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A unilateral tunnel-diode frequency converter Little, Warren David 1963

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A UNILATERAL TUNNEL-DIODE FREQUENCY CONVERTER by WARREN DAVID LITTLE B . A o S c , The U n i v e r s i t y of B r i t i s h Columbia, 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1963 •r In presenting th i s thesis in p a r t i a l fulf i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the Library sha l l make i t free ly avai lable for reference and study. I further agree that per-' mission for extensive copying of this thesis f o r . s c h o l a r l y purposes may be granted by the Head of my Department or by his representativeso It i s understood that copying, or p u b l i -cat ion of this thesis for f i n a n c i a l gain sha l l not be allowed without my written permission. Department of E l e c t r i c a l E n g i n e e r i n g The Univers i ty of B r i t i s h Columbia,. Vancouver 8, Canada. Date J u l y 25, 1963 ; ABSTRACT Frequency converters u s i n g pumped n o n l i n e a r conductance and pumped n o n l i n e a r capacitance w i t h the property of s i g n a l flow i n one d i r e c t i o n only are the subject of t h i s t h e s i s . The u n i l a t e r a l property can be obtained e i t h e r by a s u i t a b l e t e r m i n a t i o n at the image frequency or by quadrature pumping the conductance and c a p a c i t a n c e . Of great importance i s a frequency converter i n which the output frequency i s lower than t h a t of the i n p u t . Such a down—converter based upon a proposed image t e r m i n a t i o n method i s examined both a n a l y t i c a l l y and e x p e r i m e n t a l l y . Conditions are given which the conductance and capacitance must s a t i s f y i n order t h a t the image t e r m i n a t i o n be p a s s i v e . The c o n d i t i o n s are f u l f i l l e d by a s i n g l e t u n n e l — d i o d e . I t i s found t h a t a forward- to r e v e r s e - g a i n r a t i o of at l e a s t 20 db over a 5% bandwidth i s f e a s i b l e ; the estimated noise f i g u r e i s 3.4 db. The quadrature pumped converter i s compared with the image terminated converter and i t i s shown i n p a r t i c u l a r t h a t the former can be u n i l a t e r a l only at one frequency. ACKNOWLEDGEMENT The author wishes to thank the s u p e r v i s i n g p r o f e s s o r * Dr« M, P. Beddoesi f o r h i s help and encouragement d u r i n g the course of t h i s study. G r a t e f u l acknowledgement i s given to the B r i t i s h Columbia Telephone Company f o r a s c h o l a r s h i p awarded i n 1961 j and to the N a t i o n a l Research C o u n c i l f o r a Studentship i n 1962* The work d e s c r i b e d i n t h i s t h e s i s was supported by the N a t i o n a l Research C o u n c i l under Grant BT-68. v i TABLE OP CONTENTS Page LIST OP ILLUSTRATIONS .. v ACKNOWLEDGEMENT v i LIST OP SYMBOLS v i i 1. INTRODUCTION 1 2. MIXING-ELEMENT ANALYSIS 5 2.1 D i f f e r e n t i a l Equation of the Mi x i n g -Element ••«..••.... «.. 6 2.2 The Mixing-Element M a t r i x 9 3. FREQUENCY CONVERTER ANALYSIS 14 3.1 Network Representation of the M i x i n g -Element • •••• 14 3.2 U n i l a t e r a l Frequency Conversion 16 4. UNILATERAL DOWN-CONVERSION BY THE IMAGE TERMINATION METHOD 22 4.1 Approximations ...................... 22 4.2 Optimum Terminations 23 4.3 Converter S t a b i l i t y ........•••..*... 27 4.4 Gain w i t h Optimum Terminations ...... 30 4.5 Frequency C h a r a c t e r i s t i c s of a Computer Analyzed Model wi t h " P r a c t i c a l " Terminations ... 32 5.. NOISE 39 6. EXPERIMENTAL 40 6.1 Experimental Converter C i r c u i t and I t s Elements ..... 40 6.2 Experiment Techniques and Resu l t s ... 45 6.3 Suggestions f o r Fu r t h e r Experimental Study .«•«.««....»..•............*... 47 7. CONCLUSIONS 48 Page APPENDIX I P o s i t i v e Real D r i v i n g - P o i n t Admittance • 49 APPENDIX I I Terminating Components f o r a " P r a c t i c a l " Converter •••..»•••••. » •••• • 51 REFERENCES 53 LIST OP ILLUSTRATIONS F i g u r e Page 1- 1 Block Diagram I l l u s t r a t i n g Frequency-Converting Process 3 2— 1 Mixing—Element C i r c u i t ..... 7 2- 2 Sm a l l - S i g n a l Frequency Spectrum 11 3- 1 Mixing—Element with Terminations ....... 15 4- 1 Boundaries of a Passive Image Termination .............. 25 4-2 Input and Output Conductance v s . Normalized Input Frequency 29 4-3 Forward Gain with Optimum Terminations v s . Normalized Input Frequency 31 4-4 Equal Gain Contours of Converter with Optimum Terminations 33 4-5 Mixing—Element with '^Practical" Termin-a t i o n s ••*...».............. 34 4-6 Forward and Reverse Gain of a Converter w i t h " P r a c t i c a l " Terminations ........... 36 6-1 Experimental Converter C i r c u i t ......... 41 6-2 E q u i v a l e n t C i r c u i t and C h a r a c t e r i s t i c s of a Pumped 1N2939 Tunnel-Diode 42 6-3 Block Diagram of Converter and Measuring Equipment .............................. 45 v L i s t of Symbols F i r s t d e f i n e d i n S e c t i o n V ( t ) t o t a l v o l t a g e across n o n l i n e a r ^ conductance 2 . 1 V ( t ) conductance pump vo l t a g e 2 . 1 v ( t ) s m a l l - s i g n a l v o l t a g e 2 . 1 I ( t ) t o t a l c u r r e n t through n o n l i n e a r ^ conductance 2 . 1 g ( t ) instantaneous incremental conduct-ance 2 . 1 I ( t ) conductance pump cur r e n t 2 . 1 S P i ( t ) s m a l l - s i g n a l conductance current 2 . 1 I C p ( t ) capacitance pump cur r e n t 2 . 1 q charge on n o n l i n e a r capacitance 2 . 1 V ( t ) capacitance pump v o l t a g e 2 . 1 cp V ^ capacitance b i a s v o l t a g e 2 . 1 conductance b i a s v o l t a g e 2 . 1 i c ( t ) s m a l l - s i g n a l capacitance current 2 . 1 C ( t ) instantaneous incremental c a p a c i t -ance 2 . 1 "fc h g n n conductance harmonic ( r e a l ) 2 . 2 t h C ' n capacitance harmonic (complex) 2 . 2 t h C n n capacitance harmonic ( r e a l ) 2 . 2 V\ i ^ n v o l t a g e harmonic ( s m a l l - s i g n a l ) 2 . 2 "fch i c u r r e n t harmonic ( s m a l l - s i g n a l ) 2 . 2 WQ pump angular frequency 2 . 2 to^ r . f . angular frequency 2 . 2 fi>2 i . f . angular frequency 2 . 2 G>2 image angular frequency 2 . 2 admittance across mixing-element at frequency a. 3 . 1 v i i L i s t of Symbols F i r s t d e f i n e d i n S e c t i o n Y ^ Y ^ minus admittance of f i l t e r s 3.1 Y . . admittance parameters of the 1 ^ mixing—element matrix 3.1 forward transducer g a i n 3.2 G^2^ reverse transducer g a i n 3.2 a n o n l i n e a r f a c t o r 4.1 P l o a d i n g f a c t o r 4.1 normalized frequency 4.1 Y . converter input admittance 4.2 i n Y Q U ^ . converter output admittance 4.2 F converter noise f i g u r e 5 A UNILATERAL TUNNEL—DIODE FREQUENCY CONVERTER 1. INTRODUCTION Most modern hig h frequency communication r e c e i v e r s use the heterodyne p r i n c i p l e , ^ ^ i n which power at the c a r r i e r frequency of the incoming s i g n a l gives r i s e to power at a lower intermediate frequency which i s more s u i t a b l e f o r a m p l i f i -c a t i o n and d e t e c t i o n . This frequency conversion i s c a r r i e d out i n the down-converter stage of the r e c e i v e r . The noise p e r f o r -mance of the down-converter i s of prime importance since i t (2) determines the u l t i m a t e s e n s i t i v i t y of the r e c e i v e r . ' U n t i l (3) the advent of the tunnel-diode i n 1957, c r y s t a l - d i o d e s were (4) used almost e x c l u s i v e l y as the frequency changing elements. The best c r y s t a l - d i o d e converters have conversion l o s s e s of 3 to 4 d e c i b e l s and noise f i g u r e s of 4 to 5 d e c i b e l s * ' In (6) 1960 Chang, et a l ' showed t h a t tunnel-diode frequency converters w i t h 22 d e c i b e l s of c o n v e r s i o n g a i n and noise f i g u r e s of 3 d e c i b e l s were p o s s i b l e . To the author's knowledge a l l tunnel—diode down-converters that have been s t u d i e d are r e c i p r o c a l . This p r o p e r t y i s u n d e s i r a b l e because the input i s i n no way i s o l a t e d from the output and as a r e s u l t the input (o) noise i s i n c r e a s e d by noise i n the output c i r c u i t . - ' * For a r e c i p r o c a l transducer the forward g a i n i s equal to the reverse g a i n . 