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Construction, theory, and application of non-linear titanate condensers Hobson, John Peter 1950

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CONSTRUCTION, THEORY, AND APPLICATION OF NON-LINEAR TITANATE CONDENSERS  by JOHN PETER HOBSON  A T h e s i s Submitted I n P a r t i a l F u l f i l m e n t of t h e Requirements f o r the Degree o f MASTER OF APPLIED SCIENCE IN ENGINEERING PHYSICS  THE UNIVERSITY OF BRITISH COLUMBIA. AUGUST, 19^0  THE UNIVERSITY O F BRITISH C O L U M B I A VANCOUVER. CANADA DEPARTMENT O F PHYSICS  September 15, 1950.  Dr. L. W. Dunlap, Librarian, U n i v e r s i t y o f B r i t i s h Columbia. Dear Or. Dunlap: T h i s l e t t e r w i l l c e r t i f y t h a t the t h e s i s o f Mr. John P e t e r Eobson has been c a r e f u l l y s t u d i e d by t h e unders i g n e d , and t h a t the t h e s i s meets the r e q u i r e d standards and an a b s t r a c t has been approved by t h e Department. Yours s i n c e r e l y ,  A. M. crooker Acting-Head o f the Department  A. J . Dekker Associate Professor of Physics AMC:lc  ABSTRACT  A method f o r t h e making o f non l i n e a r barium tifcanate condensers f o r auaio f r e q u e n c i e s i s d e s c r i b e d .  Preliminary-  measurements on these condensers are g i v e n . An i d e a l i z e d theory f o r the behaviour o f the non l i n e a r condensers i n a c a r r i e r a m p l i f i e r c i r c u i t i s developed. A c a r r i e r a m p l i f i e r b u i l t on t h i s p r i n o i p l e i s d e s c r i b e d . t h e o r e t i c a l l y p o s s i b l e power a m p l i f i c a t i o n f o r t h i s of 1 8 0 i s d e r i v e d .  Experimental  c a r r i e r a m p l i f i e r are given.  amplifier  r e s u l t s obtained w i t h t h e  A power a m p l i f i c a t i o n of 7 ° was  obtained. C o n c l u s i o n s on the p o s s i b l e l i n e a r condensers are drawn.  A  a p p l i c a t i o n s o f non  ACKNOWLEDGEMENTS  The a u t h o r i s i n d e b t e d t o t h e N a t i o n a l Researoh C o u n c i l o f Canada from whom he r e c e i v e d a b u r s a r y f o r the year  194-9-50. The -work: d e s c r i b e d i n t h i s t h e s i s has been  carried  out under t h e Defense Research Board of Canada which has f i n anced t h e p r o j e o t and has employed t h e author f o r the summers of  1948, 1949, 1950. The author thanks D r . A. Van d e r Z i e l and D r . A.J*  Dekker o f t h e P h y s i o s Department  o f the U n i v e r s i t y o f B r i t i s h  Columbia f o r t h e i r kindness and a s s i s t a n c e d u r i n g t h e vrork.  TABLE  OP  CONTENTS  CHAPTER  PAGE  I  INTRODUCTION  II  MAKING OF NON LINEAR CONDENSERS  1 FOR AUDIO  FREQUENCIES III  THE HYSTERESIS LOOP AND BASIC EQUATION OF THE NON LINEAR CONDENSER  IT  4  9  DETERMINATION OF COEFFICIENTS IN THE BASIC EQUATION  1J  V  THEORY OF A DIELECTRIC CARRIER AMPLIFIER  20  VI  EXPERIMENTAL RESULTS FROM CARRIER AMPLIFIER  28  VII  CONCLUSIONS  37  BIBLIOGRAPHY  41  ILLUSTRATIONS Plate I  II  IU  XV V  Diagram  F a c i n g Page  1.  High Temperature Furnace  2.  Block Diagram o f Furnace C o n t r o l  3.  Condenser E l e c t r o d e  4.  Condenser Mount  5.  Condenser Stand  6.  Hysteresis Circuit  7.  H y s t e r e s i s Loops  8.  "Variation of H y s t e r e s i s Loop w i t h Frequency  9.  Equivalent C i r c u i t  10.  5  7  10  of Non L i n e a r Condenser  E q u i v a l e n t C i r c u i t f o r 3rd Harmonic 15  Current Measurement 11.  11  C i r c u i t f o r 3rd Harmonic C u r r e n t Measurement  12.  T y p i c a l Waveform i n 3rd Harmonic Current Measurement  VI  13.  C i r c u i t f o r Fundamental Current Measurement  14.  Vector diagram f o r Fundamental C u r r e n t  17  Measurement VII  15.  3 r d Harmonic Current vs Fundamental V o l t a g e  16.  Fundamental C u r r e n t vs Fundamental V o l t a g e  17.  C a p a c i t y and R e s i s t a n c e vs Fundamental  18  Voltage  VIII  18.  C o e f f i c i e n t s vs Fundamental V o l t a g e  19.  V a r i a b l e Frequency Audio O s c i l l a t o r Amplifier  and 19  YIH  IX  X  (Cont'd.) 20.  C i r c u i t f o r 2nd Harmonic Measurement  21.  2nd Harmonic C u r r e n t vs D.C. B i a s V o l t a g e  22.  Equivalent C i r c u i t f o r Carrier  23.  Carrier Amplifier  24.  T y p i c a l Waveform o f C a r r i e r  Amplifier  Circuit Amplifier  Output  XI  2^.  Vector Diagram f o r Low Frequency Input  26.  Output Sideband Current vs Low Frequency Voltage  27-  Output Sideband C u r r e n t vs Frequency o f Low Frequency  28.  Output Power vs Load R e s i s t o r (Correct  29.  Tuning)  Output Power vs Load R e s i s t o r (Incorrect  XII  Tuning)  30.  Output Power vs H i g h Frequency V o l t a g e  31.  Low Frequency Input R e s i s t i v e C u r r e n t vs High Frequency V o l t a g e  32.  Low Frequency C a p a c i t i v e C u r r e n t vs High Frequency V o l t a g e  33•  Power Gain vs H i g h Frequency V o l t a g e  CONSTRUCTION, THEORY AND APPLICATION OF NON LINEAR TITANATE CONDENSERS  CHAPTER I INTRODUCTION A non l i n e a r c i r c u i t element i s one whose inductance, r e s i s t a n c e , transconductance  or capacitance i s not a constant  but depends on the v o l t a g e a p p l i e d .  Such elements a r e a l l mixers,  i . e . i f two v o l t a g e s are a p p l i e d t o them then c e r t a i n c u r r e n t s flow which depend s i m u l t a n e o u s l y on both, v o l t a g e s . o f non l i n e a r c i r c u i t  elements i s w i d e l y  This property  used.  The magnetic a m p l i f i e r which uses a non l i n e a r  induc-  tance has been d i s c u s s e d t h e o r e t i c a l l y and e x p e r i m e n t a l l y i n many papers  (1,2,3,4). The  c r y s t a l r e c t i f i e r o r non l i n e a r r e s i s t a n c e has been  w i d e l y used and thoroughly i n v e s t i g a t e d as a h i g h frequency mixer (5,6). Note:  The r e f e r e n c e s a r e t y p i c a l b u t n o t complete.  1.  Lamm, A.U. "The Transductor, D.C. p r e s a t u r a t e d R e a c t o r " . holm E s s e l t e A k t i e b o l a g 1 9 4 3 .  2.  B o y a j i a n , A. "Theory o f D-C E x c i t e d I r o n Core R e a c t o r s and Regu l a t o r s " . A.I.E.E.Trans, V o l . 4 3 , p . 9 1 9 , June 1 9 2 4 . C h i c a g o , 1 1 1 .  3.  C a s t e l l i n i , R.R. "The Magnetic A m p l i f i e r " , Proc.I.R.E., No 2 , pp 1 5 1 - 1 5 8 . Feb. 1 9 5 0 New York.  4.  Greene, W.E. " A p p l i c a t i o n s of Magnetic A m p l i f i e r s " . E l e c t r o n i c s , Sept. 1 9 4 7 .  5.  H e r o l d , E.W. "Frequency M i x i n g i n Diodes". No 1 0 , p 5 7 5 , O c t . 1 9 4 3 , New York.  6.  T o r r e y , H.C. and Whitmer, C A . " C r y s t a l R e c t i f i e r s " . M.I.T. R a d i a t i o n Laboratory S e r i e s . McGraw-Hill, New York, 1 9 4 8 .  Proc.I.R.E.,  Stock-  Vol.38,  Vol.31,  The non l i n e a r transconductance i s u n i v e r s a l l y used as a modulator. The non l i n e a r c a p a c i t a n c e has not however had wide use or t h e o r e t i c a l treatment because non l i n e a r condensers w i t h marked non l i n e a r p r o p e r t i e s have u n t i l r e c e n t l y been i m p o s s i b l e t o make. But r e c e n t l y i t has been found t h a t some of t h e compounds o f t i t a n i u m , n o t a b l y barium t i t a n a t e (BaTi03) a r e f e r r o e l e c t r i c a t room temperature, i.e;. a graph o f charge a g a i n s t v o l t a g e f o r a s u i t a b l e condenser h a v i n g BaTi03 as d i e l e c t r i c i s not a s t r a i g h t l i n e but i s a h y s t e r e s i s l o o p , s i m i l a r i n shape t o a h y s t e r e s i s loop of a ferromagnetic m a t e r i a l .  A condenser o f t h i s type i s  c a l l e d a non l i n e a r condenser - i t s c a p a c i t y i s a f u n c t i o n o f the voltage across i t . The p h y s i c a l t h e o r y f o r t h e marked d i e l e c t r i c behaviour of some of the compounds o f t i t a n i u m has r e c e i v e d much r e c e n t a t t e n t i o n (7,8,9,10). The r e s e a r c h d e s c r i b e d i n t h i s t h e s i s i s on t h e use o f non l i n e a r condensers w i t h BaTiOJ d i e l e c t r i c s , i n c i r c u i t s . 7.  Von H i p p e l , A., Breckenridge, R.G., Chesley, F.G., and L a s z l o Tisza. '•High D i e l e c t r i c Constant Ceramics". I n d . and Eng. Chem. Vol.38. No 11. pp 1097-1109, Nov. 1946, E a s t o n , Pa.  8.  Jonker, G.H. and Van Santen, J.H. " P r o p e r t i e s o f Barium T i t a n a t e i n Connection w i t h i t s C r y s t a l S t r u c t u r e " . S c i e n c e , Vol.109,  No 2843, PP 632-635, June 1949. 9.  Kay, H.F. and Rhodes, R.G. "Barium T i t a n a t e C r y s t a l s " . N a t . V o l . 160 p 126. J u l y 1947. London.  10. Wul, B.N. and Goldman, I . M . , " D i e l e c t r i c Constant o f Barium T i t a n a t e as a F u n c t i o n o f S t r e n g t h o f an A l t e r n a t i n g F i e l d , " Compt. Rend. Acad. S c i . V o l . 4 9 , PP 177-180, Oct. 1945  -3-  Dr. A. Tan der Z i e l , the d i r e c t o r o f the work, has a p p l i e d g e n e r a l mixer t h e o r y to non l i n e a r condensers i n an a u d i o c a r r i e r a m p l i f i e r e i r c u i t ( l l ) and i n a h i g h frequency (10 c y c l e s ) mixer c i r c u i t (12).  mega-  Very l i t t l e p u b l i s h e d e x p e r i m e n t a l  work on non l i n e a r condensers i n c i r c u i t s was found.  