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A silicon-diode-bridge parametric amplifier for low frequencies Sang, Marie Emmanuel Fok Ning Yow 1962

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A SILICON-DIODE-BRIDGE PARAMETRIC AMPLIFIER FOR LOW FREQUENCIES by M. EMMANUEL FOK NING YOW SANG B. Sc., Dunelm, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In the Department of E l e c t r i c a l Engineering We accept th i s thesis as conforming to the required standard Members of the Department of E l e c t r i c a l Engineering The University of B r i t i s h Columbia September, 1962 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Electrical flnginQftring The University of British Columbia, Vancouver 8, Canada. Date September 19. 1962 . i ABSTRACT The v a r a c t o r p r o p e r t i e s of s i l i c o n — d i o d e r e c t i f i e r s i n the low-frequency r e g i o n are i n v e s t i g a t e d . A t h e o r e t i c a l a n a l y s i s of a two-diode bri d g e i s made, and the r e s u l t s e x perimentally v e r i f i e d u s i n g a matched p a i r of these diodes and a pump frequency of 455 Kc. The t h e o r e t i c a l transducer power gain i s found to be a f u n c -t i o n of ( PU"»F ^ q u e n c y . } 2 p t h a m p l i f i e r b u i l t , a xupper 3-db frequency 7 ^ ' transducer power gain of 12.6 db has been achieved w i t h a band-width from 0 to 8 Kc. With the diodes pl a c e d i n a t h e r m o s t a t i c a l l y c o n t r o l l e d oven, the d-c d r i f t i s + 30 \xv per hour. The noise f i g u r e of the parametric a m p l i f i e r bridge i s about 3 db. A study of p o s s i b l e sources of noise i s made. V ACKNOWLEDGEMENT This p r o j e c t was c a r r i e d out under the sponsorship of the N a t i o n a l Research C o u n c i l . The author wishes to express h i s g r a t e f u l a p p r e c i a t i o n f o r the generous a s s i s t a n c e and guidance g i v e n by P r o f e s s o r P. K. Bowers, s u p e r v i s o r of the p r o j e c t . Acknowledgement i s a l s o given to Dr. P. Noakes, grantee of the p r o j e c t and to other members 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 f o r t h e i r a s s i s t a n c e during the course of t h i s r e s e a r c h . The author i s indebted t;o the Canadian Government f o r the Commonwealth s c h o l a r s h i p awarded him during the years 1960-1962. i i T A B L E OP CONTENTS P a g e A b s t r a c t » . . • i L i s t o f I l l u s t r a t i o n s . . . . . . . . . . . i v A c k n o w l e d g e m e n t . . . . . . . v 1 . I n t r o d u c t i o n 1 2 . V a r a c t o r D i o d e s 4 2 . 1 S e l e c t i o n o f D i o d e s 4 2 . 2 T h e o r y 7 2 . 3 E f f e c t i v e S e r i e s R e s i s t a n c e o f D i o d e . . . . . . 10 3 . T h e o r y o f P a r a m e t r i c A m p l i f i e r B r i d g e . . . . . . . . 13 3 . 1 T h e B r i d g e 13 3 . 2 C i r c u i t A n a l y s i s o f B r i d g e 13 3 . 3 G a i n a n d B a n d w i d t h 18 3 . 3 . 1 . G a i n o f a n I d e a l B r i d g e C i r c u i t 18 3 . 3 . 2 G a i n o f a P r a c t i c a l C i r c u i t 2 3 3 . 4 N o i s e 2 5 4 . D e s i g n a n d C o n s t r u c t i o n o f C i r c u i t . • 2 8 4 . 1 R e q u i r e m e n t s . . . . . . . . . . . . 28 4 . 2 P r a c t i c a l B r i d g e C i r c u i t . . . 30 4 . 2 . 1 B a l a n c i n g t h e B r i d g e . . . . . . . . . . 30 4 . 3 B a n d - P a s s H i g h - F r e q u e n c y A m p l i f i e r . 3 3 : 4 . 4 P h a s e - S e n s i t i v e D e m o d u l a t o r 3 5 4 . 5 D i f f e r e n c e A m p l i f i e r a n d L o w - P a s s F i l t e r . . . . 38 4 . 6 M e t h o d s U s e d t o M e a s u r e N o i s e 4 0 4 . 7 P r e l i m i n a r y M e a s u r e m e n t s 42 5 . M e a s u r e m e n t s . , • 4 3 5 . 1 O p t i m u m O p e r a t i n g B i a s 4 3 i i i P a g e 5 . 2 G a i n a n d B a n d w i d t h 4 3 5 . 3 D r i f t 4 5 5 . 4 Hum 46 5 . 5 N o i s e . . . . 46 5 . 6 G a i n a n d N o i s e as a F u n c t i o n o f Pump V o l t a g e . . 4 9 5 .7 P o s s i b l e S o u r c e s o f N o i s e . 50 6 . C o n c l u s i o n s 54 6 . 1 Summary o f R e s u l t s o f ' M e a s u r e m e n t s 54 6 . 2 S i l i c o n - D i o d e B r i d g e ; a s a bowi-Fre4u.1e.ncy P a i r a m e ' t r i c A m p l i f i e r . 5 4 6 . 3 S u g g e s t i o n s f o r F u r t h e r W o r k . 55 R e f e r e n c e s . 56 i v L I S T O F I L L U S T R A T I O N S F i g u r e Page 2 . 1 Measurement of Jfcbode,. Leakage;,..Current 5 2 . 2 Measurement of Capacitance of Reversed-Biased Diodes . 5 2 . 3 Leakage Current vs Voltage Curves f a r .Six .Diodes . . . 6 2 . 4 Capacitance C vs Bias V o l t a g e V of Two Matched Diodes 8 2 . 5 Log C vs Log V Curve 11 2 i 6 V a r i a t i o n of a with V 12 3 . 1 A c t u a l Bridge C i r c u i t w i t h Load , . 14 3 . 2 E f f e c t i v e A-C Bridge C i r c u i t 1 5 3 . 3 Thevenin's E q u i v a l e n t C i r c u i t 1 5 3 . 4 E q u i v a l e n t C i r c u i t at f 20 P 3 . 5 E q u i v a l e n t C i r c u i t at f I n c l u d i n g P r a c t i c a l L i m i t a t i o n s ? . . , 2 4 4 . 1 Block Diagram of the Parametric A m p l i f i e r 29 4 . 2 P r a c t i c a l Bridge C i r c u i t Showing B i a s i n g and B a l a n c i n g Schemes . . 3 1 4 . 3 Band-Pass High-Frequency A m p l i f i e r 34 4 . 4 P h a s e - S h i f t e r and P h a s e - S e n s i t i v e Demodulator . . . . 36 4 . 5 V o l t a g e Waveforms Showing Demodulation . . . . . . . . 37 4 . 6 D i f f e r e n c e A m p l i f i e r and Low-Pass F i l t e r 39 4 . 7 Noise Measuring C i r c u i t 41 5 .1 O p t i m i z a t i o n Curve f o r B i a s V o l t a g e 44 1 1. INTRODUCTION The fundamental p r i n c i p l e of n o n - l i n e a r reactance ampli-f i c a t i o n , f i r s t observed by M. Faraday as f a r back as 1831, may be b r i e f l y s t a t e d as f o l l o w s : - 'The energy of o s c i l l a t i o n i n a system may be i n c r e a s e d by pumping energy i n t o i t from an a - c source whose frequency d i f f e r s from the fundamental frequency of the o s c i l l a t o r ' . A survey of the h i s t o r y of parametric transducers can be 1 2 obtained i n the l i t e r a t u r e . ' The i n t e r e s t i n t h i s type of amp-l i f i c a t i o n has been r e v i v e d i n recent years by the development of low-loss v a r i a b l e capacitance diodes. There are two main reasons f o r t h i s renewed i n t e r e s t . Since displacement c u r r e n t s r a t h e r than conduction c u r r e n t s are u t i l i z e d i n the a m p l i f y i n g process, the a m p l i f i e r s w i l l e x h i b i t low noise p r o v i d e d the d i e l e c t r i c i s not lossy.- Because of the small power d i s s i p a t i o n i n v o l v e d and the robust nature of semiconductors, the diodes are expected to have extremely long l i f e . The t h e o r e t i c a l aspects of parametric a m p l i f i c a t i o n were considered by A. Van der Z i e l , H. Suhl, J . M. Manley and H. E. 2 3 Rowe, H. Heffner and G. Wade. ' E a r l y i n 1958, the low-noise p r o p e r t i e s of parametric a m p l i f i e r s i n the microwave r e g i o n were v e r i f i e d e x p e r i m e n t a l l y by M. Uenohara, R. S. Engelbrecht, B. I 4 Salzberg, H. Heffner and K. L. Kotzebue. ' Since then, r a i j i d progress has been made i n the f i e l d . I t should be emphasized that most of the work i n t h i s f i e l d has been c a r r i e d out s u c c e s s f u l l y at very h i g h f r e q u e n c i e s , mainly i n the microwave r e g i o n . The i n e v i t a b l e questions asked weres- Can these v a r a c t o r diodes s t i l l produce power g a i n with low n o i s e at low f r e q u e n c i e s ? I f so, how does the low-frequency noise ( i n c l u d i n g d-c d r i f t ) com-pare wi t h t h a t of other d-c a m p l i f i e r s ? These are i n t e r e s t i n g problems to study, p a r t i c u l a r l y as there i s need f o r good low-frequency a m p l i f i e r s . Low-level d-c s i g n a l s or s i g n a l s of low frequency are f r e q u e n t l y encountered i n many f i e l d s of s c i e n c e and engineering, f o r i n s t a n c e , i n i n s t r u -mentation, i n d u s t r i a l c o n t r o l equipment, d i g i t a l and analogue com-puters, b i o l o g y , medicine and n u c l e o n i c s . The l i m i t a t i o n s t h a t determine the minimum d e t e c t a b l e s i g n a l are ( l ) random noise and (2) d r i f t of the quiescent o p e r a t i n g p o i n t as a r e s u l t of temperature e f f e c t s . D r i f t i s a s e r i o u s disadvantage i n d-c a m p l i f i e r s s i n c e i t cannot be d i s t i n g u i s h e d from an-actual s i g n a l . Most vacuum-tube and t r a n s i s t o r a m p l i f i e r s designed to handle such small s i g n a l s have poor noise f i g u r e s and are s u b j e c t to d r i f t . The low input impedance of t r a n s i s t o r s makes matching to a high impedance source d i f f i c u l t . Mechanical choppers have a short l i f e - t i m e while the performance of t r a n s i s t o r choppers i s impaired by the switching t r a n s i e n t s . I t i s p e r t i n e n t here to mention a few e x c e p t i o n a l amplifiers,, namely the cascode and the f i e l d e f f e c t t r a n s i s t o r , which are f r e e from most of these t r o u b l e s . Both have low noise f i g u r e s and high input impedances. But the f i e l d e f f e c t t r a n s i s t o r i s s t i l l i n an e a r l y stage of development and i s t h e r e f o r e expensive. 3 The purpose of t h i s t h e s i s i s f i r s t 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 u s i n g o r d i n a r y s i l i c o n r e c t i f i e r s as v a r a c t o r diodes at low f r e q u e n c i e s , and then t o analyze, d e s i g n and b u i l d a two-diode b r i d g e a m p l i f i e r f o r small d-c and low-frequency s i g n a l s and study i t s g a i n , bandwidth, as w e l l as the l i m i t a t i o n s caused by n o i s e , d r i f t and other p r a c t i c a l c o n s i d e r a t i o n s . A band-pass high-frequency a m p l i f i e r and a p h a s e - s e n s i t i v e demodulator were found t o be e s s e n t i a l i n the experimental v e r i f i c a t i o n o f the p r o p e r t i e s of the b r i d g e . Other sporadic work performed on low-frequency parametric a m p l i f i c a t i o n should be mentioned. In J u l y 1959, B e l l Telephone L a b o r a t o r i e s , i n an i n t e r n a l memorandum, r e p o r t e d some work done 5 on a v a r a c t o r diode as a d-c a m p l i f i e r . The l i m i t a t i o n imposed by the in p u t c i r c u i t was not d e a l t w i t h i n t h a t r e p o r t and w i l l be shown t o be an important f a c t o r i n t h i s t h e s i s * In December 1961, an a r . t i c l e was p u b l i s h e d on a s i n g l e - d i o d e a m p l i f i e r w i t h 6 a frequency response of 3 c.p.s. to 200 Kc . 