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Magnetic amplifiers 1951

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MAGNETIC: AMPLIFIERS by JOHN CHARLES TEBBY A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE: i n t h e Department of ELECTRICAL ENGINEERING Weeaercept t h i s t h e s i s aa c o n f o r m i n g t o t h e s t a n d a r d r e q u i r e d : f r o m candidaAess f o r t h e degree of- MASTER OF APPLIED SCIENCE MeSsbe'rs: of t h e Department o f ELECTRICAL ENGINEERING THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1 9 5 1 MAGNETIC. AMPLIFIERS ABSTRACT D i r e c t c u r r e n t c o n t r o l l e d t r a n s d i i c t o r s , now known as m a g n e t i c a m p l i f i e r s , w e r e u s e d s u c c e s s f u l l y b y t h e Germans i n t h e 1939-1945 war f o r t h e a m p l i f i c a t i o n and m i x i n g o f powers down t o a m i c r o w a t t . T h e d e s i r e t o know more a b o u t t h e i r o p e r a t i o n s t h a n has b e e n g i v e n i n v a r i o u s p e r i o d i c a l s - : p r o m pted t h i s t h e s i s . An e x p l a n a t i o n o f m a g n e t i c a m p l i f i e r a c t i o n i s g i v e n and t h e waveforms.'-.of f l u x and o u t p u t c u r r e n t p r e d i c t e d . The b e h a v i o u r of a a . c i r c u i t s c o n t a i n i n g a : m a g n e t i c a m p l i f i e r i s t h e n e x p l a i n e d b y ; t h e u s e o f waveforms and a m a t h e m a t i c a l i n v e s t i g a t i o n i s made.A u s e f o r a m a g n e t i c a m p l i f i e r i n a n o s c i l l a t o r t a n k c i r c u i t i s d e s c r i b e d . G r a p h s o f t h e c h a r a c t e r i s t i c s o f t h e m a g n e t i c a m p l i f i e r and p h o t o g r a p h s of t h e waveforms a r e shown and t h e o b s e r v e d waveforms compared w i t h t h e p r e d i c t e d o n e s . A s e n s i t i v i t y t e s t on t h r e e c i r c u i t s c o n t a i n i n g : a magnetic., a m p l i f i e r i s d e s c r i b e d : and r e s u l t s g i v e n , s h o w i n g t h a t b y p l a c i n g a a c o n d e n s e r across?, t h e s u p p l y w i n d i n g s r i n c r e a s e s t h e s e n s i t i v i t y c o n s i d e r a b l y b u t d e c r e a s e s ^ t h e power a m p l i f i c a t i o n . The p r o p e r way t o s e c u r e a t h o r o u g h u n d e r s t a n d i n g o f t h e p r o p e r t i e s of t h e m a g n e t i c a m p l i f i e r b y t h e u s e of an o s c i l l o s c o p e i s d i s c u s s e d and t h e t h e s i s c o n c l u d e s b y l i s t i n g t h e a d v a n t a g e s and d i s a d v a n t a g e s o f a m a g n e t i c a m p l i f i e r . The m a t h e m a t i c a l a p p r o a c h t o t h e s o l u t i o n o f t h e m a g n e t i c a m p l i f i e r p r o b l e m u s i n g F r o l i c h ' s e q u a t i o n f o r t h e m a g n e t i z a t i o n c u r v e i s g i v e n i n a s p e c i a l appendix*. J o h n C.Tebby U.B.C. A p r i l , 1951 ACKNOWLEDGEMENT The a u t h o r wishes?;, t o e x p r e s s s h l s indebtedness? t o t h o s e who have a s s i s t e d him towasrds=. the? c o m p l e t i o n of t h ' l s s i n v e s t i g a t i o n ; e s p e c i a l l y t o P r o f e s s o r W.B. C o u l t h a r d f o r h i s : - e x p e r t guidance?, and encouragement^ Acknowledgement i s r a l s o made t o t h e B r i t i s h . Columbian Telephone Company L i m i t e d whose s c h o l a r s h i p made t h e y e a r ' s work p o s s i b l e . J ohn C.Tebby CONTENTS I I n t r o d u c t i o n Page 1 I I Theory o f t h e M a g n e t i c A m p l i f i e r 3 I I I M a t h e m a t i c a l I n v e s t i g a t i o n 7 I V C i r c u i t Theory A. A magnetic a m p l i f i e r i n s e r i e s 13 w i t h a r e s i s t a n c e and c a p a c i t y B. A magnetic a m p l i f i e r i n s e r i e s w i t h a r e s i s t a n c e and a c a p a c i t y 14 i n p a r a l l e l v / i t h t h e s u p p l y w i n d i n g s . C, A magnetic a m p l i f i e r u s e d as t h e i n d u c t a n c e i n an o s c i l l a t o r c i r c u i t 18 V A p p a r a t u s 19 V I P r o c e d u r e 20 V I I E x p e r i m e n t a l R e s u l t s 21 V I I I D i s c u s s i o n : 34 I X C o n c l u s i o n 36 Appendix: A - S o l u t i o n of t h e s e r i e s c i r c u i t 38 Appendix: B - The use o f F r o l i c h 1 s magnet- 39 i z a t i o n e q u a t i o n A p p e n d i x C - L i s t o f Symbols 41 X'- B i b l i o g r a p h y 43 LIST OF ILLUSTRATIONS Page F i g . l - A magnetic a m p l i f i e r w i t h the c o n t r o l windings 3 connected i n s e r i e s o p p o s i t i o n F i g . 2 - Waveforms of the magnetic a m p l i f i e r of F i g . l 4 Fig« 3 - The approximate BH curve 7 F i g . 4 - A magnetic a m p l i f i e r w i t h the c o n t r o l 7 windings omitted Fig«5 - Waveforms of the magnetic a m p l i f i e r of F i g , 4 8 F i g . 6 - A magnetic a m p l i f i e r w i t h the c o n t r o l windings 9 added F i g . 7 - The approximate BH curve showing i t s d i v i s i o n 9 l n t o ^ a n d ^ when Cj[ Is p o s i t i v e F i g # 8 - Waveforms of the magnetic a m p l i f i e r of F i g . 6 10 w i t h ^ p o s i t i v e F i g . 9 - Approximate BH curve showing d i v i s i o n i n t o (JrJ[ 11 an&(fiCwhenL/ i s negative Fig. 1 0 - Waveforms of the magnetic a m p l i f i e r of F i g . 6 11 w i t h negative Fig. 1 1 - Magnetic a m p l i f i e r having a condenser i n 13 series^ w i t h the load r e s i s t a n c e Fig. 1 2 - Magnetic a m p l i f i e r having a condenser i n 14 p a r a l l e l w i t h the supply windings Fig. 1 3 - Waveforms of the magnetic a m p l i f i e r of Fig. 1 2 16 Fig.14 - O s c i l l a t o r w i t h an i r o n cored inductance 18 i n the tank c i r c u i t Graphs of Magnetic A m p l i f i e r C h a r a c t e r i s t i c s Fig. 1 5 - Output current p l o t t e d against c o n t r o l current 22 f o r v a r ious values of load r e s i s t a n c e 29 30 Photographs of Waveforms Page F i g , 2 5 - Waveform of l o a d c u r r e n t f o r a 150 ohm l o a d 27 F i g , 2 6 - Waveform of l o a d c u r r e n t f o r a 100 ohm l o a d 28 F i g , 2 7 - Waveform of l o a d c u r r e n t f o r a 7.5 mfd. condenser i n p a r a l l e l and of not e q u a l t o C% F i g , 2 8 - Waveform of l o a d c u r r e n t f o r a 7.5 mfd. condenser i n p a r a l l e l and tf equail t o c 0, F i g . 