2 The main t o p i c of t h i s t h e s i s i s the development of a u n i l a t e r a l tunnel—diode down—converter* Previous s t u d i e s of tunnel-diode converters have considered the n o n l i n e a r conductance of the diodes as the s o l e frequency c o n v e r t i n g element* In a d d i t i o n to a n o n l i n e a r conductance however, tunnel—diodes have a n o n l i n e a r j u n c t i o n (3) capacitance I ' and t h i s * as w i l l be shown, can p l a y a very important p a r t i n the frequency c o n v e r t i n g p r o c e s s . For g e n e r a l i t y , equations are developed f o r a converter having separate n o n l i n e a r conductance and n o n l i n e a r capacitance elements* The equations are then a p p l i e d to tunnel-diodes i n which the two n o n l i n e a r elements are d i r e c t l y i n p a r a l l e l * The concept of frequency conversion i s i n t r o d u c e d by c o n s i d e r i n g the b l o c k diagram of F i g u r e 1—1. The frequency c o n v e r t e r , p r o p e r t c o n s i s t s of blocks 1, 2 and 3. The combination of blocks 3. and 2 i s d e f i n e d as the mixing-element* I t i s the a c t i v e p o r t i o n of the c o n v e r t e r . The input s i g n a l i s a sinewave of frequency f^ and the pump i s another sinewave of frequency fQ. The n o n l i n e a r i t i e s cause v o l t a g e or c u r r e n t harmonics to be developed at f ^ and f^y and at sums and d i f f e r e n c e s of the harmonic f r e q u e n c i e s * The output frequency f2 i s one of these f r e q u e n c i e s — f o r examples f, - f . In g e n e r a l , the converter e x h i b i t s * For a u n i l a t e r a l transducer the forward g a i n i s f i n i t e and the reverse g a i n i s zero* ** Corresponding to o p e n — c i r c u i t -terminations, v o l t a g e harmonics w i l l be produced: s h o r t — c i r c u i t t e r minations w i l l l e a d to c u r r e n t harmonics. 3 s i g n a l source 1 F i g u r e 1-1. harmonic loa d (1) r n o n l i n e a r elements (2) "1 -Mixing-element _ J Block Diagram I l l u s t r a t i n g Frequency-Converting Process. b i l a t e r a l but not r e c i p r o c a l g a i n . In t h i s t h e s i s , i t i s shown t h a t a s p e c i a l t e r m i n a t i o n at one normally unused harmonic (image) produces zero g a i n i n one d i r e c t i o n . The theory of u n i l a t e r a l frequency conversion, together w i t h experimental techniques and f i n d i n g s , are d i s c u s s e d i n the * Gain i s l o o s e l y b i l a t e r a l . An input s i g n a l at f, produces an output response at f 2 ; and v i c e - v e r s a an input s i g n a l at ± 2 produces an output at f^t the input and output s i g n a l s are separated by frequency r a t h e r than d i s t a n c e but usage has i t t h a t the g a i n from f ^ to f2 i s d e s c r i b e d i n terms of d i r e c t i o n * The c o n v e r t e r , having g a i n i n both such d i r e c t i o n s , i s thus b i l a t e r a l • 4 chapters t h a t f o l l o w . In Chapter 2 a mathematical d e s c r i p t i o n -of the mixing-element i s developed. This d e s c r i p t i o n i s i n the form of an admittance matrix r e l a t i n g s m a l l - s i g n a l v o l t a g e and c u r r e n t harmonics. I t i s d e r i v e d , using F o u r i e r S e r i e s methods, from the d i f f e r e n t i a l equation of the mixing-element. In Chapter 3 i t i s shown t h a t f o r the case under study the mixing—element i s e q u i v a l e n t to a t h r e e — p o r t network. The th r e e — p o r t i s reduced to a two—port. Using the s h o r t - c i r c u i t admittance matrix of the reduced network, two methods by which the converter may be made u n i l a t e r a l are o u t l i n e d . A converter made u n i l a t e r a l by proper t e r m i n a t i o n of an image c i r c u i t i s s t u d i e d i n Chapter 4. This study i s based upon the matrices e s t a b l i s h e d i n Chapter 2 and Chapter 3 and i n c l u d e s important r e a l i z a b i l i t y * s t a b i l i t y and g a i n c o n d i t i o n s * I t i s found that with optimum t e r m i n a t i o n s ( S e c t i o n 4.2) wide-band u n i l a t e r a l frequency c o n v e r s i o n w i t h g a i n can be obtained but t h a t w i t h simple t e r m i n a t i o n s the u n i l a t e r a l p r o p e r t y i s r e a l i z e d only over a narrow frequency band. Chapter 5 contains a very b r i e f noise a n a l y s i s of the co n v e r t e r . In Chapter 6 an experimental converter i s d i s c u s s e d . The r e s u l t s obtained w i t h t h i s c o n v e r t e r , and problems a s s o c i a t e d w i t h i t , are presented and compared wi t h the theory. Suggestions f o r f u r t h e r experimental work are also g i v e n . 2. MIXING-ELEMENT ANALYSIS E s s e n t i a l to the a n a l y s i s of a frequency converter i s a s u i t a b l e mathematical d e s c r i p t i o n of the n o n l i n e a r mixing-element (Figure 2-1 ) . Assuming t h a t s i g n a l v o l t a g e s are small compared to those of the pump, i t i s shown i n t h i s chapter t h a t s i g n a l v o l t a g e s and cu r r e n t s are r e l a t e d through a l i n e a r d i f f e r e n t i a l equation. By c o n s i d e r i n g the harmonic content of the s m a l l — s i g n a l power t h a t i s made a v a i l a b l e by the mixing (7) a c t i o n , and by a p p l y i n g the P r i n c i p l e of Harmonic Balance, ' 7 a s o l u t i o n of the d i f f e r e n t i a l equation i s l a t e r o b tained. In g e n e r a l , the s o l u t i o n i s i n the form of a n by n matrix r e l a t i n g the n v o l t a g e harmonics to the n cu r r e n t harmonics. In t h i s work 3 harmonics are considered so that the mathematical d e s c r i p t i o n of the mixing—element i s i n the form of a 3 by 3 ma t r i x . The mixing-element to be i n v e s t i g a t e d c o n s i s t s of a n o n l i n e a r conductance g(v) i n p a r a l l e l w i t h a n o n l i n e a r capacitance C ( v ) . These n o n l i n e a r i t i e s are assumed to be those a s s o c i a t e d w i t h semiconductor diodes where g(v) i s the incremental conductance and C(v) i s the j u n c t i o n c a p a c i t a n c e . Ohmic l o s s e s and s e r i e s l e a d inductances of the diodes are negl e c t e d on the understanding t h a t the a n a l y s i s w i l l be s t r i c t l y v a l i d only f o r r e l a t i v e l y low fre q u e n c i e s ( l e s s than about 150 mc.). These s e r i e s components can be accounted f o r by c o n s i d e r i n g 6 2.1 D i f f e r e n t i a l E q u a t i o n of the Mixing-Element A d i f f e r e n t i a l equation r e l a t i n g the s i g n a l c u r r e n t -I i ( t ) to the s i g n a l v o l t a g e v ( t ) of the mixing-element i s to be d e r i v e d (equation 2-10)s t h i s equation i s used i n S e c t i o n 2.2 to r e l a t e v o l t a g e and cur r e n t harmonics. The mixing—element c i r c u i t of Fig u r e 2-1 c o n s i s t s of two n o n l i n e a r elements wi t h t h e i r r e s p e c t i v e b i a s and pump vo l t a g e sources i n t e r c o n n e c t e d i n a system of f i l t e r s . The pump v o l t a g e s are s i n u s o i d a l at frequency fg» F i l t e r s F Q b l o c k frequency f ^ and d-c, but pass a l l other f r e q u e n c i e s . F i l t e r s F Q pass fQ and d-c, but bl o c k a l l other f r e q u e n c i e s . With t h i s c i r c u i t arrangement the s i g n a l v o l t a g e v ( t ) and s i g n a l c u r r e n t i ( t ) are i n f l u e n c e d by the pump sources only through t h e i r a c t i o n on the n o n l i n e a r elements. From Fig u r e 2-1 the vo l t a g e across the n o n l i n e a r conductance i s V ( t ) = -V (t ) + v ( t ) g gP •..