4 2. VARACTOR DIODES 2.1 S e l e c t i o n of Diodes The main c h a r a c t e r i s t i c s of v a r a c t o r diodes are ( l ) a h i g h Q, i . e . , a high leakage r e s i s t a n c e , (2) the s i z e of capacitance under reverse b i a s i n r e l a t i o n to the frequency range of i n t e r e s t s (3) the degree of n o n l i n e a r i t y of the capacitance as a f u n c t i o n of b i a s v o l t a g e . For the design of the parametric a m p l i f i e r b r i d g e , a matched p a i r of v a r a c t o r diodes i s d e s i r a b l e . These are a v a i l a b l e commer-c i a l l y but they are expensive and normally of low capacitance. I f o r d i n a r y s i l i c o n r e c t i f i e r diodes could be found to have the above-mentioned c h a r a c t e r i s t i c s they could be used i n s t e a d f o r economical reasons. A batch of ten s i l i c o n diodes, IN2484, was i n v e s t i g a t e d and t h e i r c h a r a c t e r i s t i c s measured i n the f o l l o w i n g manner:-( i ) With the arrangement shown i n F i g . 2.1, a known r e v e r s e -b i a s v o l t a g e i s a p p l i e d to the diode placed i n s e r i e s w i t h a 1 Megohm r e s i s t o r R^. The p o t e n t i a l at Q i s then read on the h i g h l y s e n s i t i v e ( 2 0 0 uv/ c m ) o s c i l l o s c o p e and t h i s v o l t a g e d i v i d e d by the p a r a l l e l combination of R^ and the 10 Megohm probe input r e s i s t a n c e gives the leakage c u r r e n t f l o w i n g through the diode. Proper s h i e l d i n g and grounding are e s s e n t i a l to ensure r e l i a b i l i t y of measure-ments. Under r e v e r s e - b i a s v o l t a g e s of 0 to 9 v o l t s , the —8 leakage c u r r e n t s of the diodes are of the order of 10~ amperes and t h e i r dynamic shunt r e s i s t a n c e s l i e i n the 8 9 range 2x10 to 5x10 ohms ( F i g . 2.3). 5 r X R x = 2 K R 2 = 1 M Q R, " X " T e k t r o n i x -° 502 H i g h S e n s i t i v i t y "° O s c i l l o s c o p e D = I N 2 4 8 4 P = 1 0 M , 11 \x\it P r o b e P i g . 2 . 1 M e a s u r e m e n t o f D i o d e L e a k a g e C u r r e n t O s c i l l a t o r l O K c - l M c R, 1_ R , T e k t r o n i x _o 515A I n p u t I m p e d a n c e I 'M, 4 0 u u f R x = 1M R 2 = 2 K C1 = l O O u u f C 2 = 14 - 6 4 u u f D = 1 N 2 4 8 4 P i g . 2 . 2 M e a s u r e m e n t o f C a p a c i t a n c e o f R e v e r s e - B i a s e d D i o d e s 6 <^ V ( v o l t s ) Leakage C u r r e n t vs V o l t a g e Curve f o r S i x Diodes 7 ( i i ) To m e a s u r e t h e d i o d e c a p a c i t a n c e a n d f i n d how i t v a r i e s w i t h r e v e r s e - b i a s v o l t a g e , t h e c i r c u i t a r r a n g e m e n t o f F i g . 2 . 2 i s u s e d . ( a ) W i t h d i o d e D i n p l a c e a n d a t m i n i m u m , t h e d - c b i a s i s a d j u s t e d t o a s u i t a b l e v a l u e . ( b ) A p p l y o s c i l l a t o r v o l t a g e a n d a d j u s t u n t i l o s c i l l o s c o p e s h o w s v p e a k - t o - p e a k s i g n a l o f e x a c t l y k n o w n , v a l u e . ( c ) Remove d i o d e a n d i n c r e a s e Cg u n t i l o s c i l l o s c o p e s h o w s e x a c t l y t h e same s i z e . t r a c e . ( d ) The i n c r e a s e i n i n s t e p ( c ) y i e l d s t h e d i o d e c a p a c i t a n c e f o r t h e k n o w n d - c b i a s . The e x p e r i m e n t was c a r r i e d o u t f o r a - c s i g n a l s r a n g i n g f r o m 0 . 1 t o 2 v o l t s w h i l e t h e d - c b i a s was k e p t f i x e d a t - 2 v o l t s . The v a l u e o f c a p a c i t a n c e m e a s u r e d r e m a i n e d c o n s t a n t . The e x p e r i m e n t was r e p e a t e d a s t h e f r e q u e n c y was v a r i e d f r o m 10 Kc t o 1 Mc a n d t h e c a p a c i t a n c e f o r a c e r t a i n d - c b i a s w a s f o u n d t o be i n d e p e n d e n t o f f r e -q u e n c y . A p a i r o f t h e d i o d e s t h a t m a t c h c a p a c i t i v e l y a s c l o s e l y a s p o s s i b l e w e r e s e l e c t e d ( F i g . 2 . 4 ) . The d i o d e c a p a c i t a n c e v a r i e s f r o m 120 t o 30 \x\xf a s t h e b i a s v o l t a g e i s a l t e r e d f r o m 0 t o - 9 v o l t s . T h i s i s a n a d e q u a t e v a r i a t i o n f o r p a r a m e t r i c a m p l i f i c a t i o n . W i t h s u c h a h i g h l e a k a g e r e s i s t a n c e i n s h u n t w i t h t h i s s i z e o f c a p a c i t a n c e , t h e b a c k - b i a s e d d i o d e p r e -s e n t s a h i g h i m p e d a n c e t o s i g n a l s o f l o w f r e q u e n c i e s f r o m 0 t o 10 Kc a n d o p e r a t e s a s a g o o d v o l t a g e - s e n s i t i v e d e v i c e f o r t h i s r a n g e o f f r e q u e n c i e s . 2 . 2 T h e o r y Now t h e c a p a c i t a n c e C o f a s i l i c o n d i o d e u n d e r r e v e r s e - b i a s e d 2 c o n d i t i o n s h a s b e e n d e r i v e d t h e o r e t i c a l l y a n d v e r i f i e d e x p e r i m e n t a l l y . 8 I t i s given by the formula:-C = C + C. = C + K(V, + V ) ~ n P 0 P h where C = p a r a s i t i c capacitance of case ( t y p i c a l l y , 0 . 3 - 3 |ip,f) P C. = j u n c t i o n capacitance of diode "V"k = b a r r i e r height at zero b i a s , or contact p o t e n t i a l ( 0 . 3 - 0 . 7 v f o r s i l i c o n ) V = d-c r e v e r s e - b i a s v o l t a g e K = a constant depending on the area of the j u n c t i o n and c o n c e n t r a t i o n of doping, n = exponent depending on the type of j u n c t i o n For a l i n e a r l y - g r a d e d j u n c t i o n , n = i For an abrupt or step j u n c t i o n , n = •— In g e n e r a l , the diode j u n c t i o n has a charge d i s t r i b u t i o n which l i e s between those two l i m i t s . T h erefore | < n < | • Next i t i s important to f i n d the optimum oper a t i n g c o n d i t i o n s f o r the diode when a small s i g n a l v o l t a g e i s a p p l i e d . As shown l a t e r i n S e c t i o n 3 . 2 , i t i s the q u a n t i t y - ^ ^ t h a t matters. We s h a l l c a l l i t a. I f C = K V~ n, a = n and i s independent of the b i a s v o l t a g e V. I f C = K(V^ + V ) ~ , a = y +y, which tends to n as V tends to h i n f i n i t y . Therefore we should use a very l a r g e b i a s V. I f C = C p + K(V h + V ) ~ n , there i s an optimum value of V f o r maximum a and t h i s a w i l l be somewhat l e s s than n. Since a =-^ r ^ = - ^ ( i p g V) ' ^ c a n ^ e plowed- a g a i n s t 1 0 l o g V, a n d t h e p o i n t w h e r e t h e s l o p e ( - a ) i s a maximum w i l l y i e l d t h e o p t i m u m o p e r a t i n g b i a s V ( P i g . 2 . 5 ) . P r o m t h e g r a p h o b t a i n e d f o r one o f t h e m a t c h e d p a i r o f d i o d e s , t h e o p t i m u m b i a s V l i e s b e t w e e n 1 a n d 2 . 5 v o l t s a n d i s f o r t u n a t e l y n o t c r i t i c a l . The m a x -imum v a l u e o f a i s 0 . 3 7 5 . The v a l u e s o f a f o r d i f f e r e n t r e v e r s e - b i a s v o l t a g e s w e r e c o m p u t e d f r o m t h e g r a p h o f F i g . 2 . 5 a n d t h e n p l o t t e d . The v a r i a t i o n o f a w i t h V c a n b e s e e n more r e a d i l y o n t h i s g r a p h ( F i g . 2 . 6 ) . 2 . 3 E f f e c t i v e S e r i e s R e s i s t a n c e o f D i o d e I n S e c t i o n 2 . 1 , a p a i r o f r e c t i f i e r d i o d e s 1 N 2 4 8 4 w e r e s e l e c t e d w i t h m a t c h i n g c a p a c i t a n c e c h a r a c t e r i s t i c s a n d a h i g h l e a k a g e r e s i s t a n c e . S i n c e t h e d i o d e s a r e t o b e u s e d i n a b r i d g e a r r a n g e -m e n t , t h e i r c a p a c i t a n c e s s h o u l d b e made e q u a l , a n d s o s h o u l d be t h e i r e f f e c t i v e s e r i e s r e s i s t a n c e s i n o r d e r t o a c h i e v e b a l a n c e o f t h e b r i d g e ( F i g . 4 . 2 ) . T h i s e f f e c t i v e s e r i e s r e s i s t a n c e i n c l u d e s t h e s e r i e s r e s i s t a n c e due t o t h e b u l k o f t h e s e m i c o n d u c t o r m a t e r i a l a n d t h e c o n t a c t r e s i s t a n c e o f t h e j u n c t i o n , a n d t h e s h u n t r e s i s t a n c e d u e t o t h e l e a k a g e c u r r e n t . I t i s o f i n t e r e s t t o f i n d t h e v a l u e s o f t h e e f f e c t i v e s e r i e s r e s i s t a n c e s a s t h e y a r e s i g n i f i c a n t f o r t h e g a i n c a l c u l a t i o n s i n S e c t i o n 3 . 3 . 2 . T h e p a r a m e t r i c a m p l i f i e r b r i d g e , a s d e p i c t e d i n F i g . 4 . 2 was u s e d t o d e t e r m i n e t h e e f f e c t i v e s e r i e s r e s i s t a n c e o f e a c h d i o d e . One d i o d e a t a t i m e i s r e p l a c e d b y a v a r i a b l e c a p a c i t o r i n s e r i e s w i t h a r e s i s t a n c e a n d t h e s e t w o c o m p o n e n t s a r e a d j u s t e d u n t i l b a l -a n c e i s o b t a i n e d . T h e e f f e c t i v e s e r i e s r e s i s t a n c e f o r t h e t w o d i o d e s was f o u n d t o b e 7 5 a n d 1 0 0 ohms r e s p e c t i v e l y . 0 1 2 3 4 5 6 7 V ( v o l t s ) P i g . 2.6 V a r i a t i o n of q w i t h V 13 3 THEORY OP P A R A M E T R I C A M P L I F I E R B R I D G E 3 . 1 The B r i d g e The m o s t s a t i s f a c t o r y m e t h o d f o u n d o f u s i n g v a r a c t o r d i o d e s a s l o w - f r e q u e n c y a m p l i f i e r s i s i n a t w o - d i o d e b r i d g e a s s h o w n i n F i g u r e 3 . 