Applied Science, Faculty of
Electrical Engineering, Department of
DSpace
UBCV
Bennett, William Frederick
Master of Applied Science - MASc
University of British Columbia
The determination of the amplitude-response characteristic is an important means of checking a network design. However, this measurement is usually a time-consuming procedure and at best does not yield a continuous curve. A device which would produce a continuous curve corresponding to the amplitude-response characteristic would be extremely useful in network design and development. It is shown that the system response to a frequency-modulated signal can be made to approximate the amplitude-response characteristic if the frequency is varied slowly enough so that the "quasi-stationary" conditions exist. The physical realization of this slowly varying frequency requires an oscillator with an extremely large frequency range, controllable by one circuit parameter. The greatest difficulties involved in the design of this oscillator were the development of a simple and stable subtractor and the synthesis of the frequency-determining networks. A mathematical analysis was made to determine the characteristics of the network necessary to produce a logarithmic relation between the oscillator frequency and the control position. The audio-frequency sweep generator was constructed using networks designed to approximate the required characteristics and when tested proved to have a satisfactory output waveform. Any improvement in the oscillator performance would require a better approximation to the specified network characteristics.
AN AUDIO-FREQUENCY SWEEP-GENERATOR by WILLIAM FREDERICK BENNETT B.A.Sc., U n i v e r s i t y o f B r i t i s h C o l u m b i a , 19^2 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE -REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e De p a r t m e n t o f o f E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as 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 o r c a n d i d a t e s f o r t h e d e g r e e o f MASTER OF APPLIED SCIENCE THE UNIVERSITY OF BRITISH COLUMBIA J a n u a r y , 19f>9 i ABSTRACT The d e t e r m i n a t i o n o f t h e a m p l i t u d e - r e s p o n s e c h a r a c t e r -i s t i c i s an i m p o r t a n t means o f c h e c k i n g a n e t w o r k d e s i g n . How-e v e r , t h i s measurement i s u s u a l l y a t i m e - c o n s u m i n g p r o c e d u r e and at b e s t does n o t y i e l d a c o n t i n u o u s c u r v e . A d e v i c e w h i c h would p r o d u c e a c o n t i n u o u s c u r v e c o r r e s p o n d i n g t o t h e a m p l i t u d e -r e s p o n s e c h a r a c t e r i s t i c w o u l d be e x t r e m e l y u s e f u l i n n e t w o r k d e s i g n and d e v e l o p m e n t . I t i s shown t h a t t h e s y s t e m r e s p o n s e t o a f r e q u e n c y -m o d u l a t e d s i g n a l c a n be made t o a p p r o x i m a t e t h e a m p l i t u d e -r e s p o n s e c h a r a c t e r i s t i c i f t h e f r e q u e n c y i s v a r i e d s l o w l y enough so t h a t t h e " q u a s i - s t a t l o n a r y " c o n d i t i o n s e x i s t . The p h y s i c a l r e a l i z a t i o n o f t h i s s l o w l y v a r y i n g f r e q u e n c y r e q u i r e s an o s c i l l a t o r w i t h an e x t r e m e l y l a r g e f r e q u e n c y r a n g e , c o n t r o l l a b l e b y one c i r c u i t p a r a m e t e r . The g r e a t e s t d i f f i c u l t i e s i n v o l v e d i n t h e d e s i g n o f t h i s o s c i l l a t o r were t h e d e v e l o p m e n t o f a s i m p l e and s t a b l e s u b t r a c t o r and t h e s y n t h e s i s o f t h e f r e q u e n c y - d e t e r m i n i n g n e t -w o r k s . A m a t h e m a t i c a l a n a l y s i s was made t o d e t e r m i n e t h e c h a r a c t e r i s t i c s o f t h e n e t w o r k n e c e s s a r y t o p r o d u c e a l o g -a r i t h m i c r e l a t i o n between t h e o s c i l l a t o r f r e q u e n c y and t h e c o n t r o l p o s i t i o n . The a u d i o - f r e q u e n c y sweep g e n e r a t o r was c o n s t r u c t e d u s i n g n e t w o r k s d e s i g n e d t o a p p r o x i m a t e t h e r e q u i r e d c h a r a c t e r -i s t i c s and when t e s t e d p r o v e d t o h a v e a s a t i s f a c t o r y o u t p u t waveform. Any improvement i n t h e o s c i l l a t o r p e r f o r m a n c e would r e q u i r e a b e t t e r a p p r o x i m a t i o n t o t h e s p e c i f i e d n e t w o r k c h a r a c t e r i s t i c s . In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of E l e c t r i c a l Engineering The University of B r i t i s h Columbia, Vancouver 8, Canada. Date 27 January, 1959. I I T ABLE OP CONTENTS Page L i s t o f I l l u s t r a t i o n s „• >• . . . * . . •* , .- * *• . • . I l l Acknowledgement •*- ;• •* . -.• . . i v 1. I n t r o d u c t i o n * •;• * V 1 2. C o m p a r i s o n o f V a r i o u s T y p e s o f Sweep F u n c t i o n s >• * . 4 3. C h o i c e o f O s c i l l a t o r T y p e * v * •* * i . * * * * V « 7 i f i The P r i n c i p l e o f O p e r a t i o n , . . . . . « * . . . . . 10 The S u b t r a c t o r D e s i g n . . . *• . . . <. . » « * .•**<. l i f 6. T h e Network D e s i g n . . . . 19 7. The D e s i g n o f t h e A m p l i t u d e - C o n t r o l l i n g C i r c u i t r y . . 23 8. A c c u r a c y T e s t s • . • . < . * * * • . » * v . 27 9^ C o n c l u s i o n s *• *• * ; * . - . . * * > • w i, •»• ^ » * « • * * 30s A p p e n d i x A. A n a l y s i s o f Network Response t o a S i g n a l o f V a r y i n g F r e q u e n c y .• . . •• . . . . . . . . 31 A p p e n d i x B* R-C Ne t w o r k A p p r o x i m a t i o n o f a C o n s t a n t R e a c t a n c e . . . . 3£ L i s t o f R e f e r e n c e s . .- . . . . . . . * . . . *• . . . . ^ I l l L I S T OP ILLUSTRATIONS F i g u r e Page 1. The O s c i l l a t o r B r i d g e A r r a n g e m e n t . . . . . . . . . 10 2. The E q u i v a l e n t C i r c u i t o f t h e O s c i l l a t o r B r i d g e . . 3* The S u b t r a c t i o n C i r c u i t . v ; w •; * * i * '.- . i£ The C omplete O s c i l l a t o r C i r c u i t . . . . . t o f o l l o w 18 5. The F i f t h - D e g r e e R-C N etwork D e s i g n . * . . . . . w 20 6. The F o U r t h - D e g r e e R-C N e t w o r k D e s i g n .• . . * . * . 21 7. The T h e r m i s t o r I n s t a l l a t i o n . ; . . . . . . . . . . 8. The A m p l i t u d e - S t a b i l l z i n g C i r c u i t . . . . t o f o l l o w 26 9. The O s c i l l a t o r O u t p u t . . . . . . . . . . t o f o l l o w 27 10. H e a t h k i t A m p l i f i e r R e s p o n s e C u r v e . . . . t o f o l l o w 27 11. A m p l i f i e r R e s p o n s e C u r v e w i t h T r e b l e G a i n V a r i a t i o n v v t o f o l l o w 27 12. F r e q u e n c y C a l i b r a t i o n o f O s c i l l a t o r . . . t o f o l l o w 28 13. O u t p u t Amplitude o f O s c i l l a t o r . . . . . . t o f o l l o w 28 l i j . . O s c i l l a t o r D i s t o r t i o n . . * . . s . . . . t o f o l l o w 28 l ^ . The D e s i r e d C o n f o r m a l M a p p i n g v . . •. . . . v . -* . 3£. 16. M a p p i n g f o r E q u a l - R i p p l e A p p r o x i m a t i o n . . * . . >< 31. 