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An analog-to-digital conversion circuit using a stack of tunnel diodes each constructed from the same… Strong, James Thomas 1965

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AN ANALOG-TO-DIGITAL CONVERSION CIRCUIT USING A STACK OP ' TUNNEL DIODES EACH CONSTRUCTED FROM THE SAME MATERIAL by JAMES THOMAS STRONG B.E., Nova S c o t i a T e c h n i c a l C o l l e g e , 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT 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 E n g i n e e r i n g ¥e accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d 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 August 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s , , I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada D a t e August S o , ( ?6f ABSTRACT Th i s t h e s i s d e s c r i b e s a m a t h e m a t i c a l - g r a p h i c a l a n a l y s i s and some a n a l o g computer s i m u l a t i o n s t u d i e s t h a t were c a r r i e d out t o determine the f e a s i b i l i t y of a proposed c i r c u i t t o be used f o r a n a l o g — t o — d i g i t a l c o n v e r s i o n . The c i r c u i t a n a l y s e d and s i m u l a t e d c o n t a i n s a s t a c k of t u n n e l d i o d e s w h i c h a re c o n s t r u c t e d from the same type of semi-conductor m a t e r i a l . The s w i t c h i n g c h a r a c t e r i s t i c s of t h i s c i r c u i t are c o n t r o l l e d p r i m a r i l y by the r a t i o s and the v a l u e s of the c a p a c i t a n c e s w h i c h shunt the i n d i v i d u a l t u n n e l d i o d e s and t o a l e s s e r e x t e n t by the i n t e r d i o d e c a p a c i t a n c e s . T h i s i s r e v e a l e d i n a study of the e f f e c t s of d i f f e r e n t c i r c u i t parameter v a r i a t i o n A two t u n n e l diode s t a c k c i r c u i t (two b i t s o f i n f o r m a -t i o n c a p a c i t y ) i s a n a l y s e d by s t u d y i n g the n a t u r e of the s w i t c h -i n g t r a j e c t o r i e s i n the p r o x i m i t y of the s i n g u l a r p o i n t s of the eq u a t i o n s d e s c r i b i n g -the c i r c u i t o p e r a t i o n . Three d i f f e r e n t modes of o p e r a t i o n , each of w h i c h d i f f e r s i n the manner i n w h i c h the 11 s t a t e i s r e a c h e d , are r e v e a l e d f o r t h i s c i r c u i t * The a n a l y s i s i n d i c a t e s a f e a t u r e of the c i r c u i t w h i c h can be used to determine the f i n a l s t a t e of the c i r c u i t b e f o r e s t e a d y s t a t e con-d i t i o n s have been r e a c h e d . An e x t e n s i o n of the two t u n n e l diode s t a c k c i r c u i t to one c o n t a i n i n g t h r e e t u n n e l d i o d e s y i e l d e d e i g h t s t a b l e and a c c e s s i b l e s t a t e s . T h i s i n d i c a t e s t h a t the c i r c u i t proposed w i l l be a b l e t o r e a l i z e 2 n s t a t e s w i t h n t u n n e l d i o d e s . I t i s shown t h a t d i f f e r e n t i n t e r d i o d e c a p a c i t a n c e c o n n e c t i o n s w i l l f a c i l i t a t e the achievement of t h i s r e s u l t . TABLE OF CONTENTS L i s t of I l l u s t r a t i o n s . . . . . . . . L i s t of Tables L i s t of S p e c i a l Symbols and Terms Acknowledgement 1. INTRODUCTION 2. ANALYSIS OF AN ANALOG-TO-DIGITAL CONVERSION CIRCUIT UTILIZING A TWO TUNNEL DIODE STACK CONFIGUATION .... 2.1 Reason f o r A p p l y i n g a S i n g u l a r P o i n t A n a l y s i s to the C i r c u i t C o n t a i n i n g a Two Tunnel Diode S t a c k 2.2 The E q u i v a l e n t C i r c u i t of a Tunnel Diode to be Used i n S i n g u l a r P o i n t A n a l y s i s 2.3 S i n g u l a r P o i n t A n a l y s i s of the Two Tunnel Diode S t a c k C i r c u i t 2.4 S i n g u l a r P o i n t A n a l y s i s A p p l i e d t o a P h y s i c a l 3. ANALOG COMPUTER STUDIES OF AN ANALOG-TO-DIGITAL CONVERSION CIRCUIT UTILIZING A TWO TUNNEL DIODE 3.1 Analog Computer S i m u l a t i o n of the Two Tunnel Diode S t a c k C i r c u i t 3.2 Modes of O p e r a t i o n 3.3 Advance D e t e r m i n a t i o n of Output 3.4 V a r i a t i o n of C i r c u i t Parameters 3.4.1 V a r i a t i o n of & I System STACK 3.4.2 Simultaneous V a r i a t i o n s of and v 3.4.3 V a r i a t i o n of I B 3.4.4 Simultaneous V a r i a t i o n of 3.4.5 V a r i a t i o n of t i v Page 3 . 4 . 6 V a r i a t i o n of CQ, , and .. . 4 2 3.5 Inductance E f f e c t s 4 5 4. ANALOG COMPUTER SIMULATION STUDIES OF A THREE BIT ANALOG-TO-DIGITAL CONVERSION CIRCUIT USING A TUNNEL DIODE STACK CONFIGURATION 4 8 4.1 S i m u l a t i o n of a Three Tunnel Diode Stack C i r c u i t 4 8 4.2 I n t e r d i o d e C a p a c i t a n c e E f f e c t s 4 9 5. SUMMARY AND CONCLUSIONS 5 2 APPENDIX I ANALOG COMPUTER CIRCUIT FOR SIMULATION OF TWO TUNNEL DIODE STACK CIRCUIT . 5 5 APPENDIX I I ANALOG COMPUTER CIRCUITS USED TO STUDY INDUCTANCE EFFECT IN A T¥0 TUNNEL DIODE CIRCUIT ..: 58 A I I . 1 Two Tunnel Diode S t a c k C i r c u i t w i t h Inductance i n S e r i e d w i t h Tunnel Diodes .... 5 8 A I I . 2 Two Tunnel Diode S t a c k C i r c u i t w i t h Inductance i n - S e r i e s w i t h CQ 6 1 A I I . 3 Two Tunnel Diode S t a c k C i r c u i t w i t h Inductance and Damping R e s i s t a n c e 6 2 APPENDIX I I I ANALOG COMPUTER CIRCUIT FOR SIMULATION OF A THREE TUNNEL DIODE STACK CIRCUIT AND INTERDIODE CAPACITANCE EFFECTS ASSOCIATED WITH THIS CIRCUIT 6 7 REFERENCES 7 1 i v V LIST OP ILLUSTRATIONS F i g u r e Page 1- 1 Tunnel Diode S t a c k C o n f i g u r a t i o n ...... .» 1 2- 1 Proposed A n a l o g — t o - D i g i t a l C o n v e r s i o n C i r c u i t U t i l i z i n g a Tunnel Diode S t a c k 4 2-2 Large S i g n a l E q u i v a l e n t C i r c u i t of a Tunnel Diode.. 7 2-3 (a) S i m p l i f i e d Large S i g n a l E q u i v a l e n t C i r c u i t of a Tunnel Diode . . *. 8 (b) P l o t of J u n c t i o n C u r r e n t as a F u n c t i o n of J u n c t i o n V o l t a g e 8 2—4 C i r c u i t C o n t a i n i n g Two Tunnel Diode S t a c k to be A n a l y s e d 9 2-5 E q u i v a l e n t C i r c u i t of Two Tunnel Diode Stack to be A n a l y s e d 9 2—6 I l l u s t r a t i o n of A p p r o x i m a t i o n of Tunnel Diode S t a t i c C h a r a c t e r i s t i c Curve 12 2-7 P l o t I l l u s t r a t i n g Areas of V 2 V ^ P l a n e 18 2- 8 S o l u t i o n Curves i n the P r o x i m i t y of the S i n g u l a r P o i n t s of a Two Tunnel Diode S t a c k C i r c u i t 20 3- 1 I l l u s t r a t i o n of Input P u l s e 1^ 23 3-2 (a) I l l u s t r a t i o n of Tunnel Diode and R e s i s t o r C h a r a c t e r i s t i c Curves 24 (b) Composite C h a r a c t e r i s t i c Curve Developed when R e s i s t o r Shunts Tunnel Diode 24 (c) C i r c u i t w h i c h Produces Curve of 3.2(b) ....... 24 3-3 (a) I l l u s t r a t i o n of Tunnel Diode C h a r a c t e r i s t i c Curve and Magnitude of B i a s C u r r e n t I 25 (b) Composite C h a r a c t e r i s t i c Curve f o r Tunnel Diode Shunted by B i a s Source I . 25 (c) C i r c u i t w h i c h Produces Curve 3.3(b) 25 3-4 (a) I l l u s t r a t i o n of Mode 1 O p e r a t i o n 27 (b) I l l u s t r a t i o n of Mode 2 O p e r a t i o n 27 (c) I l l u s t r a t i o n of Mode 3 O p e r a t i o n 27 v F i g u r e Page 3-5 F a m i l y of Mode 1 O p e r a t i o n Curves 28 3-6 F a m i l y of Mode 2 O p e r a t i o n Curves 29 3-7 F a m i l y of Mode 3 O p e r a t i o n Curves 30 3-8 I l l u s t r a t i o n of the Manner i n whi c h Four S t a b l e S t a t e s are Obtained i n the Time Domain f o r Mode 1 O p e r a t i o n 32 3—9 I l l u s t r a t i o n of the Manner i n which Four S t a b l e S t a t e s are Ob t a i n e d i n the Time Domain f o r Mode 2 O p e r a t i o n 33 3-10 I l l u s t r a t i o n of the Manner i n which Four S t a b l e S t a t e s are Ob t a i n e d i n the Time Domain f o r Mode 3 O p e r a t i o n 34 3-11 V a r i a t i o n of A V a l u e s as a F u n c t i o n of I . • • 38 3-12 V a r i a t i o n of A V a l u e s when & I i s Decreased and & I i s I n c r e a s e d S i m u l t a n e o u s l y 39 3-13 V a r i a t i o n of A V a l u e s as a F u n c t i o n of I g ....... 41 3-14 V a r i a t i o n of A V a l u e s as a F u n c t i o n of 42 3-15 V a r i a t i o n of A V a l u e s as a F u n c t i o n of 43 3-16 V a r i a t i o n of A Values; as a F u n c t i o n of C^(=C2) . • 43 3- 17 V a r i a t i o n of A V a l u e s as a F u n c t i o n of 44 4— 1 I l l u s t r a t i o n of the Manner i n whi c h E i g h t S t a b l e S t a t e s are Obta i n e d i n the Time Domain f o r Three Tunnel Diode C i r c u i t 50 A I - I A n a l o g C i r c u i t of Two Tunnel Diode S t a c k C i r c u i t 57 A I I - l Two Tunnel Diode S t a c k C i r c u i t w i t h Inductance i n S e r i e s w i t h Tunnel Diodes 58 A I I - 2 Ana l o g C i r c u i t f o r Two Tunnel Diode Stack C i r c u i t w i t h Inductance i n S e r i e s w i t h Tunnel Diodes .. 60 A I I - 3 Two Tunnel Diode S t a c k C i r c u i t w i t h Inductance i n S e r i e s w i t h CQ * 61 A I I - 4 Ana l o g C i r c u i t f o r a Two Tunnel Diode Stack C i r c u i t w i t h Inductance i n S e r i e s w i t h CQ...... 63 A I I — 5 .Two Tunnel Diode S t a c k C i r c u i t w i t h Inductance and Damping R e s i s t a n c e ; 64 v i F i g u r e Page A I I - 6 Analog C i r c u i t f o r a Two Tunnel Diode Stack C i r c u i t w i t h Inductance and Damping R e s i s t a n c e . 66 A I I I - l Three Tunnel Diode Stack C i r c u i t C o n t a i n i n g One A d d i t i o n a l I n t e r d i o d e C a p a c i t o r , C^ 67 A I I I - 2 A n a l o g C i r c u i t f o r a Three Tunnel Diode S t a c k C i r c u i t w i t h an A d d i t i o n a l I n t e r d i o d e C a p a c i t o r , C 4 70 v i i LIST OF TABLES Table Page 2-1 V a l u e s of r , and r 0 f o r A p p r o x i m a t i o n of f ( v j and f ( v ) A i 18 2- 2 D e s c r i p t i o n of S i n g u l a r i t i e s of C i r c u i t A n a l y s i s .. 19 3— 1 V a l u e s of C a p a c i t a n c e s and T h r e s h o l d C u r r e n t s f o r Mode 1, Mode 2, and Mode 3 C i r c u i t s ............... 