 " + < - 1 > r " k " 1 ° 2 ( p - k - l ) + ( - l ) r ' k c 2 ( r - k ) S B l " - 5 , 1 9 » a q u a r t i c of the form b r / 2 - l + S - b r/2-1 + T " b r / 2 - l + U - b r / 2 - l + T = 0 5 ' 2 0 > i s produced and S, T, U and V are r e a l c o e f f i c i e n t s . The c o e f f i c i e n t b^yg ^  i s a r e a l root of t h i s q u a r t i c equation. S e l e c t i o n of the c o r r e c t root must be made by ref e r e n c e to the d e f i n i t i o n of r e f l e c t i o n c o e f f i c i e n t i n Equation 5:8 and the expression f o r the impedance at the te r m i n a l s of a symmetrical network, Equation 5:11. At f i r s t glance, the above procedure seems very s t r a i g h t -forward, but i t so turns out th a t there i s a choice of sign f o r bg, the constant term of the expression f o r B', which i n f l u e n c e s the si g n of a l l subsequent b's. Moreover, should the choice of s i g n f o r be i n c o r r e c t i t i s p o s s i b l e t h a t none of the f o u r r o o t s of the q u a r t i c equation c o n t a i n i n g 33. b r / 2 - l w i l l be t h e c o r r e c t r o o t . B e s i d e s t h e e x t r a c o m p l i c a t i o n i n t h e d e s i g n p r o c e s s d u e t o t h e c h o i c e o f s i g n s m e n t i o n e d , i t a l s o seems p o s s i b l e t h a t t h e d e s i g n p r o c e d u r e f o r d i s t r i b u t e d a m p l i f i e r s h a v i n g i d e n t i c a l n a t u r a l modes i n g r i d a n d p l a t e n e t w o r k s a n d e a c h n e t w o r k s y m m e t r i c a l a b o u t t h e m i d - p o i n t o f i t s l e n g t h may n o t b e c o n v e r g e n t . To show t h a t t h e r e m a r k s c o n c e r n i n g t h e a b o v e p r o c e d u r e h o l d t r u e a n a c t u a l d e s i g n o f a l o w - p a s s a m p l i f i e r s u b j e c t t o t h e s p e c i f i c a t i o n s d i s c u s s e d i s g i v e n b e l o w . F i g u r e 11 s h o w s t h e n e t w o r k c o n f i g u r a t i o n a n d t h e e l e m e n t v a l u e s f r o m w h i c h t h e i n t i a l N ( p ) was o b t a i n e d . Figure 11. I n i t i a l Element Values f o r a Two-Tube A m p l i f i e r Containing Symmetrical Networks. 34. The choice of a low-pass network configuration makes P (p)= 1 i n Equation 5:3, since there are no f i n i t e zeros of S transmission i n the network. After forming N(p), N(p)N(-p), and evaluating N ( j ) N ( - j ) , these expressions were substituted into Equation 5:18 to give ( l - p 2 B ' 2 ) 2 = e Q 2 - 2 e 0 b 0 2 p 2 + [ b Q 4 - Ae^^ p 4 + . . . T o u 2/, 2 o u . \ , „ K 2,2~1 16 , 3 18 , 4 20 + L 2 b 4 ( b2 + 2 b 0 b 4 ) + 4 b 2 b 4 J P + 4 b 2 b 4 P + b 4 P ' * 1 - 2p 2 - 7p 4 - (8+ 2 0c Q )p 6 + ( l 6 - 2 0 c 2 ) p 8 - 2 0 ( c 4 p 1 0 + c 6 p 1 2 + c 8 p 1 4 + c 1 0 p 1 6 + c 1 2 p 1 8 + c 1 4 p 2 0 ) . . . 5:21, where B' = b Q + b 2 p 2 + b 4 p 4 5:22, and c{jic(-[i ^ 0 + C 2 p 2 + C 1 2 p 1 2 + C 1 4 p U 5 i 2 3 < As mentioned above, P (p) = 1, which makes e n = 1. g u Therefore, by subst i tut ing eg = 1 into 5:21 and then equating coe f f i c i ents on both sides of the equation, the fol lowing resu l t s are obtained. F i r s t a pos i t ive choice of b^ gives bQ = 1, bg = 2 , and through subsequent manipu-la t ions the quart ic equation containing b 4 becomes b „ 4 - 4 b . 3 + 8b 2 - 8b. - 36 = 0. 4 4 4 4 There are two rea l and two complex roots for b 4 , but only the rea l roots are possible choices for b 4 since the coe f f i c i ents of p i n B' must be r e a l . The values for b 4 are b. = -1.3075 and b. = 3.3075 which, when combined with 4 4 ' bQ and b 2 , give the two expressions for B ' , 35. and B' = 1 + 2 p 2 - 1.3075p 4, B' = 1 + 2 p 2 + 3.3075p 4. Had the other choice of s i g n f o r DQ been made, then we would f i n d DQ = -1, bg = -2 and the q u a r t i c equation would become b 4 + 4b. 3 + 8b 2 + 8b - 36 = 0. 