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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 Thesis Submitted In P a r t i a l Fulfilment of the Requirements f o r the Degree of MASTER OF APPLIED SCIENCE IN ENGINEERING PHYSICS THE UNIVERSITY OF BRITISH COLUMBIA. AUGUST, 19^0 T H E U N I V E R S I T Y O F BRITISH C O L U M B I A VANCOUVER. CANADA DEPARTMENT O F PHYSICS September 15, 1950. Dr. L. W. Dunlap, L i b r a r i a n , University of B r i t i s h Columbia. Dear Or. Dunlap: This l e t t e r w i l l c e r t i f y that the thesis of Mr. John Peter Eobson has been c a r e f u l l y studied by the under- signed, and that the thesis meets the required standards and an abstract has been approved by the Department. Yours sincerely, A. M. crooker Acting-Head of the Department A. J . Dekker Associate Professor of Physics AMC:lc ABSTRACT A method f o r the making of non l i n e a r barium tifcanate condensers f o r auaio frequencies i s described. Preliminary- measurements on these condensers are given. An idealized theory f o r the behaviour of 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 amplifier b u i l t on t h i s p r i n o i p l e i s described. A t h e o r e t i c a l l y possible power am p l i f i c a t i o n f o r t h i s amplifier of 1 8 0 i s derived. Experimental r e s u l t s obtained with the c a r r i e r a m p l i f i e r are given. A power ampl i f i c a t i o n of 7 ° was obtained. Conclusions on the possible applications of non l i n e a r condensers are drawn. ACKNOWLEDGEMENTS The author i s indebted to the National Researoh Council of Canada from whom he received a bursary f o r the year 194-9-50. The -work: described i n t h i s thesis has been carried out under the Defense Research Board of Canada which has f i n - anced the projeot and has employed the author f o r the summers of 1948, 1949, 1950. The author thanks Dr. A. Van der Z i e l and Dr. A.J* Dekker of the Physios Department of the Un i v e r s i t y of B r i t i s h Columbia f o r t h e i r kindness and assistance during the vrork. TABLE OP CONTENTS CHAPTER PAGE I INTRODUCTION 1 II MAKING OF NON LINEAR CONDENSERS FOR AUDIO FREQUENCIES 4 II I THE HYSTERESIS LOOP AND BASIC EQUATION OF THE NON LINEAR CONDENSER 9 IT DETERMINATION OF COEFFICIENTS IN THE BASIC EQUATION 1J V THEORY OF A DIELECTRIC CARRIER AMPLIFIER 2 0 VI EXPERIMENTAL RESULTS FROM CARRIER AMPLIFIER 28 VII CONCLUSIONS 37 BIBLIOGRAPHY 41 ILLUSTRATIONS Plate Diagram Facing Page I 1 . High Temperature Furnace 5 2. Block Diagram of Furnace Control I I 3 . Condenser Electrode 7 4. Condenser Mount 5 . Condenser Stand I U 6. Hysteresis C i r c u i t 10 7. Hysteresis Loops 8. "Variation of Hysteresis Loop with Frequency XV 9 . Equivalent C i r c u i t of Non Linear Condenser 11 V 10 . Equivalent C i r c u i t for 3rd Harmonic Current Measurement 15 11. C i r c u i t for 3rd Harmonic Current Measurement 12 . Typical Waveform i n 3rd Harmonic Current Measurement VI 13 . C i r c u i t for Fundamental Current Measurement 17 14. Vector diagram for Fundamental Current Measurement VII 15 . 3rd Harmonic Current vs Fundamental Voltage 18 16. Fundamental Current vs Fundamental Voltage 17. Capacity and Resistance vs Fundamental Voltage 18. C o e f f i c i e n t s vs Fundamental Voltage VIII 19. Variable Frequency Audio O s c i l l a t o r and Amplifier 19 Y I H (Cont'd.) 20. C i r c u i t f o r 2nd Harmonic Measurement 21. 2nd Harmonic Current vs D.C. Bias Voltage IX 22. Equivalent C i r c u i t for Carrier Amplifier 23. C a r r i e r Amplifier C i r c u i t X 24. Typical Waveform of Carrier Amplifier Output 2^. Vector Diagram for Low Frequency Input XI 26. Output Sideband Current vs Low Frequency Voltage 27- Output Sideband Current vs Frequency of Low Frequency 28. Output Power vs Load Resistor (Correct Tuning) 29. Output Power vs Load Resistor (Incorrect Tuning) XII 30. Output Power vs High Frequency Voltage 31. Low Frequency Input Resistive Current vs High Frequency Voltage 32. Low Frequency Capacitive Current vs High Frequency Voltage 33• Power Gain vs High Frequency Voltage 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, resistance, transconductance or capacitance i s not a constant but depends on the voltage applied. Such elements are a l l mixers, i . e . i f two voltages are applied to them then c e r t a i n currents flow which depend simultaneously on both, voltages. This property of non l i n e a r c i r c u i t elements i s widely used. The magnetic amplifier which uses a non l i n e a r induc- tance has been discussed t h e o r e t i c a l l y and experimentally 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 or non l i n e a r resistance has been widely used and thoroughly investigated as a high frequency mixer ( 5 , 6 ) . Note: The references are t y p i c a l but not complete. 1 . Lamm, A.U. "The Transductor, D.C. presaturated Reactor". Stock- holm Esselte Aktiebolag 1 9 4 3 . 2. Boyajian, A. "Theory of D-C Excited Iron Core Reactors and Reg- ul a t o r s " . A.I.E.E.Trans, V o l . 4 3 , p . 9 1 9 , June 1 9 2 4 . Chicago,111 . 3 . C a s t e l l i n i , R.R. "The Magnetic Amplifier", Proc.I.R.E., V o l . 3 8 , No 2, pp 1 5 1 - 1 5 8 . Feb. 1950 New York. 4 . Greene, W.E. "Applications of Magnetic Amplifiers". E l e c t r o n i c s , Sept. 1 9 4 7 . 5 . Herold, E.W. "Frequency Mixing i n Diodes". Proc.I.R.E., Vol.3 1 , No 1 0 , p 5 7 5 , Oct.1943, New York. 6. Torrey, H.C. and Whitmer, C A . "Crystal R e c t i f i e r s " . M.I.T. Radiation Laboratory Series. McGraw-Hill, New York, 1 9 4 8 . 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 capacitance 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 with marked non l i n e a r properties have u n t i l recently been impossible to make. But recently i t has been found that some of the compounds of titanium, notably barium titanate (BaTi03) are f e r r o e l e c t r i c at room temperature, i.e;. a graph of charge against voltage f o r a suitable condenser having 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 hysteresis loop, s i m i l a r i n shape to a hysteresis loop of a ferromagnetic material. A condenser of t h i s type i s ca l l e d a non l i n e a r condenser - i t s capacity i s a function of the voltage across i t . The physical theory f o r the marked d i e l e c t r i c behaviour of some of the compounds of titanium has received much recent attention (7,8,9,10). The research described i n t h i s thesis i s on the use of non l i n e a r condensers with BaTiOJ d i e l e c t r i c s , i n c i r c u i t s . 7. Von Hippel, A., Breckenridge, R.G., Chesley, F.G., and Laszlo T i s z a . '•High D i e l e c t r i c Constant Ceramics". Ind. and Eng. Chem. Vol .38. No 11. pp 1097-1109, Nov. 1946, Easton, Pa. 8. Jonker, G.H. and Van Santen, J.H. "Properties of Barium Titanate i n Connection with i t s Crys t a l Structure". Science, Vol.109, No 2843, PP 632-635, June 1949. 9. Kay, H.F. and Rhodes, R.G. "Barium Titanate C r y s t a l s " . Nat. Vol. 160 p 126. July 1947. London. 10. Wul, B.N. and Goldman, I . M . , " D i e l e c t r i c Constant of Barium Titanate as a Function of Strength of an Al t e r n a t i n g F i e l d , " Compt. Rend. Acad. S c i . Vol .49, PP 177-180, Oct. 1945 - 3 - Dr. A. Tan der Z i e l , the d i r e c t o r of the work, has applied general mixer theory to non li n e a r condensers i n an audio c a r r i e r amplifier e i r c u i t ( l l ) and i n a high frequency (10 mega- cycles) mixer c i r c u i t (12). Very l i t t l e published experimental work on non l i n e a r condensers i n c i r c u i t s was found. Donley (13) did some q u a l i t a t i v e experiments on non l i n e a r condensers, with BaTiOj and SrTiOj d i e l e c t r i c s , of less 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 at 20 megacycles, and a frequency modulator at 40 megacycles, but his r e s u l t s could not be used to check Van der Z i e l ' s theory. The primary object of t h i s research was to 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 analysed by Van der Z i e l i n order to check experimentally h i s analysis i n d e t a i l , and to revise i t i f necessary. The secondary purpose was to evaluate the usefulness of non l i n e a r condensers i n these and other c i r c u i t s . There follows a b r i e f summary of the work done: Non l i n e a r condensers suitable for use i n audio frequency c i r c u i t s were developed. F i r s t r e s u l t s showed that the condensers made were more suitable f o r experimental measurements i n a s l i g h t l y d i f f e r e n t amplifying c i r c u i t to that analysed by Van der Z i e l . An 11. Van der Z i e l , A. Report to Defense Research Board of Canada. University of B r i t i s h Columbia. Jan. 1949. 