17. The P r o c e s s o f Z e r o - S h i f t i n g . . . v . . t o f o l l o w 39 ACKNOWLEDGEMENT The author expresses his indebtedness to the members of the Department of El e c t r i c a l Engineering at the University of British Columbia, especially to Dr* A »D. Moore for his guidance throughout the research project and to Dr. F. Noakes for arranging for assistance whenever It was needed. This research was jtiade possible by the British Columbia Electric Railway Company Limited Grant to the Ele c t r i c a l Engineering Department. The major portion of the research work was done while the author was working under an appointment as a National Research Council Research Assistant In the summer of 19^8. AN AUDIO-FREQUENCY SWEEP GENERATOR 1. INTRODUCTION The u n i t t o be d e s c r i b e d i s a v a r i a b l e - f r e q u e n c y a u d i o -o s c i l l a t o r so a r r a n g e d t h a t t h e f r e q u e n c y v a r i e s w i t h t i m e a c c o r d i n g t o some m a t h e m a t i c a l r e l a t i o n w h i l e t h e o u t p u t a m p l i -t u d e r e m a i n s c o n s t a n t . W i t h t h i s a r r a n g e m e n t i t i s p o s s i b l e t o sweep a c e r t a i n f r e q u e n c y r a n g e i n a g i v e n p e r i o d o f t i m e . S u c h an o s c i l l a t o r can be u s e d i n c o n j u n c t i o n w i t h a l o n g - p e r s i s t e n c e c a t h o d e - r a y t u b e t o o b t a i n t h e a m p l i t u d e -r e s p o n s e c h a r a c t e r i s t i c o f a n e t w o r k by a p p l y i n g t h e s i g n a l f r o m t h e o s c i l l a t o r t o t h e n e t w o r k and c o n n e c t i n g t h e Output o f t h e n e t w o r k t o t h e v e r t i c a l - d e f l e c t i o n p l a t e s o f t h e c a t h o d e -r a y t u b e . W i t h t h e o s c i l l o s c o p e t i m e - b a s e s y n c h r o n i z e d w i t h t h e s w e e p y t h e a b s c i s s a o f t h e d i s p l a y i s t h e n a f u n c t i o n o f f r e q u e n c y and t h e o r d i n a t e i s p r o p o r t i o n a l t o t h e a m p l i t u d e r e s p o n s e . N o r m a l l y t h e s e c h a r a c t e r 1 s t l e s a r e o b t a i n e d p o i n t by p o i n t ; t h a t i s * t h e i n p u t i s s e t at one f r e q u e n c y and t h e ; a m p l i f i c a t i o n o r a t t e n u a t i o n i s m e a s u r e d , and t h e n t h e f r e q u e n c y i s c h a n g e d and t h e o p e r a t i o n i s r e p e a t e d . T h i s p r o -c e s s y i e l d s a number o f p o i n t s and t h e c u r v e i s i n t e r p o l a t e d b etween them. Many e x t r a p o i n t s a r e r e q u i r e d f o r n e t w o r k s 1 2 h a v i n g a b r u p t changes I n t h e a m p l i t u d e c h a r a c t e r i s t i c and t h e w hole o p e r a t i o n i s q u i t e t i m e - c o n s u m i n g . The a m p l i t u d e - r e s p o n s e c h a r a c t e r i s t i c o f a n e t w o r k c a n be o b t a i n e d i n a few s e c o n d s w i t h t h e o s c i l l a t o r d e s c r i b e d h e r e . A l s o , v a r i a t i o n s i n t h e a m p l i t u d e r e s p o n s e o f a n e t w o r k due t o c h a n ges i n t h e component v a l u e s c a n be shown as s u c c e s s i v e t r a c e s on t h e c a t h o d e - r a y t u b e . T h i s same method o f t e s t i n g n e t w o r k r e s p o n s e i s w i d e l y u s e d i n t h e . r a d i o - f r e q u e n c y s p e c t r u m , where t h e r a d i o - f r e q u e n c y sweep g e n e r a t o r i s now a s t a n d a r d p i e c e o f e l e c t r o n i c t e s t e q u i p m e n t . S e v e r a l p a p e r s h a v e a p p e a r e d d e s c r i b i n g a t y p e o f a u d i o - f r e q u e n c y sweep g e n e r a t o r ' i n w h i c h t h e O u t p u t s i g n a l o f a r a d i o - f r e q u e n c y sweep g e n e r a t o r i s h e t e r o d y n e d down t o t h e a u d i o - f r e q u e n c y r a n g e . T h i s s y s t e m i s s u s c e p t i b l e t o f r e q u e n c y ' ' 3 d r i f t and i s i n f e r i o r t o a s y s t e m s u g g e s t e d by F * B . A n d e r s o n . I n t h e A n d e r s o n O s c i l l a t o r t h e r e l a t i o n between f r e q u e n c y and t i m e c a n be p r o d u c e d by means o f c o n s t a n t - s p e e d m o t o r - d r i v e n p o t e n t i o m e t e r s . The m a t h e m a t i c a l r e l a t i o n s h i p w i l l t h e r e f o r e depend upon t h e p o t e n t i o m e t e r r a t i o and f r e q u e n c y , and i t h a p p e n s t h a t t h e p o t e n t i o m e t e r r a t i o i s a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e l o g a r i t h m o f f r e q u e n c y . P r e v i o u s s y s t e m s o f t h e b e a t - f r e q u e n c y t y p e h a v e u s e d o t h e r f u n c t i o n a l r e l a t i o n -it 2 s h i p s } f o r example, a s i n u s o i d a l , o r a l i n e a r d e p e n d e n c e . I n t h i s i n v e s t i g a t i o n a l o g a r i t h m i c r e l a t i o n a p p e a r e d t o h a v e two i m p o r t a n t a d v a n t a g e s . A l o g a r i t h m i c f r e q u e n c y r e l a t i o n combined w i t h a l i n e a r t i m e - b a s e on t h e c a t h o d e - r a y t u b e w i l l p r o d u c e an 3 a b s c i s s a on t h e f i n a l d i s p l a y w h i c h i s d i r e c t l y p r o p o r t i o n a l t o t h e l o g a r i t h m o f f r e q u e n c y , t h e most a p p r o p r i a t e method o f d i s p l a y i n g n e t w o r k r e s p o n s e c u r v e s . A l s o , s i n c e e a c h d e c a d e o f f r e q u e n c y i s c o v e r e d i n t h e same i n t e r v a l o f t i m e , t h e r e s -p o nse t o c i r c u i t s o f t h e same Q, i n e a c h d e c a d e w i l l be t h e same. I n t h e r e s e a r c h r e p o r t e d h e r e , a sweep o s c i l l a t o r o f t h e A n d e r s o n t y p e h a s b e en c o n s t r u c t e d f o l l o w i n g a more r i g o r -ous m a t h e m a t i c a l d e s i g n t h a n t h e o r i g i n a l . Improvements h a v e a l s o b e en made i n t h e o s c i l l a t o r c i r c u i t c o n f i g u r a t i o n . T h i s o s c i l l a t o r h a s y i e l d e d r e s u l t s b e t t e r i n some r e s p e c t s t h a n t h o s e O b t a i n e d by A n d e r s o n . 2. COMPARISON OP VARIOUS TYPES OF SWEEP FUNCTIONS S e v e r a l a n a l y s e s h a v e been made o f t h e r e s p o n s e o f a n e t w o r k t o a f r e q u e n c y - m o d u l a t e d s i g n a l . However, most o f t h e s e a n a l y s e s h a v e t h e r e s t r i c t i o n t h a t t h e f r e q u e n c y d e v i a t i o n must be s m a l l compared t o t h e c e n t r e f r e q u e n c y , i . e . , t h e m o d u l a t i o n d e p t h must be much l e s s t h a n u n i t y ' v Van d e r P o l has u s e d H e a v l s i d e ' s o p e r a t i o n a l c a l c u l u s t o a n a l y s e t h e same p r o b l e m w i t h o u t e x p l i c i t l y i m p o s i n g t h i s r e s t r i c t i o n on t h e m o d u l a t i o n g d e p t h . A r e c e n t l y p u b l i s h e d p a p e r by Baghdady c o n t a i n s a more r i g o r o u s m a t h e m a t i c a l p r o o f Of t h e s e p r e v i o u s a n a l y s e s . T h i s p r o o f I s r e p r o d u c e d i n A p p e n d i x A and t h e r e s u l t s show t h a t t h e q u a s i - s t a t i o n a r y s o l u t i o n I s a s a t i s f a c t o r y a p p r o x i -m a t i o n i f t h e f o l l o w i n g i n e q u a l i t y i s f u l f i l l e d ; 1 2 0 " ( t ) « 1 , (2-1) max max where 0 f ( t ) i s t h e m o d u l a t i n g f u n c t i o n d e s c r i b i n g t h e d e p e n d -ence o f CJ upon t i m e , and Z ( j w ) I s t h e s y s t e m t r a n s f e r f u n c t i o n ; T h i s r e s u l t w i l l be u s e d t o d e t e r m i n e t h e r e s t r i c t i o n s n e c e s s a r y f o r t h e q u a s i - s t a t i o n a r y s o l u t i o n * As m e n t i o n e d p r e v i o u s l y , t h e r e a r e t h r e e t y p e s o f sweep t h a t h a v e been u s e d f o r s u c h an o s c i l l a t o r * Of t h e s e , t h e s i n u s o i d a l sweep does n o t seem t o o f f e r any p a r t i c u l a r a d v a n t a g e and h e n c e w i l l not be c o n s i d e r e d . F o r t h e l i n e a r sweep, s w e e p i n g f r o m a h i g h f r e q u e n c y t o a low, t h e e x p r e s s i o n f o r t h e m o d u l a t i n g f u n c t i o n i s 2 2 0 ( t ) = -fck o ) o t k Then, from Equation ( 2 - 1 ) , » - | k O J 0 max. ( 2 - 2 ) A logarithmic sweep from a high frequency to a low frequency can be expressed as t = -ln( to / OJ 0 ) -t or OJ = OJ Q e A more complete expression for the frequency is t <*> = w 0 +d0/dt = OJ 0 e " c w o where, in this case 0(t) = -0 OJ • t . e u o - O J t o The condition for the quasi-stationary state is Z(.j OJ j OJ ) 2 -cw t ( 2 - 3 ) max Therefore to obtain the same quasi-stationary condi-tions for either the linear or logarithmic sweep applied to a given impedance, -k O J 2 2 * o jj£a> 2 e " ^ e* or k = c e " c W o t v (2-4) TO obtain the same rate of change of frequency for both cases at t = 0 , It is necessary that k = c This of course requires that the time-base for the logarithmic sweep must be several times longer than that for the linear sweep to maintain the quasi-*- stationary conditions at the high frequencies. But, as t —>-oe, k —>• 0 , which indicates that for the quasi-stationary conditions to hold at the low frequencies, the 6 time-base for the linear case must be an infin i t e number of times longer than that for the logarithmic case. The log-arithmic sweep therefore seems to be the better type of sweep . . . . . . . . . . ^ mathematically. It also has the other advantages mentioned previously, that i s , i t provides a simple means of obtain-ing a logarithmic frequency scale on the f i n a l presentat ion, arid an equal amount of the sweep time Is spent on each decade of frequency. 