4 4 4 4 T h i s equation i s s i m i l a r to t h a t given above except that the c o e f f i c i e n t s of the odd-power terms of b^ have changed s i g n . There are two complex and two r e a l r o o t s f o r the q u a r t i c j the r e a l r o o t s are b^ = -3.3075 and b^ = +1.3075. The f a c t t h a t the r e a l r o o t s of the two q u a r t i c equations are equal except f o r a change i n t h e i r signs i s due to the changes i n the signs of the c o e f f i c i e n t s of the odd powers of b„. 4 Here again then, are two p o s s i b l e expressions f o r B', i . e . , B' = -1 - 2 p 2 + 1.3075p 4, and B' = -1 - 2 p 2 - 3.3075p 4. Since B 1 i s known, i t i s now p o s s i b l e to f i n d D (p) = A + pB from Equation 5:13 r e w r i t t e n i n the form A 2 - p 2 B 2 = 1 - p 2 B ' 2 5:24. Prom the f a c t t h a t there are f i v e r e a c t i v e elements i n the network, the polynomial D (p) i s known to be of the S 36. f i f t h degree. Thus, A and B are polynomials of the f o u r t h t degree, having a l l c o e f f i c i e n t s p o s i t i v e . The s o l u t i o n of Equation 5:24 f o r a p a r t i c u l a r choice of B 1 w i l l always produce a polynomial D (p) with p o s i t i v e 8 c o e f f i c i e n t s and the c o e f f i c i e n t f o r the h i g h e s t power of p w i l l be equal i n magnitude to the c o e f f i c i e n t of the highest power of_p i n B 1. This f a c t becomes s i g n i f i c a n t i f an i n s p e c t i o n of the o p e n - c i r c u i t and s h o r t - c i r c u i t impedance expressions i n Equation 5:12 i s made. That i s , to r e a l i z e a network of the c o n f i g u r a t i o n f i r s t assumed, where the f i r s t component i n the network i s an inductance, the open-c i r c u i t and s h o r t - c i r c u i t impedance expressions must both be of a form such t h a t the numerator polynomial i s one degree i n p g r e a t e r than i t s corresponding denominator polynomial. Since A 1 = 0, the only way i n which these impedance f u n c t i o n s can assume t h i s form i s f o r the co-e f f i c i e n t of the h i g h e s t power terra of £ i n B 1 to be equal i n magnitude, but opposite i n si g n , to the c o e f f i c i e n t of the h i g h e s t power of p i n B. These f a c t s reduce the number of p o s s i b l e expressions f o r B' to B 1 = 1 + 2 p 2 - 1.3075p 4, or B« = - 1 - 2p 2 - 3.3075p 4. Since there seems to be no way of determining which of these expressions g i v e s the b e t t e r r e s u l t , the only a l t e r -n a t i v e i s to use both expressions to obtain t h e i r c o r r e -sponding D (p) = A + pB from Equation 5:24. 37. The f i r s t choice l i s t e d f o r B' gives D (p) = 1.3075p5+4.9871p4+7.5110p3+6.4982p2+3.7412p+l, S and the combination of these expressions to form the open-c i r c u i t impedance r e s u l t e d i n the network element values shown i n Figure 12. Figure 12. G r i d or P l a t e Network a f t e r One Cycle of I t e r a t i o n f o r the Choice, B'= l+2p 2-1.3075p 4. For the second choice l i s t e d f o r B', the expression f o r the n a t u r a l modes i s D (p)= 3;3075p5+7,2342p4+9.9113p3+7.9337p2+4.1070p+l, S and the combination of the expressions gives the open-c i r c u i t impedance which produces the network element values i n Figure 13. f 9 0 0 0 »"* O t » 0,9144 *~rt m rrrnrxi v. » n->-«» » » » v»» 3.278 0.9144 1° 1 1.554~j~ 1.554"p Figure 1 3 . G r i d or P l a t e Network a f t e r One Cycle of I t e r a t i o n f o r the Choice, B' =-1-2p 2-3.3075p 4. 