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Study of ferroresonance with application to digital logic Reed, Albert James 1968

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A STUDY OF FERRORESONANCE WITH APPLICATION TO DIGITAL LOGIC  by  ALBERT JAMES REED B . A . S c , U n i v e r s i t y of B r i t i s h Columbia, 1964  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n the Department of Electrical  We accept t h i s  Engineering  t h e s i s as conforming  required  to the  standard  Research S u p e r v i s o r Members of the Committee  Head of the Department , . Members of the Department of E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY  OF BRITISH COLUMBIA  In presenting this thesis  in 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 shall make it freely available Study. thesis  for reference and  I further agree that permission for extensive  copying of  this  for scholarly purposes may be granted by the Head of my  Department or by h.iis representatives. or publication of this thesis  It is understood that  copying  for financial gain shall not be allowed  without my written permission.  Department of The University of B r i t i s h Columbia Vancouver 8, Canada  i  ABSTRACT  A s e r i e s resonant L-C c i r c u i t  i n which e i t h e r the  i n d u c t o r o.r the c a p a c i t o r i s n o n l i n e a r and which i s e x c i t e d by a s i n u s o i d a l v o l t a g e of a f i x e d frequency may s t e a d y - s t a t e responses.  One  have  two  of these responses i s c h a r a c t e r -  i z e d by a h i g h amplitude o s c i l l a t i o n ; the other by a low one.  I f the amplitude or frequency of the d r i v i n g s i g n a l i s  v a r i e d s l o w l y , the response may to the other s t a t e . called  suddenly change or "jump"  As a r e s u l t , t h i s phenomenon has been  jump resonance,  or f e r r o r e s o n a n c e .  Because the h i g h and low resonant s t a t e s could be  con-  s i d e r e d as a 0 and 1 b a s i s f o r d i g i t a l l o g i c o p e r a t i o n s , i t was  the purpose  of t h i s work to study the phenomenon and to  i n v e s t i g a t e the p o s s i b i l i t y of u s i n g i t i n the d e s i g n of digital logic  elements.  Equations''which e x h i b i t the necessary f e a t u r e s were s t u d i e d on an analogue  computer.  The r e s u l t s of the study  were used as d e s i g n c r i t e r i a f o r the c o n s t r u c t i o n of an a c t u a l c i r c u i t and a l s o as a b a s i s f o r an approximate study.  analytical  The a n a l y t i c a l study uses the R i t z method to f i n d  u s e f u l f e a t u r e s of the responses.  The r e s u l t s of p r e v i o u s  users of t h i s method have been extended  to i n c l u d e equations  with both second d e r i v a t i v e c o u p l i n g and non-symmetrical  non-  linearities. Based  on the above s t u d i e s , a prototype c i r c u i t  was  designed which has some of the b a s i c p r o p e r t i e s of c o n v e n t i o n a l  ii  flip-flop circuits. circuit  One of the main f e a t u r e s of t h i s  i s that i t i s almost  e n t i r e l y made of r e a c t i v e  components and as a r e s u l t has very low power consumption. The o p e r a t i o n of the c i r c u i t i s us^d t o v e r i f y the v a l i d i t y of the approximations  made i n both the analogue s i m u l a t i o n  and the a n a l y t i c a l study.  The r e s u l t s obtained from the  analogue study, the R i t z a n a l y s i s , and the prototype compare f a v o r a b l y with each other. f u t u r e work are g i v e n .  circuit  Some suggestions f o r  iii TABLE OP CONTENTS Page ABSTRACT  ....  TABLE OF CONTENTS  i i i i  LIST OF ILLUSTRATIONS LIST OF SYMBOLS  v . .  ACKNOWLEDGEMENT  v  ... .  v i i  •  ix  1.  INTRODUCTION  1  2.  ANALOGUE SIMULATION  4  2.1  Preamble  4  2.2  Computer C i r c u i t s  4  2.3  D i s c u s s i o n of Analogue Computer R e s u l t s  2.4 3.  (a)  Form of S o l u t i o n s  (b)  E f f e c t of V a r y i n g the R e s i s t a n c e  (c)  E f f e c t of V a r y i n g the Coupling  . 10 10  .  11  ...  12  Summary of the Analogue S i m u l a t i o n Results  12  CIRCUIT ANALYSIS AND DESIGN  15  3.1  Preamble  15  3.2  The R i t z Method as A p p l i e d t o Forced Oscillations  15  3'3.  Development of the R i t z C o n d i t i o n s  ....  16  3-4  Q u a l i t a t i v e D i s c u s s i o n of S o l u t i o n s ...  20  3.5  Features  22  3.6  D i s c u s s i o n of the Uncoupled Case  24  (a)  Frequency C h a r a c t e r i s t i c s  24  • Ob)  Amplitude C h a r a c t e r i s t i c s  24  of the S o l u t i o n s  iv Page .(c)  4.  5.  25  E f f e c t of Damping  3.7  The E f f e c t of Coupling  3-8  Source Impedance C o n s i d e r a t i o n s  . ...  25 27  EXPERIMENTAL RESULTS  31  4.1  Basic C i r c u i t . Configuration  31  4.2  Output Waveform  31  4«3  Power Consumption  32  4.4  Bistability  32  4.5  Use as a Memory Device  55  4.6  Use as a Counter  53  4.7  Switching  58  4.8  Summary  41 42  CONCLUSIONS  APPENDIX A:  APPENDIX B: REFERENCES  Polynomial Approximation t o the f ( q ) C h a r a c t e r i s t i c f o r the S i l i c o n Capacitor  45  Determination icients  48  of the R i t z  Coeff-  50  V  • . LIST OF ILLUSTRATIONS  Figure .2.1  ^ S a  B a s i c Coupled C i r c u i t  ...  2.2  S i l i c o n Capacitor  2.3  Quality Factor  2.4  F o r c i n g Function  2.5  Simulation  2.6  Sample Analogue Computer S o l u t i o n  14  2.7  Sample Analogue Computer S o l u t i o n  14  2.8  Sample Analogue Computer S o l u t i o n  14  2.9  Sample Analogue Computer S o l u t i o n  2.10  Sample Analogue Computer S o l u t i o n  14  2.11  Sample Analogue Computer S o l u t i o n  14  3.1  T y p i c a l Frequency Responses  21  3.2  Frequency Response f o r Uncoupled Case .  23  3.3  Amplitude Response f o r Uncoupled Case .  23  3.4  E f f e c t of Damping on Frequency Response  26  3.5  E f f e c t of Damping on Amplitude Response  26  3.6  E n t i r e Approximate S o l u t i o n f o r Equation 3.17  28  E f f e c t of Coupling on I d e n t i c a l S o l u t i o n Pairs  30  3.8  E f f e c t of C o u p l i n g on Solution Pairs  30  4.1  Memory Device C o n f i g u r a t i o n  4.2  Counter Device C o n f i g u r a t i o n  4.3  Operation of Counter  3.7  Characteristics  5 6  f o r S i l i c o n Capacitor  ..  Generator  6 8  of Basic Coupled C i r c u i t  ...  ...,  Non-identical  9  14  35 ........... ,  35 37  e  vi  Figure  Page  4.4  Waveforms of Two  4.5  • Switching Process  Consecutive Counters  '37 39  4.6  Switching Waveform  A. l  Normalized f'(q) C h a r a c t e r i s t i c  A 2 ,  D e v i a t i o n Between f(q,) and the P o l y nomial Approximations  B. l  Flow Diagram of Program to Ritz Coefficients  s  ..  '.. ' .........  37 46 • 4&  Determine 49  LIST OP SYMBOLS  A,B,C  constants  D  damping c o e f f i c i e n t  E  a function  a,c,d,e,h  coefficients  F„,F F„ o s' c  functions  G ,0f ;Go s c  functions  G-  amplitude  f ,g  functions  k  a subscript  K  a  o f  L M  Yj  coefficient  constant  inductance coupling  coefficient  OJ/K  p  amplitude  q  a variable representing  q  an approximation to q  f  nonlinearity coefficient  y k  a set of f u n c t i o n s  1  s t  Z x,y,u,v  p/K  coefficient  2  a v a r i a b l e , time w t  variables  char,  •, dx/dt d x/dt 2  2  angular a  frequency  constant  resistance voltage  viii  ACKNOWLEDGEMENT  G r a t e f u l acknowledgement i s given t o the N a t i o n a l Research C o u n c i l of Canada f o r f i n a n c i a l  support  i n the form o f a Bursary i n 1964-65 and a studentship i n 1965-66. The author wishes t o thank h i s s u p e r v i s o r , Dr. A.C. Soudack f o r h i s guidance  throughout  the course o f  t h i s work. Thanks are a l s o g i v e n t o Dr. M.S. Davies f o r r e a d i n g the manuscript  and f o r h i s u s e f u l comments and  suggestions, and to Dr. G. C h r i s t e n s e n f o r h i s h e l p f u l discussions.  1  1.  INTRODUCTION  The phenomenon of ferroresonance  was observed f o r the  f i r s t time i n 1906 d u r i n g the t u n i n g of r a d i o t r a n s m i t t e r s . Since, then i t has appeared i n the l i t e r a t u r e p e r t a i n i n g to power s y s t e m s , e l e c t r o n i c s , ^ ^ Ferroresonance  and n o n l i n e a r mechanics.  can occur i n a d r i v e n s e r i e s r e s o n a n t ' c i r c u i t  which c o n s i s t s of an i n d u c t o r and a c a p a c i t o r , one of which i s nonlinear.  