3, CHOICE OP OSCILLATOR TYPE I n c h o o s i n g an O s c i l l a t o r t h e f r e q u e n c y r a n g e was p r o v i s i o n a l l y s e t a t $ cps t o 1 Mc. T h i s f r e q u e n c y r a n g e d e f i n i t e l y c o v e r s t h e a u d i o s p e c t r u m and p r o v i d e s a r e a s o n -a b l e s a f e t y m a r g i n f o r f u t u r e d e s i g n s . F i r s t c o n s i d e r a t i o n was g i v e n t o t h e c o n v e n t i o n a l o s c i l l a t o r c o n f i g u r a t i o n s . However, t h e n o r m a l L-C t y p e o f O s c i l l a t o r h a s a f r e q u e n c y r a n g e o f about f o u r t o one by v a r y i n g e i t h e r t h e i n d u c t a n c e o r t h e c a p a c i t a n c e . R e s i s t a n c e - c a p a c i t a n c e o s c i l l a t o r s s u c h as t h e 9 l O 11 12 13 Wien b r i d g e , b r i d g e d - T , p a r a l l e l - T , and v a r i o u s o t h e r s , ' a r e c a p a b l e o f much h i g h e r f r e q u e n c y r a t i o s o f t h e O r d e r o f one t h o u s a n d t o one. But b o t h t h e r e s i s t a n c e and t h e c a p a c i t a n c e must be changed t o p r o d u c e t h i s r a n g e * I n o r d e r t o i n c r e a s e t h e f r e q u e n c y v a r i a t i o n e v e n f u r t h e r , t h e f i r s t and most l o g i c a l s t e p seemed t o be t h e u s e o f a b e a t - f r e q u e n c y o s c i l l a t o r . The f r e q u e n c y v a r i a t i o n c o u l d t h e n be o b t a i n e d e i t h e r by a m e c h a n i c a l v a r i a t i o n o f one o f t h e element v a l u e s o r by t h e u s e o f a r e a c t a n c e - t u b e m o d u l a t o r . The r e a c t a n c e - t u b e m o d u l a t o r i s t h e o n l y c o n v e n t i o n a l means o f p r o d u c i n g f r e q u e n c y - ^ m o d u l a t i o n t h a t p e r m i t s an a v e r a g e f r e -q u e n c y d e v i a t i o n . U s i n g t h e d e s i g n c r i t e r i a o f A u g u s t Hund^" and 13 C.F. S h e a r e r , s u c h a b e a t - f r e q u e n c y o s c i l l a t o r w i t h a r e a c t a n c e - t u b e m o d u l a t o r was c o n s t r u c t e d and t e s t e d , , The r e a c t a n c e - t u b e m o d u l a t o r y i e l d e d e x c e l l e n t r e s u l t s ? h owever, t h e s t a b i l i t y o f t h e b e a t - f r e q u e n c y o s c i l l a t o r was e x t r e m e l y 7 8 poor. A l l efforts to improve the st a b i l i t y to the necessary one part in a million proved useless. Resistance-stabilization, reactance-stabilization, frequency-feedback, and frequency-measuring were a l l tried with l i t t l e or no success. The major trouble experienced with a l l these systems was the large frequency range over which they were expected to operate. With a reduction In the required frequency range i t may have been possible to design a stable beat -frequency oscillator. After considerable experimentation with various means of st abilizat ion, the beat -frequency type of oscillator was abandoned In favour of a rather unusual type of R-C oscillator described by F.B. Anderson. This oscillator apparently f u l f i l l e d a l l the requirements Of the audio-frequency oscillator, but the conversion to a sweep oscillator required the use of motor-driven sweep-potentiometers Instead of vacuum-tube sweep circuits as was possible In the beat-frequency oscillator. This disadvantage was considered minor because of the increase in frequency s t a b i l i t y possible with this circuit arrangement. The increase in frequency s t a b i l i t y is not the only advantage. The linearity of the curve of logarithmic f r e -quency versus potentiometer setting depends only on the design of the frequency-determining networks. On the other hand, in the beat-frequency oscillator, the shape of the curve depends upon the shape of the transconductance curve of the tube used in the react ance-tube modulator. The only requirement on the tubes in this new circuit arrangement that they have sufficient amplification to produce o s c i l lat ion j only passive network elements are used to deter-mine the oscillator calibration. if. THE PRINCIPLE OF OPERATION An exact description of the principle of operation of this oscillator is quite d i f f i c u l t due to the characteristics of the associated networks. Therefore, a simplification w i l l be made to aid Ih the description. This simplification w i l l be corrected in the bourse of the analysis. Figure 1. The Oscillator Bridge Arrangement Figure 1 shows the basic circuitry of the o s c i l ^ lator and the associated phasor diagram* In this c i r c u i t , the right arm of the bridge is a non-inductive potentiometer and the left arm is a const ant-react ance network. The resistor E± Is the series equivalent resistance of the generator and V is the open-circuit voltage of the generator. The const ant-react ance network is assumed to have a high impedance so that is effectively in phase with V. Then, for a particular setting of 10 t h e p o t e n t i o m e t e r , t h e r e w i l l be some f r e q u e n c y s u c h t h a t t h e v o l t a g e V a - V b w i l l be at r i g h t a n g l e s t o t h e a p p l i e d v o l t a g e V. I f t h e v o l t a g e V a - V b i s t h e n a p p l i e d t o a n o t h e r s i m i l a r b r i d g e a f u r t h e r 9 0 - d e g r e e p h a s e s h i f t w i l l be p r o d u c e d . W i t h 180 d e g r e e s o f p h a s e s h i f t i n t r o d u c e d e l s e w h e r e i n t h e c i r c u i t r y , one o f t h e n e c e s s a r y C o n d i t i o n s f o r o s c i l l a t i o n h a s b e e n f u l -f i l l e d . By I n t r o d u c i n g a m p l i f i e r s i n t o t h e c i r c u i t t o I s o l a t e t h e two b r i d g e s , t h e y may be d e s i g n e d t o p r o v i d e b o t h t h e e x t r a l 8 0 - d e g r e e p h a s e s h i f t and a l s o t h e a m p l i f i c a t i o n n e c e s s a r y t o overcome t h e a t t e n u a t i o n o f t h e n e t w o r k s , f u l f i l l i n g t h e s e c o n d c o n d i t i o n f o r o s c i l l a t i o n . T h i s t h e n i s t h e p r i n c i p l e o f o p e r a t i o n o f t h e o s c i l l a t o r . As was m e n t i o n e d p r e v i o u s l y , a s i m p l i f i c a t i o n was u s e d t o d e s o r i b e t h e o p e r a t i o n . A m a t h e m a t i c a l a n a l y s i s w i l l now be made t o d e t e r m i n e t h e n e c e s s a r y c h a r a c t e r i s t i c s o f t h e n e t -work. P o r t h e a n a l y s i s , t h e c i r c u i t w i l l be c o n v e r t e d , u s i n g T h e v e n i n » s T h e o r e m , t o t h a t shown I n F i g u r e 2. R + Rj_ Rj+R F i g u r e 2. The E q u i v a l e n t C i r c u i t o f t h e O s c i l l a t o r B r i d g e 12 R e f e r r i n g t o F i g u r e 2, the v o l t a g e Vb can be expressed as V b = k ( R 0 + Z) I , where k Is the, potentiometer r a t i o measured from the lower t e r -m i n a l . Then the a t t e n u a t i o n o f t h e network i s Vo = V ar-y D = R 0 - k ( R a +, M = RQ:,Y |1 " - k. V? V« Z + R Q + Rj- 1 + R» Y where R» = R Q + RJ > and Y = 1/Z = G + J B . Now, u s i n g two stages Of a m p l i f i c a t i o n A and two i d e n t i c a l networks * the c o n d i t i o n s f o r o s c i l l a t i o n are ( A V 0 / V ) 2 = -1 f the n e g a t i v e s i g n i s s u p p l i e d by the a d d i t i o n a l 180 degrees o f phase s h i f t i n t r o d u c e d elsewhere i n t h e c i r c u i t * The c o n d i t i o n s f o r each network are t h e r e f o r e V 0/V» = + j/A = i j a , where a = l / A . I n t h i s case where the networks are t o have an R-C r e a l i z a t i o n , V 0 must l e a d V-1 by 90 degrees, i . e . , V Q A 1 = J/A = j a v Then, a + aR»G = R Q B ( 1-k) , and - aR»B = R Q G ( l - k ) -k T h e r e f o r e k = " a .-.<* R':G + RQB, and k = -go. •<*+•«• M B i + R G G S o l v i n g f o r the a t t e n u a t i o n , .