(2-1) The s i g n a l v o l t a g e v ( t ) i s small compared to the pump v o l t a g e V ( t ) and t h e r e f o r e the t o t a l c u r r e n t I (t ) through the gP g 5 conductance can be expanded i n a two-term Ta y l o r S e r i e s . I ( t ) = I g V ( t ) V ( t ) L gP v J + g ( t ) v ( t ) ,.*(2^2) where •ct) = & I [T J ...(2-3) V = V ( t ) g gP 7 a i i ( t ) i ( t ) v (t) gp i ( t g + F p i ( t ) c v ' P / \ I ( t ) 7 i ( t ) V „ ( t ) £ v ( t ) g(v) g p c o s ( 2 T i f 0 i j ) Conductance C(v) v ( t ) v ( t ) cp v ' |V c pcos(27xf 0t+^ cb I Capacitance j I ' I pump and b i a s | pump and bias | I c i r c u i t I s m a l l - s i g n a l c i r c u i t c i r c u i t I I I I Fig u r e 2-1* Mixing-Element C i r c u i t . also, y*> = i g p t t ) + i g ( t ) ...(2-4) t h e r e f o r e I ( t ) + i ( t ) = I gP g V ( t ) . gp : + g ( t ) v ( t ) ...(2-5) This equation can be separated i n t o l a r g e and s m a l l - s i g n a l components, V*> =1 v (t) . gp . ...'(2-6) and 8 i ( t ) = g ( t ) v ( t ) . o.. ( 2—7 ) The s m a l l - s i g n a l c u r r e n t — v o l t a g e r e l a t i o n s h i p f o r the pumped n o n l i n e a r c a p a c i t o r i s d e r i v e d i n a s i m i l a r manner by-expanding the charge on the c a p a c i t o r i n a two^-term T a y l o r S e r i e s about the pump v o l t a g e * The r e s u l t i n g l a r g e and s m a l l - s i g n a l equations are I (t) cp x ' I T * V (t) cp • • • (2—8 ). and i ( t ) c v ' 3 at C ( t ) v ( t ) ..(2-9) The s m a l l - s i g n a l d i f f e r e n t i a l equation r e l a t i n g v ( t ) and i ( t ) f o l l o w from equations (2-7) and (2-9). Since i ( t ) = i ( t ) + i ( t ) , these equations can be combined to give 6 C i ( t ) = g ( t ) v ( t ) + | ST C ( t ) v ( t ) ....(.2-10) I t should be noted that t h i s i s a l i n e a r d i f f e r e n t i a l equation w i t h time v a r y i n g c o e f f i c i e n t s * Since t h i s d i f f e r e n t i a l equation i s l i n e a r j the s m a l l — s i g n a l frequency converter w i l l be a l i n e a r 'device. 2.2 The Mixing-Element M a t r i x A s o l u t i o n of d i f f e r e n t i a l equation (2-10) under c e r t a i n assumptions and r e s t r i c t i o n s i s r e q u i r e d . I t i s evident from equation (2—3) t h a t the instantaneous incremental conductance g ( t ) i s a f u n c t i o n of the s i n u s o i d a l pump v o l t a g e V ( t ) . Therefore, g ( t ) i s a p e r i o d i c f u n c t i o n of time w i t h gP fundamental frequency fQ. Since the instantaneous incremental capacitance C ( t ) i s s i m i l a r l y p e r i o d i c , i t f o l l o w s t h a t both g ( t ) and C ( t ) can be expanded i n F o u r i e r S e r i e s , 2 j n t t n t g„ e U ..,(2-11) n z=~^-cO and 3 n^n't C ( t ) = > C n' e U ...(2-12) n = — OO where <o„ = 2uf 0 0 Consider the e f f e c t of an input s i g n a l i 2 ( t ) , at angular frequency tt^r being a p p l i e d to the mixing—element (between t e r m i n a l s a-b of F i g u r e 2-1). This s i g n a l w i l l "mix" with the components of g ( t ) and C ( t ) to produce harmonics at angular f r e q u e n c i e s o» n = n <D0 ± <D2 n = l , 2, .• By t e r m i n a t i n g the mixing—element p r o p e r l y , power at the v a r i o u s f r e q u e n c i e s can be i s o l a t e d . I f terminations are provided such t h a t power can flow only at fr e q u e n c i e s given by n = 1, the r e s u l t i n g frequency converter i s s a i d to be of the fundamental-(4) mode type* ' A 3 by 3 matrix f o r the mixing-element of such a converter w i l l be developed. Let the angular f r e q u e n c i e s at which s m a l l - s i g n a l power can e x i s t be w^ , to^ a n < ^ w 3 where and a) a>1 = wQ + tt. b) fl>3 = <o0 - « 2 ...(2-13) This frequency spectrum i s shown i n F i g u r e 2-2. The s m a l l - s i g n a l v o l t a g e and c u r r e n t can be expressed (o) by the f o l l o w i n g s eries} 3 v ( t ) = > V. e + V.r e + high-order v o l t a g e harmonics i = 1 ...(2-14) i ( t ) = jt t . t -3»,t I. e 1 + I . * e 1 i = 1 + high-order c u r r e n t harmonics ...(2-15) Since power i s r e s t r i c t e d to angular f r e q u e n c i e s (o^t a n d to^t e i t h e r the high-order v o l t a g e harmonics, or the high-order c u r r e n t harmonics must be absent. Which i s absent depends upon 11 A Power ( 0 , 6) 0 (0 > 1 frequency-F i g u r e 2-2* S m a l l - S i g n a l Frequency Spectrum, the t e r m i n a t i o n s of the mixing—element, S u b s t i t u t i o n of equations ( 2 - l l ) to (2-15) i n t o equation (2-10) and equating the c o e f f i c i e n t s of e ) i = 1,' 2, 3, y i e l d s , i n matrix form, the equation fox the mixing—element• I * 3 * 0 + l^V «i *>iV g 2 + D^CJJ* « - l + ^ 2 ° - ! * «0 + «l + Jft)2Cl1 V * 3 ...(2-16) or tt • H W ...(2-17) 12 In general the o f f - d i a g o n a l F o u r i e r C o e f f i c i e n t s of matrix Y are complex. However, with conductance pump vo l t a g e as shown i n F i g u r e 2-1; V (t) = V . + V . cos ( « n t ) , gP gb gp 0 ' g ( t ) i s an even f u n c t i o n of time w i t h g r e a l and g = g , 6 n 6 n 6 - n A l s o f o r V ( t ) = V , + V cos (<D_t + © ) , cp 7 cb cp v 0 ' C ( t ) leads g ( t ) by 0 radians so t h a t C can be r e p l a c e d by C . where C i s r e a l and C = C . n n n —n M a t r i x Y can now be w r i t t e n i n terms of r e a l conduct-ance c o e f f i c i e n t s g^, g^ and a n ( ^ r e a l capacitance c o e f f i c i e n t s CQ> C ^ and a s f o l l o w s ! Y = 1- 0" g x + 3«2 C1 6 g l + 3 w C i e g 0 + 3 « 2 C 0 3© g 2 + 3 « 1 C 2 e 8 l + J' W2 C1 e J20 3© l 0 " 3*>3C0 ...(2-18) This mixing-element matrix can be used to analyze the frequency c o n v e r t i n g p r o p e r t i e s of a n o n l i n e a r conductance, a n o n l i n e a r capacitance, or a p a r a l l e l combination of the two. The 3 by 3 matrix can be reduced to a 2 by 2 matrix by 13 e l i m i n a t i n g row i and column i i f the converter i s terminated so t h a t no power e x i s t s a t frequency co^. The parameters g Q and C n are F o u r i e r C o e f f i c i e n t s and as such are determined fromi g n 2% 2TZ -jn» nt g ( t ) e d ( » 0 t ) = g n ( V ,V ) gb' gp' ...(2-19) and r 2% C n " 27 J 0 - 3 n ( t t n t +0) C ( t ) e U d(tt f tt) = C (V , ,V ) ' 0 n v cb' cp ...(2-20) Curves of g (V , ,V ) f o r tunnel-diodes and of C (Y . ,V ) 6 n v gb* gp 7 n v cb' cp f o r v a r a c t o r - d i o d e s are obtainable from the l i t e r a t u r e • * ( l ^ ) The parameters are a l l p o s i t i v e except g^ and g^ of the t u n n e l -diode which are negative over a l i m i t e d range of b i a s and pump v o l i a g e s V , and V r e s p e c t i v e l y . S SP 3. FREQUENCY CONVERTER ANALYSIS The mixing-element d e s c r i b e d by Y-matrix (2-18) i s to be used i n a u n i l a t e r a l down—converter * In the f i r s t s e c t i o n of t h i s chapter a 3-port network e q u i v a l e n t of the mixing-element matrix i s d e f i n e d * The 3-port i s reduced to a 2-port and the Y-matrix of the l a t t e r network i s determined. In the second s e c t i o n , two s p e c i a l cases of the 2-port are considered, and methods by which each case may be made u n i -l a t e r a l are o u t l i n e d . 3.1 Network Re p r e s e n t a t i o n of the Mixing-Element Equations (2-16) to (2—18) d e s c r i b e the mixing-element when power at the three f r e q u e n c i e s <o^ , co^ and OJ^ i s allowed to e x i s t . The form of these equations i s the same as the nodal equations of a l i n e a r 3—port network. The mixing-element can t h e r e f o r e be represented as i n Fig u r e 3-1. The s i g n a l f r e q u e n c i e s at p o r t s 1 ,2 and 3 of t h i s network are Oi^, (a^ and ft>2 r e s p e c t i v e l y . Using the 3-port e q u i v a l e n t of the mixing-element i n a (9) n o n i n v e r t i n g down-converter, ' po r t s 1, 2 and 3 are designated r e s p e c t i v e l y s (1) r . f . input p o r t (2) i . f . output p o r t (3) image p o r t In subsequent a n a l y s i s * s u b s c r i p t s 1, 2 and 3 w i l l be a s s o c i a t e d with the r . f . , i . f * and image p o r t s , r e s p e c t i v e l y . 