12. Van der Z i e l , A. "On the Mixing Properties of Non Linear Condensers", Jour. App. Phys. Vol .19, No 11, pp 999-1006, Nov. 1948. Lancaster, Pa* 13. Donley, H.L., '^Effect of F i e l d Strength on D i e l e c t r i c Properties of Barium Strontium Titanate"., R.C.A. Rev. Vol . VIII, No 3, PP 539-553, Princeton, New Jersey, Sept. 1947. adaption of h i s theory was made and the conclusions of the theory thus changed have been checked experimentally with good but not complete agreement. An account of the main audio frequency pro- per t i e s of these non l i n e a r condensers i s now possible. No measurements have been done on the high frequency mixer, although the c i r c u i t was roughly b u i l t , but the r e s u l t s of the work at low frequencies should greatly a s s i s t work at high frequencies. CHAPTER I I MAZING OF NON LINEAR CONDENSERS FOR AUDIO FREQUENCIES At the beginning of the work i t was known that BaTiOj properly prepared has a marked hysteresis loop at 4800 volts per cm. (14) The problem was to use t h i s knowledge to make a condenser of capacity large enough f o r audio c i r c u i t s (.001 to .01 mfds), showing marked non l i n e a r i t y at voltages not exceeding JOOv A.C, having a breakdown voltage at le a s t above the voltage at which non l i n e a r i t y was marked.", and having some means of removing the heat generated by hysteresis losses. The preparation developed for condensers s a t i s f y i n g these requirements has three main stages: 1. Preparation of a Sample of Bulk D i e l e c t r i c The method used was very s i m i l a r to that used by Yon 14. See Ref. 7. Hippel and h i s co-workers (15) with a few minor changes. Barium titanate i n dry powder form (obtained from Titanium A l l o y Manufacturing Co.) was pressed i n a l / 2 t t diam. press at 60000 l b s / s q . i n . to a thickness of about .5 mm. The discs were then placed on platinum f o i l i n an alundum crucible and passed through the following temperature cycle i n a high temperature furnace: An increase of 100°C per hour f o r nearly 13-1/2 hours to a temperature of 1350°C. A constant temperature of 1350°C for 6 hours. A decrease i n temperature at the cooling rate of the furnace. (The furnace took 36 hours to cool from 1350°C to room temperature). After the s i n t e r i n g cycle the d i e l e c t r i c was a hard yellow brown, b r i t t l e disc which had shrunk about 20% of i t s o r i g i n a l s i z e . The furnace used (diag.l) was made at the begin- ning of the work and was controlled with a Wheelco Chronatrol P o t e n t i o t r o l Model 23241, which controlled the power through a Superior E l e c t r i c Co. Powerstat No 1156. A disc which may be cut for any 24 hour temperature cycle i s the master control f o r the Pot e n t i o t r o l which controls the temperature i n s i d e the furnace to about 5°C. I t was found necessary to pl o t the r e l a t i o n between temperature at the sample and that given by the furnace thermo- couple since these are not p h y s i c a l l y at the same place. A blook diagram of the furnace and control i s given i n diagram 2. 2. Grinding the d i e l e c t r i c to 0.1 mm. 15. See Ref. 7. P L A T E I Alwndv/m O m u i t S t e e l Casing 3 i i - 0 - C c ! l CH Tnlw5atio« Fa«tn<J Rat^ if - Sfe=J £bj&m<j Ahwwfc/m Muff la. 2 A Icsrv̂ V I1V Q S < I "Jv»b«. Pt (O Rh as above F U R N A C E . D i a g r a m I FURNACE CONTROL. 5 0 Amps A . C . S Tun peicdvwa. - 6 - To reduoe the voltage necessary f o r marked non- l i n e a r i t y , to reduce the hysteresis losses, and to increase the capacity, as t h i n a d i e l e c t r i c as possible i s required. The technique used was borrowed from geologists who used i t to prepare t h i n rock samples. One side of the sintered d i s c was ground f l a t on a glass plate with carborundum and water and then polished on another glass plate with aluminum oxide and water. A t h i n f i l m of beeswax was melted on the f l a t side of the disc and allowed to harden. Grind- ing of the f l a t side was repeated. P a r t i c l e s of carborundum became embedded i n the wax which was forced into t i n y depressions on the surface of the d i s c . Grinding was continued u n t i l a l l wax except that i n the depressions had been removed, f i n a l removal of the wax being done on aluminum oxide. A coat of Dupont Conductive Coating No 4351 s i l v e r electrode paint was applied to the whole f l a t surfaee and allowed to dry. The di s c was heated to 600°C, held there f o r 10 minutes, and allowed to cool. This operation f i r e d the electrode to the d i e l e c t r i c . The wax technique was introduced to plug t i n y holes i n the d i e l e c t r i c to prevent flow of wet s i l v e r paint into them. The disc was next mounted s i l v e r e d side down with warm Canada balsam on a 1 " x 2" glass microscope s l i d e and the d i e l e c t - r i c ground down and polished u n t i l thickness was .1mm. The waxing procedure was repeated and the upper electrode applied to the disc while s t i l l on the s l i d e . This electrode was applied i n 6 t r i a n g l e s (diag . 3 ) from which 6 condensers were made. This made a semi- variable condenser from the whole disc and permitted a broken down -7- section to be removed. A f t e r the s i l v e r paint was dry the disc was c a r e f u l l y removed by heating from i t s monnt, cooled, and dipped i n benzene to remove the Canada balsam and the upper electrode f i r e d as before. Mechanically the disc could have been ground to . 0 3 mm. and s t i l l handled without breakage but i t was found that i f the discs were thinner than . 1 mm. e l e c t r i c a l breakdown had either occurred before any voltage was applied or d i d occur at low v o l t - ages f o r a large percentage of the condensers made. This break- down was attributed to t i n y holes and flaws i n the d i e l e c t r i c into which the electrode paint flowed or which i n some other way caused breakdown. The waxing technique reduced the l i m i t i n g thickness from .2 mm. to .1 mm. In an e f f o r t to further reduce the l i m i t i n g thickness electrodes of t i n f o i l and gold were t r i e d without success. A search of the l i t e r a t u r e yielded a paper by Howatt and co-workers (16) on the f a b r i c a t i o n of t h i n ceramic sheets f o r capacitors. In t h i s paper a method was given for making sheets of minimum thickness .15 mm. having a breakdown voltage f o r BaTiC-3 o f about four times that found i n the above condensers. Their method included i n the o r i g i n a l mix before s i n t e r i n g bonding materials to remove flaws i n the f i n a l product. Their method however was complex and required s k i l l . I t was decided therefore to obtain whatever r e s u l t s could be obtained with the present technique before combining the two 1 6 . Howatt, G.N., Breckenridge, B.G., and Brownlow, J.M., "Fabrication of Thin Ceramic Sheets for Capacitors", Jour. Am. Cer. S o c , Y o l . 3 0 , pp 237-242, 1 9 4 7 . P L A T E 1 1 Fatting Pa^C- 7 C O N D E N S E R E L E C T & O & E '• ni i I I I I I II i n H U I I I I I II w i i ' i ' m "•"inn M T U M I ' •n'l'i i i" 1 " -Hf* C O N D E N S E R M O U N T GaR-Og OiclectVic Tin .fiiil Sleek Wowntjnq Holes OLaqraTn 3 Diacjra'm 4* C O M D E W S £ R S T A N D Copper Biack Coppei 0a.Tr DteiectVic D i a q T a m 5 - 8 - techniques. 