1 . The b i a s r e s i s t o r s R^ a n d Rg a r e made l a r g e c o m p a r e d w i t h t h e r e a c t a n c e s o f t h e d i o d e c a p a c i t a n c e s . C ^ a n d C^g a r e b l o c k i n g c a p a c i t o r s c h o s e n t o be much l a r g e r t h a n t h e d i o d e c a p a c i t a n c e s . , W i t h i d e n t i c a l d i o d e s t h e b r i d g e i s b a l a n c e d a n d no v o l t a g e a p p e a r s b e t w e e n p o i n t X a n d g r o u n d . Now w h e n a s m a l l p o s i t i v e d - c s i g n a l v o l t a g e i s a p p l i e d t o t h e d i o d e s a t X , t h e c a p a c i t a n c e o f one d i o d e D^ w i l l b e i n c r e a s e d a n d t h a t o f Dg d e c r e a s e d , s o t h a t a f r a c t i o n o f t h e pump v o l t a g e p r o p o r t i o n a l t o t h e s i g n a l v o l t a g e i s o b t a i n e d a t X . I f a n e g a t i v e s m a l l s i g n a l v o l t a g e i s a p p l i e d ^ t h e h - f o u t p u t v o l t a g e i s s t i l l p r o p o r t i o n a l t o t h e s i g n a l v o l t a g e b u t i s 1 8 0 ° o u t o f p h a s e w i t h t h e pump v o l t a g e . I t i s p o s s i b l e t o b u i l d a o n e - d i o d e b r i d g e , b u t a t w o - d i o d e b r i d g e i s ; m o r e a d v a n t a g e o u s ; i n "tto© "f^ il^owi-ng'-way's. A o n e - d i o d e b r i d g e i s v e r y s e n s i t i v e t o t h e pump v o l t a g e b e c a u s e d r i f t s i n c a p a c i t a n c e o r i n l e a k a g e c u r r e n t o f a s i n g l e d i o d e c a n n o t b e b a l a n c e d o u t b y a f i x e d c a p a c i t o r . W i t h a m a t c h e d p a i r o f d i o d e s , d r i f t s i n c h a r a c t e r i s t i c s , e . g . o f f z e r o , t e n d t o c a n c e l . A l s o t h e o u t p u t i s d o u b l e d w h e n u s i n g t w o d i o d e s i n s t e a d o f o n e . 3 . 2 C i r c u i t A n a l y s i s o f B r i d g e A s s u m e C-^  a n d Cg t o b e t h e s m a l l s i g n a l c a p a c i t a n c e s o f d i o d e s D1 a n d D g ( F i g . 3 . 2 ) . 15 P i g . 3.2 E f f e c t i v e A-C B r i d g e C i r c u i t a . A V . ^ C 1 + C 2 = 2C tt F i g « 3'3 Thevenin's E q u i v a l e n t C i r c u i t 16 L e t z^ = 1/JOJC^ a n d . z 2 as l / j w C 2 T h e n 2 . V « ( Z ; L + z 2 ) i + Z ] L i Q 2 e0 = ( z 2 * z l > 1 ' z l *<> S o l v i n g f o r i a n d i y i e l d s e_ + V ^ i = _2 £ Z 2 , z 2 - z l 21 + z 2 o z^ z 2 p z^ z 2 o T h e o p e n - c i r c u i t v o l t a g e i s o b t a i n e d b y p u t t i n g i = o z 2 - Z l v e = • . V oc zi + z 2 P S u b s t i t u t i n g e Q = o g i v e s t h e s h o r t - c i r c u i t c u r r e n t Z 2 - Z l v 2_ — _ _ _ _ _ _ _ - y SC Z^ Zg P e o c The o u t p u t i m p e d a n c e ZQ = —— s c z l • Z P 1 T h e r e f o r e z = ^ o - z-_ + z 2 J _ J _ z l Z 2 S u b s t i t u t e z , = —=—i , z 0 = '1 ~ jcoC^ ' "2 ~ JOJC 2 (c 1 - c 2) e o c = {C1 + C 2 ) ' T p a n d Z o = j w (C- \ •<*-_) The n e x t s t e p i s t o c o n s i d e r t h e f i r s t o r d e r e f f e c t d u e t o a s m a l l s i g n a l v o l t a g e A V . I f t h e a s s u m p t i o n i s made t h a t t h e s i g n a l v o l t a g e ( A V ) i s s m a l l c o m p a r e d w i t h t h e d - c r e v e r s e - b i a s 17 v o l t a g e across each diode, then i t i s only necessary to conside r the way i n which the small s i g n a l capacitance v a r i e s l i n e a r l y w ith the a p p l i e d s i g n a l v o l t a g e at the ope r a t i n g v o l t a g e V. I f C = capacitance, at the b i a s v o l t a g e V AC = change i n capacitance due to an input s i g n a l AV T h e n C± = C - AC C 2 = C + AC AC<<C, si n c e A V « V . The o p e n - c i r c u i t v o l t a g e becomes oc 2 A C v _p_ ac = - - 2 C " * V p = " C * W •AV v ac C " 3V . AV . = a . AV . where a = - 77 V dC c " dT = d ( l o g C d ( l o g V The Thevenin's E q u i v a l e n t c i r c u i t f o r the brid g e i s now ob-t a i n e d i n a simple form ( P i g . 3.3) The pump v o l t a g e can be i n c r e a s e d without d r i v i n g the diodes i n t o the conducting r e g i o n u n t i l i t s peak value reaches approx-imately the r e v e r s e — b i a s v o l t a g e . I f V i s then the r.m.s. value of the pump v o l t a g e , and V the d-c b i a s v o l t a g e , J2 . V = V v P The optimum o p e n - c i r c u i t v o l t a g e becomes oc max , a . AV 18 3.3 Gai n and Bandwidth The b r i d g e a c t s as a b u f f e r stage between the s i g n a l source and the f o l l o w i n g h i g h - f r e q u e n c y a m p l i f i e r . I t produces no v o l -tage g a i n , but i t p r o v i d e s an i n s e r t i o n power g a i n which i s im-p o r t a n t when c o n s i d e r i n g the n o i s e f i g u r e of the whole a m p l i f i e r . T h e r e f o r e , the g a i n which i s m a i n l y of i n t e r e s t here i s the t r a n s -ducer power g a i n G. T h i s i s d e f i n e d a s the r a t i o of W^ , the power d e l i v e r e d t o the l o a d , t o ¥ g , t h e power a v a i l a b l e from the s o u r c e . Thus i f v o l t s i s the output v o l t a g e a c r o s s a l o a d R^ r e s u l t i n g from the a p p l i c a t i o n of a s m a l l s i g n a l whose amplitude i s V g v o l t s and source r e s i s t a n c e R , t h e n s r A \ ! VL I " A W* = K I V « 1 S ' / s L o o k i n g i n t o the b r i d g e , the source sees the diode c a p a c i -t a n c e s i n p a r a l l e l w i t h each o t h e r and w i t h the c o u p l i n g c a p a c i t o r C c. ( f i g . 3.1). T h e r e f o r e the time c o n s t a n t due t o the source r e s i s t a n c e R and the t o t a l c a p a c i t a n c e (2C + C ) l i m i t s the upper s c 3 - db f r e q u e n c y fg» whence f 1  2 ~ 2TTR (2C H- C ) s v ' c' I n o r d e r t o ensure r e a s o n a b l e g a i n , C w i l l be found t o be c comparable t o 2C. 3.3.1 G a i n of. an I d e a l B r i d g e C i r c u i t L e t us assume t h a t we have a b r i d g e c i r c u i t u s i n g d i o d e s w i t h g i v e n C and g i v e n a; assume a l s o t h a t t h i s b r i d g e i s t o be pumped a t a g i v e n f r e q u e n c y f and used t o a m p l i f y s m a l l s i g n a l s w i t h P f f r e q u e n c y components from 0 t o f 0 . The r a t i o -J^- = (3 i s assumed t o be * .2 19 l a r g e . The s i g n a l s are assumed t o o r i g i n a t e i n a source V w i t h s i n t e r n a l impedance R . R i s regarded as a d j u s t a b l e . (An a l -s s t e r n a t i v e e q u i v a l e n t problem would t a k e R g as f i x e d and make C an a d j u s t a b l e p a r a m e t e r ) . The h-f s i g n a l s from the b r i d g e a r e cou p l e d t h r o u g h c a p a c i t o r C c i n t o ,an a m p l i f i e r w i t h i n p u t impedance R-^ . R-^  i s a t our d i s -p o s a l , s i n c e we c o u l d use i d e a l t r a n s f o r m e r s t o change the e f f e c -t i v e a m p l i f i e r impedance t o any d e s i r e d v a l u e . I n order t o t r a n s -f e r most h-f power i n t o R-^ , an i n d u c t o r L s h o u l d be p l a c e d i n s e r i e s w i t h C . c The problem then i s t h i s : G iven C, a, f , f g j f i n d the optimum v a l u e s of R , R T, C and L i n order t o o b t a i n maximum * s s L ' c t r a n s d u c e r g a i n G f o r the b r i d g e , i . e . , t o maximize the r a t i o of power d e l i v e r e d t o the a m p l i f i e r t o the power a v a i l a b l e from the s i g n a l s o u r c e . At f r e q u e n c y f , the e q u i v a l e n t c i r c u i t i s t h a t of F i g . 3.4 ( a ) . For maximum g a i n , we want R as l a r g e as p o s s i b l e , s i n c e s t h i s makes V l a r g e r ( f o r a g i v e n power absorbed from the s o u r c e ) and s i n c e i t a l s o shunts l e s s of the pump fr e q u e n c y s i g n a l t o ground. The l a r g e s t p e r m i s s i b l e v a l u e i s s e t by bandwidth con-s i d e r a t i o n s . At v e r y low f r e q u e n c i e s , the i n p u t impedance seen by the s i g n a l source i s v i r t u a l l y i n f i n i t e , a l l of V appears a t the b r i d g e . At h i g h e r f r e q u e n c i e s , the impedance i s t h a t of (2C + C ) and t h i s w i l l b e g i n t o reduce the v o l t a g e a p p l i e d t o the bridgeo To o b t a i n a bandwidth f Q > R must be no g r e a t e r t h a n 1/2* f Q (2C + C ). We t h e r e f o r e put R g = 2 7 t f ( 2 C T T T T 1 I T w h e r e * = C c / 2 C > P i g . 3 . 4 E q u i v a l e n t C i r c u i t s a t f 21 and note t h a t a t f r e q u e n c y f , R w i l l be a v e r y l a r g e impedance, p s T — — t i m e s t h a t of 2C . I t i s t h e n c o n v e n i e n t t o change the e q u i v a l e n t c i r c u i t t o the form shown i n F i g . 3.4 ( b ) , w i t h t h e h e l p of Thevenin's theorem. S i n c e R g i s l a r g e , the new v o l t a g e source w i l l be v i r t u a l l y the same as the o l d one, and C w i l l be almost equal t o 2 C . The s e r i e s r e s i s t o r r w i l l be a s m a l l one, s w i t h impedance i — ~ i i t i m es t h a t of C , i . e . , • p s 1 + u 's ~ p . 2n f . 2C p From the r e v i s e d e q u i v a l e n t c i r c u i t i t i s e a s i l y seen t h a t f o r maximum power t r a n s f e r i n t o the l o a d (1) L s h o u l d be chosen so as t o tune out the s e r i e s com-b i n a t i o n of C and C , i . e . , L = — 1 + l / p , — S 2 4 i T f . 2C P (2) R^ s h o u l d be made equal t o r g . Me t h e n get f o r t h e t r a n s d u c e r power g a i n G:-r l \ l 2 / \ G = - 9 V r /4R I s I ' s a V > 2 s 12 J2 V 2 /4R s s « R s  2 r s 2 Q 2 2(1 + u ) 2 22 I n t h i s l a s t e x p r e s s i o n , a and 8 are g i v e n , but p i s s t i l l t o be determined, and s h p u l d c l e a r l y be a s s m a l l as p o s s i b l e . However, i f C i s made s m a l l , L would have t o be l a r g e , and the c r e s u l t i n g tuned c i r c u i t would have a \ery h i g h Q and would no-t-a l l o w us t o pass t h e s i d e bands f + formed by a m p l i t u d e -modulated s i g n a l coming from the b r i d g e . Thus i n order t o keep the bandwidth f i x e d at f ^ , t h e Q cannot be l a r g e r t h a n f ^ / 2 f 2 = [3/2. The s m a l l e s t p o s s i b l e u i s the n g i v e n by 2 1 (3 = Q T s + R L (_L + J_) 2u f V2C ^ C p c 8 n f r C -P. d + i ) 1 + u 1 T h e r e f o r e , p = 1 S u b s t i t u t i n g t h i s i n t o t he e x p r e s s i o n f o r the t r a n s d u c e r g a i n , we o b t a i n t h e f o l l o w i n g r e s u l t : -The l a r g e s t p o s s i b l e t r a n s d u c e r g a i n i s g i v e n by G = | a 2 B 2 = 36.5 (15.6 db) and t h i s i s o b t a i n e d when the a d j u s t a b l e parameters a re C = 2C = 120 p f c r R - I/811 f Q C = 66.5 K L = l / 4 n 2 f 2 C = 2.05 mh RT = f Q / 2 T t ' f 2 C = 128 ohms. L 2' p 2 3 ( t h e n u m e r i c a l v a l u e s r e s u l t f r o m s u b s t i t u t i n g f o r t h e f i x e d p a r a m e t e r s t h e v a l u e s a p p r o p r i a t e f o r o u r d i o d e s : - C = 60 p f , a = 0 . 