• '. - Ro B a ~ R 0 R* +1 + (R 0 + R«) G + R QR« G2 If Rr then a = *o B2 + B (R 0 R )' + G 2 . ... (4-1) Let Then G ay (R0R') I \ 2 a = RA G^„ X , o av ' ... (4-2) (4-3) where X Is the reactance in the square bracket in Equation (If 1). Therefore, the impedance Z In parallel with a resis-tance (R R 1 ) 8 should together have a constant reactance. $. THE SUBTRACTOR DESIGN As mentioned previously, amplifiers w i l l be needed In the oscillator to provide both isolation between the bridges and also the necessary forward gain. Prom the form of the outputs of the bridge i t can be seen that a sub-tract or circuit would be suitable. Anderson has used such a circuit in his design, but It was felt that some Improvements could be made. F i r s t * he has chosen pentodes for the active elements in his subtractor. Since good balance In the output of a subtractor depends upon the size of the common cathode resistor, the use of pentodes tends to defeat this objectIve, because the screen-grid-to-cathode clrouit is effectively in parallel with the cathode resistor. Also, the use of the bridge network as the plate load of one half of the subtractor w i l l tend to cause unbalance. A modified form of the subtractor (Figure 3) was designed for the sweep generator. The following Is an analysis of Its operation under the assumption that the two tubes in the subtractor are not alike. . . . . . | : • •' : The action of this subtract or Is described mathe-matically by the following matrix equation: "(u + 6u ) (V-L-VJ (Rk) (RL+^p - 6 r n+R t) ( H~ 6> ) ( V 2 - V K ) (R L + r p + 6 r p + Rfc) (Rk) *1 r2 14 13 +300 volts (;a+6uv).v RI t.-.H .2.- = u ' 2 ' Figure 3 . - - p i + -"P^ = r V k = Rk ( I x + I 2) The Subtraction Circuit The solution for the output voltage Is given by the equation. -V D u ( V l >v 2 ) [RL + r p + 2( u+l)R k] '\i(V1-V2) ( 6 r p - 6u 2Rk) - |i{V1+V2) ( 6 r p ^ 6 > i 2 R k ) + 6u (V]+V2) [RL+ rp+2( u+l)R k], . . . (3-D where D is the determinant of the system matrix* 16 In order to establish design c r i t e r i a and the effects of tube unbalance, It is desirable to separate the in-phase and out-of-phase components of amplifier gain; In doing so, only the first-Order effects of the incremental terms w i l l be con-sidered* 16 In the system Of symmetrical components Used by Gray, A + i s the ratio of the in-phase components at the output to those at the input, and A_ is the ratio of the out-of-phase components at the output to those at the input* Then Af = 6u _ 6 r p + 2Rk 6u , A- ~ > R L + > P +'2Rk( u+ 1) If (R L + r p) « 2(u.+ l ) R k , and 6 r « 2Rk 6u y then A. 6ji- _ 2Rk 6u- . A - > 2Rk( >+ 1) Equation (3-2) indicates that a subtract or In this circuit configuration and Using the above restrictions Is independent of minor tube variations. The Incremental terms may therefore be eliminated from the original equations for a more detailed investigation of the operation of the subtractor. Further analysis shows that -R RL R L + PP + + X> 17 and Therefore,. If 2R.( u + 1) » (R + r ) the in-phase rejection ratio is much greater than unity, that i s , A-K » i It can also be seen by inspection that one half of any load connected between the plates of the subtracter w i l l be In parallel with each of the plate load resistors for the out-of-phase components. This of course w i l l affect the expression for the out -of-phase gain, Equation . The bridge network described i n Chapter 4 w i l l be connected between the plates of the subtractor. In order to provide both a reasonable subtracter design and a reasonable network design an out-of-phase amplifi-cation of five was chosen. A low value of R^ was desired both to satisfy the restrictions of Equation (5>-f?) and also to maintain the plates of the subtractor at a low impedance level;. This low impedance level was desirable since both the potentiometer and the network were to be connected to the subtracter plates. A vacuum tube with a high transconductance was therefore chosen, and the network was designed to have a high impedance so that any loading effect due to the network would be negligible. 18 With the subtracter designed to these requirements the in-phase amplification was approximately 0.£. This circuit arrangement worked quite satisfactorily as an oscillator, but to improve the balance between the two outputs It was neces-sary to return the cathode resistor to -300 volts* This increase in the cathode resistor reduced the in-phase gain to a negli-gible value. The f i n a l subtractor design can be seen in Figure If , which shows the complete oscillator c i r c u i t . The grids of both subtractors have been returned to a common point on a voltage divider so that a fine adjustment of the operating point of the vacuum tubes Is possible. +300 volts" A • 7K 2w. 4„7K 2w. 1 ,4. 7K 2w. 4.7K 2w. O.luf 10K 3 OK » — W A / v V W Z 1M< 100K 12AT7 Output #1 * -_ O.lufX Hi 0.1 uf. 10K. 4\ 30K X T i>—ww*/v-4>- Z IM IM, ~ \ 12AT7 O.luf 16. 3K lOw. -300 volts 16.3K J lOw. > — I Note: A l l Resistor values are in ohms. Figure 4„ The Complete O s c i l l a t o r C i r c u i t 6. IRE NETWORK DESIGN As shown in Chapter if, the network must approximate a constant reactance. It was desired to have an R-G realization fpr this impedance function. A design method for such a net -17 work has been given by Tuttle, both for the Chebyshev and Butterworth approximations * In this problem the Chebyshev, Or equal-ripple, approximation seemed to be most appropriate. Appendix B gives Tuttle«s derivation for the Chebyshev approxi-mation, which leads to the reactance function. The correspond-ing impedance function must be obtained and this is most con-veniently done using Bode's method. Bode's method consists of finding the partial-fraction expansion of the odd part of Z(s), which Is quite readily obtained from the reactance expres-sion. In this case Z(s) is obtained by simply doubling the coefficients Of the left-half-plane poles. It can also be readily shown that the impedance function is minimum resistive and therefore i t Is unnecessary to check the actual function. In this particular application, a frequency range of f>cps to 1 Mc was chosen and It was decided to approximate the necessary impedance with a fifth-degree network. With the impedance level scaled to unity at jjepsf the resulting reactance function was: +6,413,016. s(s2-237.]Z62)ls2-lElf) ( s 2 „54,725 2 )< s 2 -833, 377*) jX(s) = ' (s2-S7732)(s2-926.952Ks2-14,0S82)(s2-2'13, 1882)(s2-3,488,77p2) 19 20 The Impedance f u n c t i o n was then obtained u s i n g Bode's method and t h e admittance G f t V, p r e v i o u s l y d e f i n e d i n Chapter If, was removed. P o r the c i r c u i t v a l u e s used, G a y = 31.32 pmhos. The r e s u l t i n g impedance f u n c t i o n was m u l t i p l i e d by the f a c t o r aR'. Th i s s c a l i n g f a c t o r i s d e r i v e d d i r e c t l y from Eq u a t i o n (if-3) and has t h e val u e Of 3800 ohms i n t h i s c i r c u i t . The r e s u l t i n g imped-ance f u n c t i o n was then expanded i n t o t h e f i r s t Cauer canonic form shown i n F i g u r e $. m "*41uuf p — v w v w — . 13,277 .0) ~475uuf rAWVW—• 25,226 .0) 4120uuf 1—VWWW—. 67,150 .0) ".0296uf p-VWVW1—1 729,200 .0) ~ 132 uf oj = ohms F i g u r e $. The F i f t h - D e g r e e R-C Network Design Two such networks were c o n s t r u c t e d and t e s t e d i n the o s c i l l a t o r . The r e s u l t s were v e r y p r o m i s i n g , but when the a m p l i t u d e - s t a b i l i z i n g network (to be d e s c r i b e d l a t e r ) was in t r o d u c e d , d i s t o r t i o n was produced at f r e q u e n c i e s correspond-i n g t o the p o l e p o s i t i o n s and o s c i l l a t i o n s ceased i n the v i c i n i t y of th e z e r o s . T h i s i n d i c a t e s t h a t the e r r o r i n the approximating Impedance was so l a r g e t h a t the magnitude o f the loop g a i n of the o s c i l l a t o r could not be c o n t r o l l e d e f f i c i e n t l y with the s t a b i l i z i n g network b e i n g used. Consid-e r a t i o n was then g i v e n t o the use o f one f i f t h - d e g r e e network 21. and one' f o u r t h - d e g r e e n e t w o r k so d e s i g n e d t h a t t h e p o l e s o f one ne t w o r k w o u l d o c c u p y a p p r o x i m a t e l y t h e same p o s i t i o n I n t h e c o m p l e x - f r e q u e n c y p l a n e as t h e z e r o s o f t h e o t h e r n e t w o r k . The e r r o r s o f t h e two n e t w o r k s would t h e n t e n d t o c a n c e l r a t h e r t h a n add. The d e s i g n o f t h i s f o u r t h - d e g r e e n e t w o r k was some-what a r b i t r a r y . I t was f i n a l l y d e c i d e d t o c h o o s e t h e f r e q u e n c y l i m i t s o f t h e f o u r t h - d e g r e e n e t w o r k e q u a l t o t h e f r e q u e n c i e s c o r r e s p o n d i n g t o t h e ex t r e m e p o l e s o f t h e f i f t h - d e g r e e n e t w o r k . I n t h i s manner, t h e f r e q u e n c i e s o f t h e f i r s t and l a s t "minima" o f t h e f o u r t h - d e g r e e n e t w o r k a r e s e t e q u a l t o t h e f r e q u e n c i e s o f t h e f i r s t and l a s t maxima r e s p e c t i v e l y o f t h e f i f t h - d e g r e e n e t w o r k . The r e s u l t i n g r e a c t a n c e f u n c t i o n was: 2 2 2 2 2 ? jX(s) = +291. 