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 d i s c s . Another method t r i e d to plug the holes was to separate by the t h i n d i s c without electrodes two l i q u i d s which formed a prec i p i t a t e as they met i n the holes of the d i e l e c t r i c . This method was used successfully with CaC03 but i t s breakdown voltage i s not s u f f i c i e n t l y high above that obtained by the waxing method. I t i s thought that t h i s method could be developed i f a material with a high breakdown value and which could be e a s i l y p r e c i p i t a t e d could be found. 3. Attachment of Leads and mounting of Condensers. The condensers were mounted on a small copper block 1-1/8" x l / 2 " x 3 / l 6 " with two countersunk holes 7/8" apart. A sheet of t i n f o i l was placed on the block and the disc, the oxide on the s i l v e r coat having been removed with emery, placed on the t i n f o i l with the divided electrode side up. A t h i n copper wire with a l i t t l e t i n f o i l wrapped on the end was placed on each upper electrode. The whole was heated u n t i l the t i n melted and sealed the d i s c to the block and a lead to each upper electrode. Solder may not replace t i n i n t h i s operation because solder tends to dissolve the s i l v e r electrode. Beeswax was melted to the upper electrode to seal the leads, (diag. 4 f o r d e t a i l s ) When a con- denser i s to be used i t Is screwed to a s o l i d copper (diag.5) which may be put i n a beaker of water for cooling. A summary of the properties of an average condenser of t h i s type i s now given: • 9 - Thickness . 1 mm. Area of 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. Operating Volts 1 5 0 v A.C. R.M.S. Expected t o t a l capacity of 6 sections at low voltage . 0 0 8 mfd. Temperature r i s e caused by hysteresis heating very small on copper mount and base. I f copper base i s not used hysteresis heating causes temperature of d i e l e c t r i c to r i s e above 1 2 0°C (Curie Point f o r BaTiOj) and a l l non li n e a r properties are l o s t . Voltage necessary f o r marked non l i n e a r i t y 8 0 v A.C. R.M.S. These properties s a t i s f i e d a l l the requirements and i t was decided to proceed with tes t i n g before an attempt to improve these condensers was made. I t was o r i g i n a l l y planned to test con- densers made of various titanate compositions but time has not permitted t h i s to 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 hysteresis loop. The hysteresis loops of the condensers made were observed at d i f f e r e n t A.C. and D.C. voltages across the condensers i n Sawyer and Tower's c i r c u i t ( 1 7 ) with a biassing addition (diag 6 ) . F u l l 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 . Minneapolis, Minn. - 1 0 - sized tracings from the scope face are given i n diagram 7 f o r a t y p i c a l specimen at 60 c.p.s. 60 c.p.s. was used for convenience. The approximate hysteresis loss for a given loop was calculated as follows: Let 7 be the D.C. Voltage required across the v e r t i c a l plate terminals to move the spot 1 inch. Let H be the corresponding horizontal voltage. I t i s assumed that a l l the applied voltage appears across N.L.C. . • l n 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. T0~7 l t t Horizontal d e f l e c t i o n means H(Ri + R 2 ) - 20 H vol t s Kg across N.L.C. Now V - 27, H = .54 • • 1 sq.in. of scope face corresponds to ( 2 0 H)(V) = ( 2 0 ) ( 5 4)(27) = 2 .9 joules. io7 io7 1 0 ? Consider the loop at 150v A.C. 0 Bias. Measured area = . 2 5 2 sq.ins. Energy Loss per cycle « ( . 2 5 2 ) ( 2 . 9) joules. (lO?) At 60 c.p.s. Power Loss - ( . 2 5 2 ) ( 2 . 9 ) ( 6 0 ) *; .045-watts. (lb>) At 10000 c.p.s. Power Loss - ( . 2 5 2 ) ( 2^9)(10000) •» 7.5 watts. (lO?) This l a t t e r c a l c u l a t i o n has assumed the hysteresis loop has the same area at 10000 c.p.s. as at 60 c.p.s. This was found to be nearly true (diag.8). Thus 7.5 watts i s approximately the power to be dissipated i n such a condenser i f i t i s to be run at 150 v 10000 c.p.s. This power must be supplied by the c a r r i e r of the c a r r i e r a m p l i f i e r . P L A T E H i HYST£R£5IS CIRCUIT ISOV D .C- T 1~F . . • • SNkq 23 Q^Jsov D-C. V a r i a c Step Up-t!ww*»'fov-w«Mr 2 o > .91 ;25D A.C. M.L.C hrr C.R.O -j i H'hJr c i A . c ; V O L T M E T E R So I O O (50 sro . HYSTER£SI3.. LOOPS IOO i s o s o IOO ISO IOO ISX) o ocvoLTM ares SO •OO •so Dlaqram 7 WgjATjOPj OF LOOP WITH FREQUENCY Dlacjva*m 8 -11 2. Basic Equation of the Non Linear Condenser. The problem i s to place the information contained i n the given hysteresis loops i n an equation which can be used to calculate the performance of the condensers i n c i r c u i t s . The problem w i l l be solved i f an expression f o r the current flowing i n a non l i n e a r condenser can be written as a function of the voltage across i t . Without bias the hysteresis loops are symmetrical about the horizon- t a l a x i s . Hence the current flowing i f a sine wave of voltage i s applied w i l l contain only odd harmonics. The non l i n e a r condenser without bias i s represented by a condenser C + i n p a r a l l e l with a r e s i s t o r R + (diag. 9 ) . C + has an equation: Q, - aV + bV? + cV^ (1) I e= ad7 + 3bV2<iv + ^eV^dV dT dT dT R + has an equation: I R - dV + eV3 + fV? .(2) where a,b,c,d,e,f ... are functions of the maximum value of A.C. Voltage (Vmax) applied to the condenser. a,b,c are s l i g h t l y dependent on frequency and d,e,f are strongly dependent on frequency. Thus t o t a l current into condensers i s I - adV + 3bV2dV + ^cV^-dV + dV + eV? + fV^ (3) dT dT dT The c o e f f i c i e n t s a,b,c,d, etc. could be found by an expansion of (3) into fundamental and harmonic currents for a sine wave of applied voltage and a Fourier analysis of an observed current wave. The exact solution of the problem involves tedious experimental P L A T E I V NON-LINEAR COMDEMSEfi WITHOUT ©IAS N . L . C . F O R C FOR D i A c j r a T n 9 0.2' and t h e o r e t i c a l work. In the theory developed i n the present research (3) has been approximated t o: I = adV + 2bV2dV (4) dt dt where a and b are functions of Ymax. The neglection of dV represents a serious quantitative error i n the theory but not a serious q u a l i t a t i v e error. The present work was concerned i n the main with the qu a l i t a t i v e r e s u l t s of ( 4 ) . Thus the model used of the non l i n e a r condenser consists of a non l i n e a r capacity with the equation: Q, = aV + bY? where b must be negative from the shape of the hysteresis loop. I f a bias i s applied to the condenser even power terms are also introduced into (3) and even harmonic currents flow. I f the shape at the hysteresis loop can be al t e r e d u n t i l i t i s almost rectangular as has been done with ferromagnetics, then with suitable bias the model of non l i n e a r condenser could be al t e r e d to a capac- i t y with the equation: Q, m aV + bV 2 I = adV + 2b7dV (5) dt d? This was the model used by Tan der Z i e l i n his two papers ( 1 8 ) and ( 1 9 ) . Since the condensers which had been made were more r e a d i l y represented by (4) than by (5) even with large bias, i t 1 8 . See Ref. 1 1 . 1 9 . See Ref. 1 2 . -13 was decided to work out Van der Z i e l ' s theory using (4) rather than (j?)and attempt to b u i l d a c a r r i e r amplifier based on (4). This t h e o r e t i c a l and experimental work w i l l be described l a t e r . The measurements f o r the c a l c u l a t i o n of the c o e f f i c i e n t s a and b i n (4) as functions of Vjaax at 9000 c.p.s. w i l l next be given. This frequency was chosen as the c a r r i e r frequency rather than 10,000 c.pi-s. because some of the parts for the c a r r i e r amplifier tuned more r e a d i l y to 9000 c.p.s. CHAPTER 17 DETERMINATION OF COEFFICIENTS IN BASIC EQUATION 1 . Derivation of Equations for C o e f f i c i e n t s . The c o e f f i c i e n t s a and b could be found by the general method mentioned above but the method given below i s somewhat simpler. I The neglection of condenser losses i n (4) w i l l not a f f e c t the v a l i d i t y of the calculations f o r a and b since these are concerned only with oapacitive currents (provided d i s con- stant) . I f a voltage Vsin wt Is applied, to the non l i n e a r con- denser the r e s u l t i n g current i s : I ( t ) = a ay(t) + 3b ( V ( t ) ) 2 dV(t) dt at « waVcos wt + 3/4 bwV^cos wt - 3/4 bwY5cos3wt»..