38. Of the networks shown i n F i g u r e s 12 and 13, the b e t t e r choice from the viewpoint of a t t a i n i n g the higher g a i n -bandwidth f a c t o r i s t h a t shown i n Fig u r e 12, since i t contains the l a r g e r shunt c a p a c i t a n c e s . A f t e r choosing t h i s network and using i t to form N(p) i n order to begin the second c y c l e of i t e r a t i o n i t i s found that some of the c o e f f i c i e n t s of p i n the polynomial B' are complex. Therefore, s i n c e the co-e f f i c i e n t s of p i n B' must be r e a l f o r p h y s i c a l - r e a l i z a b i l i t y , i t i s apparent t h a t the network i n Figure 13 cannot be used i n the i t e r a t i o n p rocess. By using the other network i n F i g u r e 12 to form N(p) i t turns out t h a t the c o e f f i c i e n t s of j> i n ^ ' are r e a l . However, during t h i s second c y c l e of i t e r a t i o n , the same s i t u a t i o n a r i s e s as i n the f i r s t c y c l e ; i . e . , which choice of the p o s s i b l e two choices f o r b^ should be taken to form B 1 ? Because of the f a c t t h a t i t r e q u i r e s e i g h t s i g n i f i c a n t f i g u r e s i n a l l the computations i n order to give element values accurate to f o u r f i g u r e s , i t takes s e v e r a l days to perform the c a l c u l a t i o n s f o r only one c y c l e of i t e r a t i o n . . Hence, i t was decided t h a t a t the p o i n t i n each c y c l e of the design procedure where the choice of b^ a r i s e s , the higher negative r e a l r o o t would be chosen. By c a r r y i n g out the c a l c u l a t i o n s through s i x c y c l e s of i t e r a t i o n , the networks shown i n F i g u r e 14 were r e a l i z e d . These networks show the element values obtained a f t e r each su c c e s s i v e c y c l e . Besides the networks t h a t r e s u l t e d from the choice of b. as the higher negative r e a l r o o t , those r 1 T 2 r 1 1 1 ^ . . 2~[~ . 2 j | 1 I n i t i a l Element Values t o r n nr » f r < n w B' P'» P » YTmrtO 'ft o m i \ I 0.9144j_ 3.278 [ 0.91441 > 1 | 1 .553J 1.553 "|~ Pirst Cycle. a a wr»<Tnrr>B a a » rostra t a a O " » T »•» . 0 . 9388^4.775 J_0.9388 1.297"|" 1.297]" Second Cycle. 1.592_[_5.592J_1.592 1 .136T 1.136 > J 3 6 J Third Cycle. • 3.556j[3.927 J_ 3.556 1 | 0 ' 9 6 8 1 T 0 ' 9 6 8 i|~ Fourth Cycle. 2.971_[_6.860_|_ 2.971 | V ;0.8760"[~0.8760|" 0.5244]_ 1.692^ 0.5244] 2.37 l j " 2.371"p First Cycle (choice of b^ as Lower Nega-tive Real Root). 3.199J_ 1.416_|_3.199 WfTTi fo.6558~p.6558~|~ Third C y l e (choice of b^ as Lower Nega-tive Real Root). Fi f t h Cycle. 6.666_L 2.487j_6.666 1.073^ 1" 1.07.3]" Networks that Resuited from the Choice of as the Higher Negative Real Root. Final Cycle. Figure 14. Network Element Values at Successive Stages of the Iteration Procedure. 39. networks t h a t r e s u l t e d from the other choice of b. are shown 4 f o r the f i r s t and t h i r d c y c l e s . No evidence of the convergence of the values of the components was present from the networks r e a l i z e d thus f a r . Because of t h i s f a c t and the f a c t t h a t the c a l c u l a t i o n s necessary to c a r r y out more c y c l e s of i t e r a t i o n would take weeks, i t seemed a d v i s a b l e to terminate any f u r t h e r i n v e s t i -g a t i o n of the problem. A c t u a l l y , these c a l c u l a t i o n s should be c a r r i e d out i n order to prove or disprove