636 s(s -633.318 ) (s -14057. 303 ) (s -312,019.73) -2 2 •2 2 .2 2 (s -128.75 )(s -2986.5 ) (s -66,166.3 ) (s -1,534,804.5 ) Once a g a i n t h e impedance f u n c t i o n was o b t a i n e d and expanded i n t o t h e C a u e r r e a l i z a t i o n shown i n F i g u r e 6. -WWW-11, 564.0) 80|iuf A V W W 19,387.0) 113-TM4tf A W W V — 49,830.0) ".012UX •WWW-422,020.0). .0832U* OJ = ohms F i g u r e 6. The F o u r t h - D e g r e e R-C Network D e s i g n A l t h o u g h t h i s c o m b i n a t i o n o f n e t w o r k s y i e l d e d good r e s u l t s , t h e a m p l i t u d e o f o s c i l l a t i o n was s t i l l n o t c o n s t a n t . The variation in amplitude was of course due to the error in the approximating function and corresponded to the location of the poles and zeros in the approximating function. It was therefore considered necessary to add an amplitude-stabilizing network to the oscillator. 7. THE DESIGN OF THE AMPLITUDE-CONTROLLING CIRCUITRY As m e n t i o n e d p r e v i o u s l y , t h e o u t p u t o f t h e o s c i l l a t o r h as a wide v a r i a t i o n i n a m p l i t u d e due t o t h e e r r o r i n t h e a p p r o x i m a t i n g f u n c t i o n . S i n c e t h e f r e q u e n c y r a n g e o f t h i s o s c i l l a t o r i s so l a r g e t h e o n l y r e a s o n a b l e method o f c o n t r o l l i n g t h e a m p l i t u d e i s w i t h t h e r m i s t o r s . An attempt t o u s e t h e same t h e r m i s t o r u s e d by A n d e r s o n , W e s t e r n E l e c t r i c t y p e 21A. showed t h a t v e r y l i t t l e c o n t r o l was e x e r c i s e d o v e r t h e a m p l i t u d e o f -I Q o s c i l l a t i o n s . U s i n g t h e method o f B o l l m a n and K r e e r , ° i t was f o u n d t h a t t h e c h a r a c t e r i s t i c s o f t h a t p a r t i c u l a r t h e r m i s t o r were u n s u i t a b l e f o r t h i s a p p l i c a t i o n . An a n a l y s i s was p e r f o r m e d t o d e t e r m i n e t h e r e q u i r e d t h e r m i s t o r c h a r a c t e r i s t i c s . T h i s a n a l y s i s was o n l y a p p r o x i -mate b e c a u s e i t i n v o l v e d e x t r a p o l a t i o n o f c u r v e s and a s s i g n i n g an i n t e r n a l Impedance t o t h e o s c i l l a t o r . The v a l u e o f t h e c o l d r e s i s t a n c e R Q o f t h e t h e r m i s t o r f o u n d by t h i s a n a l y s i s was 20,000 ohms. The t h e r m i s t o r s e l e c t e d was t h e V i c t o r y E n g i n e e r i n g C o r p o r a t i o n t y p e 5>1A22. T h i s t h e r m i s t o r h a s t h e f o l l o w i n g c h a r a c t e r i s t i c s : C o l d R e s i s t a n c e R Q (23°C ) = 100,000 £ 2 3 $ ,: D i s s i p a t i o n C o n s t a n t 6 = 0.09 m i l l i - w a t t s / o c , R e s i s t a n c e R a t i o R o ^° C ^ = 10.3 , R 0 (30°c ) Time C o n s t a n t T = 0 . 3 sees.,' T e m p e r a t u r e C o e f f i c i e n t a (23°C) = -k.k%/°V • 23 24 Nowy f o l l o w i n g B o l l m a n and K r e e r ' s a n a l y s i s and u s i n g t h e s e n u m e r i c a l v a l u e s , P = (273 + ffl).(273) I n 50 S = 273 • + • 25 • "P. R 0°C R Q 5o°c, = 4113 >• 0.0725 , o ( R o P o > * = (273 + 25) s = 1.944 m i l l i - w a t t s , 13.942 v o l t s , 7^172 ( I O ) 3 amps^ 1. The o p e r a t i n g p o i n t c h o s e n on t h e g e n e r a l i z e d t h e r m i s t o r c h a r a c t e r i s t i c s was: v» = 0 . 6 5 , and u = 1 . 5 , o r V = 9; 062 v o l t s , and I = 0.209 m i l l i - a m p s . T h i s o p e r a t i n g p o i n t was c h o s e n on t h e R/R Q c u r v e c o r r e s p o n d -i n g t o a r a t i o o f 0.010> w h i c h l e a d s t o a v a l u e f o r R o f 1000 ohms. I n o r d e r t o c o n t r o l t h e a m p l i t u d e o f t h e o s c i l l a t i o n s i t was n e c e s s a r y t o c o n n e c t t h e t h e r m i s t o r I n t h e c i r c u i t i n s u c h a way t h a t t h e t h e r m i s t o r was e f f e c t i v e l y i n p a r a l l e l w i t h t h e o s c i l l a t o r p l a t e - l o a d r e s i s t o r s . I t was a l s o n e c e s -s a r y t o p r o v i d e t h e b i a s i n g c u r r e n t . T he t h e r m i s t o r s were f i n a l l y c o n n e c t e d as shown i n F i g u r e 7. ( 25 +300 volts F i g u r e 7. The T h e r m i s t o r I n s t a l l a t i o n I n t h i s c i r c u i t t h e t h e r m i s t o r and t h e s l o p e -a d j u s t i n g r e s i s t o r R a r e c o n n e c t e d i n s e r i e s w i t h a c u r r e n t -l i m i t i n g r e s i s t o r b etween t h e p l a t e and g r o u n d . The c U r r e n t -l i m i t i n g r e s i s t o r i s a d j u s t e d t o a v a l u e t h a t g i v e s t h e d e s i r e d o p e r a t i n g c o n d i t i o n s and i s , o f c o u r s e , v e r y h e a v i l y b y - p a s s e d . As w i l l be s e e n f r o m F i g u r e 8, t h e r e a r e two s u c h t h e r m i s t o r n e t w o r k s , one f o r e a c h p l a t e . By u s i n g t h e two t h e r m i s t o r s i t s h o u l d be p o s s i b l e t o m a i n t a i n b o t h p l a t e s i g n a l s c o n s t a n t , and c o n s e q u e n t l y t h e p l a t e - t o - p l a t e s i g n a l a l s o c o n s t a n t . W i t h t h e t h e r m i s t o r s i n t r o d u c e d i n t o t h e c i r c u i t r y i n t h i s manner, t h e t o t a l v a r i a t i o n i n t h e a m p l i -t u d e o f t h e o s c i l l a t o r o u t p u t was +_ 1 db. T h i s v a r i a t i o n .in t h e o u t p u t c o u l d be e l i m i n a t e d by a l t e r i n g t h e o p e r a t i n g 26 p o i n t o f t h e t h e r m i s t o r s s l i g h t l y ; h o w ever t h i s a l s o i n t r o -d uced c o n s i d e r a b l e d i s t o r t i o n i n t o t h e o u t p u t s i g n a l . S i n c e t h e o s c i l l a t o r i s f o l l o w e d by a b u f f e r a m p l i f i e r , i t was d e c i d e d t o l e a v e t h e s e t h e r m i s t o r s a t t h e i r d e s i g n o p e r a t i n g p o i n t and t o d e s i g n t h e b u f f e r a m p l i f i e r so t h a t an a d d i t -i o n a l s e t o f t h e r m i s t o r s c o u l d be i n s t a l l e d i n t h a t c i r c u i t . The b u f f e r a m p l i f i e r i s i n r e a l i t y a n o t h e r s u b -t r a c t o r , as shown i n F i g u r e 8, I t was f e l t t h a t t h i s was t h e b e s t way t o o b t a i n t h e o u t p u t s i g n a l f r o m t h e o s c i l l a t o r , s i n c e i n t h i s manner a b a l a n c e d p u s h - p u l l s i g n a l i s a v a i l a b l e f o r e x t e r n a l u s e . A t h e r m i s t o r was u s e d a g a i n i n p a r a l l e l w i t h e a c h p l a t e - l o a d r e s i s t o r . The c o n s t a n t - c u r r e n t p e n t o d e i n t h e c a t h o d e c i r c u i t y w h i c h r e q u i r e s a s e p a r a t e h e a t e r s u p p l y , was n e c e s s a r y i n o r d e r t o i n c r e a s e t h e i n - p h a s e r e j e c t i o n o f t h e c i r c u i t . A two-gang r h e o s t a t was a l s o i n c l u d e d i n t h e c a t h o d e c i r c u i t i n o r d e r t o a d j u s t t h e s i g n a l l e v e l , and g i v e s a t h r e e t o one c o n t r o l o v e r t h e a m p l i t u d e o f t h e o u t p u t s i g n a l . The s e c o n d s e t o f t h e r m i s t o r s , a l t h o u g h an i n e f f i -c i e n t method o f a c c o m p l i s h i n g t h e end r e s u l t , d i d make t h e o u t p u t v o l t a g e c o n s t a n t . W i t h a r e d e s i g n o f t h e f r e q u e n c y -d e t e r m i n i n g n e t w o r k s , t o be d i s c u s s e d l a t e r , t h e t h e r m i s t o r s i n t h e b u f f e r a m p l i f i e r c o u l d be e l i m i n a t e d and an a l t e r -n a t i v e d e s i g n f o r t h e a m p l i f i e r , w h i c h does n o t r e q u i r e t h e c o n s t a n t - c u r r e n t p e n t o d e i n t h e c a t h o d e c i r c u i t y c o u l d be u s e d . Such a r e d e s i g n w o u l d p r o d u c e b o t h a s i m p l e r and a more e c o n o m i c a l o s c i l l a t o r . . Note: A l l Resistor values are in ohms. A l l capacitor values are in microfarads. F