31 3-2 V a r i a t i o n of X** a n t^ C i r c u i t S w i t c h i n g Time w i t h Change i n I g 3- 3 V a r i a t i o n of A V a l u e s w i t h Change of C i r c u i t P a r a -meters 45 4- 1 V a r i a t i o n of T h r e s h o l d V a l u e s w i t h C^ 51 v i i i LIST OF SPECIAL SYMBOLS AND TERMS Symbol F i r s t D e f i n e d i n S e c t i o n I Diode Peak P o i n t C u r r e n t 2.2 P I Diode V a l l e y P o i n t C u r r e n t 2.2 v J V Diode Peak P o i n t V o l t a g e 2.2 P 6 V v Diode V a l l e y P o i n t V o l t a g e 2.2 V Diode Forward V o l t a g e 2.2 f ( v ) Diode v v s . i S t a t i c C h a r a c t e r i s t i c 2.2 t h I . mn S t a t e T h r e s h o l d C u r r e n t V a l u e 3. Amn t h A Range of mn S t a t e 3. mn & & I D i f f e r e n c e i n the Peak C u r r e n t s of Two p Diodes 3.4 £ I D i f f e r e n c e i n the V a l l e y C u r r e n t s of v Two Diodes 3.4 Term Tunnel Diode Stack 1. T h r e s h o l d C u r r e n t 3. Mode 1 O p e r a t i o n 3.2 Mode 2 O p e r a t i o n 3.2 Mode 3 O p e r a t i o n 3.2 i x X ACKNOWLEDGEMENT The author would l i k e to express h i s s i n c e r e thanks and a p p r e c i a t i o n t o Dr. M.P. Beddoes, the s u p e r v i s i n g p r o f e s s o r of t h i s p r o j e c t , f o r h i s encouragement, h e l p f u l s u g g e s t i o n s , and p a t i e n c e d u r i n g the course of t h i s r e s e a r c h . The a u t h o r would a l s o l i k e to express h i s indebtedness t o Dr. J.S. MacDonald of the Mas s a c h u s e t t s I n s t i t u t e of Technology, Cambridge, M a s s a c h u s e t t s , U.S.A., f o r a v e r y i l l u m i n a t i n g and h e l p f u l p r i v a t e communication. The work d e s c r i b e d i n t h i s t h e s i s was supporte d by the N a t i o n a l Research C o u n c i l of Canada under Grant BT-68. x 1. INTRODUCTION The invention of the tunnel d i o d e ^ ^ produced a switch-ing device with the following c h a r a c t e r i s t i c s : very high switching speed, low power dis s i p a t i o n , and small physical dimen sions. These characteristics make the tunnel diode very attrac-tive for application i n d i g i t a l c i r c u i t s . One particular group of c i r c u i t s in which the tunnel diode can be used i s the group of c i r c u i t s used for analog-to-digital conversion. For this application, several a u t h o r s ^ ^ ' ^ ^ ' ^ ^ have proposed a c i r c u i t containing a number of tunnel diodes i n series; this type of c i r c u i t i s known as a tunnel diode "stack". The stack configura tion i s i l l u s t r a t e d i n Figure 1-1. D, D. D n D n Figure 1-1. Tunnel Diode Stack Configuration. In the stack configuration a l l the tunnel diodes are operated as voltage dependent bistable devices. The low voltage state and the high voltage state of each,tunnel diode are referred to as the '0* state and the '1' state, respectively. Thus the steady state of the tunnel diode stack may be presented in binary notation. For example, the binary notation 1011..., infers that D i s i n the high state, D , i s in the low state, n 6 ' n-1 ' 2 etc. When a tunnel diode stack i s used i n an analog-to—digital conversion c i r c u i t , the binary notation representing the steady state voltages of the diodes i n the stack also represents the magnitude of the analog signal driving the c i r c u i t . For such an application Diode 1 (D^) represents the lowest or "least s i g n i f i c a n t " b i t . (2) Renton and Rabinovici developed c r i t e r i a to govern the shape of the s t a t i c c h a r a c t e r i s t i c s of n voltage controlled negative resistance devices (called 'S' type negative resistance devices) i n series or n current controlled negative resistance devices ('N' type) i n p a r a l l e l , when i t i s required that these configurations produce 2 n d i s t i n c t and accessible states. When these c r i t e r i a are applied to the tunnel diode stack configura-tion i t i s seen that the number of tunnel diodes which may be used i n the stack: i s r e s t r i c t e d to approximately four. This r e s t r i c t i o n i s due to the fact that one of the c r i t e r i a requires that each successive diode of the stack must have a forward voltage equal to twice the forward voltage of the preceeding diode. (3) Salama investigated the dynamic char a c t e r i s t i c s of a c i r c u i t containing a two tunnel diode stack when the input sampling speed was i n the range of the switching speed of the tunnel diodes. This study pointed out the fact that'the junction capacitances shunting the tunnel diodes play a major role i n determining the operational c h a r a c t e r i s t i c s of thi s c i r c u i t . This thesis proposes a means of obtaining analog—to-d i g i t a l conversion using a tunnel diode stack configuration i n which the tunnel diodes are constructed from the same type of semiconductor m a t e r i a l and i n w h i c h 2 s t a t e s can be o b t a i n e d u s i n g 'n' t u n n e l d i o d e s . T h i s e f f e c t i s a c h i e v e d w i t h a n o v e l c i r c u i t i n w h i c h the c a p a c i t a n c e s t h a t shunt the d i o d e s must be s u i t a b l y chosen. There are t h r e e main s e c t i o n s i n t h i s t h e s i s . A m a t h e m a t i c a l i n v e s t i g a t i o n i n t o the dynamic c h a r a c t e r i s t i c s of a c i r c u i t c o n t a i n i n g a two t u n n e l diode s t a c k i s p r e s e n t e d i n Chapter 2. By a p p l y i n g a c o m b i n a t i o n of a n o n l i n e a r a n a l y s i s t e c h n i q u e and a g r a p h i c a l t e c h n i q u e to the s i n g u l a r i t i e s of the e q u a t i o n s d e s c r i b i n g the o p e r a t i o n of the c i r c u i t , the c h a r a c -t e r i s t i c s of the system s w i t c h i n g t r a j e c t o r i e s i n the p r o x i m i t y of the s i n g l u l a r p o i n t s are d e s c r i b e d . T h i s a n a l y s i s proves to be a v e r y v a l u a b l e t o o l i n u n d e r s t a n d i n g the s w i t c h i n g c h a r a c t e r -i s t i c s of the c i r c u i t . By s i m u l a t i n g the c i r c u i t s t u d i e d i n Chapter 2 on a PACE 231R a n a l o g computer, i t was p o s s i b l e to obse'rve and r e c o r d the e f f e c t s of d i f f e r e n t parameter v a r i a t i o n s on the c i r c u i t o p e r a t i o n . The r e s u l t s of t h i s i n v e s t i g a t i o n were complemented by the a n a l y t i c a l t o o l s p r e s e n t e d i n Chapter 2. These r e s u l t s as w e l l as a s t u d y of the e f f e c t s of p a r a s i t i c i n d u c t a n c e on the c i r c u i t o p e r a t i o n are p r e s e n t e d i n Chapter 3. A t h r e e t u n n e l diode s t a c k c o n f i g u r a t i o n was a l s o s i m u l a t e d on the a n a l o g computer. The r e s u l t s of the i n v e s t i g a -t i o n of the dynamic c h a r a c t e r i s t i c s of t h i s c i r c u i t are p r e s e n t e d i n Chapter 4. A l s o i n c l u d e d i n t h i s c h a p t e r i s an i n v e s t i g a t i o n of the e f f e c t s of p l a c i n g an a d d i t i o n a l i n t e r d i o d e c a p a c i t a n c e i n the c i r c u i t . 4 2. ANALYSIS OF AN ANALOG-TO-DIGITAL CONVERSION CIRCUIT UTILIZING A TWO TUNNEL DIODE STACK CONFIGURATION Renton and R a b i n o v i c i have i l l u s t r a t e d , by means of E q u a t i o n ( 2 - l ) , the o b s t a c l e c o n f r o n t e d when a t t e m p t i n g t o o b t a i n 2 n d i s t i n c t and a c c e s s i b l e s t a t e s from n t u n n e l d i o d e s connected i n a s t a c k c o n f i g u r a t i o n , V- = '.2^m_1V- , f o r m = 2 to n (2-1) f,m f , l t h where V„ i s the f o r w a r d v o l t a g e of the m t u n n e l d i o d e . S i n c e f,m e the f o r w a r d v o l t a g e of a t u n n e l diode i s d i r e c t l y r e l a t e d to the band gap v o l t a g e of the semiconductor m a t e r i a l from w h i c h the d e v i c e i s c o n s t r u c t e d , the number of d i f f e r e n t t y p e s of diod e s r e q u i r e d f o r the s t a c k i s equal t o the number of d i g i t s a v a i l a b l e . At p r e s e n t t h i s number i s l i m i t e d t o f o u r . T h i s t h e s i s proposes a n o v e l c i r c u i t w hich w i l l c i r c u m -v e n t the c o n s t r i c t i n g f o r w a r d v o l t a g e c r i t e r i o n ; see F i g u r e 2-1. F i g u r e 2-1. Proposed A n a l o g - t o - D i g i t a l C o n v e r s i o n C i r c u i t U t i l i z i n g a Tunnel Diode S t a c k . 5 1 ^ i s the i n p u t c u r r e n t p u l s e w h i c h i s a sample of the a n a l o g s i g n a l and I g i s the b i a s c u r r e n t s o u r c e . By c o n t r o l l i n g the c h o i c e o f c a p a c i t o r s C^ C Q •-3 w h i c h shunt each t u n n e l diode and a l s o by c o n t r o l l i n g the v a l u e o f C Q ( t h e feedback c a p a c i t o r ) , i t i s p o s s i b l e to govern the s w i t c h i n g c h a r a c t e r i s t i c s o f the c i r c u i t t o the e x t e n t t h a t the R e n t o n - R a b i n o v i c i f o r w a r d v o l t a g e c r i t e r i o n may be r e l a x e d . T h i s c h a p t e r p r e s e n t s a m a t h e m a t i c a l - g r a p h i c a l a n a l y s i s o f the c i r c u i t i l l u s t r a t e d i n F i g u r e 2-1 where n = 2 . T h i s a n a l y s i s a i d s t h e u n d e r s t a n d i n g o f the c i r c u i t o p e r a t i o n and g i v e s i n s i g h t i n t o the manner i n w h i c h the o p e r a t i o n i s a f f e c t e d by the v a r i a t i o n o f d i f f e r e n t c i r c u i t p a r a m e t e r s . The a c t u a l e f f e c t s o f the parameter v a r i a t i o n s w i l l be p r e s e n t e d i n Chap-t e r 3. 2.1 , Reason f o r A p p l y i n g a S i n g u l a r P o i n t A n a l y s i s t o the C i r c u i t C o n t a i n i n g a Two Tunnel Diode S t a c k . There are t h r e e ways i n w h i c h the d u a l problem of ( l ) e x p l a i n i n g the o p e r a t i o n of a c i r c u i t and (2) e x p r e s s i n g the c i r c u i t parameters i n t e r d e p e n d e n c y may be t a c k l e d * The f i r s t i s an a n a l y t i c a l approach bu t o f t e n t h i s approach i s t o o t e d i o u s . The second approach i s a g r a p h i c a l t e c h n i q u e i n which the system s o l u t i o n c u r v e s c a l l e d " t r a j e c t o r i e s " are p l o t t e d . These t r a j e c t o r i e s may i l l u s t r a t e the v a r i a t i o n o f one p a r a -meter as a f u n c t i o n o f time o r t h e y may i l l u s t r a t e the v a r i a t i o n o f one parameter as a f u n c t i o n o f a n o t h e r parameter. The main 6 disadvantage with this approach i s that i t i s d i f f i c u l t to obtain a generalized c r i t e r i o n covering the system behaviour under a l l conditions, although one particular case can be well i l l u s t r a t e d . The t h i r d approach to the problem i s a combination of an a n a l y t i c a l and graphical technique. If the operation of a c i r c u i t can be described by a set of d i f f e r e n t i a l equations, then a s t u d y ^ ^ ' ^ ^ of the singular points of these equations i s very worthwhile, since i t i s these singular points which are basic in determining the nature of the solutions of these equations. By graphically representing the solution curves i n the proximity of each singular point i t i s possible to i l l u -strate how one or more s i n g u l a r i t i e s a f f e c t the shape of the solution curves. A s i n g u l a r i t y analysis i s both rigorous and i l l u s t r a t i v e since (l) the description of the s i n g u l a r i t i e s holds for a l l cases and (2) their influence on the solution curves may be i l l u s t r a t e d for a l l cases at once. The main r e s t r i c t i o n on t h i s type of analysis i s that only two para-meters can be i l l u s t r a t e d at once since the solution curve i s only plotted i n two dimensions. Because of these advantages of t h i s approach and because the r e s t r i c t i o n stated does not apply for the case to be discussed, a si n g u l a r i t y analysis i s very applicable for the study of a two tunnel diode stack c i r c u i t . The complete mathematical theory of the singular point (5) analysis i s described i n Cunningham (Chapter 5) and shall not be repeated here. 7 2.2 The E q u i v a l e n t C i r c u i t of a Tunnel Diode t o