(6) Since only fundamental and 3rd harmonic currents are present a and -14- b can be determined i n terms of fundamental and 3rd 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 of Vsin wt with zero impedance to 3rd harmonic. This i s d i f f i c u l t to make but an approximation to i t (diag 10) was constructed. The equivalent c i r c u i t from which calculations are made i s given i n diagram 11. Yoltage applied to non l i n e a r condenser i s : Vsin wt + V j Sin ( 3wt + 0) where « V Substitution of thi s voltage into I ( t ) - adV(t) + 3b ( V ( t ) ) 2 dV(t) dT ~~oT gives currents at fundamental and 3rd harmonic. Main terms at fundamental frequency are: In j t ) » waVcos wt + 3/4 bw v3cos wt Main terms at 3rd harmonic frequency are: I ^ t ) • 3aw V j cos (3wt + 0) - 3bwv3 00s 3wt In complex notation: I x = j (waV + 3/4- bw V?) (7) 13 - j (3aw V 5 - 3/4 bw V 2 V +) but I3 = -Vj "IT zll = J'(3aw V 5 - 3/4 bw V 2 V*). (8) R where V - V e ^ , V3 - VjeJ^wt+tf), Y + - VeJ3wt From (7) - waV + 3/4 bwV? (71) From (8) i t can be shown that: 1+ 9a2w2R «? R2 b2 w2y6 ( 9 j IS" - "v§2 -15- (7^) and (9) may now be solved simultaneously to give: (Because Y j « T some expressions nave been eliminated) a - \ n 2 - 4mt\ where m 2m w 2 R 2 V 2 2 I ; L R 2 W V t = i^2 - v 5 2 .(10) where b = -q. - J q 2 * 4pr 2p ^ P 8 3 ^ W 2 R 2 V 8 ) q = |1 wR 2Y 5 2 I-jV 5 j ) .(11) r = V j 2 V 2 + 91"!* V^R* ) (10) and (11) are the required expressions for a and b i n terms of quantities whose measurement next w i l l be described. When the measured quantities were substituted i n these expressions one of the two possible values for a and b was eliminated because b had to be negative. A l l values appearing i n the above equations are peak values. Volts, Amps, Ohms were used. Since the flow of fundamental current i n the above c a l - culations i s independent of the fundamental current may be measured i n a separate c i r c u i t to that shown above. This was done. The following measurements were done on a t y p i c a l con- denser of capacity that desired f o r the c a r r i e r amplifier (.004j>mfd). 2. Measurement of 3rd Harmonic Current flowing i n a non l i n e a r condenser with varying Fundamental Voltage. Diagram 10 gives the c i r c u i t used. 2^2 P L A T E V Faci«<, CUSCUIT PQR 3RD H A R M O N I C CURfigSsiT MEASUREMENT £ N.L.C. 28 mh Series TunaJ •ha 3-rd Hotwowc 0 S O O O ê -s. D i a g r a m 10 EQUIVALENT CIRCUIT OF ABOVS AT. 3 R D HARMONtC FReQUE&Sy T M.UC TYP ICAL WAVEFORM Q&SE.RVEP A T Z ( 0 ' A 6 t o ) O bssrvad Assumed 3-«d H e r D i a q T a m 12. - 1 6 - The circuit, was tuned to p a r a l l e l resonance at fundamental frequency by c 1 . The voltage at Z was observed on a calibrated o s c i l l o - scope which was part of the tuned c i r c u i t , a correction made for the presence of the fundamental and the 3rd harmonic current (I^) through the series tuned c i r c u i t calculated. A t y p i c a l wave form observed at Z i s shown i n diagram 12 with the harmonic composition assigned to i t . The peak d e f l e c t i o n was observed and converted to v o l t s . One h a l f of the peak fundamental voltage calculated at Z was subtracted and the remaining voltage converted to R.M.S. value. D i v i s i o n by the reactance of the c o i l gave the required value of 3rd harmonic current. Example: R.M.S. Volts at E at 9000 c.p.s. - 70 Oscilloscope S e n s i t i v i t y » 44 Volts per inch. Max Observed Deflection = l / 2 Total D e f l e c t i o n on Scope « 1.6 i n s . Peak Voltage - ( 1 . 6 )(44) = 70.4 volts Calculated fundamental peak voltage across 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. •» 64 .2 v o l t s R.M.S. 3rd Har. Current - ( 6 4 . 2 ) ( 1 0 0 0)(l) ( 1 . 4 1 4 H W L )(3) « 9 . 6 m-as The r e s u l t s f o r voltages at E from 0 to 150 are given with the re s u l t s of the next sections i n diagram 15. -17- 3. Measurement of Fundamental ourrent flowing i n non l i n e a r condenser with varying Fundamental voltage. Diagram 13 gives the c i r c u i t . The Voltages at E and Y were observed on a Cossor double beam oscilloscope and the capacitive and resistance com- ponents of fundamental current flowing i n the non li n e a r conden- ser calculated from the vector diagram 14. Example: R.M.S. Volts at E at 9000 c.p.s. - 70 © from double beam oscilloscope - 27.5® V - lb.O v I = (V y)(wC) =30.6 m-as Let V x - V E - (V y) cos 0 - 70 - 18 cos 27-5 - 54.0 V 2 - (T y ) S i n e = l b s i n 27.5 - 8.3 Tan 0 - V2 - 8.3 = .1537 7T 5370" 0 = 8.7° • y r = 9 O - e - 0 = 53-8° Resistive Current I R - I cos"v|r > (30.6)(,59) - l b . l m-as Capacitive Current I 0 * I sin-\^= (30 . 6)(.807) - 24 . 7 m-as Voltage across condenser = V 2 cosee 0 » 8jjŜ  = 55 v o l t s The r e s u l t s from these observations and calculations are given i n diagram 16. The values of capacity and resistance f o r the non l i n e a r condenser are shown i n diagram 17. I t can be seen that the neglection of the condenser resistance w i l l i n t r o - duce a serious quantitative error i n the theory. Further at low voltages t h i s resistance i s non l i n e a r and w i l l introduce a further error. P L A T E V I CIRCUIT frOR FUNDAMENTAL CU8R6WT M6ASUHEMENT Dvac|ram 13 VECTOR DIAGRAM FOR fUMDAMENTAL, CUftftgMT MEASUREMENT - 1 8 - 4. Calculation of a and b. The data for the c a l c u l a t i o n of a and b from formulas (10) and (11) has now been assembled. The constants used are given here: w = ( 5 . b 5 ) ( l 0 4 ) w2 = ( 3 . 2 ) ( l 0 9 ) R - 80 V j = IjR I i i l3» V, are taken from diagrams 15 and 16 and converted to peak value s. The values of a and b f o r voltages from 0 to 120 at 9000 c.p.s. are given i n diagram 18. Addition to Chapter IV. The following two sections are added to Chapter IV because they are o f f the main l i n e of t h e o r e t i c a l and experimental r e s u l t s discussed i n th i s thesis but are most clos e l y connected to the discussion of Chapter IV and should not be omitted. 5 . Audio O s c i l l a t o r and Power Amplifier. This unit was designed as the source of the c a r r i e r for the c a r r i e r a m p l i f i e r . I t has a push p u l l output capable of d e l i v e r i n g 18 watts and i s variable from 5000 c.p.s. to 15000 c.p.s. Since i t i s a standard o s c i l l a t o r and amplifier i t s design w i l l not be discussed. C i r c u i t i s given i n diagram 19. One h a l f of the output was used for measurements described i n sections 2 and 3 of thi s chapter. 