I f the n o n l i n e a r c h a r a c t e r i s t i c i s symmetrical,  such a c i r c u i t can be approximated by D u f f i n g ' s  x + C O x + hx q  equation:  = G- cos cot  (l-l)  C l o s e r approximations u s i n g h i g h e r order terms can be found i n ' (5) Hayashi.  This equation has the property that under c e r t a i n  c o n d i t i o n s i t has two s t a b l e s o l u t i o n s near resonance, one a l a r g e amplitude o s c i l l a t i o n phenomenon i s c a l l e d  and the other a s m a l l one.  This  ferroresonance.  The b i s t a b l e m u l t i v i b r a t o r , or f l i p - f l o p , i s one of the most u s e f u l e l e c t r o n i c devices employed i n d i g i t a l computers.  An i d e n t i f y i n g f e a t u r e of a b i s t a b l e m u l t i v i b r a t o r  i s that i t c o n s i s t s of a p a i r of two-state symmetrically  devices  arranged  so as to allow only two s t a b l e s t a t e s of the  complete c i r c u i t .  For example, i n a t r a n s i s t o r f l i p - f l o p ,  e i t h e r t r a n s i s t o r may be on or o f f but they are arranged so that when one i s on i t keeps the other o f f .  I t was proposed to i n v e s t i g a t e the p o s s i b i l i t y ' o f making t;. f l i p - f l o p type computer component u s i n g the f e r r o resonant'.regions  of n o n l i n e a r L-C c i r c u i t s .  amplitude o s c i l l a t i o n could represent small one an o f f s t a t e .  The l a r g e  an on s t a t e and the  Some work i n t h i s area has been  done by other workers such as I s b o r n ^ ^ , 6  Gremer^^, and  (8) Ozawa  .  However, t h e i r work was based on the assumption  o  of a symmetrical c h a r a c t e r i s t i c f o r the n o n l i n e a r element. More recent developments i n semiconductors, p a r t i c u l a r l y the advent of v a r a c t o r diodes, have made i t necessary s i d e r the problem a l l o w i n g non-symmetrical The work presented  t o con-  characteristics.  here i n c l u d e s the choice of a  p a r t i c u l a r type of n o n l i n e a r c a p a c i t o r , an analogue s i m u l a t i o n to o b t a i n a f i n a l c i r c u i t c o n f i g u r a t i o n and to give approximate design values t o the components, an approximate mathematical a n a l y s i s of the c i r c u i t , and the b u i l d i n g of an o p e r a t i v e prototype  unit.  The c i r c u i t f i n a l l y used can be  d e s c r i b e d by the f o l l o w i n g  equations:  q^ + 2DKg(q ) + K f ( 2  1  q i  )  + Mq  2  - p sin?  , 0  (1-2) q  + 2DKg(q ) + K f ( q ) + Mq^' - p s i n S 2  2  2  These equations method.  are analyzed  2  = 0  u s i n g the R i t z or R i t z - G a l e r k i n  Some of K l o t t e r ' s ^ m e t h o d s have been extended  to equations  with second d e r i v a t i v e c o u p l i n g .  General  algebraic  c o n d i t i o n s r e l a t i n g the response and frequency  to the d r i v i n g  amplitude  are d e r i v e d f o r t h i s type of equation.  These r e s u l t s are a p p l i e d to the s p e c i f i c case of a with n e g l i g i b l e damping and with an asymmetrical f u n c t i o n which can be approximated f(q)  = q + uq  circuit  restoring  by: (l-3)  2  Parametric e x c i t a t i o n u s i n g the v a r i a b l e capacitance p r o p e r t i e s of m a t e r i a l s such as barium achieved^^  and i t was  thought  t i t a n a t e has been  that components of t h i s  type  could be used as the n o n l i n e a r elements i n the s y n t h e s i s of the proposed  circuit.  However, they were deemed u n s u i t a b l e  f o r use." at the present time due to t h e i r cost and w i t h i n the needed capacitance t o l e r a n c e . commercially  available silicon  scarcity  Instead, low c o s t ,  c a p a c i t o r s were used to demon-  s t r a t e the p r i n c i p l e s of o p e r a t i o n and were found to be q u i t e satisfactory. This t h e s i s c o n s i s t s of a d i s c u s s i o n of the analogue computer s i m u l a t i o n i n Chapter a n a l y s i s i n Chapter  2, the development of the R i t z  3? some d i s c u s s i o n of the c i r c u i t  design  and r e s u l t s i n Chapter 4, and some suggestions f o r f u t u r e study i n the c o n c l u d i n g Chapter  5.  4  2. 2-1  ANALOGUE SIMULATION Preamble An  e l e c t r o n i c analogue computer c o n s i s t s of a c o l l e c t i o n  of u n i t s , each of which i s designed t o produce an output that i s a p a r t i c u l a r l i n e a r or n o n l i n e a r  f u n c t i o n of the i n p u t s .  These u n i t s are r e a d i l y i n t e r c o n n e c t e d equations or to simulate  to solve mathematical  the behaviour of a p h y s i c a l system.  A convenient f e a t u r e i s o n - l i n e c o n t r o l , that i s the f a c i l i t y w i t h which changes i n parameters of the equations can be made manually d u r i n g the a c t u a l operation 2-2  or s o l u t i o n of an equation.  Computer C i r c u i t s The  following  circuit  shown i n Figure 2-1 can be d e s c r i b e d  by the  equations: d q  dq £ + R — - + ± — q <MT dt f (v ) 2  V sincot = L  d q - M § 2  2  1  °1  2  d q dq 1 + R — - + - — q_ - M ^ <MT dt f(v ) &t 2 ?  V sincot = L  (2-1)  2 d q  d  Z  c  The  voltage-capacitance  Transitron SC-5 s i l i c o n  c h a r a c t e r i s t i c , f ( v ) i s that of a capacitor  (Figure 2—2)  and was  obtained  from the manufacturer's s p e c i f i c a t i o n s and d i r e c t measurement. The  frequency of the d r i v i n g s i n u s o i d was chosen t o c o i n c i d e  with that of the' highest 2 x 1 0 r/s. 6  q u a l i t y f a c t o r (Figure 2-3)  at about  An estimate was made of the maximum probable  Figure  2. 2  S i l i c o n Capacitor  C h a r a c t e r i st i c  F i g u r e 2.3  Quality Factor f o r S i l i c o n Capacitors  v a l u e of a l l the v a r i a b l e s and of t h e i r f i r s t and second derivatives.  Amplitude and time s c a l i n g t r a n s f o r m a t i o n s  were then made on (2-1) so that no v a r i a b l e or i t s d e r i v a t i v e s would exceed u n i t y and a l s o so that the o p e r a t i n g would be about two c y c l e s / s e c .  frequency  These operations are necessary -  to prepare an equation f o r s o l u t i o n on an analogue computer. The s i n u s o i d a l f o r c i n g f u n c t i o n was obtained by s o l v i n g (12) a Van der P o l type of equation: x - 10(A  2  - x  2  - (M)  x + co x = 0  (2-2)  2  CO  x(0) = A,  x(0)  =0  This equation has the s o l u t i o n x = A cos cot  (2-3)  and has the p r o p e r t y that i f perturbed, the s o l u t i o n w i l l q u i c k l y r e t u r n t o (2-3), and i t w i l l not decay due to leakage i n the computer c a p a c i t o r s or other n o n - i d e a l f a c t o r s . This can r e a d i l y be seen by an examination  of the s i g n  of  the damping c o e f f i c i e n t i n (2-2).  The c i r c u i t  to  s o l v e 2-2 i s shown i n Pigure 2-4-  The c i r c u i t used to s o l v e  the time and amplitude  used  s c a l e d (2-1) i s shown i n Pigure 2-5.  There were v a r i o u s minor m o d i f i c a t i o n s and a d d i t i o n s to t h i s circuit  t o allow s i g n changes of M/L, i n c l u s i o n of source  impedance, v a r i o u s input d i s t u r b a n c e s , and some p r o t e c t i v e prec a u t i o n s , but these are a l l omitted i n Pigure 2-5 f o r the sake of  clarity.  Figure, 2.4  Forcing Function  Generator  F i g u r e 2.5  S i m i l a t i o n of Basic- Coupled C i r c u i t  10  2-3  D i s c u s s i o n of Analogue Computer Results  (a)  Form of S o l u t i o n s The  r e s u l t s obtained  v e r y u s e f u l i n determining circuit.  on the Pace 231-P- computer proved  component values  f o r the  final 2-6  Examples of s o l u t i o n s are given i n F i g u r e s  through 2-11.  From the r e s u l t s , s e v e r a l f e a t u r e s are apparent.  F i r s t l y , i t can be seen that the s o l u t i o n i s approximately a b i a s s e d s i n u s o i d with the same frequency as the d r i v i n g .function i n most cases. separately. occur. he two 2-1.  