be Used i n the S i n g u l a r P o i n t A n a l y s i s . For l a r g e s i g n a l a n a l y s i s a p p l i c a t i o n s the t u n n e l d iode can be s a t i s f a c t o r i l y r e p r e s e n t e d by i t s v o l t a g e depen-dent j u n c t i o n c a p a c i t a n c e , a n o n l i n e a r c u r r e n t source w h i c h i s a f u n c t i o n of the v o l t a g e a c r o s s the j u n c t i o n , a s e r i e s r e s i s t a n c e r e p r e s e n t i n g the b u l k and c o n t a c t r e s i s t a n c e s , and an i n d u c t o r . The c i r c u i t c o n f i g u r a t i o n used f o r t h i s r e p r e -s e n t a t i o n i s i l l u s t r a t e d i n F i g u r e 2-2. b u l k and c o n t a c t r e s i s t a n c e l e a d i n d u c t a n c e j u n c t i o n c a p a c i t a n c e f ( v ) n o n l i n e a r c u r r e n t source F i g u r e 2-2. Large S i g n a l E q u i v a l e n t C i r c u i t of a Tunnel Diode. S i n c e the magnitudes of R , and L are v e r y s m a l l and s s a l s o s i n c e the c a p a c i t a n c e i s much l e s s dependent on j u n c t i o n v o l t a g e t h a n the j u n c t i o n c u r r e n t , the s i m p l i f i e d c i r c u i t shown i n F i g u r e 2-3(a) can a d e q u a t e l y r e p r e s e n t the t u n n e l diode used i n the d e s i g n t o be a n a l y s e d . The c u r r e n t produced by the n o n l i n e a r c u r r e n t source i s p l o t t e d as a f u n c t i o n of v o l t a g e i n F i g u r e 2—3(b). r V 1-R s L s C(v) i = f ( v ) (b) F i g u r e 2-3 ( a ) . S i m p l i f i e d Large S i g n a l E q u i v a l e n t C i r c u i t of a Tunnel Diode. ( b ) . P l o t of J u n c t i o n C u r r e n t F u n c t i o n of J u n c t i o n V o l t a g e . The i v e r s u s v curve p l o t t e d i n F i g u r e 2.3(b) i s r e f e r r e d t o as the " s t a t i c c h a r a c t e r i s t i c " of a t u n n e l d i o d e . The symbols used to i d e n t i f y the d e f i n i n g p o i n t s of t h i s c h a r a c t e r i s t i c are shown i n t h i s f i g u r e arid are e x p l a i n e d below. I = Diode Peak P o i n t 0 u r r e n t P I v = Diode V a l l e y P o i n t C u r r e n t V = Diode Peak P o i n t V o l t a g e V v = Diode V a l l e y P o i n t V o l t a g e V-, = Diode Forward V o l t a g e a t I 2.3 S i n g u l a r P o i n t A n a l y s i s of the Two Tunnel Diode S t a c k C i r c u i t . A s i n g u l a r p o i n t a n a l y s i s w i l l be c a r r i e d out f o r the c i r c u i t shown i n F i g u r e 2-4. 9 F i g u r e 2-4. C i r c u i t C o n t a i n i n g Two Tunnel Diode S t a c k to be S t u d i e d . U s i n g the e q u i v a l e n t c i r c u i t shown i n F i g u r e 2-3(a) to r e p r e s e n t D^ and B^f the c i r c u i t of F i g u r e 2-4 can be i l l u s t r a t e d as i n F i g u r e 2-5. The c u r r e n t s and v o l t a g e s t h a t w i l l be used i n the a n a l y s i s are shown i n t h i s diagram. F i g u r e 2-5. E q u i v a l e n t C i r c u i t of Two Tunnel Diode S t a c k C i r c u i t t o be A n a l y s e d . The t u n n e l d i o d e s i n t h i s c i r c u i t are c o n s t r u c t e d of the same type of semiconductor m a t e r i a l and t h e r e f o r e V .c^. V ~, V and V„ , — V _ The peak c u r r e n t p,1 p,2' v, 1 Y.2' f , l f , 2 ^ and the v a l l e y c u r r e n t r e l a t i o n s h i p s between the t u n n e l d i o d e s are g i v e n by the i n e q u a l i t i e s ; 10 I -1 < I 0 p , l p,2 ^ , 1 ^ , 2 (2.2) In r e a l i t y 1^ i s a c u r r e n t p u l s e which has a d e f i n i t e r i s e time and f a l l t i m e , but f o r t h i s a n a l y s i s i t w i l l be assumed t o be a p e r f e c t r e c t a n g u l a r p u l s e . At q u i e s c e n c e , s i n c e I . and I., are z e r o , o n l y I_, f l o w s i n t o the s e c t i o n of the c i r c u i t c o n t a i n i n g the t u n n e l d i o d e s and t h e r e f o r e , (2.3) f ( v x ) = f ( v 2 ) at t = 0 C o n s i d e r the e q u a t i o n s d e s c r i b i n g the o p e r a t i o n of t h i s c i r c u i t * dv 1 ~d¥ = < X l + V " f < V dv \ = ( I , + I B ) - f ( v 2 ) 2 d t C i ^ l + ^ l ' 1 °0 d t + d t , = I 0 (2.4) (2.5) (2.6) A = I 0 + I l (2.7) By rearrangement and s u b s t i t u t i o n of the p r e v i o u s f o u r e q u a t i o n s the f o l l o w i n g e x p r e s s i o n f o r 1^ i s d e v e l o p e d : C C J A ~ a I B + C 2 f < V + C T ^ V (2.8) where; a = 0 and 6 = 1 + a 11 From E q u a t i o n s (2.4) and ( 2 . 8 ) , dv, d t q [ C 2 ( I A + V ~ <C0 + C l } f ( V + °0 f ( v 2 } _ (2.9) where K ]. = + Cfa + S i m i l a r l y by m a n i p u l a t i n g E q u a t i o n s (2.5) and (2.8) and u s i n g the same e x p r e s s i o n s f o r , a, and |3, the e x p r e s s i o n f o r dv^/dt becomes dv _2 d t C 1 ( I A + V + C 0 f ( V " ( C 0 + C 1 > f ( v 2 > (2.10) T a k i n g the r a t i o of E q u a t i o n (2.9) and E q u a t i o n (2.10) y i e l d s the f o l l o w i n g e x p r e s s i o n : K 7 [ C 2 ( I A + C 0 f ( v 2 ) - ( C 0 + C 2 ) f ( v l } ] dv dv 2 t j L - [ C l ( l A + I B ) + C 0 f ( V l ) - ( C 0 + C l ) f ( v 2 ) ] (2.11) By d e f i n i t i o n , the s i n g u l a r i t i e s of E q u a t i o n ( 2 . 1 l ) o c c u r when v^ and v ^ are such t h a t dv^/dt = 0 and d v 2 / d t — 0 s i m u l t a n e o u s l y . I n o t h e r words, c 2 ( i A + i B ) + c 0 f ( v 2 ) - ( C Q + c 2 ) f ( V l ) = 0 C 1 ( I A + V + C 0 f ( V " ( C 0 + C l > f ( v 2 } =' 0 (2.12) a t the same t i m e . S o l v i n g E q u a t i o n s (2.12) s i m u l t a n e o u s l y y i e l d s the f o l l o w i n g v a l u e s of f ( v ^ ) and f ( v 2 ) a t a s i n g u l a r p o i n t : f ( V l ) = f ( v 2 ) = I A + I B (2.13) The equilibrium points of the system or of one particu-l a r element of the system are graphically i l l u s t r a t e d by the intersections of the system's or the elements*s characteristic curve with the appropriate load l i n e . In this case the system load l i n e i s an i n f i n i t e resistance load line since the c i r c u i t i s being fed from current sources. For the purpose of this analysis i t i s acceptable to approximate the tunnel diode s t a t i c c h a r a c t e r i s t i c curve by a f i v e l i n e a r segment curve. This approximation i s shown i n Figure 2-6. Figure 2-6. I l l u s t r a t i o n of Approximation of Tunnel Diode Static Characteristic Curve. Let v be the value of v at a s i n g u l a r i t y , l e t v_ 1S J_ S be the value of v •at a s i n g u l a r i t y , and also f(v. ) and f(v_ ) be the respective tunnel diode currents at a s i n g u l a r i t y * Next consider a point on the V 2 V 1 P-*-ane "that i s very near a singular point. Since the f(v^) and the ^(v^) functions have been approximated by l i n e a r segments, the values of f(v^) and f(v^) at t h i s point near the s i n g u l a r i t y may be expressed i n the following way: 13 d f ( v - ) f ( v . ) = f ( v . ) + . 1 1' v I s dv. d f ( v j f (v 2) = f ( v 2 s ) + a I s 2s D e f i n i n g — 1 d f ( v x ) 1 —z and — = d v l r 2 (2.14) becomes f ( v 2 ) = f ( v 2 g ) + r b dv„ (2.14) E q u a t i o n (2.14a) F o r t h i s p a r t i c u l a r problem the magnitudes of r ^ and r 2 w i l l have f i v e d i f f e r e n t v a l u e s , one c o r r e s p o n d i n g t o each l i n e a r segment. Now c o n s i d e r E q u a t i o n (2.9) and E q u a t i o n (2*10) a t a p o i n t near a s i n g u l a r i t y . E q u a t i o n (2.9) becomes d ( T ls> . d ( V 1 d t d t S ^ A + V + C 0 f < V 2 s > -< C6 + C2> f ( v l s > C Q V b ( C 0 + C 2 } a d ( v l s ) 1 C 2 ^ A + V + °0 F ^ 2 8 ) - ( C 0 + C 2 ) f ^ 1 . ) = o by d e f i n i t i o n a t the s i n g u l a r i t y . T h e r e f o r e , the e q u a t i o n i s reduced to the f o l l o w i n g form: dv a 1 d t ~ K, C 0 V b ( C 0 + C 2 ) a (2.15) 14 By similar manipulation of Equation (2.10) for a point close to a singularity, the equation becomes dt 0 a ( c 0 + C l ) (2.16) x l 2 Taking the r a t i o of Equation (2.15) and Equation (2.16) yields the expression dv j dv. a V b L r 2 ( c 0 + c 2 ) r l V a [ V a _ r l ( c 0 + C l ) r 2 V b i n t h e same f o r m a s t h e n a m e l y , dv c u + dv d u a u + b v (2.17) (2.18) Therefore, for this problem the values of the constants a, b, c, and d are given below: K, * r ' '0 ) b = '0 K l r l (2.19) c = K f * l r 2 d _I ( Co + % i r i The values of the roots and \^ of the solution curves in the proximity of a sin g u l a r i t y are given by the expression (\v >»2) = 1 j ( a + d) ± |^(a + d)^ + 4(bc - ad) (2.20) 15 T h e r e f o r e , r l ( C 0 + C l ) + r 2 ( C 0 + C 2 ) 2 K l r l r 2 r l ^ C 0 + C l ) + r 2 ( C 0 + C 2 )  K l r l r 2 K l r l r 2 (2.21) Si n c e the t u n n e l diode s t a t i c c h a r a c t e r i s t i c c o n t a i n s b o t h r e g i o n s of p o s i t i v e r e s i s t a n c e and r e g i o n s of n e g a t i v e r e s i s t a n c e , i t i s i m p o r t a n t to i n v e s t i g a t e the type of s i n g u l a r i t y t h a t would e x i s t f o r the f o l l o w i n g c a s e s : r ^ and x^ are both n e g a t i v e , r ^ and x^ are of o p p o s i t e p o l a r i t y , and r ^ and x^ are b o t h p o s i t i v e . By s u b s t i t u t i n g i n the v a l u e s of a, b, c, and d i n t o the term 4(bc - a d ) , the f o l l o w i n g e q u a l i t y r e s u l t s : 4(be - ad) •= -K l r l r 2 (2.22) I f e i t h e r r ^ or x^ i s n e g a t i v e then E q u a t i o n s (2.22) s t a t e s t h a t the term 4(bc - ad) w i l l be p o s i t i v e . T h i s b e i n g the c a s e , and \^ w i l l be r e a l and o p p o s i t e i n s i g n and the u n s t a b l e s i n g u l a r i t y formed w i l l be a saddle p o i n t . C o n s i d e r n e x t the s i n g u l a r i t i e s t h a t l i e i n r e g i o n s where b o t h r ^ and ^ are n e g a t i v e . From the e x p r e s s i o n , (a + d) '= VC 1 , C 0 + C 2 (2.23) ^2 A l i t i s seen t h a t when b o t h r ^ and r ^ are n e g a t i v e the whole e x p r e s s i o n i s p o s i t i v e s i n c e n e i t h e r CQ, C^,nor C 2 can be a n e g a t i v e q u a n t i t y . S i n c e b o t h (a +. d) and 4(bc - ad) are n e g a t i v e f o r t h i s r e g i o n , the s i n g u l a r i t y formed w i l l be e i t h e r an unstable node or an unstable focus. Therefore, i t follows from the conclusions following Equations (2.22) and (2.23) that, i f a s i n g u l a r i t y f a l l s i n a region where either r^ or x^ i s negative, or where both r^ arid x^ are negative, then the solution curves i n the proximity of this s i n g u l a r i t y w i l l be unstable. The requirements for a vortex to be formed are (l) (a + d) =0 and (2) 4(bc - ad)<0 simultaneously. If either r^ or x^ i s negative, then condition (2) cannot be met. If either r^ and r ^ are positive, or i f both r^ and x^ are negativ condition (l) cannot be met. Therefore, for t h i s c i r c u i t , a vortex cannot exist anywhere on the "^v^ plane. The s i n g u l a r i t i e s that occur i n regions where both r-^  and r 2 are positive w i l l be stable since (a + d)<0 and 4(bc - ad)<0. These s i n g u l a r i t i e s w i l l be stable f o c i i f -(a + d) >4(bc — ad) or they w i l l be stable nodes i f -(a + d) 2<4(bc - ad). For this c i r c u i t , the value of Ig i s chosen to create three s i n g u l a r i t i e s per tunnel diode during steady state condi-tions; i . e . 