15 Image C i r c u i t s i g n a l source s i g n a l c i r c u i t f l 3 S l 1 F l i n F M 3 • 3 V '3 n l-= Two-Port Mixing-Element 3-port mixing-element Source used i n reverse g a i n c a l c u l a t i o n s _ output c i r c u i t t — Z Z L T j B 'out F i g u r e 3-1* Mixing-Element with Terminations. The 3-port mixing-element (Figure 3-1) can be reduced to the 2-port by t e r m i n a t i n g the image p o r t with an admittance X^. The 2-port i s shown w i t h c u r r e n t sources and t e r m i n a l admittances as w e l l as the r e q u i r e d f i l t e r s . To be s p e c i f i c , the f i l t e r s are shown to be of the type that s h o r t - c i r c u i t high-order v o l t a g e harmonics* That i s , f i l t e r F^ allows only a. v o l t a g e at frequency (o^ to develop across i t s t e r m i n a l s . The equations of the 2—port are obtained from equation (2-16) and matrix (2-18) by e l i m i n a t i n g V_* and I _ * 16 through the r e l a t i o n I ^ * = -Y^V^** ^ n e equations can be put i n the form; Y Y x l l L12 Y Y 21 x22 ...(3-1) where X 1 2 = gn + j(0-,C 1~0 T i 2 ,= S i + >>1 C1 e j© T21 = g l + 3«2 Ci e" Y 2 2 = g n + j» 0C 2"0 ( g 2 4- ^ C 2 e J 2 Q ) ( g 2 - e - 3 2 0 g 2 + j * ^ eJ 2^) ( g l - J a > 3C 1 e " ^ V + g 0 " J<o3C0 *1 + ^ 2 ^ 1 g o - j». 3C 2 ej 2 ° ) ( g l + j « 0 C 1 e j e V + g 0 " J' t t 3 C0 g l + jO) 2C 1 e j 9 ) (g n - j a ^ C , e ^ V + g 0 " 3 « 3 C 0 '3V1 (3-2) These parameters are the b a s i s of f u r t h e r a n a l y s i s . 3.2 U n i l a t e r a l Frequency Conversion The frequency conv e r t i n g p r o p e r t i e s of a converter can be determined from a knowledge of the g a i n of the d e v i c e * (13) The transducer g a i n 1 d e f i n e d as, _ power d e l i v e r e d to l o a d  rT ~~ a v a i l a b l e power from source , w i l l be used f o r t h i s purpose. From t h i s d e f i n i t i o n the forward g a i n G,j ,^ 2 ^ s g i v e n 17 by, T12 4G, 4 G1 G2 '21 |T1 + T n ) I I 2 + Y 2 2 - Y 1 2 I i a I 2 = 0 ...(3-3) and the reverse g a i n G ^ i by 2 G, 'T21 4G, 4 G1 G2 | T12l I, = 0 Y l +- T11 T 2 + Y22 " Y12 Y21 ...(3-4) For a u n i l a t e r a l down—converter G ^ l i s zero and G ^ 2 i s f i n i t e . This i m p l i e s t h a t Y^2 = 0 and Y^ ^ 0. From Y-parameters (3—2) two methods of o b t a i n i n g u n i l a t e r a l down-conversion are pl a c e d i n evidence. Method I. Quadrature Pumping This method, a p p l i c a b l e i n p a r t i c u l a r to converters with a s h o r t - c i r c u i t image t e r m i n a t i o n (Y^ = OO), depends upon proper choice of the pump phase angle 9. With I 3 = 0 0 the Y matrix of equation (3-1) becomes. 18 11 21 12 22 g 0 + O ^ O g l + j * ^ e"J'e g l + j < 0 l C l e g 0 + d « 2 c 0 j© . . . ( 3 -5 ) For u n i l a t e r a l g a i n T 12 = g i + ^ i 0 ! 6 3© = 0, Therefore, i f and 1. 9 = +2" radians 2. gj^  +' tt1C1 = 0 the c o n v e r t e r w i l l be u n i l a t e r a l . Since 0 i s the phase angle by which the capacitance pump v o l t a g e leads the conductance pump v o l t a g e , the two n o n l i n e a r elements must be pumped i n time quadrature. T h i s c o n d i t i o n can only be s a t i s f i e d i f the two n o n l i n e a r elements are p h y s i c a l l y separate, i . e . , not j o i n e d d i r e c t l y together as i s the case f o r the no n l i n e a r conductance and the n o n l i n e a r capacitance of a s i n g l e t u nnel-diode. The u n i l a t e r a l f e a t u r e obtained from t h i s method i s present only at the band—centre frequency s i n c e the balance * The theory of u n i l a t e r a l frequency conversion by the quad-r a t u r e pumping method, f o r a c r y s t a l - d i o d e and a v a r a c t o r -diode used as an up - c o n v e r t e r j has been o u t l i n e d i n the literature*(H)»(12) The method can e q u a l l y w e l l be a p p l i e d to a tunnel-diode and a v a r a c t o r — d i o d e f o r which u n i l a t e r a l down— convers i o n with g a i n i s p o s s i b l e * r e q u i r e d by c o n d i t i o n 2 above can only be s a t i s f i e d at one frequency. Method I I . Image Termination This method f o r o b t a i n i n g u n i l a t e r a l down-conversion depends upon proper choice of the image t e r m i n a t i o n T^. The case i n which 0 = 0 w i l l be d i s c u s s e d i n d e t a i l s i n c e f o r t h i s case the n o n l i n e a r conductance and the n o n l i n e a r capacitance can be t h a t of a s i n g l e tunnel—diode. The T-parameters* f o r © = 0, from equations(3-2) become: T i 2 = S i + i°>ici T21 = S i + J"2 C1 T 2 2 = S 0 + J « 2 C 0 -(*2 + J W l C 2 ) f s 2 - i»3c2) V + g 0 " - 3 « 3 C 0 fg 2 + J f l> 1c 2)(g 1 - ^ 3 C l ) T 3* + g 0 • " J' W3 C0 ( g2 " J W 3 C 2 ) (si + J*2 Cl) V + S 0 • " J t o3 C0 V + s 0 " - J t t3 C0 ...(3-6) The down-converter w i l l be u n i l a t e r a l provideds ( g 2 . + 3 « 1 C 2 ) ( g 1 - j ^ C j ) ? 3* + g Q - j « 3 C 0 T i 2 = s i + d-iCj. - v ^ = o 20 The image t e r m i n a t i o n Y^ r e q u i r e d to s a t i s f y t h i s i d e n t i t y i s : ( g 2 ~ 3 « 1 C 2 ) ( g 1 +' j ^ C j ) + ' ' * 3 = - g 0 - 3" 3C 0 + ^ ' n ^ •••O-?) g x - 3 « 1 C 1 This p a r t i c u l a r image t e r m i n a t i o n gives u n i l a t e r a l down-conversion. can be r e a l i z e d , and equation (3-7) s a t i s f i e d , f o r the case where the capacitance terms are zero: but both Y.^ a n ( i v a n i s h simultaneously. Capacitance terms as w e l l as conductance terms are e s s e n t i a l f o r u n i l a t e r a l conversion u s i n g t h i s method. When the g and C parameters that make up are those of pumped semiconductor diodes i t can be shown t h a t cannot be a p o s i t i v e r e a l d r i v i n g - p o i n t admittance, (see Appendix I ) . With tunnel-diode g parameters the r e a l p a r t of Y-j can be p o s i t i v e ( S e c t i o n 4.2b) so that Y^ can be approximated with l i n e a r passive elements. Since the admittance can only be approximated, u n i l a t e r a l g a i n by t h i s method, as f o r the quadrature pumped converter, can only be obtained at the band-centre frequency. The p a r a l l e l combination of a r e s i s t o r and a c a p a c i t o r i s a cl o s e enough approximation to Y^ to give 20 db. of r e j e c t i o n over a 5fo bandwidth ( S e c t i o n 4.5). I t i s p o s s i b l e to e x p l a i n the inherent narrow-band fea t u r e of the quadrature and image t e r m i n a t i o n methods. Roughly speaking, these methods depend upon the reverse g a i n due to the conductance element to cancel t h a t due to the capacitance element. For t h i s c o n d i t i o n , a net forward g a i n s t i l l e x i s t s because the forward and reverse gains due to the (14) capacitance are not equal* The g a i n provided by the (4) conductance i s frequency independent whereas t h a t of the (o) capacitance i s frequency dependent* Hence the balance r e q u i r e d f o r the u n i l a t e r a l f e a t u r e can only be obtained at one frequency. For the quadrature pumped converter one cannot, by a l t e r i n g the t e r m i n a t i o n s * overcome t h i s b a s i c l i m i t a t i o n ; whereas, f o r the image-terminated converter, zero reverse g a i n can be obtained over as l a r g e a bandwidth as the of equation (3-7) can be s y n t h e s i z e d . 22 4. UNILATERAL DOWN-CONVERSION BY THE IMAGE TERMINATION METHOD The f i r s t p o r t i o n of t h i s chapter i s devoted to the study of a converter with optimum ter m i n a t i o n s (Sections 4.2 to 4.4). Such t e r m i n a t i o n s can be r e a l i z e d only w i t h an i n f i n i t e number of elements, but they enable the t h e o r e t i c a l study of the converter to be made under the best p o s s i b l e c o n d i t i o n s . Such t o p i c s as g a i n and s t a b i l i t y are co n s i d e r e d . In the l a s t p a r t of t h i s chapter, a more p r a c t i c a l converter w i t h simple t e r m i n a t i o n s i s d i s c u s s e d . A model of t h i s converter was analyzed on the computer i n order to o b t a i n frequency response curves. 