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UBC Theses and Dissertations

Identification and parameter optimizaiton of linear systems with time delay Robinson, William Reginald 1968

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A STUDY OF FERRORESONANCE WITH APPLICATION TO DIGITAL LOGIC by ALBERT JAMES REED B.A.Sc, University 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 E l e c t r i c a l Engineering We accept th i s thesis as conforming to the required standard Research Supervisor Members of the Committee Head of the Department , . Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and Study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by h.iis representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of Brit ish Columbia Vancouver 8, Canada i ABSTRACT A series resonant L-C c i r c u i t i n which either the inductor o.r the capacitor i s nonlinear and which i s excited by a sinusoidal voltage of a fixed frequency may have two steady-state responses. One of these responses i s character ized by a high amplitude o s c i l l a t i o n ; the other by a low one. If the amplitude or frequency of the d r i v i n g signal i s varied slowly, the response may suddenly change or "jump" to the other state. As a r e s u l t , t h i s phenomenon has been called jump resonance, or ferroresonance. Because the high and low resonant states could be con sidered as a 0 and 1 basis f o r d i g i t a l l o g i c operations, i t was the purpose of t h i s work to study the phenomenon and to investigate the p o s s i b i l i t y of using i t i n the design of d i g i t a l l o g i c elements. Equations''which exhibit the necessary features were studied on an analogue computer. The results of the study were used as design c r i t e r i a f o r the construction of an actual c i r c u i t and also as a basis for an approximate a n a l y t i c a l study. The a n a l y t i c a l study uses the Ritz method to fi n d useful features of the responses. The results of previous users of t h i s method have been extended to include equations with both second derivative coupling and non-symmetrical non- l i n e a r i t i e s . Based on the above studies, a prototype c i r c u i t was designed which has some of the basic properties of conventional i i f l i p - f l o p c i r c u i t s . One of the main features of thi s c i r c u i t i s that i t i s almost e n t i r e l y made of reactive components and as a result has very low power consumption. The operation of the c i r c u i t i s us^d to v e r i f y the v a l i d i t y of the approximations made i n both the analogue simulation and the a n a l y t i c a l study. The results obtained from the analogue study, the Ritz analysis, and the prototype c i r c u i t compare favorably with each other. Some suggestions for future work are given. i i i TABLE OP CONTENTS Page ABSTRACT .... i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS v LIST OF SYMBOLS . . v... . v i i ACKNOWLEDGEMENT • i x 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 Discussion of Analogue Computer Results . 10 (a) Form of Solutions 10 (b) E f f e c t of Varying the Resistance . 11 (c) E f f e c t of Varying the Coupling ... 12 2.4 Summary of the Analogue Simulation Results 12 3 . CIRCUIT ANALYSIS AND DESIGN 15 3.1 Preamble 15 3.2 The Ritz Method as Applied to Forced O s c i l l a t i o n s 15 3 ' 3 . Development of the Ritz Conditions .... 16 3-4 Qualitative Discussion of Solutions ... 20 3 . 5 Features of the Solutions 22 3 . 6 Discussion of the Uncoupled Case 24 (a) Frequency Characteristics 24 • Ob) Amplitude Characteristics 24 i v Page .(c) Effect of Damping 25 3.7 The Effect of Coupling . 25 3-8 Source Impedance Considerations ... 27 4. 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 B i s t a b i l i t y 32 4.5 Use as a Memory Device 55 4.6 Use as a Counter 53 4.7 Switching 58 4.8 Summary 41 5. CONCLUSIONS 42 APPENDIX A: Polynomial Approximation to the f(q) Characteristic f o r the S i l i c o n Capacitor 45 APPENDIX B: Determination of the Ritz Coeff i c i e n t s 48 REFERENCES 50 V • . LIST OF ILLUSTRATIONS Figure ^ a S e .2.1 Basic Coupled C i r c u i t ... 5 2.2 S i l i c o n Capacitor Characteristics 6 2.3 Quality Factor for S i l i c o n Capacitor .. 6 2.4 Forcing Function Generator 8 2.5 Simulation of Basic Coupled C i r c u i t ... 9 2.6 Sample Analogue Computer Solution 14 2.7 Sample Analogue Computer Solution 14 2.8 Sample Analogue Computer Solution 14 2.9 Sample Analogue Computer Solution ..., 14 2.10 Sample Analogue Computer Solution 14 2.11 Sample Analogue Computer Solution 14 3.1 Typical Frequency Responses 21 3.2 Frequency Response for Uncoupled Case . 23 3.3 Amplitude Response for Uncoupled Case . 23 3.4 Ef f e c t of Damping on Frequency Response 26 3.5 Effect of Damping on Amplitude Response 26 3.6 Entire Approximate Solution for Equation 3.17 28 3.7 Effect of Coupling on Identical Solution Pairs 30 3.8 E f f e c t of Coupling on Non-identical Solution Pairs 30 4.1 Memory Device Configuration 35 4.2 Counter Device Configuration ........... 35 4.3 Operation of Counter , 37 v i Figure Page 4.4 Waveforms of Two Consecutive Counters .. '37 4 . 5 • Switching Process 39 4 .6 Switching Waveform '.. ' 37 A. l Normalized f'(q) Characteristic ......... 46 A s 2 , Deviation Between f(q,) and the Poly nomial Approximations • 4& B. l Flow Diagram of Program to Determine Ritz Coefficients 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 c o e f f i c i e n t s F„,F o fF„ functions o s' c G ,0f ;G- functions o s c G- amplitude c o e f f i c i e n t f ,g functions k a subscript K a constant L inductance M coupling c o e f f i c i e n t Yj OJ/K p amplitude c o e f f i c i e n t q a variable representing char, q an approximation to q f1 nonlinearity c o e f f i c i e n t y k a set of functions s p/K 2 t a variable, time Z w t x,y,u,v variables •, v i i i d x/dt d 2 x / d t 2 a n g u l a r f r e q u e n c y a c o n s t a n t r e s i s t a n c e v o l t a g e ACKNOWLEDGEMENT Grateful acknowledgement i s given to the National Research Council of Canada for f i n a n c i a l support i n the form of a Bursary i n 1964-65 and a studentship i n 1965-66. The author wishes to thank his supervisor, Dr. A.C. Soudack for his guidance throughout the course of this work. Thanks are also given to Dr. M.S. Davies for reading the manuscript and fo r his useful comments and suggestions, and to Dr. G. Christensen for his helpful discussions. 1 1 . INTRODUCTION The phenomenon of ferroresonance was observed for the f i r s t time i n 1906 during the tuning of radio transmitters. Since, then i t has appeared i n the l i t e r a t u r e pertaining to power s y s t e m s , e l e c t r o n i c s , ^ ^ and nonlinear mechanics. Ferroresonance can occur i n a driven series resonant'circuit which consists of an inductor and a capacitor, one of which i s nonlinear. If the nonlinear 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 Duffing's equation: x + CO q x + hx = G- cos cot ( l - l ) Closer approximations using higher order terms can be found i n ' (5) Hayashi. This equation has the property that under certain conditions i t has two stable solutions near resonance, one a large amplitude o s c i l l a t i o n and the other a small one. This phenomenon i s called ferroresonance. The bistable multivibrator, or f l i p - f l o p , i s one of the most useful electronic devices employed i n d i g i t a l com puters. An i d e n t i f y i n g feature of a bistable multivibrator i s that i t consists of a pair of two-state devices arranged symmetrically so as to allow only two stable states of the complete c i r c u i t . For example, i n a tr a n s i s t o r f l i p - f l o p , either t r a n s i s t o r may be on or off but they are arranged so that when one i s