1^ = 1 ^ = 0 . Since there are two tunnel diodes, there are nine s i n g u l a r i t i e s . Four of these s i n g u l a r i t i e s wil^L occur i n regions where r^ and x^ are positive and are therefore stable. These s i n g u l a r i t i e s couid be either stable nodes or stable f o c i . Four other s i n g u l a r i t i e s w i l l occur i n regions where either r^ or x^ i s negative and therefore these s i n g u l a r i t i e s are unstable. The s i n g u l a r i t i e s occurring i n these regions are saddle points. One si n g u l a r i t y w i l l occur when both r, and x0 are negative and thus i t i s unstable: i t w i l l be e i t h e r an u n s t a b l e node or an u n s t a b l e f o c u s . S i n c e t h e r e i s no i n d u c t a n c e i n the c i r c u i t c o n s i d e r e d , no o s c i l l a t i o n s can occur and t h e r e f o r e , no f o c i or v o r t e x e s can be formed. I f an i n d u c t i v e element were to e x i s t , o s c i l l a -t i o n s would be e x p e c t e d t o occur near some of the s i n g u l a r i t i e s . Thus, f o r l a r g e i n d u c t a n c e s , the o p e r a t i o n of t h i s c i r c u i t would be a l t e r e d from t h a t o u t l i n e d i n t h i s a n a l y s i s . F or i n s t a n c e , i f the magnitude of the o s c i l l a t i o n s b u i l t up to a l a r g e v a l u e near one of the u n s t a b l e s i n g u l a r i t i e s , one of the t u n n e l d i o d e s might end up a t an i n c o r r e c t s t a t e . 2.4 S i n g u l a r P o i n t A n a l y s i s A p p l i e d t o a P h y s i c a l System. For the purpose of i l l u s t r a t i n g t h i s a n a l y t i c a l -g r a p h i c a l t e c h n i q u e of s t u d y i n g the b e h a v i o u r of a c i r c u i t , c o n s i d e r a two t u n n e l diode c i r c u i t w h i c h has the f o l l o w i n g p a rameters: I = 1,1 mA., I , = 0.40 mA., I 0 = 1.2 mA., p , l v , l p,2 I v 2 = -0.23 mA,, I g = 0.45 mA., C Q = 12 pF., = 2pF., and d>2 = 1*1 P-^ * s t a t i c c h a r a c t e r i s t i c c u r v e s f o r and w i l l be approximated w i t h f i v e l i n e a r segments h a v i n g the v a l u e s of r ^ , r 2> and v o l t a g e i n c r e m e n t s g i v e n i n Table 2—1. The axes of the v 2 v ^ plane are each d i v i d e d i n t o f i v e d i f f e r e n t r e g i o n s which c o r r e s p o n d to the f i v e l i n e a r segments of the approximate f ( v ) c u r v e s . Each of the t w e n t y - f i v e areas on the ^2V1 P ^ a n e be i d e n t i f i e d by q u o t i n g the two r e g i o n s of the axes w h i c h bound i t ( F i g u r e 2-7). For example, the a r e a bounded by 0 < V 2 60 mV. and 60 mV. *^ v^ <^  200 mV. i s known as A r e a ( 1 - 2 ) . Each of the n i n e s i n g u l a r i t i e s w i l l be i d e n t i f i e d by the same number as the a r e a i n w h i c h i t l i e s . 18 Diode 1 Diode 2 A r e a V o l t a g e Range (mV*) ( n ) V o l t a g e Range (mV.) r 2 ( A ) 1 0 — 60 +54.3 0 — SO +50 2 60 — 200 -244 60 — 200 -151 3 200 — 300 -769 200 — 400 -417 4 300 — 400 +358 400 — 440 +131 5 400 — +114 440 — +35.7 Table 2-1. V a l u e s of r . and r f o r A p p r o x i m a t i o n of f ( v ) and f ( v 2 ) . 1 ^ 1 1-5 1-4 1-3 1-2. I - I Z - 5 z - 4 Z-3 2 — 2 2^1 3 — 5 3 - 4 -3 — 3 3 — e 3 - I I 4-4 4-5 4-3 4-2 4-1 5 - 5 5 - 4 -5-3 5 - 2 5-1 4 3 F i g u r e 2-7* P l o t I l l u s t r a t i n g Areas of V 2 v i •* > l a n e* Table 2—2 g i v e s a d e s c r i p t i o n of the n i n e s i n g u l a r i t i e s of t h i s system. 19 A r e a C o o r d i n a t e s ( v 2 — v.^ mV. Type of S i n g u l a r i t y S t a b i l i t y 1-1 15 — 15 node s t a b l e 1-3 15 — 240 saddle u n s t a b l e 1-4 15 — 330 node s t a b l e 2-1 175 — 15 saddle u n s t a b l e 2-3 175 — 240 node u n s t a b l e 2-4 175 — 330 saddle u n s t a b l e 5-1 435 — 15 node s t a b l e 5-3 435 — 240 saddle u n s t a b l e 5-4 435 — 330 node s t a b l e Table 2-2. D e s c r i p t i o n of S i n g u l a r i t i e s of C i r c u i t A n a l y s i s . The s i n g u l a r i t i e s and the s o l u t i o n c u r v e s i n the p r o x i m i t y of these s i n g u l a r i t i e s are p l o t t e d i n F i g u r e 2-8. As p o i n t e d out e a r l i e r i n t h i s c h a p t e r , these s i n g u l a r i t i e s and s o l u t i o n curves e x i s t i n these p o s i t i o n s o n l y f o r steady s t a t e c o n d i t i o n s , i . e . when 1^ = 1^ = 0. When.an i n p u t i s a p p l i e d t o the c i r c u i t the p o s i t i o n s of these s i n g u l a r i t i e s are a l t e r e d . Because the f ( v ) c h a r a c t e r i s t i c c u r v e s are n o n l i n e a r , the p o s i t i o n s of the s i n g u l a r i t i e s move i n a v e r y i n t r i c a t e manner. T h e r e f o r e , i t would n ot be p r a c t i c a l t o attempt t o p l o t a com-p l e t e s e t of s o l u t i o n c u r v e s . I n o r d e r t o d e s c r i b e , q u a l i t a t i v e l y , the i n f l u e n c e of the d i f f e r e n t s i n g u l a r i t i e s on the shape of a s o l u t i o n curve f o r a g i v e n i n p u t , i t i s b e s t t o c o n s i d e r how the p a t h of a v 2 v i p o i n t i s c o n s t r u c t e d a f t e r the i n p u t i s a p p l i e d . C o n s i d e r t h a t the c i r c u i t i s i n the 00 s t a t e and t h e r e f o r e the v p o i n t remains a t the s t a b l e node i n A r e a ( l - l ) . When an i n p u t p u l s e i s a p p l i e d t o the c i r c u i t , the v-v. p o i n t l e a v e s the node i n F i g u r e 2-8. S o l u t i o n Curves i n the P r o x i m i t y of the S i n g u l a r P o i n t s of a Two Tunnel Diode Stack C i r c u i t . 21 A r e a ( l - l ) s i n c e the e q u i l i b r i u m p o i n t of the system has been moved. The f i r s t s e c t i o n of the p a t h a l o n g which v^v^ p o i n t moves i s p r i m a r i l y determined by the r a t i o s of CQ, C^, and C^ w i t h each o t h e r s i n c e i t i s these r a t i o s t h a t determine how the charge, and thus the v o l t a g e , i s d i s t r i b u t e d throughout the c i r c u i t . I f the l o a d l i n e p o s i t i o n becomes such t h a t i t does not i n t e r s e c t Segment 1 of b o t h f ( v ) c h a r a c t e r i s t i c c u r v e s , then i t i s s a i d t h a t the A r e a ( l - l ) s i n g u l a r i t y becomes a " v i r t u a l s i n g u l a r i t y " . T h i s s i n g u l a r i t y w i l l t hen be found i n A r e a ( l - 2 ) or Area(2—2) depending upon the v a l u e of 1^. As the V 2 Y 1 P0-*-11^ moves out of one are a i n t o a n o t h e r , i t c o n t i n u e s to seek an e q u i l i b r i u m p o i n t and thus the pa t h i t t r a c e s out i s p r i m a r i l y d e t e r m i n e d by whether or not the e q u i l i b r i u m p o i n t n e a r -by i s s t a b l e . - As F i g u r e 2-8 shows, the s o l u t i o n curves near an u n s t a b l e p o i n t are f o r c e d t o a v o i d t h a t p o i n t and so the ^2Yl p o i n t i s f o r c e d t o end up e v e n t u a l l y on one of the f o u r s t a b l e p o i n t s . I f the P0^11'*' approaches a saddle or an u n s t a b l e node near a t —*> + oo asymptote i t w i l l be f o r c e d i n one d i r e c t i o n ; i f i t approaches the t o o asymptote i t w i l l be f o r c e d t o go i n another d i r e c t i o n . The manner in which an u n s t a b l e p o i n t i s approached i s governed by the c h o i c e of CQ, C^, C^ and the r e l a t i v e magnitudes of f ( v ) and f ( v 0 ) . 22 3. ANALOG COMPUTER STUDIES OF AN ANALOG-TO-DIGITAL CONVERSION CIRCUIT UTILIZING A TWO TUNNEL DIODE STACK For an a n a l o g — t o - d i g i t a l c o n v e r t e r , the change of i n p u t q u a n t i t y ( v o l t a g e or c u r r e n t or charge) needed to change the output by the l e a s t s i g n i f i c a n t b i t must be the same t h r o u g h -out the e n t i r e i n p u t range. T h i s c r i t e r i o n w i l l be used i n t h i s c h a p t e r t o e v a l u a t e ( l ) the f e a s i b i l i t y of the c i r c u i t a n a l y s e d i n Chapter 2 as an a n a l o g — t o - d i g i t a l c o n v e r t e r and (2) the e f f e c t s on the o p e r a t i o n of the c i r c u i t when d i f f e r e n t parameters are v a r i e d . S i n c e a two t u n n e l diode s t a c k c i r c u i t i s to be s t u d i e d i n t h i s ^ c h a p t e r , t h e r e w i l l be o n l y f o u r s t a b l e s t a t e s to d i s c u s s , namely, the 00, 01, 10, and 11 s t a t e s . The l e a s t v a l u e of 1^ t h a t i s r e q u i r e d t o make the c i r c u i t end up i n the m n ^ s t a t e w i l l be d e s i g n a t e d I . . I . w i l l a l s o be known ° Amn Amn t h as the mn s t a t e t h r e s h o l d c u r r e n t v a l u e * The range of i n p u t t h v a l u e s which w i l l produce the mn s t a t e w i l l be d e s i g n a t e d by A . mn In some cases i t may not be p o s s i b l e to make a l l of the ranges e q u a l and thus some form of compensation w i l l have to be used. One range which can e a s i l y be compensated i s the 00 range (AQQ). T h i s may be done by add i n g an a p p r o p r i a t e pedes-t a l t o the i n p u t p u l s e s . I n orde r to a v o i d a d d i t i o n a l compensa-t i o n c i r c u i t r y w h i c h would c o m p l i c a t e the o v e r a l l d e v i c e , i t i s n e c e s s a r y t o ensure t h a t the r e m a i n i n g ranges, w i t h the e x c e p t i o n of the 2 n - 1 range, be e q u a l . 23 3.1 A n a l o g Computer S i m u l a t i o n of the Two Tunnel Diode S t a c k C i r c u i t . The c i r c u i t proposed i n Chapter 2 and i l l u s t r a t e d by F i g u r e 2-5 was s i m u l a t e d on the PACE 231R a n a l o g computer. The a n a l o g c i r c u i t f o r t h i s s i m u l a t i o n i s g i v e n i n Appendix I . I n o r d e r t h a t the s i m u l a t e d i n p u t p u l s e , I , be A r e p r e s e n t a t i v e of an a c t u a l p u l s e , a p u l s e w i t h a r i s e time of 0.5 n.se.o. and a f a l l time of 1.0 n s e e , was used, ( l n s e c . .