i g u r e 8. The A m p l i t u d e - S t a b i l i z i n g C i r c u i t 8. ACCURACY TESTS A c c u r a c y t e s t s o f b o t h a dynamic and a s t a t i c n a t u r e h a v e been made. The f i r s t t e s t s shown a r e d y n a m i c , i . e . , t h e sweep was i n o p e r a t i o n . The r e s u l t s o f t h e s e t e s t s a p p e a r i n F i g u r e s 9,- 10, and 11* F i g u r e 9 shows t h e o u t p u t o f t h e o s c i l l a t o r as a f u n c t i o n o f f r e q u e n c y . F i g u r e 10 i s t h e a m p l i t u d e -r e s p o n s e c h a r a c t e r i s t i c o f a H e a t h k i t a u d i o - f r e q u e n c y a m p l i f i e r , t o g e t h e r w i t h t h e a c t u a l m e a s u r e d r e s p o n s e * F i g u r e 11 shows a t y p i c a l a p p l i c a t i o n o f t h e sweep o s c i l l a t o r where t h e r e s p o n s e s f o r two d i f f e r e n t s e t t i n g s o f t h e t r e b l e g a i n c o n t r o l a r e s u p e r i m p o s e d . The r e s p o n s e c u r v e s i n t h e s e f i g u r e s a r e so s c a l e d t h a t t h e o r d i n a t e s c o r r e s p o n d t o a r e f e r e n c e l e v e l and t h e c o r r e s p o n d i n g h a l f - p o w e r l e v e l . The r e s p o n s e c u r v e s were o b t a i n e d w h i l e t h e o s c i l -l a t o r was s w e e p i n g t h e f r e q u e n c y s p e c t r u m at a r a t e o f two-t h i r d s o f a sweep p e r m i n u t e . T h i s sweep r a t e a p p e a r s t o be t h e l i m i t on t h e o s c i l l a t o r o p e r a t i o n and i s d e t e r m i n e d by t h e t i m e - C o n s t a n t o f t h e t h e r m i s t o r s i n t h e a m p l i t u d e - c o n t r o l l i n g c i r c u i t r y . As d i s c u s s e d l a t e r , an improvement i n t h e d e s i g n o f t h e f r e q u e n c y - d e t e r m i n i n g n e t w o r k s w o u l d r e d u c e t b e amount o f c o n t r o l t h e t h e r m i s t o r s would h a v e t o e x e r c i s e and t h e n t h e sweep r a t e c o u l d p r o b a b l y be i n c r e a s e d . The n e x t t e s t s a r e o f a s t a t i c n a t u r e . F i g u r e 12 shows t h e c a l i b r a t i o n c u r v e o f t h e o s c i l l a t o r , r e l a t i n g f r e q u e n c y and p o t e n t i o m e t e r r a t i o . A l s o I n c l u d e d i n t h i s 27 28 f i g u r e i s a curve f o r a s i m i l a r network designed u s i n g Anderson*s method. F i g u r e 13 shows the output amplitude of the o s c i l l a t o r both w i t h and without the t h e r m i s t o r s i n the b u f f e r a m p l i f i e r . The t o t a l harmonic d i s t o r t i o n at the out-put of t h e o s c i l l a t o r i s shown i n F i g u r e II4.. A c l o s e r i n s p e c t i o n of these r e s u l t s shows t h a t the curve of frequency versus potentiometer r a t i o , the d i s t o r t i o n curve, and the amplitude curve without the b u f f e r - a m p l i f i e r t h e r m i s t o r s a l l undergo a sudden d e v i a t i o n at approximately the same frequency. T h i s frequency a l s o corresponds t o the p o s i t i o n i n the p o l e - z e r o p l o t of the product of the two reactance f u n c t i o n s where two zeros are a d j a c e n t . T h e r e f o r e , i t can be assumed t h a t the present network designs are not t h e b e s t . To overcome these d i f f i c u l t i e s , i t may be p o s s i b l e t o d e s i g n both networks s i m u l t a n e o u s l y , working with the t o t a l loop g a i n o f the o s c i l l a t o r * T h i s method has been t r i e d , but no d i r e c t s o l u t i o n t o the problem has been found. Two o t h e r more p r a c t i c a l s o l u t i o n s are p o s s i b l e . The f i r s t s o l u t i o n i s to assume that the f i f t h - d e g r e e network i s s u i t a b l e and t o design a fourth-degree network u s i n g the p o l e - z e r o l o c a t i o n s of the f i f t h - o r d e r network as the z e r o - p o l e l o c a t i o n s f o r the f o u r t h - o r d e r network as f a r as p o s s i b l e . Such a f o u r t h -degree network would not g i v e a Chebyshey approximation, but p o s s i b l y i t would reduce the t o t a l e r r o r i n the approximation. The second s o l u t i o n i s t o assume the f i f t h - o r d e r network s u i t a b l e and then t o design e i t h e r a f o u r t h - o r a o o L 1 ; 1 : 1 I I I 1 10 100 1000 10,000 100,000 1,000,000 Frequency,- cycles per second F i g u r e 12. F r e q u e n c y C a l i b r a t i o n : o f O s c i l l a t o r r V » > * » X H MX » » X X M M » fc— R e f e r e n c e L e v e l w i t h B u f f e r T h e r m i s t o r s * » » X X * » » » M x W a > x R e f e r e n c e L e v e l w i t h o u t B u f f e r T h e r m i s t o r s -L 10 100 1000 10,000 F r e q u e n c y , c y c l e s p e r s e c o n d F i g u r e 13. O u t p u t A m p l i t u d e o f O s c i l l a t o r 100,000 1,000,000 31 I I I I I I 1 10 100 - . . 1000 10,000 100,000 1,000,000 F r e q u e n c y , c y c l e s per. s e c o n d F i g u r e 14. O s c i l l a t o r D i s t o r t i o n 29 sixth-degree network using Tuttle»s method as was done in this system. Then, by a process of iteration, make the appropriate poles of the fourth-or the sixth-order network coincide with the f i r s t and last zeros of the fifth-order network. This method would also reduce the total error, but even though both networks would be equal-ripple the total effect might not be. Although both these practical solutions have cer-tain advantages, further work should f i r s t be done on design-ing both networks simultaneously. As mentioned previously, this has been attempted in the present investigation, but no solution was found. 9. CONCLUSIONS A l t h o u g h f u r t h e r d e v e l o p m e n t a l work i s r e q u i r e d on t h e networks,- t h i s o s c i l l a t o r h a s s e v e r a l a d v a n t a g e s o v e r t h e one d e s c r i b e d by A n d e r s o n . Of t h e s e a d v a n t a g e s t h e most s i g n i f i c a n t I s t h e change i n t h e s u b t r a c t o r d e s i g n . T h i s s u b t r a c t o r h a s o p e r a t e d s u c c e s s f u l l y at 600 k i l o c y c l e s w i t h o u t any c o m p e n s a t i n g n e t w o r k s s u c h as t h o s e needed b y A n d e r s o n i n t h e Ca t h o d e c i r c u i t . A n o t h e r a d v a n t a g e o f t h i s o s c i l l a t o r i s t h e improvement I n t h e r e l a t i o n s h i p between l o g a r i t h m i c f r e q u e n c y and p o t e n t i o -m e t e r r a t i o . W i t h a b e t t e r d e s i g n o f t h e f r e q u e n c y -d e t e r m i n i n g n e t w o r k s , t h e m a j o r d i s a d v a n t a g e o f t h e p r e -s e n t o s c i l l a t o r w i l l be c o r r e c t e d . T h i s d i s a d v a n t a g e i s t h e I n c r e a s e i n t h e t o t a l h a r m o n i c d i s t o r t i o n at t h e low f r e q u e n c i e s . T h i s o s c i l l a t o r h a s a l s o b e e n o p e r a t e d as a sweep o s c i l l a t o r , t h e sweep b e i n g s u p p l i e d by a d-c m o t o r . The e x p e r i m e n t a l r e s u l t s o f t h i s a r r a n g e m e n t were g o o d ; t h e o n l y v a r i a t i o n f r o m t h e a m p l i t u d e - r e s p o n s e c h a r a c t e r i s t i c o f t h e t e s t n e t w o r k o c c u r r e d at t h e low f r e q u e n c i e s where t h e h a r m o n i c d i s t o r t i o n o f t h e o s c i l l a t o r c a u s e d a s p u r i o u s r e s p o n s e . I n c o n c l u s i o n , i t c a n be s t a t e d t h a t an a u d i o - f r e q u e n c y sweep g e n e r a t o r o f new d e s i g n h a s been p r o d u c e d , and t h a t s u g -g e s t i o n s f o r t h e c o r r e c t i o n o f t h e one d i s a d v a n t a g e h a v e been p r o p o s e d . 30 APPENDIX A.