Subharmonics w i l l be  discussed  Secondly, i t i s c l e a r that jump resonance  can  This means that f o r the same d r i v i n g f u n c t i o n , there p o s s i b l e s t a t e s of each s i d e of the c i r c u i t This f e a t u r e i s shown i n F i g u r e 2-6  of F i g u r e  where the upper t r a c e  i s the input d r i v i n g f u n c t i o n which i s kept constant,  and  lower t r a c e i s p r o p o r t i o n a l to the v o l t a g e across one  of the  s i l i c o n capacitors. s t a t e s was  The  jump or t r a n s i t i o n between the  the  two  i n i t i a t e d by a small pulse a p p l i e d to a m p l i f i e r  which simulated of F i g u r e 2-1,  a pulse i n the b i a s of one  can  of the  A22  capacitors  thus causing a momentary change i n the average  capacitance. F i g u r e s 2-7,  2-8  2-9  and  show the p o s s i b i l i t y of the  (5) existence  of subharmonics  .  I t was  somewhat d i f f i c u l t  to  i n i t i a t e t h i s mode of resonance, but by v a r i o u s o n - l i n e d i s turbances of the s o l u t i o n , subharmonics of t h i s type could made to e x i s t .  Once i n i t i a t e d , they were maintained  be  indefinitely.  11  However, because of the peaked shape of F i g u r e 2-3, i t i s most u n l i k e l y that any subharmonics could p e r s i s t i n the a c t u a l circuit  of F i g u r e 2-1 because of the g r e a t e r damping at h i g h e r  and lower frequencies-,  a f e a t u r e not i n c l u d e d i n the analogue  model. (b)  v  E f f e c t of V a r y i n g the R e s i s t a n c e The  2-1  c o e f f i c i e n t r e p r e s e n t i n g the r e s i s t a n c e of F i g u r e  was r e a d i l y v a r i e d by changing potentiometers  of F i g u r e 2-5.  Q13 and Q18  I t was found that f o r the f i r s t d e r i v a t i v e  c o e f f i c i e n t value of above about 0.2 there was no resonance.  This value represented  jump  a series resistance  17 ohms i n the c i r c u i t being modelled.  of-about  Progressively smaller  values of t h i s c o e f f i c i e n t gave an i n c r e a s e i n the r a t i o between h i g h and low s t a t e s and a l s o a decrease i n the s w i t c h i n g time  between the two s t a t e s .  However, there was a l s o an i n -  crease i n the time r e q u i r e d f o r the t r a n s i e n t modulation t o damp out.  This f e a t u r e i s shown i n F i g u r e 2-10 which i s the  t r a n s i t i o n between high and low s t a t e s with a damping c o e f f i c i e n t of 0.04. velope  Note that the modulation of the c a r r i e r en-  a f t e r the jump i s very s i m i l a r to the overshoot  underdamped l i n e a r second order system. jump occurs  i n an  Note a l s o that the  i n s i x c y c l e s of the c a r r i e r i n s t e a d of the eleven  c y c l e s i n F i g u r e 2-6 where the damping c o e f f i c i e n t i s 0.05 and the t h i r t e e n c y c l e s i n F i g u r e 2-11 where the damping c o e f f i c i e n t i s 0.07.  Thus i t i s seen that to b u i l d the d e s i r e d  circuit,  12  the l o s s must be kept low enough to allow ferroresonance,  but  not so low as to permit excessive t r a n s i e n t modulation of the output waveform during switching i n t e r v a l s . (c)  E f f e c t of Varying the Coupling One of the main reasons f o r studying a model of the  c i r c u i t (Figure 2-1) on the analogue computer rather than using the a c t u a l c i r c u i t was the ease with which the M/L be v a r i e d .  Potentiometers  between the branches.  r a t i o could  P55 and P56 c o n t r o l l e d the coupling  Amplitude and frequency responses were  obtained f o r various values of the M/l r a t i o .  I t was found  that up to a p o i n t , i n c r e a s i n g t h i s r a t i o increased the region i n which the e n t i r e coupled c i r c u i t could be symmetrically b i s t a b l e ; that i s , when one branch of the c i r c u i t i s i n the high s t a t e , the other w i l l be i n the low one (see Figure  2-11).  However, i f t h i s r a t i o was increased beyond about 0.3, the two branches became t i g h t l y coupled so that they would both be i n e i t h e r the high s t a t e or the low s t a t e - - a symmetrically b i s t a b l e s t a t e was not p o s s i b l e .  Hence i t i s seen that i f the  M/L  r a t i o i s too s m a l l , the c i r c u i t must be very accurately tuned to be b i s t a b l e at a l l , but i f the r a t i o i s too l a r g e the desired symmetrically b i s t a b l e state cannot e x i s t .  An M/L value of 0.1  was found to be a good compromise. -v.  2-4  Summary of the Analogue Simulation Results The s i m u l a t i o n of various c i r c u i t s on the analogue  computer provided i n v a l u a b l e groundwork f o r the chapters to  13  follow.  F i r s t l y , the b a s i c c i r c u i t  was determined.  c o n f i g u r a t i o n t o be used  S e c o n d l y , t h e g e n e r a l shape and f e a t u r e s of '  the output waveform were found.  Knowledge of t h e s e i s  n e c e s s a r y f o r t h e R i t z a n a l y s i s i n the next c h a p t e r .  Also,  study o f t h e analogue c i r c u i t gave the d e s i g n c r i t e r i a f o r M, L, and R t h a t a r e used l a t e r i n t h i s work.  The most i m p o r t a n t  r e s u l t of t h i s s i m u l a t i o n , a l b e i t i n t a n g i b l e , was the i n t u i t i v e u n d e r s t a n d i n g which i t gave of the c i r c u i t .  14 INPUT  INPUT  1/wwvwwwwwwwvwwwwwv RESPONSE  RESPONSE  F i g u r e 2.6  F i g u r e 2.9  INPUT  INPUT  RESPONSE  RESPONSE  111, „  F i g u r e 2.7  . F i g u r e 2.10  INPUT INPUT  AAAAAAAAAAAAAAAAAAAAAAA/^yWWWW  RESPONSE AT X  RESPONSE RESPONSE  'f  F i g u r e 2.8  AT Y  APPLIED  PULSE  F i g u r e 2.11  Sample Analogue Computer S o l u t i o n s  15 3. 3-1  CIRCUIT ANALYSIS AND  DESIGN  Preamble When c o n s i d e r i n g  convenient to deal with  the behaviour of l i n e a r systems, i t i s sinusoidal  which are again s i n u s o i d a l . superposition,  inputs  and t h e i r  As a r e s u l t o f t h e p r i n c i p l e o f  t h e r a t i o b e t w e e n o u t p u t and i n p u t  of frequency i s a b a s i s system behaviour.  outputs  as a  f o r a complete d e s c r i p t i o n  However, i n n o n l i n e a r  function  of the  systems t h i s p r i n c i p l e  (13) c a n n o t be u s e d  ^' and o t h e r m e t h o d s , w h i c h a r e u s u a l l y  a p p r o x i m a t i o n s , must be f o u n d . the  R i t z ^ ^ or R i t z - G a l e r k i n  is  capable of handling a large  the  method w i l l  4  The m a i n d i f f i c u l t y  involved  In the following  analysis,  be u s e d .  T h i s method  v a r i e t y of steady state  cases.  i n u s i n g t h e R i t z method i s t h a t  a p p r o x i m a t e f o r m o f t h e s o l u t i o n must be known o r a s s u m e d .  As a r e s u l t , t h i s method w i l l  n o t f i n d unknown f e a t u r e s  of the  s o l u t i o n s u c h as h i g h e r o r l o w e r h a r m o n i c s i f t h e y a r e n o t included  i n t h e assumed s o l u t i o n .  c o m p u t e r s t u d y show t h a t  The r e s u l t s o f t h e a n a l o g u e  a biassed sinusoid  at the f o r c i n g  f r e q u e n c y i s a good 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 w a v e f o r m , and  s u c h a s o l u t i o n c a n be w e l l a p p r o x i m a t e d w i t h  the R i t z  method. 3-2  The R i t z M e t h o d a s Ap-plied  t o Forced  Oscillations (9)  The R i t z m e t h o d , as s u m m a r i z e d by K L o t t e r on t h e f o l l o w i n g p r i n c i p l e .  Let the d i f f e r e n t i a l  , i s based equation  d e s c r i b i n g a system be: E[q(t)]  =0  (3-D  and r e p l a c e the s o l u t i o n , q ( t ) , by an assumed form: ' m 'q ( t ) = 2 ]  a  ^ (t)  k  k  (3-2)  for a < t < b  k=l where the l | ^ ( t ) are an a p p r o p r i a t e l y chosen s e t of l i n e a r l y k  independent f u n c t i o n s .  Unless q ( t ) i s the exact s o l u t i o n ,  E [ q ( t ) ] w i l l not v a n i s h .  The R i t z c o n d i t i o n s f o r the system  are c o n d i t i o n s on a" such that k  b  J  E [q(t)J  = 0  ^ (t)dt k  (3-3)  (k=l,2,...m)  a 3-3  Development of the R i t z Conditions The equations  to be considered  d e s c r i b e the c i r c u i t of  F i g u r e 2 - 1 and are of the form: EJqJ  = q^ + 2DKg(q ) + 1  Aiq-J  + Mq  2  - p sin  Z =0 (3-4a)  E[q ] 2  = q  + 2DKg(q ) + K f ( q ) 2  2  2  2  + Mq^ - p s i n Z  =0  • (3-4b) where  & = cot.  We can choose our ^  k  to give assumed approximate s o l u t i o n s of  the form:  '  q  = C, + A  .  sin£  - B, cos %  (3~5a)  q  where C  and  1  = C  2  a  r  + A  2  included  e  in the f or g functions. 'q  sin 2  2  i  = -to A  By d e f i n i n g a u x i l i a r y  ±  c  o  s  2  t o allow  i  3  ~  5  b  )  asymmetry  that i = 1,2  (3-6a)  i = 1,2  (3-6b)  ;• i .