on i t keeps the other o f f . It was proposed to investigate 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 using the f e r r o - resonant'.regions of nonlinear L-C c i r c u i t s . The large amplitude o s c i l l a t i o n could represent an on state and the small one an off state. Some work i n thi s area has been done by other workers such as Isborn^ 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 for the nonlinear element. More recent developments i n semiconductors, p a r t i c u l a r l y the advent of varactor diodes, have made i t necessary to con sider the problem allowing non-symmetrical c h a r a c t e r i s t i c s . The work presented here includes the choice of a pa r t i c u l a r type of nonlinear capacitor, an analogue simulation to obtain a f i n a l c i r c u i t configuration and to give approximate design values to the components, an approximate mathematical analysis of the c i r c u i t , and the building of an operative prototype unit. The c i r c u i t f i n a l l y used can be described by the following equations: q^ + 2DKg(q1) + K 2 f ( q i ) + Mq2 - p s i n ? , 0 (1-2) q 2 + 2DKg(q 2) + K 2 f ( q 2 ) + Mq^ ' - p si n S = 0 These equations are analyzed using the Ritz or Ritz-Galerkin method. Some of K l o t t e r ' s ^ m e t h o d s have been extended to equations with second derivative coupling. General algebraic conditions r e l a t i n g the response to the d r i v i n g amplitude and frequency are derived for t h i s type of equation. These results are applied to the s p e c i f i c case of a c i r c u i t with n e g l i g i b l e damping and with an asymmetrical restoring function which can be approximated by: f(q) = q + uq 2 ( l - 3 ) Parametric excitation using the variable capacitance properties of materials such as barium titanate has been a c h i e v e d ^ ^ and i t was thought that components of this type could be used as the nonlinear elements i n the synthesis of the proposed c i r c u i t . However, they were deemed unsuitable for use." at the present time due to t h e i r cost and s c a r c i t y within the needed capacitance tolerance. Instead, low cost, commercially available s i l i c o n capacitors were used to demon strate the p r i n c i p l e s of operation and were found to be quite s a t i s f a c t o r y . This thesis consists of a discussion of the analogue computer simulation i n Chapter 2, the development of the Ritz analysis i n Chapter 3? some discussion of the c i r c u i t design and results i n Chapter 4, and some suggestions for future study i n the concluding Chapter 5. 4 2 . ANALOGUE SIMULATION 2-1 Preamble An electronic analogue computer consists of a c o l l e c t i o n of units, each of which i s designed to produce an output that i s a p a r t i c u l a r l i n e a r or nonlinear function of the inputs. These units are re a d i l y interconnected to solve mathematical equations or to simulate the behaviour of a physical system. A convenient feature i s on-line control, that i s the f a c i l i t y with which changes i n parameters of the equations can be made manually during the actual operation or solution of an equation. 2-2 Computer C i r c u i t s The c i r c u i t shown i n Figure 2-1 can be described by the following equations: d 2q dq d 2 q 2 V sincot = L £ + R — - + ± — q - M § <MT dt f (v ) 1 °1 (2-1) 2 2 d q ? dq d q V sincot = L 1 + R — - + - — q_ - M ^ <MT dt f(v ) d &tZ c 2 The voltage-capacitance c h a r a c t e r i s t i c , f(v ) i s that of a Transitron SC -5 s i l i c o n capacitor (Figure 2—2) and was obtained from the manufacturer's specifications and direct measurement. The frequency of the dr i v i n g sinusoid was chosen to coincide with that of the' highest quality factor (Figure 2-3) at about 2 x 1 0 6 r/s. An estimate was made of the maximum probable Figure 2. 2 S i l i c o n Capacitor Characteris t i c Figure 2.3 Quality Factor for S i l i c o n Capacitors value of a l l the variables and of t h e i r f i r s t and second derivatives. Amplitude and time scaling transformations were then made on (2-1) so that no variable or i t s derivatives would exceed unity and also so that the operating frequency would be about two cycles/sec. These operations are necessary - to prepare an equation for solution on an analogue computer. The sinusoidal forcing function was obtained by solving (12) a Van der Pol type of equation: x - 10(A 2 - x 2 - ( M ) x + co2x = 0 (2-2) CO x(0) = A, x(0) =0 This equation has the solution x = A cos cot (2-3) and has the property that i f perturbed, the solution w i l l quickly return to (2-3), and i t w i l l not decay due to leakage i n the computer capacitors or other non-ideal factors. This can readi l y be seen by an examination of the sign 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 used to solve 2-2 i s shown i n Pigure 2-4- The c i r c u i t used to solve the time and amplitude scaled (2-1) i s shown i n Pigure 2-5. There were various minor modifications and additions to this c i r c u i t to allow sign changes of M/L, inclusion of source impedance, various input disturbances, and some protective pre cautions, but these are a l l omitted i n Pigure 2-5 for the sake of c l a r i t y . Figure, 2.4 Forcing Function Generator Figure 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 Discussion of Analogue Computer Results (a) Form of Solutions The results obtained on the Pace 231-P- computer proved very useful i n determining component values for the f i n a l c i r c u i t . Examples of solutions are given i n Figures 2-6 through 2-11. From the r e s u l t s , several features are apparent. F i r s t l y , i t can be seen that the solution i s approximately a biassed sinusoid with the same frequency as the d r i v i n g .function i n most cases. Subharmonics w i l l be discussed separately. Secondly, i t i s clear that jump resonance can occur. This means that for the same dr i v i n g function, there can he two possible states of each side of the c i r c u i t of Figure 2-1. This feature i s shown i n Figure 2-6 where the upper trace i s the input d r i v i n g function which i s kept constant, and the lower trace i s proportional to the voltage across one of the s i l i c o n capacitors. The jump or t r a n s i t i o n between the two states was i n i t i a t e d by a small pulse applied to amplifier A22 which simulated a pulse i n the bias of one of the capacitors of Figure 2-1, thus causing a momentary change i n the average capacitance. Figures 2-7, 2-8 and 2-9 show the p o s s i b i l i t y of the (5) existence of subharmonics . It 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 various on-line d i s  turbances of the solution, subharmonics of this type could be made to e x i s t . Once i n i t i a t e d , they were maintained i n d e f i n i t e l y . 