3 7 5 , f = 4 5 5 K c , ± 2 = 10 K c ) . 3 . 3 . 2 G a i n o f a P r a c t i c a l C i r c u i t I n t h e l a s t s e c t i o n we h a v e c o m p u t e d t h e t r a n s d u c e r g a i n i d e a l l y p o s s i b l e , a n d t h e p a r a m e t e r s r e q u i r e d t o a c h i e v e t h i s . I n a p r a c t i c a l c i r c u i t , t h e r e w i l l b e some r e d u c t i o n i n t h e e x p e c t e d g a i n a s a r e s u l t o f t h e f o l l o w i n g t h r e e f a c t o r s : -( 1 ) P r a c t i c a l c o m p o n e n t v a l u e s may d i f f e r s l i g h t l y f r o m o p t i m u m v a l u e s . T h u s i n o u r c a s e R = 7 2 ! K , C = 1 4 0 p f s c ( 2 ) The d i o d e s h a v e some e f f e c t i v e s e r i e s r e s i s t a n c e , a n d i n t h e p r o c e s s o f b a l a n c i n g t h e b r i d g e , a n a d d i t i o n a l s m a l l r e s i s t o r i s p l a c e d i n s e r i e s w i t h one o f t h e m i n o r d e r t o b r i n g b o t h s e r i e s r e s i s t a n c e s t o t h e s a m e v a l u e , r = 1 0 0 o h m s . ( 3 ) T h e i n d u c t o r L i n e v i t a b l y h a s s e r i e s r e s i s t a n c e R-^. W i t h a Q o f 6 0 , R^ = 98 ohms a n d t h e r e f o r e c a n n o t be n e g l e c t e d . A s a r e s u l t o f t h e s e f a c t o r s , t h e e q u i v a l e n t c i r c u i t s a r e a s s h o w n i n F i g s . 3 . 5 ( a ) a n d ( b ) , w h e r e e ' e„ = a V / J2 oc oc s " C s 2C = 120 p f R ' = £ + r = 50 + 118 = 168 ohms d s T h e n R^ + R ' = 2 6 6 o h m s . The t r a n s i s t o r i n p u t i m p e d a n c e ( R ^ ) o f 2 7 0 ohms was t h e r e f o r e a g o o d m a t c h . L i s s t i l l c h o s e n t o t u n e o u t t h e s e r i e s c a p a c i t o r s , a n d t h e • a V g \ r e s u l t a n t v o l t a g e a c r o s s t h e l o a d i s now = « — . ^ — - j — ^ — - R t V L 1 a n d h e n c e t h e t r a n s d u c e r p o w e r g a i n i s 24 r r . / 2 - A A V 2C tt O C R I — -C L _ R , -j ^ —^TJ^F^—wv R-( a ) R ' oc c' C L R, (b) V T > R L L r = 1 0 0 ohms C 6 0 u u f R = s 72 K C = 1 2 0 u u f R l -98 ohms C c = 1 4 0 u u f R' = 164 ohms L = 2 . 1 mh \- 2 7 0 ohms f P 4 5 5 K c P i g , 3 , 5 E q u i v a l e n t C i r c u i t s a t f I n c l u d i n g P r a c t i c a l L i m i t a t i o n s 2 5 G </\ G = —5 V V 4 R s s o 2 a R R T _ s L  ( R L + R± + R « ) 2 = 19 ( o r 1 2 . 8 d b ) T h i s c o m p a r e s f a v o u r a b l y w i t h t h e 1 5 . 6 db i d e a l l y a v a i l a b l e , w i t h t h e o p t i m u m c o m p o n e n t v a l u e s , a n d v e r y w e l l w i t h t h e e x p e r i -m e n t a l l y m e a s u r e d g a i n o f 1 2 . 6 d b . 3 . 4 N o i s e I n o u r m e a s u r e m e n t s we h a v e d i s t i n g u i s h e d b . e t w e e n ( 1 ) s l o w d r i f t s i n t h e o u t p u t a n d ( 2 ) r a p i d f l u c t u a t i o n s h a v i n g t h e a p p e a r -a n c e o f r a n d o m n o i s e . The s l o w d r i f t s r e s u l t f r o m c h a n g e s i n t h e d i o d e c h a r a c t e r -i s t i c s o v e r many m i n u t e s o r h o u r s , : a n d a r e n o r m a l l y l a r g e i n a m p -l i t u d e c o m p a r e d w i t h t h e r a p i d f l u c t u a t i o n s . No t h e o r e t i c a l e s t i m a t e o f t h e m a g n i t u d e o f t h e s e d r i f t s was a t t e m p t e d . I n t h i s s e c t i o n o n l y t h e r a p i d f l u c t u a t i o n s a r e c o n s i d e r e d . E v e n i f t h e p a r a m e t r i c a m p l i f i e r w e r e p e r f e c t l y n o i s e l e s s a n d o f v e r y h i g h g a i n , some f l u c t u a t i o n s i n o u t p u t w o u l d be o b s e r v e d . T h e s e a r e t h e r e s u l t o f t h e r m a l f l u c t u a t i o n s i n t h e r e s i s t o r R s u s e d t o s i m u l a t e a s o u r c e w i t h f i n i t e i n t e r n a l r e s i s t a n c e . T h e n o i s e p o w e r a v a i l a b l e f r o m s u c h a r e s i s t o r i s p r o p o r -t i o n a l t o b a n d w i d t h B . I n o u r c a s e , n o i s e m e a s u r e m e n t s w e r e made w i t h B = 5 K c . A t t e m p e r a t u r e T = 2 9 0 ° K t h e a v a i l a b l e n o i s e p o w e r i s t h e n , N = :kTB =s 1 . 3 8 x 1 Q " 2 3 x 290 x , 5 x 1 0 ? = 1 . 9 4 x lG'17 , w a t t , 26 w h e r e k i s t h e B o l t z m a n n c o n s t a n t ( i n j o u l e s / ° K ) T h e p o w e r d e l i v e r e d t o t h e f i r s t t r a n s i s t o r i s t h i s a m o u n t m u l t i p l i e d b y t h e t r a n s d u c e r p o w e r g a i n G o f t h e b r i d g e . The n o i s e p o w e r c o m i n g f r o m t h e o u t p u t o f t h e c o m p l e t e a m p l i f i e r w i l l be g r e a t e r b y t h e g a i n o f t h e s u c c e e d i n g s t a g e s . I n p r a c t i c e we o b s e r v e a n o u t p u t n o i s e p o w e r l a r g e r t h a n t h e a m o u n t c o m p u t e d w i t h t h e a b o v e f o r m u l a ; t h e r a t i o o f t h i s o b s e r v e d n o i s e t o t h a t c o m p u t e d a s c o m i n g f r o m R may b e d e f i n e d as t h e s n o i s e f i g u r e F o f t h e w h o l e i n s t r u m e n t . A l t e r n a t i v e l y we c a n c o n s i d e r n o i s e v o l t a g e s a t t h e i n p u t o f t h e d i o d e b r i d g e . We c a n f i r s t c o m p u t e t h e t h e r m a l n o i s e v o l t a g e s d u e t o R as a = »/4kTBR = 2 . 3 6 LXV. The o b s e r v e d ° s n v s ^ o u t p u t n o i s e v o l t a g e o f t h e c o m p l e t e a m p l i f i e r c a n t h e n be d i v -i d e d b y t h e o v e r a l l v o l t a g e g a i n t o g i v e t h e n o i s e v o l t a g e e^ e f f e c t i v e l y a t t h e i n p u t . F i s t h e n c o m p u t e d a s p - < ° X > 2 A v e r y s m a l l p a r t o f t h e e x c e s s n o i s e c a n be a t t r i b u t e d t o t h e f i n i t e g a i n o f t h e b r i d g e n e t w o r k a n d t h e n o i s e o f t h e t r a n s -2 i s t o r a m p l i f i e r t o f o l l o w . I t i s r e a d i l y s h o w n t h a t F = F± + ( F 2 - l ) / g w h e r e F ^ i s t h e n o i s e f i g u r e o f t h e b r i d g e n e t w o r k b y i t s e l f , F g i s t h e n o i s e f i g n t r e tit t h e - c o m p l e t e s u c c e e d i n g t r a n -s ' s ^ b r a m p l i f i e r } a n d g i i t h a . a y a i l a b l © g a i n o f t h e b r i d g e . I n o u r c a s e , g = G = 1 8 . E v e n w i t h a p e r f e c t b r i d g e ( F , = l ) , 27 F would be greater than 1. I f , = 10 we would measure F = 1 H- ^ 1 . 5 (or 1.76 db) Experimental evidence suggests that i n our case Fg was ac tua l ly less than 10. Yet measured noise f igure gave F = 3 db. This must be a t tr ibuted to noise generated i n the parametric ampli f ier i t s e l f making F^ greater than uni ty . Noise i n such amplif iers has been studied extensively at microwave frequencies 2 7 8 ' ' , but l i t t l e i s known about the i r low-frequency behaviour. The possible sources of noise are discussed i n Section 5.7. 28 4 , DESIGN AND CONSTRUCTION OP CIRCUIT 4 . 1 Requirements To investigate the properties of the parametric amplifier bridge and v e r i f y the theoreti c a l results obtained i n Chapter 3 , the following building blocks are neededs- (see Pig. 4 . 1 ) „ (i) A pump source i n the form of a sine-wave h-f generator, (Radiometer MSllh) ( i i ) A balanced bridge, as already outlined i n Section 3 . 1 ( i i i ) A band pass h-f amplifier to pass both side-bands fP - V (iv) A, phase-shifting network to compensate for any phase difference that may exist between the band-pass h-f amp-l i f i e r output and the reference c a r r i e r so as to demod-ulate e f f e c t i v e l y , (v) A phase-sensitive demodulator to detect the bridge mod-ulated output after amplification by the h-f amplifier, (yi) A d-c difference amplifier including a low-pass f i l t e r to amplify the demodulated output, ( v i i ) An audio-freq.uency amplifier followed by a thermal m i l l i -ammeter i n order to measure r.m.s. noise current. Parts (i) to (vi) cons'.ti.tut-e a complete d-c to d-c amplifier with the parametric amplifier as a low-noise f i r s t stage. The choice of a pump frequency i s arbitrary except that i t has to be much greater than the upper 3-db frequency to ensure reasonable gain for the bridge. The upper 3-db frequency i s chosen to be abdut 8 Kc. It i s then convenient to select 4 5 5 Kc as the pump frequency because components such as i . f , transformers,, c o i l s and.tran-sistors operating at that frequency are,readily available. Apart from the bridge and the phase-^sensitive demodulator, the other Pump S o u r c e P h a s e -S h i f t e r T h e r m a l M i l l i a m m e t e r S i g n a l I n P a r a m e t r i c A m p l i f i e r B r i d g e B a n d - P a s s H - F A m p l i f i e r P h a s e -S e n s i t i v e D e m o d u l a t o r D i f f e r e n c e A m p l i f i e r + L o w - P a s s F i l t e r A u d i o -F r e q u e n c y A m p l i f i e r A 1 2 = 0 . 1 3 A 2 3 = 1 . 1 x 1 0 d - c " 3 4 39 a - c 21 ) ^ P a s s e d 45 d - c 7 . 2 a - c ^ *g I U n b y p a s s e d F i g . 4 . 1 B l o c k D i a g r a m o f t h e P a r a m e t r i c A m p l i f i e r 30 b u i l d i n g b l o c k s a r e made o f s t a n d a r d c i r c u i t s . B u t a s h o r t d e s -c r i p t i o n w i l l be g i v e n t o p o i n t o u t t h e s a l i e n t f e a t u r e s a n d t h e p r a c t i c a l d i f f i c u l t i e s . 4 . 2 P r a c t i c a l B r i d g e C i r c u i t The c i r c u i t r y o f t h e b r i d g e i s shown i n F i g . 4 . 