= 1,2  (3-7a)  + coB^ sin£>  s i n S + </B  (  f o rpossible  I tfollows  = coA c o s 5  q  - ^  cos^  functions .2%  Fj(C.,  A  A . , B.) i  I  f(q .)d3 J  0 ,2jt P  s  ( C  i' V  A  V  = \  f(5i)sin5 dg  i = 1,2 ' ( 3 - 7 b )  f ( q ) c o s S 6.Z  i = 1,2  0 n * 2  P ( C . , A., B.) 1  c  l '  i  1/  l '  (3-7c)  «7  0  1 />2JC  G ^ ( A , B., OO) = i ±  g(q\)d3  i  =  1,2  (3-8a)  ^0  r2it 2JC G-^(A., B., co) = | . i  j '0  g ^ s i n g  d g i = 1,2  .  (3-8b)  g(^_)cosg d £ i = 1,2 0 the R i t z  (3-8c)  conditions:  5  >2JC Z7C  •0  B[qJ  aZ  = 0  1 = 1,2  (3-9a)  I  E [ q j sing  d£  = 0  i = 1,2  (3-9b)  ^  E [ q ] cos g  d£  = 0  i = 1,2  (3-9c)  2ft  0  ±  0  '  give r i s e  to the f o l l o w i n g set of algebraic equations:  P  o  F  1  +  2  D  G  o =0  + 2DG-  1  s F  1  c  s  + 2DG  w h e r e S = p/K The  1  c  -  (3-10a)  2  7  My]  -  Y) A.  7,  1  + yi B. 2  y  2  A. - S  0/  i=l,2  (3-10b)  3  + M r) B .  J  1  2  (3-10c)  =0  j  a n d YJ = to/K.  f(q) characteristic f o r the s i l i c o n capacitors  u s e d may be a p p r o x i m a t e d  ( s e e A p p e n d i x A) b y :  f(q) = q + uq Using  (3-ll)  2  ( 3 - 1 1 ) , we o b t a i n t h e f o l l o w i n g R i t z c o n d i t i o n s  f o r t h e coupled, non-symmetric F F  F  I f we e v a l u a t e F  1  1  o 1  s 1  c  equations: (3-12a)  + 2DG = 0 o 1  + 2DG  1  1=1,2  - Y 7 A . - M y ? A .-s 2  s  J  1  J  2  j  + 2 D G + * B.'+M Y) B. c J I J j 1  2  from equation  2  0  3=1,2  0  (3-12b) (3-12c)  ( 3 - 7 c ) we o b t a i n :  c (3-13)  19 I t i s e v i d e n t now, (i.e. B»0),  t h a t i f we assume n e g l i g i b l e  equation  damping  ( 3 - 1 2 c ) i s s a t i s f i e d f o r B. = B. = 0.  F o r t h i s c a s e , o u r s i m p l i f i e d R i t z c o n d i t i o n s become:  t  i =  1,2  j =  1.2 (3-14b)  i ^ 3 S o l v i n g f o r (1 i n (3-12h) and s u b s t i t u t i n g i n ( 3 - 1 2 a ) equations  of the xc 4  4  + bx  + cx + dxy + e y  2  2  + c y + d x y + ex  2  2  + hy + a  0  (3-15)  + hx + a  0  (3-16)  d = M ^ J  x = A.  MJ9  y = A,  2u  4-2u<  4 -1  b =  = £3  4  J  2 a =  (3-15) and v .[4ur  r  Mr? s 2 2  h =  Making the s u b s t i t u t i o n s x = u + v , subtracting  yields  form:  A + by y" where  (3-14a)  y = u - v and a d d i n g  (3-16) u n c o u p l e s 2  2u< and  t h e e q u a t i o n s and g i v e s :  + 2(b-e) u + c - h j  = 0  (3-17)  2 2 2 2 2 2 2 + 4 u ( r -u ) + b r + c u + d ( 2 u - r ) + e r + h u + a = 0 (3-18)  where r v = 0.  p  o  = u  p  + v .  Equation  (3-17) i s s a t i s f i e d i f  I f v £ 0, s o l v i n g f o r r  i n (3-17) and u s i n g t h i s  value  20  in  (3-16)  g i v e s a p o l y n o m i a l i n powers o f u: u [ - 4 ] + u [ 2 ( d + e - b)] 6  + u [2h]  4  + u  2  [3e  2  - b  + u  [(f -  + ^  [h - c]  + 2bd - 2dc - 2be + 4 a ] .i  2  f).(h  16 L  2  3  - c)]  = 0 (3-19)  J  When v = o, x = y a n d e q u a t i o n  (3-15.) may be s o l v e d  directly.  When v ^ o t h e R i t z c o e f f i c i e n t s a r e d e t e r m i n e d f r o m t h e r o o t s o f (3-19).  3-4  Qualitative To  it  Discussion of  g e t an h e u r i s t i c i d e a  i suseful  equation -1  °i i s zero,  i n front  ative for C  =  + 1 - 2\fk ±v  1  2u  must a l s o  be z e r o b e c a u s e t h e r e i s no  o r dc term.  Thus, o n l y t h e p o s i t i v e  o f t h e r a d i c a l i n (3-20) i s m e a n i n g f u l .  i n ( 3 - 1 4 b ) , and s o l v i n g f o r  1  2  Solving  (3-20)  ( i = 1,2!  sign leads t o extraneous roots. i  solutions,  gives  o s c i l l a t i o n t o cause t h e b i a s sign  o f what t o e x p e c t f o r  t o e x a m i n e ( 3 - 1 4 a ) a n d ( 3 - 1 4 h ) more c l o s e l y .  f o r C. i n t h e f i r s t  Now i f  Solutions  1  Using (3-20), 2 Yj y i e l d s :  o 2, fIil - 2u A. 2 - S  The n e g substituting  (3-21)  r  (1 + M j l )  i  I n t h e s p e c i a l c a s e where M = 0, t h e s e e q u a t i o n s r e d u c e t o (Q)  exactly  t h e same f o r m a s g i v e n b y K l o t t e r  equations.  w  for  uncoupled  F i g u r e 3-1(a) i s a t y p i c a l f r e q u e n c y r e s p o n s e p l o t o f  ( 3 - 2 1 ) f o r M = 0.  F o r H / 0, t h e t e r m l / ( l  + M ^)  expands o r  F i g u r e 3.1  T y p i c a l Frequency Responses  c o n t r a c t s t h i s curve  2 Yj a x i s .  a l o n g the  F i g u r e 3 - 1 ( h ) d e p i c t t h i s f e a t u r e and  The  curves  shown i n  a r e drawn a s s u m i n g t h a t  A^  A. 7-^ i s s l i g h t l y g r e a t e r t h a n - 1 . 3 i A. A. B e c a u s e ( l + M 7—) i s . s m a l l e r t h a n ( l + M -r^-) , t h e c u r v e f o r . J i 2 i = 2, j = 1 i s e x p a n d e d a l o n g t h e YJ a x i s more t h a n t h e c u r v e • i s g r e a t e r t h a n A.  and  that M  A  f o r i = 1,  j = 2.  i s greater than A l W i t h M 7— 2  Thus t h e p o s s i b l e o p e r a t i n g r e g i o n where D  i s i n c r e a s e d as shown i n F i g u r e  A^  3-1(c).  A  p o s i t i v e , the p o s s i b l e o p e r a t i n g r e g i o n i s decreased.  A  3-5  Features  of the  Solutions  A more t h o r o u g h and Ritz coefficients directly. equations.  i s obtained  There are Four are  t h e c a s e x = y.  quantitative determination  The  by s o l v i n g ( 3 - 1 6 ) and  obtained  from the s o l u t i o n of  other twelve  are  ( 3 - 1 9 ) and  the t r a n s f o r m a t i o n r  these  (3-16) f o r  s o l u t i o n s of the 2  the  (3-19)  s i x t e e n root p a i r s r e s u l t i n g from  2 order  of  sixth  2  = u  + v  i n (3-18).  e v e r , b e c a u s e o f t h e symmetry i n v o l v e d , f o r e a c h o f s i x x, s o l u t i o n p a i r s r e s u l t i n g from (3-19) t h e r e solution.  The  equations  nomial  r o o t s , (see Appendix B ) .  f  o  r  The  T h i s method was  and  s, t h e n o r m a l i z e d  t h e c o u p l i n g term M were f o u n d .  w h i c h come f r o m t h e r e a l r o o t s o f  digital  f o u n d t o be  R i t z c o e f f i c i e n t s w e r e d e t e r m i n e d and  ence on t h e d r i v i n g a m p l i t u d e  x  extracting poly-  r e l i a b l e f o r t h i s w o r k t h a n t h e M u l l e r , Newton, o r methods.  y  i s an i d e n t i c a l y,  were s o l v e d on an I.B.M. 7040  computer u s i n g L a g u e r r e ' s m e t h o d ^ " ^ '  How-  more  Bairstow their  frequency  dependYJ ,  Examples of the s o l u t i o n s  ( 3 - 1 6 ) and  (3-19) a r e  given  23 x  X  4-  *  X  X. N  \  \^ \  \  \  M= 0  \  \  1  2  _  —  i  —  .2  1  .4  F i g u r e 3.2  .6  i  1.0  1.2  1-4  1.8  2.0  NORMALIZED FREQUENCY  F r e q u e n c y R e s p o n s e f o r U n c o u p l e d Case  DRIVING  F i g u r e 3*3  1.6  AMPLITUDE  A m p l i t u d e R e s p o n s e f o r U n c o u p l e d Case  •in F i g u r e s  3-2  through 3-8.  Complex r o o t s h a v e no  o f t h e U n c o u p l e d C a s e (M =  3-6  Discussion  (a)  Frequency C h a r a c t e r i s t i c s In Figure  of this,  3-5  nonlinear  0)  ''it i s seen t h a t the  circuit  frequency  i s quite unlike that  l i n e a r c o u n t e r p a r t - t w o s e r i e s r e s o n a n t L-C is  shaped  so t h a t a l t h o u g h t h e  response i s everywhere f i n i t e .  the  circuit  frequency.  can  by'negative slope  be  shown^^'*^  regions  circuits.  i n the  that solutions -|x|  and  the  coupled case, the a 1-1  state  circuit  ( b o t h x and  d r i v i n g t e r m ) ; a 0-0  (b)  un-  opposite  ( b o t h x and  s t a t e ; and  i n phase t o  y s m a l l and  a 0-1  the  i n phase  with  state.  Amplitude C h a r a c t e r i s t i c s Figure  on t h e  3-3  shows t h a t t h e  dependence of the  d r i v i n g amplitude i s also quite u n l i k e that  cuit's l i n e a r counterpart.  The  p o s s i b l e response.  3-2  Again, the u n s t a b l e  shown w i t h b r o k e n l i n e s .  responses of the  dependence i s examined at  f r e q u e n c y t h a t i s known f r o m F i g u r e  are  shown i n  exist i n four d i f f e r e n t states:  y l a r g e and  t h e d r i v i n g t e r m ) ; a 1-0  unstable  I t i s seen then, t h a t f o r the  can  state  driving  These s o l u t i o n s  ones r e s u l t i n g f r o m e x t r a n e o u s r o o t s a r e figures.  mentioned,  characterized  - s plane are  hence cannot e x i s t i n a p h y s i c a l system.  curve  is lossless,  s t a t e at a given  and  broken l i n e s i n the  The  A l s o , as p r e v i o u s l y  e x i s t i n more t h a n one  I t can  response  of i t s  system described  the  p h y s i c a l meaning.  