11 However, because of the peaked shape of Figure 2-3, i t i s most un l i k e l y that any subharmonics could persist i n the actual c i r c u i t of Figure 2-1 because of the greater damping at higher and lower frequencies-, a feature not included i n the analogue model. (b) v Effect of Varying the Resistance The c o e f f i c i e n t representing the resistance of Figure 2-1 was re a d i l y varied by changing potentiometers Q13 and Q18 of Figure 2-5. It was found that for the f i r s t derivative c o e f f i c i e n t value of above about 0.2 there was no jump resonance. This value represented a series resistance of-about 17 ohms i n the c i r c u i t being modelled. Progressively smaller values of thi s c o e f f i c i e n t gave an increase i n the r a t i o between high and low states and also a decrease i n the switching time between the two states. However, there was also an i n  crease i n the time required for the transient modulation to damp out. This feature i s shown i n Figure 2-10 which i s the t r a n s i t i o n between high and low states with a damping coeff i c i e n t of 0.04. Note that the modulation of the c a r r i e r en velope a f t e r the jump i s very similar to the overshoot i n an underdamped l i n e a r second order system. Note also that the jump occurs i n six cycles of the c a r r i e r instead of the eleven cycles i n Figure 2-6 where the damping c o e f f i c i e n t i s 0.05 and the thirteen cycles i n Figure 2-11 where the damping co e f f i c i e n t i s 0.07. Thus i t i s seen that to build the desired c i r c u i t , 12 the loss must be kept low enough to allow ferroresonance, but not so low as to permit excessive transient modulation of the output waveform during switching i n t e r v a l s . (c) Effect of Varying the Coupling One of the main reasons for studying a model of the c i r c u i t (Figure 2-1) on the analogue computer rather than using the actual c i r c u i t was the ease with which the M/L r a t i o could be varied. Potentiometers P55 and P56 controlled the coupling between the branches. Amplitude and frequency responses were obtained for various values of the M/l r a t i o . It was found that up to a point, increasing t h i s r a t i o increased the region i n which the entire coupled c i r c u i t could be symmetrically bistable; that i s , when one branch of the c i r c u i t i s i n the high state, 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 either the high state or the low state - - a symmetrically b i  stable state was not possible. Hence i t i s seen that i f the M/L r a t i o i s too small, the c i r c u i t must be very accurately tuned to be bistable at a l l , but i f the r a t i o i s too large the desired symmetrically bistable 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 simulation of various c i r c u i t s on the analogue computer provided invaluable groundwork for the chapters to 13 f o l l o w . F i r s t l y , the basic c i r c u i t c o n f i g u r a t i o n to be used was determined. Secondly, the general shape and features of ' the output waveform were found. Knowledge of these i s necessary f o r the R i t z a n a l y s i s i n the next chapter. A l s o , study of the analogue c i r c u i t gave the design c r i t e r i a f o r M, L, and R that are used l a t e r i n t h i s work. The most important 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 understanding which i t gave of the c i r c u i t . 14 INPUT 1/wwvwwwwwwwvwwwwwv INPUT R E S P O N S E R E S P O N S E F i g u r e 2.6 F i g u r e 2.9 INPUT INPUT R E S P O N S E F i g u r e 2.7 R E S P O N S E 111, „ . F i g u r e 2.10 INPUT INPUT AAAAAAAAAAAAAAAAAAAAAAA/^yWWWW R E S P O N S E AT X R E S P O N S E R E S P O N S E AT Y F i g u r e 2.8 ' f A P P L I E D P U L S E F i g u r e 2.11 Sample Analogue Computer Solu t i o n s 15 3 . CIRCUIT ANALYSIS AND DESIGN 3 - 1 Preamble When c o n s i d e r i n g the b e h a v i o u r of l i n e a r systems, i t i s c o n v e n i e n t t o d e a l w i t h s i n u s o i d a l i n p u t s and t h e i r o u t p u t s w h i c h a r e a g a i n s i n u s o i d a l . As a r e s u l t o f the p r i n c i p l e of s u p e r p o s i t i o n , t h e r a t i o between output and i n p u t as a f u n c t i o n o f f r e q u e n c y i s a b a s i s f o r a complete d e s c r i p t i o n o f the system b e h a v i o u r . However, i n n o n l i n e a r systems t h i s p r i n c i p l e (13) cannot be used ^' and o t h e r methods, which 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 found. I n t h e f o l l o w i n g a n a l y s i s , t h e R i t z ^ 4 ^ o r R i t z - G a l e r k i n method w i l l be used. T h i s method i s c a p a b l e of h a n d l i n g a l a r g e v a r i e t y of s t e a d y s t a t e c a s e s . The main d i f f i c u l t y i n v o l v e d i n u s i n g the R i t z method i s t h a t t h e approximate form of the s o l u t i o n must be known or assumed. As a r e s u l t , t h i s method w i l l not f i n d unknown f e a t u r e s o f the s o l u t i o n such as h i g h e r o r l o w e r harmonics i f t h e y are not i n c l u d e d i n t h e assumed s o l u t i o n . The r e s u l t s of the analogue computer s t u d y show t h a t a b i a s s e d s i n u s o i d a t 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 the s o l u t i o n waveform, and such a s o l u t i o n can be w e l l approximated w i t h the R i t z method. 3 - 2 The R i t z Method as Ap-plied t o F o r c e d O s c i l l a t i o n s ( 9 ) The R i t z method, as summarized by K L o t t e r , i s based on the f o l l o w i n g p r i n c i p l e . L e t t h e d i f f e r e n t i a l e q u a t i o n describing a system be: E[q(t)] = 0 ( 3 - D and replace the solution, q ( t ) , by an assumed form: ' m 'q (t) = 2 ] a k ^ k ( t ) f o r a < t < b (3-2) k=l where the l | ^ k ( t ) are an appropriately chosen set of l i n e a r l y independent functions. Unless q(t) i s the exact solution, E [q(t)] w i l l not vanish. The Ritz conditions for the system are conditions on a"k such that b J E [q(t)J ^ k ( t ) d t = 0 (k=l,2,...m) (3-3) a 3-3 Development of the Ritz Conditions The equations to be considered describe the c i r c u i t of Figure 2 - 1 and are of the form: E J q J = q^ + 2DKg(q 1) + Aiq-J + Mq2 - p s i n Z = 0 (3-4a) E[q 2] = q 2 + 2DKg(q 2) + K 2 f ( q 2 ) + Mq^ - p s i n Z = 0 • (3-4b) where & = cot. We can choose our ^ k to give assumed approximate solutions of the form: ' . q = C, + A sin£ - B, cos % (3~5a) q 2 = C 2 + A 2 s i n 2 - ^2 c o s ( 3 ~ 5 b ) where C 1 and a r e i n c l u d e d t o a l l o w f o r p o s s i b l e asymmetry i n t h e f o r g f u n c t i o n s . I t f o l l o w s t h a t 'q = coAi cos 5 + coB^ sin£> i = 1,2 (3 -6a) q = -to A ± s i n S + </Bi c o s ^ i = 1,2 (3-6b) By d e f i n i n g a u x i l i a r y f u n c t i o n s .2% A 0 P s ( C i ' V V = \ f(5i)sin5 dg i = 1,2 ' (3 -7b ) Fj(C., A . , B.) i I f(qJ.)d3 ;• i .= 1,2 (3-7a) ,2jt A 0 n 2* P 1(C., A., B.) c l ' l ' 1/ f(q i)cosS 6.Z i = 1,2 (3 -7c) «7 0 />2JC G^(A ±, B., OO) = i 1 g(q\)d3 i = 1,2 (3-8a) ^ 0 r2it G-^(A., B., co) = | . i j g ^ s i n g dg 2JC i = 1,2 '0 . (3-8b) g(^_)cosg d£ i = 1,2 0 (3 -8c) t h e R i t z c o n d i t i o n s : >2JC 5 Z7C B[qJ aZ = 0 1 = 1,2 (3-9a) •0 I2ft E [ q j s i n g d£ 0 = 0 i = 1,2 (3-9b) ^ E [ q ± ] cos g d £ = 0 0 ' i = 1,2 ( 3 - 9 c ) g i v e r i s e t o the f o l l o w i n g s e t of a l g e b r a i c e q u a t i o n s : P o + 2 D G o = 0 (3-10a) F 1 + 2DG-1 - Y)2A. - M y ] 2 A. - S s s 7 1 7 , 3 F 1 + 2DG 1 + y i 2 B. + M r) 2 B . = 0 c c y 1 J j 0/ i = l , 2 (3-10b) (3-10c) where S = p/K and YJ = to/K. The f ( q ) c h a r a c t e r i s t i c f o r t h e s i l i c o n c a p a c i t o r s used may be approximated (see Appendix A) by: f ( q ) = q + u q 2 ( 3 - l l ) U s i n g ( 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 the c o u p l e d , non-symmetric e q u a t i o n s : F 1 + 2DG 1 = 0 o o F 1 + 2DG 1 - Y 7 2 A . - M y ? 2 A .-s s s J 1 J j F 1 + 2DG 1 + *2B.'+M Y)2B. c c J I J j 0 0 (3-12a) 1=1,2 3=1,2 (3-12b) (3-12c) I f we e v a l u a t e F 1 from e q u a t i o n (3-7c) we o b t a i n : c ( 3 - 1 3 ) 19 I t i s e v i d e n t now, t h a t i f we assume n e g l i g i b l e damping ( i . e . B » 0 ) , e q u a t i o n (3-12c) 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 , our 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: i = 1,2 t j = 1.2 i ^ 3 (3-14a) (3-14b) 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 (3-12a) y i e l d s e q u a t i o n s of the form: c 4 A where x 4 + b x 2 + cx + dxy + e y 2 + hy + a 2 2 y" + by + cy + dxy + ex + hx + a x = A . d = M ^ J 0 0 (3-15) (3-16) y = A , b = 4-4 -1 = £3 2u< 2 h = MJ9 2u Mr? 2 s  J 2 a = 2u< Making the s u b s t i t u t i o n s x = u + v , y = u - v and a d d i n g and s u b t r a c t i n g (3-15) and (3-16) uncouples the e q u a t i o n s and g i v e s : v . [ 4 u r 2 + 2(b-e) u + c - h j = 0 (3-17) 4 2 2 2 2 2 2 2 r + 4u ( r -u ) + b r + cu + d(2u - r ) + e r + hu + a = 0 (3-18) p o p where r = u + v . E q u a t i o n (3-17) i s s a t i s f i e d i f v = 0. 