2 9 - Waveform of l o a d c u r r e n t f o r a 6.5 mfd. 31 condenser i n p a r a l l e l and t n o t e q u a l to*'*. F i g , 3 0 - Waveform of l o a d c u r r e n t f o r a 2.5 mfd. 32 condenser I n s e r i e s F i g . 3 1 - Waveform showing t h e t i m e l a g i n t h e a m p l i f i e r r e s p o n s e t o a change i n thet 33 c o n t r o l c u r r e n t Page F i g , 1 6 - I n d u c t a n c e of t h e magnetic a m p l i f i e r 22 p l o t t e d a g a i n s t c o n t r o l c u r r e n t F i g , 1 7 - Output c u r r e n t p l o t t e d a g a i n s t c o n t r o l c u r r e n t f o r a 3,5 mfd,condenser i n s e r i e s 23 and p a r a l l e l F i g . 1 8 - Output c u r r e n t p l o t t e d a g a i n s t c o n t r o l c u r r e n t f o r a 100 m i l l i h e n r y choke I n 23 s e r i e s w i t h t h e l o a d r e s i s t a n c e F i g . 1 9 - Output v o l t a g e p l o t t e d a g a i n s t c o n t r o l c u r r e n t f o r a 6,5 mfd,condenser i n s e r i e s 24 and p a r a l l e l F i g , 2 0 - Output v o l t a g e p l o t t e d a g a i n s t c o n t r o l c u r r e n t f o r a 10 mfd.condenser i n s e r i e s ' 24 and p a r a l l e l F i g . 2 1 - Output v o l t a g e p l o t t e d a g a i n s t c o n t r o l c u r r e n t f o r v a r i o u s v a l u e s of c a p a c i t y 25 i n p a r a l l e l F i g . 2 2 - Output v o l t a g e p l o t t e d a g a i n s t c o n t r o l c u r r e n t f o r v a r i o u s v a l u e s of c a p a c i t y 25 i n s e r i e s : F i g . 2 3 - Outout c u r r e n t p l o t t e d a g a i n s t c o n t r o l c u r r e n t f o r a 10 mfd.condenser i n s e r i e s 26 and p a r a l l e l F i g , 2 4 - Output c u r r e n t p l o t t e d a g a i n s t c o n t r o l c u r r e n t f o r a 6.5 mfd.condenser i n s e r i e s : 26 and p a r a l l e l MAGNETIC AMPLIFIERS I INTRODUCTION The-purpose of t h i s i n v e s t i g a t i o n i s t o d e v e l o p a- t h e o r y f o r a - c i r c u i t c o n t a i n i n g ; RLO i n - s e r i e s o r i m p a r a l l e l w i t h as n o n l i n e a r element s u c h as? m. magnetic; a m p l i f i e r . S i n c e l i t t l e work has? been dome om t h e t h e o r y o f theemagnetic ampl- i f i e r t h e i n i t i a l i n v e s t i g a t i o n w i l l c o n c e r n t h e magnetic; a m p l i f i e r . The terms magnetic a m p l i f i e r , s a t u r a b l e r e a c t o r , i n d u c t o r , t r a n s d u c t o r and a m p l i s t a t have been u s e d t o denote? i r o n - c o r e d i n d u c t o r s i n w h i c h t h e a*c.impedance I s c o n t r o l l - e d by means o f an a u x i l i a r y d . o . e x c i t a t i o n . T h e y have b e e n known and a p p l i e d f o r some f i f t y y e a r s b e i n g u s e d m a i n l y i n : t h e dimming o f l i g h t s i n t h e a t r e s . T h e y had a l s o been u s e d t o a v e r y l i m i t e d e x t e n t i n heavy c u r r e n t and l i g h t c u r r e n t power c o n t r o l . I n t h e l a t t e r two c a s e s s i n c e 1935 much p r o g - r e s s hasEbeen made on t h e i r d e s i g n p a r t i c u l a r l y i n E n g l a n d . D u r i n g t h e 1939-1945 war c o n s i d e r a b l e p r o g r e s s was made i n Germany on t h e development of s i n g l e phase a m p l i f i e r s f o r s e r v o a p p l i c a t i o n s , w h i l e i n Sweden t h e main work seems t o have been on t h r e e - p h a s e i n d u c t o r s s u i t a b l e f o r l a r g e power o u t p u t s . I n t h e German l o n g range r o c k e t an I n d u c t o r w e i g h i n g t h r e e pounds w i t h a c o r e v o f Mumetal m a i n t a i n e d t h e a l t e r - n a t o r f r e q u e n c y t o w i t h i n ± ^ . A p p l i c a t i o n s were a l s o f o u n d i n a u t o - p i l o t s , s e r v o mechanisms and b l i n d a p p r o a c h systems.From v a r i o u s a c c o u n t s i t w o u l d appear t h a t a l t h o u g h German t e c h n i c - i a n s o b t a i n e d e x c e l l e n t r e s u l t s t h e y g a i n e d most of t h e i r 2 knowledge t h r o u g h t r i a l and e r r o r n o t by t h e u s e of t h e o r e t i c a l d e s i g n p r i n c i p l e s . Many, a r t i c l e s d e a l i n g w i t h t h e magnetic a m p l i f i e r have been p u b l i s h e d however t h e m a j o r i t y o f them a r e non-math- e m a t i c a l . N o n e , t h i s a u t h o r i n c l u d e d , h a s b e e n - a b l e t o d e v i s e a n o n - l i n e a r method f o r t h e s o l u t i o n o f t h e magnetic a m p l i f i e r . A g e n e r a l s o l u t i o n by n o n - l i n e a r methods does n o t e x i s t , h o w e v e r s t e p - b y - s t e p i n t e g r a t i o n y i e l d s ^ a p a r t i c u l a r s o l u t i o n - : a p p l y i n g o n l y t o a p a r t i c u l a r c u r v e . F o r t h i s r e a s o n a l l o f t h e a u t h o r s have e l i m i n a t e d t h e s o l u t i o n by n o n * - l i n e a r methods i n f a v o r o f an a p p r o x i m a t e s o l u t i o n by l i n e a r methods.In o r d e r t o o b t a i n t h e s o l u t i o n s b y l i n e a r methods assumptions must be made.The p u b l i s h e d a r t i c l e s ? have f o r t h e most p a r t made t h e f o l l o w i n g a p p r o x i m a t i o n s t o c o n v e r t t h e n o n - l i n e a r e q u a t i o n s t o l i n e a r ones: 1. The BH c u r v e can be ap p r o x i m a t e d t o be made up o f t h r e e s t r a i g h t l i n e s , a r e g i o n o f c o n s t a n t p e r m e a b i l i t y f o l l - owed b y complete s a t u r a t i o n . 