6. Measurement of 2nd Harmonic current flowing i n biassed non l i n e a r condensers. PLATE VII ^ ^ i s ALL£K&PH§ FOR XYP^Ai . . . ; .CQWDEH^R^; I TOTAL* FUNDAMENTAL VoLTS DLaqram J5, O 5C FWWDAMEWTAL VoLTS D'uiqraro 16 3 V I V O FUNDAMENTAL \fc)LTS Dlaqram 17 FUNDAWENTAL \&LTS •~" Piaqram IB . •19- A f t e r i t had been decided to b u i l d an amplifier based on equation (4) t h i s measurement assumed secondary importance since i t would have been the decisive measurement of an ampli f i e r based on equation (j?). Both sides of the push p u l l output of the audio amp- l 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 possible i n the group of condensers available and placed i n series from plate to plate of the 6L6S. Adjustment of R and C balanced the condensers with respect to the fundamental and the fundamental current flowing through series tuned c i r c u i t 1 could be reduced to zero. 3rd harmonic current flowing from X to ground could not also be matched with respect to the 3rd harmonic. The voltage E to D was held f i x e d during the experiment at 200v R.M.S. at 9000 c.p.s. and the voltage B to ground observed for each value of bias on the condensers. The 2nd harmonic current from both condensers flowed through c i r c u i t 2 and could be c a l - culated when the voltage at B was known. The c i r c u i t had to be retimed for each bias s e t t i n g . The r e s u l t s are given i n diagram 21. The non coincidence of the curves f o r increasing and decreasing bias i s not considered s i g n i f i c a n t . I t may be caused by s l i g h t permanent e l e c t r i f i c a t i o n . The small residual 2nd harmonic current at zero bias had a 90° phase s h i f t to the current r e s u l t i n g from biassed condensers and was p a r t l y caused by de- tuning of the main c i r c u i t ( t h i s point 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. V I I I Dlaqram 19 CIRCUIT FOR 2NQHAftMOMtC CURRENT -MBASVftMWV N.4..C. "*" Oiacpftim 20 2 N D HAftWfcONlC CURRENT EC. © I A S VQLTASe. • • * • « « ' ' * *, » • -60 --|o _ - ^ 0 >.C Stag oa*t» Condenser Duaqrcurn 21 -20 The condensers were found to have a D.C. resistance of about 3 Megohms and to give 1 microamp of D.C. r e c t i f i e d current when lOOv A.C. was across them at 9000 c.p.s. The r e s i d u a l 2nd harmonic current at zero bias., was however small and was not further investigated. CHAPTER V THEORY 0E A DIELECTRIC CARRIER AMPLIFIER The theory given follows the pattern of Van der Z i e l 1 s analysis (20) but i s applied to a condenser having: Q. = aV + bV3 rather than Q. = aV + bV 2 1. P r i n c i p l e of Amplification A large voltage at high frequency (ang f r e q . w) and a small voltage at low frequency (ang f r e q . p) are applied to a non l i n e a r condenser of the type described above. Currents of frequency 2w + p and 2w - p flow. The p o t e n t i a l power contained i n these side bands i s greater than the low frequency input power. The p o t e n t i a l power amplification may be r e a l i z e d i n a suitable c i r c u i t by the addition of a small current at angular frequency 2w at the correct phase and the detection of the r e s u l t i n g mod- ulated s i g n a l . Some of the c a r r i e r power has been transferred to the low frequency s i g n a l . 2. Derivation of the currents flowing when two signals are applied to a non l i n e a r condenser. 20. See Ref.11. -21 Let Y Sin wt and. P S i n pt ( V » P , w>^p) be applied to a non l i n e a r condenser having K t ) = a a v(t) 4- 5 b ( v ( t ) ) 2 a v(t) a t a t In t h i s case V(t) - V Sin v/t + P Sin pt By substitution i t can be shown that: I ( t ) = aw V cos wt + ap P cos pt + ^ b j ^ l . wY^ 0os wt + wP2V cos vrt BiwY^ Cos Jwt - wY2P cos ( 2w + p)t — r 4 — — r - pPY2 cos ( 2w + p ) * + wV2p cos ( 2w - p)t - pPV2 Cos( 2W-P)" 6 4 - wP̂ V" cos (w + 2p)"fc - pVP 2 cos(w + 2p)* - wP2y cos(w" 2p)* ""4" 2 2 + pVP 2 cos (w - 2p)t + ppy2 cos pt + pP? cos pt 2 T ~ - pP^ cos 3ptJ Of these the mixed currents of greatest size are: 3bf-w72P cos(2w + p)* - pPV*2 cos(2w + p)* L ~~T~ 4 + wV2P cos( 2w - p)* - pPV 2 cos(2w - p)*) 2". 4 - 1 Since w » p these become approximately: 3b( -wVfP cos ( 2w + p)"t + wV2P eos( 2w - p)*) (12) 2 2 These are the currents upon which amplification i s based. Since they are inaepenaent of the sign of V a fcalancea c i r c u i t may be aesignea where the high frequency i s appliea to the conaensers i n push p u l l , the low frequency i n p a r a l l e l , ana these output currents fea into a loaa i n p a r a l l e l . The high frequency may thus be eliminated from the output c i r c u i t . The equivalent c i r c u i t of such a c a r r i e r amplifier i s given i n diagram 22. PLATE IX EQUIVALENT CIRCUIT O P CARRIER AMPLIFIER N.L.C I , »̂ ft« Psm pfr Xa w.L.cr̂ DLaqraw 22 B Is. C A 9 R I 6 R AMPLlPtER C I R C U I T a«gtta*rtinoci ite Hoar 4 " 4 " ' ^ I t 85 K ieoc.p-& HH KfctX. -s •8 ' 1 aoo DlaqmTn 23 Serashvtied •am ±L , I' -22' 3. Analysis of Carrier Amplifier Equivalent Circuit. (1) Assumptions made in c i r c u i t : (a) The non linear 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 is R2« The low frequency impedance between A and B i s high and capacitive ( i t w i l l be considered infinite) 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 internal impedance. Matching of practical generators w i l l be considered later after the impedance seen by the ideal generators have been found. (e) The 3rd harmonic voltage from D to E i s neglected. (2) Circuit Voltages. Across either non linear condenser there are six main voltages. ( V » P , w » P ) complex form 1. V Sin wt V - 2. P Sin(pt +^) P = Pe3(pt + f) 3. VxSinU 2w + p)t + 0) Vl= Y i ej((2w + p)t + 4. V ^ S i n U 2w - p)t + fifS V - V l l e J ( ( 2 w - p)t • r jfL) 5. VgSinU 2w + p)t + ©) v 2 = Y 2 e j ( ( 2w + p)t + ©) 6. V ^ S i n U 2w - p)t + 01) Vglas V 2 1e(( 2w - p)t + ei) (3) Derivation of Mixed Currents. These are applied to a condenser having: -23- I - a dY + 3bV 2 dY I t dT Prom the f i r s t term the currents i n complex form are: ja(wV + p p + (2w + p ) ( Y x + Y 2) + (2w - pKY-j 1 + Y 2 1 ) ) (13) From the second term the main currents i n trigonometric form are: ( V 1 , T 2 , V 1 1 , V 2 ? - . « v ) 3bfwY3 cos wt * wV3 cos 3wt + pPY2 cos (pt +4") + wY^P cos (( 2w - p)t - ^ r ) + WY2Y I C O S ( p t + 0) + w Y 2 j_i C O S (pt - 01) + w Y 2 y ? cos (pt + ©) 2~ ~ T " ~ T ~ ~ + W Y 2 Y P 1 cos(pt - 61) + (2w + p)YiY 2 cos((2w+p)t+#) - ( 2w+p)ViY2cos( pt- 2 I ZX ) ("~2T) + ( 2w - pjV-^V 2 C O S ( ( 2 W - p)t +'&-) - ( 2vy - p j Y ^ Y ^ o s (pt - j f 1 ) + ( 2w + P ) Y Q Y 2 c o s ( ( 2 w + p)t + e) - ( 2w + p)YpY 2 cos (pt + e) ( 2 ) ( 4 . ) + ( 2 w - pjv^v 2 cos ( ( 2 w - p)t + e1) - (2w - p) Y9H* cos (pt - e 1 )! ( 2 ) * ( 4 ) J C o l l e c t i n g a l l currents of angular frequency P from both terms and using complex notation: Ip = japP + J | b p l ! ( 2P + Y ± 1 + + + Y 2 1 + + - TX* - Y 2 + ) (14) where Y x + - V-je^P*^) y l + + . V l l e j ( p t - . j ^ - ) V2l++= Y 2 l e ^ P t - @ 1 ) S i m i l a r l y for angular frequency 2w+P: l2w+p - J ' a ( 2 w + p ) ( Y i +Y2) .+ i 3 b Y 2(-wP^+ ( 2w + p)(Y x + Y 2 ) ) # v . ( l $ ) -24- where P+ = Pe^^2w + p)t + T*) For angular frequency 2w - p I|w-p » ja(2w-p)(V 1 1+Y 2 1) + j|bY 2(wP 1 + + +( 2w-p)( V ^ + V ^ ) ) (l6) where pl++ - Pe^( ( 2w-p)t-^) (4) Derivation of output Power. From c i r c u i t : T i = - I i U i + 2jwLl) ) V 2 » -I2R2 ) Y1 = -2I 2(R 1 + 2jwLi) j (17) Since I - L • 2 I 2 because c i r c u i t i s balanced Substitution of (17) i n t o (15) with the assumption 2w + f « 2w y i e l d s the equation f o r I 2 at 2w + P*: Ij, - - j 3 bw V 2 P+ 2 l l - 4w2Lx( 2a + 3bY2j + jw( 2Rx + R 2 U 2*+%W) . r ( l 8 ) I f the absolute value of I 2 i s to be a maximum than: 1 - 4w 2L 1( 2a + 3bV 2) - 0 2wL]_ - 1 2w( 2a+3bT2) (19) I f (19) be used i n (18) I 2 = - 3 b Y 2 P+ 2(2Ri+R 2)(2a+3bV 2) (20) S i m i l a r l y for I 2 at 2w - p i . e . I 2 * I 2 1 - 5b V 2 P 1** ( a ) 2( 2Ri+R 2)( 2a+3bV2) The maximum possible power available from both side band frequencies for an input voltage P i n R i and both R 2 can be calculated to be: -25- Po = 9b 2 V4 p 2 .(22) 2( 2Ei+R2)( 2a+3bV2)2 I 2 and I2 1 nave'been converted to R.M.S. for t h i s derivation. ( 22) gives the maximum possible power available for reconversion to angular frequency P . Of t h i s power the amount available i n S-|_ i s : P = 9b2 P2 y4 H I ( 2 3 ) ( 2R 1+R 2 ) 2 ( 2 a + 3 b V 2 ) 2 This i s the p r a c t i c a l l y useful output power. (5) Derivation of Input Currents. From ( 2) where x = From ( 21) Thus I 2 = -xP + JbYJL 2( 2Ri+R 2)( 2a+3bV2) I g 1 = x P! + + IT - 2Io = -2xP + IT. 1 - 212 1 - 2 x P 1 + + From (17) V 2 - xR^"1" T g 1 - -x R 2 P 1 + + V! = 2x(R x + 2jwLi)P + Vj} m -2x(R]_ + 2jwL ! ) p l + + From t h i s i t can be shown that: V 2 + 8 8 2R2P ) Y l + - 2x(R 1+2jwL 1)P 7^1++ m -2x(Rx+2jwLi)P Substitution of (24) int o (14) y i e l d s : Ip - P( j(ap+2 bV 2p) - j 3 bpY2 x(2Ri + R 2 + 4 J W L T J ) 2 2 .