Donley  (13)  d i d some q u a l i t a t i v e experiments on non l i n e a r condensers, w i t h B a T i O j and S r T i O j d i e l e c t r i c s , o f l e s s than 100 mmfds i n a f r e quency t r i p l i n g c i r c u i t , a mixer a t 20 megacycles, and a f r e q u e n c y modulator a t 40 megacycles, but h i s r e s u l t s c o u l d not be used to check Van der Z i e l ' s t h e o r y . The primary o b j e c t o f t h i s r e s e a r c h was t o b u i l d the c i r c u i t s t h e o r e t i c a l l y a n a l y s e d by Van der Z i e l i n o r d e r to check e x p e r i m e n t a l l y h i s a n a l y s i s i n d e t a i l , and t o r e v i s e i t i f n e c e s s a r y . The secondary purpose was  t o e v a l u a t e the u s e f u l n e s s of  non l i n e a r condensers i n these and o t h e r c i r c u i t s . There f o l l o w s a b r i e f summary of the work done: Non l i n e a r condensers s u i t a b l e f o r use i n audio f r e q u e n c y c i r c u i t s were developed.  F i r s t r e s u l t s showed t h a t the condensers  made were more s u i t a b l e f o r experimental measurements i n a  slightly  d i f f e r e n t a m p l i f y i n g c i r c u i t t o t h a t a n a l y s e d by Van der Z i e l .  An  11.  Van der Z i e l , A. Report t o Defense Research Board o f Canada. U n i v e r s i t y o f B r i t i s h Columbia. J a n . 1949.  12.  Van der Z i e l , A. "On the M i x i n g P r o p e r t i e s o f Non L i n e a r Condensers", J o u r . App. Phys. Vol.19, No 11, pp 999-1006, Nov. 1948. L a n c a s t e r , Pa*  13.  Donley, H.L., '^Effect o f F i e l d S t r e n g t h on D i e l e c t r i c P r o p e r t i e s o f Barium S t r o n t i u m T i t a n a t e " . , R.C.A. Rev. V o l . V I I I , No 3, PP 539-553, P r i n c e t o n , New J e r s e y , Sept. 1947.  a d a p t i o n o f h i s t h e o r y was made and t h e c o n c l u s i o n s o f the theory thus changed have been checked e x p e r i m e n t a l l y w i t h good but not complete  agreement.  An account of the main audio frequency p r o -  p e r t i e s o f these non l i n e a r condensers  i s now p o s s i b l e .  No measurements have been done on the h i g h frequency mixer, a l t h o u g h the c i r c u i t was r o u g h l y b u i l t , but t h e r e s u l t s o f the work a t low f r e q u e n c i e s should g r e a t l y a s s i s t work a t h i g h frequencies.  CHAPTER I I MAZING OF NON LINEAR CONDENSERS FOR AUDIO FREQUENCIES A t the b e g i n n i n g o f the work i t was known t h a t  BaTiOj  p r o p e r l y prepared has a marked h y s t e r e s i s l o o p a t 4800 v o l t s p e r cm. (14)  The problem was t o use t h i s knowledge t o make a  condenser  of c a p a c i t y l a r g e enough f o r audio c i r c u i t s (.001 t o .01 mfds), showing marked non l i n e a r i t y a t v o l t a g e s not exceeding JOOv A . C , having a breakdown v o l t a g e a t l e a s t above the v o l t a g e a t which non l i n e a r i t y was marked.", and h a v i n g some means of removing the heat generated by h y s t e r e s i s l o s s e s . The p r e p a r a t i o n developed f o r condensers  satisfying  these requirements has t h r e e main s t a g e s : 1.  P r e p a r a t i o n of a Sample o f Bulk D i e l e c t r i c The method used was very s i m i l a r t o t h a t used by Yon  14. See R e f . 7.  H i p p e l and h i s co-workers (15) w i t h a few minor  changes.  Barium t i t a n a t e i n d r y powder form ( o b t a i n e d from T i t a n i u m A l l o y M a n u f a c t u r i n g Co.) was p r e s s e d i n a l / 2 p r e s s a t 60000 l b s / s q . i n .  t o a t h i c k n e s s o f about  d i s c s were then p l a c e d on p l a t i n u m f o i l  t t  diam.  .5 mm.  The  i n an alundum c r u c i b l e  and passed through the f o l l o w i n g temperature c y c l e i n a h i g h temperature f u r n a c e : An i n c r e a s e o f 100°C per hour f o r n e a r l y 13-1/2 hours to a temperature o f 1350°C. A constant temperature o f 1350°C f o r 6 h o u r s . A decrease i n temperature a t the c o o l i n g r a t e o f the f u r n a c e . (The f u r n a c e took 36 hours t o c o o l from 1350°C t o room temperature). A f t e r the s i n t e r i n g c y c l e the d i e l e c t r i c was a hard y e l l o w brown, b r i t t l e d i s c which had shrunk about 20% of i t s original size.  The f u r n a c e used ( d i a g . l ) was made a t t h e b e g i n -  n i n g o f t h e work and was c o n t r o l l e d w i t h a Wheelco C h r o n a t r o l P o t e n t i o t r o l Model 23241, which c o n t r o l l e d the power through a S u p e r i o r E l e c t r i c Co. Powerstat No 1156.  A d i s c which may be c u t  f o r any 24 hour temperature c y c l e i s the master c o n t r o l f o r the P o t e n t i o t r o l which c o n t r o l s t h e temperature i n s i d e the f u r n a c e t o about 5°C.  I t was found n e c e s s a r y t o p l o t the r e l a t i o n between  temperature a t the sample and that g i v e n by the f u r n a c e thermocouple s i n c e these are n o t p h y s i c a l l y a t the same p l a c e . diagram o f the furnace and c o n t r o l i s g i v e n i n diagram 2. 2.  G r i n d i n g t h e d i e l e c t r i c t o 0.1 mm.  15. See Ref. 7.  A blook  PLATE I  Fa«tn<J Rat^ if  - Sfe=J £bj&m<j Alwndv/m S t e e l  Ahwwfc/m Muff la.  O m u i t  Casing  3ii-0-Cc!l  CH  2A  Tnlw5atio«  Icsrv^  VI1VQS<I  Pt  (O  "Jv»b«.  Rh  as above  FURNACE. D i a g r a m  I  FURNACE CONTROL.  S 5 0 Amps A . C .  Tun peicdvwa.  -6To reduoe the v o l t a g e n e c e s s a r y f o r marked nonl i n e a r i t y , to reduce the h y s t e r e s i s l o s s e s , and t o i n c r e a s e the c a p a c i t y , as t h i n a d i e l e c t r i c as p o s s i b l e i s r e q u i r e d . The it  technique used was borrowed from g e o l o g i s t s who used  to prepare t h i n r o c k samples. One  s i d e o f t h e s i n t e r e d d i s c was ground f l a t on a g l a s s  p l a t e w i t h carborundum and water and then p o l i s h e d on another g l a s s p l a t e w i t h aluminum oxide and water. melted on the f l a t ing  o f the f l a t  A t h i n f i l m o f beeswax was  s i d e o f the d i s c and allowed to harden.  s i d e was r e p e a t e d .  P a r t i c l e s o f carborundum became  embedded i n the wax which was f o r c e d i n t o t i n y depressions surface o f t h e d i s c .  G r i n d i n g was continued  t h a t i n the depressions wax being  on the  u n t i l a l l wax except  had been removed, f i n a l  done on aluminum o x i d e .  Grind-  removal o f the  A coat o f Dupont Conductive  C o a t i n g No 4 3 5 1 s i l v e r e l e c t r o d e p a i n t was a p p l i e d to the whole flat  surfaee  and allowed to dry.  The d i s c was heated t o 600°C,  h e l d t h e r e f o r 10 minutes, and allowed t o c o o l . f i r e d the e l e c t r o d e t o the d i e l e c t r i c . introduced  to plug t i n y holes  This  operation  The wax technique was  i n the d i e l e c t r i c to prevent  flow  o f wet s i l v e r p a i n t i n t o them. The  d i s c was next mounted s i l v e r e d s i d e down w i t h warm  Canada balsam on a 1 " x 2 " g l a s s microscope s l i d e and t h e d i e l e c t r i c ground down and p o l i s h e d u n t i l t h i c k n e s s was .1mm.  The waxing  procedure was r e p e a t e d and t h e upper e l e c t r o d e a p p l i e d t o the d i s c while s t i l l (diag.3)  on t h e s l i d e .  T h i s e l e c t r o d e was a p p l i e d i n 6 t r i a n g l e s  from which 6 condensers were made.  T h i s made a semi-  v a r i a b l e condenser from the whole d i s c and permitted  a broken down  -7s e c t i o n t o be removed. A f t e r the s i l v e r p a i n t was  dry the d i s c was  carefully  removed by h e a t i n g from i t s monnt, cooled, and dipped i n benzene to remove the Canada balsam and the upper e l e c t r o d e f i r e d  as  before. M e c h a n i c a l l y the d i s c c o u l d have been ground to . 0 3 and s t i l l handled without d i s c s were t h i n n e r than  breakage but i t was  . 1 mm.  o c c u r r e d b e f o r e any v o l t a g e was ages f o r a l a r g e percentage down was  mm.  found t h a t i f the  e l e c t r i c a l breakdown had  either  a p p l i e d or d i d occur a t low  o f the condensers made.  This  volt-  break-  a t t r i b u t e d to t i n y h o l e s and f l a w s i n the d i e l e c t r i c  which the e l e c t r o d e p a i n t flowed or which i n some o t h e r way  into  caused  breakdown.  The waxing technique reduced the l i m i t i n g t h i c k n e s s  from  t o .1  .2 mm.  mm.  I n an e f f o r t to f u r t h e r reduce  the l i m i t i n g t h i c k n e s s  e l e c t r o d e s of t i n f o i l and g o l d were t r i e d without s u c c e s s .  A  search of the l i t e r a t u r e y i e l d e d a paper by Howatt and co-workers (16)  on the f a b r i c a t i o n o f t h i n ceramic  t h i s paper a method was .15  mm.  sheets f o r c a p a c i t o r s .  In  g i v e n f o r making sheets of minimum t h i c k n e s s  having a breakdown v o l t a g e f o r BaTiC-3  t h a t found i n the above condensers.  o  f  about f o u r times  T h e i r method i n c l u d e d i n the  o r i g i n a l mix b e f o r e s i n t e r i n g bonding m a t e r i a l s t o remove f l a w s i n the f i n a l p r o d u c t . skill.  I t was  T h e i r method however was  complex and r e q u i r e d  decided t h e r e f o r e t o o b t a i n whatever r e s u l t s c o u l d  be obtained with the p r e s e n t technique before combining 16.  the  two  Howatt, G.N., Breckenridge, B.G., and Brownlow, J.M., " F a b r i c a t i o n of T h i n Ceramic Sheets f o r C a p a c i t o r s " , J o u r . Am. Cer. S o c , Y o l . 3 0 , pp 237-242, 1 9 4 7 .  