2. The r e a c t a n c e o f t h e t r a n s d u c t o r w i t h z e r o c o n t r o l i s so l a r g e compared w i t h t h e r e s i s t a n c e o f t h e l o a d t h a t t h e d i f f e r e n c e i n phase between t h e a p p l i e d v o l t a g e and t h a t a c r o s s s t h e t r a n s d u c t o r i s n e g l i g i b l e . » 3. The r e c t i f i e r s a r e p e r f e c t and t h e b a c k l e a k a g e i s z e r o . W i t h t h e p r e c e d i n g a p p r o x i m a t i o n s i t hasr.been p o s s i b l e t o p r e d i c t i n advance t h e c h a r a c t e r i s t i c s o f a p a r t i c u l a r mag- n e t i c a m p l i f i e r . T h i s has: c o n t r i b u t e d g r e a t l y t o t h e p r o g r e s s t h a t has been made i n t h e p a s t few y e a r s . THEORY OF THE MAGNETIC AMPLIFIER The most i m p o r t a n t component of a magnetic a m p l i f i e r i s t h e i n d u c t o r "which c o n s i s t s of two a.c.chokes w i t h a superimposed d . e . w l n d i n g as shown i n F i g . 1 . I n F l g . l an i n d u c t o r i s shown where t h e d . c . w i n d i n g s a r e co n n e c t e d i n s e r i e s oppos- i t i o n . vVW—| As t h e i n d u c t o r d . c . w i n d i n g s a r e c o n n e c t e d i n s e r i e s o p p o s i t i o n t h e superimposed d . c . e x c i t a t i o n w i l l cause one c o r e t o s a t u r a t e i n t h e n e g a t i v e f l u x d i r e c t i o n and l a t e r i n t h e c y c l e t h e o t h e r c o r e w i l l s a t u r a t e i n t h e p o s i t i v e f l u x d i r e c t i o n as shown i n F i g . 2 . The v o l t a g e a p p l i e d t o t h e a . c . w i n d i n g s d u r i n g t h e p e r i o d 0 t o < i s p o s i t i v e and t h e f l u x i n b o t h c o r e s i n c r e a s e s u n t i l e v e n t u a l l y c o r e 1 becomes s a t u r a t e d p o s i t i v e l y and remains so u n t i l t h e v o l t a g e e n t e r s t h e n e x t h a l f c y c l e . 4; S u p p l y V o l t a g e Load C u r r e n t F l u x Waveforms FIG. 2 Core 1 has no impedance when s a t u r a t e d and I n c r e a s e s t h e c u r r e n t t h r o u g h c o r e 2 w h i c h a l s o "becomes s a t u r a t e d and t h e r e f o r e t h e whole o f t h e s u p p l y v o l t a g e between <»c and^? i s t r a n s f e r r e d t o t h e l o a d r e s i s t a n c e . B e t w e e n 0 and«<the o u t p u t v o l t a g e a c r o s s t h e l o a d r e s i s t a n c e i s s m a l l because b e i n g u n s a t u r a t e d t h e t r a n s d u c t o r r e a c t a n c e i s l a r g e compared t o t h e l o a d r e s i s t a n c e . D u r i n g t h e n e x t h a l f c y c l e t h e a c t i o n i s r e p e a t e d w i t h t h e d i f f e r e n c e t h a t i t i s c o r e 2 t h a t s a t u r a t e s i n t h e n e g a t i v e f l u x d i r e c t i o n . Because t h e a p p l i e d v o l t a g e i s s i n u s o i d a l t h e f l u x waveforms must be s i n u s o i d a l and each c o r e must c o n t r i b u t e h a l f t h e f l u x i . e . t h e s u p p l y v o l t a g e w i l l d i v i d e e v e n l y between each core.When we add d . c . e x c i t a t i o n one f l u x w i l l b e i n c r e a s e d and t h e o t h e r w i l l be d e c r e a s e d because o f t h e r e v e r s a l of t h e w i n d i n g s . T h e f l u x d e n s i t i e s may be r e p r e s e n t e d by t h e f o l l o w i n g two e q u a t i o n s : B0S//fGjt"Bm N e g l e c t i n g h y s t e r e s i s e f f e c t s , t h e m a g n e t i z a t i o n c u r v e f o r S t a l l o y I s r e p r e s e n t e d by t h e f o l l o w i n g e q u a t i o n ; H=aB+bB+"cB+c/B impere turns per cm. When we have a m a t e r i a l r e q u i r i n g a low number o f ampere-turns t o s a t u r a t e i t we have t o add 9 t h . o r 1 1 t h . powers and t h e e q u a t i o n i s a l t e r e d . B y p u t t i n g e q u a t i o n 1 i n t o e q u a t i o n 3 we g e t t h e e x c i t a t i o n f o r c o r e 1 asr o b t a i n e d by H a u f f e 1 ! , h, - a B„ *cB^fbB*Bm *f*J?A, ^scB^ -(ihB;Bm+fc£?B„+Sc2CBZ)c<*zut - t A ' + f w * lfcB?BmJcos4ot+(}kcB?)sms-a>t The a . c . e x c i t a t i o n i s r e p r e s e n t e d b y t h e e q u a t i o n and t h u s t h e ou t p u t c u r r e n t w i l l c o n t a i n a t h i r d harmonic w h i c h a c c o u n t s f o r t h e peaky shape o f t h e waveform. The c o n t r o l c i r c u i t e x c i t a t i o n c o n s i s t s of a st e a d y s t a t e and a s e c o n d , f o u r t h e t c . h a r m o n i c w i t h t h e harmonics b e i n g c o n t r i b u t e d by t h e s u p p l y voltage.We have t h e r e f o r e : The c u r v e o f i n d u c t o r o u t p u t a g a i n s t c o n t r o l c u r r e n t shows t h a t we have a s y m m e t r i c a l c u r v e about z e r o I n p u t c u r r e n t however t h e r e i s a s m a l l m a g n e t i z i n g c u r r e n t w h i c h v a r i e s w i t h s u p p l y v a l t a g e . W i t h two t r a n s d u c t o r s i t i s d e s i r - a b l e t o have i d e n t i c a l c h a r a c t e r i s t i c s and t h i s i s accomp- l i s h e d i n two ways,by s e l f - e x c i t a t i o n o r by b i a s e x c i t a t i o n . The m a t h e m a t i c a l t h e o r y o f t h e m agnetic a m p l i f i e r i s t h e n e x t s t e p and t h i s w i l l be worked out f o r an a m p l i f i e r w i t h a r e s i s t a n c e l o a d , w i t h a condenser i n s e r i e s , f i n a l l y f o r a condenser i n p a r a l l e l w i t h t h e s u p p l y w i n d i n g s of t h e a m p l i f i e r . We w i l l n o t work out a m a t h e m a t i c a l t h e o r y f o r t h e o s c i l l a t o r i l l u s t r a t e d i n t h e l a s t p a r t o f t h e n e x t s e c t i o n 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 c i r c u i t , T h e purpose of m e n t i o n i n g t h e o s c i l l a t o r w i l l appear i n t h e t e x t of t h a t s e c t i o n . I l l MATHEMATICAL INVESTIGATION One of t h e a p p r o x i m a t i o n s t o c o n v e r t t h e n o n - l i n e a r e q u a t i o n s t o l i n e a r ones was t h a t t h e BH c u r v e c o u l d be a p p r o x i m a t e d by t h r e e s t r a i g h t l i n e s - a r e g i o n o f c o n s t a n t p e r m e a b i l i t y f o l l o w e d by complete s a t u r a t i o n . T h u s we w i l l assume a c h a r a c t e r i s t i c of t h e shape shown i n F i g . 