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 2 0 i n ( 3 - 1 6 ) g i v e s a p o l y n o m i a l i n powers of u: u 6 [ - 4 ] + u 4 [ 2 ( d + e - b)] + u 3 [ 2 h ] + u 2 [ 3 e 2 - b 2 + 2bd - 2dc - 2be + 4a ] .i + u [(f - f ) . ( h - c ) ] + ^ [ h - c ] 2 = 0 16 L J ( 3 - 1 9 ) When v = o, x = y and e q u a t i o n (3-15.) may be s o l v e d d i r e c t l y . When v ^ o the R i t z c o e f f i c i e n t s a r e determined from t h e r o o t s of ( 3 - 1 9 ) . 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 To get an h e u r i s t i c i d e a of what t o expect f o r s o l u t i o n s , i t i s u s e f u l t o examine (3-14a) and (3-14h) more c l o s e l y . S o l v i n g f o r C. i n t h e f i r s t e q u a t i o n g i v e s - 1 + °i = ±v 1 - 2\fk 2 u 1 ( i = 1 , 2 ! (3-20) Now i f i s z e r o , must a l s o be zer o because t h e r e i s no o s c i l l a t i o n t o cause the b i a s o r dc term. Thus, o n l y the p o s i t i v e s i g n i n f r o n t of the r a d i c a l i n (3-20) i s m e a n i n g f u l . The neg a t i v e s i g n l e a d s t o extraneous r o o t s . U s i n g ( 3 - 2 0 ) , s u b s t i t u t i n g 2 f o r C i i n (3-14b), and s o l v i n g f o r Yj y i e l d s : 2 1 1 ( 1 + M j l ) i Ii o 2, 2 S f l - 2u A. - r (3-21) In t h e s p e c i a l case where M = 0, thes e e q u a t i o n s reduce t o (Q) e x a c t l y t he same form as g i v e n by K l o t t e r w f o r uncoupled e q u a t i o n s . 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 response p l o t o f (3-21) f o r M = 0. F o r H / 0, the term l / ( l + M ^ ) expands o r Figure 3.1 T y p i c a l Frequency Responses 2 c o n t r a c t s t h i s curve a l o n g the Yj a x i s . The c u r v e s shown i n 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 are drawn assuming t h a t A^ A. i s g r e a t e r t h a n A. and t h a t M 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. Because ( l + M 7—) i s . s m a l l e r t h a n ( l + M -r^-) , the curve f o r . J A i 2 i = 2, j = 1 i s expanded a l o n g the YJ a x i s more t h a n the curve • f o r i = 1, j = 2. Thus the p o s s i b l e o p e r a t i n g r e g i o n where A^ i s g r e a t e r t h a n A D i s i n c r e a s e d as shown i n F i g u r e 3 - 1 ( c ) . A l W i t h M 7— 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 d e c r e a s e d . A 2 3 -5 F e a t u r e s of the S o l u t i o n s A more tho r o u g h and q u a n t i t a t i v e d e t e r m i n a t i o n of the R i t z c o e f f i c i e n t s i s o b t a i n e d by s o l v i n g (3-16) and (3-19) d i r e c t l y . There are s i x t e e n r o o t p a i r s r e s u l t i n g from t h e s e e q u a t i o n s . Four a r e o b t a i n e d from the s o l u t i o n o f (3-16) f o r t h e case x = y. The o t h e r t w e l v e are s o l u t i o n s of the s i x t h 2 2 2 o r d e r (3-19) and the t r a n s f o r m a t i o n r = u + v i n ( 3 - 1 8 ) . How ev e r , because of the symmetry i n v o l v e d , f o r each of s i x x, y 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 i s an i d e n t i c a l y, x s o l u t i o n . The e q u a t i o n s were s o l v e d on an I.B.M. 7040 d i g i t a l computer u s i n g L a g u e r r e ' s method^"^ ' f o r e x t r a c t i n g p o l y  n o m i a l r o o t s , (see Appendix B ) . T h i s method was found t o be more r e l i a b l e f o r t h i s work t h a n t h e M u l l e r , Newton, or B a i r s t o w methods. The R i t z c o e f f i c i e n t s were determined and t h e i r depend ence on the d r i v i n g a m p l i t u d e s, the n o r m a l i z e d f r e q u e n c y YJ , and t h e c o u p l i n g term M were found. Examples of the s o l u t i o n s w hich come from the r e a l r o o t s o f (3-16) and (3-19) a r e g i v e n 23 4- 2 x X X. X \ N \ \ ^ \ M= 0 _ * \ \ \ 1 — i — .2 1 .4 .6 i 1.0 1.2 1-4 NORMALIZED 1.6 1.8 2.0 FREQUENCY F i g u r e 3.2 Frequency Response f o r Uncoupled Case D R I V I N G A M P L I T U D E F i g u r e 3*3 A m p l i t u d e Response f o r Uncoupled Case •in F i g u r e s 3-2 t h r o u g h 3 - 8 . Complex r o o t s have no p h y s i c a l meaning. 3-6 D i s c u s s i o n o f the Uncoupled Case (M = 0) (a) Frequency C h a r a c t e r i s t i c s I n F i g u r e 3-5 ' ' i t i s seen t h a t the f r e q u e n c y response of t h i s , n o n l i n e a r c i r c u i t i s q u i t e u n l i k e t h a t o f i t s 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 c i r c u i t s . The curve i s shaped so t h a t a l t h o u g h the system d e s c r i b e d i s l o s s l e s s , the response i s everywhere f i n i t e . A l s o , as p r e v i o u s l y mentioned, the c i r c u i t can e x i s t i n more than one s t a t e a t a g i v e n d r i v i n g f r e q u e n c y . I t can be s h o w n ^ ^ ' * ^ t h a t s o l u t i o n s c h a r a c t e r i z e d b y ' n e g a t i v e s l o p e r e g i o n s i n the -|x| - s p l a n e a r e u n s t a b l e and hence cannot e x i s t i n a p h y s i c a l system. These s o l u t i o n s and t h e ones r e s u l t i n g from extraneous r o o t s are shown i n broken l i n e s i n the f i g u r e s . I t i s seen t h e n , t h a t f o r the un c o u p l e d c a s e , the c i r c u i t can e x i s t i n f o u r d i f f e r e n t s t a t e s : a 1-1 s t a t e ( b o t h x and y l a r g e and o p p o s i t e i n phase t o the d r i v i n g t e r m ) ; a 0-0 s t a t e ( b o t h x and y s m a l l and i n phase w i t h t h e d r i v i n g t e r m ) ; a 1-0 s t a t e ; and a 0-1 s t a t e . (b) A m p l i t u d e C h a r a c t e r i s t i c s F i g u r e 3-3 shows t h a t the dependence of t h e r esponses on t h e d r i v i n g a m p l i t u d e i s a l s o q u i t e u n l i k e t h a t of the c i r  c u i t ' s l i n e a r c o u n t e r p a r t . The dependence i s examined at a f r e q u e n c y t h a t i s known from F i g u r e 3-2 t o have more than one p o s s i b l e r e s p o n s e . A g a i n , the u n s t a b l e and e x t r a n e o u s s o l u t i o n s a r e shown w i t h broken l i n e s . I t i s seen t h a t i f t h e c i r c u i t i s 25 i n the 0-0 s t a t e and the d r i v i n g a m p l i t u d e i s g r a d u a l l y i n c r e a s e d , t h e 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 v e r t i c a l t a ngent p o i n t on the l o w e r curve i s r e a c h e d , and t h e n s u d d e n l y i n c r e a s e s t") the a m p l i t u d e and phase g i v e n hy t h e upper curve (a 1-1 s t a t e ; . From then on, the response g r a d u a l l y i n c r e a s e s w i t h i n c r e a s i n g i n p u t a m p l i t u d e . (c) E f f e c t o f Damping The e f f e c t 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 the 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 and 3-5. 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 i n - t h e 0-0 s t a t e and the i n p u t a m p l i t u d e i s g r a d u a l l y i n c r e a s e d , the response 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 phase o c c u r s as b e f o r e . I f now the a m p l i t u d e i s s l o w l y d e c r e a s e d , the response d e c r e a s e s s l o w l y u n t i l the o t h e r v e r t i c a l tangency p o i n t i s reached a t which time the c i r c u i t r e v e r t s t o the 0-0 s t a t e . I n the r e g i o n between the two v e r t i c a l t angent p o i n t s , a l l f o u r s t a t e s a r e p o s s i b l e . The change of s t a t e w i t h d r i v i n g a m p l i t u d e i s d i s c u s s e d i n 3-8 as a means of l i m i t i n g the number of s t a b l e s t a t e s . I f the c i r c u i t i s more h e a v i l y damped so t h a t no v e r t i c a l t angent p o i n t s o c c u r , the two s i d e s o f the c i r c u i t cannot be i n d i f f e r e n t s t a t e s . • 3-7 The E f f e c t of C o u p l i n g • I t i s u s e f u l f i r s t t o c o n s i d e r a t y p i c a l r esponse p l o t as shown i n F i g u r e 3-6. T h i s p l o t i s the e n t i r e s o l u t i o n from 26 6 - X.Y 4 - \NV ^ V x - / 2 - yJJ > • • i • . 