the convergence of the design procedure f o r an a m p l i f i e r of the type con-s i d e r e d . But i n view of the f a c t s mentioned concerning the time and labour r e q u i r e d when us i n g a hand c a l c u l a t o r , the knowledge to be gained d i d not warrant the expenditure, at l e a s t not u n t i l the process becomes s u f f i c i e n t l y w e l l under-stood to j u s t i f y f u r t h e r study using a high-speed d i g i t a l computer. The above design concluded the work c a r r i e d out on the c o n v e n t i o n a l types of d i s t r i b u t e d a m p l i f i e r s . In the f o l l o w i n g Chapter, a s p l i t - b a n d m o d i f i c a t i o n of the d i s t r i -buted a m p l i f i e r i s i n v e s t i g a t e d . 40. VI A SPLIT-BAND DISTRIBUTED AMPLIFIER Consider a s i n g l e - s t a g e d i s t r i b u t e d a m p l i f i e r c o n t a i n -i n g n tubes with a g a i n per tube of G e f f e c t i v e over a band-width B. As shown i n Fig u r e 15a, the delay l i n e s f o r t h i s a m p l i f i e r are terminated i n the impedance ZQ; t h e r e f o r e , f o r an i n p u t v o l t a g e s i g n a l E ^ Q , the output voltage i s zn \ /z/Y - n ' S m z 0 E i n ' o ' - o y \ " z o , making the t o t a l g a i n of the a m p l i f i e r a _ out _ B 0 6:1b, G T " E . n " 4 E , = -E. f — ^ — \ n.gm(^-l= " A u x n ..... 6:1a, out i n l z r t + z r t N s \2z,J 4 or -n.gm z n = , 0 6:1c. 4 Now consid e r an a m p l i f i e r arranged as shown i n Figure 15b. I t could be c a l l e d a s p l i t - b a n d , divided-band or a multi-channel a m p l i f i e r because each of the s i g n a l paths between the delay l i n e s i s used to amplify only those f r e q u e n c i e s l y i n g w i t h i n l / n t h p a r t of the t o t a l bandwidth B. That i s , i f the a m p l i f i e r i s a low-pass device with a bandwidth from zero to f c y c l e s per second, then the f i r s t channel a m p l i f i e s the frequency band from zero to f . / n , the second channel a m p l i f i e s the band from f /n to 2f /n, the c c t h i r d from 2 f c / n to 3f^/n and so on up to the nth channel, which a m p l i f i e s the frequency band from ~ ^ ^ c "t° f • The output of each channel i s connected to a delay l i n e which corresponds to the p l a t e l i n e of a d i s t r i b u t e d a m p l i f i e r , and which serves to add the s i g n a l s as i n a d i s t r i b u t e d z A r t i f i c i a l Line out Plate A r t i f i c i a l Line Grid A r t i f i c i a l Line <f) <f> <f) • • • 1 • i 1 ! I I I I • • D A A A D • O 4 C) () • • I I I I I I • • • • () o A r t i f i c i a l Line Figure 15a. Single-Stage Distributed Amplifier. Figure 15b. Split-Band Distributed Amplifier. 41. a m p l i f i e r . Although the problem of adding the s i g n a l s i n the delay-l i n e to give an output s i g n a l of s u i t a b l e amplitude and phase c h a r a c t e r i s t i c has to be considered, and could p o s s i b l y be the g r e a t e s t drawback i n t r y i n g to r e a l i z e such an a m p l i f i e r , i t i s f i r s t necessary to show whether or not any d e f i n i t e ad-vantages i n g a i n and bandwidth c h a r a c t e r i s t i c s can be obtained. An e v a l u a t i o n df these c h a r a c t e r i s t i c s can be made by examining them f o r one channel i n the a m p l i f i e r , since i t i s assumed that both g a i n and bandwidth are equal f o r a l l channels. Let us assume that the maximum p o s s i b l e gain per tube i n a s i n g l e - s t a g e d i s t r i b u t e d a m p l i f i e r of bandwidth B i s G. Now, suppose an a m