3 . The f i r s t c i r c u i t t o be a n a l y z e d w i l l be t h e one o f F i g . 4 where we have two c o r e s w i t h no a d d i t i o n a l w i n d i n g s as? i n t h e c o n v e n t i o n a l a m p l i f i e r . B Ems/nujt i F i g . 4 Lb) CO Supply- V o l t a g e Load Ourrefct •£ F l u x i n C o r e ! F l u x i n Core: 2 C o n s i d e r i n g F i g . 5 a , a s t h e v o l t a g e a t p o i n t A o f F i g . 4 i n c r e a s e s s i n u s o i d a l l y f r om 0 t o p o i n t a,the v a l u e o f <£«. i s v e r y low b e c a u s e of t h e h i g h r e a c t a n c e o f core; 1, t h e r e f o r e : t h e v o l t a g e d r o p - E L O A ( 3 ~ a c r o s s t h e r e s o s t a n c e i s low and /o* ait- la shown i n F i g . 5 c . B e t w e e n 0 and ar, t h e f l u x waveform w i l l be s i n u s o i d a l asr g£ = ^ ^ s , n t u t .At ar, c o r e 1 s a t u r a t e s t h e r e f o r e ^ -o ~ £>» T h i s s a t u r a t e d c o n d i t i o n e x i s t s u n t i l t h e v o l t a g e d r o p a c r o s s t h e s u p p l y w i n d i n g I s no l o n g e r h&gh enough t o m a i n t a i n t h e s a t u r a t i o n c u r r e n t and t h e r e f o r e -^0 .When t h i s o c c u r s t h e r e a c t a n c e o f c o r e 1 becomes v e r y h i g h and a g a i n approx- i m a t e l y t h e whole of t h e s u p p l y v o l t a g e i s aciross c o r e 1 f r o m b, t o -rf . D u r i n g t h e n e x t h a l f c y c l e c o r e 2 f o l l o w s t h e c o n d i t i o n s p r e v i o u s l y o u t l i n e d f o r c o r e 1 and c o r e 1 i s i n e r t . C o n s i d e r n e x t t h e c i r c u i t o f F i g . 6 w h i c h i s t h e same as t h a t o f F i g . 4 w i t h &- d . c . c o n t r o l w i n d i n g . T h i s d # c . f l u x a l t e r s t h e BH c u r v e as shown i n F i g . 7 (2) EL sin tot A/a Cone 1 No. Cone Z xc AQ V W \ A - F i g . 6 <f / ; 1 * * i AT As shown i n F i g . 7 t h e r e q u i r e d a . c . m a g n e t i z i n g c u r r e n t f o r s a t u r a t i o n w i l l be l e s s t h a n f o r t h e p r e v i o u s The c i r c u i t o f F i g . 6 w i l l have t h e e q u a t i o n as? f o l l o w s : To*** As t h e v o l t a g e a t p o i n t A o f F i g . 6 i n c r e a s e s s i n u s o i d a l l y f r o m 0 t o p o i n t a^ t h e v a l u e of ̂ i s v e r y low and t h u s E l o d w i l l be s m a l l and t h e r e f o r e S i n c e ^ i s a c o n s t a n t , ^ w i l l c o n s i s t of a s i n u s o i d a l f l u x and a s t e a d y v a l u e f l u x as shown i n F i g . 8 c . IP) (?) S u p p l y V o l t a g e Load C u r r e n t F l u x i n Coree 1 F l u x i n Core 2 F/G a A t p o i n t a , t h e c o r e s a t u r a t e s and t h u s (//is a c o n s t a n t . S i n e e e M ^ O j Sf/tiot' w h i c t L l B s h o w n i n F l g . 8 b . A s t h e s u p p l y v o l t a g e p a s s e s f r o m a. f t o b ^ t h e c o r e w i l l r e m a i n s a t u r a t e d , h o w e v e r a t b ; t h e s u p p l y v o l t a g e i s n o t h i g h enough t o m a i n t a i n s a t u r a t i o n c o n d i t i o n s i . e . ^Jfj^O and t h e r e f o r e I s v e r y low and t h e f l u x d e c r e a s e s t o t h e s t e a d y d . c . v a l u e . F o r t h e n e x t h a l f c y c l e t h e p r o c e s s i s . r e p e a t e d on c o r e 2. When i s n e g a t i v e t h e BH c h a r a c t e r i s t i c w i l l be a l t e r e d as i n F i g . 9 and t h e a . c . m a g n e t i z i n g c u r r e n t r e q u i r e d f o r s a t u r a t i o n w i l l be l a r g e r t h a n p r e v i o u s l y . T h e waveforms w i l l be shown i n F i g . 1 0 . Note t h e l e n g t h o f t h e s a t u r a t e d p o r t i o n of t h e f l u x wave and how i t has d e c r e a s e d f r o m t h e l e n g t h when l j was p o s i t i v e . The c o r e s w i l l s a t u r a t e l a t e r i n t h e c y c l e and u n s a t u r a t e e a r l i e r i n t h e c y c l e . F o r t h i s c o n d i t i o n t h e e q u a t i o n s a r e : 9 , - & r # CIRCUIT THEORY C o n s i d e r a c i r c u i t a s ^ F l g . l l where we have a cond- e n s e r and a r e s i s t a n c e I n s e r t e d f o r t h e l o a d . /r\£^~, 5/ncot —1|— ( c j y - z — The; g e n e r a l e q u a t i o n s a r e : T h e : l o a d v o l t a g e w i l l b e a p p r o x i m a t e l y a e r o f o r t h e p e r i o d 0 t o ar-of F i g . 8 a : and f o r t h a t p e r i o d /0* 4 * W J » and t h e f l u x , w i l l have t h e waveform p r e v i o u s l y shown i n F i g . 8c.From a . t o b . t h e c o r e i s s a t u r a t e d and t h e r e f o r e : w h i c h g i v e s us T?*<*L*~£ Jl'*&= t h e s t e a d y s t a t e s o l u t i o n : f o r w h i c h i s g i v e n i n Appendix: A and i s From bjto-TTthe f l u x waveform w i l l be s i n u s o i d a l as: i n F i g . 8 c , C o n s i d e r n e x t t h e magnetic a m p l i f i e r w i t h a r e s i s t - ance loa d . A condenser i s c o n n e c t e d aerosiss t h e a - i c . w i n d i n g s . F o r t h e c i r c u i t see F i g . 1 2 . -vww (^) Fig*12 The g e n e r a l e q u a t i o n s : a r e : ca * C^C^ As t h e v o l t a g e a t p o i n t A (Fig.12) i n c r e a s e s f r o m 0 t o a ; ( F i g . 