2 a n d n e e d s v e r y l i t t l e d e s c r i p t i o n . The p u m p i n g a n d b i a s i n g s c h e m e s a r e s e l f -e x p l a n a t o r y . S i n c e t h e maximum a v a i l a b l e v o l t a g e f r o m t h e pump s o u r c e i s 1 v o l t r . m . s , , t h e t r a n s f o r m e r was d e s i g n e d t o s t e p u p t h i s v o l t a g e . S p e c i a l c a r e i s n e e d e d w h e n w i n d i n g t h e t r a n s -f o r m e r t o o b t a i n a g o o d c e n t r e t a p . T h e b i a s i n g r e s i s t a n c e s R g , Rg a n d t h e r e a c t a n c e s o f t h e b l o c k i n g c a p a c i t o r s C g , C^ a r e made a t l e a s t t e n t i m e s l a r g e r t h a n t h e r e a c t a n c e o f e a c h d i o d e c a p a c i -t a n c e C a t t h e o p e r a t i n g b i a s v o l t a g e . The s m a l l d - c o r a - c s i g n a l s t o be a m p l i f i e d are f e d i n t o t h e b r i d g e a t t h e common i n p u t t e r m i n a l 1 . 4 . 2 . 1 B a l a n c i n g t h e B r i d g e The b r i d g e b a l a n c e w i l l d e p e n d t o a l a r g e e x t e n t o n t h e b a l a n c e o f t h e c a p a c i t a n c e s o f t h e d i o d e s a n d t h a t o f t h e i r e f f e c t i v e s e r i e s r e s i s t a n c e s , a n d t o a l e s s e r e x t e n t o n t h e t r a n s f o r m e r s u p p l y i n g a pump v o l t a g e o f t h e same a m p l i t u d e b u t o f o p p o s i t e p o l a r i t y t o t h e t w o d i o d e s , a n d o n t h e g e o m e t r i c a l a r r a n g e m e n t o f t h e w h o l e b r i d g e c i r c u i t . C a p a c i t i v e b a l a n c e i s a c h i e v e d w h e n t h e c a p a c i t a n c e s o f t h e t w o d i o d e s a r e e q u a l . A s e x p l a i n e d i n S e c t i o n 3 . 1 , u n b a l a n c i n g t h e b r i d g e b y c h a n g i n g t h o s e c a p a c i t a n c e s p r o d u c e s t h e u s e f u l o u t -p u t o f t h e b r i d g e , b u t i t i s e s s e n t i a l t h a t t h e o u t p u t be z e r o w h e n no s i g n a l i s a p p l i e d . A t r i m m i n g c a p a c i t o r C^ o f a p p r o p r i a t e s i z e p l a c e d i n s h u n t w i t h t h e d i o d e o f s m a l l e r c a p a c i t a n c e w i l l c o m p e n s a t e 31 From Pump S o u r c e _ R 1 5 R 1 4 1 ' V W 1 R 1 5 1 I R l = 2K R l l = C 5 = 1-3. 5uuf R 2,R 3 68K R12 = ^® ohms C 6 - 140puf R., R -4' o = IK R 1 3 - IK C 7 , C g m l u f ( p o l y s t y r e n e ) R 6 = 30 ohms R 1 4 = IK C 9 = 2\xt R 7 = 100 ohms R 1 5 = 0.5M L l = 2-3mh R 8 22 ohms C, = 0.002pf M = step—up R 9 = 72K C 2 = 33ppf transformer Is 3 R10 = 10K C3,.C4 = 7 5 0 u u f P i g . 4. 2 P r a c t i c a l Bridge C i r c u i t Showing B i a s i n g and 32 f o r any c a p a c i t i v e unbalance. ( P i g . 4.2). Another method i s to apply a s l i g h t l y l a r g e r b i a s v o l t a g e to the diode having a l a r g e r c a p a c i t a n c e . R e s i s t i v e unbalance i s due to the diodes having d i f f e r e n t e f f e c t i v e s e r i e s r e s i s t a n c e s . T h i s type of unbalance can be com-pensated f o r by having a s m a l l ;resistance (combination of Rg, R^ and Rg) of app r o p r i a t e s i z e p l a c e d i n s e r i e s with the diode having a smaller e f f e c t i v e s e r i e s r e s i s t a n c e . Unbalance may be caused by some discrepancy i n the amplitude and phase of the pump v o l t a g e s u p p l i e d by the transformer. T h i s can al s o be c o r r e c t e d by proper adjustment of both the trimming c a p a c i -t o r and s e r i e s r e s i s t a n c e . B a l a n c i n g to e l i m i n a t e the fundamental pump frequency does not n e c e s s a r i l y e l i m i n a t e the higher harmonics which are present due to the n o n l i n e a r i t y of the diode c h a r a c t e r i s t i c s . The r e s u l t i n g higher frequency components are subsequently reduced to a t o l e r a b l e amount by the tuned c i r c u i t s of the h-f a m p l i f i e r . Stray c o u p l i n g produces some r e s i d u a l s i g n a l at pump frequency. This e f f e c t i s made almost n e g l i g i b l e by proper geometrical arrange-ment of the bridge c i r c u i t elements, and can i n any case be compen-sated f o r by s l i g h t unbalance i n the b r i d g e . In p r a c t i c e the procedure f o r b a l a n c i n g the bridge i s as f o l l o w s : -( i ) The b i a s v o l t a g e s a p p l i e d to the two diodes are made equal. The pump v o l t a g e i s set at the d e s i r e d v a l u e . U s u a l l y , the peak value of the pump vo l t a g e i s made app-roxi m a t e l y equal to the bias v o l t a g e of 1.8 v o l t s f o r maximum g a i n . 33 ( i i ) Looking at the output at t e r m i n a l 3, the trimming capa-c i t o r Cg i s adjusted so as to reduce the output s i g n a l to a minimum. ( i i i ) The trimming potentiometer i s next adjusted to f u r t h e r reduce the output s i g n a l at t e r m i n a l 3. ( i v ) Steps ( i i ) and ( i i i ) are repeated s u c c e s s i v e l y u n t i l a f i n e r c o n t r o l to balance the capacitances i s needed. Instead of step ( i i ) , the " H e l i p o t " R ^ suppl y i n g d-c s i g n a l s to the bridge may now be used to provide f i n e r adjustment of the c a p a c i t a n c e s . The output s i g n a l at 3 i s soon reduced to zero when balance i s achieved, (v) The d-c b i a s v o l t a g e s a p p l i e d to the diodes can a l s o be used to e f f e c t small changes i n t h e i r c a p a c i t a n c e s , but t h e i r main purpose i s to provide the best dynamic balance pf the bridge as explained l a t e r i n S e c t i o n 5.7 ( 2 ) . 4.3 Band-Pass High-Frequency A m p l i f i e r F o l l o w i n g the bridge i s a two-stage tuned high-frequency a m p l i f i e r ( F i g . 4.3), wit h a pass band of 36 Kc centred about the pump frequency- of 455 Kc. The main problem encountered i n the c o n s t r u c t i o n of t h i s c i r c u i t i s i n s t a b i l i t y due to p o s i t i v e feedback. While s h i e l d i n g and c a r e -f u l l ayout of the c i r c u i t helped to e l i m i n a t e p r a c t i c a l l y a l l ex-t e r n a l feedback, the i n t e r n a l feedback of the t r a n s i s t o r s r e s u l t e d i n the input impedance of the a m p l i f i e r having a negative r e a l com-ponent causing unwanted o s c i l l a t i o n s . A f t e r c o n s i d e r i n g v a r i o u s 9 10 methods of s t a b i l i z a t i o n ' , the simple way of i n s e r t i n g s t a b i l i z i n g r e s i s t o r s Rg, R^Q at the output t e r m i n a l s was used with some r e -du c t i o n i n g a i n and i n c r e a s e i n bandwidth. -10V Input from P a r a m e t r i c A m p l i f i e r B r i d g e © 3 Output t o Demodu-l a t o r -R 1 » R 2 33K R 3 , R 4 = 47 O i l R 5 , R 6 = 5.6K R 7 > R 8 = IK R 9 , R 1 0 = 10K = i 6 0 u u f c 2 = iso^t C 3 , C 4 = O . l u * C 5 , C 6 , C 7 = 0.05^iuf Cg = 10uf T-^Tg = 2N247 4 . 4 P h a s e - S e n s i t i v e D e m o d u l a t o r T h e a m p l i f i e d h - f o u t p u t i s a s i g n a l w h o s e p h a s e w i t h r e f e r -e n c e t o t h e pump v o l t a g e d e p e n d s o n -fee p o l a r i t y o f t h e i n p u t s i g -n a l , a s e x p l a i n e d i n S e c t i o n 3 . 1 . To e n s u r e a maximum d e m o d u l a t e d o u t p u t , s y n c h r o n o u s d e t e c t i o n m u s t be u s e d . A p h a s e - s h i f t e r a n d a p h a s e - s e n s i t i v e d e m o d u l a t o r w i l l t h e r e f o r e be r e q u i r e d ( P i g . 4 . 4 ) , . I n o r d e r t o c o m p e n s a t e f o r p h a s e - s h i f t s i n t h e t u n e d a m p l i f i e r , t h e p h a s e - c o m p e n s a t i n g n e t w o r k i s i n s e r t e d b e t w e e n t h e c a r r i e r s o u r c e a n d t h e p h a s e - s e n s i t i v e d e t e c t o r . A " l o n g - t a i l e d p a i r " c o n s i s t i n g o f t r a n s i s t o r s T-^  a n d Tg i s u s e d t o c o n v e r t t h e r e f e r e n c e s i n e - w a v e i n t o t w o t r a i n s o f s q u a r e p u l s e s o f 1 v o l t a m p l i t u d e a n d 1 8 0 ° o u t o f p h a s e w i t h e a c h o t h e r . T h i s e n s u r e s v e r y s h a r p s w i t c h i n g o f t r a n s i s t o r s T g a n d T ^ . To e x p l a i n t h e a c t i o n o f t h e d e m o d u l a t o r ( s e e F i g s . 4 . 4 a n d 4 . 5 ) , l e t u s f i r s t c o n s i d e r t h e c a s e o f z e r o i n p u t s i g n a l b e i n g a p p l i e d t o t h e p a r a m e t r i c a m p l i f i e r b r i d g e a t t e r m i n a l 1 . I t i s h e l p f u l a t t h i s s t a g e t o i m a g i n e t h a t t h e s m o o t h i n g c a p a c i t o r Cg i s a b s e n t . T r a n s i s t o r s T g a n d T ^ a r e a l t e r n a t e l y s w i t c h e d o n a n d o f f a t t h e c a r r i e r r a t e . When t h e r e i s no s i g n a l , t h e y p r o d u c e p u l s e s a t X , Y ( w i t h d o t t e d f l a t t o p s i n F i g . 4 , 5 ( d ) , ( e ) ) w h o s e h e i g h t s d e p e n d o n t h e q u i e s c e n t c u r r e n t f l o w i n g u p t h e c o l l e c t o r o f T g a n d t h e r e s i s t o r s Rg o r R g . The v a r i a b l e r e s i s t o r R_, c o m -p e n s a t e s f o r a d i f f e r e n c e b e t w e e n t h e v a l u e s o f Rg a n d R g , a n d a l l t h e p u l s e s t h e n h a v e t h e same h e i g h t . T h e o u t p u t w h i c h i s t a k e n b e t w e e n X a n d Y s h o u l d b e z e r o a f t e r f i l t e r i n g o u t t h e c a r -r i e r , i t s h a r m o n i c s a n d o t h e r h i g h e r f r e q u e n c i e s g e n e r a t e d . Now w h e n a s m a l l s t e p s i g n a l v o l t a g e i s a p p l i e d t o t h e b r i d g e , t h e a m p l i f i e d h - f s i g n a l a t P , t h e b a s e o f T - , w i l l b e t h a t s h o w n 36 <j Output S i g n a l t o Y - D i f f e r e n c e A m p l i f i e r ^ = 470 ohms R 2,R 7 = 500 ohms • R 3 s R 4 j R l 9 = 1 K R5° R6 , R14^| R l K s R 1 8 ) = """^ ^ o n m s R 8 , R 9 ' R 1 1 = 3.9K R 1 0 ' R 1 2 2.2K R 1 3 = 220K R 1 6 = 27K R 1 7 = 5.6K C l j C 2 = 0.001|if C 3 ' C 4 ' C 5 ' C 8 ' -°6 " °7' C9 = <10-T 1 , T 2 , T 3 , T 4 = 20u-f 0.0020* O.OSnf O.lnf 2N1309 2N247 P i g . 4.4 P h a s e - S h i f t e r and P h a s e - S e n s i t i v e Demodulator 37 (a) (e) V ( f ) (a (b (c (d (e (f Large pump or c a r r i e r v o l t a g e Small step input s i g n a l v o l t a g e to bridge A m p l i f i e d modulated h-f output v o l t a g e Voltage waveform at X (Cg absent) Voltage waveform at Y (Cg absent) A m p l i f i e d demodulated output a f t e r f i l t e r i n g P i g . 