• = -9 \ 10 s e c o n d ) . These v a l u e s c o r r e s p o n d to the output of a T e k t r o n i x 111 p u l s e g e n e r a t o r . F i g u r e 3-1 i l l u s t r a t e s the p u l s e , 1^, and time d e s i g n a t i o n s a s s o c i a t e d w i t h i t . F i g u r e 3-1 0 Ofi i.o n (n«-i.o) time (Lnsec.) I l l u s t r a t i o n of the Input P u l s e I The a c t u a l v a l u e s of the c i r c u i t parameters were chosen i n a h e u r i s t i c manner. However, i t was known from the works of Renton and R a b i n o v i c i , and Salama t h a t the f o l l o w i n g c u r r e n t c r i t e r i a s h o u l d be obeyed: I n < I 0 and I i > I „• ! J p , l P,2 v , l v,2 The v a l u e s of CQ* C^f and were chosen t o y i e l d AQ^ = A^Q f o r . s e t v a l u e s of I ,> I n * I ,, I 0 , and t ~ . Having found p * l ' p,2' v , l ' v,2' 2 e 24 s u i t a b l e c o m b i n a t i o n s of CQ, C^> and C^, a l l the c i r c u i t p a r a -meters were v a r i e d i n e i t h e r d i r e c t i o n about these v a l u e s and the c o r r e s p o n d i n g e f f e c t s on the c i r c u i t o p e r a t i o n were s t u d i e d * . "\ While e x p e r i m e n t i n g w i t h the parameter v a l u e s w h i c h would y i e l d AQ^ - k-^Qt i t w & s found t h a t the magnitudes of I , » and I o t as quoted i n the m a n u f a c t u r e r s ' s p e c i f i c a t i o n s , d i d not l i e i n the range n e c e s s a r y to produce the d e s i r e d r e s u l t s * I n o r d e r t o i n c r e a s e the e f f e c t i v e v a l u e of I , a V fl r e s i s t o r was shunted a c r o s s D^. I n o r d e r to decrease the e f f e c -t i v e v a l u e of I 0 , a D.C* b i a s source was p l a c e d i n p a r a l l e l w i t h Dg*. F i g u r e 3-2 and F i g u r e 3-3 i l l u s t r a t e t hese e f f e c t s , r e s p e c t i v e l y . v v (a) (b) (c) F i g u r e 3—2 ( a ) * I l l u s t r a t i o n of Tunnel Diode and R e s i s t o r C h a r a c t e r i s t i c C urves. (b) . Composite C h a r a c t e r i s t i c Curve Developed when R e s i s t o r Shunts Tunnel Diode. (c) . C i r c u i t which Produces Curve 3 - 2 ( b ) * (a) F i g u r e 3-3 (a) (b) 25 (c) I l l u s t r a t i o n of Tunnel Diode C h a r a c t e r i s t i c Curve and Magnitude of B i a s C u r r e n t I . Composite C h a r a c t e r i s t i c Curve f o r Tunnel Diode Shunted w i t h a B i a s Source I . ( c ) . C i r c u i t w hich Produces Curve 3-3(b)« 3.2 Modes of O p e r a t i o n * D u r i n g the course of the experiments d e s i g n e d t o o b t a i n d i f f e r e n t c o m b i n a t i o n s of c a p a c i t a n c e v a l u e s t h a t would y i e l d AQ^  = I^O ^ 0 T "^ne ^ u n n e i diode s t a c k c i r c u i t y t h r e e d i f f e r e n t modes of o p e r a t i o n were d i s c o v e r e d . These modes are c a l l e d Mode 1, Mode 2r and Mode 3. The t h r e e modes are s i m i l a r i n the manner i n w h i c h the 01 s t a t e i s o b t a i n e d but d i f f e r from each o t h e r i n the manner i n whi c h the 11 s t a t e i s o b t a i n e d . I n a d d i t i o n , Mode 1 and Mode 2 d i f f e r from Mode 3 i n the manner i n wh i c h t h e 10 s t a t e i s o b t a i n e d . W i t h the a i d of F i g u r e 3.-^4 ( a ) , '(b)* and (c) the manner i n which the c i r c u i t s w i t c h e s f o r each of the t h r e e modes w i l l be e x p l a i n e d * Mode 1 For I A "^AOl* V l a n d V 2 a s s u m e l ° w v a l u e s and e v e n t u a l l y r e t u r n t o t h e i r quiescent v a l u e s d e s c r i b e d by the p o s i t i o n of the 26 00 s t a b l e node. F o r I ^ Q J < 1^ < I^Q.' v ^ i n c r e a s e s t o the h i g h v o l t a g e v a l u e w h i l e the v a l u e of v^ remains i n the low range. I n t h i s case the f i n a l v a l u e s of v^ and v ^ are det e r m i n e d by the p o s i t i o n of the 01 s t a b l e node. F o r < 1 ^ ^ v 2 a t t a i n s a h i g h v a l u e w h i l e v ^ s t a y s i n the low v o l t a g e range* The end p o i n t v a l u e s of v^ and v ^ are determined by the p o s i t i o n of the 10 s t a b l e node* F o r V I remains i n the low range u n t i l v 2 has assumed a h i g h v a l u e a f t e r w hich time v^ r i s e s t o a h i g h v a l u e . The f i n a l v a l u e s of v^ and v 2 are d e s c r i b e d by the p o s i t i o n of the 11 s t a b l e node ( F i g u r e 3-4(a))» Mode 2 Fo r 1^<=Z I ^ Q > the s w i t c h i n g t r a j e c t o r i e s f o r t h i s mode are s i m i l a r t o thos e d e s c r i b e d f o r Mode 1 o p e r a t i o n . F o r ^ < ^ " A l l * ^ e t r a j e c t o r i e s are s i m i l a r t o those of Mode 1 but i n t h i s case the magnitude of v o l t a g e w h i c h v^ a t t a i n s i n i t s t r a n s i e n t c o n d i t i o n i s h i g h e r t h a n i n the p r e v i o u s mode. Fo r 1^ >^j^]_]_» v i a n d v 2 approach t h e i r end s t a t e v a l u e s s i m u l t a n -e o u s l y ( F i g u r e 3 — 4 ( b ) ) * Mode 3 As s t a t e d e a r l i e r , f o r I < ^XIO ^ e p i t c h i n g t r a j e c -t o r i e s f o r t h i s mode of o p e r a t i o n are v e r y s i m i l a r t o those d e v e l o p e d i n Mode 1 and Mode 2 o p e r a t i o n s . When I . . R . < I . < I . , , , v_ * A10 A A l l 2 remains a t a low v a l u e u n t i l v^ has o b t a i n e d a h i g h v a l u e . A f t e r t h i s has o c c u r e d , v 2 i n c r e a s e s t o the h i g h s t a t e v a l u e w h i l e v^ decr e a s e s t o i t s low s t a t e v a l u e . F o r 1^ > v 2 r e m a i n s a t a low v a l u e u n t i l v ^ has i n c r e a s e d t o a l a r g e magnitude and then v 2 i n c r e a s e s t o i t s h i g h s t a t e v a l u e ( F i g u r e 3 - 4 ( c ) ) . 27 100 200 300 400 500 6 V2(mV) (a) 100 200 300 400 500 v2Uv.) (b) 100 200 300 400 500 VJmV) ( c ) F i g u r e 3-4 ( a ) . I l l u s t r a t i o n of Mode 1 O p e r a t i o n . (b) . I l l u s t r a t i o n of Mode 2 O p e r a t i o n . (c) . I l l u s t r a t i o n of Mode 3 O p e r a t i o n . F a m i l i e s o f cu r v e s f o r Mode 1, Mode 2, and Mode 3 ty p e s of o p e r a t i o n have been i l l u s t r a t e d i n F i g u r e s 3—5. 3—6. and 3—7 r e s p e c t i v e l y , i n o r d e r t o show the p o s i t i o n s and the e f f e c t s of the s i n g u l a r i t i e s d e s c r i b e d i n Chapter 2. The t u n n e l d i o d e s , the v a l u e of the b i a s c u r r e n t I g , arid the v a l u e of a r e "^ne same f o r each of the t h r e e c a s e s . The o n l y parameters t h a t have been v a r i e d i n these examples are the v a l u e s of CQ, C^, C^y and the magnitude of I^» The magnitude of 1^ was v a r i e d over a range to y i e l d a l l f o u r s t a t e s . F o r the t h r e e examples chosen I , = 1.1 mA*., IVtr =. 40 mA. I 0 = 1.2 mA., I 0 = -6.23 mA., I D = 0.45 mA., and t 0 = 3 n s e c . Table 3-1 g i v e s the v a l u e s of c a p a c i t a n c e s and t h r e s h o l d c u r r e n t s f o r each of the t h r e e c i r c u i t s . A l t h o u g h the example used here to i l l u s t r a t e Mode 3 o p e r a t i o n does n ot y i e l d AQ^ = A^Q, the t r a j e c t o r i e s are c h a r a c t e r i s t i c of t h i s type of mode. 500 VOLTAGE \fc GTMDUB) Figure 3-5. Family of Mode 1 Operation Curves. 31 Mode C Q ( p F . ) C ^pF.) C 2 ( p F . ) I A 0 1 ( m A . ) I A 1 0 ( m A . ) I A 1 1 ( m A . ) 1 12 2 1.1 1.362 2.140 2.924 2 20 4.5 4.5 2.128 3.244 4.572 3 12 1.1 6.5 1.344 2.902 3.338 Table 3-1. V a l u e s of C a p a c i t a n c e s and T h r e s h o l d C u r r e n t s f o r Mode If Mode 2t and Mode 3 C i r c u i t s . The combined s e t of examples d i s p l a y the e f f e c t s of a l l n i n e s i n g u l a r i t i e s on the c i r c u i t o p e r a t i o n : these were d i s c u s s e d i n S e c t i o n 2.4. The i n f l u e n c e of the c a p a c i t a n c e r a t i o s , e s p e c i a l l y C^/C 2, upon the manner i n whi c h the c i r c u i t o p e r a t e s i s l u c i d l y d i s p l a y e d by comparing F i g u r e s 3-5, 3-6, and 3—7. To i l l u s t r a t e the v a r i a t i o n of the c u r r e n t s and v o l t a g e s w i t h i n the c i r c u i t as a f u n c t i o n of t i m e , I A , 1^ + I g , I Q , V ^ , and v 2 were p l o t t e d v e r s u s time f o r each of the t h r e e modes of o p e r a t i o n . F i g u r e s 3—8, 3-9, and 3-10 demonstrate these p l o t s f o r Mode 1, Mode 2, and Mode 3 type of o p e r a t i o n , r e s p e c t i v e l y . 3.3 Advance D e t e r m i n a t i o n of Output. When the v a l u e of I A became v e r y c l o s e t o the v a l u e of I^o^» i t was found t h a t f o r a l l modes the time r e q u i r e d t o r e a c h s t e a d y s t a t e c o n d i t i o n s became v e r y l a r g e : f o r o t h e r t h r e s -h o l d v a l u e s and o t h e r modes the same was t r u e . T h i s e f f e c t w i l l i n c r e a s e the computation time enormously. However, a comparator t e c h n i q u e i s d e s c r i b e d here w h i c h enables a simple c i r c u i t t o be used t o g i v e advance i n f o r m a t i o n of the f i n a l d i g i t a l o u t p u t . 32 Figure 3-8. I l l u s t r a t i o n of the Manner i n which the Four Stable States are Obtained i n the Time Domain for Mode 1 Operation, 33 5-4-'A (mA) 3-2 I - I - I — i — r — i — r T (Idiv.<= 2-5n Sec.) F i g u r e 3-9. I l l u s t r a t i o n of the Manner i n which the Four Stable States are Obtained i n the Time Domain f o r Mode 2 Operation. 34 F i g u r e 3-10. I l l u s t r a t i o n of the Manner i n which the Four S t a b l e S t a t e s are Obt a i n e d i n the Time Domain f o r Mode 3 O p e r a t i o n . 35 A close look at the characteristics of a saddle point si n g u l a r i t y i l l u s t r a t e s that there are only two t r a j e c t o r i e s which actually terminate on the singular point, (See Figure 2-8). These t r a j e c t o r i e s are the asymptotes for t + oo and t — • - o o . In other words, i t would take an i n f i n i t e length of time to reach a saddle point i f the trajectory coincided with the t -> + oo asymptote. Similarly, i f the trajectory coincided with the t-*-+oo asymptote of an unstable node i t would take i n f i n i t e amount of time to move away from that unstable point. Consider the saddle point i n Area(l-3) of Figure 3—5. For - ^ o i ' V 2 V 1 P01'-11^ -*-s n e v e r placed in a position near this saddle point such that i t w i l l be forced to go toward the stable point i n A r e a ( l — 4 ) . However, as 1^ takes on values approaching the I^Q-^ value, the v^v^ point travels on a trajectory nearer to the two asymptotes which w i l l force i t to return to the 00 stable node. In other words, as 1^ tends towards I^Q-^ the time required for the c i r c u i t to reach the steady state condition i s increased. Theoretically, i t should take an i n f i n i t e amount of time for steady state conditions to be reached when 1^ = I^ Q-^  • S i m i l a r i l y , when 1^ = I^Q-^ + £> where e i s very small, i t w i l l take the c i r c u i t a very long time to reach the 01 stable node. In both the case where 1^ = I^Q^ - e a n < i where 1^ = IJ^Q^+E the value of v^ only varies a small amount from the 00 state value. In the former case, v^ r i s e s to a value nearing the v^ coordinate value of the saddle point i n Area(l-3) and then re-ceeds slowly to i t s 00 state value. In the l a t t e r case v^ rises s l i g h t l y above the v^ coordinate value for the Area(l-3) singular-i t y , then i t rises slowly to i t s high state value. From this 36 discussion i t can be seen that there i s a threshold value of voltage over which v^ must pass in order to go to i t s high state value or conversely, the threshold which v^ i s not to exceed i n order to return to i t s low state value. Therefore, i n order that the end state of the diodes be known before the c i r c u i t has reached i t s steady state condi-t i o n , a l l that i s required i s that the value of v^ be compared to the known threshold Value and an appropriate output be given from the comparator. The comparing of voltages could be carried out only a few nanoseconds after 1^ has been reduced to zero. This system of comparing diode voltages, to known threshold voltages would decrease the overall time required for readout since i t i s not necessary to wait u n t i l the stack c i r c u i t has reached quiescence. For the two tunnel diode case considered i t was found that for Mode 1 operation when 1^ < IJQQ> ^ w o u l d reach i t s high state voltage l e v e l by the time 1^ had been reduced to zero. Thus i t would not be necessary for a comparator c i r c u i t to be used for thi s p a r t i c u l a r diode. However, for other modes and for c i r c u i t s containing more diodes a comparator might be neces-sary for B^' 3.4 Variation of C i r c u i t Parameters. To gather more insight into the characteristics of the two tunnel diode stack c i r c u i t , a l l of the parameters of this i c i r c u i t were varied around the values chosen ( h e u r i s t i c a l l y ) to give the four d i s t i n c t stable states (Section 3.2). The influence of these variations on the c i r c u i t performance were studied and those quoted below are representative for a l l modes of operation. The tunnel diodes used i n these experiments had the following c h a r a c t e r i s t i c s : I , = 1.1 mA., I 0 = 1.2 mA., P > 1 P > I •• , = 0.40 mA. , I _ = —0.23 mA. The basic values for some v , l v,2 of the other parameters were = 3 n s e c , Ig = 0.45 mA., and C Q = 12 pF. 3.4.1 Variation of Si . v Let 61 = (I , — I „). In this experiment the v v,1 v,2' * value of I , was held constant at 0.40 mA. and the value of v , l I _ was varied so that Si assumed the following values: 0.63 mA. , 0.40 mA., 0.20 mA. , and 0.10 mA. As a result of decreasing the value of S I y j AQQ remained e s s e n t i a l l y constant, A.