1 3 a ) t h e r e a c t a n c e of c o r e 1 i s v e r y h i g h and t h e r e f o r e £ tsma 11, how e v e r a c u r r e n t w i l l f l o w t h r o u g h C^and t h u s a v o l t a g e drop w i l l o c c u r a c r o s s t h e l o a d . F o r t h i s c o n d i t i o n t h e c i r c u i t r e d u c e s ^ t o a - s e r i e s RC c i r c u i t and t h e e q u a t i o n s a r e : ^CL — CF t h e s t e a d y s t a t e s o l u t i o n f o r w h i c h i s (Appendix A ) c = UJ fco s/#ust - que* SM(I*'+«9 + UJ / L e t t h e v o l t a g e a c r o s s t h e core: be E I f t h e v o l t a g e a c r o s s t h e c o r e r e q u i r e d f o r t h e s a t u r a t i o n , c u r r e n t t o be s u p p l y v o l t a g e i s s u c h t h a t E i s e q u a l t o E B ^ t h e r e f o r e t h e c o r e s a t u r a t e s . U n d e r t h i s c o n d i t i o n t h e n e g l i g i b l e impedance of t h e c o r e b e i n g i m p a r a l l e l w i t h t h e capaasltance r e d u c e s t h e r e a c t a n c e o f t h e p a r a l l e l c i r c u i t t o as low v a l u e and t h e r e f o r e t h e v a l u e o f E i s low and so s u p p l y v o l t a g e reaehese s u c h a v a l u e t h a t t h e v o l t a g e a c r o s s t h e c o r e :1B no l o n g e r a b l e t o m a i n t a i n t h e c o r e i n s a t u r - a t i o n . F r o m b j t o T T t h e s u p p l y v o l t a g e i s d i v i d e d between and CL^.For t h e n e x t h a l f c y c l e t h e waveforms a r e r e p e a t e d e x c e p t t h a t c o r e 2 w i l l be i n v o l v e d i n s t e a d o f c o r e 1 . I t w i l l be notssd t h a t &&>c a n d ^ a r e added t h e r e s u l t a n t waveform of F i g . l 3 f i s a l m o s t af s i n e wave.Since t h i s i s alBO a r e p r e s e n t a t i o n of t h e l o a d v o l t a g e i t w i l l a p p r o x i m a t e a s i n e wave of t h e same fo r m as t h e c u r r e n t wave. d e v e l o p e d i s E g.From 0 t o a?, ^/hs/rfujt^VtUctL*'£ .At a-.the From a . t o b . t h e c o r e remains s a t u r a t e d u n t i l t h e S u p p l y V o l t a g e Load V o l t a g e Condenser V o l t a g e A m p l i f i e r C u r r e n t Condenser C u r r e n t Load C u r r e n t F l u x i n Core 1 * F l u x i n Core 2 F i g . 13 W i t h t h e c i r c u i t of F i g . 11 i t i m p o s s i b l e t o haves a: c o n d i t i o n where? theres i s no l o a d v o l t a g e n o r w i l l t h e c o r e be s a t u r a t e d , f o r t h i s t o o c c u r t h e r e a c t a n c e o f t h e condenser w i l l have t o be h i g h . B y c a r e f u l l y c h o o s i n g C Awe c a n have a c o n d i t i o n where t h e c o r e w i l l be s a t u r a t e d f o r o n l y as f r a c t i o n of a c y c l e t h u s g i v i n g a s h a r p c u r r e n t p u l s e . T h i s r e s u l t s when t h e v o l t a g e a c r o s s : t h e a m p l i f i e r i s h i g h enough t o cause s a t u r a t i o n and d e s a t u r a t i o n o n l y when t h e s u p p l y v o l t a g e i s n e a r i t s - p e a k . C o n s i d e r n e x t t h e c i r c u i t o f F i g . 12.With t h i s c i r c u i t i t i s p o s s i b l e t o have a - c o n d i t i o n wherea there? i s ? z e r o o u t p u t y e t t h e core: i s s a t u r a t e d . T h e c u r r e n t t h r o u g h the: c a p a e i t y w i l l be between 90*and 180° out of phase w i t h t h e c u r r e n t f l o w i n g t h r o u g h t h e s a t u r a t e d c o r e w i n d i n g , t h u s aa c o n d i t i o n c a n e x i s t where t h e s e two c u r r e n t s ? c a n c e l , h e n c e t h e l o a d c u r r e n t i s - zero.The r e a c t a n c e of t h e c a p a c i t y ina t h i s c i r c u i t w i l l have t o be h i g h o t h e r w i s e t h e c u r r e n t f l o w i n g t h r o u g h i t w i l l b e v e r y much l a r g e r t h a n t h e c u r r e n t t h r o u g h t h e a m p l i f i e r . W i th t h a t c o n d i t i o n t h e a m p l i f i e r w i l l be u n a b l e t o e x e r t any c o n t r o l o v e r t h e o u t p u t . IVC OSCILLATOR USING- AN IRON CORED INDUCTANCE C o n s i d e r n e x t t h e c i r c u i t i l l u s t r a t e d i n F i g . 1 4 , T h i s i s : a: t u n e d g r i d o s c i l l a t o r c i r c u i t w i t h a ma g n e t i c a m p l i f i e r as t h e i n d u c t a n c e i n t h e c i r c u i t . T h e purpose o f u s i n g t h e magnetic a m p l i f i e r i n t h i s c i r c u i t I s t o f i n d out how t h e i n d u c t a n c e v a r i e s as t h e d . c . c o n t r o l w i n d i n g i s e x c i t e d up t o s a t u r a t i o n . W e w i l l a l s o have f r o m t h e r e s u l t s an i n d i c a t i o n of t h e shape of t h e BH c u r v e . Let^*= f o r t h i s o s c i l l a t o r , t h e n -WJfZZ i U i U i i i D — — > — - co*C W h i l e t h i s i s n o t t h e c o r r e c t f o r m u l a t h e a p p r o x i m a t i o n w i l l s u f f i c e f o r our p u r p o s e . osc. F i g . 1 4 V APPARATUS The magnetic a m p l i f i e r u s e d i n t h e i n v e s t i g a t i o n - was a V i c k e r s E l e c t r i c E d u c a t i o n a l M a g n e t i c A m p l i f i e r ( s e e f o o t n o t e ) .The c o n t r o l c u r r e n t was measured w i t h a Weston d , c . m i l l i a m e t e r , t h e l o a d c u r r e n t w i t h a Weston aic.ammeter and t h e l o a d v o l t a g e w i t h a G e n e r a l J l a d i o vacuum tu b e v o l t m e t e r . F o r t h e o s c i l l a t o r c i r c u i t of F i g , 1 4 a 6 j 5 tube was used.The o s c i l l a t o r was c a l i b r a t e d on a C o s s o r O s c i l l o s c o p e by comparing i t s f r e q u e n c y w i t h t h a t f r o m a H e w l e t t - P a c k a r d o s c i l l a t o r • , The photographs of t h e waveforms were t a k e n on aa Westlnghouse s i x element o s c i l l o g r a p h u s i n g V e r i c h r o m e jjjf 122 f i l m . V a r i o u s v a l u e s o f r e s i s t a n c e s and condensers weree r e q u i r e d , b o t h f i x e d and v a r i a b l e , a s w e l l as s o u r c e s of d i r e c t c u r r e n t and a 110 v o l t , 6 0 c y c l e s u p p l y . F o r a d d i t i o n a l i n f o r m a t i o n on t h i s a m p l i f i e r see t h e " L a b o r a t o r y Manual of S p e c i f i c a t i o n s , I n s t r u c t i o n Notes; and E x p e r i m e n t s f o r t h e E d u c a t i o n a l M a g n e t i c A m p l i f i e r " p u b l i s h e d by t h e V i c k e r s : E l e c t r i c D i v i s i o n , 1 8 1 5 L o c u s t S t . , S t . L o u i s 3 , M i s s o u r i , U . S . A . V I PROCEDURE T e s t s were-made on t h e c i r c u i t s o f F i g s , 6 , 1 1 and 12.The graphs and photographs f o l l o w i n g a r e t h e r e s u l t s o f t h e s e t e s t s . T h e d a t a on t h e graphs and p h o t o g r a p h s states-- what c i r c u i t was u s e d and what was: v a r i e d . F o r F i g . 