4.1 Approximations The 2—port Y-parameters (equations 3-6) are a p p l i c a b l e to an image terminated converter i n which the n o n l i n e a r conduct-ance and n o n l i n e a r capacitance are d i r e c t l y i n p a r a l l e l . For the frequency range being considered and the n o n l i n e a r capacitance t h a t of a semiconductor diode, i t i s found that 0^ can be neg l e c t e d ( S e c t i o n 6.1). Under the f o l l o w i n g assumption and d e f i n i t i o n s , c 2 = 0 a A ¥ l Si A <°0 C0 '0 1 « 0 -2C-2-2 a>0 3 <o0 n o n l i n e a r i t y f a c t o r ) l o a d i n g f a c t o r ) normalized frequency) . . . ( 4 - l ) 1 1 ) 1 1 ) the Y parameters s i m p l i f i e d forms of (3-6) can be w r i t t e n i n the f o l l o w i n g 23 11 "12 g Q ( l + j i ^ p ) g 1 ( i +• j r ^ a ) 22 g 22 Y 2 1 = g l ( l + jn 2 a ) -g 0 ( i + jn 2P) -Y 3 * + g 0 ( i - jrijP) g 1 g 2 ( 1 " J ^ 3 a ) Y 3 * + g 0 ( i - ja 3 p ) g 1 g 2 ( l + j i l 2 a ) Y 3 * + g 0 ( l - jOjP) g x 2 ( l + j f ^ a M l - jOjOc) Y 3 * + g Q ( l - jA 3P) ...(4-2) These parameters w i l l be used as the b a s i s f o r a n a l y s i s of the image terminated c o n v e r t e r . 4.2 Optimum Terminations A u n i l a t e r a l down—converter t h a t has the f o l l o w i n g p r o p e r t i e s w i l l be considered to have optimum t e r m i n a t i o n s . a) No r . f . , i . f . or image a t t e n u a t i o n due to f i l t e r s . b) Wide-band u n i l a t e r a l behaviour with a passive image t e r m i n a t i o n . c) Input and output admittance r e a l over a wide band, a) Optimum F i l t e r s I t i s assumed that the f i l t e r s F^, F 2 and F 3 , (Figure 3-1) give zero a t t e n u a t i o n and zero phase s h i f t over wide pass-bands centredaround t h e i r r e s p e c t i v e centre f r e q u e n c i e s , t h a t i s , Y^ = Y^ over a wide band. These f i l t e r s w i l l be termed optimum. In a d d i t i o n , i t i s assumed that f i l t e r F^ presents i n f i n i t e admittance to s i g n a l s at f r e q u e n c i e s other than f . . 24 This l a t t e r assumption has been i m p l i c i t l y assumed i n the d e r i v a t i o n of the mixing-element matrix equations (2-16) to (2-18). b) Optimum Image Termination C o n s i d e r a t i o n of equations (3-7) and (4-1) r e v e a l s that i f (1 + jfUcx) T_ = -g (1 + j f u 0 ) + g0 — ...(4-3) the down-converter w i l l be u n i l a t e r a l f o r a l l f r e q u e n c i e s . Y^ i s not a p o s i t i v e r e a l d r i v i n g - p o i n t admittance and t h e r e -f o r e cannot be r e a l i z e d e x a c t l y . For a converter w i t h optimum te r m i n a t i o n s i t i s assumed, however, that the approximation to Y^ i s very c l o s e over a wide frequency band. For Y^ to be a passive t e r m i n a t i o n i t i s necessary that Re ^Y^J- 0. From the expansion of Y^ i t f o l l o w s that Re (Y 3) ^ 0 f o r g 0 20. a 2 f * 1 - — ± ...(4-4) 2 -i . rt 2 2 1 +iL a The e q u a l i t y of (4-4) i s p l o t t e d i n F i g u r e 4-1 as a f u n c t i o n of _ f l ^ w i t h a as a parameter. The Re (T<J) w i l l be p o s i t i v e f o r a g i v e n a only i f g 0 — i s l e s s than the value g i v e n by the graph. For a c r y s t a l -g 2 25 g 2 1.0-0.5-oc 0 a = -0.3 0 1.0 1.1 1.2 a = -1.0 1.3 1.4 —i fr» 1.5 n . -0.5" -1.0-a = -5.0 Fi g u r e 4-1. Boundaries of a Passive Image Termination (Re ( Y 3 ) = 0 ) . ( l ) g 0 \ diode i t can be shown x ' t h a t — ^ 1. This r u l e s out the g 2 p o s s i b l e use of such a diode to achieve u n i l a t e r a l c o n v e r s i o n with a passive image t e r m i n a t i o n . A tunnel-diode, however, can s a t i s f y c o n d i t i o n 4-4 over a r e g i o n of i t s c h a r a c t e r i s t i c curve. F u r t h e r a n a l y s i s w i l l t h e r e f o r e be c a r r i e d out on the under-standing t h a t the r e q u i r e d g parameters are those of a t u n n e l -d i o d e . (10) 26 c) Optimum Input and Output Terminations The input and output admittances are made r e a l by s u i t a b l y choosing and r e s p e c t i v e l y . From F i g u r e 3-1 i t f o l l o w s that w i t h optimum f i l t e r s ; X i n = T l l " Yj^L- + j B, ...(4-5) Y + Y x22 + 2 and Y o u t = T 2 2 * l 1 2 l f Y +3 B 2 — <4-6> 111 + X l Since Y^ 2 = ®t by choice of Y^, the input and output admittances w i l l be r e a l i f and 2 a *2 = ^m ( T2 2) = — «h V « 0 — ( 4 ~ 8 ) For the optimum te r m i n a t i o n s and f i l t e r s d e f i n e d i n a, b and o above, the converter equations* from equations (4-2), become; 27 I, 1 0 G 2 + g 0 - g 2 1 -XL-jfLjOc' i + n 3 2 a 2 0 /1 + a-n-2 a\ -•j 2 g l a ( G 2 + \1 - j/1-a/ ^ >0 ...(4-9) I t should be noted that Y^f and B 2 are a l l impossible to r e a l i z e with a f i n i t e number of passive elements. The converter with optimum terminations i s s t i l l of i n t e r e s t s i n c e the te r m i n a t i o n s can be synt h e s i z e d t h e o r e t i c a l l y to any r e q u i r e d degree of accuracy. A l s o , the band-centre c h a r a c t e r -i s t i c s of a converter with simple t e r m i n a t i o n s can equal those w i t h optimum t e r m i n a t i o n s . ( S e c t i o n 4.5). 4.3 Converter S t a b i l i t y I t has been e s t a b l i s h e d ( S e c t i o n 4.2b) th a t the g parameters of a tunne l - d i o d e , i n c o n j u n c t i o n with a passive image t e r m i n a t i o n , are s u i t a b l e f o r u n i l a t e r a l down-conversion. Since spurious o s c i l l a t i o n s are d i f f i c u l t to avoid i n c i r c u i t s (15) employing tunnel-diodes ' the converter s t a b i l i t y c o n d i t i o n s are of prime importance. V i t h r e f e r e n c e to the 2-port network of Fig u r e 3—1, (16) necessary c o n d i t i o n s f o r s t a b i l i t y ares (1) G± +• Be ( l i n ) >0 ...(4-10) ( 2 ) G 2 + R e (You^° For a converter with optimum terminations and f i l t e r s , 28 I. and T , are r e a l , and have values G. and G , giv e n by xn out ' i n out 6 J equations (4-5)» (4-6) and (4—9), g 0 - g 2 1 ~ 2(2 - r ^ V 1 + (2 - ^ 1 ) 2 a 2 j (4-11) G o u t = g 0 - r- f 1 >1_ >2 Si = g o 1 - (r^-i) ^ a ' ..(4-12) These f u n c t i o n s are p l o t t e d i n Fig u r e 4-2 f o r a p a r t i c u l a r s et of g and a parameters. Both G^ n and G ^ are predominantly n e g a t i v e . From (4-10), the converter w i l l be st a b l e i f and G, •+- G, > 0 1 i n ^ G G , > 0 2 out .*(4-13) f o r a l l values of frequency. -2 mrf 1 * The g parameters chosen, namely gQ = -4 mrr-, v . - ^ and g^ = 2 mrT 1 are approximately those of a 1 ma. germanium tunnel-diode b i a s e d at the p o i n t where g(v) i s a minimum and pumped w i t h a 100 mv. peak to peak source. This mode of oper a t i o n s a t i s f i e s the c o n d i t i o n g^ ^ (equation 4-4) and provides £ 1 c o n v e r s i o n g a i n with source and l o a d r e s i s t a n c e s w i t h i n the common range of 50 to 300 ohms. These g parameters are used i n f u r t h e r i l l u s t r a t i v e graphs i n t h i s chapter. For these o p e r a t i n g c o n d i t i o n s the n o n l i n e a r capacitance of the tunnel diode i s s u f f i c i e n t to give values of a between about 0 and -1.0 f o r pump fr e q u e n c i e s l e s s than about 150 mc. Values of a l e s s than —1.0 correspond to pumped elements c o n s i s t i n g of a tunnel—diode i n p a r a l l e l with a varactor-diode„ Figure 4-2. Input and Output Conductance v s . Normalized Input Frequency 30 4.4 Gain With Optimum Terminations From equations (3—3) and (4-9) the forward g a i n of the converter with optimum t e r m i n a t i o n s becomesj 16 G 1G 2 g-^oc2 G i + o-n-2a 1 - j-0- 3a T12 iG-. + e o -1 -ajTLjOC 1 J J O 2 2 \1 +fLy a / G 2 + g 0 g 2 I -...