p l i f i e r of the multi-channel type i s to be con-s t r u c t e d to pass the same bandwidth B. Then, i n the i n i t i a l stage of each channel there i s an i n c r e a s e i n the gain by a f a c t o r of n since the bandwidth f o r the channel i s —. More-— n over, i f f o u r - t e r m i n a l i n t e r s t a g i n g i s used between the i d e n t i -c a l cascaded stages i n the channel, a f u r t h e r i n c r e a s e i n gain 20 by a f a c t o r of two can be obtained. Therefore, the ga i n per tube i n the channel i s 2n times as l a r g e as the maximum gain per tube f o r the conv e n t i o n a l d i s t r i b u t e d a m p l i f i e r operating over the same bandwidth. I f there are m cascaded stages i n a channel, then (m-l) of these can have a gain per stage of 2nG, but the f i n a l cascaded s e c t i o n can only have a ga i n per tube of G. This l i m i t a t i o n i s present f o r the f i n a l stage because H. W. Bode, "Network A n a l y s i s and Feedback A m p l i f i e r Design," D. Van Nostrand Company, Inc., New York, 1945. 42. the impedance l e v e l i n the p l a t e l i n e i s determined by the o v e r a l l bandwidth B and the requirement t h a t the phase v e l o c i t y , i n the p l a t e l i n e be equal to that of the g r i d l i n e . Therefore, the o v e r a l l g a i n per channel, which i s the same as the t o t a l g a i n of the a m p l i f i e r , i s G T = ( 2 n G ) m _ 1 G = G m ( 2 n ) m " 1 6:2. But nG i s the ga i n of a conventional s i n g l e - s t a g e d i s t r i b u t e d a m p l i f i e r c o n t a i n i n g n tubes with a bandwidth B and a gain per tube G. Furthermore, ( n G ) m i s the ga i n of m such s i n g l e - s t a g e a m p l i f i e r s when cascaded. Thus i t seems th a t f o r any f i x e d value of m and n, the multi-channel a m p l i f i e r would have a gai n higher than the conve n t i o n a l d i s t r i b u t e d a m p l i f i e r ope-r a t i n g over the same bandwidth by a f a c t o r of 2 /n. To determine how la r g e the f a c t o r 2 m Vn a c t u a l l y i s , a comparison i s to be made between a conventional s i n g l e - s t a g e d i s t r i b u t e d a m p l i f i e r and a s i n g l e - s t a g e of the proposed m u l t i -channel a m p l i f i e r i n the f o l l o w i n g way: l e t the t o t a l band-width and the t o t a l g a i n be equal f o r each a m p l i f i e r and then determine what minimum number of tubes i s r e q u i r e d i n each to give the same g a i n . For the s i n g l e - s t a g e d i s t r i b u t e d ampli-f i e r , the number of tubes r e q u i r e d i s merely n = G^/G, s i n c e the maximum gai n per tube i s f i x e d bj the bandwidth. In the multi-channel a m p l i f i e r , the number of tubes i s N = mn, and to f i n d the minimum number r e q u i r e d f o r the ga i n G^ , i t i s neces-sary to proceed as f o l l o w s : G T = G m ( 2 n ) m ' 1 6:3; 43. t h e r e f o r e . - ~ r 0, M " 1 n = 1 6:4, 2G and the number of tubes i s m m^l 1 m GT ft.* in 2 G m " 1 To f i n d the minimum number of tubes, i t i s necessary to d i f f e r e n t i a t e Equation 6:5 v i t h respect to m when G and G ^ are f i x e d . A s i m p l i f i c a t i o n of the d i f f e r e n t i a t i o n i s pro-duced i f the logarithm of both sides of the equation i s f i r s t obtained. Then ln(N) = ln(m) + ^ I n ^ ) - ln(2) - j j j ^ j l n ( G ) 6:6, and the d e r i v a t i v e becomes d 4 ~ i = h - — ^ - 7 l n ( G m ) + — ^ - p l n ( G ) 6:7. d m m (m-lT T (m-1) 2 For a minimum, • ~ ^* hence, or i A - i j S z i l f 6:8, G m 2 GT m - (2+ln^-)m + 1 = 0 