-ANALYSIS OF NETWORK RESPONSE TO A SIGNAL OF VARYING FREQUENCY As m e n t i o n e d I n C h a p t e r 2, t h e most c o m p l e t e a n a l y -s i s o f a n e t w o r k r e s p o n s e t o a v a r y i n g f r e q u e n c y was f o u n d I n Baghdady's p a p e r , " L o w - D i s t o r t i o n R e p r o d u c t i o n o f F M S i g n a l s T h i s A p p e n d i x c o n t a i n s Baghdady's a n a l y s i s i n a c o n d e n s e d f o r m . The p r o b l e m c a n be r e s t a t e d as t h a t o f f i n d i n g t h e c o n d i t i o n s t h a t a g e n e r a l l i n e a r - s y s t e m f u n c t i o n must s a t i s f y I n o r d e r t o e n s u r e an a c c e p t a b l e r e p r o d u c t i o n o f a s p e c i f i e d f r e q u e n c y - m o d u l a t e d s i g n a l . As a b a s i c r e q u i r e -ment , t h e s y s t e m must r e s p o n d i n a q u a s i - s t a t i c manner; i . e . , i t s " d y n a m i c " r e s p o n s e must be c l o s e l y d e s c r i b e d by i t s " s t a t i c " r e s p o n s e as p r e d i c t e d b y a r c s t e a d y - s t a t e c i r c u i t t h e o r y . T h u s , t h e d e s i r e d r e s p o n s e i s v ( t ) = Z ( j o ) 1 ( t ) ) . I exp ft±(t) d t ] , w h i c h i s t h e same as t h e e x p r e s s i o n f o r t h e s t e a d y - s t a t e r e s -p o nse t o a c o n s t ant - f r e q u e n c y e x c i t a t i o n . The a c t u a l r e s p o n s e i s g i v e n by v ( t ) = E ( t ) exp £ j 1 ( t ) d t ] , where u)^, ( t ) = OJ C + d0/dt, t h e I n s t a n t a n e o u s f r e q u e n c y o f t h e e x c i t at i o n , and 0 ( t ) i s some a r b i t r a r y f u n c t i o n o f t i m e . The e x c i t a t i o n a p p l i e d t o t h e s y s t e m i s i ( t ) = I exp £jJu> ±(t) d t ] . 31 32 I f t h e c h a r a c t e r i s t i c u n i t i m p u l s e r e s p o n s e o f t h e s y s t e m I s h ( t ) , t h e s u p e r p o s i t i o n i n t e g r a l s t a t e s t h a t v ( t ) = f h( x ) i ( t - x ) d x . •'o The a c t u a l r e s p o n s e I s t h e r e f o r e E ( t ) = I fQh( i ) exp f - j [ coc x + 0 ( t ) -0 ( t - x ) ] d x ] , o r E ( t ) = I JQla( x ) exp {"J } * g ( t , x ) d x , v.. ( A - l ) where g ( t , T ) = exp [ j [ 0 ( t - x ) -0( t ) +x0 »(t) ] | * I f 0(t) and 0 1 ( t ) a r e c o n t i n u o u s f o r a l l t , and 0" ( t ) e x i s t s f o r a l l f i n i t e t , t h e n 0 ( t ~ x ) = 0 ( t ) - X0' ( t ) + R 2 0 , where R ^ = ^0 " ( t - Fx ) ( - x ) 2 , 0 < T < 1, and |0 " ( t - Tx )| < | 0 " ( t ) | max* T h e r e f o r e , g ( t , : c ) = exp { J R 2 0 } = 1 + R l g where and R l g = J R 2 0 e x P { ^ R 2 0 J > 0 < M < 1 > l g {2 0 S u b s t i t u t i n g t h e s e v a l u e s i n E q u a t i o n ( A - l ) , t h e e x p r e s s i o n f o r t h e v o l t a g e f u n c t i o n becomes E ( t ) = I fQh( x ) exp { - j w c x ^ ( l + R l g ) dx • o r E ( t ) = I Z ( J U i ) + I R 1 E ( t ) , where Z ( i w i ) =J0H%) exp [ - j OJ T dx R l E ( t ) = ^ R l g h( x ) exp ^-JoJixJ dx , 33 and R 1 E ( t ) < 0"(t) max Z ( j OJ. ) max ( A - 2 ) T h i s l a t t e r i n e q u a l i t y a r i s e s from the convergence p r o p e r t i e s of the Van der Pol-Stumpers expansion. T h i s expansion i s performed i n t h e f o l l o w i n g way. I f the q u a n t i t y g ( t , T ) i n E q u a t i o n ( A - l ) can be expanded i n a T a y l o r ' s s e r i e s i n powers of ( - T ) as where the I n f i n i t e s e r i e s i s u n i f o r m l y convergent f o r a l l x , then the v o l t a g e e x p r e s s i o n becomes E ( t ) = foh( x )[ ^ o a m ( - T ) m ] exp [-jo) ±x j dx . I f h( T ) i s bounded f o r a l l T , then ____ a ( - x ) ^ x) e x p f - j c o T ] m=o i s u n i f o r m l y convergent f o r a l l t ? I t can t h e r e f o r e be i n t e -g r a t e d term by terra t o y i e l d the u n i f o r m l y convergent s e r i e s E ( t ) = 22 J ( - x )mh( x ) exp {-joj xj dx . m=o " o L i f z ( j OJ_ ) = 7o h ( T ) e x p f - J u , - i . T } d T > then the v o l t a g e f u n c t i o n can be w r i t t e n as E (t) = *™ ^ ( ^ l ) m •= 0 d( j ^ i ) m C o n t i n u i n g with t h e a n a l y s i s , the r i g h t hand s i d e of Equation ( A - 2 ) t h e r e f o r e r e p r e s e n t s an upper bound on the magnitude of the e r r o r i n c u r r e d i n u s i n g Only Z ( J O J ) t o represent E ( t ) . Beeause t h e e r r o r v a r i e s with t i m e , the g e n e r a l 3k c o n d i t i o n f o r q u a s i - s t a t i o n a r y r e s p o n s e i s e m _ i - a d 20 * z"( jo) ..) d t 2 max « 1 ... (A-3) max Baghdady. h a s a l s o p r o v e d t h a t , f o r t h e p a r t i c u l a r c a s e o f p e r i o d i c m o d u l a t i o n , t h e h i g h e r o r d e r t e r m s i n t h e e x p a n -s i o n can be a r r a n g e d I n an a s y m p t o t i c s e r i e s . The s e c o n d t e r m , t h a t o f E q u a t i o n A-3, i s t h e same i n b o t h e x p a n s i o n s , and i s t h e r e f o r e t h e l a r g e s t . As was m e n t i o n e d p r e v i o u s l y , t h e v a l i d i t y o f t h i s a n a l y s i s r e q u i r e s o n l y t h a t 0 ( t ) and 0 ( t ) be c o n t i n u o u s f o r a l l t and t h a t 0 ( t ) e x i s t f o r a l l f i n i t e t . However Z(s) c a n h a v e o n l y l e f t - h a l f - p l a n e p o l e s . T h i s I s t h e same c o n d i t i o n f o r t h e q u a s i - s t a t i o n a r y r e s p o n s e a r r i v e d at by Van d e r P o l u s i n g H e a v i s i d e ' s o p e r -a t i o n a l c a l c u l u s . Baghdady h a s e x t e n d e d h i s i n v e s t i g a t i o n t o o t h e r s p e e i a l c a s e s , but t h i s i s t h e o n l y p a r t o f h i s a n a l y s i s n e e d e d i n t h e p r e s e n t a p p l i c a t i o n . APPENDIX B. R-C NETWORK APPROXIMATION OP A CONSTANT REACTANCE 17 As m e n t i o n e d i n C h a p t e r 5>, T u t t i e h a s g i v e n a d e s i g n method f o r t h e Cheby s h e y a p p r o x i m a t i o n t o a c o n s t a n t r e a c t a n c e . T h i s de'sign method i s d e r i v e d i n t h e f o l l o w i n g manner. The S c h w a r z - C h r i s t o f f e l t r a n s f o r m a t i o n i s u s e d t o o b t a i n t h e f u n c t i o n n e c e s s a r y t o map t h e l e f t h a l f o f t h e c o m p l e x - f r e q u e n c y p l a n e i n t o a r e c t a n g u l a r r e g i o n I n t h e W-plane as i n F i g u r e 1S>. E D B A; A, B' 0 2 .:• s - p l a n e W-plane F i g u r e If?. The D e s i r e d C o n f p r m a l M a p p i n g as T h i s n e c e s s a r y m a p p i n g f u n c t i o n may be e x p r e s s e d W = F ( s ) 3$ 36 and i t s d e r i v a t i v e may be w r i t t e n as dW/ds = H (s + j o J 2 ) a l ( s + jco _ ) a 2 ( s - j c o 1 ) a 3 ( s - J o ) 2 ) & ^ . Now i n t r a v e r s i n g t h e co - a x i s i n t h e c o m p l e x - f r e q u e n c y p l a n e f r o m - °o t o + 0 0 , and t h e r e c t a n g u l a r b o u n d a r y I n t h e W-plane i h t h e c o u n t e r - c l o c k w i s e s e n s e , e a c h c r i t i c a l p o i n t i s met i n s u c c e s s i o n , T h e n , a t P 1 i n t h e W - p l a n e , dW u n d e r g o e s , an a n g u l a r change o f +%•/• 2 r a d i a n s , and i n t h e s - p l a n e , ds u n d e r g o e s an a n g u l a r change o f -?w r a d i a n s . Prom a n g u l a r c o n s i d e r a t i o n s , a, = 2 — - = ~k • X -2% S i n c e t h © same a n g u l a r r e l a t i o n s h i p h o l d s at a l l f o u r c r i t i c a l p o i n t s , a l l t h e a*s h a v e t h e same v a l u e , ~|. The lncijion can t n e r W = H j 2 J (s + to 1 mapping f u n c t i o n can t h e r e f o r e be w r i t t e n as ..ds.. / f ( 8 £ + 0 ) ^ ) ^ (S 2 + C 0 2 2 ) F w h i c h i s an e l l i p t i c i n t e g r a l ; h owever, i n t e g r a l t a b l e s do n o t u s u a l l y l i s t i n t e g r a l s i n t h i s f o r m , so t h e i n t e g r a l w i l l be changed t o l/co ds W = _H_ ' f x / 1 J (1 + s 2 / ^ 2 ) ^ ( 1 + co_/ w 2 w l 2 T h i s i n t e g r a l i s t h e same as i n t e g r a l number 1 3 0 . 0 £ o f B y r d 19 and F r i e d m a n , where t h e s o l u t i o n i s g i v e n as W = ___ t n _ 1 ( s _ , k) . CO o CO 137 -1 a I n t h i s equation t n (-— , k) Is the i n v e r s e e l l i p t i c tangent f u n c t i o n of argument s^, and modulus k, where w l k = [1 - ( o ^ / o ) ^ ] Now, l e t t i n g H = co 2 and i n v e r t i n g t h e f u n c t i o n , s = OJ.' t n ( W,k ) T h i s f u n c t i o n maps the e n t i r e complex-frequency plane In a d o u b l y - p e r i o d i c manner i n t o the W-plane, as shown i n F i g u r e 16. I f by means of t h i s mapping the c o n d i t i o n s of the approximation can be s t a t e d i n a convenient manner In the W-plane, i t w i l l be p o s s i b l e t o l o c a t e the p o l e s and zeros by symmetrical r e l a t i o n s . T h e r e f o r e , r e f e r r i n g t o F i g u r e 16, the problem Is t o approximate a constant over the r e g i o n between J co 1 and j co 2 i n t h e s-planej In the -W plane, due t o • .• A v .. 