4.5 Voltage Waveforms Showing Demodulation 38 i n F i g , 4,5 (c)„ The c o l l e c t o r c u r r e n t of T^ the n v a r i e s s i n u s o i -d a l l y w i t h t h i s o u t p u t , and t h e r e s u l t i n g v o l t a g e v a r i a t i o n s are superimposed on the p r e v i o u s p u l s e s at X, Y (Fig„ 4,5 ( d ) , ( e ) ) . Note the d i s c o n t i n u i t i e s due t o the 180° phase change as the ste p s i g n a l v o l t a g e goes from a p o s i t i v e v a l u e + AV t o a n e g a t i v e one - AV. The f i l t e r e d output t a k e n between X and Y w i l l be an amp-l i f i e d s t e p s i g n a l o The same r e a s o n i n g can be extended t o any typ e of i n p u t signal„ In essence, t h e p h a s e - s e n s i t i v e demodulator p r o v i d e s f u l l -wave r e c t i f i c a t i o n . S i n c e t h e quiescent e m i t t e r p o t e n t i a l a t ..Q' i s ^-1.4 v o l t s , t h e maximum s i g n a l a m p l i t u d e o b t a i n a b l e between, 2 X and Y i s a p p r o x i m a t e l y e q u a l t o (— x 1.4 x 3.9) or 3.5 v o l t s . R^g i s a 100-ohm r e s i s t o r which i s bypassed or unbypassed by C-^ Q a c c o r d i n g t o g a i n r e q u i r e m e n t s . When R^g i s bypassed, the v o l -t age g a i n of the demodulator i s 5.4 times as l a r g e as when R^g i s unbypassed. A t w o - s e c t i o n low-pass R-C f i l t e r w i t h 8 Kc, bandwidth i s r e -q u i r e d between t h e p h a s e - s e n s i t i v e demodulator!- and the d i f f e r e n c e a m p l i f i e r . The r e s i d u a l c a r r i e r and h i g h e r f r e q u e n c i e s are thus made s m a l l enough so as not t o o v e r l o a d the f o l l o w i n g d i f f e r e n c e a m p l i f i e r . 4.5 D i f f e r e n c e A m p l i f i e r and Low-Pass F i l t e r The purpose of the d i f f e r e n c e a m p l i f i e r ( F i g . 4.6) i s two-f o l d . F i r s t , i t p r o v i d e s s u f f i c i e n t g a i n so as t o enable us t o examine t h e n o i s e generated by the a m p l i f i e r . Second, the f i n a l output t a k e n from c o l l e c t o r Tg i s a v a i l a b l e w i t h r e s p e c t t o a s u i t a b l e ground. The output can be v a r i e d by the g a i n c o n t r o l Rg. The low-pass L-C f i l t e r i s d e s i g n e d t o have a bandwidth of 39 -16v o — W W R 4 A A A A -From Phase-S e n s i t i v e Demodulator A A A r R 5 _J_C x; Rr R, •R„ R 8< X " L, -o 4 To Noise Measuring A m p l i f i e r B l 100 ohms C l = 20uf B 2 = 1.8K G 2 ' C 3 s= 0.0062uf R 3 = 3.3K C4>,C5 s 0.003uf R4' R5 4.7K T1> T2 = 2N404 R 6 500,; ohms L 120mh R-,, Rg = 22K P i g . 4.6 D i f f e r e n c e A m p l i f i e r and Lo%y-Pass F i l t e r 40 8 Kc i n order to remove the r e s i d u a l c a r r i e r , a l l higher f r e q u e n c i e s and a l l noise above 8 Kc. 4.6 Methods Used to Measure, Noise An i n v e s t i g a t i o n of n o i s e was c a r r i e d out u s i n g the n o i s e -measuring c i r c u i t shown i n F i g . 4.7. This c o n s i s t s of an amp-l i f i e r w ith a frequency response from 50 c y c l e s to 50 Kc f o l l o w e d by a thermal milliammeter which measures r.ra.s. c u r r e n t . The r.m.s. output v o l t a g e at t e r m i n a l 5 i s the product of the c u r r e n t measured and the meter r e s i s t a n c e of 1.6 K. The f o l l o w i n g two methods were used to measure n o i s e . (1) In tie absence of hum (or of an a-c input s i g n a l ) , e.g. when the b r i d g e was disconnected f r o m the high-frequency a m p l i f i e r , an accurate measurement of the n o i s e was made on Hie thermal m i l l i a -mraeter, (2) In t h e presence of hum, the meter measures the t o t a l power due to both n o i s e and hum. A crude but convenient method of d i s t i n g u i s h i n g between the amount of n o i s e and the amount of hum present i s to observe the noise waveform onan o s c i l l o s c o p e synchro-n i z e d to 60 c y c l e s . We then estimate the + 2a p o i n t s on the o s c i l l -oscope. These are the p o i n t s between which t h e s i g n a l l i e s 95.5$ 11 of the time i f the n o i s e i s gaussian. The d i s t a n c e between these two p o i n t s on the o s c i l l o s c o p e then i s 4oy i.e., i t corresponds to 4 times the r.m.s. nois e v o l t a g e . A f t e r some experience, one can d i s t i n g u i s h n o i s e from hum or any i n j e c t e d s i g n a l , and can measure the 4a p o i n t s due to the noise alone to an accuracy of about 15$. Noise or hum measured has to be r e f e r r e d back to the i n p u t . In order to ensure against any v a r i a t i o n i n g a i n , the method of d i -r e c t comparison was used. A known a-c s i g n a l i s i n j e c t e d at the 41 -22v Rr Rc F i l t e r e d Output R^ 2 o- V W f rom A m p l i f i e r ^ ft- Thermal Milliammeter Cambridge I n s t r u m e n t s R, R-^  = 3 • 3K Rg — 2•2K R 3 = 18K R 4 = 330 ohms R c = 2.2K o • •  C1 = 50p,f C 2 = 160uf Cg = 50uf T x = 2N247 R 6 = 1.6K i F i g . 4.7 Noise-Measuring C i r c u i t 42 i n p u t , s a y 1 . The o u t p u t v o l t a g e , s a y a t t e r m i n a l 5 , i s m e a s u r e d o n t h e o s c i l l o s c o p e a n d b y e s t i m a t i n g t h e 4 o p o i n t s t h e n o i s e v o l t a g e c a n be o b t a i n e d a t t h e same t i m e . T h e n b y d i r e c t c o m -p a r i s o n , t h e 1 e q u i v a l e n t n o i s e v o l t a g e a t i n p u t 1 c a n be c a l c u l a t e d . 4 . 7 P r e l i m i n a r y M e a s u r e m e n t s B e f o r e t h e g a i n , b a n d w i d t h , d r i f t a n d n o i s e o f t h e b r i d g e c a n b e m e a s u r e d , some p r e l i m i n a r y m e a s u r e m e n t s h a v e t o be made o n v a r -i o u s p a r t s o f t h e a m p l i f i e r . The i n p u t i m p e d a n c e o f t h e h i g h - f r e q u e n c y a m p l i f i e r was m e a s -u r e d b y t h e u s e o f r e s i s t o r s a n d a s m a l l h i g h - Q c o i l i n s e r i e s w i t h t h e i n p u t . A t 4 5 5 K c , t h e i n p u t i m p e d a n c e was f o u n d t o b e o f a s e r i e s c o m b i n a t i o n o f C-^  = 1 2 0 0 p f . a n d = 2 7 0 o h m s . T h e v o l t a g e g a i n s o f t h e v a r i o u s s t a g e s w e r e m e a s u r e d a s f o l l o w s : -A 4 5 5 K c s i g n a l w i t h t h e c o r r e c t p h a s e was i n j e c t e d a t t e r m i n a l 1 . The o u t p u t v o l t a g e s a t 3 was r e a d o n t h e o s c i l l o s c o p e w h i l e t h e c h a n g e i n d - c V o l t a g e a t 4 was o b s e r v e d o n a m e t e r . H e n c e t h e v o l t a g e g a i n s A^g> A ^ 3 a n < i A 2 3 w e r e c a l c u l a t e d . The n e c e s s a r y p r e -c a u t i o n was t a k e n t h a t n o n e o f t h e s t a g e s was o v e r l o a d e d o r a p p r e -c i a b l y a f f e c t e d b y t h e m e a s u r i n g i n s t r u m e n t s . S i n c e t h e d - c a n d a - c l o a d s a t t e r m i n a l 5 a r e d i f f e r e n t ( 3 . 3 K a n d 1 . 8 6 K r e s p e c t i v e l y ) , t h e n f o r a - c p u r p o s e s , we h a v e t o m u l t i p l y a l l d - c g a i n s b y 0 . 5 5 . The v o l t a g e g a i n A ^ g was e a s i l y o b t a i n e d b y f e e d i n g a 1 K c s i g n a l i n t o t e r m i n a l 4 a n d m e a s u r i n g t h e o u t p u t v o l t a g e a t t e r m i n a l 5 . A l l t h e i m p o r t a n t v o l t a g e g a i n s a r e s h o w n o n t h e b l o c k d i a g r a m i n P i g . 4 . 1 . 4 3 5 . MEASUREMENTS The c i r c u i t d e s c r i b e d i n C h a p t e r 4 w i l l now e n a b l e u s t o s t u d y t h e g a i n , b a n d w i d t h , d r i f t a n d n o i s e o f t h e b r i d g e . T h e o p t i m u m o p e r a t i n g b i a s v o l t a g e f o r t h e b r i d g e w a s f i r s t d e t e r m i n e d e x p e r i m e n t a l l y . 5 . 1 O p t i m u m O p e r a t i n g B i a s A c o n s t a n t i n p u t v o l t a g e was a p p l i e d t o t h e b r i d g e a n d t h e o u t p u t v o l t a g e a t t e r m i n a l 3 was m e a s u r e d f o r d i f f e r e n t v a l u e s o f r e v e r s e - b i a s v o l t a g e , t h e p e a k pump v o l t a g e b e i n g made a p p r o x i -m a t e l y e q u a l t o t h e b i a s v o l t a g e e a c h t i m e . A s > s h o w n i n S e c t i o n 5 . 6 , t h i s r e s u l t s i n l a r g e s t g a i n w i t h o u t e x c e s s i v e n o i s e . The r e s u l t s a r e s h o w n i n P i g . 5 . 1 a n d a r e f o u n d t o b e i n a g r e e m e n t w i t h t h o s e caxjpecjbeid f r o m m e a s u r e m e n t s o f d i o d e c h a r a c t e r i s t i c s as i n d i c a t e d i n F i g . 2 . 6 . The o p t i m u m b i a s f o r maximum g a i n w a s f o u n d t o b e 1 . 8 v o l t s . 5 . 2 G a i n a n d B a n d w i d t h The g a i n o f t h e b r i d g e was m e a s u r e d i n t w o w a y s u s i n g a 1 K c i n p u t s i g n a l . ( 1 ) A 1 m v . p e a k - t o - p e a k s i g n a l was f e d i n t o t h e b r i d g e a n d t h e o u t p u t v o l t a g e m e a s u r e d a t t e r m i n a l 3 was e q u a l t o 2 v . p e a k -t o - p e a k . V o l t a g e g a i n o f b r i d g e _ A - „ = -M = & j l m v _ 0 . 1 3 X * A 2 3 1 . 1 x 1 0 * ( 2 ) A 40 pv p e a k - t o - p e a k s i g n a l was a p p l i e d t o t h e b r i d g e a n d t h e o u t p u t v o l t a g e m e a s u r e d a t t e r m i n a l 5 a c r o s s t h e t h e r m a l m i l l i a m m e t e r was 3 . 8 v p e a k - t o - p e a k . O p t i m u m O p e r a t i n g V o l t a g e = 1 . 8 v ( f o r Maximum G a i n ) B i a s V o l t a g e = Pump V o l t a g e ( V o l t s ) ^ F i g . 5 . 1 O p t i m i z a t i o n C u r v e f o r B i a s V o l t a g e 4 5 V o l t a g e g a i n of b r i d g e = A A 1 5 _ 3.8v/40uv A 2 5 4 o 2 x 1 0 4 x 1 7 = 0 . 1 3 A 1 2 = TV~~ I s = O o , 13 +.. 0.01 T h e r e f o r e the t r a n s d u c e r power g a i n of the b r i d g e ( 0 . 1 3 ) 2 x 4 x 7 . 2 x 10 _ . = 1 8 ± 2 (or 1 2 . 6 ± 0.5 db) T h i s i s i n c l o s e agreement w i t h the v a l u e of 12.8 db c a l -c u l a t e d i n S e c t i o n 3.2.3. The o v e r a l l bandwidth a t -3 db p o i n t of the a m p l i f i e r was measured and found t o be equal t o 5*0 + 0.3 Kc. T h i s bandwidth i s due t o s e v e r a l l i m i t a t i o n s a c t i n g t o g e t h e r , e s p e c i a l l y t h e time c o n s t a n t R (C + 2C) g i v i n g an upper 3-db f r e q u e n c y of 8 Kc by s c i t s e l f . 