6:9. The s o l u t i o n of 6:9 gives , r T l n ^ In order to a s c e r t a i n which value of m w i l l give a minimum, i t i s f i r s t necessary to f i n d the range of values over which the r a t i o G ^ / G can vary. For any p r a c t i c a l G^, a m p l i f i e r G ^ w i l l be greater than G j therefore In^r- w i l l always be p o s i t i v e . Moreover, m must be greater than u n i t y since i t i s the number of cascaded stages. The choice of 4 4 , the value g i v e s values of m g r e a t e r than u n i t y i f the r a t i o ^r— i s always l a r g e r than one which t h e r e f o r e f u l f i l l s the r e q u i r e -ments mentioned above. To prove t h a t t h i s c h o i c e of m produces a minimum, an examination of the second d e r i v a t i v e + fl dm2 m2 ( m - i r must be made. T h i s examination shows that the c o n d i t i o n G T £r=- > 1 i s a l s o the one needed, along with the choice o f the ro o t making m > 1, to make the value of the second d e r i v a -t i v e p o s i t i v e . Knowing m, the value of n i s e a s i l y obtained by using Equation 6:4. To c a l c u l a t e the values of m and n, the maximum theo-r e t i c a l bandwidth f o r the a m p l i f i e r must be f i x e d by choosing G m the t o t a l g a i n r e q u i r e d and G the g a i n per tube. S e t t i n g jr- » r , the value of m becomes Inr m = 1 + + "\ Inr + (Inr)' » e • 6 • 12 < To determine n, i t i s a d v i s a b l e to s i m p l i f y Equation 6 : 4 ; i . e . , 1 ra-1 1 m^T n =.. 2G' m m-1 G because G ^ G = 0™ L Q*-1. Furthermore, from Equation 6 : 8 , r can be ' ( m-1) 2 ' , making w r i t t e n as r = e u r M M I I—} / I , • ' l 0 5 10 15 20 25 Number of Tubes (N=mn) " -Figure 16. Comparison of a Split-Band and a Single-Stage Distributed Amplifier. or m-1 ra r» ^ . • • e « e o e o « * o e * « o o « e o 6 • 13 e n ~ 2G By choosing v a r i o u s values of G- and G^, making sure t h a t Gq, G, both m and n are f i x e d and a l s o the number of tubes, N = mn. F i g u r e 16 shows a p l o t of the maximum theo-r e t i c a l g ain G^ a g a i n s t the minimum number of tubes r e q u i r e d to produce t h i s g a i n f o r a multi-channel and a s i n g l e - s t a g e d i s t r i b u t e d a m p l i f i e r . These curves show that the m u l t i -channel a m p l i f i e r would have a h i g h e r g a i n than the d i s -t r i b u t e d a m p l i f i e r c o n t a i n i n g the same number of tubes and i t would have i t s g r e a t e s t advantage when the gain per tube was very low; i . e . , when the o v e r a l l bandwidth was l a r g e . The a n a l y s i s thus f a r seems to i n d i c a t e t h a t a m u l t i -channel a m p l i f i e r has the higher gain bandwidth f a c t o r , but 21 the a n a l y s i s i s not yet complete. I t has been shown t h a t s i n g l e - s t a g e d i s t r i b u t e d a m p l i f i e r s cascaded to give an o p t i -mum ga i n (over a given bandwidth) should have a gain per stage of e_ (base of naperian l o g a r i t h m s ) . Consequently, f o r the comparison shovn i n F i g u r e 16, the d i s t r i b u t e d a m p l i f i e r should have been cascaded f o r gains l a r g e r than e_. Fig u r e 17 compares s i n g l e - s t a g e d i s t r i b u t e d a m p l i f i e r s cascaded to give an optimum ga i n ( f o r a f i x e d bandwidth) with a multi-channel a m p l i f i e r c o n t a i n i n g the same number of tubes and opera t i n g over the same bandwidth. T h i s f i g u r e shows t h a t the m u l t i -channel a m p l i f i e r s t i l l has a s l i g h t advantage over the H e w l i t t , Ginzton, Jasberg and Noe, op. c i t . , p. 966. Number of Tubes (N=mn)-~ Figure 17. Comparison of Split-Band and Cascaded Distributed Amplifiers. 