t o h a v e more t h a n and  extraneous  I t i s seen that i f the  cira one  solutions  circuit  is  25 i n t h e 0-0  s t a t e and  the d r i v i n g amplitude i s g r a d u a l l y  increased, the response  i n c r e a s e s g r a d u a l l y u n t i l the  vertical  t a n g e n t p o i n t on t h e l o w e r c u r v e i s r e a c h e d , and t h e n  suddenly  i n c r e a s e s t") t h e a m p l i t u d e and p h a s e g i v e n hy t h e u p p e r ( a 1-1  state;.  From t h e n on, t h e r e s p o n s e  curve  gradually increases  with increasing input amplitude. (c)  Effect  o f Damping  The  effect  of s l i g h t  l o s s i n the r e s o n a n t c i r c u i t can  he  (5)  shown  t o m o d i f y t h e p r e v i o u s l y d i s c u s s e d c u r v e s as d e p i c t e d  i n F i g u r e s 3-4 i n - t h e 0-0  and  3-5.  s t a t e and  the response as b e f o r e .  I t i s seen t h a t i f the c i r c u i t i s a g a i n  the i n p u t amplitude i s g r a d u a l l y i n c r e a s e d ,  i n c r e a s e s . a n d a jump i n a m p l i t u d e and p h a s e o c c u r s I f now  the amplitude i s s l o w l y decreased,  response decreases s l o w l y u n t i l the other v e r t i c a l  the  tangency  p o i n t i s reached at which time the c i r c u i t r e v e r t s to the state. all  I n t h e r e g i o n b e t w e e n t h e two v e r t i c a l  f o u r states are p o s s i b l e .  a m p l i t u d e i s d i s c u s s e d i n 3-8 of  stable  The  tangent  points,  change o f s t a t e w i t h d r i v i n g  as a means o f l i m i t i n g  t h e number  states.  I f t h e c i r c u i t i s more h e a v i l y damped so t h a t no t a n g e n t p o i n t s o c c u r , t h e two different states. 3-7  The  0-0  vertical  s i d e s o f t h e c i r c u i t c a n n o t be i n •  E f f e c t of C o u p l i n g •It i s u s e f u l f i r s t  as shown i n F i g u r e 3-6.  to consider a typical  response  plot  This p l o t i s the e n t i r e s o l u t i o n  from  26  X.Y 6-  4-  -  \ ^ V NV  x  /  yJJ  2-  >  •  . 2.  •  .4  i  •  .6  .8  i  i  l  1.0  1.2  1.4  I  1.6  NORMALIZED  F i g u r e 3.4  i 1.8  I 2.0  FREQUENCY  E f f e c t o f Damping on F r e q u e n c y R e s p o n s e  X,Y  J  I  2  4  F i g u r e 3«5  1—;  6  Effect  1  8  1—:  10  1  1  1  ,  12  14  16  18  DRIVING AMPLITUDE  p-  20  o f Damping o n A m p l i t u d e R e s p o n s e  .  equation  2  7  ( 3 - 1 9 ) as i t v a r i e s w i t h d r i v i n g a m p l i t u d e .  x, y p a i r s shown a r e l a b e l l e d  (x^,y^)  The  through (x^,yg).  six  It  should  be remembered t h a t t h e r e a r e s i x more p a i r s ( x ^ , y ^ )  through  ^ 12'"^12^ k ~k  (xg,y^).  x  The  u  unstable  l i n e s and  "these a r e t h e same as and  through  e x t r a n e o u s s o l u t i o n s a r e a g a i n shown i n b r o k e n  i t should  hence i s the  (x^,y^)  be n o t e d t h a t o n l y j_»y-j_ i s l e f t  o n l y s o l u t i o n of i n t e r e s t . .  frequency  effect  shown.  o f v a r y i n g M i s t h e n s e e n by e x a m i n i n g  p l o t s of F i g u r e s  3~7  and  3-8.  I t i s seen that  coupling w i t h M p o s i t i v e r e s u l t s i n a decreased region p o s s i b l e b i s t a b i l i t y w i t h b o t h s i d e s of the state  (0-0  or l - l ) .  possible symmetrical opposite (0-0  bistability  (0-1  or 1-0).  I t should  be p o s s i b l e t h e n ,  s t a t e s can  greater  i n the  same of  W i t h M negative,, the region  region i s  t o have the  at a p o i n t such t h a t o n l y the d e s i r e d  the•  of  Also, i t results i n a greater region  or l - l ) i s i n c r e a s e d whereas the s y m m e t r i c a l  operate  3-8  circuit  e f f e c t i s observed, that i s the unsymmetrical  decreased.  and  In the f o l l o w i n g d i s -  c u s s i o n o n l y t h e s o l u t i o n s o f i n t e r e s t w i l l be The  solid  x  circuit  symmetrical  exist.  S o u r c e Impedance Consider  Considerations  the e f f e c t of p l a c i n g a c a p a c i t o r , C  i n series s  w i t h the was  c a r r i e r source  of the  previously discussed.  reased  circuit  shown i n F i g u r e 2-1  Let the source  s u c h t h a t when t h e c i r c u i t  v o l t a g e , V,  i s i n e i t h e r t h e 0-1  s t a t e , t h e v o l t a g e a t A i s t h e same as i t was  before  be or the  which inc1-0  - = - = --Jt--zje-- -.a - j - - w - - -1-2 - -  <  DRIVING AMPLITUDE  —«r  +  7*7-  J:6___]-8__ 2"° x 6  M = 0.2 rj = 0.6 6  1  UU~/-'--„-_. Y  Y  UNSTABLE :  X  E X T R A N E O U S :.  F i g u r e 3.6  3  Y , ^  Y , X5Y  3  X  2  4  Y , X 2  6  ;  5  Y . 6  E n t i r e Approximate S o l u t i o n f o r E q u a t i o n 3.17 ro oo  29 i n s e r t i o n of the c a p a c i t o r . t o go i n t o t h e 0-0 source  I f t h e c i r c u i t were now  s t a t e , l e s s c u r r e n t w o u l d he d r a w n f r o m t h e  and t h e v o l t a g e d r o p a c r o s s  C  would decrease.  v o l t a g e a t A w o u l d i n c r e a s e a n d , as c a n he s e e n f r o m 3-5, i t w o u l d t e n d 1-0  state.  to attempt  to force the c i r c u i t  Hence t h e Figure  h a c k i n t o t h e 0-1 o r  S i m i l a r l y , i f t h e c i r c u i t w e r e t o a t t e m p t t o go  i n t o t h e 1-1 s t a t e , t h e v o l t a g e a t A w o u l d d e c r e a s e and t h e c i r c u i t w o u l d a g a i n be f o r c e d b a c k t o t h e 0-1 o r 1-0 This series  state.  c a p a c i t o r c a n t h u s be u s e d t o i n c r e a s e t h e p r e -  viously discussed  e f f e c t of p o s i t i v e  coupling.  30 X,Y 6-j  \  4M »  \  0.8 \  \ Q2\  \ 0l0  \ \-Q2  -0.8  NORMALIZED  F i g u r e 3.7  Effect  14  Effect  2.0  o f C o u p l i n g on I d e n t i c a l S o l u t i o n  NORMALIZED  F i g u r e 3.8  FREQ.  ^  1  1.6  Pairs  FREQUENCY 1.8  o f C o u p l i n g on N o n - i d e n t i c a l  2D  Solution  Pairs  31 4. 4-1  EXPERIMENTAL RESULTS Ba^'ic C i r c u i t C o n f i g u r a t i o n A basic circuit  as shown i n P i g u r e 4-1  The n o n l i n e a r c a p a c i t o r s i n t h e r e s o n a n t T r a n s i t r o n SC-5  a l s o i n the approximate  analogue study  As t h e s e c a p a c i t o r s a r e a t y p e o f d i o d e ,  c a r r i e r s i g n a l s o u r c e was  o f t h e Q - c u r v e ( P i g u r e 2-3)  kilohm  they  resistor.  a G e n e r a l R a d i o Type 1001-A  s t a n d a r d r a d i o s i g n a l g e n e r a t o r w h i c h was a t 350 k H z .  o p e r a t e d n e a r t h e peak The  t r a n s f o r m e r were wound on S i e m e n s S i f e r r i t w h i c h h a v e a n a d j u s t a b l e a i r gap, be v a r i e d .  f o r the  analytical  w e r e b a c k - b i a s s e d a t 5 v o l t s t h r o u g h a 100 The  the  s i l i c o n c a p a c i t o r s t h a t have been p r e v i o u s l y  c o m p u t e r s t u d y and 3.  constructed.  c i r c u i t s are  d i s c u s s e d and whose c h a r a c t e r i s t i c s were u s e d  of Chapter  was  i n d u c t o r s and  f e r r i t e pot  cores  p e r m i t t i n g the inductance to  T h i s f e a t u r e i s n e c e s s a r y t o tune the  resonant  c i r c u i t s t o t h e o p e r a t i n g f r e q u e n c y and a l s o t o compensate, f o r v a r i a t i o n s i n c a p a c i t a n c e between d i f f e r e n t 4-2  silicon  capacitors.  O u t p u t Waveform The  o u t p u t w a v e f o r m s were f o u n d t o be v e r y n e a r l y b i a s s e d  s i n u s o i d s a t t h e same f r e q u e n c y as t h e c a r r i e r s o u r c e , v a l i d a t i n g the use was  used, i n C h a p t e r  of the approximate 3.  thus  form of the s o l u t i o n t h a t  U n l i k e t h e r e s u l t s shown by t h e  s t u d y , h i g h e r h a r m o n i c and  subharmonic responses  e x i s t e n t o r s o s l i g h t t h a t any  analogue  were e i t h e r non-  e f f e c t t h e y m i g h t h a v e had  on  •32  the  solutions  circuit  i snegligible.  actual circuit slightly  i n C h a p t e r 3 o r on t h e a c t u a l o p e r a t i o n  T h i s apparent d i s c r e p a n c y between t h e  a n d i t s a n a l o g u e m o d e l was due t o t h e u s e o f a  s i m p l i f i e d model.  I t i s the s i m p l i f i c a t i o n ofthe  f r e q u e n c y c h a r a c t e r i s t i c s o f t h e analogue model w h i c h the  actual  ofthe  circuit  t o a c t more p r e d i c t a b l y  causes  than t h e analogue  model ( s e e Chapter 2). 