(4-14) The denominator of t h i s e x p r e s s i o n i s of the form ( S l + °in)( G2 + °out) which upon comparison with the s t a b i l i t y c o n d i t i o n s (equations 4—13) i s seen to approach zero when i n s t a b i l i t y o c c u r s . That i s , the c o n d i t i o n f o r i n f i n i t e g a i n i s the same as the c o n d i t i o n s f o r i n s t a b i l i t y . With such a device any reasonable value of g a i n can be r e a l i z e d by choice of G^ and G^. but h i g h g a i n i s obtained at the expense of s t a b i l i t y . V a r i a t i o n of forward g a i n G r p 1 2 with frequency f o r p a r t i c u l a r v a l u e s of source and l o a d conductances G-^  and G 2 r e s p e c t i v e l y , (both 8 mri- 1) i s shown i n F i g u r e 4-3. Only f o r |oc| l a r g e i s the g a i n s i g n i f i c a n t l y frequency dependent. The reverse g a i n Grj 2i °^ a converter w i t h these t e r m i n a t i o n s i s d e f i n e d to be z e r o . A set of three equal—gain contour curves i n the 31 - 5 -F i g u r e 4-3. Forward Gain w i t h Optimum Terminations v s . Normalized Input Frequency. 32 G-^  - G^ plane corresponding to three values of a with £L^ = 1«25 i s giv e n i n Fig u r e 4-4. For a = -.3 and -1.0 both G. and G . are ne g a t i v e , but f o r a = -5.0 G. i s negative i n out • • o T i n and CrQu^. p o s i t i v e : consequently the curves f o r a = -5.0 are of a d i f f e r e n t form. Since Grj,^ 1 S p r a c t i c a l l y independent of j f l ^ f o r a = -.3 and -1.0 the contours corresponding to these values are approximately v a l i d f o r a l l values of - T L ^ . 4.5 Frequency C h a r a c t e r i s t i c s of a Computer Analyzed Model with " P r a c t i c a l Terminations"  The optimum f i l t e r s and te r m i n a t i o n s s p e c i f i e d i n S e c t i o n 4.2 cannot be r e a l i z e d p r a c t i c a l l y . One step c l o s e r towards the p r a c t i c a l case i s provided by simple p a s s i v e -element terminations (Figure 4-5 elements with s u p e r s c r i p t bars) and a n t i r e s o n a n t frequency f i l t e r i n g (L^£» ^ i f 7 * * ^ e f i l t e r i n g by the l a t t e r components i s assumed adequate to suppress unwanted harmonics i n the sense d e s c r i b e d f o r optimum f i l t e r s ( S e c t i o n 4*2a). This assumption i s only v i o l a t e d when L ^ / C ^ i s a very h i g h r a t i o . The equations f o r t h i s converter are based upon the mixing-element parameters (4—2). The t e r m i n a t i n g and f i l t e r components f o r t h i s " p r a c t i c a l " converter are chosen to give the same l o a d i n g , at band-centre* as optimum terminations ( S e c t i o n 4.2). By choosing the components i n t h i s manner, the c h a r a c t e r i s t i c s of the converter at band-centre are the same as those obtained w i t h optimum t e r m i n a t i o n s . The t e r m i n a t i n g component values are g i v e n i n Appendix I I . A converter with t e r m i n a t i n g and f i l t e r components as d e s c r i b e d (Figure 4-5) was analyzed on the d i g i t a l computer. F i g u r e 4-4. Equal-Gain Contours of Converter with Optimum Terminations u> U> image c i r c u i t s i g n a l c i r c u i t t r-VWV-i G\, mixing-element 34 output c i r c u i t 1 T?2f. F i g u r e 4-5. Mixing-Element with P r a c t i c a l Terminations. Frequency responses based on the forward and reverse g a i n equations (3-3) and (3-4) r e s p e c t i v e l y , were obtained f o r v a r i o u s ^ i f / C ^ f r a t i o s and values of a. These are shown i n F i g u r e s 4-6. For small L ^ / C ^ r a t i o s the bandwidth i s very narrow. As the r a t i o i s i n c r e a s e d the bandwidth i n c r e a s e s : the upper l i m i t to the bandwidth i s set by the n e c e s s i t y to channel power at the d e s i r e d f r e q u e n c i e s . The curves of F i g u r e 4-6 f o r l a r g e s t band-width could be r e a l i z e d by u s i n g a d d i t i o n a l frequency f i l t e r i n g . The L ^ / C \ £ r a t i o to which they correspond i s l a r g e , and thus v i o l a t e s the assumption s t a t e d i n the f i r s t paragraph of t h i s s e c t i o n . , The reverse g a i n "bandwidth" seems to be p r a c t i c a l l y independent of the L i f / C i f r a t i o , i n d i c a t i n g t h at t h i s "band-35 width" i s l i m i t e d by the approximate image t e r m i n a t i o n . The narrow reverse gain "bandwidth" probably has r e p e r c u s s i o n s i n the c r i t i c a l tuning of the experimental converter, ( S e c t i o n 6.2). The response i s not g r e a t l y a f f e c t e d by a ( a measure of the n o n l i n e a r c a p a c i t a n c e ) , even though the u n i l a t e r a l property depends upon t h i s capacitance. Thus, there i s l i t t l e to be gained at these f r e q u e n c i e s by i n c r e a s i n g the e f f e c t i v e n o n l i n e a r capacitance of a tunnel-diode by b r i d g i n g i t , f o r example, with a v a r a c t o r - d i o d e . In a d d i t i o n to the response curves of F i g u r e 4-6, the g a i n contours of F i g u r e 4-4 are a l s o a p p l i c a b l e at band-centre to the converter with p r a c t i c a l t e r m i n a t i o n s . Thus, the mid-band gains of F i g u r e 4-6, f o r which G^ = 8m.Hr1 and 5^ = 8m.nl1, can a l s o be obtained from F i g u r e 4-4. •G T(db) 10 4 -10 t. -20 - 3 0 4 -30 G T(db) L i f -<:G T 2 1 a --- -5.0 = 8000 ' i f (5X = 8 m i l " , 5 2 = 8 , p = -.75, g Q = -4 mXV 1, g ; L = -2 m-TiT1, g 2 = 2 m^" 1) F i g u r e 4-6(a). Forward and Reverse Gain of a Converter w i t h P r a c t i c a l T e r m i n a t i o n s . UJ 0^  •& T(db) 10+ -10t -20T -30T - 4 0 t f T(db) H * l -20 -30 -40 G„(db) • T rT12 10f 1.2 \ GT21 -10 t I \ -20' L i f . -30" a = -1.0 = OO C i f -40" T21 / / I I l ' I I I / 'it. a = -5.0 = OO ' i f (S 1 = 8 mn."1, G 2 = 8 mA _ 1. P = --75, g Q = -4 m-TL"1, g x - -2 mn" 1, g ? = 2 m i l 1 ) u> F i g u r e 4-6(b). Forward and Reverse Gain of a Converter w i t h P r a c t i c a l Terminations, F i g u r e 4 - 6 ( c ) » Forward a,nd Reverse Gain of a Converter w i t h P r a c t i c a l Terminations. 00 3 9 5. NOISE The very important t o p i c of noise performance has not been s t u d i e d experimentally* The noise at the output i s due to two sources! the s i g n a l source and the e q u i v a l e n t diode n o i s e ; ( 8 ) and the image c i r c u i t n o i s e * From Kim, a l i m i t i n g f i g u r e f o r the f i r s t source of noise i s gi v e n by G 0 F = 1 + ^7- where G n = e q u i v a l e n t shot noise conductance* °i For the experimental diode, G Q = 10 mxC 1 and G = l O m r T 1 ; F = 3 db. A d d i t i o n a l noise due to the image c i r c u i t can be represented by a conductance across the output t e r m i n a l s . The value of t h i s conductance* based on an estimated power g a i n of u n i t y between image and output c i r c u i t s , i s G^ (2m.rL 1). Kim ts noise f i g u r e now becomes; G 0 G 3 F = 1 + + _ and has a value F = 3.4 db. These noise c a l c u l a t i o n s , based on Kim's l i m i t i n g noise f i g u r e e x p r e s s i o n , i n d i c a t e t h a t the noise performance of the u n i l a t e r a l c onverter i s s l i g h t l y poorer than the b i l a t e r a l c o n v e r t e r . This r e s u l t can be re v e r s e d i f the converter feeds i n t o a c i r c u i t producing noise power across i t s input t e r m i n a l s ; (e.g. a t r a n s i s t o r ) . In such a case, the noise at the input of the b i L a t e r a l converter i s i n c r e a s e d by feedback; and the o v e r a l l noise f i g u r e can be g r e a t e r than t h a t with the u n i l a t e r a l c o n v e r t e r . 