(24) -26- This i s the low frequency current supplied to one condenser only. For both 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: IT - 2 I p - P( 2a+3bV2) - J9b2pV*( gJRi+RgH-jwLi)) 2( 2R1+R2K 2a+3bV2) ) When I«j i s resolved 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 capacitive input currents: ( (19) i s used) XJj m 9Pb2pV4 . (25) 2w( 2R1+R2M 2a+3bV2) 2 Ic = P( jp( 2a+3bY2) - j 9 b2 p ) ( 2£) 2( 2a+3bV2) ) (6) Derivation of Power Gain From (25) the low frequency input power i s : *E - 9 b 2 P2p y* ( 2 < 7 j 4w( 2Ri+R2) ( 2a+3bT2) 2 D i v i s i o n of (22) by (27) gives the maximum possible power gain: G - g° = *>l . . . . ( 2 8 ) (28) i s the end r e s u l t of the theory. I t gives the maximum possible power gain under i d e a l conditions f o r t h i s type of amp l i f i e r . I t can be shown that the power gain but not power input or output i s independent of the tuning condition (19) In the amplifier to be described i n the next chapter w - 90 P Thus maximum possible power gain = 180 (7) Discussion of the theory. - 2 7 - Tlie theory as given y i e l d s the following conclusions: I f most of the output power i s to be used In R]_ and not i n R 2 then R 2 must be as small as possible r e l a t i v e to RT.. I f a large output power i s desired RT_ and R 2 must be small from (22) but fo r a given R 2 there i s a value of R i where a maximum of power i s used i n RT_. The low frequency input capacitive current given by (26) shows that i t i s not effected 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 strongly effected by the output c i r c u i t . As the output power increases so does the input power. Thus input c i r c u i t matching depends on the output c i r c u i t . 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 high frequency generator sees a capacitive impedance which can be found from the f i r s t term of (13) and the f i r s t term of the following expansion. I t also sees a small r e s i s t i v e com- ponent since i t must supply the power for the power gain. 4. Qualitative Effects of Inclusion of Condenser Losses i n Amplifier Equivalent C i r c u i t . A complete analysis could be done with the non l i n e a r condensers i n section 3 of t h i s chapter shunted by appropriate r e s i s t o r s . Such an analysis i s quantitatively more complete than that given above but does not bring forward r e s u l t s which cannot be predicted q u a l i t a t i v e l y by a simple discussion of the c i r c u i t . The theor- e t i c a l picture i s further 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). - 2 8 - The following general conclusions about the i n c l u s i o n of condenser losses are drawn. Since the currents flowing through the shunting r e s i s t o r s flow through L]_ the tuning condition ( 1 9 ) w i l l be a l t e r e d . The low frequency signal input impedance w i l l contain an a d d i t i o n a l resistive.element representing the energy fed by the low frequency generator' to hysteresis losses. I f the greatest power gain i s to be achieved this input power must be small r e l a t i v e to the input power of ( 2 7 ) . Hence R± and therefore R 2 should be as small as possible but given Rg, R i must not f a l l , below a c e r t a i n value. The high frequency input signal w i l l have to supply considerably more power than indicated by the i d e a l i z e d theory. It must supply the hysteresis losses for the.,.high frequency high voltage s i g n a l . These can be calculated from diagram 1.7. Losses i n condenser hysteresis at side band frequencies w i l l also reduce the power gain achievable i n p r a c t i s e . The above theory and conclusions were checked.in an amplifier which was a physical r e a l i z a t i o n of diagram 2 2 . CHAPTER VI - EXPERIMENTAL RESULTS FROM CARRIER .AMPLIFIER ; •• . 1 ; 1. C i r c u i t . The c i r c u i t was similar to that used for measurement, of 2nd harmonic current and i s given i n diagram 2 3 . As before -29- the non lin e a r condensers were balanced with respect to fund- amental by R and C. Unless otherwise stated the following setting up procedure was used i n tests on the amplifier: The high frequency was tuned to 9000 c.p.s. and the low frequency to 100 c.p.s. The c i r c u i t was tuned with c 1 . The condensers were matched u n t i l the voltage at Y was a minimum. C i r c u i t 2 was tuned u n t i l side band current was a maximum. Since a l l these adjustments are not independent they were repeated u n t i l a l l held simultaneously. About l/2 hour warm-up time was required for conditions i n the c i r c u i t to s t a b i l i z e . Since non linear condensers are highly voltage sensitive f l u c t u a t i o n s i n l i n e voltage were always obser- vable but were not s u f f i c i e n t to necessitate use of voltage regulated supplies. 2. Measurement of Output Power. The measurement of sideband current or output power was done as follows: • A calibrated oscilloscope was placed from B to ground. When the c i r c u i t was operating the side bands were observed. The condensers were biassed to a few volts u n t i l a l i t t l e less than 100 per cent modulation existed. (Theory quickly shows that biassing does produce 2nd harmonic i n the correct phase to give a normal modulated s i g n a l . - 3 0 - I f i n the expansion Q, = f ( ^ ) for a non l i n e a r condenser there i s a term proportional to Y 2 a n a y gj_ n j _ s applied then the 2nd harmonic current has the phase Sin 2 wt. From equation ( 1 2 ) the sideband currents have the phase 1 cos ( 2w + p)*. This i s the correct phase r e l a t i o n ) . A sketch of a typical'waveform at B i s given i n diagram 24. The distance x was measured and converted to v o l t s . From the theory of an amplitude modulated signal one quarter of t h i s voltage i s due to one sideband. The current at one sideband i s r e a d i l y calculated by the d i v i s i o n of t h i s voltage by the imped- ance from B to ground. The power i n one sideband i s given by the square of the current times the resistance i n series tuned c i r c u i t 2. Example: 2" t o t a l d e f l e c t i o n required 20v R.M.S. at 18000 c.p.s. . x measured = . 8 3 ins. wL = impedance B to gnd. = 2830 ax. RT_ = 110 ohms. Sideband current = ( . 8 3 ) ( 2Q)(1000) = 1.47 m-as 14)12b30) Power i n R i at 1 Sideband = ( 1.47) 2 ( 1 1 0 ) = 236 microwatts. ( 1 0 0 0 ) ( ) Power i n R-j_ at both sidebands = Power Output of Amplifier =472 microwatts. 3 . Measurement of Low Frequency input power- For t h i s measurement i t was e s s e n t i a l to remove a l l voltages except low frequency voltages at Y. The voltages at X and Y and the microammeter reading P L A T E X P a c i n g ftvtjc. 3 0 TYPICAL WAVEFORM AT AMPLIFIER OUTPUT IOO C P S . (6 m d l « ^ 2 3 ) Dlaqrarn 24 V E C T O R . D IAGRAM FOR LOW FREQUENCY I N P U T D l a q r a T n 25 were observed. YR was the voltage drop across R l . The current from X to Y as a function of V R was c a r e f u l l y plotted independ- ently. Hence the vector diagram 25 could be drawn since the lengths of a l l sides of the t r i a n g l e were known and the angle Q calculated. This gave the phase r e l a t i o n between the voltage, between Y and ground and the current flowing into the c i r c u i t at Y. The capacitive and r e s i s t i v e components of the current could be found and the power input calculated. The capacitive and r e s i s t i v e currents flowing into 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 subtracted from the res u l t s obtained with non l i n e a r condensers. The low frequency power and capacity currents flowing into the non l i n e a r condensers were thus found. Example: Y x = 2.42 volts Yy = 1.60 volts at 100 c.p.s. A = 2b.O microamps = I . 0 9 volts (from c a l i b r a t i o n curve). S = 1 (Yx + Yy + Y R ) = 2 .555 2 S- Y y = .955 S - VR = 1 . 465 VyY r - 1.745 Sin g. - f ( S - VyHS - Y R ) \ 1 / 2 . ( - 9 5 5 ) ( I . 