4-3  Power C o n s u m p t i o n An  a p p r o x i m a t e measurement o f t h e s t e a d y s t a t e  consumption of t h e basic  power  c i r c u i t was made b y c o n s i d e r i n g t h e  whole u n i t as a b l a c k - b o x , measuring t h e d r i v i n g v o l t a g e , and f i n d i n g the current  t h a t was i n p h a s e w i t h  manner:  The v o l t a g e  vertical  a x i s o f an o s c i l l o s c o p e  the  current  and be  a c r o s s t h e u n i t was d i s p l a y e d  was d i s p l a y e d  resulting Lissajous  on t h e h o r i z o n t a l a x i s .  f i g u r e the in-phase current  l e s s than t e n microwatts.  Is a result of t h i s  circuit  a c t i v e components.  The p o r t i o n  bias 4-4  Prom t h e  was d e t e r m i n e d  c i r c u i t was c a l c u l a t e d t o This extremely low l o s s  c o n s i s t i n g almost e n t i r e l y of r e of the loss a t t r i b u t a b l e tothe  r e s i s t o r was l e s s t h a n one m i c r o w a t t . Bistability The  circuit  was r e a d i l y made s y m m e t r i c a l l y  adding a series capacitor of C  on t h e  and a s i g n a l p r o p o r t i o n a l t o  t h e power d i s s i p a t e d b y t h e b a s i c slightly  i t i n the following  was c h o s e n s u c h t h a t  as d i s c u s s e d the voltage  b i s t a b l e by  i n s e c t i o n 3-8. at point  The v a l u e  A of Figure  4-1  33 was s u f f i c i e n t  t o permit  s i g n a l generator  were t u n e d  output.  to the operating  b y v a r y i n g t h e a i r gap i n i n d u c t o r s  the c i r c u i t  a n d L^.  With  a d j u s t e d i n t h i s manner, t h e u n i t was s y m m e t r i c a l l y  b i s t a b l e w i t h the amplitude f o u r times  b i s t a b i l i t y when t h e  was a t t h r e e - q u a r t e r s o f i t s maximum  Both s i d e s of t h e c i r c u i t frequency  symmetrical  of the h i g h resonant  s t a t e about  t h a t o f t h e l o w one a s shown b y t h e c e n t e r two t r a c e s  of Pigure 4-3• 4-5  Use a s a Memory  Device  To u s e a b a s i c c i r c u i t n e e d e d i s a means o f c h a n g i n g  a s a memory d e v i c e , a l l t h a t i s i t s state.  done by a p p l y i n g a n e g a t i v e . p u l s e t h r o u g h o r Y" i n F i g u r e 4 - 1 . and  a diode  at point  I f side X i s i n the high resonant  X"  state  s i d e Y i s i n t h e l o w s t a t e , a n e g a t i v e p u l s e a t X" mom-  e n t a r i l y i n c r e a s e s t h e b i a s on SC5-X. reduces i t s average capacitance the low s t a t e , a n d :  This increase i n bias  and causes s i d e X t o drop  into  b e c a u s e t h e c o u p l i n g and s e r i e s c a p a c i t o r  require that the c i r c u i t  be s y m m e t r i c a l l y b i s t a b l e a s d e s c r i b e d ,  side Y i s forced into the high state. at  This i s r e a d i l y  X" t h e n h a v e no f u r t h e r e f f e c t  To r e s t o r e t h e c i r c u i t  Successive  pulses  on t h e s t a t e o f t h e c i r c u i t .  to i t s o r i g i n a l state, a negative  pulse  must be a p p l i e d a t Y". 4-6  Use a s a C o u n t e r The c i r c u i t  c o n f i g u r a t i o n used t o demonstrate  counter  34  o p e r a t i o n o f t h e d e v i c e i s shown i n F i g u r e 4-2.  Successive  pulses applied at input point A reverse the state of the circuit  i n a manner t o be d i s c u s s e d i n S e c t i o n 4-7.  The  s w i t c h i n g p u l s e s w e r e o b t a i n e d by p a s s i n g a s q u a r e wave capacitor  D3.  and d i o d e  t h i s c i r c u i t w i t h diode t h e same v e r t i c a l  through  F i g u r e 4-3 shows t h e o p e r a t i o n o f  D3 s h o r t e d  out.  A l l the traces are at  scale of 1 v o l t / d i v i s i o n .  The u p p e r t r a c e  shows t h e i n p u t t r i g g e r i n g p u l s e s a t p o i n t A, t h e n e x t  two  t r a c e s show t h e c a r r i e r e n v e l o p e a t p o i n t s X and Y a n d t h e lower  t r a c e i s of the output  p u l s e s a t p o i n t Y' w h i c h ' a r e •  obtained., b y r e c t i f y i n g t h e s i g n a l a t Y w i t h d i o d e d i f f e r e n t i a t i n g i t w i t h C^, R^. w i t h D3 s h o r t e d  I t s h o u l d be n o t e d  o u t , b o t h p o s i t i v e and n e g a t i v e  p o i n t A, b u t o n l y t h e n e g a t i v e  D2 and  then  here that  pulses  reach  pulses trigger the c i r c u i t .  T h i s e f f e c t i s due t o t h e way t h e c i r c u i t was t u n e d and w i l l be d i s c u s s e d i n t h e n e x t s e c t i o n . The s e n s i t i v i t y pulse amplitude  of the c i r c u i t  to v a r i a t i o n s i n input  and d u r a t i o n was d e t e r m i n e d by r e p l a c i n g t h e  s q u a r e wave g e n e r a t o r  and c a p a c i t o r  Type 1217A p u l s e g e n e r a t o r .  with a General  Radio  The minimum p u l s e d u r a t i o n t h a t .  w o u l d r e l i a b l y t r i g g e r t h e c i r c u i t was a b o u t 5 m i c r o s e c o n d s o r n e a r l y two c o m p l e t e c y c l e s o f t h e c a r r i e r s i g n a l . n o t be r e a s o n a b l e  t o expect  d u r a t i o n of l e s s than  the c i r c u i t  I t would  t o respond to a pulse  one c o m p l e t e c y c l e b e c a u s e t h e t r a n s i e n t  r e s p o n s e w o u l d t h e n depend on what p o r t i o n o f t h e c y c l e was disturbed.  H o w e v e r , i f more t h a n  one o r s e v e r a l c y c l e s a r e  P i g u r e 4.1  Memory D e v i c e  P i g u r e 4.2  Counter Device  Configuration  Configuration  36 d i s t u r b e d by a p u l s e , t h e c i r c u i t s t a t e and w i l l  can a t t a i n a pseudo-steady  a c t as d e s c r i b e d i n t h e n e x t s e c t i o n .  The m i n -  imum a m p l i t u d e p u l s e t o w h i c h t h e c i r c u i t w o u l d r e s p o n d was a b o u t 0.15 v o l t s .  reliably  T h i s minimum a m p l i t u d e o f p u l s e i s a l s o  d e p e n d e n t on t h e c i r c u i t t u n i n g i n a manner t o be d i s c u s s e d i n 4-7.  Section  •To t r i g g e r s u c c e s s i v e c i r c u i t s , e i t h e r X' o r Y' o f F i g u r e 4-2 unit. units.  the output /pulses from  were u s e d a t p o i n t A o f t h e n e x t  F i g u r e 4 - 4 . s h o w s t h e e n v e l o p e a t X and Y o f two c o n s e c u t i v e The u p p e r two t r a c e s a r e o f " t h e f i r s t  c i r c u i t which i s  t r i g g e r e d b y n e g a t i v e p u l s e s f r o m a d i f f e r e n t i a t e d and r e c t i f i e d s q u a r e wave a s b e f o r e , a n d t h e l o w e r two t r a c e s a r e o f the  s e c o n d u n i t w h i c h i s t r i g g e r e d by o u t p u t p u l s e s as shown i n  the  l o w e r t r a c e o f F i g u r e 4-3-  Note t h e i r r e g u l a r i t i e s i n the  e n v e l o p e o f t h e s e c o n d u n i t t h a t o c c u r when t h e f i r s t switches. the  unit  These a r e a r e s u l t o f t h e 50 ohm o u t p u t i m p e d a n c e o f  s i g n a l g e n e r a t o r which causes f l u c t u a t i o n s i n i t s output  amplitude at switching i n s t a n t s .  An u n d e s i r a b l e outcome o f  t h i s e f f e c t i s t h e p o s s i b l e o c c u r r e n c e o f unwanted  switching  w h i c h was d e m o n s t r a t e d b y p l a c i n g a r e s i s t o r i n s e r i e s w i t h t h e g e n e r a t o r t o i n c r e a s e t h e o u t p u t i m p e d a n c e as s e e n by t h e c i r cuit.  A l t h o u g h t h e impedance o f t h e g e n e r a t o r i t s e l f  was n o t  s u f f i c i e n t to adversely a f f e c t the prototype u n i t s being studied, i f many s u c h u n i t s were b e i n g d r i v e n by a s i n g l e o s c i l l a t o r as would o c c u r i n a l a r g e r s c a l e computing d e v i c e , i t would become more i m p o r t a n t t h a t t h e o s c i l l a t o r be o f l o w o u t p u t  Figure  4.3  F i g u r e 4.6  Operation  of Counter  S w i t c h i n g Waveform  impedance. 4-7  Switching It  i s n o t t h e p u r p o s e h e r e t o a t t e m p t a. r i g o r o u s  cussion of the t r a n s i e n t behaviour  of the c i r c u i t .  dis-  I n s t e a d , an  i n t u i t i v e d i s c u s s i o n b a s e d on t h e p r e v i o u s l y d e v e l o p e d a n a l y t ical results w i l l state reversal.  be g i v e n i n o r d e r  t o e x p l a i n the process of  Suppose t h a t i n i t i a l l y t h e c i r c u i t  of Figure  4-2 h a s s i d e Y i n t h e h i g h s t a t e a n d s i d e X i n t h e l o w s t a t e as d e p i c t e d on t h e s o l i d  curves  o f F i g u r e 4-5a.  