40 6. EXPERIMENTAL An experimental study was undertaken i n order to v e r i f y the u n i l a t e r a l p r o p e r t y set f o r t h i n the theory. With t h i s i n mindj a 127 mc, to 27 mc, converter u t i l i z i n g the n o n l i n e a r conductance and the n o n l i n e a r capacitance of a s i n g l e t u n n e l -diode was co n s t r u c t e d , 6,1 Experimental Converter C i r c u i t and I t s Elements The c i r c u i t chosen f o r experimental work i s shown i n Fig u r e 6-1, Before c o n s i d e r i n g t h i s c i r c u i t i n d e t a i l , i t i s worthwhile to examine the c h a r a c t e r i s t i c s and e q u i v a l e n t c i r c u i t of the GE IN 2939 tunnel—diode used. These are shown i n F i g u r e 6-2. In theory i t was found that a b i a s p o i n t at or near the minimum value of the conductance g(v) i s d e s i r a b l e . t From the manufacturers s p e c i f i c a t i o n s and from experimental measurement, the conductance and the capacitance a t t h i s b i a s p o i n t are approximately -6,6 mn"''' and 12 p . f , r e s p e c t i v e l y . For these v a l u e s , the s e r i e s r e s i s t a n c e R = In. and the s e r i e s ' s inductance L = 5 n.h, are n e g l i g i b l e at fre q u e n c i e s below s the r . f . frequency of 127 mc* This f a c t had been p r e v i o u s l y assumed i n the theo r y . A 100 mc. pump source e f f e c t s the v a r i a t i o n of the conductance and c a p a c i t a n c e . For a pump volt a g e of 100 mv. peak to peak across the diode^ t h e o r e t i c a l values of the fundamental, the f i r s t and second harmonics of the time v a r y i n g For example, footnote o f S e c t i o n 4.3. 41 Pump s i g n a l Image L 3 r-Ofin-, 5 3 4 f 0. P g 6 ^ •AAAA-1N2939 T . D 0 '0 Output — "5 'b _zz I Pump frequency 100 mc S i g n a l frequency 127 mc Image frequency 73 mc Go = 20mrL~'1j 5 = l O r n n " 1 , G 2 = lOrn-fL - 1, 5 3 = 2ml\~1, G 6 = 200mrL _ 1, 1^ * .05 ( i . h . , L 2 — .2 ( J , » h » , — .15 f J - . h . , L Q - .1 f x . h . , C l =7-45 p . f . , C 2 = 200-250p.f ., C 3 = 7-45 C 0 = 7-45 p . f . , C b = .1 j i f , G 4 = 100 mn"1. • 1 F i g u r e 6-1. Experimental Converter C i r c u i t 42 F i g u r e 6—2. E q u i v a l e n t C i r c u i t and C h a r a c t e r i s t i c s of a Pumped IN 2939 Tunnel-Diode* conductance and capacitance wave forms are approximately? g Q = -4.0 m i l " 1 C Q = 12.0 p . f . g 1 =-2.0 mrT1 C± = 2.0 p . f . g 2 = 2.0 mn."1 C 2 = 0.5 p . f . These values give r i s e to an a and a ^ o f : "O 0! 2% x 100 x 1 0 6 x 2 x 1 0 " 1 2 n ,- Q a = — — — = • = —u.o^o g l -3 1 -2.0 x I O J a a°°0 2TX x 100 x 1 0 6 x 12 x 10 1 2 , n p = — = _ = -1.9 u -4.0 x 10 J With the parameter val u e s above, the theory p r e d i c t s t h a t u n i l a t e r a l conversion with g a i n i s p o s s i b l e . In order that c i r c u i t ( 6 — l ) be s t a b l e at the b i a s p o i n t i n the negative conductance r e g i o n * a d-c s t a b i l i t y c o n d i t i o n , together w i t h the s t a b i l i t y c o n d i t i o n s o u t l i n e d i n the theory* (equations 4—13) must be obeyed. The former i s s a t i s f i e d i f the conductance (at d—c) across the diode i s g r e a t e r than the absolute v a l u e of the diode conductance (6i6 mn. 1 ) . The G^f G^ combination provides a s u i t a b l e value, The l a t t e r c o n d i t i o n s are s a t i s f i e d w i t h a margin of s a f e t y , i f t h e r e a l p a r t of the admittance across the diode i s g r e a t e r than the absolute value of the diode conductance at a l l (15) f r e q u e n c i e s below the diode r e s i s t i v e c u t - o f f frequency. In order t h a t the admittance be l a r g e , i t i s e s s e n t i a l t h a t w i r i n g inductance be kept to a minimum. The tuned c i r c u i t s of the converter are designed to have a h i g h admittance at harmonic f r e q u e n c i e s other than t h e i r centre f r e q u e n c i e s . With such a design the converter i s equ i v a l e n t to the computer analyzed model of S e c t i o n 4.5. The c i r c u i t elements w i t h the exception of and G^ can t h e r e f o r e be determined from equations AII*-2 through AII-6. G^ and G^ are chosen to give the d e s i r e d s t a b i l i t y and g a i n . Since lumped elements at V.H.F. fre q u e n c i e s have a s s o c i a t e d p a r a s i t i c s the values g i v e n by the above equations must be adjus t e d a f t e r the components are mounted i n the a c t u a l c i r c u i t . In the experimental model, hi g h q u a l i t y components were used i n an e f f o r t to minimize p a r a s i t i c e f f e c t s . The f o u r i n d u c t o r s , each 3/4" i n diameter* were wound with \ to 3 turns of #14 t i n n e d copper w i r e . T h e i r Q at 100 mc. was above 50. With such a Q the e q u i v a l e n t c o i l conductances are small compared to the load conductances* The ad j u s t a b l e c a p a c i t o r s were of the ceramic d i s c t y p e . The components were placed on a 3/16" s h i e l d e d b rass c h a s s i s f i t t e d with standard B.N.C. * I f the r e a l p a r t of the admittance i s not s u f f i c i e n t l y l a r g e * s e l f — o s c i l l a t i o n s w i l l occur* By arr a n g i n g these o s c i l l a t i o n s to take place a t the pump frequency, i t i s p o s s i b l e to e l i m i n a t e the need f o r an e x t e r n a l pump source* Converters of t h i s s e l f -o s c i l l a t i n g type(8) were experimented with, but great d i f f i c u l t y was found i n c o n t r o l l i n g the amplitude of the o s c i l l a t i o n s . Since the adjustments of the converter f o r d i r e c t i o n a l g a i n are c r i t i c a l to begin w i t h ( S e c t i o n 6*2), the i n v e s t i g a t i o n of s e l f — o s c i l l a t i n g converters was not pursued beyond the p r e -l i m i n a r y stage* connectors i n such a manner t h a t no w i r i n g between the components was r e q u i r e d . 6*2 Experimental Techniques and R e s u l t s The equipment a v a i l a b l e f o r alignment and check-out was a l l of the 50X1 c h a r a c t e r i s t i c impedance type. To provide a match, independent of frequency* between the equipment and the c o n v e r t e r , i t was necessary to use r e s i s t i v e matching n e t -works at a l l times* These networks superimposed a d d i t i o n a l noise on the v a r i o u s s i g n a l s so that i t was not p o s s i b l e to measure the noise f i g u r e of the c o n v e r t e r . A sweep—frequency s i g n a l generator and a de t e c t o r were used to a l i g n roughly the fo u r tuned c i r c u i t s . Hewlett-Packard S i g n a l Generator (M 808D) -Crystal O s c i l l a t o r = 100 mc 0 r e s i s t i v e matching networks Detector (RCA, AR-88LP) Radar Detector (AN/APR-4) Hewlett-Packard S i g n a l Generator (M 808D) F i g u r e 6-3. Block Diagram of Converter and Measuring Equipment* 46 F i g u r e 6-3 i s a b l o c k diagram of the equipment used f o r f i n e t uning and measurement of the forward and reverse g a i n s . The very l a r g e forward g a i n p o s s i b l e i n theory was not r e a l i z e d i n p r a c t i c e . Considerable e f f o r t was expended i n a sequence of tuning, l o a d matching and b i a s p o i n t adjustments, i n order to o b t a i n 1 db. of forward g a i n and a reverse g a i n of afeout -10 db. at one spot frequency. An i n s i g h t i n t o the d i f f i c u l t y i s contained i n the f a c t t h a t with s l i g h t detuning of any of the tuned c i r c u i t s i n one way, a h i g h forward g a i n could be r e a l i z e d (25 db.)« but f o r t h i s c o n d i t i o n the converter was completely r e c i p r o c a l as f a r as measurements could be t r u s t e d . I t i s l i k e l y t h a t the detuning s u b s t a n t i a l l y a l t e r s the e f f e c t i v e value of 8 i n equation (4-3) and completely changes the value of r e q u i r e d f o r u n i l a t e r a l p r o p e r t i e s . The response curves of the Computer analyzed model (Figures 4-6) i n d i c a t e t h at the reverse g a i n has a very narrow bandwidth! t h i s f u r t h e r e x p l a i n s why the tuning and other adjustments might be d i f f i c u l t . In s h o r t , the experiments show that d i r e c t i o n a l g a i n can be obtained as p r e d i c t e d f o r the mathematical models but that the adjustments to achieve such g a i n ( i n the c i r c u i t used) are extremely c r i t i c a l . * P r e c i s e measurement of the reverse g a i n was not p o s s i b l e because of the very low s i g n a l — t o - n o i s e r a t i o of the s i g n a l t r a n s m i t t e d to the 127 mc* d e t e c t o r . Measurements were f u r t h e r hampered by the n e c e s s i t y to have the magnitude of a l l f o r c i n g s i g n a l s small compared to the pump s i g n a l (100 mv. p. to p . ) . 