PLATE CON DENSER '• ni  i  II I I  I II  in  H U I I I II  11  CONDENSER  ELECT&O&E  II  w i i ' i ' m "•"inn M T U M I '  •n'l'i  Fatting Pa^C- 7  ii" " -Hf* 1  GaR-Og OiclectVic  Sleek  Tin .fiiil  Wowntjnq Holes  Diacjra'm 4*  OLaqraTn 3  COMDEWS£R STAND Copper Biack  Coppei  MOUNT  0a.Tr  DiaqTam 5  DteiectVic  -8techniques.  The combination i t i s thought would r e a l i z e the  mechanical l i m i t of .03 mm. f o r the above mentioned  discs.  Another method t r i e d t o p l u g t h e h o l e s was t o s e p a r a t e by the t h i n d i s c w i t h o u t e l e c t r o d e s two l i q u i d s which formed a p r e c i p i t a t e as t h e y met i n the h o l e s o f t h e d i e l e c t r i c .  This  method was used s u c c e s s f u l l y w i t h CaC03 but i t s breakdown v o l t a g e i s n o t s u f f i c i e n t l y h i g h above t h a t o b t a i n e d by t h e waxing method. It  i s thought that t h i s method c o u l d be developed i f a m a t e r i a l  w i t h a h i g h breakdown v a l u e and which could be e a s i l y p r e c i p i t a t e d c o u l d be found. 3.  Attachment  of Leads and mounting  o f Condensers.  The condensers were mounted on a s m a l l copper b l o c k 1-1/8" x l / 2 " x 3 / l 6 " w i t h two countersunk h o l e s 7/8" a p a r t .  A  sheet o f t i n f o i l was p l a c e d on the b l o c k and t h e d i s c , t h e oxide on the s i l v e r coat h a v i n g been removed w i t h emery, p l a c e d on the tin  f o i l w i t h t h e d i v i d e d e l e c t r o d e s i d e up. A t h i n copper w i r e  with a l i t t l e electrode. the  t i n f o i l wrapped on the end was p l a c e d on each upper  The whole was heated u n t i l the t i n melted and s e a l e d  d i s c t o t h e b l o c k and a l e a d t o each upper e l e c t r o d e .  Solder  may n o t r e p l a c e t i n i n t h i s o p e r a t i o n because s o l d e r tends t o d i s s o l v e the s i l v e r e l e c t r o d e . e l e c t r o d e to s e a l the l e a d s ,  Beeswax was melted t o the upper (diag. 4 f o r details)  When a con-  denser i s t o be used i t I s screwed to a s o l i d copper ( d i a g . 5 ) which may be put i n a beaker o f water f o r c o o l i n g . A summary o f the p r o p e r t i e s o f an average condenser o f t h i s type i s now g i v e n :  •9-  Thickness . 1 mm. Area o f d i e l e c t r i c used approximately . 6 sq.cms. Expected breakdown 2 5 0 v A.C. R.M.S. Recommended Max. O p e r a t i n g V o l t s 1 5 0 v A.C. R.M.S. Expected t o t a l c a p a c i t y o f 6 s e c t i o n s a t low v o l t a g e . 0 0 8 mfd. Temperature r i s e caused by h y s t e r e s i s h e a t i n g v e r y s m a l l on copper mount and base.  I f copper base i s n o t  used h y s t e r e s i s h e a t i n g causes temperature  of d i e l e c t r i c  to r i s e above 1 2 0 ° C ( C u r i e P o i n t f o r BaTiOj) and a l l non l i n e a r properties are l o s t . V o l t a g e n e c e s s a r y f o r marked non l i n e a r i t y 8 0 v A.C. R.M.S. These p r o p e r t i e s s a t i s f i e d a l l the requirements and i t was d e c i d e d t o proceed w i t h t e s t i n g these condensers was made.  before an attempt t o improve  I t was o r i g i n a l l y planned t o t e s t  con-  densers made o f v a r i o u s t i t a n a t e compositions but time has not p e r m i t t e d t h i s t o be done.  CHAPTER I I I THE HYSTERESIS LOOP AND BASIC E 0 J J A T I 0 N OF THE NON LINEAR CONDENSER 1.  The h y s t e r e s i s l o o p . The h y s t e r e s i s l o o p s of the condensers made were observed  at d i f f e r e n t  A.C. and D.C. v o l t a g e s a c r o s s the condensers i n Sawyer  and Tower's c i r c u i t ( 1 7 ) w i t h a b i a s s i n g a d d i t i o n ( d i a g 6 ) .  Full  1 7 . Sawyer,C.B. and Tower,CH. "Rochelle S a l t as a D i e l e c t r i c " Phys. Rev. V o l . 3 5 No 3 PP 2 6 9 - 2 7 3 Feb. 1 9 3 0 . M i n n e a p o l i s , Minn.  -10s i z e d t r a c i n g s from the scope f a c e a r e g i v e n i n diagram 7 f o r a t y p i c a l specimen a t 60 c.p.s.  60 c.p.s. was used f o r convenience.  The approximate h y s t e r e s i s l o s s f o r a g i v e n loop was c a l c u l a t e d as f o l l o w s : L e t 7 be the D.C. V o l t a g e r e q u i r e d a c r o s s the v e r t i c a l t e r m i n a l s t o move the spot 1  plate  inch.  L e t H be the corresponding h o r i z o n t a l v o l t a g e . I t i s assumed that a l l the a p p l i e d v o l t a g e appears a c r o s s N.L.C. .• l  V e r t i c a l d e f l e c t i o n means VC = V _ coulombs on N.L.C.  n  T0~7 l  H o r i z o n t a l d e f l e c t i o n means H ( R i + R 2 ) - 20 H v o l t s Kg a c r o s s N.L.C.  t t  Now V - 27,  H = .54  • • 1 s q . i n . o f scope f a c e corresponds t o (20H)(V) = (20)(54)(27)  io7  io7  = 2.9 10?  Consider the l o o p a t 1 5 0 v A.C.  joules. 0 Bias.  Measured a r e a = . 2 5 2 s q . i n s . Energy L o s s per c y c l e « ( . 2 5 2 ) ( 2 . 9 ) j o u l e s . (lO?) A t 60 c.p.s. Power Loss - ( . 2 5 2 ) ( 2 . 9 ) ( 6 0 ) *; .045-watts.  (lb>) A t 10000 c.p.s. Power Loss - ( . 2 5 2 ) ( 2 ^ 9 ) ( 1 0 0 0 0 ) (lO?)  •» 7.5 w a t t s .  T h i s l a t t e r c a l c u l a t i o n has assumed t h e h y s t e r e s i s loop has the same area a t 1 0 0 0 0 c.p.s. as a t 60 c.p.s. to be n e a r l y t r u e ( d i a g . 8 ) .  T h i s was found  Thus 7.5 watts i s approximately the  power t o be d i s s i p a t e d i n such a condenser i f i t i s to be r u n a t 150 v 10000 c.p.s. the c a r r i e r  T h i s power must be s u p p l i e d by the c a r r i e r o f  amplifier.  PLATE HYST£R£5IS  Hi  CIRCUIT  .  . • •  SNkq  ISOV D . C -  T  23 Q ^ J s o v D-C. 1~F  Step Uptww*»'fov-w«Mr -!  Variac  ;25D  2  o  M.L.C  A.C.  hrr  >  C.R.O  -j H'hJr i c  .91  A.c;  . HYSTER£SI3.. LOOPS  V O L T M E T E R  So  I O O  i  (50  IOO  sro  SO  o  ocvoLTM ares  iso  so  IOO  •OO  Dlaqram 7  WgjATjOPj OF LOOP WITH FREQUENCY  Dlacjva*m 8  ISO  IOO  •so  ISX)  -11  2.  Basic Equation  of the Non  L i n e a r Condenser.  The problem i s t o p l a c e the i n f o r m a t i o n contained  i n the  g i v e n h y s t e r e s i s l o o p s i n an equation which can be used to c a l c u l a t e the performance of the condensers i n c i r c u i t s .  The problem w i l l  be s o l v e d i f an e x p r e s s i o n f o r the c u r r e n t f l o w i n g i n a non  linear  condenser can be w r i t t e n as a f u n c t i o n o f the v o l t a g e a c r o s s i t . Without b i a s the h y s t e r e s i s l o o p s are symmetrical t a l axis.  about the  horizon-  Hence the c u r r e n t f l o w i n g i f a s i n e wave o f v o l t a g e i s  applied w i l l  c o n t a i n o n l y odd  harmonics.  The non l i n e a r condenser without a condenser C C  i n p a r a l l e l with a r e s i s t o r R  +  has an  +  bias i s represented +  by  (diag. 9 ) .  equation:  (1)  Q, - aV + bV? + cV^ I = e  R  +  ad7 +  has an I  R  dT  3bV2<iv  dT  + ^eV^dV  dT  equation:  - dV + eV3  + fV?  .(2)  where a,b,c,d,e,f ... are f u n c t i o n s o f the maximum value of Voltage  (Vmax) a p p l i e d to the condenser.  dependent on frequency  a,b,c  are  A.C.  slightly  and d,e,f are s t r o n g l y dependent on  frequency.  Thus t o t a l c u r r e n t i n t o condensers i s I - adV  + 3bV2dV + ^cV^-dV + dV + eV? + f V ^  dT The  dT  (3)  dT  c o e f f i c i e n t s a,b,c,d, e t c . c o u l d be found by an expansion o f  (3) i n t o fundamental and harmonic c u r r e n t s f o r a s i n e wave o f a p p l i e d v o l t a g e and a F o u r i e r a n a l y s i s of an observed c u r r e n t wave. The  exact  s o l u t i o n of the problem i n v o l v e s t e d i o u s  experimental  P L A T E  I V  NON-LINEAR COMDEMSEfi WITHOUT ©IAS  N.L.C.  FOR  C  FOR  DiAcjraTn  9  0.2'  and t h e o r e t i c a l work. In  the t h e o r y developed  i n the p r e s e n t r e s e a r c h (3)  has been approximated t o : I = adV + 2bV dV  (4)  2  dt  dt  where a and b a r e f u n c t i o n s o f Ymax. The n e g l e c t i o n o f dV r e p r e s e n t s a s e r i o u s q u a n t i t a t i v e e r r o r i n the t h e o r y but not a s e r i o u s q u a l i t a t i v e e r r o r . was  concerned  The p r e s e n t work  i n the main w i t h the q u a l i t a t i v e r e s u l t s o f ( 4 ) .  Thus t h e model used o f the non l i n e a r condenser c o n s i s t s of a non l i n e a r c a p a c i t y w i t h the e q u a t i o n :  Q, = aV + bY? where b must be n e g a t i v e from the shape o f the h y s t e r e s i s l o o p . I f a b i a s i s a p p l i e d t o the condenser even power terms are a l s o i n t r o d u c e d i n t o (3) and even harmonic c u r r e n t s f l o w . I f the shape at the h y s t e r e s i s l o o p can be a l t e r e d u n t i l i t i s almost r e c t a n g u l a r as has been done with f e r r o m a g n e t i c s , then w i t h s u i t a b l e b i a s the model o f non l i n e a r condenser c o u l d be a l t e r e d t o a capaci t y w i t h the e q u a t i o n : Q, m aV + b V  2  I = adV + 2b7dV dt d? T h i s was t h e model used by Tan der Z i e l i n h i s two papers  (5) ( 1 8 ) and  (19). S i n c e t h e condensers  which had been made were more  r e a d i l y r e p r e s e n t e d by (4) than by ( 5 ) even w i t h l a r g e b i a s , i t 18. See R e f . 1 1 . 19.  See R e f . 1 2 .  -13  was decided t o work out Van der Z i e l ' s theory u s i n g (4)  rather (4).  than (j?)and attempt t o b u i l d a c a r r i e r a m p l i f i e r based on This t h e o r e t i c a l and experimental  work w i l l be d e s c r i b e d  later.  The measurements f o r the c a l c u l a t i o n o f the c o e f f i c i e n t s a and b i n (4) given.  