46. cascaded distributed amplifier. This s l i g h t advantage of the multi-channel amplifier i s far outweighed by i t s disadvantages, namely, large delay, possible f a i l u r e of a channel due to one fau l t y tube, and the d i f f i c u l t y of combining the signals i n the "plate l i n e " to give good response c h a r a c t e r i s t i c s . Therefore, i t seemed nothing of p r a c t i c a l value could be realized by pursuing t h i s investigation any further. However, i t i s possible that some modification of the c i r c u i t suggested here might prove to have advantages. YII CONCLUSIONS 47. Though none of the designs shoved any improvements i n t h e i r c h a r a c t e r i s t i c s over present-day a m p l i f i e r s , they d i d i l l u s t r a t e dravbacks a s s o c i a t e d v i t h c e r t a i n types of d i s -t r i b u t e d a m p l i f i e r s . The r e s t r i c t i o n to i d e n t i c a l n a t u r a l modes i n the g r i d and p l a t e networks f o r the f i r s t three a m p l i f i e r s considered caused t h e i r gain-bandvidth f a c t o r s to be low. A l s o , the non-minimum phase response would be an un d e s i r a b l e charac-t e r i s t i c i n c e r t a i n a p p l i c a t i o n s . These f a c t s coupled with t h a t concerning the cascading of these a m p l i f i e r s i n d i c a t e d t h a t t h e i r u s e f u l n e s s would be l i m i t e d . Por the a m p l i f i e r which had i d e n t i c a l n a t u r a l modes i n both networks, with the networks symmetrical about the mid-p o i n t s of t h e i r l e ngths, there was evidence t h a t the i t e r a t i v e s y n t h e s i s procedure was not convergent. A p o s s i b l e explana-t i o n f o r t h i s i s t h a t , a f t e r u s i n g the number of degrees of freedom necessary to f u l f i l l the r e s t r i c t i o n s on the networks, the number of degrees of freedom remaining t h a t c o u l d be used to approximate a m a x i m a l l y - f l a t amplitude response was not s u f f i c i e n t to ensure convergence. The proposed s p l i t - b a n d a m p l i f i e r had some advantage over the conv e n t i o n a l d i s t r i b u t e d a m p l i f i e r i n i t s g a i n -bandwidth f a c t o r . T h i s seemed i n s i g n i f i c a n t when compared to i t s disadvantages and the i n v e s t i g a t i o n of the a m p l i f i e r vas d i s c o n t i n u e d . One m o d i f i c a t i o n of the d i s t r i b u t e d a m p l i f i e r t hat should be examined, and could be s t u d i e d by the i t e r a t i v e 48. method, i s one i n which p a r a l l e l e d vacuum tubes are used a t those p o i n t s along the delay l i n e s where the shunt c a p a c i -tance i s l a r g e . Such a p r o j e c t should not be undertaken un l e s s adequate c a l c u l a t i n g equipment i s a v a i l a b l e , f o r as t h i s r e p o r t has shown i t i s i m p r a c t i c a l to attempt to design d i s t r i b u t e d a m p l i f i e r s using a hand c a l c u l a t o r . APPENDIX I 49. To show that two i d e n t i c a l amplifiers when cascaded as shown i n Figure A l , have a transfer function containing natural modes which l i e on the rea l frequency axis, we must f i n d what change i n the chara c t e r i s t i c s of the amplifier occurs when the plate l i n e of the i n i t i a l stage i s connected to the grid l i n e of the f i n a l stage. -Tt h T h T ()gm1 ()gms °2 °i: I T " h T Li l_n o <, ? n <g_f f i A l