4 6 5 ) = .895 2 ( Vy-v^. j -ii.-JiJ) ai, = 1270 e - 55 Resistive Current = ( 2 6 . 0 )(cos 53) = 1 5 . 6 microamps. Capacitive Current = ( 2 6 . 0)(Sin 53) = 20.6 microamps. Low frequency Power Input = ( 1 . 6 ) ( 1 5 . 6 ) = 25 microwatts. -32- From these the power, capacitive current- and r e s i s t i v e current fed into the c i r c u i t without non lin e a r condensers must be sub- tracted but the method of measurement of these i s i d e n t i c a l and i s not given. 4. .Variation of Output Sideband Current with Low Frequency Voltage. The following were held constant: High Frequency = 9000 c.p.s. High Frequency Volts per condenser - 40 and 120 R-L = 110 ohms. Low Frequency 100 c.p.s. The voltage at Y was varied and the output at B observed. Results: Diagram 26. Equation (26) predicts a l i n e a r v a r i a t i o n . The observed curve i s ce r t a i n l y very nearly l i n e a r with a tendency to be'concave down- ward which increases at the lower value of high frequency voltage. This point was not investigated further since t h i s bending might have been caused by a zero error i n the voltmeter. 3* V a r i a t i o n of Output Sideband Current with Frequency of Low Frequency Input. The following were held constant: High Frequency = 9000 c.p.s. High Frequency Volts per condenser = 40 R x = 110 ohms Low Frequency Voltage .= 1.5 v o l t The low frequency was varied and the output at B observed. Results: Diagram 27. IS . . . ^V .Q • < CD 1 1 (* ' .... ' * . y • jT1 lr..''/0-J.. r R.i«3.v=.4< • ~ : - f t t Of | r i j ~ ' . ' ' P L A T E XI Facing Pa.̂ e 32. - .1- fo © to it! •fOO. 2 © o Low FS|E^UENCY \/OLTS| •;:',] L:£ RRB̂ UEWCV :C;RS. 5 0 0 ~ 11 i- 1 ( . • • ' . - • • a . : i'r:'4 ' -t-4 '-.•.wcov^ft, a .a - 3 3 - Equation ( 2 6 ) predicts a horizontal straight l i n e . ' The curvature of the observed curve i s more l i k e l y to have occurred from experimental than t h e o r e t i c a l error. I t was not investigated further. 6 . Variation of Output Power with R T _ . The following were held constant: High Frequency = 9 0 0 0 c.p.s. High Frequency Volts per condenser = 40 and 1 2 0 . Low Frequency = 1 0 0 c.p.s. Low Frequency Volts =1.5 R 2 = D.C. Resistance of one half of T = 7 0 ohms. R^ v/as varied and output at B observed. Results: Diagram 2 8 . Equation (23) i s also plotted and has used values of a and b given i n diagram 1 8 . The experiment was repeated f o r L - i = 0 i n which case ( 2 3 ) becomes: P = 9 w 2 b 2 V4 P 2 Ri , , - — - ( 2 9 ) l+( SRx+Rg) 2( 2 a + 3 b V 2 ) 2 W 2 Experiment and theory are given for t h i s case i n diagram 2 9 . In the l a t t e r case agreement between theory and practise i s good but i n the former agreement i s poor. The absolute agree- ment between theory and practise within about 20f» i s not considered important since experimental error could possibly be t h i s great. But i n diagram 28 the errors greatly exceed t h i s f i g u r e . No explanation i s given f o r t h i s divergence. I t would seem incorrect to discard the theory e n t i r e l y since agreement i s good i n diagram 29 and c e r t a i n l y the experimental curves are greatly d i f f e r e n t i n -34- diagram 28 to diagram 29 and the g e n e r a l q u a l i t a t i v e f e a t u r e s o f the t h e o r e t i c a l curves show the same d i f f e r e n c e s . Nor w i l l the c o n s i d e r a t i o n of condenser l o s s e s save e q u a t i o n (23) f o r t h i s w i l l lower r a t h e r than r a i s e the t h e o r e t i c a l c u r v e s . I t i s thought t h a t the next approach to t h i s problem should be the c a r e f u l a l t e r a t i o n of LT_ and the p l o t t i n g of output pov/er as a f u n c t i o n of as LT_ i s changed from zero to the t u n i n g c o n d i t i o n o f ( 19) which i s c o n s i d e r e d to h o l d f o r diagram 2 8 . The apparatus would have to be a l t e r e d f o r t h i s because a t p r e s e n t L i i s n ° t e a s i l y c o n t r o l l e d . S ince t h e f a c t o r between theory and p r a c t i s e from diagram 28 a p p a r e n t l y depends on V, the v a r i a t i o n of output power as a f u n c t i o n of V f o r f i x e d R^ was next done. R i = 110 was chosen s i n c e the power output was ; g r e a t e s t f o r small r e s i s t a n c e s , t h i s being the r e s i s t a n c e of tuned c i r c u i t 2 without,any added r e s i s - t ance. 7« V a r i a t i o n of Output Power w i t h High Frequency V o l t a g e . The f o l l o w i n g were h e l d constant:' High Frequency = 9000 c.p.s. Low Frequency = 100 c.p.s. Low Frequency v o l t s = 1 . 3 Rx = 110 ohms R2 = .70 ohms. The h i g h frequency v o l t a g e was v a r i e d and the output a t B observed. R e s u l t s : Diagram 3 0 . E q u a t i o n (23) i s a g a i n p l o t t e d . - 3 5 - Th e shape of the. • two curves agrees w e l l except at low voltages. The divergence here may be assigned u n t i l extension of the theory to TDhe neglection i n the theory of the non l i n e a r i t y of the resistance component of the condensers at low voltages shown in diagram 1 7 . This i s the reason the low voltage t h e o r e t i c a l points are not weighted as heavily i n diagram 3 0 . Diagram 3 0 generally supports the v a l i d i t y of equation ( 2 3 ) . The low frequency input c i r c u i t was next investigated. 8. V a r i a t i o n of low Frequency Input Currents with High Frequency Voltage. The following were held constant: High Frequency =9000 c.p.s. Low Frequency = 1 0 0 c.p.s. Low Frequency Volts = 1 . 6 R l = 1 1 0 ohms R2 = 70 ohms. Results: Diagrams 3 1 and 3 2 . Equations ( 2 5 ) and ( 2 6 ) are plotted for comparison i n diagrams 3 1 and 3 2 respectively. The wide divergence between theory and experiment i n diagram 3 1 was expected since the low frequency input r e s i s - t i v e current w i l l be very sensitive to the neglection of conden- ser losses. At low high-frequency voltages the t h e o r e t i c a l points must again be l i g h t l y regarded. I t i s impossible to t e l l how much of the discrepancy would be eliminated by i n c l u s i o n of condenser losses. The negative slope of the experimental curve at low high frequency volts i s probably caused by non l i n e a r i t y PLATE XII o u Jro /op /as- Mi6H F R E Q U E N C Y V O L T S ° u & lop w «5" M i G H F R E Q U E N C Y V O L T S Dcacjmm 30 0 tacrram 31 O 50 foQ 12S HISH FREQUENCY VOLTS DtaqTuro 32, fOO I2S HI&H FREQUENCY VOLTS Dtacjr-a.m 33 - 3 6 - of condenser resistance. The agreement "between theory and practise for. the capacitive input current probably occurs because these currents i n theory are almost independent of condenser losses. 9 . Power Gain as a function of High Frequency Voltage. The experimental curve of diagram 33 was obtained by passing a smooth curve through the input r e s i s t i v e current plots and through the output power plots (not given but similar to diagram J>Q), and obtaining the r a t i o of output to input power. .The corresponding t h e o r e t i c a l curve i s a horizontal l i n e at height 135>f ( l80)( 2R\ )\, on the v e r t i c a l axis (from no loss theory). (R2 +2Ri)J dropping of the experimental curve to zero at low- v o l t s was expected since 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 losses at low frequency, but the subsequent behaviour of the curve cannot have any t h e o r e t i c a l explanation u n t i l an exhaustive analysis of non l i n e a r condensers has been done. However 70 of the t h e o r e t i c a l l y available 135 i s not considered unreasonable. 1 0 . General Discussion of Results. Although at least one experimental r e s u l t (diag . 