5.3 D r i f t D r i f t s i n c a p a c i t a n c e or i n leakage c u r r e n t a r e m a i n l y due t o the temperature dependence of t h e c o n t a c t p o t e n t i a l s of the d i o d e s . Unequal d r i f t s i n t h e di o d e c h a r a c t e r i s t i c s w i l l r e s u l t i n a d-c d r i f t . The l a t t e r was reduced c o n s i d e r a b l y by p l a c i n g the two di o d e s s i d e by s i d e i n an oven a t a temperature of 32.5°C. T h i s was m a i n t a i n e d to> within± 0.5°C, as l o n g as the room temper-a t u r e d i d not exceed 25°C. A f t e r a warm-up p e r i o d of about an hour, d r i f t i n output i s 46 e q u i v a l e n t t o a change a t the i n p u t of about ± 30 uv per hour. I t i s thought t h a t t h i s may be p a r t l y due t o the b i a s v o l t a g e s a p p l i e d t o the diode s changing by 0.01$, and may t h e r e f o r e be reduced a p p r e c i a b l y by c a r e f u l s t a b i l i z a t i o n of t h e b i a s voltages 0 5.4 Hum When the a m p l i f i e r was f i r s t s e t up, c o n s i d e r a b l e hum due t o 60 c y c l e s and i t s harmonics was encountered and i n v e s t i g a t e d . I t was found t h a t some hum was due t o e x t e r n a l p i c k u p and some o r -i g i n a t e d from the pump so u r c e . The hum was g r e a t l y reduced by f h e l f a l l o w i n g t h r e e ways:-(1) P r o p e r s h i e l d i n g (2) C a r e f u l dynamic b a l a n c e of the b r i d g e as d i s c u s s e d f u l l y i n S e c t i o n 5.7(2) (3) E x t e r n a l m o d u l a t i o n (about 0.1$) of the pump v o l t a g e w i t h 60 c y c l e s h a v i n g the r i g h t phase. 5.5 N o i s e (1) B r i d g e D i s c o n n e c t e d The b r i d g e was d i s c o n n e c t e d from the h i g h - f r e q u e n e y a m p l i f i e r and a 270-ohm r e s i s t o r was p l a c e d a t the i n p u t of the h-f a m p l i -f i e r t o s i m u l a t e the output impedance of t h e b r i d g e . The t h e r m a l m i l l i a m m e t e r then read 0.27 rav r.m.s. c o r r e s p o n d i n g t o (0„27 ma x 1.6 K) or 0.43v a t t e r m i n a l 5., T h i s was co n f i r m e d by l o o k i n g on o s c i l l o s c o p e a t t e r m i n a l 5. R.M.S. n o i s e v o l t a g e g e n e r a t e d a t t e r m i n a l 2 _ 0.43v  A 2 5 4 7 = 0 . 4 3 v 3 . 9 x 1 0 6 = 0 . 1 1 + 0 . 0 1 u v T h i s r e s u l t was c h e c k e d i n two w a y s s -( a ) T h e 4a p o i n t s w e r e e s t i m a t e d o n a n o s c i l l o s c o p e a t t a c h e d t o t e r m i n a l 4 a n d f r o m t h i s , t h e n o i s e v o l t a g e a t t e r m i n a l 4 was a p p r o x i m a t e l y e q u a l t o 2 5 mv r . m . s . . • N o i s e a t t e r m i n a l 2 = 2 5 mv a - c A 2 4 2 5 mv .5 2 . 3 x 10 = 0 . 1 1 + 0 . 0 1 u v ( b ) D i r e c t C o m p a r i s o n A 4 5 5 K c , 1 . 0 5 u v s i g n a l w i t h t h e c o r r e c t p h a s e f e d i n t o t e r m i n a l 2 c a u s e d a d - c o u t p u t o f 4 0 0 mv a t t e r m i n a l 4 . T h e c o r r -e s p o n d i n g a - c c h a n g e a t 4 w o u l d h a v e b e e n 0 . 5 5 x 4 0 0 = 2 2 0 m v . T h e r e f o r e 2 5 mv r . m . s . a t t e r m i n a l 4 c o r r e s p o n d s a t t e r m i n a l 2 2 5 t o a n o i s e v o l t a g e o f 220" X ^LV 2= 0 . 1 2 + 0 . 0 1 \iv Now i f t h e t r a n s i s t o r w e r e n o i s e l e s s , t h e o n l y n o i s e g e n e r a t e d w o u l d b e t h e r m a l n o i s e b y t h e p a r a l l e l c o m b i n a t i o n o f t h e i n p u t i m p e d a n c e o f 2 7 0 ohms o f t h e h - f a m p l i f i e r a n d t h e 2 7 0 ohm t e r m i n a t i o n . T h e c o r r e s p o n d i n g n o i s e v o l t a g e = y 7 4 k T B R ^J4 x 1 . 3 8 x 1 0 ~ 2 3 x 5 x 1 0 3 x 1 3 5 = 0 . 0 7 4 u v . • ? 0 1 1 3 T h e r e f o r e n o i s e f i g u r e o f t r a n s i s t o r a m p l i f i e r 23 ( 0 * 0 7 4 ) = 2 . 3 + 0 . 2 ( o r 3 . 6 + " 0 . 4 d b ) T h i s i s s u r p r i s i n g l y l o w , a n d one s u s p e c t s t h a t t h e i n p u t 4 8 i m p e d a n c e o f t h e a m p l i f i e r may n o t be a c o n s t a n t 2 7 0 ohms o v e r t h e b a n d o f 10 K c . I n a n y c a s e , i t w i l l be s e e n t h a t t h e m e a s u r e d n o i s e p o w e r d u e t o t h e t r a n s i s t o r a m p l i f i e r i s o n l y 8 $ o f t h a t d u e t o t h e b r i d g e when i n o p e r a t i o n , a n d h e n c e may be n e g l e c t e d . ( 2 ) B r i d g e C o n n e c t e d W i t h t h e b r i d g e c o n n e c t e d t o t h e h i g h - f r e q u e n c y a m p l i f i e r , n o i s e was m e a s u r e d u s i n g t h e f o l l o w i n g t w o m e t h o d s ( i ) D i r e c t c o m p a r i s o n u s i n g a - c i n p u t A 1 K c , 4 0 u v p e a k - t o - p e a k s i g n a l was a p p l i e d t o t h e b r i d g e g i v i n g a n o u t p u t e q u a l t o 4 v o l t s p e a k - t o - p e a k o n o s c i l l o s c o p e a t t e r m i n a l 5 . The c o r r e s p o n d i n g m e t e r r e a d i n g w a s 0 . 9 7 ma r . m . s . When t h e i n p u t s i g n a l was r e d u c e d t o z e r o , t h e o s c i l l o -s c o p e s h o w e d a b o u t 2 v p e a k - t o - p e a k o f 60 c y c l e s . T h e m e t e r t h e n r e a d 0 . 4 9 ma r . m . s . T h e r e f o r e i f o n l y 1 K c w e r e p r e s e n t , m e t e r w o u l d h a v e r e a d (J^ x 0 . 9 7 ) o r 0 . 8 4 ma r . m . s . ^ w h i c h c o r r e s p o n d s t o ( 0 . 8 4 x 1 . 6 x 2 J2 ) o r 3 . 8 v o l t s . T h i s c h e c k s w i t h t h e 4 v o l t s p e a k - t o - p e a k a s s e e n o n o s c i l l o s c o p e . B u t w i t h i n p u t s i g n a l e q u a l t o z e r o , r e a d i n g o n m e t e r = 0 . 3 3 m a . 0 33 T h e r e f o r e , t h e e q u i v a l e n t n o i s e p l u s hum a t i n p u t 1 = n * f l A x uv = 5 . 5 u v r . m . s . ( i i ) N o i s e e s t i m a t e d o n o s c i l l o s c o p e u s i n g 4 o p o i n t s W i t h a m e t e r r e a d i n g o f 0 . 3 3 ma w h e n i n p u t s i g n a l i s z e r o , we e x p e c t t o s e e ( 4 x 1 . 6 x 0 . 3 3 ) o r 2L2 v o l t s o n t h e o s c i l l o s c o p e f o r 4 3 p o i n t s . The o s c i l l o s c o p e was s y n c h r o n i z e d t o 60 c . p . s . , a n d c a r e f u l m e a s u r e m e n t s o f v i s i b l e n o i s e w e r e m a d e . ' T h e s e g a v e 4 f l = 1 . 2 _ 0 . 2 v o l t s a t t e r m i n a l 5 . H e n c e 6* •= 0 . 3 + 0 . 0 5 v o l t s , a n d • 49 meter reading due to t h i s noise should have been Y^QK o r ^.187 ± 0 . 0 2 ma. Therefore some of meter power i s s t i l l 60 c y c l e s or harmonics / 2 g' i amounting to /O.33 - 0.187 or 0.265 ma r.m.s. On t h i s b a s i s , we o b t a i n noise at input 1 = °' 3g | v ° , 0 5 Y x 40 nv = 3.15 + 0.5 uv r.m.s. 0 265 Hum pickup at best dynamic balance must have been (Q*I'S'Y^ x 3.15, or 4.5 (iv r.m.s. The o p e n - c i r c u i t noise v o l t a g e due to the 72 K input r e s i s t o r over a 5 Kc bandwidth = 2.36 uv as computed i n S e c t i o n 3.4. Therefore, 9 o "I c i r\ c n o i s e f i g u r e of a m p l i f i e r = (—- 2 36 * ^  ar 2 ± 0.5 (or 3 + 1 db) This i s a l s o the noise f i g u r e of the parametric a m p l i f i e r bridge s i n c e the noise c o n t r i b u t i o n from the ..subsequent stages has been shown to be n e g l i g i b l e . T h i s i s r e l a t i v e l y low noise f i g u r e f o r a low-frequency a m p l i f i e r . We s h a l l look i n t o the l i k e l y sources of noise i n the S e c t i o n 5.7. 5.6 Gain and Noise as a Fu n c t i o n of Pump Voltage A 1 Kc,80 uv peak-to-peak s i g n a l was i n j e c t e d at t e r m i n a l 1 i n t o the b r i d g e . With an o s c i l l o s c o p e at t e r m i n a l 5 5 both the 1 Kc component and the noise could be seen and measured. The d-c bi a s a p p l i e d to the diodes was kept constant at 1.8 v o l t s . The pump vo l t a g e was v a r i e d and the r e s u l t s obtained are t a b u l a t e d belowj-50 V ( p e a k ) pump V . o u t a t 5 p - t - p A 1 5 4o ( v o l t s ) N o i s e a t 1 (uv) 1 . 3 v ' 4 . 4 v 5 . 5 x 1 0 4 0 . 7 0 3 . 2 1 . 8 8 1 . 0 x 1 0 5 1 . 5 0 3 . 7 2 . 3 10 1 . 2 5 x 1 0 5 4 . 0 8 . 0 When t h e pump v o l t a g e V ^ i s l e s s t h a n t h e b i a s v o l t a g e V , b o t h t h e g a i n a n d t h e o u t p u t n o i s e i n c r e a s e w i t h t h e pump v o l t -a g e i n s u c h a way t h a t t h e e q u i v a l e n t i n p u t n o i s e : r e m a i n s a l m o s t c o n s t a n t . When V i s g r e a t e r t h a n V , i . e . , t h e d i o d e s a r e f o r w a r d -b i a s e d , t h e g a i n i n c r e a s e s s l i g h t l y b u t t h e n o i s e a t i n p u t g o e s up t r e m e n d o u s l y . The o p t i m u m p e r f o r m a n c e o f t h e a m p l i f i e r , a s f a r as g a i n t o g e t h e r w i t h n o i s e i s c o n c e r n e d , i s a c h i e v e d w h e n t h e pump v o l -t a g e i s j u s t l e s s t h a n t h e b i a s v o l t a g e . 5 . 7 P o s s i b l e S o u r c e s o f N o i s e ( l ) The t r a n s i s t o r s t a g e s a n d t h e r e s i s t a n c e o f t h e t u n i n g i n d u c t o r (= 98 ohms) p r o d u c e some n o i s e . The a m o u n t o f n o i s e was m e a s u r e d w i t h t h e pump v o l t a g e o f f a n d t h e b r i d g e s t i l l c o n n e c t e d t o t h e h i g h - f r e q u e n c y a m p l i f i e r . The n o i s e r e m a i n e d t h e s a m e , i . e . , 0 . 2 7 ma o n m e t e r a s w h e n t h e b r i d g e was d i s c o n n e c t e d a n d r e p l a c e d b y a 2 7 0 - o h m r e s i s t o r a t t h e i n p u t o f t h e h - f a m p l i f i e r . T h u s , t h e n o i s e c o n t r i b u t i o n f r o m t h e t r a n s i s t o r a m p l i f i e r a n d t h e i r e s i s t a n c e o f t h e i n d u c t o r c a n o n l y be a v e r y s m a l l f r a c t i o n o f t h e n o i s e o b s e r v e d . 51 (2) N o i s e may be due t o a m p l i t u d e - m o d u l a t i o n of the pump v o l t a g e . T h e o r e t i c a l l y we expect the b r i d g e t o be s e n s i t i v e t o AM i f the diode dynamic c h a r a c t e r i s t i c s are unbalanced and we expect the s e n s i t i v i t y t o a l t e r i f d i f f e r e n t d-c b i a s v o l t a g e s are a p p l i e d t o the di o d e s ( c f . F i g - 2.4). An e x p e r i m e n t a l i n -v e s t i g a t i o n showed t h a t the s e n s i t i v i t y t o AM c o u l d be reduced by h a v i n g s l i g h t l y d i f f e r e n t b i a s v o l t a g e s , and i n our case i t was found t h a t t h e optimum b i a s v o l t a g e s were 1.8 v and 1.82 v a p p l i e d t o d i o d e s and D g r e s p e c t i v e l y ( F i g . 