2. .4 .6 .8 i 1.0 i 1.2 l 1.4 I 1.6 i 1.8 I 2.0 NORMALIZED FREQUENCY F i g u r e 3.4 E f f e c t of Damping on Frequency Response X,Y J I 1—; 1 1—: 1 1 1 , p - 2 4 6 8 10 12 14 16 18 2 0 DRIVING AMPLITUDE F i g u r e 3«5 E f f e c t of Damping on Ampl i t u d e Response . 2 7 e q u a t i o n (3-19) 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 . The s i x x, y p a i r s shown a r e l a b e l l e d (x^,y^) t h rough ( x ^ , y g ) . I t s h o u l d be remembered t h a t t h e r e are s i x more p a i r s (x^,y^) t h rough ^ x12'"^12^ ku~k "these a r e the same as (x^,y^) t h r o u g h ( x g , y ^ ) . The u n s t a b l e and e x t r a n e o u s s o l u t i o n s are a g a i n shown i n broken l i n e s and i t s h o u l d be noted t h a t o n l y xj_»y-j_ i s l e f t s o l i d and hence i s the o n l y s o l u t i o n of i n t e r e s t . . I n the f o l l o w i n g d i s  c u s s i o n o n l y the s o l u t i o n s of i n t e r e s t w i l l be shown. The e f f e c t of v a r y i n g M i s t h e n seen by e xamining t h e • f r e q u e n c y p l o t s of F i g u r e s 3~7 and 3-8. I t i s seen t h a t g r e a t e r c o u p l i n g w i t h M p o s i t i v e r e s u l t s i n a d e c r e a s e d r e g i o n o f 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 o f the c i r c u i t i n the same s t a t e (0-0 or l - l ) . A l s o , i t r e s u l t s i n a g r e a t e r r e g i o n of p o s s i b l e s y m m e t r i c a l b i s t a b i l i t y (0-1 or 1-0). W i t h M n e g a t i v e , , the o p p o s i t e e f f e c t i s observed, t h a t i s the u n s y m m e t r i c a l r e g i o n (0-0 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 r e g i o n i s d e c r e a s e d . I t s h o u l d be p o s s i b l e t h e n , t o have the c i r c u i t o p e r a t e a t a p o i n t such t h a t o n l y the d e s i r e d s y m m e t r i c a l s t a t e s can e x i s t . 3-8 Source Impedance C o n s i d e r a t i o n s C o n s i d e r 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 s e r i e s s w i t h the c a r r i e r s o u r c e of the c i r c u i t shown i n F i g u r e 2-1 which was p r e v i o u s l y d i s c u s s e d . L e t the s o u r c e v o l t a g e , V, be i n c  r e a s e d such t h a t when the c i r c u i t i s i n e i t h e r the 0-1 or 1-0 s t a t e , the v o l t a g e a t A i s the same as i t was b e f o r e the DRIVING AMPLITUDE < —«r + 7*7-- =- =--Jt--zje-- -.a - j - - w - - -1-2 - - J:6___]-8__ 2"° x6 M = 0.2 rj = 0.6 Y 6 Y1 UU~/-'--„-_. ; U N S T A B L E : X 3 Y 3 , ^ Y 4 , X 5 Y 5 E X T R A N E O U S :. X 2 Y 2 , X 6 Y 6 . Figure 3.6 Entire Approximate Solution f o r Equation 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 . I f the c i r c u i t were now to attempt t o go i n t o the 0-0 s t a t e , l e s s c u r r e n t would he drawn from the s o u r c e and the v o l t a g e drop a c r o s s C would d e c r e a s e . Hence the v o l t a g e a t A would i n c r e a s e and, as can he seen from F i g u r e 3-5, i t would te n d t o f o r c e the c i r c u i t hack i n t o the 0-1 o r 1-0 s t a t e . S i m i l a r l y , i f the c i r c u i t were t o attempt t o go i n t o the 1-1 s t a t e , t h e v o l t a g e a t A would d e c r e a s e and the c i r c u i t would a g a i n be f o r c e d back t o the 0-1 or 1-0 s t a t e . T h i s s e r i e s c a p a c i t o r can thus be used t o i n c r e a s e the p r e  v i o u s l y d i s c u s s e d e f f e c t of p o s i t i v e c o u p l i n g . 30 X,Y 6-j 4 - \ \ \ \ \ M » 0 . 8 \ Q 2 \ 0 l 0 \ - Q 2 - 0 . 8 ^ N O R M A L I Z E D F R E Q . 2.0 F i g u r e 3.7 E f f e c t 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 P a i r s N O R M A L I Z E D F R E Q U E N C Y 1 1.8 2D 1 4 1.6 F i g u r e 3.8 E f f e c t of C o u p l i n g on N o n - i d e n t i c a l S o l u t i o n P a i r s 31 4. EXPERIMENTAL RESULTS 4-1 Ba^'ic C i r c u i t C o n f i g u r a t i o n A b a s i c c i r c u i t as shown i n P i g u r e 4-1 was c o n s t r u c t e d . The n o n l i n e a r c a p a c i t o r s i n the r e s o n a n t c i r c u i t s a r e the 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 t h a t have been p r e v i o u s l y 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 used f o r the analogue computer s t u d y and a l s o i n the approximate a n a l y t i c a l s t u d y of Chapter 3. As t h e s e c a p a c i t o r s a r e a type o f d i o d e , t h e y were 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 k i l o h m r e s i s t o r . The c a r r i e r s i g n a l s o u r c e was a G e n e r a l Radio 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 o p e r a t e d near the peak of the Q-curve ( P i g u r e 2-3) a t 350 kHz. The i n d u c t o r s and t r a n s f o r m e r were wound on Siemens S i f e r r i t f e r r i t e pot c o r e s w h i c h have an a d j u s t a b l e a i r gap, p e r m i t t i n g t h e i n d u c t a n c e t o be v a r i e d . 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 t h e r e s o n a n t c i r c u i t s t o the 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 s i l i c o n c a p a c i t o r s . 4-2 Output Waveform The output waveforms were found 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 the same f r e q u e n c y as the c a r r i e r s o u r c e , thus v a l i d a t i n g the use o f the approximate form of t h e s o l u t i o n t h a t was used, i n Chapter 3. U n l i k e the r e s u l t s shown by the analogue s t u d y , h i g h e r harmonic and subharmonic responses were e i t h e r non e x i s t e n t o r so s l i g h t t h a t any e f f e c t t h e y might have had on •32 t h e s o l u t i o n s i n Chapter 3 or on the a c t u a l o p e r a t i o n of the c i r c u i t i s n e g l i g i b l e . T h i s apparent d i s c r e p a n c y between the a c t u a l c i r c u i t and i t s analogue model was due t o the use o f a s l i g h t l y 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 of t h e f r e q u e n c y c h a r a c t e r i s t i c s of the analogue model w h i c h causes the a c t u a l c i r c u i t t o a c t more p r e d i c t a b l y t h a n the analogue model (see Chapter 2). 4-3 Power Consumption An approximate measurement o f the st e a d y s t a t e power consumption of t h e b a s i c c i r c u i t was made by c o n s i d e r i n g the whole u n i t as a b l a c k - b o x , measuring the d r i v i n g v o l t a g e , and f i n d i n g the c u r r e n t t h a t was i n phase w i t h i t i n the f o l l o w i n g manner: The v o l t a g e a c r o s s the u n i t was d i s p l a y e d on the v e r t i c a l a x i s o f an o s c i l l o s c o p e and a s i g n a l p r o p o r t i o n a l t o the c u r r e n t was d i s p l a y e d on the h o r i z o n t a l a x i s . Prom the r e s u l t i n g L i s s a j o u s f i g u r e t h e in - p h a s e c u r r e n t was determined and the power d i s s i p a t e d by t h e b a s i c c i r c u i t was c a l c u l a t e d t o be s l i g h t l y l e s s t h a n t e n m i c r o w a t t s . T h i s e x t r e m e l y low l o s s I s a r e s u l t o f t h i s c i r c u i t c o n s i s t i n g almost e n t i r e l y of r e  a c t i v e components. The p o r t i o n of t h e l o s s a t t r i b u t a b l e t o the b i a s 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 . 