47 6,3 Suggestions f o r F u r t h e r Experimental Study The experimental study vas hindered by high-frequency e f f e c t s and tuning problems. These d i f f i c u l t i e s could be overcome i n two ways; one* use d i s t r i b u t e d elements (and work at a s t i l l higher frequency); or two, use a tunnel-diode i n p a r a l l e l w i t h a v a r a c t o r — d i o d e and work at a lower frequency. Although h i g h frequency experimental work would l e a d d i r e c t l y to p r a c t i c a l a p p l i c a t i o n s , work at low f r e q u e n c i e s would promote a b e t t e r understanding of the phenomena i n v o l v e d i n u n i l a t e r a l c o n v e r s i o n . Furthermore* at low f r e q u e n c i e s , i t should be p o s s i b l e to o b t a i n wide-ba.nd u n i l a t e r a l g a i n since the optimum t e r m i n a t i o n s ( S e c t i o n 4,2) c o u l d be c l o s e l y approximated. In a d d i t i o n , an experimental study of a quadrature pumped converter i s r e q u i r e d to i n d i c a t e whether such a converter i s p r a c t i c a l ; the study would a l s o provide a r e a l i s t i c comparison of the two methods. Future experimenting should a l s o be done w i t h c i r c u i t s s u i t a b l e f o r noise f i g u r e measurements. The v a r a c t o r — d i o d e i s r e q u i r e d as the n o n l i n e a r capacitance of a tunnel—diode i s i n s u f f i c i e n t a t low f r e q u e n c i e s . 7. CONCLUSIONS c This study has shown t h a t a frequency converter w i t h a passive image t e r m i n a t i o n can provide u n i l a t e r a l down-conversion w i t h g a i n . For t h i s type of conversion the mixing-element must have both no n l i n e a r capacitance and n o n l i n e a r conductance, such as that of a tunnel-diode ( S e c t i o n 4.2). With simple t e r m i n a t i o n s , the u n i l a t e r a l f e a t u r e i s present only over a narrow frequency band ( S e c t i o n 4.5), but w i t h more complex t e r m i n a t i o n s , wide-band u n i l a t e r a l g a i n can be achieved* ( S e c t i o n 4.4)• The noise mechanism of the converter i s extremely complex, but approximate c a l c u l a t i o n s i n d i c a t e t h a t i t s noise f i g u r e i s only s l i g h t l y worse than that of a b i l a t e r a l converter (Chapter 5). U n i l a t e r a l down—conversion wi t h g a i n can a l s o be obtained by the quadrature pumping method. Converters using t h i s method requi r e fewer parameters than image terminated converters* but the u n i l a t e r a l property i s p o s s i b l e only over a narrow bandwidth. APPENDIX I 49 P o s i t i v e Real D r i v i n g - P o i n t Admittance This appendix deals with the c o n d i t i o n s that the image te r m i n a t i n g admittance must s a t i s f y i n order to be a p o s i t i v e r e a l d r i v i n g - p o i n t admittance. The image t e r m i n a t i n g admittance r e q u i r e d to give wide-band u n i l a t e r a l g a i n i s , from equation (3-7) i Y , i . 0 + ( g2 ~ j W l C 2 ) (Sl + > 3 C l ) X 3 = " g0 -J*3°0 + ~ «1 - 3 <°1 C1 This admittance can be w r i t t e n i n terms of the complex frequency v a r i a b l e s by l e t t i n g s = jco^ a n < i a l s o n o t i n g that Y 3 = g r (*z-*o) + 4 « o \ (gic2-goci) 2 « l C l ( « 2 - go) + * 1 2 ( C2 " Co) + V C 1 2 ( C2 " Co) + 3 2 C x 2 ( g 2 - g Q) + 2 g l C l ( c 2 - C 0) I 2 ( C2 - Co) + 3 »o(Cl*2 ~ C 2 g l ) f g l + s C l )  gi 2 + ^ o2^2 + s 2 g i c i + s2 Gi 2 . . • ( A I - 1 ) For t h i s to be a p o s i t i v e r e a l d r i v i n g - p o i n t admittance i t i s at (17) l e a s t necessary t h a t the f o l l o w i n g c o n d i t i o n s be s a t i s f i e d ; ' + s + s 3 0, 2 50 (1) c l g 2 - c 2 g l = 0 (2) C 2 * C 0 (3) g1C]_ ^ 0 C o n d i t i o n (2) cannot be s a t i s f i e d w i t h semiconductor diodes or (9) other known n o n l i n e a r capacitance elements. ' This i s s u f f i c i e n t to prove t h a t cannot be a p o s i t i v e r e a l d r i v i n g - p o i n t admittance. 51 APPENDIX II Terminating Components f o r a " P r a c t i c a l " Converter The t e r m i n a t i n g components are chosen to give the same lo a d i n g as optimum terminations ( S e c t i o n 4.2) at band-centre. The equations d e s c r i b i n g the converter (equations 4-2) can be w r i t t e n i n the f o l l o w i n g s i m p l i f i e d form; Y l + Y l l T 1 2 "21 Y + Y 2 22 ...(AII - 1 ) Let the normalized angular r . f . * i . f * and image band-centre f r e q u e n c i e s be denoted by -H-^ O* "^ "20 a n < 1 "^ "30 r e s P e c " k i v e l y * The r e q u i r e d l o a d i n g w i l l be r e a l i z e d i f the components of the converter (Figure 4—5) are the f o l l o w i n g v a l u e s s L i f C i f = -n. 2io i = 1, 2 , 3 ...(AII - 2 ) ^ 1 0 c i = - ^ 1 1 ) Q i = x l i o = - a i o P « o + g 2 a 2 2 1 +-nr30oc ...(AII-3) 52 - f L20 C2 " I n^22y n 2 =XL 2 f 2. • ^ 2 0 ^ 0 + go ^ 1 0 + r i 2 0 (AII-4) G 3 + a^ 3 0c 3 =• T 3 from which = n 3 0 G 3 = " g 0 + g 2 / 2 \ 1 + X l 1 0 2 a 2 rL30C3 = "So^ O + 2g 0« i +r\ 0 2a 2 .....(AII-6) The inductance and capacitance values given by these equations are normalized with r e s p e c t to the pump frequency* Should .TL^QC^ be negative, an i n d u c t i v e t e r m i n a t i o n i s r e q u i r e d , and ^ ^QC^ i s r e p l a c e d by -1 xO I These equations determine a l l but the f i l t e r components u n i q u e l y . I f the L ^ / C ^ r a t i o $ which i s a s s o c i a t e d w i t h the bandwidth^ i s al s o s p e c i f i e d * a l l components can be determined* 53 REFERENCES 1. Torrey, H.C. and Whitmer, C.A.* C r y s t a l R e c t i f i e r s , McGraw-H i l l Book Compaay* Inc.* New York and London, (1948). 2. van der Z i e l , A», Noiseg P r e n t i c e - H a l l , Inc., Englewood C l i f f s , N.J.* (1956)* 3. E s a k i , L., "New Phenomenon i n Narrow P-N J u n c t i o n s " , Phys* Rev. L e t t e r s . V o l * 109, p. 603, (1958). 4. Edwards, C.F., "Frequency Conversion by Means of a Nonlinear Admittance"* B e l l Sys* Tech. J . . V o l . 35, pp. 1403-16, (November* 1956). 5* S t e r z e r , F. and P r e s s e r * A** "Stable Low-Noise Tunnel-Diode Frequency Converters"* RCA Rev,. V o l . 23, pp. 3-28, (Mar* 1962). 6* Chang* K*K*N** He i l m e i e r * G*H* and Prager, H.J«, "Low-Noise Tunnel-Diode Down Converters Having Conversion Gain"* Proc* IRE* V o l * 48, pp* 854-58, (May, I960)* 7* Cunningham* W*J.* I n t r o d u c t i o n to Nonlinear A n a l y s i s , McGraw^ H i l l Book Company* Inc., New York and London, ( 1 9 9 8 ) * 8* Kim* C.S.* "Tunnel-Diode Converter A n a l y s i s " * IRE Trans* PGED* V o l . ED-8* pp* 394-404, (Sept. 1961). 9* B l a c k w e l l * L*A. and Kotzebue* K*L.* Semiconductor-Diode  Parametric A m p l i f i e r s , P r e n t i c e - H a l l , Inc., Englewood C l i f f s * N*J.* (1961)* 10* P u c e l , R.A., "Theory of the E s a k i Diode Frequency Converter"* S o l i d - S t a t e E l e c t r o n i c s , V o l . 3, pp. 167-208, (Nov.-Dec. 1961)* 11* Englebrecht, R.S.* "Parametric Energy Conversion by Nonlinear Admittances' 1* Proc* IRE, V o l . 50, pp. 312-21, (Mar. 1962)* 12* Ross* P.W* and S k a l n i k * J*F** ''Parametric Frequency Converters V i t h A r b i t r a r y Pumping Angles", Proc. IRE, V o l . 51* p. 239, (Jan* 1963)* 13* L i n v i l l e , J.G. and Gibbons* J.F., T r a n s i s t o r s and A c t i v e C i r c u i t s , McGraw-Hill Book Company, Inc.* New York* Toronto and London* ( l 9 6 l ) . 54 14* Manley, J.M., and Rowe* H.E.* "Some General P r o p e r t i e s of Nonlinear Elements - Part I , General Energy R e l a t i o n s " * Proc. IRE., V o l . 44, pp. 904-913, (J u l y 1956). 15* Sommers, H.S., "Tunnel-Diodes as High Frequency Devices"* Proc. IRE. V o l . 47, pp. 1201-06, (Ju l y 1959). •. 16* Gartner, W.W.* T r a n s i s t o r s ; P r i n c i p l e s , Design and A p p l i c a t i o n s * D. Van Nostrand Company, Inc., Pr i n c e t o n , N.J., ( i 9 6 0 ) . 17. G u i l l emin* E.A.* Synthesis of Passive Networks. John Wiley and Sons, Inc.* New York, (1959)• 

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