e g » if L2 I L l JC2 "pi SR ^1 T T 2 R« Figure A l . A Cascaded Distributed Amplifier. If we assume that the networks A and B i n Figure A2 rep-resent the plate and g r i d networks, respect ive ly , of the ampl i f ier i n Figure A l , tne problem essent ia l ly involves f inding what the transfer impedance z ^ i becomes when 1 i s connected to 1' i n the diagram of Figure A2. A and B are i d e n t i c a l react ive ladder networks and are terminated i n equal resistancesR; therefore, these networks have the fol lowing e q u a l i t i e s : (a) z 1 2 = z 2 1 = z-pg, = z 2 « i i » (b) z i ; i = z l t l , . Figure A2. Identical Reactive Ladder Networks. n12 n l l If z , 0 and z , , are defined as -s— and - 3 — , respect ively 12 11 d 1 2 d n » then the open-c ircu i t voltage at 1 i s T T ^ 1 21 d 2 1 By Thevenin's Theorem, the current flowing into l 1 when 1 and 1' are connected i s 51. But E, = In , Z and hence E, '2« ~ n. d 12 12 1 ( n 1 2 } 2 n n d n Therefore, since z 22' ~ I ' i t becomes o z 22 • ~ 2 n n d n This means that the transfer impedance ^22' c o n " k a i n s both the symmetrical natural modes of the network contained i n d ^ and the anti-symmetrical natural modes produced by n ^ . Natural modes produced by n ^ w i l l l i e on the w-axis since the networks are purely reac t ive . Therefore, peaks in the amplitude charac ter i s t i c w i l l occur at these frequencies, destroying the maximally-f lat amplitude response. BIBLIOGRAPHY 52, 1. Bode, H. W., "A General Theory of E l e c t r i c Wave F i l t e r s , " Journal of Mathematics and Physics . XIII , 275-362, (1934). 2. Bode, H. W., "Network Analys is and Feedback Ampl i f ier Design," D. Van Norstrand Company, Inc . , New York, 1945. 3. Caryotakis , G. A . , "Iterative Methods i n Ampli f ier Inter-stage Synthesis," TR-No. 86 (Contract N6onr 25107), Stanford Univers i ty (Electronics Research Laboratory), May, 1955. 4. Darl ington, S . , "Synthesis of Reactance Four-Poles which Produce Prescribed Insertion Loss Charac ter i s t i c s ," Journal of Mathematics and Physics . XXVIII, 257-353, (1939). 5. Demuth, Howard B . , "An Invest igation of the I terat ive Synthesis of Dis tr ibuted Ampl i f i er s ," TR-No. 77 (Contract N6onr 25107), Stanford Univers i ty (Electronics Research Laboratory) , August, (1954). 6. Elmore, W. C , "The Transient Response of Damped Linear Networks with P a r t i c u l a r Regard to Wideband Ampl i f i ers ," Journal of Applied Physics . XIX, 55-63, (1948). 7. Ginzton, E . L . , Hewlett, W. R . , Jasberg, J . H . , and Noe, J . D . , "Distributed Ampl i f i ca t ions ," Proceedings of the Inst i tute of Radio Engineers. XXXVI, 956-959, TI948T. 8. Horton, W. H . , "A Further Theoret ica l and Experimental Invest igation of the P r i n c i p a l of Dis tr ibuted A m p l i f i -cat ion ," Ph.D. D i s ser ta t ion , Stanford Univers i ty , 1951. 9. Moore, A. D . , "Synthesis of Dis tr ibuted Ampli f iers for Prescribed Amplitude Response," TR-No. 53 (Contract N6onr 25107), Stanford Univers i ty (Electronics Research Laboratory) , 1952. 10. P e r c i v a l , W. S . , "Thermionic Valve C i r c u i t s , " B r i t i s h Patent No. 460,562, July 24, 1935, January 25, 1937. 11. Pederson, D. 0 . , "The Analys is and Synthesis of D i s -tr ibuted Ampl i f iers with Ladder Networks," TR-No. 34 (Contract N6onr 25107), Stanford Univers i ty (Electronics Research Laboratory) , May, 1951. 12. Va l l ey , G. E . J r . , and Wallman, H , , "Vacuum Tube Ampli-f i e r s , " New York:McGraw-Hill Book C o . , 1948. 

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