A  negative  pulse applied t o the nonlinear capacitors increases the bias, reducing frequency  t h e average capacitance, and a m p l i t u d e  l i n e s i n F i g u r e 4-5a.  and t h u s s h i f t i n g t h e  response curves  a s shown b y t h e b r o k e n  At t h i s i n s t a n t , t h e only s t a b l e con-  f i g u r a t i o n i s t h e 0-0 s t a t e and s o Y s t a r t s t o d e c r e a s e . c a u s e l e s s c u r r e n t i s b e i n g drawn t h e v o l t a g e a c r o s s and  the d r i v i n g amplitude  begins  and  Y t o move a s i n d i c a t e d .  When t h e p u l s e i s removed i t i s s e e n r e g i o n and d e c r e a s i n g .  Because l e s s c u r r e n t i s b e i n g drawn by t h e c i r c u i t , capacitor C  drops  to increase, causing points X  ( F i g u r e 4-5b) t h a t Y i s now i n a n u n s t a b l e  drop across  Be-  i s decreased, causing  the voltage  the driving  s amplitude  t o be i n c r e a s e d , and t h u s c a u s i n g X t o i n c r e a s e .  B e c a u s e t h e r e g i o n i s u n s t a b l e , Y must c o n t i n u e X i s forced past  t o decrease  until  t h e v e r t i c a l t a n g e n c y p o i n t , T-^, on t h e a m p l i -  tude response curve  (Figure 4-5h(2)),  and t h e c i r c u i t a t t a i n s  t h e complement o f i t s o r i g i n a l s t a t e ( F i g u r e 4 - 5 c ) .  The n e x t  Output Amplitude  Output Amplitude  Output Amplitude  v>i  pulse causes a repeat  of t h e p r o c e s s .  evident that the amplitude the  circuit  Tu.  From F i g u r e 4-5a  of the p u l s e r e q u i r e d t o  i s d e p e n d e n t on how  v e r t i c a l tangency p o i n t than X i s to the lower a negative  c u r v e up t h e f r e q u e n c y r e a d i l y than w i l l curve  one,  axis.  By  use  low  preceding  H o w e v e r , i t c a n be ( F i g u r e s 2-6,  positive pulses  a  the  pulses.  d i s c u s s i o n uses the r e s u l t s of the  and  2-11)  and  a  Ritz  transient-analysis. results  a l s o the r e s u l t s from the a c t u a l  t h a t the s w i t c h i n g process  takes  place  s e v e r a l c y c l e s o f t h e c a r r i e r s i g n a l so t h a t a t  p a r t i c u l a r i n s t a n t t h e c i r c u i t may The  be  quasi-steady  state.  ( F i g u r e 4-6)  t a k e s p l a c e o v e r a b o u t 10  t h a t was  considered  unit KHz  e x h i b i t s t h e same t y p e  •  from  any  in a  c y c l e s o f t h e 350  apparent i n r e s u l t s obtained  analogue computer s t u d y .  t o be  s w i t c h i n g of the p r o t o t y p e  c a r r i e r s i g n a l as d e s c r i b e d a b o v e and overshoot  at  for  between  seen from the analogue computer  ( F i g u r e 4-6)  Slowly over  frequency  s t a t e s i s l e s s i n t h i s r e g i o n so i t i s b e t t e r t o  a n a l y s i s which i s a s t e a d y - s t a t e - not  circuit  more  o p e r a t i n g the c i r c u i t  i t i s p o s s i b l e t o use  t h e f i r s t method w i t h n e g a t i v e The  shown  frequency  t r i g g e r the c i r c u i t  s w i t c h i n g , However, the d i f f e r e n c e i n a m p l i t u d e h i g h and  as  a p o s i t i v e pulse which s h i f t s the  down t h e f r e q u e n c y  higher frequency  upper  pulse which s h i f t s the  axis w i l l  trigger  f a r the o p e r a t i n g p o i n t i s from  A l s o , i t i s seen that i f Y i s c l o s e r t o the  i n F i g u r e 4-5a,  i t is  the  of  4-8  Summary This chapter  prototype  has d e a l t w i t h t h e c o n s t r u c t i o n o f a  low l o s s b i s t a b l e c i r c u i t  properties of a conventional ferroresonant  circuits  flip-flop.  I t was shown how  c o u l d be u s e d a s memory d e v i c e s o r  cascaded t o form counters. circuit  t h a t h a s many o f t h e  Results obtained  from t h e a c t u a l  compared f a v o r a b l y w i t h a n a l o g u e a n d a n a l y t i c a l s o l u -  t i o n s and t h e u s e o f t h e assumed a p p r o x i m a t e s o l u t i o n u s e d i n t h e R i t z a n a l y s i s was shown t o be j u s t i f i e d . d e s c r i p t i o n of the s w i t c h i n g process  Finally,  was g i v e n w h i c h c o m b i n e d  a consideration of the R i t z r e s u l t s with observable o f t h e a n a l o g u e and c i r c u i t  a  s w i t c h i n g waveforms.  features  42 5.  CONCLUSIONS  The p u r p o s e o f t h i s w o r k was t o s t u d y phenomena a r d t o i n v e s t i g a t e t h e p o s s i b i l i t y p r i n c i p l e i n the design  of d i g i t a l l o g i c  jump r e s o n a n c e of using the  elements.  The s y s t e m  c h o s e n f o r s t u d y was d e s c r i b e d by a p a i r o f n o n l i n e a r order d i f f e r e n t i a l  equations  w h i c h were c o u p l e d  second  by second  d e r i v a t i v e t e r m s a n d were d r i v e n by a s i n u s o i d a l f o r c i n g function.  K l o t t e r ' s w'ork.on t h e R i t z method o f a n a l y s i s was  extended t o study  t h e p r e v i o u s l y mentioned coupled  with asymmetrical  nonlinearities.  to  symmetrical  equations  K l o t t e r ' s w o r k was l i m i t e d  n o n l i n e a r i t i e s in- equations  w i t h c o u p l i n g and  t h e c o u p l i n g was o n l y i n t h e d e p e n d e n t v a r i a b l e .  The a n a l y s i s  y i e l d e d a l g e b r a i c c o n d i t i o n s w h i c h w e r e s o l v e d on a d i g i t a l computer t o o b t a i n frequency  and a m p l i t u d e  responses of the  model. The r e s u l t s those  obtained  o f t h e above s t u d y  from the study  compared f a v o r a b l y w i t h  o f a m o d e l on a n a n a l o g u e computer,,  and  i n f o r m a t i o n f r o m t h e two s t u d i e s was u s e d i n t h e d e s i g n  and  construction of a prototype  low  and h i g h r e s o n a n t  represent type  b i s t a b l e c i r c u i t which used  s t a t e s o f jump r e s o n a n t  a 0 and 1 b a s i s f o r l o g i c a l  circuits to  operations.  The p r o t o -  c i r c u i t s h a d many o f t h e f e a t u r e s o f c o n v e n t i o n a l  flip-  f l o p s s u c h a s b e i n g a b l e t o s t o r e a n d c o u n t b i n a r y numbers a n d drive other u n i t s . they  However, u n l i k e c o n v e n t i o n a l  r e q u i r e d an ac d r i v i n g source  flip-flops,  as w e l l as a dc b i a s  supply.  43 The  dc b i a s s u p p l y  c o u l d he e l i m i n a t e d f r o m t h e c i r c u i t  the advent o f n o n l i n e a r barium t i t a n a t e . have a n o n l i n e a r  capacitors with a d i e l e c t r i c  characteristic.  Another problem inherent  and  These f e r r o e l e c t r i c  I t was v e r y  devices  amplifiers.  i n the device  s t u d i e d was t h e  d e p e n d e n t on t h e c a r r i e r  a l s o on t h e l o s s o f t h e c i r c u i t .  computer study  such as  S u c h c a p a c i t o r s w o u l d be o f l o w l o s s , a n d  have been s u c c e s s f u l i n p a r a m e t r i c  s w i t c h i n g time.  with  frequency  I t was shown b y t h e a n a l o g u e  t h a t s w i t c h i n g would n o t occur r e l i a b l y i n l e s s  t h a n f i v e c y c l e s o f t h e c a r r i e r s i g n a l and t h a t t e n c y c l e s w o u l d be more u s e f u l a s a d e s i g n minimum.  The i m p l i c a t i o n  h e r e i s t h a t t h e d r i v i n g s o u r c e must c o n s i s t o f a n o s c i l l a t i o n a t a f r e q u e n c y a t l e a s t t e n t i m e s t h e d e s i r e d maximum s w i t c h i n g rate.  The c u t o f f f r e q u e n c y o f t h e n o n l i n e a r  maximum o p e r a t i n g  capacitor sets the  frequency of the resonant c i r c u i t  t h e -maximum s w i t c h i n g s p e e d .  and t h e r e f o r e  F o r low speed a p p l i c a t i o n s con-  v e n t i o n a l l u m p e d e l e m e n t s c a n be c o n n e c t e d t o g e t h e r c i r c u i t s s u c h a s was done w i t h t h e p r o t o t y p e s  t o form  described  i n this  report, but f o r higher  operating frequencies  increasingly difficult  t o contain the s i g n a l s i n or near the  e l e m e n t s and l i n e s . to use conventional  logic  i t w o u l d become  One way o f a v o i d i n g t h i s p r o b l e m w o u l d be microwave d e v i c e s  waveguides and r e s o n a n t c a v i t i e s . l a r g e a n d cumbersome.  such as c o a x i a l c a b l e s ,  However, t h e s e a r e q u i t e  A n o t h e r p o s s i b l e way o f i n c r e a s i n g t h e  s w i t c h i n g s p e e d w o u l d be t o make t h e s i z e o f t h e d e v i c e  small  44 compared t o one w a v e l e n g t h . is  still  workers  The  u n d e r d e v e l o p m e n t and i n that f i e l d  developed  technique of m i c r o m i n i a t u r i z a t i o n  i t w o u l d be u s e f u l f o r f u t u r e  to c o n s i d e r m i n i a t u r i z i n g the  circuits  here.  