~, increased, A . « decreased. It was found that when SI =0.10 mA., 01 ' 1 0 v ' A , ~ was reduced to zero and therefore for this value of 61 the 10 v c i r c u i t would not y i e l d four states (Figure 3 - l l ) . The way i n which the c i r c u i t behaves to a change i n S I i s determined by the manner i n which the s i n g u l a r i t i e s are affected. When I 0 i s increased the section of the f(v~) for V < T < Y „ i s altered: the section 0 < v -< V i s unaltered. P — — f P This l a t t e r condition implies that AQQ i s not changed. The al t e r a t i o n w i l l be r e f l e c t e d i n the value of r 2 , the positions of the s i n g u l a r i t i e s on the v 1,^ P l a n e » a n d i n the values of X-^ and X 2» etc. These changes w i l l be reflected, f i n a l l y , i n altered values of AQ^ and A ^ Q . 38 C 0«I2 pP C, = &OpE C e = | | PF. <fly (mA) F i g u r e 3-11» V a r i a t i o n of A Values as a F u n c t i o n of o"l . 3.4.2 Simultaneous V a r i a t i o n of o'I y and 0*1^. Let 61 = ( l „ - I -, ) . For t h i s experiment I 0 P P,2 p , l 7 * p,2 and I ~ were v a r i e d while I , and I - r e m a i n e d constant* v,2 p , l v , l 61 was v a r i e d from 0*1 mA. up to 0.5 mA. while o r I v was v a r i e d from 0.63 mA* to 0.23 mA. ¥hen & I was i n c r e a s e d and I decreased; AQQ remained e s s e n t i a l l y constant, AQ^ i n c r e a s e d , and A^Q decreased (Figure 3-12). The v a r i a t i o n s i n the A v a l u e s can be i n t e r p r e t e d i n the same manner as f o r the previous case. The curves f o r AQ-^ and A^Q i n ,this case d i f f e r from those f o r AQ^ and A-^Q i n F i g u r e 3—11 because the mode of o p e r a t i o n of the c i r c u i t was changed from Mode 1 to Mode 3 i n the i n t e r v a l between o'l^ = 0.63 and = 0.43 mA* By comparing the e f f e c t s of v a r y i n g (f I T 39 alone w i t h the e f f e c t s of v a r y i n g Si and Si s i m u l t a n e o u s l y , v p i t i s seen t h a t the v a r i a t i o n of & I has much more e f f e c t on the c i r c u i t o p e r a t i o n than the v a r i a t i o n of Si . T h i s statement P assumes t h a t t h e r e i s no change i n the mode of o p e r a t i o n f o r the c i r c u i t . 1-5-(mA) l-O-0-5-0 C 0= 12 pF. C,= 2PF. C 2 = I.I pF-M O D E CHANGE A IQ, •5 -6 -7 ^Iy (mA-) T ~ A" I P (mA-) F i g u r e 3-12. V a r i a t i o n of A V a l u e s when Si Decreases and £ I I n c r e a s e s . 3.4.3 V a r i a t i o n of I g . The v a l u e o f the b i a s c u r r e n t was i n c r e a s e d from 0.45 mA. , to 1.00 mA and the f o l l o w i n g A values changed i n the >10 f o l l o w i n g manner: AQQ d e c r e a s e d , A ^ i n c r e a s e d , and A N n d e c r e a s e d ( F i g u r e 3-13). 40 I t was a l s o found t h a t f o r h i g h e r v a l u e s of I g the s w i t c h i n g time of the system d e c r e a s e d near the t h r e s h o l d c u r r e n t v a l u e s . T h i s c h a r a c t e r i s t i c may be e x p l a i n e d by a c o n s i d e r a t i o n of the s i n g u l a r i t i e s and the X v a l u e s . When I g i s i n c r e a s e d the s i n g u l a r i t i e s of the system are moved s i n c e the quie s c e n t l o a d l i n e now i n t e r s e c t s the f (y^) and the f (vg) curves a t h i g h e r p o i n t s on the c u r r e n t s c a l e . F o r some v a l u e s of I g the s i n g u l a r i t i e s w i l l f a l l on d i f f e r e n t segments of the approximate f ( v ) curve and thus the v a l u e of r w i l l change. T h i s change w i l l cause a c o r r e s p o n d i n g change i n the v a l u e of X^ and X 2» I n the p a r t i c u l a r case s t u d i e d , the s i n g u l a r i t i e s change segments (and thus Areas) f o r o n l y D^. The r e s u l t i n g changes i n X-^  and X 2 when I g was i n c r e a s e d from 0.45 mA. t o 0.65 mA. are g i v e n i n Table 3-2. Area (AjX 10~ 9) ( X 2 x 10~ 9) S w i t c h i n g Time Ig= .45 mA. Ig= .65 mA. Ig= .45 mA. Ig= .65 mA. 1-1 -0.7490 -0.7490 -12.48 -12.48 no change 1-3,2 +0.0975 +0.3418 -6.772 -6.086 de c r e a s e d 1-4,5 -0.1805 -0.4674 -7.855 -9.569 decreased 2-1 +0.6905 +0.6905 -4.468 -4.468 no change 2-3,2 +2.697 +3.513 +0.081 +0.196 decrease 2-4,5 +1.693 +0.9575 -0.2764 -1.542 decrease 5-1 -0.8605 -0*8605 -15.22 -15.2.2 no change 5-3,2 +0.0961 +0.3267 -9.617 -8.917 decrease 5-4,5 -0.1857 -0*5061 -10.70 -12.38 decrease Table 3-2. V a r i a t i o n of X-^ , X 2 , and C i r c u i t S w i t c h i n g Time w i t h Change i n !„. 41 1-5 1-0-A 0 5 •35 — i — •45 •55 i •65 •75 •85 — i — • 9 5 1.05 C 0 = I 2 P R C, = * P F . Ca = 1-lpF ±B(mA.') Fi g u r e 3-13. V a r i a t i o n of A Values as a F u n c t i o n of I g . 3.4.4 Simultaneous V a r i a t i o n of 61 and I - . ¥ith I B = 0*65 mA., S I assumed the f o l l o w i n g v a l u e s : 0.63 mA., 0.40 mA. , arid 0.20 mA. For each of these v a l u e s of <^I V» s t a t e 11 was i n t e r s p e r s e d i n the range f o r st a t e 10 and thus the output sequence was i n c o r r e c t . I t was found t h a t Ig could be i n c r e a s e d to 0*625 niA. from 0.45 mA. and the c o r r e c t output sequence would be preserved f o r (Si = 0.40 mA. 3.4.5. V a r i a t i o n of t 2 * For t h i s experiment the values of capacitances were the f o l l o w i n g : C Q = 20 pF., C;L = 4. 5 pF., and C"2 = 4.5 pF* For t 2 = 1 nsec. the 01 st a t e d i d not appear. When t 2 was i n c r e a s e d from 3 nsec. to 5 n s e c , A Q Q decreased, A Q ^ decreased, A ^ Q decreased, and the mode of o p e r a t i o n changed from Mode 2 to Mode 3. The decrease i n t h r e s h o l d v a l u e s f o r an i n c r e a s e i n t ~ 42 was expected s i n c e the energy p l a c e d i n t o the c i r c u i t i s d i r e c t l y p r o p o r t i o n a l t o the a r e a of the i n p u t p u l s e . 3.4.6 V a r i a t i o n of CQ* and C^-To i n v e s t i g a t e the e f f e c t s of changing the v a l u e s o f the d i f f e r e n t c a p a c i t a n c e s i n the c i r c u i t , one c a p a c i t a n c e was v a r i e d w h i l e the o t h e r two were h e l d c o n s t a n t . F o r an a d d i t i o n a l s t u d y , C^ and C^ were e q u a l and were s i m u l t a n e o u s l y v a r i e d . When CQ and C^ were h e l d c o n s t a n t and C^ v a r i e d , i t was found t h a t AQQ remained e s s e n t i a l l y c o n s t a n t , AQ^ i n c r e a s e d , and A^Q d e c r e a s e d . F o r t h i s s tudy CQ = 12 pF., C-^  = 2 pF., and C 2 was v a r i e d from 1 pF. t o 4.5 pF. S i n c e the mode of o p e r a t i o n changed from Mode 1 t o Mode 3 w h i l e C^ v a r i e d from 1.2 pF. t o 2.0 pF. t h e r e was an a l t e r a t i o n i n the r a t e of change of AQ^ and A 1 Q ( F i g u r e 3 - 1 4 ) . n C 0=I2 PF; C, = 2?F. Ct is varied I 2. 3 4 C £(pF) F i g u r e 3-14. V a r i a t i o n of A V a l u e s as a F u n c t i o n of C 4 3 For CQ and C^ constant and increasing the value of C^, AQQ arid A^Q increased and AQ^ decreased (Figure 3 - 1 5 ) . A 15-1 " " c e =ie P F C2= 4.5PF. C| is varied A (mA) 10-— i -4 — i — 4,5 C,(PF) Figure 3 - 1 5 . Variation of A Values as a Function of C ^ When CQ was constant and C^ and were equal and varied simultaneously", AQQ and A^Q increased and AQ^ decreased i n magnitude (Figure 3 — 1 6 ) . L . — © — A (mA.) •o--r -4- — i — 4v5 C 0 = | £ p F . C ,=C2 varied C,= C . ( P F ) Figure 3 - 1 6 . Variation of A Values as a Function of C-^  44 Figure 3 — 1 7 i l l u s t r a t e s the fact that a l l A values increase i n magnitude for an increase i n C Q when C ^ and C 2 are constant. i£ . 20 C0(PF:) Figure 3 — 1 7 . Variation i n A Values as a Function of C Q . These results indicate that a va r i a t i o n i n the value of one or more capacitances i n thi s c i r c u i t has a decided effect on the magnitude of the di f f e r e n t A ' s . A change i n the value of C Q has much less e f f e c t on these A values than a corresponding change i n either C ^ or C 2 » However, as the graphs i l l u s t r a t e , a v a r i a t i o n of 1 0 $ i n either C ^ or C 2 changes A Q ^ and A ^ Q i n opposite directions by about 40%. Therefore, i t would be advis-able to use tunnel diodes with low junction capacitance and then shunt the diodes with larger fixed capacitors. 4 5 Table 3-3 summarizes a l l the r e s u l t s of i n d i v i d u a l parameter v a r i a t i o n s * Parameter V a r i e d by i c\oL Bate of Change of A's (mA./unit) l\j/0 Aoo A 0 1 A 1 0 Ll d e c r e a s e d V 0 /mA* +1 /mA. - 0 * 7 5 /mA. o*I v d e c r e a s e d o l p i n c r e a s e d 0 /mA. +1.5 /mA. - 0 * 6 3 /mA. Lg i n c r e a s e d +1.8 /mA. +0.8 /mA. - 0 . 6 /mA. t g i n c r e a s e d - 0 . 7 /nsec. - 0 . 2 5 /nsec. - 0 * 6 /nsec. i n c r e a s e d + 0 . 1 7 /pF. - 0 . 7 3 /pP. + 0 . 7 8 /pP. i n c r e a s e d + 0 . 0 6 /pF. + 0 . 3 4 /pP. - 0 . 4 9 /pP. (C^=C^) i n c r e a s e d + 0 . 1 0 /pP. - 0 . 2 5 /pF. + 0 . 2 6 /pF. CQ i n c r e a s e d + 0 . 0 7 /pP. +0*05. /pP. + 0 * 0 8 / p F . Table 3 - 3 . V a r i a t i o n s o f A V a l u e s w i t h Change of D i f f e r e n t C i r c u i t P a r a m e t e r s . 