4-4 B i s t a b i l i t y The c i r c u i t was r e a d i l y made s y m m e t r i c a l l y b i s t a b l e by a d d i n g a s e r i e s c a p a c i t o r as d i s c u s s e d i n s e c t i o n 3-8. The v a l u e of C was chosen such t h a t the v o l t a g e a t p o i n t A o f F i g u r e 4-1 33 was s u f f i c i e n t t o p e r m i t s y m m e t r i c a l b i s t a b i l i t y when the s i g n a l g e n e r a t o r was a t t h r e e - q u a r t e r s of i t s maximum o u t p u t . Both s i d e s of the c i r c u i t were tuned t o the o p e r a t i n g f r e q u e n c y by v a r y i n g the a i r gap i n i n d u c t o r s and L^. With t h e c i r c u i t a d j u s t e d i n t h i s manner, the 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 t h e a m p l i t u d e of t h e h i g h r e s o n a n t s t a t e about f o u r t i m e s t h a t of the low one as shown by the c e n t e r two t r a c e s o f P i g u r e 4 - 3 • 4 - 5 Use as a Memory D e v i c e To use a b a s i c c i r c u i t as a memory d e v i c e , a l l t h a t i s needed i s a means of ch a n g i n g i t s s t a t e . T h i s i s r e a d i l y 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 a d i o d e a t p o i n t X" o r Y" i n F i g u r e 4 - 1 . I f s i d e X i s i n the h i g h r e s o n a n t s t a t e and s i d e Y i s i n the low 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. T h i s i n c r e a s e i n b i a s r educes i t s average c a p a c i t a n c e and causes s i d e X t o drop i n t o t h e low s t a t e , : a n d because the c o u p l i n g and s e r i e s c a p a c i t o r r e q u i r e t h a t 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 as d e s c r i b e d , s i d e Y i s f o r c e d i n t o the h i g h s t a t e . S u c c e s s i v e p u l s e s a t X" t h e n have no f u r t h e r e f f e c t on the s t a t e o f the c i r c u i t . To r e s t o r e the c i r c u i t t o i t s o r i g i n a l s t a t e , a n e g a t i v e p u l s e must be a p p l i e d a t Y". 4 - 6 Use as a Counter 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 c o u n t e r 34 o p e r a t i o n of the d e v i c e i s shown i n F i g u r e 4-2. S u c c e s s i v e p u l s e s a p p l i e d a t i n p u t p o i n t A r e v e r s e the s t a t e of the c i r c u i t 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 were o b t a i n e d by p a s s i n g a square wave th r o u g h c a p a c i t o r and d i o d e D3. F i g u r e 4-3 shows the o p e r a t i o n of t h i s c i r c u i t w i t h d i o d e D3 s h o r t e d out. A l l the t r a c e s a re a t the same v e r t i c a l s c a l e of 1 v o l t / d i v i s i o n . The upper t r a c e shows the 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, the next two t r a c e s show the c a r r i e r envelope a t p o i n t s X and Y and the l o w e r 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., by r e c t i f y i n g the s i g n a l a t Y w i t h d i o d e D2 and the n d i f f e r e n t i a t i n g i t w i t h C^, R^. I t s h o u l d be n o t e d here t h a t w i t h D3 s h o r t e d o u t , both p o s i t i v e and n e g a t i v e p u l s e s r e a c h p o i n t A, but o n l y t h e n e g a t i v e p u l s e s t r i g g e r t h e c i r c u i t . T h i s e f f e c t i s due t o the way the c i r c u i t was tuned and w i l l be d i s c u s s e d i n the next s e c t i o n . The s e n s i t i v i t y of the c i r c u i t t o v a r i a t i o n s i n i n p u t p u l s e a m p l i t u d e and d u r a t i o n was determined by r e p l a c i n g the square wave g e n e r a t o r and c a p a c i t o r w i t h a G e n e r a l Radio Type 1217A p u l s e g e n e r a t o r . The minimum p u l s e d u r a t i o n t h a t . would r e l i a b l y t r i g g e r the c i r c u i t was about 5 microseconds or n e a r l y two complete c y c l e s of the c a r r i e r s i g n a l . I t would not be r e a s o n a b l e t o expect the c i r c u i t t o respond t o a p u l s e d u r a t i o n of l e s s t h a n one complete c y c l e because the t r a n s i e n t response would t h e n depend on what p o r t i o n of the c y c l e was d i s t u r b e d . However, i f more th a n one or s e v e r a l c y c l e s a re P i g u r e 4.1 Memory De v i c e C o n f i g u r a t i o n P i g u r e 4.2 Counter D e v i c e C o n f i g u r a t i o n 36 d i s t u r b e d by a p u l s e , the c i r c u i t can a t t a i n a pseudo-steady s t a t e and w i l l a c t as d e s c r i b e d i n the ne x t s e c t i o n . The min imum a m p l i t u d e p u l s e t o which the c i r c u i t would respond r e l i a b l y was about 0.15 v o l t s . 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 dependent on the 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 S e c t i o n 4-7. •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 , t he output /pulses from e i t h e r X' o r Y' o f F i g u r e 4-2 were used a t p o i n t A of the ne x t u n i t . F i g u r e 4 -4 .shows the envelope a t X and Y o f two c o n s e c u t i v e u n i t s . The upper two t r a c e s a re o f " t h e f i r s t c i r c u i t w hich i s t r i g g e r e d by n e g a t i v e p u l s e s from 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 square wave as b e f o r e , a n d the l o w e r two t r a c e s a re of the second u n i t w h i c h i s t r i g g e r e d by output 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 the i r r e g u l a r i t i e s i n the envelope of t h e second u n i t t h a t o c c u r when the f i r s t u n i t s w i t c h e s . These are a r e s u l t o f the 50 ohm output impedance o f the 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 a m p l i t u d e a t s w i t c h i n g i n s t a n t s . An u n d e s i r a b l e outcome of t h i s e f f e c t i s the p o s s i b l e o c c u r r e n c e o f unwanted s w i t c h i n g w h i c h was demonstrated by 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 the g e n e r a t o r t o i n c r e a s e the output impedance as seen by the c i r  c u i t . A l t h o u g h the impedance o f the g e n e r a t o r i t s e l f was not s u f f i c i e n t t o a d v e r s e l y a f f e c t the p r o t o t y p e u n i t s b e i n g s t u d i e d , i f many such 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 the o s c i l l a t o r be of low output Figure 4.3 Operation of Counter Figure 4.6 Switching Waveform impedance. 4-7 S w i t c h i n g I t i s not the purpose here t o attempt a. r i g o r o u s d i s  c u s s i o n of t h e t r a n s i e n t b e h a v i o u r o f the c i r c u i t . 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 based 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  i c a l r e s u l t s w i l l be g i v e n i n o r d e r t o e x p l a i n t h e p r o c e s s of s t a t e r e v e r s a l . Suppose t h a t i n i t i a l l y the c i r c u i t of F i g u r e 4-2 has s i d e Y i n the h i g h s t a t e and s i d e X i n t h e low s t a t e as d e p i c t e d on the s o l i d c u r v e s o f F i g u r e 4-5a. A n e g a t i v e p u l s e a p p l i e d t o the n o n l i n e a r c a p a c i t o r s i n c r e a s e s the b i a s , r e d u c i n g t h e average c a p a c i t a n c e , and thus s h i f t i n g t he f r e q u e n c y and a m p l i t u d e response c u r v e s as shown by the broken l i n e s i n F i g u r e 4-5a. At t h i s i n s t a n t , the o n l y s t a b l e con f i g u r a t i o n i s the 0-0 s t a t e and so Y s t a r t s t o d e c r e a s e . Be cause l e s s c u r r e n t i s b e i n g drawn the v o l t a g e a c r o s s drops and t h e d r i v i n g a m p l i t u d e b e g i n s t o i n c r e a s e , c a u s i n g p o i n t s X and Y t o move as i n d i c a t e d . When the p u l s e i s removed i t i s seen ( F i g u r e 4-5b) t h a t Y i s now i n an u n s t a b l e 