16 t h e c i r c u i t o f F i g . 14 was connected" up and v a r i o u s v a l u e s of c o n t r o l c u r r e n t were used t o v a r y t h e f r e q u e n c y . F o r each v a l u e o f c o n t r o l c u r r e n t t h e f r e q u e n c y was measured.This was a c c o m p l i s h e d "by p l a c i n g t h e o s c i l l - a t o r o u t p u t on one s e t o f t h e C o s s o r o s c i l l o s c o p e d e f l e c - t i o n p l a t e s and t h e o u t p u t o f t h e H e w l e t t - P a c k a r d o s c i l l a t o r on t h e o t h e r s e t of d e f l e c t i o n p l a t e s - of t h e same beam.The H e w l e t t - P a c k a r d o s c i l l a t o r f r e q u e n c y wasp t h e n a l t e r e d u n t i l b o t h f r e q u e n c i e s ? were t h e same as i n d i c a t e d by t h e c i r c l e , o v a l o r s t r a i g h t l i n e waveform b e i n g m o t i o n - l e s s > t h e shape of t h e waveform depending on t h e phase d i s p l a c e m e n t and v o l t a g e d i f f e r e n c e between t h e two o u t p u t s * A t e s t comparing t h e s e n s i t i v i t i e s ? o f t h e c i r c u i t s o f F i g g . 6 , 1 1 and 12 was made.R^ and C „ were 150 ohms and 10 m i c r o f a r a d s r e s p e c t i v e l y . B y u s i n g t h e graphs o f F i g s . 15 and 23 t h e mid p o i n t o f t h e s t e e p e s t p o r t i o n of each g r a p h was found.The c o n t r o l c u r r e n t was t h e n v a r i e d 2 ma.around t h i s v a l u e , i . e . 1 1 ma.,and t h e change i n o u t p u t c u r r e n t measured. A t e s t t o f i n d t h e t i m e l a g of t h e a m p l i f i e r i n r e s p o n d i n g t o a change i n c o n t r o l c u r r e n t was made.The c o n t r o l c u r r e n t was a b r u p t l y changed f r o m 0 ma.to 13 ma. and a p h o t o g r a p h made w h i l e t h i s change was t a k i n g p l a c e . V I I EXPERIMENTAL RESULTS The graphs and photographs f o l l o w i n g t h i s page show t h e r e s u l t s o f t h e v a r i o u s t e s t s made on t h e magnetic a m p l i f i e r , O t h e r t e s t s were c a r r i e d out w i t h i n t e r m e d i a t e v a l u e s and a l s o v a l u e s o u t s i d e t h e ranges: i n d i c a t e d however t h e r e s u l t s o b t a i n e d were v e r y s i m i l a r t o t h o s e f o r t h e ones shown and t h u s t h e y were n o t i n c l u d e d . The s e n s i t i v i t y t e s t s on t h e c i r c u i t s o f F i g s , 1 1 , 6 and 12 gave t h e f o l l o w i n g r e s u l t s : 1. F o r t h e c i r c u i t o f F i g . 6 , a change i n c o n t r o l c u r r e n t f r o m -5ma.to-7ma.gave a change i n l o a d c u r r e n t f r o m .40 amps.to .36 amps. 2. F o r t h e c i r c u i t o f F i g . 1 1 , a change I n c o n t r o l c u r r e n t f r o m -14 ma.to - 16 ma.gave a change i n l o a d c u r r e n t f r om .22 amps.to .14 amps. 3. F o r t h e c i r c u i t o f F i g . 1 2 , a change i n c o n t r o l c u r r e n t f r o m -11 ma.to -13 ma.gave a change i n l o a d c u r r e n t f r o m .18 t o .32 amps. The photograph,Fig.31»made t o f i n d t h e t i m e l a g o f t h e a m p l i f i e r i n r e s p o n d i n g t o a change i n c o n t r o l c u r r e n t shows t h a t t h e a m p l i f i e r r e q u i r e d one c y c l e of t h e 60 c y c l e t i m i n g w a v e , i . e . l / 6 0 , s e c o n d , t o respond f u l l y t o t h e change.   -*0 -So -so -io o ,o ;.o .:'•*> -'-n ->o •-• -%o •"->  - ? 0 - 3 0 - ^ 0 -10 O lO Z0 SO 4-0 Waveform showing load current w i t h a r e s i s t a n c e load F i g . 2S- Top Waveform C o n t r o l current Load current Load voltage Load r e s i s t a n c e -5 ma. .415 amps.rms. 78 v o l t s rms. 150 ohms. Bottom Waveform Sine wave of 60 c y c l e supply frequency Waveform of load current with a 100 ohm resistance as load Top Waveform Control current Load current Load voltage Load resistance 0 ma. . 5 8 amps.rms, 71 volts rms. 100 ohms Bottom Waveform 60 cycle sine wave of supply frequency WavefDim showing l o a d c u r r e n t w i t h a; condenser i n p a r a l l e l F i g . 27 Top Waveform C o n t r o l c u r r e n t - 0 ma. Load current: - .37 amps.rms. Load v o l t a g e - 73 v o l t s rms. Load r e s i s t a n c e - 150 ohms Condenser of 7.5 mfd.in p a r a l l e l C u r r e nt i n condenser b r a n c h - .31 amps.rms. Current i n a m p l i f i e r b r a n c h - .54 amps.rms. Bottom Waveform Sine wave of 60 c y c l e supply frequency Waveform of load current w i t h a.condenser i n p a r a l l e l 30 F i g . 3S> Top Waveform C o n t r o l c u r r e n t - -10 ma. Load current - .19 amps.rms. Load voltage - 31 v o l t s rms. Load r e s i s t a n c e - 150 ohms. Condenser of 7.5 mfd.in p a r a l l e l Current i n condenser branch - .36 amps.rms. Current i n a m p l i f i e r branch - .36 amps.rms. Bottom Waveform Sine wave of 60 c y c l e supply frequency Waveform of load current with a: condenser in parallel 3( Fig, 2 1 Top Waveform Control current - -25 ma. Load current - .34 amps.rms. Load voltage - 94 volts rms. Load resistance - 150 ohms Condenser of 6.5 mfd.in parallel Bottom Waveform Sine wave of 60 cycle supply frequency 5; WavefDum of load cu r r e n t w i t h a: condenser i n s e r i e s F i g . 30 Top Waveform C o n t r o l current - -25 ma. Load current - .27 amps.rms. Load vol t a g e - 91 v o l t s rms. Load r e s i s t a n c e - 150 ohms Condenser of 2.5 mfd.in s e r i e s Bottom Wave form 60 c y c l e s i n e wave of supply frequency Wavefroms showing the time l a g i n the a m p l i f i e r response t o a change i n c o n t r o l c u r r e n t F i g . 31 Top Waveform 60 c y c l e s i n e wave-supply frequency Middle Wavefrom c o n t r o l current waveform showing the second harmonics. Bottom Waveform load current C o n t r o l current changed from 0 ma.to 13 ma. Load current changed from . 1 amp.to . 