3.5 Inductance E f f e c t s * As a means o f i n v e s t i g a t i n g the e f f e c t s of p a r a s i t i c i n d u c t a n c e on the o p e r a t i o n of the c i r c u i t b e i n g d i s c u s s e d , two d i f f e r e n t a n a l o g c i r c u i t s Were employed. Each of thes e c i r c u i t s c o n t a i n e d the two t u n n e l d i o d e s as shown i n F i g u r e 2-5 and i n a d d i t i o n each had an i n d u c t o r p l a c e d i n one s e c t i o n of the c i r c u i t * The f i r s t c i r c u i t s i m u l a t e d had an i n d u c t o r p l a c e d i n s e r i e s w i t h the t u n n e l d i o d e s and the second c i r c u i t had an i n d u c t o r p l a c e d i n s e r i e s w i t h CQ. These a n a l o g c i r c u i t s are g i v e n i n Appendix I I * 46 Fo r t h i s i n v e s t i g a t i o n a l l t h r e e modes of o p e r a t i o n were s i m u l a t e d and t h r e e v a l u e s of i n d u c t a n c e , namely, 1 nH.;, 10 nH., and 30 nH were used w i t h each mode. I n a d d i t i o n , two v a l u e s of CQ were used f o r each mode and i n d u c t a n c e v a l u e j these v a l u e s were 12 pF., and 20 pF. I t was found t h a t f o r s m a l l v a l u e s of i n d u c t a n c e ( l nH.), damped o s c i l l a t i o n s would occur i n a l l modes o f o p e r a -t i o n and the range of 1^ over which t h e y o c c u r r e d depended upon the mode. F o r Mode 1 and Mode 2 the damped o s c i l l a t i o n s o c c u r r e d near the I ^ Q p o i n t and f o r Mode 3 o p e r a t i o n the damped o s c i l l a -t i o n s o c c u r r e d near I J ^ Q I a n d "*"A11* ^ o r l A R S E R v a l u e s of i n d u c -tance (10 nH. and 30 nH.), s u s t a i n e d o s c i l l a t i o n s o c c u r r e d over ranges c e n t e r e d a t these t h r e s h o l d s and damped o s c i l l a t i o n s o c c u r r e d over g r e a t e r ranges i n the p r o x i m i t y of these t h r e s h o l d p o i n t s . Both the damped o s c i l l a t i o n s and the s u s t a i n e d o s c i l l a -t i o n s were superimposed on the v^ and curves p r e v i o u s l y o b t a i n e d f o r an i n d u c t a n c e f r e e c i r c u i t , c . f . F i g u r e s 3-8, 3-9, 3-10. Comparison of the r e s u l t s from the c i r c u i t i n w h i c h the i n d u c t a n c e was i n s e r i e s w i t h the t u n n e l d i o d e s w i t h the r e s u l t s of the c i r c u i t i n wh i c h the i n d u c t a n c e was i n s e r i e s w i t h CQ i n d i c a t e d t h a t o s c i l l a t i o n s o c c u r r e d over s l i g h t l y l a r g e r ranges w i t h the l a t t e r c i r c u i t . I t was found t h a t f o r i n c r e a s i n g v a l u e s of i n d u c t a n c e , a l l the t h r e s h o l d c u r r e n t v a l u e s f o r Mode 1 and Mode 2 d e c r e a s e d and t h a t f o r Mode 3> d e c r e a s e d w h i l e I A r k , and I . , , i n c r e a s e d . F o r an i n c r e a s e A10 A01 A l l i n the v a l u e .of CQ* the ranges over w h i c h the damped o s c i l l a t i o n s and the s u s t a i n e d o s c i l l a t i o n s o c c u r s was i n c r e a s e d . 47 In o r d e r t o reduce the magnitude of the o s c i l l a t i o n s , a damping r e s i s t o r was p l a c e d i n the c i r c u i t i n s e r i e s w i t h CQ. The a n a l o g c i r c u i t used t o s i m u l a t e t h i s c i r c u i t i s g i v e n i n Appendix I I . The v a l u e of the damping r e s i s t o r used was t w i c e the v a l u e r e q u i r e d t o s t o p s u s t a i n e d o s c i l l a t i o n s . For t h i s v a l u e , the t r a n s i e n t o s c i l l a t i o n s were reduced t o the p o i n t which would make advance d e t e r m i n a t i o n of the output f e a s i b l e . F o r the Mode 1 c i r c u i t ( C Q = 12 pF. , =2pP., C 2 = 1.1 p F . ) , w i t h L = 10 nH. a v a l u e of r e s i s t a n c e (R) of 15iT gave s a t i s -f a c t o r y r e s u l t s . F o r Mode 2 c i r c u i t ( C Q = 12 pF. , C-^  =4.5 pF. , C 2 = 4.5 p F . ) , w i t h L = 10 nH. a v a l u e of R of 8X1 was s u f f i c i e n t . F o r Mode 3 c i r c u i t ( C Q = 12 pF., C1 = 1.1 pF., C 2 =4.5 p F . ) , w i t h L = 10 nH. R was chosen to be 15-TL . For i n c r e a s i n g v a l u e s of i n d u c t a n c e l a r g e r v a l u e s of R are r e q u i r e d . 48 4. ANALOG COMPUTER STUDIES OF A THREE BIT ANALOG-TO-DIGITAL CONVERSION CIRCUIT USING A TUNNEL DIODE STACK CONFIGURATION A t h r e e b i t t h r e e t u n n e l diode s t a c k c i r c u i t , o p e r a t i n g i n a s i m i l a r f a s h i o n t o the two b i t two t u n n e l diode s t a c k c i r c u i t , was i n v e s t i g a t e d . The t h r e e t u n n e l diode c i r c u i t was s i m u l a t e d on a PACE 231R a n a l o g computer and the r e s u l t s of the b r i e f s t u d i e s c a r r i e d out are d e s c r i b e d i n t h i s c h a p t e r . 4.1 S i m u l a t i o n of a Three Tunnel Diode Stack C i r c u i t . The a n a l o g c i r c u i t used to s i m u l a t e the t h r e e t u n n e l d i o d e s t a c k c i r c u i t i s g i v e n i s Appendix I I I . T h i s c i r c u i t was a l s o used f o r the study of i n t e r d i o d e c a p a c i t a n c e e f f e c t s . The purpose of the experiment was to a s c e r t a i n whether e i g h t s t a b l e s t a t e s c o u l d be o b t a i n e d u s i n g a t h r e e t u n n e l diode s t a c k . The f o l l o w i n g s e t of c i r c u i t parameters y e i l d e d these e i g h t s t a b l e s t a t e s : I , = 1.1 mA., T „ = 1.2 mA., I _ = 1.4 mA., p , l p,2 P»3 ' I , = 0.40 mA., I 0 = -0.23 mA., I = -1.05 mA., C n = 12 pF., C± = 2.8 pF., C 2 =' 2.5 pF., C 3 = .5 pF. , and t 2 = 3 nse c . T h i s c i r c u i t d i d not y i e l d e q u a l A v a l u e s but f u r t h e r e x p e r i m e n t a t i o n would undoubtedly y i e l d the d e s i r e d e q u a l A v a l u e r e s u l t . F i g u r e 4-1 i l l u s t r a t e s the e i g h t s t a b l e s t a t e s o b t a i n e d by t h i s c i r c u i t , by d e p i c t i n g p l o t s of v^, v ,•  and v^ v e r s u s time f o x d i f f e r e n t v a l u e s of i n p u t . The v a l u e of the t h r e s h o l d c u r r e n t s are as f o l l o w s : -"-^ 001 = ^*681 mA. , I ^ Q ^ Q = 1.203 mA. , A011 = 1.357 mA., I A 1 0 0 = 1.407 mA., I A 1 0 1 = 1.490 mA., I n i n = 1.551 mA., and I . i n n = 2.193 mA. A110 A l l l 49 I t was found d u r i n g the s e r i e s of experiments w i t h the t h r e e t u n n e l diode c i r c u i t t h a t i f the v a l u e s of C^, C^, f ( v ^ ) , and f ( v 2 ) t h a t y i e l d e d AQ^ = A^Q f o r the two t u n n e l diode case were u s e d , then these e q u a l increments would be p r e s e r v e d . Because of t h i s f a c t i t i s seen t h a t l a r g e r c i r c u i t s can be b u i l t t o g i v e the d e s i r e d r e s u l t by w o r k i n g w i t h t u n n e l d i o d e s i n groups of two r a t h e r than w o r k i n g w i t h the complete s t a c k a t one t i m e . S i n c e the b e h a v i o u r of the t h r e e t u n n e l diode c i r c u i t i s an e x t e n s i o n of the two t u n n e l diode c i r c u i t , i t w i l l be p o s s i b l e t o u t i l i z e the comparator t e c h n i q u e , p r e v i o u s l y d e s -c r i b e d , as a means of advance output d e t e r m i n a t i o n . As F i g u r e 4-1 i n d i c a t e s the s w i t c h i n g time of t h i s c i r c u i t i s somewhat l o n g e r t h a n t h a t of the two t u n n e l diode c a s e . T h i s i s t o be exp e c t e d s i n c e t h e r e are more s t o r a g e d e v i c e s i n the former c i r c u i t . T h i s f a c t does not p r e c l u d e the c i r c u i t from b e i n g used as an u l t r a f a s t a n a l o g - t o - d i g i t a l c o n v e r t e r . 4.2 I n t e r d i o d e C a p a c i t a n c e E f f e c t s . By p l a c i n g a f i x e d c a p a c i t o r from the top of to the t o p of an e x t r a c i r c u i t parameter i s c r e a t e d . T h i s parameter, c a l l e d C^, can be used to a i d i n a c q u i r i n g a c i r c u i t w hich w i l l y i e l d e q u a l A v a l u e s u s i n g d i o d e s and c a p a c i t a n c e s t h a t would not o t h e r w i s e g i v e t h i s r e s u l t . The c i r c u i t e q u a t i o n s used to d e s i g n an a n a l o g c i r c u i t t o s i m u l a t e t h i s e f f e c t are g i v e n i n Appendix I I I . 50 F i g u r e 4—1. I l l u s t r a t i o n o f the Manner i n which E i g h t S t a b l e S t a t e s are O b t a i n e d i n the Time Domain f o r Three Tunnel Diode C i r c u i t . 51 The presence of i n the c i r c u i t has the e f f e c t of slowing,down the s w i t c h i n g of and r e l a t i v e t o the s w i t c h -i n g of As an example of the e f f e c t i v e n e s s of i n the c i r c u i t , c o n s i d e r the v a r i a t i o n i n t h r e s h o l d c u r r e n t v a l u e s f o r a c i r c u i t where I. a = 1.40 mA., I - = -.96 mA., D, and p,3 v,3 ' 1 D 2 are the same as b e f o r e , C Q = 12 pF., C.^  = 2.7 pF., C 2 = 2.5 pF., = 0.4 pF., and i s v a r i e d from 0 pF. to 0.8 pF. See Table 4-1. T h r e s h o l d c 4 = 0 pF. C, = 0 4 .8 pF. IA001 0.696 mA. 0.659 mA. IA010 1.274 mA. 1.296 mA. •'"AO 11 1.371 mA. 1.390 mA. IA100 1.454 mA. "^101 1.507 mA. ^ 1 1 0 2.230 mA. 1.658 mA. : IA111 2.289 mA. 2.246 mA. Table 4-1. V a r i a t i o n of T h r e s h o l d V a l u e s w i t h C^ As Table 4-1 i n d i c a t e s , does not a f f e c t a l l of the t h r e s h o l d c u r r e n t v a l u e s but i t i s obvious t h a t t h i s a d d i t i o n a l parameter proves to be a v e r y u s e f u l t o o l i n attempt-i n g t o d e s i g n a s u i t a b l e c i r c u i t f o r a n a l o g - t o - d i g i t a l c o n v e r -s i o n . 52 5. SUMMARY AND CONCLUSIONS The main purpose of this study has been to investigate the characteristics of a novel tunnel diode c i r c u i t i n order to ascertain whether or not i t could function as an analog-to— d i g i t a l conversion c i r c u i t . The novelty associated with this c i r c u i t i s the fact that 2 n states can be obtained using n tunnel diodes constructed from the same type of semiconductor material. Prom the sin g u l a r i t y analysis of the two tunnel diode c i r c u i t case and the analog computer simulation studies for the two tunnel diode c i r c u i t and the three tunnel diode c i r c u i t , the following summary of results and conclusions can be made: (1) . Three modes of operation exist for the two tunnel diode case and these modes are determined by the ratio of the capacitances which shunt the tunnel diodes. For c i r c u i t s containing more than two tunnel diodes, more modes would be developed* (2) . The switching time for this c i r c u i t v a ries. The longest switching occurs at the current threshold values for di f f e r e n t states. For the c i r c u i t s explored this time was approximately 40 nsec. A means of avoiding the need to wait u n t i l the c i r c u i t has reached i t s steady state condition i n order to obtain the output has been proposed. By using this scheme i t i s conceivable that i t would be possible to obtain the output i n 10 nsec. after a 4 nsec. input pulse had been applied. 