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 , the v o l t a g e drop a c r o s s c a p a c i t o r C i s d e c r e a s e d , c a u s i n g t h e d r i v i n g s a m p l i t u d e t o be i n c r e a s e d , and thus c a u s i n g X t o i n c r e a s e . Because 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 t o decrease u n t i l X i s f o r c e d p a s t t h e v e r t i c a l tangency p o i n t , T-^ , on the a m p l i  tude response curve ( F i g u r e 4 - 5 h ( 2 ) ) , and the c i r c u i t a t t a i n s the 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 4-5c). The nex t Output Amplitude Output Amplitude Output Amplitude v>i pulse causes a r e p e a t of the p r o c e s s . From F i g u r e 4-5a i t i s e v i d e n t t h a t the a m p l i t u d e of the p u l s e r e q u i r e d t o t r i g g e r the c i r c u i t i s dependent on how f a r the o p e r a t i n g p o i n t i s from Tu. A l s o , i t i s seen t h a t i f Y i s c l o s e r t o t h e upper v e r t i c a l tangency p o i n t than X i s t o the l o w e r one, as shown i n F i g u r e 4-5a, a n e g a t i v e p u l s e which s h i f t s the f r e q u e n c y curve up t h e f r e q u e n c y a x i s w i l l t r i g g e r the c i r c u i t more r e a d i l y t h a n w i l l a p o s i t i v e p u l s e which s h i f t s t h e f r e q u e n c y curve down the f r e q u e n c y a x i s . By o p e r a t i n g the c i r c u i t a t a h i g h e r f r e q u e n c y i t i s p o s s i b l e t o use p o s i t i v e p u l s e s f o r 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 between the h i g h and low 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 use the f i r s t method w i t h n e g a t i v e p u l s e s . The p r e c e d i n g d i s c u s s i o n uses the r e s u l t s of the R i t z a n a l y s i s w h i c h i s a s t e a d y - s t a t e - not a t r a n s i e n t - a n a l y s i s . However, i t can be seen from th e analogue computer r e s u l t s ( F i g u r e s 2-6, and 2-11) and a l s o the r e s u l t s from the a c t u a l c i r c u i t ( F i g u r e 4-6) t h a t the s w i t c h i n g p r o c e s s t a k e s p l a c e Slowly over s e v e r a l c y c l e s of the c a r r i e r s i g n a l so t h a t a t any p a r t i c u l a r i n s t a n t the c i r c u i t may be c o n s i d e r e d t o be i n a q u a s i - s t e a d y s t a t e . The s w i t c h i n g of the p r o t o t y p e u n i t ( F i g u r e 4-6) t a k e s p l a c e over about 10 c y c l e s of the 350 KHz c a r r i e r s i g n a l as d e s c r i b e d above and e x h i b i t s t h e same type of overshoot t h a t was apparent i n r e s u l t s o b t a i n e d from the analogue computer s t u d y . • 4-8 Summary T h i s c h a p t e r 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 of a p r o t o t y p e low l o s s b i s t a b l e c i r c u i t t h a t has many of the p r o p e r t i e s of a c o n v e n t i o n a l f l i p - f l o p . I t was shown how f e r r o r e s o n a n t c i r c u i t s c o u l d be used as memory d e v i c e s or cascaded t o form c o u n t e r s . R e s u l t s o b t a i n e d from t h e a c t u a l c i r c u i t compared f a v o r a b l y w i t h analogue and a n a l y t i c a l s o l u  t i o n s and the use o f the assumed approximate s o l u t i o n used i n the 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 . F i n a l l y , a d e s c r i p t i o n of the s w i t c h i n g p r o c e s s was g i v e n w h i c h combined a c o n s i d e r a t i o n o f the R i t z r e s u l t s w i t h o b s e r v a b l e f e a t u r e s of t h e analogue and c i r c u i t s w i t c h i n g waveforms. 42 5 . CONCLUSIONS The purpose of t h i s work was t o s t u d y jump resonance phenomena a r d t o 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 the p r i n c i p l e i n the d e s i g n of d i g i t a l l o g i c elements. The system chosen f o r s t u d y was d e s c r i b e d by a p a i r of n o n l i n e a r second o r d e r d i f f e r e n t i a l e q u a t i o n s which were c o u p l e d by second d e r i v a t i v e terms and 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 f u n c t i o n . K l o t t e r ' s w'ork.on t h e R i t z method of a n a l y s i s was extended t o s t u d y the p r e v i o u s l y mentioned c o u p l e d e q u a t i o n s w i t h a s y m m e t r i c a l n o n l i n e a r i t i e s . K l o t t e r ' s work was l i m i t e d t o s y m m e t r i c a l n o n l i n e a r i t i e s i n - e q u a t i o n s w i t h c o u p l i n g and the c o u p l i n g was o n l y i n the dependent 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 which were s o l v e d on a d i g i t a l computer t o o b t a i n f r e q u e n c y and a m p l i t u d e r esponses of the model. The r e s u l t s of the above s t u d y compared f a v o r a b l y w i t h t h o s e o b t a i n e d from the s t u d y o f a model on an analogue computer,, and i n f o r m a t i o n from the two s t u d i e s was used i n the d e s i g n and c o n s t r u c t i o n o f a p r o t o t y p e b i s t a b l e c i r c u i t w hich used low and h i g h r e s o n a n t s t a t e s o f jump r e s o n a n t c i r c u i t s t o r e p r e s e n t a 0 and 1 b a s i s f o r l o g i c a l o p e r a t i o n s . The p r o t o  type c i r c u i t s had many of the f e a t u r e s of c o n v e n t i o n a l f l i p - f l o p s such as b e i n g a b l e t o s t o r e and count b i n a r y numbers and d r i v e o t h e r u n i t s . However, u n l i k e c o n v e n t i o n a l f l i p - f l o p s , t h e y r e q u i r e d an ac d r i v i n g s o u r c e as w e l l as a dc b i a s s u p p l y . 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 from the c i r c u i t w i t h t h e advent o f n o n l i n e a r c a p a c i t o r s w i t h a d i e l e c t r i c such as barium t i t a n a t e . Such c a p a c i t o r s would be o f low l o s s , and have a n o n l i n e a r c h a r a c t e r i s t i c . These f e r r o e l e c t r i c d e v i c e s have been s u c c e s s f u l i n p a r a m e t r i c a m p l i f i e r s . A nother problem i n h e r e n t i n the d e v i c e s t u d i e d was the s w i t c h i n g t i m e . I t was v e r y dependent on the c a r r i e r f r e q u e n c y and a l s o on the l o s s of the c i r c u i t . I t was shown by the analogue computer s t u d y t h a t s w i t c h i n g would n ot o c c u r 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 of the 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 would be more u s e f u l as a d e s i g n minimum. The i m p l i c a t i o n h ere i s t h a t the d r i v i n g source must c o n s i s t of an 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 ti m e s the d e s i r e d maximum s w i t c h i n g r a t e . The c u t o f f f r e q u e n c y of the n o n l i n e a r c a p a c i t o r s e t s the maximum o p e r a t i n g f r e q u e n c y o f the re s o n a n t c i r c u i t and t h e r e f o r e t h e -maximum s w i t c h i n g speed. 