The m a i n i n c e n t i v e f o r f u t u r e w o r k a l o n g t h e s e l i n e s i s --the i n h e r e n t l o w power c o n s u m p t i o n  of the d e v i c e s .  B e c a u s e most  o f t h e components needed a r e of a r e a c t i v e n a t u r e , l o s s i s p r i m a r i l y due  t o c a p a c i t o r l e a k a g e and  the i n d u c t i v e elements.  copper  and  "Devices of the type developed  w o u l d f i n d a p p l i c a t i o n i n s i t u a t i o n s where l o g i c a l a r e r e q u i r e d and w h e r e l o w power c o n s u m p t i o n I n summary, t h e w o r k p r e s e n t e d h e r e 1.  core l o s s i n here  operations  i s necessary. includes:  E x t e n d i n g p r e v i o u s w o r k on t h e R i t z method o f  analysis  so as t o p e r m i t s t u d y o f c o u p l e d n o n l i n e a r d i f f e r e n tial 2.  e q u a t i o n s which have a s y m m e t r i c a l  An a n a l o g u e  computer s t u d y of a model of a  ferroresonant 3.  The  and  bistable  circuit.  d e s i g n and  . circuit,  nonlinearity.  c o n s t r u c t i o n o f a p r o t o t y p e o f t h e above  some s u g g e s t i o n s f o r f u t u r e s t u d y .  • -.  45  j  APPENLIX A  Polynomial Approximation  t o the f(q) Characteristic f o r the  Silicon  Capacitor  The v o l t a g e - c a p a c i t a n c e  characteristic  ofthe  T r a n s i t r o n SC-5 s i l i c o n c a p a c i t o r s ( F i g u r e 2-2) was t a b u l a t e d •and u s i n g t h i s d a t a a p o l y n o m i a l in  approximation  ( 3 - 4 ) was made u s i n g a l e a s t s q u a r e s f i t t i n g  to the f(q) procedure.  A b r i e f o u t l i n e of that procedure f o l l o w s . The c h a r g e on t h e c a p a c i t o r c a n be d e s c r i b e d b y q = v-C(v)  (A-l)  The v o l t a g e v , a n d t h e c o r r e s p o n d i n g  c a p a c i t a n c e , C ( v ) , were  f o u n d a t 52 p o i n t s i n t h e v o l t a g e r a n g e 0 t o -30 v o l t s . Quadratic  a n d c u b i c l e a s t s q u a r e s f i t s were made a t -5 v o l t  bias to v = f(q) = a  Q  + a-^q + a q 2  + ....  (A-2)  The r e s u l t s a r e t a b u l a t e d i n F i g u r e s A - l , A - 2 , a n d T a b l e A - l . B o t h t h e q u a d r a t i c and t h e c u b i c a p p r o x i m a t i o n s istic  to the character-  a r e w e l l w i t h i n t h e 20% v a r i a t i o n among components  that i s claimed  by t h e m a n u f a c t u r e r .  r a t i c approximation  As a r e s u l t , t h e q u a d -  i s used i n t h e a n a l y s i s i n Chapter I I I .  Coefficient  a  o  a  ^2  l  a  Standard Deviation  3  Approximation Quadratic  1.65 x I O "  1  2.30 x 1 0 "  2  2.68 x 1 0 ~  5  Cubic  1.43 x 1 0 "  1  2.32 x 1 0 "  2  2.70 x 1 0 ~  5  Table A . l  0.1737 -8.6  x IO"  1 0  0.1722  48 . APPENDIX B  Determination  Pigure  of the R i v z C o e f f i c i e n t s  B - l I s the flow diagram d e p i c t i n g the  cedures f o l l o w e d i n determining coefficients  the dependence of the R i t z  on t h e v a r i o u s p a r a m e t e r s o f i n t e r e s t .  gram i s q u i t e v e r s a t i l e and  p o r t i o n s of i t are  L a g u e r r e ' s method o f f i n d i n g p o l y n o m i a l , s t a r t i n g value  s t e p , but  w i t h no  pro-  (15)  roots  gives  quick  for distinct roots.  L a g u e r r e s t e p r e q u i r e s more c a l c u l a t i o n t h a n one or Bairstow  The  self-checking.  '  c o n v e r g e n c e f r o m any  pro-  One  M u l l e r , Newton,  a p r i o r i approximation  to  the  z e r o e s i t more t h a n c o m p e n s a t e s f o r t h i s by t h e r e d u c t i o n t h e number o f i t e r a t i o n s n e e d e d . vergence i s cubic; otherwise  I f the r o o t i s s i m p l e ,  i t is linear.  The  actual  in con-  Fortran  p r o g r a m u s e d i s a m o d i f i e d f o r m o f t h e p r o g r a m "LAG-ERE" w h i c h (l6) was  w r i t t e n by  to the  J . Stevens  computing center's  i n 1966 library.  and  The  t h i s p r o g r a m a l l o w i t t o be u s e d t o f i n d d e g r e e o f a c c u r a c y and the  and  by  s i n c e b e e n added  m o d i f i c a t i o n s made t o small roots with a  These m o d i f i c a t i o n s were  the p r o p e r t i e s of polynomials  (3-13) and  the requirement that the r o o t - f i n d i n g technique  operative  high  a l s o t o c h e c k i t s a c c u r a c y by r e c o n s t r u c t i n  c o e f f i c i e n t s from the r o o t s .  n e c e s s i t a t e d by  has  over a wide range of c o e f f i c i e n t  values.  (3-17) be  ( START ) DIMENSION ETC. INPUT .PARAMETERS/ SOLVE FOURTH ORDER POLYNOMIAL FOR IDENTICAL ROOTS  FORM AUXILIARY EQUATIONS AND DETERMINE COEFICIENTS OF POLYNOMIALS SOLVE SIXTH ORDER POLYNOMIAL FOR NONIDENTICAL ROOTS  ORDER AND OUTPUT Xj.Yj PAIRS  ISOLATE COMPLEX ROOTS  YES  RECONSTRUCT POLYNOMIAL COEFICIENTS TEST: WITHIN REQUIRED TOLERANCE NO  TEST: WITHIN REQUIRED TOLERANCE  YES  DETERMINE Xj.Yj PAIRS F R O M SOLUTIONS  OUTPUT ERROR .MESSAGE,  NO ISOLATE COMPLEX ROOTS  OUTPUT ERROR .MESSAGE,  NO  TEST: DESIRED RANGE OBTAINED  ORDER AND OUTPUT Xj,Yj PAIRS INCREMENT PARAMETERS  YES (  Figure B . l  STOP  )  Flow Diagram o f Program t o Determine R i t z  Coefficients  50 REFERENCES  1.  R o u e l l e , E., " C o n t r i b u t i o n a. 1 ' e t u d e e x p e r i m e n t a l e de l a f e r r o - r e s o n a n c e " , Revue G-enerale de 1 ' E l e c t r i c i t e , 36: pp. 7 1 5 - 7 3 8 , 7 6 3 - 7 8 0 , 7 9 5 - 8 1 9 , 8 4 1 - 8 5 8 , 1934. .  2.  R u d e n b e r g , R., T r a n s i e n t P e r f o r m a n c e o f E l e c t r i c S y s t e m s . M c G r a w - H i l l , 1950.  3.  L e n k u r t E l e c t r i c Co., S e l e c t e d A r t i c l e s f r o m The L e n k u r t D e m o d u l a t o r (Second E d i t i o n ) , "The V a r a c t o r D i o d e " , pp. 667-677, 1966.  4.  S t o k e r , J . , N o n l i n e a r V i b r a t i o n s . N.Y.,  5.  Hayashi, C , Nonlinear O s c i l l a t i o n s i n Physical M c G r a w - H i l l , 1964-  6.  Isborn, C ,  "Ferroresonant F l i p - f l o p s " , A p r i l 1962, pp. 121-123-  7.  Gremer, C ,  "The N o n l i n e a r R e s o n a n t T r i g g e r P a i r " , T r a n . A I E E . C o m m u n i c a t i o n s and E l e c t r o n i c s , V o l . 26, S e p t . 1 9 5 6 , pp. 4 0 4 - 4 0 7 .  8.  Ozawa, T.,  N o n l i n e a r R e s o n a n c e Computer Components, T e c h . R e p o r t No. 1 3 0 6 - 1 , S t a n f o r d E l e c t r o n i c s L a b o r a t o r i e s , A p r i l 1963-  9.  K L o t t e r , K., " S t e a d y S t a t e V i b r a t i o n s i n S y s t e m s H a v i n g A r b i t r a r y R e s t o r i n g and A r b i t r a r y Damping F o r c e s " , P r o c . o f t h e Symp. on N o n l i n e a r C i r c u i t A n a l y s i s , V o l . 2, P o l y t e c h n i c I n s t i t u t e of- B r o o k l y n , N.Y., pp. 2 3 4 - 2 5 7 , 1 9 5 3 .  1 0 . K L o t t e r , K., " S t e a d y Multi-Loop Engineers, No. 4, pp.  Power  1950. Systems,  Electronics,  State O s c i l l a t i o n s i n Nonlinear C i r c u i t s " , Trans. I n s t , of Radio P r o f . Group on C i r c u i t T h e o r y , CT-1, 1 3 - 1 8 , Dec. 1954.  1 1 . • Z e n i t i , K., S e k i g u t i , S., and T a k a s i m a , M., " P a r a m e t r i c E x c i t a t i o n u s i n g V a r i a b l e Capacitance of F e r r o e l e c t r i c M a t e r i a l s " , J o u r n a l of the I n s t . o f E l e c t , and C o m m u n i c a t i o n E n g i n e e r s o f J a p a n , V o l . 4 1 , No. 3, M a r c h 1958, pp. 239-244. 1 2 . J a c k s o n , A., A n a l o g C o m p u t a t i o n , I960.  M c G r a w - H i l l , pp.  182-185,  51 13.  C u n n i n g h a m , ¥., I n t r o d u c t i o n t o N o n l i n e a r A n a l y s i s , McGraw-Hill, 1958.  14.  R i t z , W.,  U b e r e i n e neue M e t h o d e z u r L o s i n g g e w i s s e r V a r i a t i o n s p r o b l e m e I e r mathematischen P h y s i k " , C r e l l e s J o u r , f . d . r e i n e u . ang. M a t h . , V o l . r35, pp. 1-61, 1909.  15.  Parlett,  B., " L a g u e r r e ' s M e t h o d A p p l i e d t o t h e M a t r i x Eigenvalue Problem", Mathematics of Computation, V o l . 18, 1964, pp. 464 f f .  16.  Stevens,  J . , "LAGERE", U n p u b l i s h e d F o r t r a n p r o g r a m , U.B.C. C o m p u t i n g C e n t e r L i b r a r y , 1966.  

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