2 2 2 X X X 0) 2 2 2 X X X 2 2 2 -X X X 2 2 2 o o—-e-r-2 2 2 -*—x x 2 2 2 -e—o o 2 2 2 - x — x — x — 2 2 2 -x—x—-x-2 2 2 O i O 0 K 2 l : 2 'c i IX X 2 2 2 Q O O 2 2 2 -x—*—* 2 2 2 - x — x — x — 2 , 2 2 e^-—-e-—-e— 2 2 2 2 2 2 o 0 0 2 2 2 -x—*•—x— u s-plane W-plane F i g u r e 16. Mapping f o r E q u a l - R i p p l e Approximation 38 the double p e r i o d i c i t y , the approximation i s along t h e a l t e r -nate h o r i z o n t a l l i n e s corresponding to t h i s I n t e r v a l . F o r con-venience the constant to be approximated w i l l be i n i t i a l l y zero. Proceeding with the s o l u t i o n , a number of p o l e s (ipi) determined by the degree o f accuracy d e s i r e d i n the a p p r o x i -mation are l o c a t e d i n the W-pIane on t h e h o r i z o n t a l l i n e s corresponding t o the a - a x i s . To a v o i d c e r t a i n d i f f i c u l t i e s i n the s o l u t i o n , a l l p o l e s and zeros are doubled so t h a t the s o l u t i o n ' w i l l y i e l d the square of the odd part of the imped-ance f u n c t i o n on the co - a x i s . Each complete mapping of the complex-frequency plane w i l l t h e r e f o r e c o n t a i n l\.n p o l e s , a l l arranged i n p a i r s . To complete the p o l e - z e r o p l o t , an equal number o f z e r o s , a l s o arranged i n p a i r s , w i l l be l o c a t e d on the approximation i n t e r v a l . I n o r d e r t o y i e l d a p h y s i c a l l y -r e a l i z a b l e s o l u t i o n , poles must not be allowed t o occur at the o r i g i n or at i n f i n i t y . As can be seen i n F i g u r e 16, an equal spacing of the p o l e s and zeros i n the W-plane g i v e s an equal r i p p l e along both the c h a i n of zeros and the c h a i n of p o l e s , The a r r a y of p o l e s and zeros f o r n equal t o t h r e e seen In F i g u r e 16 can be expressed by some mathematical f u n c t i o n , -P(s) P(-s) . I f a constant equal to the n e g a t i v e of the v a l u e of the f u n c t i o n at the o r i g i n i s now added t o the f u n c t i o n , the zeros w i l l s p l i t and move along the l i n e s where the phase o f the f u n c t i o n i s an odd i n t e g e r m u l t i p l e of u , f i r s t t o the saddle p o i n t s between the zeros and then t o 39 t h e mapping o f t h e o - a x i s . W i t h t h i s a r r a n g e m e n t o f t h e ' p o l e s and z e r o s , a l l t h e z e r o s w i l l a r r i v e at t h e o - a x i s w i t h t h e same v a l u e o f added c o n s t a n t . T h i s p r o c e s s i s I l l u s t r a t e d i n F i g u r e 17, t o g e t h e r w i t h t h e a s s o c i a t e d f u n c t i o n i n t h e s - p l a n e and t h e m a g n i t u d e on t h e OJ - a x i s . The p o l e s and z e r o s must be r e d u c e d t o s i n g l e o r d e r , s i n c e J X ( s ) i s t h e d e s i r e d f u n c t i o n and t h i s r e d u c t i o n i s e q u i v a l e n t t o s e p a r a t i n g t h e p o l e s and z e r o s I n t o t h o s e b e l o n g -i n g t o J X ( s ) and t h e o t h e r s b e l o n g i n g t o J X ( - s ) . F r o m F i g u r e 17, t h e p o s i t i o n s o f t h e p o l e s and z e r o s o f t h e d e s i r e d odd p a r t o f t h e impedance f u n c t i o n i n t h e W-plane a r e t h e n + r (K/2n) + JO , where r = 0,1,2 , 3 , ... 2n , and K i s t h e c o m p l e t e e l l i p t i c i n t e g r a l , t h a t i s , t h e v a l u e o f W c o r r e s p o n d i n g t o s e q u a l t o i n f i n i t y . The p o s i t i o n s o f t h e p o l e s and z e r o s i n t h e c o m p l e x - f r e q u e n c y p l a n e a r e g i v e n by s r = + 10 t n ( K r / 2 n , k) . To expand t h i s e l l i p t i c f u n c t i o n , e q u a t i o n s numbers 126.01 and 105>0.01 o f B y r d and F r i e d m a n w i l l b e u s e d * T he f u n c t i o n t h e n becomes u) x s i n X-tt s i n 3^ + q ° s i n $ x - q 1 2 s i n 7 \ .+ . . . s r = — X : v r •• v \r. • - -! •'-k ' 3 cos X + q 2 cos 3X + q^ Cos $k + q 1 2 cos 7 \ + I n t h i s e q u a t i o n X = r %/ 4 n r a d i a n s , q i s t h e nome o f t h e e l l i p t i c f u n c t i o n , and k» i s t h e c o m p l e m e n t a r y m o d u l u s , where k* = ( 1 - k ) 3 . A l l t h a t r e m a i n s i s t o e v a l u a t e t h e nome q. 2 . I., 2 2 2 2 2 X K, X < 2 2 2 X X K > 2 2 2. 4 -P(s)p(s) 0). \7~ 2 2 2 .2 .. 2 X 31 K 2 2 2 K X » A -Q(s)Q(s) CO A oo 2222 xoxox 2 at 2 22222 «• xoxox v A J . A-X(s)X(s) s-plane W-plane Magnitude along oo - axis Figure 17. The Process of Zero-Shifting By d e f i n i t i o n , q = e ^ 20 A s e r i e s expansion f o r t h e norae, g i v e n i n Jahnke and Emde, q = (t + 2t^ + l £ t 9 + l ^ O t 1 3 + ...) • i where t = h 1 - k » 8 , 1 - k>'$ and k 1 ' = V u 2 ' T h i s expansion does not converge v e r y r a p i d l y f o r s m a l l v a l u e s of k», so an a l t e r n a t i v e expansion i s used. T h i s expansion i s based on the r e l a t i o n q . = e K A ' so t h a t l o g 1 0 (q) . l o g 1 Q ( q » ) = ( n / e ) 2 = 1.8615228 . The expansion f o r q ' i s qt = (t» + 2t • ^ + l £ t » 9 + l50t« 1 3 + ....). 0 2, where t» = £ l - ( l - k ' ^ y * , l + ( l - k » 2 ) a 2 1 as i n d i c a t e d by O b e r h e t t i n g e r and Magnus. L I S T OF REFERENCES 1. Sherman, J.B., A u d i o F r e q u e n c y C u r v e T r a c e r . P r o c . I.R.E., June 1938, pp. 700-712. 2. C a r l i s l e , S.F. and A.B. M u n d e l l , F r e q u e n c y R e s p o n s e C u r v e T r a c e r . E l e c t r o n i c s , Aug. 191+1, pp. 22-23. 3. A n d e r s o n , F.B., S e v e n L e a g u e O s c i l l a t o r . P r o c . I.R.E., Aug. 1951, pp. 881-890. i|. F r a n t z , W.J., The T r a n s m i s s i o n o f a F r e q u e n c y M o d u l a t e d Wave t h r o u g h a N e t w o r k , P r o c . I.R.E., M a r c h 1946, P P . II1 |P - 1 2 5P. 5. C a r s o n , J.R. and T.C. F r y , V a r i a b l e F r e q u e n c y E l e c t r i c C i r c u i t T h e o r y , B.S.T.J., O c t , 1937, pp. 513-51+0. 6. Weber, E . , L i n e a r T r a n s i e n t A n a l y s i s , V o l . 2. J o h n W i l e y & Sons I n c . New Y o r k . I9I+6. 7. v a n d e r P o l , B., The F u n d a m e n t a l P r i n c i p l e s o f F r e q u e n c y M o d u l a t i o n . J o u r n a l o f t h e I .E.B. ' p a r t I I I . May I9I+6, P P . 153-158. 8. Baghdady, E . J . , T h e o r y o f L o w - D i s t o r t i o n R e p r o d u c t i o n o f FM S i g n a l s i n L i n e a r S y s t e m s . I .R.E., T r a n s . P.G.C.T., S e p t . 1958, pp. 202-21]+. 9. Edson,. W.A., Vacuum Tube O s c i l l a t o r s . J o h n W i l e y & Sons I n c . New Y o r k . 1953. 10. S u l z e r , P.O., Wide Range R-C O s c i l l a t o r . E l e c t r o n i c s , S e p t . 1950, pp. 88-89. 11. S c o t t , H.H., A New T y p e o f S e l e c t i v e C i r c u i t and Some A p p l i c a t i o n s . P r o c . I.R.E., F e b . 1938,• pp. 226-235. 12. W i l l o n e r , G. and F . T i h e l k a , A Phase S h i f t O s c i l l a t o r w i t h Wide-Range T u n i n g . P r o c . I .R.E., S e p t . 191+8, pp. 1096-1100. 13. Ames, M.E., Wide Range D e y i a b l e O s c i l l a t o r . E l e c t r o n i c s , May 191+9, pp. 96-100. llf. Hund, A., R e a c t a n c e Tube M o d u l a t o r s I n F-M A p p l i c a t i o n s . E l e c t r o n i c s , O c t . 191+2, pp. 68-71. 15. S h e a r e r , C.F., F r e q u e n c y M o d u l a t o r . P r o c . I.R.E., F e b . 191+0, pp. 66-67. 16. G r a y , T.S., A p p l i e d E l e c t r o n i c s . S e c o n d E d i t i o n . J o h n W i l e y & Sons I n c . New Y o r k . 1954* 17. T u t t l e , D.P. J r . , N etwork S y n t h e s i s . J o h n W i l e y & Sons I n c . New Y o r k . 1958. 18. B o l l n i a n , J.H., and J.G. K r e e r , The A p p l i c a t i o n o f T h e r m i s t o r s t o C o n t r o l N e t w o r k s . P r o c . I.R.E., J a n . 19^ 0, pp. 20-26. 19. B y r d , P.P. and M.D. F r i e d m a n , Handbook o f E l l i p t i c I n t e g r a l s f o r E n g i n e e r s and P h y s i c i s t s . S p r i n g e r - V e r l a g . B e r l i n . 1954* 20. J a h n k e , E . and F . Emde, T a b l e s o f F u n c t i o n s w i t h F o r m u l a e and C u r v e s . F o u r t h E d i t i o n . D o v e r Pub11cat i o n s . New Y o r k , 1945. 21. O b e r h e t t i n g e r , F . and W. Magnus, Anwendung d e r E l l i p t i s c h e n F u n c t i o n e n i n P h y s i k und T e c h n l k . S p r i n g e r - V e r l a g , B e r l i n . 1949.
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Electrical and Computer Engineering
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An audio-frequency sweep-generator
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