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 lumped elements can be connected t o g e t h e r t o form l o g i c c i r c u i t s such as was done w i t h the p r o t o t y p e s d e s c r i b e d i n t h i s r e p o r t , but f o r h i g h e r o p e r a t i n g f r e q u e n c i e s i t would become i n c r e a s i n g l y d i f f i c u l t t o c o n t a i n the s i g n a l s i n o r nea r the elements and l i n e s . One way of a v o i d i n g t h i s problem would be t o use c o n v e n t i o n a l microwave d e v i c e s such as c o a x i a l c a b l e s , waveguides and r e s o n a n t c a v i t i e s . However, these a r e q u i t e l a r g e and cumbersome. Another p o s s i b l e way of i n c r e a s i n g the s w i t c h i n g speed would be t o make the s i z e of the d e v i c e s m a l l 44 compared t o one w a v e l e n g t h . The t e c h n i q u e of m i c r o m i n i a t u r i z a t i o n i s s t i l l under development and i t would be u s e f u l f o r f u t u r e workers i n t h a t f i e l d t o c o n s i d e r m i n i a t u r i z i n g the c i r c u i t s d e v e l o p e d h e r e . The main i n c e n t i v e f o r f u t u r e work 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 low power consumption o f the d e v i c e s . Because most of the components needed are 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 copper and c o r e l o s s i n t h e i n d u c t i v e e lements. " D e v i c e s of the type developed here would 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 o p e r a t i o n s a r e r e q u i r e d and where low power consumption i s n e c e s s a r y . I n summary, the work p r e s e n t e d here i n c l u d e s : 1. E x t e n d i n g p r e v i o u s work on the R i t z method of a n a l y s i s so as t o p e r m i t s t u d y of c o u p l e d n o n l i n e a r d i f f e r e n  t i a l e q u a t i o n s which have a s y m m e t r i c a l n o n l i n e a r i t y . 2 . An analogue computer s t u d y of a model of a b i s t a b l e f e r r o r e s o n a n t c i r c u i t . 3 . The d e s i g n and c o n s t r u c t i o n o f a p r o t o t y p e of the above . c i r c u i t , and 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 . • - . j 45 APPENLIX A P o l y n o m i a l A p p r o x i m a t i o n t o the f ( q ) C h a r a c t e r i s t i c f o r t h e S i l i c o n C a p a c i t o r The v o l t a g e - c a p a c i t a n c e c h a r a c t e r i s t i c of the 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 a p p r o x i m a t i o n t o the f ( q ) i n (3-4) was made u s i n g a l e a s t squares f i t t i n g p r o c e d u r e . A b r i e f o u t l i n e of t h a t procedure f o l l o w s . The charge on the c a p a c i t o r can be d e s c r i b e d by q = v-C(v) ( A - l ) The v o l t a g e v, and the 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 found a t 52 p o i n t s i n the v o l t a g e range 0 t o -30 v o l t s . Q u a d r a t i c and c u b i c l e a s t squares f i t s were made a t -5 v o l t b i a s t o v = f ( q ) = a Q + a-^q + a 2 q + .... (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, and Table A - l . B oth the q u a d r a t i c and the c u b i c a p p r o x i m a t i o n s t o the c h a r a c t e r  i s t i c a r e w e l l w i t h i n the 20% v a r i a t i o n among components t h a t i s c l a i m e d by the m a n u f a c t u r e r . As a r e s u l t , t he quad r a t i c a p p r o x i m a t i o n i s used i n the a n a l y s i s i n Chapter I I I . C o e f f i c i e n t A p p r o x i m a t i o n a o a l ^2 a 3 Stan d a r d D e v i a t i o n Q u a d r a t i c Cubic 1.65 x I O " 1 1.43 x 1 0 " 1 2.30 x 1 0 " 2 2.32 x 1 0 " 2 2.68 x 1 0 ~ 5 2.70 x 1 0 ~ 5 -8.6 x I O " 1 0 0.1737 0.1722 Table A . l 48 . APPENDIX B D e t e r m i n a t i o n of the R i v z C o e f f i c i e n t s P i g u r e B - l I s the f l o w diagram d e p i c t i n g the p r o  cedures f o l l o w e d i n d e t e r m i n i n g the dependence of the R i t z c o e f f i c i e n t s on the v a r i o u s parameters of i n t e r e s t . The p r o  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 s e l f - c h e c k i n g . ' (15) L a g u e r r e ' s method of f i n d i n g p o l y n o m i a l , r o o t s g i v e s q u i c k convergence from any s t a r t i n g v a l u e f o r d i s t i n c t r o o t s . One 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 M u l l e r , Newton, or B a i r s t o w s t e p , but w i t h no a p r i o r i a p p r o x i m a t i o n t o the z e r o e s i t more t h a n compensates f o r t h i s by the r e d u c t i o n i n t h e number of i t e r a t i o n s needed. I f the r o o t i s s i m p l e , con vergence i s c u b i c ; o t h e r w i s e i t i s l i n e a r . The a c t u a l F o r t r a n program used i s a m o d i f i e d form of the program "LAG-ERE" which ( l 6 ) was w r i t t e n by J . Stevens i n 1966 and has s i n c e been added t o the computing c e n t e r ' s l i b r a r y . The m o d i f i c a t i o n s made t o t h i s program a l l o w i t t o be used t o f i n d s m a l l r o o t s w i t h a h i g h degree of a c c u r a c y and a l s o t o check 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 t h e c o e f f i c i e n t s from the r o o t s . These m o d i f i c a t i o n s were n e c e s s i t a t e d by the p r o p e r t i e s of p o l y n o m i a l s (3-13) and (3-17) and by t h e r e q u i r e m e n t t h a t t h e r o o t - f i n d i n g t e c h n i q u e be o p e r a t i v e over a wide range of c o e f f i c i e n t v a l u e s . ( START ) DIMENSION ETC. INPUT .PARAMETERS/ FORM AUXILIARY EQUATIONS AND DETERMINE COEFICIENTS OF POLYNOMIALS SOLVE SIXTH ORDER POLYNOMIAL FOR NON- IDENTICAL ROOTS TEST: WITHIN REQUIRED TOLERANCE YES NO OUTPUT ERROR .MESSAGE, SOLVE FOURTH ORDER POLYNOMIAL FOR IDENTICAL ROOTS ORDER AND OUTPUT Xj.Yj PAIRS DETERMINE Xj.Yj PAIRS FROM SOLUTIONS NO ISOLATE COMPLEX ROOTS ISOLATE COMPLEX ROOTS TEST: DESIRED RANGE OBTAINED YES ( STOP ) YES RECONSTRUCT POLYNOMIAL COEFICIENTS TEST: WITHIN REQUIRED TOLERANCE NO OUTPUT ERROR . M E S S A G E , ORDER AND OUTPUT Xj,Yj PAIRS INCREMENT PARAMETERS F i g u r e B . l Flow Diagram o f Program t o Determine R i t z C o e f f i c i e n t s 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'etude 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. 715-738, 763-780, 795-819, 841-858, 1934. . 2. Rudenberg, R., T r a n s i e n t Performance o f E l e c t r i c Power Systems. 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 from The L e n k u r t Demodulator (Second E d i t i o n ) , "The V a r a c t o r Diode", 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., 1950. 5. H a y a s h i , C , N o n l i n e a r O s c i l l a t i o n s i n P h y s i c a l Systems, M c G r a w - H i l l , 1964- 6. I s b o r n , C , " F e r r o r e s o n a n t F l i p - f l o p s " , E l e c t r o n i c s , A p r i l 1962, pp. 121-123- 7. Gremer, C , "The N o n l i n e a r Resonant T r i g g e r P a i r " , T ran. AIEE. Communications and E l e c t r o n i c s , V o l . 26, Sept. 1956, pp. 404-407. 8. Ozawa, T., N o n l i n e a r Resonance Computer Components, Tech. Rep o r t No. 1306-1, 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., "Steady S t a t e V i b r a t i o n s i n Systems Ha 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 . of the 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. 234-257, 1953. 10. K L o t t e r , K., "Steady S t a t e O s c i l l a t i o n s i n N o n l i n e a r M u l t i - L o o p C i r c u i t s " , Trans. I n s t , of Radio  E n g i n e e r s , P r o f . Group on C i r c u i t Theory, CT-1, No. 4, pp. 13-18, Dec. 1954. 11. • Z e n i t i , K., S e k i g u t i , S., and Takasima, 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 C a p a c i t a n c e o f 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 Communication E n g i n e e r s o f Japan, V o l . 41, No. 3, March 1958, pp. 239-244. 12. J a c k s o n , A., A n a l o g Computation, M c G r a w - H i l l , pp. 182-185, I 9 6 0 . 51 13. Cunningham, ¥., 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 , M c G r a w - H i l l , 1958. 14. R i t z , W., Uber e i n e neue Methode 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. Math., V o l . r35, pp. 1-61, 1909. 15. P a r l e t t , B., "Laguerre's Method A p p l i e d t o t h e M a t r i x E i g e n v a l u e Problem", Mathematics of Computation, V o l . 18, 1964, pp. 464 f f . 16. S t e v e n s , J . , "LAGERE", U n p u b l i s h e d F o r t r a n program, U.B.C. Computing Ce n t e r L i b r a r y , 1966. 

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