as f u n c t i o n s o f Vjaax a t 9000 c.p.s. w i l l next be  T h i s frequency was chosen as t h e c a r r i e r frequency  rather  than 1 0 , 0 0 0 c.pi-s. because some of t h e p a r t s f o r the c a r r i e r a m p l i f i e r tuned more r e a d i l y t o 9000 c.p.s.  CHAPTER 17 DETERMINATION OF COEFFICIENTS IN BASIC EQUATION 1.  D e r i v a t i o n o f Equations  for Coefficients.  The c o e f f i c i e n t s a and b c o u l d be found by t h e g e n e r a l method mentioned above but the method g i v e n below i s somewhat simpler.  I The n e g l e c t i o n o f condenser l o s s e s i n (4)  w i l l not  a f f e c t the v a l i d i t y o f t h e c a l c u l a t i o n s f o r a and b s i n c e  these  are concerned o n l y w i t h o a p a c i t i v e c u r r e n t s ( p r o v i d e d d i s constant) . I f a v o l t a g e V s i n wt I s a p p l i e d , t o the non l i n e a r condenser the r e s u l t i n g c u r r e n t i s : I ( t ) = a ay(t) + 3b ( V ( t ) ) dt  2  dV(t)  at  « waVcos wt + 3/4 bwV^cos wt - 3/4  bwY5cos3wt»..(6)  Since o n l y fundamental and 3 r d harmonic c u r r e n t s a r e p r e s e n t a and  -14b can be determined i n terms o f fundamental and 3 r d harmonic currents i n a suitable c i r c u i t .  An i d e a l c i r c u i t would have a  generator o f V s i n wt w i t h zero impedance  to 3rd harmonic.  This  i s d i f f i c u l t t o make but an approximation t o i t ( d i a g 10) was constructed.  The e q u i v a l e n t c i r c u i t from which c a l c u l a t i o n s a r e  made i s g i v e n i n diagram 11. Y o l t a g e a p p l i e d t o non l i n e a r condenser i s : V s i n wt + V j S i n ( 3wt + 0) where  «  V  S u b s t i t u t i o n of t h i s voltage into I(t)  - adV(t) + 3b ( V ( t ) ) d V ( t ) 2  dT  ~~oT  g i v e s c u r r e n t s a t fundamental and 3rd harmonic. Main terms a t fundamental frequency a r e : I n j t ) » waVcos wt + 3 / 4 bw v3cos wt Main terms a t 3rd harmonic f r e q u e n c y a r e : • 3aw V j cos (3wt + 0) - 3bwv3 00s 3wt  I^t) In  complex I  x  13  but  notation:  = j (waV + 3/4- bw V?) - j (3aw V  5  (7)  - 3 / 4 bw V  2  V ) +  I3 = - V j "IT  zll J'(3aw V - 3/4 bw V V*). R where V - V e ^ , V3 - V j e J ^ w t + t f ) , Y =  (8)  2  5  - waV + 3/4 bwV?  From (7)  +  - VeJ3wt (7 ) 1  From (8) i t can be shown t h a t : 1+  9a w2R «? 2  R2 2 2y6 b  w  IS" - "v§2  ( j 9  -15(7^)  and (9)  may  now  (Because Y j « T a where  m  2R2V2  i^  L  R  2  .(10)  W V  -  2  -q. - J q  =  4mt\  2  2 I ;  b  some e x p r e s s i o n s nave been e l i m i n a t e d )  \n 2m w  t =  be s o l v e d s i m u l t a n e o u s l y t o g i v e :  v 2  5  2  * 4pr  2p  ^  where P  8  3  ^ W  R 2 V  2  q = |1 w R Y 2  = Vj V 2  r  (10) and (11) of  2  .(11)  )  8  2 5  I-jV  + 91"!*  5  2^2 V^R*  j  )  )  are the r e q u i r e d e x p r e s s i o n s f o r a and b i n terms  q u a n t i t i e s whose measurement next w i l l be d e s c r i b e d .  When the  measured q u a n t i t i e s were s u b s t i t u t e d i n these e x p r e s s i o n s one the two p o s s i b l e v a l u e s f o r a and b was  of  e l i m i n a t e d because b had  to be n e g a t i v e . A l l v a l u e s appearing i n the above equations a r e peak values.  V o l t s , Amps, Ohms were used. S i n c e the flow o f fundamental c u r r e n t i n the above  c u l a t i o n s i s independent of  the fundamental c u r r e n t may  measured i n a separate c i r c u i t to t h a t shown above.  cal-  be  T h i s was  done.  The f o l l o w i n g measurements were done on a t y p i c a l condenser 2.  o f c a p a c i t y t h a t d e s i r e d f o r the c a r r i e r a m p l i f i e r (.004j>mfd).  Measurement o f 3rd Harmonic Current f l o w i n g i n a non condenser w i t h v a r y i n g Fundamental V o l t a g e . Diagram 10 g i v e s the c i r c u i t  used.  linear  PLATE V CUSCUIT  PQR 3 R D H A R M O N I C  0 Faci«<,  CURfigSsiT  MEASUREMENT  £  S O O O e^-s.  28 mh  N.L.C.  •ha  Series TunaJ  3-rd H o t w o w c  Diagram  E Q U I V A L E N T CIRCUIT O F  10  ABOVS A T . 3 R D  HARMONtC  FReQUE&Sy  T M.UC  T Y P I C A L WAVEFORM  Q&SE.RVEP  ATZ(0'A6to)  A s s u m e d 3-«d H e r  O bssrvad  DiaqTam  12.  -16The c i r c u i t , was tuned t o p a r a l l e l resonance a t fundamental frequency by c . 1  The v o l t a g e a t Z was observed on a c a l i b r a t e d  oscillo-  scope which was p a r t o f the tuned c i r c u i t , a c o r r e c t i o n made f o r the  presence o f the fundamental and the 3rd harmonic c u r r e n t ( I ^ )  through t h e s e r i e s tuned c i r c u i t c a l c u l a t e d .  A t y p i c a l wave form  observed a t Z i s shown i n diagram 12 with the harmonic composition assigned to i t . to v o l t s .  The peak d e f l e c t i o n was observed and converted  One h a l f o f the peak fundamental v o l t a g e c a l c u l a t e d a t  Z was s u b t r a c t e d and the remaining v o l t a g e converted to R.M.S. value.  D i v i s i o n by t h e r e a c t a n c e o f the c o i l gave t h e r e q u i r e d  value o f 3rd harmonic  current.  Example: R.M.S. V o l t s a t E a t 9000 c.p.s. - 70 O s c i l l o s c o p e S e n s i t i v i t y » 44 V o l t s p e r i n c h . Max Observed D e f l e c t i o n = l / 2 T o t a l D e f l e c t i o n on Scope « 1 . 6 i n s . Peak V o l t a g e - ( 1 . 6 ) ( 4 4 ) = 70.4 v o l t s C a l c u l a t e d fundamental peak v o l t a g e a c r o s s c o i l »(D(70)(1.414) - 12.4 v o l t s  to")  Peak Obs. - 1/2 peak fundamental «* Peak 3rd Bar. •» 6 4 . 2 v o l t s R.M.S. 3rd Har. C u r r e n t - ( 6 4 . 2 ) ( 1 0 0 0 ) ( l )  « 9 . 6 m-as  ( 1 . 4 1 4 H W L )(3)  The r e s u l t s f o r v o l t a g e s a t E from 0 t o 150 a r e given w i t h the r e s u l t s o f the next s e c t i o n s i n diagram 1 5 .  -17-  3.  Measurement o f Fundamental o u r r e n t f l o w i n g i n non l i n e a r condenser w i t h v a r y i n g Fundamental v o l t a g e . Diagram 13 g i v e s the c i r c u i t . The V o l t a g e s a t E and Y were observed  on a Cossor  double beam o s c i l l o s c o p e and the c a p a c i t i v e and r e s i s t a n c e components o f fundamental c u r r e n t f l o w i n g i n the n o n l i n e a r ser  conden-  c a l c u l a t e d from t h e v e c t o r diagram 14.  Example:  R.M.S. V o l t s a t E a t 9000 c.p.s. - 70 © from double beam o s c i l l o s c o p e - 27.5® V  - lb.O v  I  = (V )(wC) =30.6 m-as  Let  y  V V  x  2  - V  - ( V ) cos 0 - 70 - 18 cos 27-5 - 54.0  E  y  - (T y ) S i n e = l b s i n 27.5 - 8.3  Tan 0 - V2 - 8.3  7T  5370"  = .1537  0 = 8.7°  • y r = 9 O - e - 0 = 53-8° R e s i s t i v e Current I  - I cos"v|r > (30.6)(,59) - l b . l m-as  R  C a p a c i t i v e Current I  0  * I sin-\^= (30 . 6)(.807) - 24 . 7 m-as  Voltage across condenser = V  2  cosee 0 » 8jjS^ = 55 v o l t s  The r e s u l t s from these o b s e r v a t i o n s and c a l c u l a t i o n s are g i v e n i n diagram 16.  The values of c a p a c i t y and r e s i s t a n c e  f o r t h e non l i n e a r condenser are shown i n diagram 17.  I t can be  seen t h a t the n e g l e c t i o n o f the condenser r e s i s t a n c e w i l l duce a s e r i o u s q u a n t i t a t i v e e r r o r i n the t h e o r y .  intro-  Further a t low  v o l t a g e s t h i s r e s i s t a n c e i s non l i n e a r and w i l l i n t r o d u c e a f u r t h e r error.  PLATE VI CIRCUIT frOR F U N D A M E N T A L CU8R6WT M6ASUHEMENT  Dvac|ram 13  VECTOR DIAGRAM FOR fUMDAMENTAL, CUftftgMT  MEASUREMENT  -18-  4.  C a l c u l a t i o n of a and b. The d a t a f o r the c a l c u l a t i o n o f a and b from formulas  (10)  and (11)  has now been  assembled.  The constants used are g i v e n h e r e :  w =  (5.b5)(l0 ) 4  w2 = ( 3 . 2 ) ( l 0 9 ) R - 80 V j = IjR Iii  l3»  V, are taken from diagrams 15  and 16 and c o n v e r t e d to peak  value s. The v a l u e s o f a and b f o r v o l t a g e s from 0 t o 120 a t 9000 c.p.s. are g i v e n i n diagram  18.  A d d i t i o n to Chapter IV. The f o l l o w i n g two s e c t i o n s are added t o Chapter IV because t h e y are o f f the main l i n e of t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s d i s c u s s e d i n t h i s t h e s i s but are most c l o s e l y connected to the d i s c u s s i o n of Chapter IV and should n o t be omitted. 5.  Audio O s c i l l a t o r and Power A m p l i f i e r . This u n i t was  the c a r r i e r a m p l i f i e r .  designed as the source o f the c a r r i e r f o r I t has a push p u l l output capable o f  d e l i v e r i n g 18 watts and i s v a r i a b l e from 5000 c.p.s. to 15000 c.p.s. S i n c e i t i s a s t a n d a r d o s c i l l a t o r and a m p l i f i e r i t s d e s i g n w i l l not be d i s c u s s e d .  C i r c u i t i s g i v e n i n diagram 1 9 .  One h a l f o f the  output was used f o r measurements d e s c r i b e d i n s e c t i o n s 2 and 3 o f t h i s chapter. 6.  Measurement o f 2nd Harmonic current f l o w i n g i n b i a s s e d l i n e a r condensers.  non  ^  PLATE VII  FOR X Y P ^ A i . . . ; .CQWDEH^R^;  ALL£K&PH§  I  FUNDAMENTAL VoLTS  DLaqram J5,  3V  FUNDAMENTAL  O  TOTAL*  5C FWWDAMEWTAL VoLTS  D'uiqraro 16  I V O  \fc)LTS  Dlaqram 17  FUNDAWENTAL  \&LTS  •~" Piaqram IB .  ^  i  •19A f t e r i t had been d e c i d e d t o b u i l d an a m p l i f i e r based on e q u a t i o n (4) t h i s measurement assumed secondary importance s i n c e i t would have been t h e d e c i s i v e measurement of an a m p l i f i e r based on equation (j?). Both s i d e s of the push p u l l output of the a u d i o ampl i f i e r were used i n the c i r c u i t of diagram 20.  