4 7 amp.rms. Load voltage changed from 50 v o l t s t o 120 v o l t s rms. Load r e s i s t a n c e - 150 ohms. Note the time l a g i n the bottom waveform which i s seen t o be one c y c l e of 60 c y c l e supply frequency. V I I I DISCUSSION I n m a g n e t i c a m p l i f i e r d e s i g n t h e maximum o u t p u t w i t h i n t h e l i n e a r r a n g e o f t h e a m p l i f i e r c u r v e i s a n i m p o r t a n t f a c t o r , u n d e r c e r t a i n c o n d i t i o n s more so t h a n t h e a m p l i f i c a t i o n f a c t o r . I n d e t e r m i n i n g t h i s r a n g e t h e p r o p e r - t i e s o f t h e c o r e m a t e r i a l and t h e n a t u r e o f t h e l o a d a r e i m p o r t a n t . More and b e t t e r c o r e m a t e r i a l s a r e c o n s t a n t l y b e i n g p r o d u c e d and t h e a s s u m p t i o n t h a t t h e BH curve c a n be a p p r o x i m a t e d b y t h r e e s t r a i g h t l i n e s i s b e c o m i n g more and more c o r r e c t . T h e knee o f t h e c u r v e i s g r a d u a l l y b e i n g e l i m i n a t e d a s F i g . 1 6 shows and i s a l m o s t a b s e n t f r o m t h e m a t e r i a l o f t h e c o r e i n t h e m a g n e t i c a m p l i f i e r u s e d i n t h e i n v e s t i g a t i o n . The p e r f o r m a n c e o f t h e a m p l i f i e r u n d e r d i f f e r e n t l o a d p a r a m e t e r s h a s b e e n i n v e s t i g a t e d a n d i t has: b e e n f o u n d t h a t t h e s e n s i t i v i t y was i n c r e a s e d b y t h e a d d i t i o n o f a s e r i e s o r p a r a l l e l c o n d e n s e r . T h i s a d d i t i o n h a s a p r o f o u n d e f f e c t on t h e shape o f t h e o u t p u t wave c a u s i n g a v a r i a t i o n f r o m a s i n e wave t o a v e r y p e a k y wave and a l s o c a u s i n g a n o u t p u t wave a p p a r e n t l y t w i c e t h e f r e q u e n c y o f t h e s u p p l y v o l t a g e . T h e p r o d u c t i o n o f an o u t p u t s i n e wave w o u l d b e v e r y u s e f u l i f t h e o u t p u t were t o b e f e d t o t h e r m i o n i c v a l v e s f o r t h e n e x t s t a g e o f c o n t r o l . T h e p r o d u c t i o n o f a v e r y peaky wave f r o m t h e s e r i e s c i r c u i t o f F i g . 1 1 m i g h t b e o f u s e where a l a r g e p u l s e o f power was w a n t e d w i t h o u t t h e u s e o f t h e r m i o n i c v a l v e s . T h e r a n g e o f t h e o u t p u t impedance o f t h e a m p l i f i e r i s I n t h e r a n g e 25 t o 2500 ohms and above t h a t t h e o u t p u t power i s t o o low t o b e o f any u s e . The s i z e o f t h e m a g n e t i c a m p l i f i e r u s e d i n t h i s i n v e s t i g a t i o n and t h e c o m p a r a b l e s i z e o f a t r a n s f o r m e r o f t h e same r a t i n g shows t h a t t h e m a g n e t i c a m p l i f i e r i s a l m o s t t h r e e t i m e s t h e s i z e o f t h e t r a n s f o r m e r . T h i s w o u l d b e d e t r i m e n t a l i f l a r g e power o u t p u t s w e r e d e s i r e d " a t 60 c y c l e s . T h i s - c a n b e overcome b y r a i s i n g t h e s u p p l y f r e q u e n c y where p o s s i b l e s i n c e t h e s i z e o f t h e m a g n e t i c a m p l i f i e r i s p r o p o r t i o n a l t o t h e t h e r e c i p r o c a l o f t h e f r e q u e n c y . The i n v e s t i g a t i o n c a r r i e d o u t h a s p r o v e n how d i f f i c u l t i t i s t o a n a l y z e t h e p e r f o r m a n c e o f a m a g n e t i c a m p l i f i e r . A s p e c i a l m a t h e m a t i c a l a p p r o a c h has s t i l l t o be d e v i s e d ' b e f o r e any p r o p e r t h e o r e t i c a l d e s i g n p r i n c i p l e s c a n b e f o u n d * T h e Germans i n u s i n g t r i a l a nd e r r o r methods o f d e s i g n have p r o v e n t h i s t o b e t r u e . T h u s a t p r e s e n t t h e o n l y way t o a r r i v e a t a r e a l u n d e r s t a n d i n g o f t h e p r o p e r t i e s o f t h e m a g n e t i c a m p l i f i e r i s t o u s e a n o s c i l l - o g r a p h t o t r a c e t h e c o u r s e of t h e c u r r e n t s . IX CONCLUSION I n c o n c l u s i o n i t may be i n t e r e s t i n g t o r e a c p i t u l a t some o f t h e c h a r a c t e r i s t i c s o f p r a c t i c a l m a g n e t i c a m p l i f i e r s T h i s i s done i n o r d e r t h a t p o t e n t i a l u s e r s may j u d g e f o r t h e m s e l v e s t h e i r a d a p t a b i l i t y t o v a r i o u s f i e l d s . T h e s e c h a r a c t e r i s t i c s a r e : 1. They a r e v e r y r u g g e d , \ t f i t h no m o v i n g p a r t s . 2. They h a v e no f i l a m e n t s n o r do t h e y r e q u i r e a power p a c k . 3. They do n o t r e q u i r e t i m e t o h e a t u p and a r e r e a d y f o r a c t i o n i m m e d i a t e l y t h e s w i t c h i s c l o s e d . 4. T h e r e i s no i n t e r n a l h e a t l o s s . 5.. The i n p u t and o u t p u t c i r c u i t s c a n b e c o m p l e t e l y i s o l a t e d . 6. D . C . i n f o r m a t i o n c a n b e c o n v e r t e d t o a . c . i n f o r m a t i o n and a m p l i f i e d a l s o . 7. S i g n a l s a t d i f f e r e n t v o l t a g e l e v e l s c a n b e m i x e d . 8. T h e i r s i z e i s t h r e e t i m e s t h a t o f a t r a n s f o r m e r o f t h e same power r a t i n g . 9 . T h e i r s i z e v a r i e s as t h e r e c i p r o c a l o f t h e s u p p l y f r e q u e n c y . 10. They t a k e f r o m 1 t o 100 c y c l e s o f t h e s u p p l y f r e q u e - n c y t o r e s p o n d c o m p l e t e l y t o a change i n c o n t r o l c u r r e n t . 11. Power g a i n s o f up t o 10,000 c a n be o b t a i n e d . 12. They have a low i n p u t i m p e d a n c e , . 1 t o 3000 ohms, and a r e s u i t a b l e f o r a m p l i f i c a t i o n o f t h e r m o c o u p l e o u t p u t , p h o t o c e l l o u t p u t etc.-. 13. T h e i r o u t p u t impedance i s 25 t o 2500 ohms t h u s t h e y c a n be u s e d where t h e l o a d i s t o b e a . c o r d . c . m o t o r s , h e a t e r s , r e c o r d i n g a p p a r a t u s e t c . 