4.2). I n o r d e r t o t r a c e , a n y n o i s e due t o AM, we want t o f i n d out how much e x t e r n a l m o d u l a t i o n i s needed t o produce the same amount of n o i s e power a t the o u t p u t , and t h e n check whether t h i s i amount or any f r a c t i o n of i t , i s p r e s e n t i n the pump s o u r c e . A 2 l / 2 $ of 400c/s m o d u l a t i o n was r e q u i r e d t o double t h e power output a t meter. T h e r e f o r e i f a l l the n o i s e power measured was due t o AM, 0.75$ of AM m o d u l a t i o n would be r e q u i r e d . But the n o i s e m o d u l a t i n g t h e pump v o l t a g e was measured as 0.014$ of 60 c y c l e s and l e s s t h a n 0.002$ of o t h e r m o d u l a t i o n . T h e r e f o r e a m p l i t u d e - m o d u l a t i o n of the pump source cannot cause a s i g n i f i -c ant p a r t o f t h e observed n o i s e . (3) F r e q u e n c y - M o d u l a t i o n of t h e pump v o l t a g e i s another p o s s i b l e source of n o i s e . The procedure d e s c r i b e d above f o r i n -v e s t i g a t i n g AM was a l s o used t o f i n d out whether the n o i s e was caused by FM. The fr e q u e n c y c o n t r o l was changed from 450 t o 460 Kc and the d-c output measured a t t e r m i n a l 4 a l t e r e d by 0„2v. For a f a s t change i n frequ e n c y of 10 Kc peak-to-peak, the v o l t a g e at t e r m i n a l 5 would have been (0.2 x 0.55x17) or 1.9 v o l t s peak-to-peak. S i n c e the n o i s e found was 0.3v r.m.s., the amount of 52 10 x 0 3 PM r e q u i r e d t o cause 0.3V r.m.s. of n o i s e i s ( 19* ) o r 1.6Kc,.which corresponds <to 0.35$ r.m.s. But the amount of FM as seen on the 502 o s c i l l o s c o p e ( w i t h a 20x expander) was l e s s t h a n 0.02$ peak-to-peak. T h e r e f o r e o n l y an i n s i g n i f i c a n t f r a c t i o n of the n o i s e observed-can be a t t r i b u t e d t o FM« (4) N o i s e may be p i c k e d up i n d u c t i v e l y or c a p a c i t a t i v e l y by the b r i d g e from e x t e r n a l c i r c u i t s e i t h e r (a) at 455 Kc or (b) 0-5 Kc v i a the i n p u t c i r c u i t t o the b r i d g e . (a) The experiment d e s c r i b e d above i n (1) showed t h a t any n o i s e p i c k e d up a t 455 Kc was an i n s i g n i f i c a n t f r a c t i o n of the n o i s e observed. (b) The h i g h impedance of t h e i n p u t c i r c u i t makes i t ex-t r e m e l y s e n s i t i v e t o n o i s e . T h i s was reduced a p p r e c i a b l y by c a r e f u l s h i e l d i n g , but as a r u l e , one cannot say t h a t i t was c o m p l e t e l y e l i m i n a t e d . (5) The f e r r i t e core of the i n d u c t o r may generate n o i s e of i t s own. But i t i s u n l i k e l y t h a t t h i s type of n o i s e can account f o r the t o t a l n o i s e observed. (6) F i n a l l y n o i s e may be produced by the d i o d e s . The v a r i o u s p o s s i b l e causes a r e s -( i ) Thermal n o i s e o r i g i n a t i n g i n the diode s e r i e s or shunt r e s i s t a n c e s , ( i i ) Shot n o i s e generated by c a r r i e r s found i n the d e p l e t i o n r e g i o n of the diode j u n c t i o n s . 5 ( i i i ) N o i s e a r i s i n g from changes i n s a t u r a t i o n c u r r e n t . ( i v ) N o i s e r e s u l t i n g from random changes i n capa-5 c i t a n c e s of the d i o d e s . 53 V a r a c t o r d i o d e s , PC 117, which have s i m i l a r c h a r a c t e r i s t i c s as the r e c t i f i e r d i o d e s used, were t r i e d and gave no s i g n i f i c a n t improvement on the n o i s e performance of the b r i d g e . However,, p l o s e r i n v e s t i g a t i o n i s suggested u s i n g more a c c u r a t e methods of m e a s u r i n g . n o i s e . I t can be concluded t h a t the n o i s e observed i s e i t h e r p i c k e d up at the h i g h impedance i n p u t of the b r i d g e , or i s i n h e r e n t i n the diodes themselves^. 54 6. CONCLUSIONS 6.1 Summary of Re s u l t s of Measurements I t has been shown t h a t under r e v e r s e - b i a s v o l t a g e s from 0 to 9 v o l t s , s i l i c o n r e c t i f i e r s (1N2484) have ( l ) low leakage c u r r e n t s —8 of the order of 10~ ampere (2) high dynamic shunt r e s i s t a n c e s of the order of 2 x 10 ohms (3) capacitances which vary with voltage; from 120 to 30 uuf (4) e f f e c t i v e s e r i e s r e s i s t a n c e s f o r r . f . of the order of 100 ohms. These c h a r a c t e r i s t i c s make them very a t t r a c t i v e as v a r a c t o r diodes f o r a parametric a m p l i f i e r . An optimum oper a t i n g b i a s of 1.8 v o l t s f o r maximum gain of the parametric a m p l i f i e r bridge was found: (1) from measurements of the diode capacitance c h a r a c t e r i s t i c s as a f u n c t i o n of b i a s v o l t a g e and determining the maximum value of a =-_/T°^ -HI (a — 0.375): ° d v l o g V; max (2) from measurement of gain as a f u n c t i o n of b i a s v o l t a g e . Using a pump frequency of 455 Kc, the transducer power g a i n of the parametric a m p l i f i e r was found to be 12.6 ± 0.5 db at a bandwidth (at -3 db p o i n t ) of 8 Kc. (The o v e r a l l bandwidth i n c l u -ding r . f . and a.f. stages was 5 K c ) . The e q u i v a l e n t input d r i f t was ± 30 uv per hour when diodes were placed i n a t h e r m o s t a t i c a l l y c o n t r o l l e d oven. At optimum dynamic balance, hum pickup was 4.5 uv r.m.s. The noise f i g u r e of the parametric a m p l i f i e r bridge was 3 + 1 db. 6.2 S i l i c o n - D i o d e Bridge as a Low-Frequency Parametric A m p l i f i e r The use of o r d i n a r y s i l i c o n r e c t i f i e r diodes as v a r a c t o r s f o r am p l i f y i n g low frequencies has been proved f e a s i b l e and there seems to be no advantage i n us i n g diodes s p e c i a l l y f a b r i c a t e d f o r 55 v a r a c t o r use. The t r a n s d u c e r power g a i n of the s i l i c o n - d i o d e b r i d g e i s p r o -2 2 p o r t i o n a l t o a (3 where |3 i s the r a t i o of t h e pump fr e q u e n c y t o the upper 3-db frequ e n c y of the. s i g n a l t o be a m p l i f i e d . A s a t -i s f a c t o r y power g a i n has been a c h i e v e d and can be i n c r e a s e d by u s i n g a higher.v-pump' tr, equency oi?:: by ^lowering^th©-bandwidth. With the two-diode arrangement, the b r i d g e i s r e l a t i v e l y i n -s e n s i t i v e t o d r i f t s i n diode c h a r a c t e r i s t i c s . The d r i f t measured i s comparable t o the d r i f t of the b e s t a v a i l a b l e d-c a m p l i f i e r s , and c o u l d p r o b a b l y be reduced by u s i n g more s t a b l e b i a s s u p p l i e s . The n o i s e of the p a r a m e t r i c a m p l i f i e r b r i d g e i s r e l a t i v e l y low f o r a d-c a m p l i f i e r a n d seems t o o r i g i n a t e i n t h e d i o d e s . Thus one can conclude t h a t v a r a c t o r d i o d e s have the same a t t r a c -t i v e p r o p e r t i e s a t low f r e q u e n c i e s as they do a t microwave f r e -q u e n c i e s , i . e . , t h e y produce power g a i n w i t h low n o i s e , and t r a n s -l a t e the s i g n a l i n t o a frequency-band where f u r t h e r a m p l i f i c a t i o n i s e a s i e r t o a c h i e v e . 6.3 S u g g e s t i o n s f o r f u r t h e r work F u r t h e r i n v e s t i g a t i o n i s suggested:-(1) i n t o t he exact source of the observed n o i s e and of the d-c d r i f t s , , (2) i n t o t he d i s t r i b u t i o n of t h e n o i s e over i t s f r e q u e n c y spectrum u s i n g more a c c u r a t e methods f o r measuring n o i s e ? (3) i n t o t he use of h i g h e r pump f r e q u e n c i e s . 56 R E F E R E N C E S 1 . M u m f o r d , W. W . , "Some N o t e s o n t h e H i s t o r y o f P a r a m e t r i c T r a n s d u c e r s " , P r o c . I R E , V o l . 4 8 , pp 8 4 7 -8 5 3 , May 1 9 6 0 . 2 . B l a c k w e l l , L . A . , a n d K o t z e b u e , K . L . , S e m i c o n d u c t o r - D i o d e P a r a m e t r i c A m p l i f i e r s , P r e n t i c e - H a l l , N . J . . 1 9 6 1 . 3 . M o u n t , E . , a n d B e g g , B . , " P a r a m e t r i c D e v i c e s a n d M a s e r s , A n A n n o t a t e d B i b l i o g r a p h y , " I R E T r a n s , o n  M i c r o w a v e T h e o r y a n d T e c h n i q u e s V o l . M T T - 8 , p 2 2 2 , M a r c h I 9 6 0 . 4 . R e e d , E . D . , " T h e V a r i a b l e - C a p a c i t a n c e P a r a m e t r i c A m p l i f i e r " , I R E T r a n s , o n E l e c t r o n D e v i c e s , V o l . E D - 6 , pp 2 1 6 - 2 2 4 , A p r i l 1 9 5 9 . 5 . J o s e p h s , H . C , " T h e V a r a c t o r D i o d e as a DC A m p l i f i e r , " I n t e r n a l M e m o r a n d u m , B e l l T e l e p h o n e L a b o r a t o r i e s , J u l y 1 9 5 9 . 6 . R o v e t i , D . , " D i o d e A m p l i f i e r h a s T e n - G i g o h m I n p u t I m p e d a n c e s , " E l e c t r o n i c s , pp 3 8 - 4 0 , D e c e m b e r 2 2 , 1 9 6 1 . 7 . R o b i n s o n , B . J . , " T h e o r y o f V a r i a b l e - C a p a c i t a n c e P a r a m e t r i c A m p l i f i e r s " , P r o c . I E E , V o l . 1 0 9 , P t . C , N o . 15 pp 1 9 8 - 2 0 7 , M a r c h 1 9 6 2 . 8 . U e n o h a r a M . , " N o i s e C o n s i d e r a t i o n o f t h e V a r i a b l e C a p a c i t a n c e P a r a m e t r i c A m p l i f i e r , " P r o c . I R E , V o l . 4 8 , pp 1 6 9 - 1 7 9 , F e b r u a r y , 1 9 6 0 . 9 . S h e a , R . P., T r a n s i s t o r C i r c u i t E n g i n e e r i n g , J o h n W i l e y a n d S o n s I n c . , New Y o r k , 1 9 5 7 . 1 0 . H o l m e s , D . D . a n d S t a n l e y , T . 0 . , " S t a b i l i t y C o n s i d e r a t i o n s i n T r a n s i s t o r I n t e r m e d i a t e - F r e q u e n c y A m p l i f i e r s " , T r a n s i s t o r s I , R . C . A . L a b . . P r i n c e t o n , N . J . , M a r c h . 1 9 5 6 . 1 1 . S c h w a r t z , M . , I n f o r m a t i o n T r a n s m i s s i o n . M o d u l a t i o n , a n d N o i s e , M c G r a w - H i l l , N o w Y o r k , 1 9 5 9 . : ~~ 

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