2 8 ) cannot be explained by the theory even with a-''qualitative d i s - cussion of condenser losses, nevertheless the conclusions concerning the design of an amplifier, of the type described, drawn at the end of Chapter 5 have been confirmed by experiment;, '> The magnitudes of the various input and output impedances must s t i l l be determined by experiment i f exact magnitudes are necessary. However the general method given i n t h i s thesis -37- predicts magnitudes to within about a factor of two. I t must be emphasized however that neither theory nor experiment has been'exhaustive. Power gain has been checked at only one output load r e s i s t o r , and the frequency of the high frequency voltage has been kept at a l l times at 9000 c.p.s. This was done because of the number oftuning changes necessary i f th i s frequency was changed. CHAPTER VII CONCLUSIONS 1. Possibly the most important conclusion from the work done i s that non l i n e a r condensers- can In practise be used as power amplifiers as predicted by theory. 2. The greatest disadvantage of the condensers used i n t h i s research was t h e i r high losses. These losses make their theor- e t i c a l analysis complex and reduce t h e i r usefulness. The losses resulted i n : (a) Bulky cooling equipment. Because of the high temperature dependence of the 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 achievable power gain. (c) A very low o v e r a l l e f f i c i e n c y defined as t o t a l output power divided by t o t a l input power. This was of the order of .1%. - Two main l i n e s of development could reduce these losses. -36- (a) The d i e l e c t r i c could be made thinner. Two possible methods were suggested i n Chapter I I , Sec.2. I f a given capacity and a given f i e l d strength i n the d i e l e c t r i c are required then the losses vary as the square of the d i e l e c t r i c thickness. A reduction of the present d i e l e c t r i c thickness by a factor of three which i s considered possible for audio frequency con- densers would reduce the losses by a factor of nine. (b) The area of the hysteresis loop might be reduced by physical and chemical research. Combinations of materials might be superior to barium t i t a n a t e . 3.. As the condenser losses are reduced a more detailed check of the theory as given w i l l be possible. In i t s present form i t i s s u f f i c i e n t for broad p r a c t i c a l design of an- amp l i f i e r . On the other hand the theory could be extended without great d i f f i c u l t y to include the condenser losses. I t should be mentioned .that ;the. extension of .Van der Z i e l ' s t h e o r e t i c a l method was done independently by the author afte r Dr. Van der Z i e l had l e f t the University of B r i t i s h Columbia, and has not been completely checked by anyone. 4. Van der Z i e l ' s analysis (21) using Q, ,= aV + bV 2 ' \ ; as the basic non l i n e a r condenser equation gave as a t h e o r e t i c a l l y possible power amplification (w) 2 as against 2(w) derived i n t h i s •'.(F) (?) thesis for a non l i n e a r condenser with the equation Q, = aV + bV^ 21. See Ref. 11. In p r i n c i p l e then the development of condensers obeying the former r e l a t i o n (rectangular hysteresis loops) would seem more pro:- mising. In the present condensers however the c o e f f i c i e n t of Y 2 i n the expansion for Q, i s small (Chapter IY, Section 6 ) , and the input power at low frequency to overcome feedback would be much . smaller r e l a t i v e to the hysteresis component of input power than i n the amplifier tested. However the greater possible gain might overcome t h i s to give a greater o v e r a l l than that achieved i n t h i s work. However disadvantages would be the necessity of about 100 v o l t s of bias and the need to feed the low frequency input i n push p u l l . Unfortunately time did not permit experi- ments on t h i s type of amplifier. 5. The use of non l i n e a r condensers at high frequencies was not investigated. That they can be used at high frequencies 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 voltages and a very t h i n chip as d i e l e c t r i c to give him about 100 m.m.fds. or l e s s . He did not report overheating. 6. Since power amplification has been demonstrated possible with non l i n e a r condensers feed back may i n p r i n c i p l e be used to i n - crease the amplification. I t could be increased u n t i l low f r e - quency o s c i l l a t i o n s were present provided a frequency control was supplied. 7. D i e l e c t r i c amplifiers have a l l the advantages of magnetic amplifiers of ruggedness, and i n d e f i n i t e l i f e t i m e ( t h i s has not been v e r i f i e d ) , and the lack of heaters. 22. See Ref. 13. In p r i n c i p l e , i n analogy with 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. amplifier the d i e l e c t r i c amplifier might be most usef u l . The input power necessary to bias non l i n e a r condensers would be small since no hysteresis losses would be present. The the o r e t i c a l gains are i n f i n i t e for both types of d i e l e c t r i c amp- l i f i e r mentioned i n section 4 of this chapter, and a very high, power gain might be achieved even with the present condensers which apparently have a r e l a t i v e l y low D.C. resistance. The non l i n e a r i t y property might be used d i r e c t l y as an automatic control. The temperature dependence of the properties of non line a r condensers and the jump to a l i n e a r condenser at the Curie temperature may also f i n d a p p l i c a t i o n . -41- BIBLIOGRAPHY 1 . Boyajian A. "Theory of D.C. Excited Iron Core Reactors and Regulators". Transactions of the American Institute of E l e c t r i c a l Engineers. V o l . 4 3 , p 9 1 9 , June 1924. . Chicago, 1 1 1 . 2. C a s t e l l i n i N.R. "The Magnetic Amplifier". Proceedings of the I n s t i t u t e of Radio Engineers. V o l . 3 8 , No 2 , pp 1 5 1 - 1 5 8 , Feb. 1 9 5 0 , New York. 3 . Donley, H.L. "Effect of F i e l d Strength of D i e l e c t r i c Properties of Barium Strontium Titanate". Radio Corporation of America Review V o l . 8 , No 3 , PP 5 3 9 - 5 5 3 , Sept.1947, Princeton,N.J. 4 . Greene W.E. "Applications, of Magnetic A m p l i f i e r s " E l e c t r o n i c s , Sept. 1947- 5 . Herold W.E. "Frequency Mixing i n Diodes". Proceedings of the Ins t i t u t e of Radio Engineers. Vol 3 1 , No 1 0 , p 5 7 5 , Oct.1943, New York. 6. Howatt G.N. Breckenridge R.G., and Brownlow J.M., "Fabrication of t h i n Ceramic Sheets for Capacitors" Journal of the American Ceramic Society, Vol 30, pp 237-242, 1 9 4 7 . 7. Jonker G.H. and Van Santen J.H. "Properties of Barium Titanate i n Connection with i t s Crystal Structure" Science Vol 1 0 9 , No 2843, PP 6 3 2 - 6 3 5 , June 1 9 4 9 . 8 . Kay H.F. and Rhodes R.G. "Barium Titanate C r y s t a l s " Nature, Vol l 6 0 , p 1 2 6 , July 1 9 4 7 , London. 9 . Lamm A.U. "The transductor, D.C. presaturated Reactor". Stockholm.Esselte aktiebolag 1 9 4 3 . -42- 1 0 . Sawyer C.B. and Tower C H . "Rochelle S a l t as a D i e l e c t r i c " . P h y s i c a l Review, V o l 3 5 , No 3 , PP 269 - 2 7 3 , Feb. 1 9 3 0 . M i n n e a p o l i s , Minn. 1 1 . Torrey H.C. and Whitmer C.A. " C r y s t a l R e c t i f i e r s " . Massachusetts I n s t i t u t e o f Technology R a d i a t i o n L a b o r a t o r y S e r i e s . McGraw-Hill, New York, 1 9 4 8 . 1 2 . Van der Z i e l A. Report to the Defense Research Board of Canada. U n i v e r s i t y o f B r i t i s h Columbia. January 1 9 4 9 • 1 3 . 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 of Non L i n e a r Condensers". J o u r n a l of A p p l i e d P h y s i c s . Vol. 19,'No 1 1 , pp 9 9 9 T 1 0 0 6 , N O V. 1948. L a n c a s t e r , Pa. 14. Von H i p p e l A., Breckenridge R.G., Chesley F.G., and L a s z l o T i s z a . "High D i e l e c t r i c Constant Ceramics". I n d u s t r i a l and E n g i n e e r i n g Chemistry. V o l 3 8 , No 1 1 , pp 1 0 9 7 - 1 1 0 9 , Nov. 1 9 4 6 . Easton, Pa. 1 5 . Wul B.M. and Goldman I.M., " D i e l e c t r i c Constant of Barium T i t a n a t e as a F u n c t i o n of 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 " . Comptes Rendues des Academies de S c i e n c e s . V o l 4 9 , pp 1 7 7 - 1 8 0 , Oct. 1 9 4 5 .

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