14. They a r e i n g e n e r a l low-impedance c u r r e n t - o p e r a t e d d e v i c e s , a n d as s u c h a r e complementary t o e l e c t r o n i c e q u i p m e n t and a r e n o t t o b e c o n s i d e r e d as r i v a l s . Q u i t e o f t e n i t i s f o u n d t h a t a c o m b i n a t i o n o f a: m a g n e t i c a m p l i f i e r and v a l v e s i s t h e b e s t s o l u t i o n . APPENDIX: A The c i r c u i t sr.; r e f e r r e d t o t h i s Appendix, approximate t o a-. s e r i e s r R C c i r c u i t aeross c: aa s i n u s o i d a l i n p u t . L e t i be t h e c u r r e n t t h r o u g h t h e c i r c u i t , R and C thee v a l u e s . of t h e res i s t a n c . e e a n d c a p a c i t a n c e ; r e s p e c t i v e l y and E s i n t t h e a p p l i e d v o l t a g e . T h e n -"̂ 31 APPENDIX B THE USE OF FROLICH'S EQUATION F o r any of t h e magnetic a m p l i f i e r c i r c u i t s when t h e v a r i a t i o n o f ^ c a n be e x p r e s s e d as a s i m p l e f u n c t i o n of t h e c u r r e n t s i n v o l v e d t h e r e l a t i o n s h i p between c u r r e n t and t i m e may be. e s t a b l i s h e d m a t h e m a t i c a l l y a s f o l l o w s : The b a s i c r e l a t i o n s h i p i s : Ho + 4-£LG) - <^ Sect- o r Ht+Ldc'i C J£ = ̂  Ss/fuif A ( l ) however F r o l i c h ' s e q u a t i o n , n e g l e c t i n g h y s t e r e s i s ; e f f e c t s , w i l l g i v e t h e r e l a t i o n s h i p between f l u x and c u r r e n t 2 t o a r e a s o n a b l e degree o f a c c u r a c y . F r ' d l i c h ' s e q u a t i o n I s 4>- w h e r eM a n d n a r e e m p i r i c a l l y d e t e r m i n e d c o n s t a n t s ; f o r t h e p a r t i c u l a r m a gnetic c i r c u i t . i s t h e f l u x p roduced i n t h e c o r e by a d.c. c u r r e n t and an a . c . m a g n e t i z i n g c u r r e n t . T h e r e f o r e , Since/ - /VCfj L = S u b s t i t u t i n g i n e q u a t i o n s / g i v e s - H° *• j^;t- jfr ~(t7*4)3"jfr'" m L e t us now c o n s i d e r b r i e f l y t h e p o s s i b i l i t i e s o f f i n d i n g a n a l y t i c a l e x p r e s s i o n s f o r t h e r e s u l t s g i v e n i n u s i n g F r o l i c h ' s e x p r e s s i o n f o r t h e m a g n e t i z a t i o n c u r v e , t h e magnetic problem can be d e s c r i b e d by T h i s e q u a t i o n i s n o t a s t a n d a r d t y p e ( B e r n o u l l i , R i c e a t i , e t c . ) and t h e s o l u t i o n s a r e t h e r e f o r e n o t l i k e l y t o be e l e m e n t a r y f u n c t i o n s . H o w e v e r , t h e r i g h t s i d e i s c o n t i n - uous i n i and £ and p o s s e s s e s a c o n s t a n t p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o 6 J f f o m P i c a r d s theorem, then,we know t h a t a u n i q u e s o l u t i o n e x i s t s and can be f o u n d by t h e method o f s u c c e s s i v e a p p r o x i m a t i o n s . T h i s method c o n s i s t s i n c h o o s i n g any z e r o - o r d e r a p p r o x i m a t i o n t o t h e s o l u t i o n , s a y ^25i t h e n d e f i n i n g 4^, t h r o u g h t o £*t*> ' / ^ y ' ^ "fa s""->t-*e«-']aCt- P i c a r d ' s theorem ensures t h e convergence of t h i s s e r i e s of f u n c t i o n s t o t h e s o l u t i o n . E x p r e s s i o n s o t h e r t h a n F r o l i c h ' s can be t r e a t e d i n a s i m i l a r manner.In any event t h e p r o c e d u r e would p r o b a b l y r e q u i r e t h e use of computing machines o r n u m e r i c a l methods. APPENDIX: C L i s t of Symbolsr a,b-,c,d - c o e f f i c i e n t s of m a g n e t i z a t i o n e q u a t i o n B e - a m p l i t u d e of s i n u s o i d a l f l u x d e n s i t y h ( - e x c i t a t i o n : i n c o r e 1 H - m a g n e t i z i n g f o r c e i n c;ore amperes turns/cm. E^- a m p l i t u d e of t h e s u p p l y v o l t a g e E - v o l t a g e drop a c r o s s ? t h e magnetic a m p l i f i e r E j - v o l t a g e a c r o s s t h e magnetic: a m p l i f i e r r e q u i r e d f-or s a t u r a t i o n . ^uTa v°ltage drop a c r o s s the^ l o a d E^- c o n t r o l c i r c u i t v o l t a g e 4» - l o a d c u r r e n t c - c u r r e n t i n t h e condenser b r a n c h 4L- c u r r e n t i n t h e magnetic a m p l i f i e r s u p p l y windings? ^Ar c u r r e n t i n t h e magnetic a m p l i f i e r c o n t r o l w i n d i n g s c c . o fff, where i f i s t h e s u p p l y f r e q u e n c y a, - p o i n t on t i m e base of s u p p l y v o l t a g e where core : s a t u r a t e s b, - p o i n t on time base of s u p p l y v o l t a g e where c.ore c desaturates^' N^- number of t u r n s on each s u p p l y w i n d i n g N^- number of t u r n s on each c o n t r o l w i n d i n g N - t o t a l number of t u r n s r e q u i r e d t o produce st e a d y component of, f l u x - d e n s i t y t h e f "hxysif i f t h e r e were o n l y one w i n d i n g t o t a l f l u x i n t h e c o r e f l u x produced by t h e a . c . m a g n e t i z i n g c u r r e n t f l u x produced by t h e c o n t r o l c u r r e n t f l u x i n c o r e 1 f l u x i n core: 2 t o t a l r e s i s t a n c e i n t h e s u p p l y c i r c u i t t o t a l r e s i s t a n c e i n t h e c o n t r o l c i r c u i t c a p a c i t y added i n s e r i e s o r i n p a r a l l e l X BIBLIOGRAPHY H a u f f e , G . , " B e i t r a g z u r q u a l i t a t i v e n T h e o r i e g l e i c h s t r o m - v o r m a g n e t i s l e r t e r E I s e n k e r n - D r o s s e l s p u l e n , " A r c h i v f u r Elektro.teehnik,1939,33,page 41. Kurtz,E.B.and C o r c o r a n G . F . , " I n t r o d u c t i o n t o E l e c t r i c T r a n s i e n t s " , J o h n W i l e y and Sons,Inc.,New York,1935» page 230. Reddick,H.W.and M i l l e r , F , H . , " A d v a n c e d Mathematics F o r E n g i n e e r s " , J o h n W i l e y and Sons,New York,1946, page 165. Coulthard,W.B.','Transients i n E l e c t r i c C i r c u i t s " , S i r I s a a c Pitman and Sons Ltd,,London,1946. Carter,G.W., , fThe S i m p l e C a l c u l a t i o n of E l e c t r i c a l T r a n s i e n t s " , U n i v e r s i t y Press,Cambridge,1944.

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