The two  condensers  were matched as c l o s e l y as was p o s s i b l e i n the group o f condensers a v a i l a b l e and p l a c e d i n s e r i e s from p l a t e t o p l a t e o f the 6L6S. Adjustment  of R and C balanced t h e condensers with r e s p e c t t o the  fundamental and t h e fundamental c u r r e n t f l o w i n g through s e r i e s tuned c i r c u i t 1 c o u l d be reduced to z e r o .  3rd harmonic  current  f l o w i n g from X to ground c o u l d not a l s o be matched w i t h r e s p e c t to the 3 r d  harmonic. The v o l t a g e E t o D was h e l d f i x e d d u r i n g the experiment  a t 200v R.M.S. at 9000 c.p.s. and the v o l t a g e B t o ground observed f o r each value o f b i a s on the condensers.  The  from both condensers flowed through c i r c u i t c u l a t e d when the v o l t a g e a t B was known. retimed f o r each b i a s s e t t i n g .  2nd harmonic  2 and c o u l d be  current cal-  The c i r c u i t had t o be  The r e s u l t s are given i n diagram  21. The non c o i n c i d e n c e of the  curves f o r i n c r e a s i n g and  d e c r e a s i n g b i a s i s not c o n s i d e r e d s i g n i f i c a n t . by s l i g h t permanent e l e c t r i f i c a t i o n . harmonic  I t may  be caused  The s m a l l r e s i d u a l  c u r r e n t a t zero b i a s had a 90° phase  2nd  s h i f t to t h e c u r r e n t  r e s u l t i n g from b i a s s e d condensers and was p a r t l y caused by det u n i n g o f the main c i r c u i t ( t h i s p o i n t i s not yet understood) and p a r t l y by a small amount of D.C.  r e c t i f i c a t i o n i n the condensers.  VIII  Dlaqram 19 CIRCUIT FOR 2NQHAftMOMtC  "*" 2ND  HAftWfcONlC  •  -60  CURRENT  -MBASVftMWV  N.4..C.  Oiacpftim 20 CURRENT  •  *  --|o _  •  -^0 >.C Stag  «  EC. © I A S  «  '  '  *  *,  »  oa*t» Condenser  Duaqrcurn 21  VQLTASe.  •  -20  The  condensers were found t o have a D.C.  r e s i s t a n c e o f about  3 Megohms and  to give 1 microamp of D.C.  lOOv A.C.  a c r o s s them a t 9000 c.p.s.  was  harmonic c u r r e n t further  at zero bias., was  r e c t i f i e d c u r r e n t when The r e s i d u a l  2nd  however small and was  not  investigated.  CHAPTER V THEORY 0E A DIELECTRIC CARRIER AMPLIFIER The a n a l y s i s (20)  t h e o r y given f o l l o w s the p a t t e r n o f Van but  r a t h e r than 1.  Q. = aV +  bV3  Q. = aV +  bV  2  P r i n c i p l e of A m p l i f i c a t i o n at h i g h frequency (ang  small v o l t a g e a t low f r e q u e n c y (ang  2w - p f l o w .  i n these s i d e bands i s g r e a t e r  The  low  Some of the  D e r i v a t i o n of the  See  a  Currents of contained  frequency i n p u t power.  at angular f r e q u e n c y  c a r r i e r power has  been t r a n s f e r r e d  frequency s i g n a l .  a p p l i e d t o a non 20.  a  c o r r e c t phase and the d e t e c t i o n o f the r e s u l t i n g mod-  ulated s i g n a l .  2.  and  be r e a l i z e d i n a s u i t a b l e  c i r c u i t by the a d d i t i o n of a small c u r r e n t  to the  above.  p o t e n t i a l power  than the low  p o t e n t i a l power a m p l i f i c a t i o n may  2w a t the  f r e q . w)  f r e q . p) are a p p l i e d to  l i n e a r condenser of the type d e s c r i b e d  frequency 2w + p and  The  1  i s a p p l i e d to a condenser h a v i n g :  A l a r g e voltage  non  der Z i e l s  Ref.11.  currents  f l o w i n g when two  l i n e a r condenser.  signals  are  -21  Let  Y S i n wt and. P S i n p t ( V » P , w>^p)  be a p p l i e d t o a non l i n e a r condenser Kt)  = a av(t)  at  having  4- 5 b ( v ( t ) ) a v ( t ) 2  at  In t h i s case V ( t ) - V S i n v/t + P S i n p t By s u b s t i t u t i o n i t can be shown t h a t : I(t)  = aw V cos wt + ap P cos p t + ^ b j ^ l . wY^ o s wt + 0  wP2V cos vrt BiwY^  — r  C  o s Jwt - wY2P  —  4 —  r  cos ( 2w + p ) t  - pPY2 cos ( 2w + p ) * + wV2p cos ( 2w - p ) t - pPV2 o s ( 2W-P)" 4  6  C  - wP^V" cos (w + 2p)"fc - p V P ""4" 2 + pVP  cos(w + 2p)* - wP2y  2  2  cos (w - 2 p ) t + ppy2 cos p t + pP?  2  T~  2  cos(w" p)* 2  cos p t  - pP^ cos 3ptJ Of these t h e mixed c u r r e n t s o f g r e a t e s t s i z e a r e : 3bf-w7 P cos(2w + p ) * - pPV* L ~~T~ 4 2  cos(2w + p ) *  2  + wV P cos( 2w - p ) * - p P V 2". 4 2  Since w » p these become  cos(2w - p ) * ) -  2  1  approximately:  3b( -wVfP cos ( 2w + p)"t + wV P eos( 2w - p ) * ) 2 2 These are t h e c u r r e n t s upon which a m p l i f i c a t i o n i s based.  (12)  2  they are inaepenaent  Since  o f t h e s i g n o f V a fcalancea c i r c u i t may be  aesignea where the h i g h frequency i s a p p l i e a t o t h e conaensers i n push p u l l , the low frequency i n p a r a l l e l , ana these output c u r r e n t s fea  into a loaa i n p a r a l l e l .  The h i g h frequency may thus be  e l i m i n a t e d from t h e output c i r c u i t .  The e q u i v a l e n t c i r c u i t of  such a c a r r i e r a m p l i f i e r i s g i v e n i n diagram 22.  PLATE IX EQUIVALENT  O P CARRIER  CIRCUIT  AMPLIFIER  N.L.C  I,  ft«  ^»  B Psm pfr  Xa  Is.  w.L.cr^  DLaqraw 22 AMPLlPtER  CA9RI6R a«gtta*rtinoci ite  Hoar  4  KfctX.  CIRCUIT -s  " 4 " ' I t HH 85 K  ^  1  ±L  ,  I'  ieoc.p-&  •8 '  •am  Serashvtied  aoo  DlaqmTn 23  -22'  3.  Analysis of Carrier Amplifier Equivalent C i r c u i t .  (1) Assumptions made i n c i r c u i t : (a) The non l i n e a r condensers are identical..  Hence no  fundamental or 3rd harmonic currents flow from A to B. (b) The 2nd harmonic impedance between A and B i s R]_+j2wL-|_ and between B and E and B and D i s R2«  The low frequency  impedance between A and B i s high and capacitive ( i t w i l l be considered i n f i n i t e ) and between B and E and B and D i s low ( i t w i l l be considered zero) (c) The low frequency generator presents a high impedance to 2nd harmonic and no side band currents flow into the  low  frequency generator. (d) Both generators have zero i n t e r n a l impedance.  Matching  of p r a c t i c a l generators w i l l be considered l a t e r a f t e r the impedance seen by the ideal generators have been found. (e) The 3rd harmonic voltage from D to E i s neglected. (2)  C i r c u i t Voltages. Across either non l i n e a r condenser there are s i x main  voltages. ( V » P ,  w»P)  complex form  1.  V S i n wt  2.  P Sin(pt  3.  VxSinU 2w + p)t + 0)  4.  V ^ S i n U 2w - p)t +  5.  VgSinU 2w + p)t + ©)  6.  V ^ S i n U 2w - p)t + 01)  V P = Pe3(pt + f)  +^)  Vl= Y j ( ( 2 w + p)t +  fifSV -  ie  V l  l J ( ( 2 w - p)t •r jfL) e  v = Y j ( ( 2w + p ) t + ©) 2  Vglas  2 e  V e ( ( 2w - p ) t + e i ) 1  2  (3) Derivation of Mixed Currents. These are applied to a condenser having:  -23I - a dY + 3bV  2  It  dY dT  Prom the f i r s t term the currents i n complex form a r e : ja(wV + p  + (2w + p ) ( Y + Y ) + (2w - p K Y - j + Y 2 ) )  p  1  x  (13)  1  2  From the second term the main currents i n trigonometric form a r e : (V ,T ,V ,V ?-. « 1  1  2  1  2  v  )  3bfwY3 cos wt * wV3 cos 3wt + pPY2 cos (pt +4") + WY2YI COS (pt  2~  + 0)  + wY2j_i C O S (pt - 0 )  1  2  cos(pt - 61) + (2w + p ) Y i Y I ZX )  + ( 2w + P ) Y Q Y  2  + (2w ( 2  c o s ( ( 2 w + p ) t + e)  2  )  ?  cos  (pt +  ©)  ~T~~ 2  - ( 2vy - p j Y ^ Y ^ o s ( p t - j f ) 1  - ( 2w + p)YpY cos ( p t + e) 2  (  pjv^v c o s ( ( 2 w - p ) t + e )  4 .  )  - (2w - p) ( 4 )  1  2  cos((2w+p)t+#) - ( 2w+p)ViY cos( pt("~2T)  2  ( 2w - pjV-^V 2 C O S ( ( 2 W - p ) t +'&-)  (  + wY2y  1  ~ T "  + WY2YP +  cos (( 2w - p ) t - ^ r )  + wY^P  ) *  Y H* 9  e )! J  cos ( p t -  1  C o l l e c t i n g a l l currents of angular frequency P from both terms and u s i n g complex n o t a t i o n : Ip = japP + J | b p l ! ( 2P + where Y  + x  Y  1  +  +  ±  + Y  1 + + 2  -  T* X  - Y )  (14)  +  2  - V-je^P*^)  yl++.  V l  l j(pt-.j^-) e  V l++= Y l ^ P t  2  2  @ 1 )  e  S i m i l a r l y f o r angular frequency 2w+P: l2w+p - J ' ( 2 w + p ) ( Y i +Y ) .+ i 3 b Y 2 ( - w P ^ + ( 2w + p ) ( Y + Y ) ) . ( l $ ) a  2  x  2  # v  -24where P+ = Pe^^2w + p ) t For  + T  *)  angular frequency 2w - p  I|w-p » j a ( 2 w - p ) ( V + Y ) + j | b Y ( w P 1  1  1  2  +( 2w-p)( V ^ + V ^ ) )  1 + +  2  (l6)  where pl++ - Pe^( ( 2 w - p ) t - ^ ) (4) D e r i v a t i o n o f output Power. From c i r c u i t : Ti = - I i U i V  + 2jwLl)  » -I2R2  2  Y  )  = -2I (R  1  2  + 2jwLi)  1  I - L • 2 I because  Since  )  2  S u b s t i t u t i o n of (17)  (17)  j  c i r c u i t i s balanced  i n t o (15) w i t h the assumption  2w + f « 2w y i e l d s the e q u a t i o n f o r I Ij,  -  - j 3  bw  V  2  a t 2w + P*:  P+  2  2 l l - 4w2L ( 2a + 3bY2j + jw( 2Rx + R 2 U I f the a b s o l u t e value o f I i s t o be a maximum t h a n : x  2*+%W) . r ( l  2  1 - 4 w L ( 2a + 3bV ) - 0 2  2  1  1  2wL]_ -  (19)  2w( 2a+3bT2) I f (19) be used i n (18) I  - 3 b Y P+ 2(2Ri+R )(2a+3bV )  =  2  2  (20)  2  2  Similarly for I I2  1  -  2  5b  a t 2w - p V  2  i.e. I  2  *  P **  (  1  2( 2Ri+R )( 2a+3bV ) 2  2  The maximum p o s s i b l e power a v a i l a b l e from both s i d e band f r e q u e n c i e s f o r an i n p u t v o l t a g e P i n R i and both R c a l c u l a t e d t o be:  2  can be  a  )  -25-  Po = I  2  and I 2  1  9 b V4 p 2 2( 2Ei+R2)( 2a+3bV2)2 2  .(22)  nave'been converted to R.M.S. f o r t h i s d e r i v a t i o n .  ( 22) g i v e s the maximum p o s s i b l e  power a v a i l a b l e f o r r e c o n v e r s i o n  t o angular f r e q u e n c y P . Of t h i s power the amount a v a i l a b l e i n S-|_ i s : P =  9b2 P2 y4  (23)  H I  ( 2R +R ) (2a+3bV2) 2  2  1  2  This i s the p r a c t i c a l l y u s e f u l output power. (5) D e r i v a t i o n of Input From ( 2)  Currents. I  = -xP  2  +  JbYJL  where x =  2( 2Ri+R )( 2a+3bV2) 2  From ( 21)  Ig  Thus  IT  1  V  P!  + +  - 2Io = -2xP  IT. From (17)  = x  1  - 212  -  1  +  2xP  1 + +  - xR^" " 1  2  Tg  1  - -x R 2 P  V!  = 2x(R  Vj}  m -2x(R]_  x  1 + +  + 2jwLi)P +  +  2jwL!)pl  + +  From t h i s i t can be shown t h a t :  V  + 2  Yl  +  7^1++  88  2R2P  ) .(24)  - 2x(R +2jwL )P 1  1  m -2x(Rx+2jwLi)P  S u b s t i t u t i o n of (24) i n t o (14) y i e l d s : Ip - P( j(ap+2 b V p ) - j 3 b p Y 2 ( 2 R i + R 2 2 2  x  2  + 4  JWLTJ)  -26-  T h i s i s the low frequency current only.  s u p p l i e d t o one condenser  For b o t h condensers t h i s must be m u l t i p l i e d by 2.  Substituting f o r x gives: 2a+3bV ) - J9b2pV*( gJRi+RgH-jwLi))  I T - 2 I - P(  2  p  2( 2R1+R2K 2a+3bV2)  )  When I«j i s r e s o l v e d i n t o i t s r e a l and imaginary components there r e s u l t s the low frequency r e s i s t i v e and c a p a c i t i v e i n p u t c u r r e n t s : ( ( 1 9 ) i s used) XJj m 9Pb2pV4 2w( 2R1+R2M 2a+3bV2) 2 I c = P( jp( 2a+3bY2)  - j 9 b2 p  (25)  .  )  (£) 2  2( 2a+3bV2) ) (6) D e r i v a t i o n o f Power Gain From (25) the l o w f r e q u e n c y i n p u t power i s : *E - 9 b  2  P2p y *  ( 2 < 7 j  4w( 2Ri+R ) ( 2a+3bT2) 2 2  D i v i s i o n o f (22) by (27) g i v e s t h e maximum p o s s i b l e power g a i n : G  (28)  -  g°  ....(28)  = *>l  i s t h e end r e s u l t o f t h e t h e o r y .  I t g i v e s the maximum  p o s s i b l e power g a i n under i d e a l c o n d i t i o n s f o r t h i s type o f amplifier.  I t c a n be shown t h a t the power g a i n but n o t power  i n p u t o r output i s independent o f t h e t u n i n g c o n d i t i o n ( 1 9 ) In the a m p l i f i e r to be d e s c r i b e d i n t h e next chapter w - 90 P Thus maximum p o s s i b l e power g a i n = 180 (7) D i s c u s s i o n of the t h e o r y .  -27-  Tlie theory as given y i e l d s the f o l l o w i n g conclusions: I f most o f the output power i s to be used In R]_ and not in R  2  then R  must be as small as p o s s i b l e r e l a t i v e t o RT.. I f a  2  l a r g e output power i s d e s i r e d RT_ and R f o r a given R  2  2  must be small from (22) but  there i s a value o f R i where a maximum of power i s  used i n RT_. The low frequency input c a p a c i t i v e current given by (26) shows that i t i s not e f f e c t e d by the output c i r c u i t . The low frequency input r e s i s t i v e current i s s t r o n g l y e f f e c t e d by the output c i r c u i t . As the output power increases so does the input power. the output c i r c u i t .  Thus input c i r c u i t matching depends on  The impedances seen by the low frequency  generator are given by ( 2 5 ) and ( 2 6 ) and a p r a c t i c a l  generator  must be matched to these. The h i g h frequency generator sees a c a p a c i t i v e impedance which can be found from the f i r s t term o f (13) and the f i r s t term of the f o l l o w i n g expansion.  I t a l s o sees a s m a l l r e s i s t i v e com-  ponent since i t must supply the power f o r the power g a i n . 4.  Q u a l i t a t i v e E f f e c t s of I n c l u s i o n of Condenser Losses i n Amplifier Equivalent C i r c u i t .  A complete a n a l y s i s could be done with the non l i n e a r condensers i n s e c t i o n 3 of t h i s chapter shunted by appropriate  resistors.  Such an a n a l y s i s i s q u a n t i t a t i v e l y more complete than that given above but does not b r i n g forward r e s u l t s which cannot be p r e d i c t e d q u a l i t a t i v e l y by a simple d i s c u s s i o n of the c i r c u i t .  The theor-  e t i c a l p i c t u r e i s f u r t h e r complicated by the non l i n e a r i t y of the r e s i s t o r s at low voltages (diag 17).  -28The  f o l l o w i n g g e n e r a l c o n c l u s i o n s about the  inclusion  of condenser l o s s e s are drawn. Since the c u r r e n t s f l o w i n g through the flow through L]_ the t u n i n g c o n d i t i o n ( 1 9 ) The low frequency  shunting  resistors  w i l l be a l t e r e d .  s i g n a l input impedance w i l l  contain  an a d d i t i o n a l r e s i s t i v e . e l e m e n t r e p r e s e n t i n g the energy f e d by the low frequency  generator' to h y s t e r e s i s l o s s e s .  power g a i n i s t o be achieved r e l a t i v e to the  I f the  t h i s i n p u t power must be  i n p u t power o f ( 2 7 ) .  Hence R± and  should be as s m a l l as p o s s i b l e but given Rg,  greatest  small  therefore  R i must not  R  2  fall,  below a c e r t a i n v a l u e . The h i g h frequency  i n p u t s i g n a l w i l l have to  supply  c o n s i d e r a b l y more power than i n d i c a t e d by the i d e a l i z e d  theory.  I t must supply the h y s t e r e s i s l o s s e s f o r the.,.high frequency voltage s i g n a l .  These can be  1.7.  c a l c u l a t e d from diagram  Losses i n condenser h y s t e r e s i s at s i d e band  high  frequencies  w i l l a l s o reduce the power g a i n a c h i e v a b l e i n p r a c t i s e . The  above theory and  a m p l i f i e r which was  c o n c l u s i o n s were checked.in  a p h y s i c a l r e a l i z a t i o n of diagram  an  22.  CHAPTER VI - EXPERIMENTAL RESULTS FROM CARRIER .AMPLIFIER ; ••  1.  .  1  ;  Circuit. The  c i r c u i t was  s i m i l a r to t h a t used f o r measurement,  of 2nd harmonic c u r r e n t and  i s g i v e n i n diagram 2 3 .  As  before  -29the  non l i n e a r condensers were balanced w i t h r e s p e c t t o f u n d -  amental by R and C.  Unless otherwise s t a t e d the f o l l o w i n g  s e t t i n g up procedure was used i n t e s t s on t h e a m p l i f i e r : The h i g h frequency was tuned t o 9000 c.p.s. and the low frequency t o 100 c.p.s. The c i r c u i t was tuned w i t h c . 1  The condensers were matched u n t i l the v o l t a g e a t Y was a minimum. Circuit  2 was tuned u n t i l s i d e band c u r r e n t was a  maximum. S i n c e a l l these adjustments a r e not independent they were r e p e a t e d u n t i l a l l h e l d s i m u l t a n e o u s l y . About l / 2 hour warm-up time was r e q u i r e d f o r c o n d i t i o n s i n the c i r c u i t to s t a b i l i z e .  Since non l i n e a r condensers are h i g h l y  v o l t a g e s e n s i t i v e f l u c t u a t i o n s i n l i n e v o l t a g e were always o b s e r vable but were not s u f f i c i e n t regulated 2.  t o n e c e s s i t a t e use o f v o l t a g e  supplies.  Measurement of Output Power. The measurement of sideband c u r r e n t or output power  was done as f o l l o w s :  •  A c a l i b r a t e d o s c i l l o s c o p e was p l a c e d from B t o ground. When the c i r c u i t was o p e r a t i n g the s i d e bands were observed. condensers were b i a s s e d t o a few v o l t s u n t i l a l i t t l e  The  l e s s than  100 per cent modulation e x i s t e d . (Theory q u i c k l y shows that b i a s s i n g does produce 2nd harmonic  i n the c o r r e c t phase to g i v e a normal modulated  signal.  -30-  I f i n the expansion Q, = f ( ^ ) f o r a non l i n e a r condenser there i s a term p r o p o r t i o n a l t o Y harmonic  2 a  n  a  y gj_  n  j _ a p p l i e d then the 2nd s  current has the phase S i n 2 wt.  From equation ( 1 2 )  sideband c u r r e n t s have the phase 1 cos ( 2w + p ) * .  the  This i s the  c o r r e c t phase r e l a t i o n ) . A s k e t c h of a typical'waveform a t B i s given i n diagram 24.  The d i s t a n c e x was measured and converted to v o l t s .  the theory of an amplitude modulated voltage i s due t o one sideband.  From  s i g n a l one q u a r t e r of t h i s  The c u r r e n t a t one sideband i s  r e a d i l y c a l c u l a t e d by the d i v i s i o n of t h i s v o l t a g e by the impedance from B t o ground.  The power i n one  sideband i s g i v e n by the  square of the current times the r e s i s t a n c e i n s e r i e s tuned  circuit  2. Example:  2" t o t a l d e f l e c t i o n r e q u i r e d 20v R.M.S. a t 1 8 0 0 0 c.p.s. . x measured = . 8 3 i n s . wL = impedance B t o gnd. = 2830 ax. RT_ = 110  ohms.  Sideband c u r r e n t = ( . 8 3 ) ( 2 Q ) ( 1 0 0 0 ) 14)12b30) Power i n R i a t 1 Sideband = ( 1 . 4 7 )  = 1.47 2  (1000)  m-as  (110) (  = 236 microwatts. )  Power i n R-j_ at b o t h sidebands = Power Output of A m p l i f i e r =472 microwatts. 3.  Measurement of Low Frequency input powerFor t h i s measurement i t was  e s s e n t i a l to remove a l l  v o l t a g e s except low frequency v o l t a g e s at Y. The v o l t a g e s at X and Y and the microammeter r e a d i n g  PLATE TYPICAL WAVEFORM  X AT  P a c i n g ftvtjc. 3 0  AMPLIFIER  OUTPUT (6 m d l « ^ 2 3 )  IOO C P S .  Dlaqrarn 24 VECTOR.  DIAGRAM  FOR  LOW F R E Q U E N C Y  DlaqraTn  25  INPUT  Y R was the voltage drop across R l .  were observed.  from X to Y as a f u n c t i o n of V ently.  R  The current  was c a r e f u l l y p l o t t e d independ-  Hence t h e v e c t o r diagram 25 c o u l d be drawn s i n c e the  lengths of a l l s i d e s of the t r i a n g l e were known and the angle Q calculated.  T h i s gave t h e phase r e l a t i o n between the v o l t a g e ,  between Y and ground and the c u r r e n t f l o w i n g i n t o t h e c i r c u i t a t Y.  The c a p a c i t i v e and r e s i s t i v e components o f the c u r r e n t could  be found and the power i n p u t c a l c u l a t e d .  