53 (3) . for c i r c u i t * containing mora than two tunnel diode*, additional' interdiode capacitances can be used to aid in tha dosign of a circuit suitable for analog-to-digital conversion* (4) . Parasitic inductance can cause incorrect operation of the c i r c u i t . However, t h i s problem can be overcome by p l a c i n g s u f f i c i e n t r e s i s t a n c e i n the c i r c u i t t o dampen these u n d e s i r a b l e e f f e c t s . (5) . Compared t o o t h e r parameter v a r i a t i o n s , a v a r i a t i o n i n the c a p a c i t a n c e s t h a t shunt the i n d i v i d u a l t u n n e l d i o d e s causes the g r e a t e s t e f f e c t on the o p e r a t i o n of the c i r c u i t * To reduce t h i s problem low j u n c t i o n c a p a c i t a n c e t u n n e l d i o d e s s h o u l d be used and f i x e d c a p a c i t a n c e s used t o shunt these d i o d e s . (6) . A summary o f the e f f e c t s of a v a r i a t i o n i n the p a r a -meters of a two t u n n e l d i o d e s t a c k c i r c u i t i s g i v e n i n Table 3-3. (7) * I n d e s i g n i n g d i o d e c h a r a c t e r i s t i c s w h i c h w i l l y i e l d the appropriate A v a l u e s so t h a t the c i r c u i t w i l l produce the p r o p e r a n a l o g - t o - d i g i t a l c o n v e r s i o n , i t i s p o s s i b l e t o work w i t h p a i r s o f d i o d e s r a t h e r t h a n a complete s t a c k o f d i o d e s a t one t i m e * T h i s a r i s e s from the f a c t t h a t by s t a c k i n g s u c c e s s i v e d i o d e s , the A r e l a t i o n s h i p o f one diode w i t h the n e x t , f o r a g i v e n s e t o f c a p a c i t a n c e s i s not a l t e r e d a p p r e c i a b l y . (8) . On the b a s i s t h a t recommendations ( 2 ) , ( 4 ) , and (5) \ 5 4 are implemented, this study has indicated that the proposed c i r c u i t would be a feasible means of producing high speed analog-to-digital conversion. 55 APPENDIX I A I . Analog Computer C i r c u i t f o r S i m u l a t i o n of Two Tunnel Diode S t a c k C i r c u i t . The c i r c u i t i l l u s t r a t e d i n F i g u r e 2-5 was s i m u l a t e d on the PACE 231R a n a l o g computer and the a n a l o g c i r c u i t used f o r t h i s purpose i s shown i n F i g u r e A 1-1. The t h r e e major e q u a t i o n s f o r t h i s c i r c u i t may be w r i t t e n as f o l l o w s : dv 1 d t c f ^ l + h - f ( v l > ] dv where _2 d t x l a = 1 r C, [ i x + i B - f(v 2j] i A - a i B + ' ^ f ( v 1 ) + ^ f ( v 2 ) C l C 2 and (2.16) (2.17) (2.21) 1 + a f ( v ^ ) and f ( v 2 ) are the e f f e c t i v e c h a r a c t e r i s t i c c urves produced by t u n n e l d i o d e s D^and D 2, r e s p e c t i v e l y , as o u t l i n e d i n S e c t i o n 3.1. The parameters of the a c t u a l c i r c u i t were s c a l e d a p p r o p r i a t e l y to f a c i l i t a t e s i m u l a t i o n . Lower case l e t t e r s d e s i g n a t e r e a l time parameters w h i l e upper case l e t t e r s d e s i g n a t e the computer s c a l e d parameters. The s c a l i n g f a c t o r s are as f o l l o w s : " = 1 0 < \ o l t s > m V -i = .2(1 n r ) mA. v v o l t s ' t = 1 0 ~ 9 ( T ) sec sec 56 —12 C n i s i n picofarads ( l pF. = 10~ Farad) By scaling Equations (2.16), (2.17), and (2.2l) the following analog c i r c u i t equations r e s u l t : dV ( 1 . 3 ) Since the maximium value of t ^ was chosen to be 20 nsec. and since the amplifiers of the analog computer were rated for a maximium input or output of 100 v o l t s , then the slope factor K, was chosen to be 4. In the analog c i r c u i t F(V^) and P(V 2) were set up on function generators with I . = 4.4 mA., and I „ = 4.8 mA. P»l P»2 and the appropiate scaling was employed by using potentiometers marked & a n d S2* respectively. F i g u r e AI--1 A n a l o g C i r c u i t of Two Tunnel Diode S t a c k . 58 APPENDIX I I A I I . A n a l o g Computer C i r c u i t s Used to S i m u l a t e Inductance E f f e c t s i n a Two Tunnel Diode S t a c k C i r c u i t . A I I . 1 Two Tunnel Diode S t a c k C i r c u i t w i t h Inductance i n S e r i e s w i t h Tunnel Diodes. The c i r c u i t t o be s i m u l a t e d i s shown i n F i g u r e A I I - l . X A X l B x0 * 7 c 1 = L A f ( V l ) c 2 + a ) f K } I "B F i g u r e A I I - l . Two Tunnel Diode S t a c k C i r c u i t w i t h Inductance i n S e r i e s w i t h Tunnel Diodes. The e q u a t i o n s d e s c r i b i n g the o p e r a t i o n of t h i s c i r c u i t are as f o l l o w s : f l l d t = of [ * i + A B - f ( V ] ( I I . 1 ) 59 dv _2 dt dv _3 dt + 1 B f ( v 2 ) = L d 2 ( i x + i B ) dt ' i . = = c 0 d.(v 1 + v 2 + v 3 ) dt ( I I . 2 ) ( I I . 3 ) ( I I . 4 ) ( I I .5) By s u b s t i t u t i o n and rearrangement of the above f i v e e q u a t i o n s the f o l l o w i n g e q u a t i o n r e s u l t s : ,2/. ( i l + i B ) 1 dt' C 0 L c c i A _ a i B _ ( i + a ) i i + 5 f i ' f ( v 1 ) + ^ f ( v 2 ) ( I I . 6 ) where * = °1 °2 E q u a t i o n s ( I l . l ) , ( I I . 2 ) , and ( I I . 6 ) are the e q u a t i o n s used t o d e s i g n the a n a l o g c i r c u i t . U s i n g the same s c a l i n g f a c t o r s as were s t a t e d i n Appendix I , these t h r e e e q u a t i o n s become: dV, dT 20 r j dV, d 2 ^ + I f i) dT 3 f [ I± + I , - F ( V 2 ) ] ( I I . 7 ) ( I I . 8 ) dT 10  C 0 L c c I A _ a I B - (1 + a ) l 1 . + s a p ( V 1 ) + 7 ^ P ( V 2 ) ( I I . 9 ) Figure AII-2 Analog C i r c u i t for Two Tunnel Diode Stack C i r c u i t with Inductance i n Series with the Tunnel Diodes. 61 where C"n i s i n p i c o f a r a d s and L i s i n n a n o h e n r i e s . F i g u r e A l t - 2 i l l u s t r a t e s the an a l o g c i r c u i t used f o r t h i s s i m u l a t i o n , A I I . 2 Two t u n n e l Diode Stack C i r c u i t w i t h Inductance i n S e r i e s w i t h C a p a c i t o r CQ, The c i r c u i t t o be s i m u l a t e d i s i l l u s t r a t e d i n F i g u r e A I I - 3 . F i g u r e A I I - 3 . Two Tunnel Diode Stack C i r c u i t w i t h Inductance i n S e r i e s w i t h CQ The e q u a t i o n s d e s c r i b i n g the o p e r a t i o n of t h i s c i r c u i t are as f o l l o w s : dv 1 d t dv 2 d t dv _2 d t dv 57 [ h + i B - f ( f ( v 2 ) ] = L d t k 0 0 0 d t " C i . = 0 *1 + *(> ( I I . 1 ) ( I I . 2 ) (11.10) (II..11) ( I I . 4 ) 62 d ( v L + v Q ) d(v 1 + v 2 ) dt dt (11.12) By substitution arid rearrangement of the above six equations the following equation r e s u l t s : ( p i + (i, • i B ) .l!vV.±il (11.13) *2<JA - *1> dt 1 L ^0 f ( v 2 ) Equations ( l l , l ) , (II.2), and (II.13) are the equations used to design the analog c i r c u i t to simulate the c i r c u i t of Figure A II-3. Using the same scaling factors as were stated in Appendix If these three equations become: dV 1 dT 20 C dV, dT 2 10 L dT 3 i +^ (i, + i B). ifigiiil °1 °2 °0 F(V 2) (II.7) (II.8) F ^ ) (11.14) where C i s i n picofarads and L i s i n nanohenries. n • : * Figure A II—4 i l l u s t r a t e s the analog c i r c u i t used for this simulation. A II.3 Two Tunnel Diode Stack C i r c u i t with Inductance and Damping Resistor. The c i r c u i t to be simulated i s shown i n Figure A II-5 Figure AII-4 Analog C i r c u i t for a Two Tunnel Diode Stack C i r c u i t with Inductance in Series with C n. 64 — R > VR "1 B 2 V J-C '0 '2 ( f .0 '=2 © f U 2 > 0 1 B Figure A I I - 5 . Two Tunnel Diode Stack C i r c u i t with Inductance and Damping Resistance. The equations describing the operation of this c i r c u i t are as follows: dt dt dt dv dv _R dt •7 [ h + i B - f ( v i ) ] d 2 i = L 0 dt' 9. - l£L dt " c 0 d i = R 0 dt d ( v R + V L + V _ d ( v l + v 2 } dt ~ dt (II. 1 ) (II.2 ) ( 1 1 . 1 0 ) ( 1 1 . 1 1 ) ( 1 1 . 1 5 ) ( 1 1 . 1 6 ) By substitution and rearrangement of tfie above six equations yi e l d s the following equation: 65 a 2 ( i - V 1 L ( 1 + 1) ( i x + i B ) . 1 W 1 ^ A " 1 ! * E d ( i A - i l ) '0 dt (11.17) E q u a t i o n s ( H . l ) , ( I I . 2 ) , and ( I I . 1 7 ) are the e q u a t i o n s which are used to d e s i g n the a n a l o g c i r c u i t to s i m u l a t e the c i r c u i t of F i g u r e A I I - 5 . The same s c a l i n g f a c t o r s t h a t were s t a t e d i n Appendix I>are used f o r t h i s c i r c u i t and thus these t h r e e e q u a t i o n s become: dT d v l dT " 20 = °1 av 2 dT " 20 = C 2 1 0 3 L <• " A " [ h + h - F ( v 2 > ( I I . 7 ) (11.8) F(V 2) '0 , R d ( l A - I l ) L dT (11.18) F i g u r e A I I — 6 i l l u s t r a t e s the a n a l o g c i r c u i t used f o r t h i s s i m u l a t i o n * Figure AII-6 Analog C i r c u i t for a Two Tunnel Diode Stack C i r c u i t with Inductance and Damping Resistance. 67 APPENDIX III A III Analog Computer C i r c u i t for Simulation of a Three Tunnel Diode Stack C i r c u i t and Interdiode Capacitance Effects Associated with this C i r c u i t . The c i r c u i t to be simulated i s shown in Figure A I I I - l . If C^ i s removed from this figure, the c i r c u i t becomes a simple three tunnel diode stack c i r c u i t . Because of this f a c t , the analog c i r c u i t for this simulation can also be used to simulate the simple three tunnel diode c i r c u i t by removing the paths which relay the effects of C. to other sections of the c i r c u i t . _ p , , Figure A I I I - l . Three Tunnel Diode Stack C i r c u i t Containing One Additional Interdiode Capacitor, C^. The equations describing the operation of this c i r c u i t are as follows: 68 d v l 1 dt ~ C 1 _ *1 d v2 1 i — . dt - c 2 dv 3 x dt - C 3 *1 d ( V l + v 2 ) *2 dt d ^ V l + v2 + V 3 ^ dt -B ~ f ( V B "  f^ 2 ) -2 + i B - f (v 3) 10 30 (IH.I) (III.2) (III.3) (IH.4) (III.5) (III.6) By substitution and rearrangement of the above six equations yields the following equations: *2 = °4 1 + a ' C l ' C 2 XA ~ a l B + (III.7) f ( v 1 ) f ( v 2 ) '0 6 + c ; f ( v 3 } (III.8) where and C0°4 Equations ( i l l . l ) , ( I I I . 2 ) , ( i l l . 3 ) , ( i l l . 7 ) , and ( i l l . 8 ) are the equations which w i l l be used to design the analog c i r c u i t to simulate the c i r c u i t of Figure A I I I - l . By using the same scaling factors outlined in Appendix I these 69 equations become: I„ = 1 1 + a d v l 20 a;T • = C l 20 dT " = C 2 dV 3 20 ~df = ~ C 3 C4 << 1 L 3 h 4 lB " F ( V h + 1 B ~ F ( V 2 } I 1 + I 2 + I B - P ( V 3 ) ' C l C2 (III.9) (III.10) (III.11) (III.12) X l " a I B + F ^ ) P ( V 2 ) e + ^ P ( V 3 ) (III.13) Figure A III.2 i l l u s t r a t e s the analog c i r c u i t used for this simulation. Figure AIII-2 Analog C i r c u i t for a Three Tunnel Diode Stack C i r c u i t with an Additional Interdiode Capacitor, C.. 71 REFERENCES 1. E s a k i , L., "New Phenomenon i n Narrow P-N J u n c t i o n s " , Phys. Rev. L e t t e r s , V o l . 109, p. 603, 1958. 2. Renton, C , and R a b i n o v i c i , B. , "Composite C h a r a c t e r i s t i c s of N e g a t i v e R e s i s t a n c e D e v i c e s and T h e i r A p p l i c a t i o n s i n D i g i t a l C i r c u i t s " , P r o c . I.R.E., V o l . 50, pp. 1648-55, J u l y , 1962. 3. Salama, C.A.T., "The S t a t i c and Dynamic C h a r a c t e r i s t i c s of S e r i e s Connected Tunnel Diodes and T h e i r A p p l i c a t i o n i n D i g i t a l C i r c u i t s " , M.A.Sc. T h e s i s , Department of E l e c t r i c a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, December, 1962. 4. K i y o n o , T., I k e d a , K., and I c h i k i , H., " A n a l o g - t o - D i g i t a l C o n v e r s i o n U t i l i z i n g an E s a k i - D i o d e S t a c k " , I.R.E. T r a n s a c t i o n s on E l e c t r o n i c Computers. V o l . EC-11, No. 16, pp. 791-2, December 1962. 5. Cunningham, ¥. J . , I n t r o d u c t i o n to 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 Book Company, Inc . , New York and London, 1958. 6. G i b s o n , J.E., N o n l i n e a r Automatic C o n t r o l , M c G r a w - H i l l Book Company, I n c . , New York and London, 1963. 7. S c h u l l e r , M., and G a r t n e r , N.V., "Large s i g n a l t u n n e l diode t h e o r y " , P r o c . I.R.E.. V o l 49, pp 1268-78, August 1961. 

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