The c a p a c i t i v e and  r e s i s t i v e c u r r e n t s f l o w i n g i n t o the c i r c u i t without the non l i n e a r condensers were measured i n the same way and s u b t r a c t e d from the r e s u l t s obtained w i t h non l i n e a r condensers.  The low frequency  power and c a p a c i t y c u r r e n t s f l o w i n g i n t o the non l i n e a r condensers were thus found. Example:  Y  x  = 2.42 v o l t s  Yy = 1.60 v o l t s  at 100 c.p.s.  A = 2b.O microamps = I . 0 9 v o l t s (from c a l i b r a t i o n S = 1 (Yx + Yy + Y R ) = 2  S- Y S-  Sin  = .955  y  VR =  VyY  r  1.465  - 1.745  g. - f ( S - V y H S - Y R ) \  2 ( ai, =  1270  R e s i s t i v e Current C a p a c i t i v e Current  e -  1  /  2  j  Vy-v^.  Low frequency  2.555  .  ( -955)(I.465)  = .895  -ii.-JiJ)  55  = ( 2 6 . 0 ) ( c o s 53) = 1 5 . 6 microamps. = (26.0)(Sin  53) = 2 0 . 6 microamps.  Power Input = ( 1 . 6 ) ( 1 5 . 6 )  = 25 microwatts.  curve).  -32From these the power, c a p a c i t i v e current- and r e s i s t i v e f e d i n t o t h e c i r c u i t without non l i n e a r  current  condensers must be sub-  t r a c t e d but t h e method of measurement o f these i s i d e n t i c a l and i s not g i v e n . 4. . V a r i a t i o n of Output Sideband Current w i t h Low Frequency V o l t a g e . The f o l l o w i n g were h e l d c o n s t a n t : High Frequency = 9000 c.p.s. High Frequency V o l t s per condenser - 40 and 120 R-L = 110 ohms. Low Frequency 100 c.p.s. The v o l t a g e a t Y was v a r i e d and the output a t B observed. Results:  Diagram 26.  Equation (26) p r e d i c t s a l i n e a r v a r i a t i o n .  The observed curve i s  c e r t a i n l y very n e a r l y l i n e a r with a tendency t o be'concave  down-  ward which i n c r e a s e s at t h e lower value of h i g h frequency v o l t a g e . T h i s p o i n t was n o t i n v e s t i g a t e d f u r t h e r s i n c e t h i s bending might have been caused by a zero e r r o r i n the v o l t m e t e r . 3*  V a r i a t i o n of Output Sideband Current w i t h Frequency o f Low Frequency Input.  The f o l l o w i n g were h e l d c o n s t a n t : High Frequency = 9000 c.p.s. High Frequency V o l t s per condenser = 40 R  x  = 110 ohms  Low Frequency Voltage .= 1.5 v o l t The low frequency was v a r i e d and the output a t B observed. R e s u l t s : Diagram  27.  Facing Pa.^e 32.  P L A T E XI IS -  1  ...^V.Q  .1-  '  (*  fo l ..''/0-J.. r  •  <  CD  ....  '  * .  y  :  f  R.i«3.v=.4<  r  • jT  • ~ -  1  t  t  1  © to it!  Of |r  i  j  ~'.'  '  •fOO.  Low FS|E^UENCY \/OLTS|  •;:',]  500  2©o  L:£ RRB^UEWCV :C;RS. ~  11  i1  (.••'.-  .  ••  a  '-.•.wcov^ft, a . a  :  i'r:'4 ' -t-4  -33-  Equation ( 2 6 ) predicts a h o r i z o n t a l s t r a i g h t The c u r v a t u r e of the observed  line.'  curve i s more l i k e l y to  have o c c u r r e d from experimental than t h e o r e t i c a l e r r o r . not i n v e s t i g a t e d 6.  I t was  further.  V a r i a t i o n o f Output Power w i t h R T _ . The f o l l o w i n g were h e l d c o n s t a n t : High Frequency = 9 0 0 0  c.p.s.  High Frequency V o l t s per condenser = 40 and 1 2 0 . Low  Frequency = 1 0 0 c.p.s.  Low  Frequency V o l t s  R  2  R e s i s t a n c e of one h a l f of T = 7 0 ohms.  = D.C.  R^ v/as v a r i e d and output a t B Results:  =1.5  observed.  Diagram 2 8 .  E q u a t i o n (23)  i s a l s o p l o t t e d and has used values of a and  g i v e n i n diagram 1 8 .  The experiment  b  repeated f o r L-i = 0 i n  was  which case ( 2 3 ) becomes: P =  9w -  2  b  2  V4 P  2  Ri  , — -  ,  (29)  l+( SRx+Rg) ( 2 a + 3 b V 2 ) 2 2 2  W  Experiment  and t h e o r y are g i v e n f o r t h i s case i n diagram 2 9 . In the l a t t e r case agreement between t h e o r y and  i s good but i n the former agreement i s poor.  The a b s o l u t e  practise agree-  ment between t h e o r y and p r a c t i s e w i t h i n about 20f» i s not c o n s i d e r e d important  s i n c e experimental  e r r o r c o u l d p o s s i b l y be t h i s g r e a t .  But i n diagram 28 the e r r o r s g r e a t l y exceed explanation i s given f o r t h i s divergence.  this figure.  No  I t would seem i n c o r r e c t  to d i s c a r d the theory e n t i r e l y s i n c e agreement i s good i n diagram 29 and  c e r t a i n l y the experimental curves are g r e a t l y d i f f e r e n t i n  -34-  diagram the  28 t o d i a g r a m  theoretical  29 and  lower r a t h e r than r a i s e  Li  i s °t n  easily  the next approach t o t h i s  a l t e r a t i o n o f LT_ a n d t h e p l o t t i n g a s LT_ i s c h a n g e d  have t o be a l t e r e d  will  problem of output  f r o m z e r o t o the  for this  because  tuning  28.  at present  controlled.  Since the factor diagram  the  for this  which i s considered to hold f o r diagram  apparatus would  of  the t h e o r e t i c a l c u r v e s .  pov/er as a f u n c t i o n o f c o n d i t i o n o f (19)  features  Nor w i l l  l o s s e s save e q u a t i o n (23)  i s thought t h a t  s h o u l d be t h e c a r e f u l  The  qualitative  c u r v e s show t h e same d i f f e r e n c e s .  c o n s i d e r a t i o n of condenser  It  the g e n e r a l  b e t w e e n t h e o r y and p r a c t i s e  28 a p p a r e n t l y d e p e n d s o n V,  from  t h e v a r i a t i o n o f o u t p u t power  as a f u n c t i o n o f V f o r f i x e d R^ was  n e x t done.  R i = 110  was  s i n c e t h e power o u t p u t was g r e a t e s t f o r s m a l l r e s i s t a n c e s , ;  being the r e s i s t a n c e  of tuned c i r c u i t  2 w i t h o u t , a n y added  chosen  this resis-  tance. 7«  Variation The  o f O u t p u t Power w i t h H i g h F r e q u e n c y V o l t a g e . f o l l o w i n g were h e l d c o n s t a n t : ' High Frequency  at  F r e q u e n c y = 100  Low  Frequency = 110  ohms  volts R2  c.p.s.  c.p.s. =1.3  = .70  ohms.  h i g h f r e q u e n c y v o l t a g e was  B observed. Results:  E q u a t i o n (23)  9000  Low  Rx The  =  Diagram  30.  i s again plotted.  varied  and t h e  output  -35-  Th e shape of the. • two voltages.  curves agrees w e l l except at low  The divergence here may  be a s s i g n e d u n t i l e x t e n s i o n of  the theory to TDhe n e g l e c t i o n i n the t h e o r y of the non l i n e a r i t y the r e s i s t a n c e component of the condensers 17.  i n diagram  of  a t low v o l t a g e s shown  T h i s i s the reason the low v o l t a g e t h e o r e t i c a l  p o i n t s are not weighted  as h e a v i l y i n diagram 3 0 .  Diagram 3 0 g e n e r a l l y supports the v a l i d i t y  o f equation  (23).  The low frequency i n p u t c i r c u i t was 8.  V a r i a t i o n of low Frequency  next  investigated.  Input Currents w i t h High  Frequency  Voltage. The f o l l o w i n g were h e l d c o n s t a n t : High Frequency  =9000 c.p.s.  Low  Frequency  = 1 0 0 c.p.s.  Low  Frequency  Volts =  Rl  = 1 1 0 ohms  Results:  1.6  R2 = 70 ohms.  Diagrams 3 1 and 3 2 .  Equations ( 2 5 ) and ( 2 6 ) are p l o t t e d f o r comparison diagrams 3 1 and  3 2 respectively.  The wide divergence between theory and 3 1 was  i n diagram  in  expected  experiment  s i n c e the low frequency i n p u t r e s i s -  t i v e c u r r e n t w i l l be v e r y s e n s i t i v e to the n e g l e c t i o n o f condenser l o s s e s .  A t low high-frequency v o l t a g e s the  p o i n t s must a g a i n be l i g h t l y regarded.  theoretical  I t i s impossible t o  tell  how much of the d i s c r e p a n c y would be e l i m i n a t e d by i n c l u s i o n of condenser  losses.  The n e g a t i v e slope of the experimental  at low h i g h frequency v o l t s i s probably caused by non  curve  linearity  PLATE XII  o  u  Mi6H  Jro FREQUENCY  /op /asVOLTS  Dcacjmm 3 0  O  50  foQ  32,  u MiGH  & lop w «5" FREQUENCY VOLTS  0 tacrram  12S  HISH FREQUENCY VOLTS DtaqTuro  °  31  fOO  I2S  HI&H FREQUENCY VOLTS Dtacjr-a.m 33  -36-  o f condenser  resistance.  The agreement "between theory and p r a c t i s e for. the c a p a c i t i v e i n p u t c u r r e n t p r o b a b l y occurs because i n theory are almost independent 9.  of condenser  these c u r r e n t s  losses.  Power Gain as a f u n c t i o n o f High Frequency V o l t a g e . The experimental curve o f diagram  33 was  obtained by  passing a smooth curve through the input r e s i s t i v e c u r r e n t p l o t s and through the output power p l o t s (not g i v e n but s i m i l a r to diagram  J>Q), and o b t a i n i n g the r a t i o o f output to i n p u t power. .The corresponding t h e o r e t i c a l curve i s a h o r i z o n t a l  l i n e at h e i g h t 1 3 5 > f ( l 8 0 ) ( 2R\ no l o s s t h e o r y ) .  )\, on the v e r t i c a l a x i s (from  (R2 2Ri)J  dropping of the  +  experimental curve to zero at low- v o l t s was  expected s i n c e the  output power i s n e g l i g i b l e r e l a t i v e to condenser l o s s e s a t low frequency, but the subsequent  behaviour of the curve cannot have  any t h e o r e t i c a l e x p l a n a t i o n u n t i l an exhaustive a n a l y s i s o f non However 70 of the  l i n e a r condensers has been done. a v a i l a b l e 135 10.  theoretically  i s not considered unreasonable.  General D i s c u s s i o n of R e s u l t s . Although at l e a s t one  experimental r e s u l t ( d i a g . 2 8 )  cannot be e x p l a i n e d by the theory even with a-''qualitative d i s c u s s i o n of condenser  l o s s e s , n e v e r t h e l e s s the c o n c l u s i o n s  concerning the d e s i g n o f an a m p l i f i e r , of the type d e s c r i b e d , drawn at the end of Chapter 5 have been confirmed by experiment;, ' > The magnitudes of the v a r i o u s i n p u t and output impedances must s t i l l be determined by experiment necessary.  i f exact magnitudes are  However the g e n e r a l method g i v e n i n t h i s  thesis  -37-  p r e d i c t s magnitudes  t o w i t h i n about a f a c t o r of two.  I t must be emphasized however t h a t n e i t h e r t h e o r y nor experiment has been'exhaustive. at o n l y one  Power g a i n has been checked  output l o a d r e s i s t o r , and the frequency o f the high  frequency v o l t a g e has been kept at a l l times a t 9000 c.p.s. was  This  done because of the number o f t u n i n g changes n e c e s s a r y i f t h i s  frequency was  changed.  CHAPTER V I I CONCLUSIONS 1.  P o s s i b l y the most important c o n c l u s i o n from the work done  i s t h a t non l i n e a r condensers- can I n p r a c t i s e be used as power a m p l i f i e r s as p r e d i c t e d by t h e o r y . 2.  The g r e a t e s t disadvantage of the condensers used i n t h i s  r e s e a r c h was t h e i r h i g h l o s s e s .  These l o s s e s make t h e i r t h e o r -  e t i c a l a n a l y s i s complex and reduce t h e i r u s e f u l n e s s . The l o s s e s r e s u l t e d i n : (a) Bulky c o o l i n g equipment.  Because  of the h i g h temperature  dependence o f t h e c h a r a c t e r i s t i c s of the non l i n e a r  condenser  constant temperatures are e s s e n t i a l . (b) Reduction of a c h i e v a b l e power g a i n . (c) A v e r y low o v e r a l l e f f i c i e n c y d e f i n e d as t o t a l output power d i v i d e d by t o t a l input power.  T h i s was  - Two main l i n e s of development  of the order of  .1%.  could reduce these l o s s e s .  -36(a) The d i e l e c t r i c were suggested a given f i e l d  could be made t h i n n e r .  i n Chapter  I I , Sec.2.  Two  p o s s i b l e methods  I f a g i v e n c a p a c i t y and  s t r e n g t h i n the d i e l e c t r i c are r e q u i r e d then  the l o s s e s vary as the square  of the d i e l e c t r i c t h i c k n e s s .  r e d u c t i o n o f the p r e s e n t d i e l e c t r i c  t h i c k n e s s by a f a c t o r  three which i s c o n s i d e r e d p o s s i b l e f o r audio frequency densers would reduce  of  con-  the l o s s e s by a f a c t o r of n i n e .  (b) The a r e a of the h y s t e r e s i s loop might be reduced and  A  chemical r e s e a r c h .  by p h y s i c a l  Combinations of m a t e r i a l s might be  s u p e r i o r to barium t i t a n a t e . 3..  As  the condenser l o s s e s are reduced  the theory as g i v e n w i l l be p o s s i b l e .  a more d e t a i l e d check of In i t s present form i t i s  s u f f i c i e n t f o r broad p r a c t i c a l d e s i g n of an- a m p l i f i e r . other hand the t h e o r y could be extended without g r e a t to i n c l u d e the condenser  On  difficulty  losses.  I t should be mentioned .that ;the. e x t e n s i o n of .Van Z i e l ' s t h e o r e t i c a l method was a f t e r Dr. Van der Z i e l had and has not been completely 4.  Van  done independently by the  left  der  author  the U n i v e r s i t y of B r i t i s h  Columbia,  checked by anyone.  der Z i e l ' s a n a l y s i s ( 2 1 ) u s i n g Q, ,= aV + b V  as the b a s i c non  2  '  \  ;  l i n e a r condenser e q u a t i o n gave as a t h e o r e t i c a l l y  p o s s i b l e power a m p l i f i c a t i o n ( w )  2  as a g a i n s t 2(w)  •'.(F) t h e s i s f o r a non l i n e a r condenser with the Q, = aV + bV^ 21.  the  See Ref.  11.  derived i n this  (?) equation  In p r i n c i p l e t h e n the development of condensers obeying the former r e l a t i o n ( r e c t a n g u l a r h y s t e r e s i s loops) would seem more pro:mising.  In the present  condensers however the  i n the expansion f o r Q, i s small (Chapter input power at low frequency  c o e f f i c i e n t of  IY, S e c t i o n 6 ) ,  and  Y  2  the  to overcome feedback would be much .  smaller r e l a t i v e to the h y s t e r e s i s component of i n p u t power than i n the a m p l i f i e r t e s t e d .  However the g r e a t e r p o s s i b l e g a i n might  overcome t h i s to g i v e a g r e a t e r o v e r a l l than t h a t achieved t h i s work. about 100  However disadvantages would be v o l t s o f b i a s and  input i n push p u l l .  in  the n e c e s s i t y o f  the need t o feed the low  frequency  U n f o r t u n a t e l y time d i d not permit  experi-  ments on t h i s type of a m p l i f i e r . 5.  The  use  of non  investigated.  l i n e a r condensers a t h i g h f r e q u e n c i e s was  That they can be used at high f r e q u e n c i e s  not  was  shown by Donley(22) i n q u a l i t a t i v e experiments at 20 megacycles and 40 megacycles.  He used low v o l t a g e s and  a v e r y t h i n c h i p as  d i e l e c t r i c t o give him about 100 m.m.fds. o r l e s s . report 6.  He d i d not  overheating.  Since power a m p l i f i c a t i o n has  been demonstrated p o s s i b l e w i t h  non l i n e a r condensers feed back may crease the a m p l i f i c a t i o n .  i n p r i n c i p l e be used to i n -  I t could be i n c r e a s e d u n t i l low  quency o s c i l l a t i o n s were present p r o v i d e d  a frequency  fre-  control  was  supplied. 7.  D i e l e c t r i c a m p l i f i e r s have a l l the advantages of magnetic  a m p l i f i e r s of ruggedness, and been v e r i f i e d ) , and 22.  See Ref.  13.  i n d e f i n i t e l i f e t i m e ( t h i s has  the l a c k of h e a t e r s .  not  In p r i n c i p l e ,  i n analogy w i t h condensers and c o i l s , i t  would seem d i e l e c t r i c could be made more e f f i c i e n t  than magnetic  amplifiers. 8.  As a D.C. a m p l i f i e r the d i e l e c t r i c a m p l i f i e r might be most  useful.  The input power necessary to b i a s non l i n e a r condensers  would be small s i n c e no h y s t e r e s i s l o s s e s would be p r e s e n t . The t h e o r e t i c a l gains are i n f i n i t e f o r both types of d i e l e c t r i c l i f i e r mentioned i n s e c t i o n 4 o f t h i s  amp-  chapter, and a very high,  power g a i n might be achieved even with the present condensers which a p p a r e n t l y have a r e l a t i v e l y low D.C. r e s i s t a n c e .  The  non l i n e a r i t y p r o p e r t y might be used d i r e c t l y as an automatic control.  The temperature dependence o f t h e p r o p e r t i e s o f non  l i n e a r condensers and the jump to a l i n e a r condenser at the C u r i e temperature may a l s o f i n d  application.  -41-  BIBLIOGRAPHY 1.  B o y a j i a n A.  "Theory o f D.C. E x c i t e d I r o n Core  R e a c t o r s and R e g u l a t o r s " .  T r a n s a c t i o n s of the American  I n s t i t u t e of E l e c t r i c a l Engineers. V o l . 4 3 , 1924.. 2.  C a s t e l l i n i N.R.  "The Magnetic A m p l i f i e r " . Proceedings  o f the I n s t i t u t e of Radio E n g i n e e r s .  3.  New  V o l . 3 8 , No  " E f f e c t of F i e l d S t r e n g t h of D i e l e c t r i c  P r o p e r t i e s o f Barium S t r o n t i u m T i t a n a t e " . America Review V o l . 8 , No Greene W.E.  3 , PP 5 3 9 - 5 5 3 ,  Sept.1947, Princeton,N.J.  1947-  H e r o l d W.E.  "Frequency M i x i n g i n Diodes". Proceedings o f  the I n s t i t u t e o f Radio E n g i n e e r s . V o l 3 1 , New 6.  Radio C o r p o r a t i o n o f  " A p p l i c a t i o n s , of Magnetic A m p l i f i e r s "  E l e c t r o n i c s , Sept. 5.  2 , pp 1 5 1 - 1 5 8 ,  York.  Donley, H.L.  4.  June  111.  Chicago,  Feb. 1 9 5 0 ,  p 919,  No 1 0 ,  p 575,  Oct.1943,  York. Howatt G.N.  Breckenridge R.G.,  and Brownlow  J.M.,  " F a b r i c a t i o n of t h i n Ceramic Sheets f o r C a p a c i t o r s " J o u r n a l of the American Ceramic S o c i e t y , V o l 30, pp 237-242, 7.  Jonker G.H.  1947.  and Van Santen J.H. " P r o p e r t i e s of Barium  T i t a n a t e i n Connection w i t h i t s C r y s t a l S t r u c t u r e " S c i e n c e V o l 1 0 9 , No 8.  2 8 4 3 , PP 6 3 2 - 6 3 5 ,  Kay H.F.  and Rhodes R.G.  Nature, V o l l 6 0 , p 1 2 6 , 9.  Lamm A.U.  June  1949.  "Barium T i t a n a t e  July 1947,  London.  "The t r a n s d u c t o r , D.C.  Stockholm.Esselte a k t i e b o l a g  1943.  Crystals"  presaturated Reactor".  -42-  10.  Sawyer C.B. a n d Tower C H . P h y s i c a l Review, V o l 35,  No  3,  "Rochelle S a l t  PP  269-273,  as a D i e l e c t r i c " .  Feb. 1930. Minneapolis,  Minn. 11.  T o r r e y H.C.  and Whitmer C.A.  Massachusetts I n s t i t u t e Series. 12.  "Crystal  Rectifiers".  o f Technology R a d i a t i o n Laboratory  M c G r a w - H i l l , New Y o r k ,  1948.  Van der Z i e l A. R e p o r t t o t h e Defense R e s e a r c h Board o f Canada.  13.  University  of B r i t i s h  Van d e r Z i e l A . Condensers".  Journal  15.  Linear  11,  Lancaster, Pa.  "High D i e l e c t r i c  and E n g i n e e r i n g C h e m i s t r y . 1946.  o f Non  o f A p p l i e d P h y s i c s . V o l . 19,'No  Von H i p p e l A., B r e c k e n r i d g e R.G., Laszlo Tisza.  1949•  January  "On t h e M i x i n g P r o p e r t i e s  pp 9 9 9 T 1 0 0 6 , N O V . 1948. 14.  Columbia.  C h e s l e y F.G., a n d  Constant Ceramics".  V o l 3 8 , No 1 1 ,  Industrial  pp 1 0 9 7 - 1 1 0 9 ,  Nov.  Easton, Pa. Wul B.M.  and Goldman I.M.,  "Dielectric  Constant o f  B a r i u m T i t a n a t e a s a F u n c t i o n o f S t r e n g t h o f an A l t e r n a t i n g Field".  Comptes Rendues d e s A c a d e m i e s de S c i e n c e s .  pp 1 7 7 - 1 8 0 ,  Oct. 1945.  Vol 49,  

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