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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.,  N o v a 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 I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n the Department of Electrical  ¥e a c c e p t  this  Engineering  t h e s i s as c o n f o r m i n g  required  to the  standard  Members o f t h e D e p a r t m e n t of E l e c t r i c a l  Engineering  THE UNIVERSITY OF B R I T I S H COLUMBIA A u g u s t 1965  In p r e s e n t i n g the  this  thesis  Columbia,  I agree that  the Library  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 . mission  f o rextensive  representatives,,  cation  of this  w i t h o u t my w r i t t e n  forfinancial  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a te  August  So,  thesis  per-  f o r scholarly  by t h e Head o f my D e p a r t m e n t o r by  It i s understood  thesis  s h a l l make i t f r e e l y  I f u r t h e r agree that  copying o f t h i s  p u r p o s e s may be g r a n t e d  D a  fulfilment of  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f  British  his  in partial  Columbia  ( ?6f  that  gain  copying o r p u b l i -  shall  n o t be a l l o w e d  ABSTRACT  This t h e s i s describes a mathematical-graphical and  some a n a l o g  c o m p u t e r s i m u l a t i o n s t u d i e s t h a t were  out t o d e t e r m i n e t h e f e a s i b i l i t y o f a p r o p o s e d c i r c u i t  analysis  carried 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 analysed  and s i m u l a t e d c o n t a i n s a s t a c k o f  t u n n e l d i o d e s w h i c h a r e c o n s t r u c t e d f r o m t h e same t y p e conductor  material.  of semi-  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 t h i s  circuit  a r e c o n t r o l l e d p r i m a r i l y by t h e r a t i o s and t h e v a l u e s o f t h e capacitances which shunt the i n d i v i d u a l  tunnel diodes  and t o a  l e s s e r extent by the i n t e r d i o d e c a p a c i t a n c e s .  This i s revealed  i n a study  parameter v a r i a t i o n  of the e f f e c t s of d i f f e r e n t A two t u n n e l d i o d e  circuit  stack 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 t h e n a t u r e ing  of the s w i t c h -  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  equations  d e s c r i b i n g -the c i r c u i t  operation.  modes o f o p e r a t i o n , e a c h o f w h i c h d i f f e r s t h e 11 s t a t e  i s reached,  Three  different  i n t h e manner i n w h i c h  are revealed f o r t h i s  circuit*  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 o f t h e c i r c u i t w h i c h c a n be u s e d t o determine the f i n a l s t a t e of the c i r c u i t before d i t i o n s have b e e n An one  s t a t e con-  reached.  e x t e n s i o n o f t h e two t u n n e l d i o d e  stack c i r c u i t to  c o n t a i n i n g three t u n n e l diodes y i e l d e d e i g h t s t a b l e and  accessible be  steady  states.  able to r e a l i z e  This i n d i c a t e s that the c i r c u i t proposed 2  n  states with n tunnel diodes.  that d i f f e r e n t interdiode capacitance the a c h i e v e m e n t o f t h i s  result.  connections  will  I t i s shown will  facilitate  TABLE OF CONTENTS  List  of I l l u s t r a t i o n s . . . . . . . .  List  of Tables  L i s t o f S p e c i a l Symbols and Terms Acknowledgement 1.  INTRODUCTION  2.  ANALYSIS OF AN ANALOG-TO-DIGITAL CONVERSION CIRCUIT U T I L I Z I N G A TWO TUNNEL DIODE STACK CONFIGUATION ....  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 t h e 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 D i o d e Stack  2.2  The E q u i v a l e n t C i r c u i t o f a T u n n e l D i o d e t o be U s e d 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 o f t h e Two T u n n e l Stack C i r c u i t  2.4  Singular Point Analysis Applied to a Physical System  Diode  ANALOG COMPUTER STUDIES OF AN ANALOG-TO-DIGITAL CONVERSION CIRCUIT U T I L I Z I N G A TWO TUNNEL DIODE STACK 3.1  A n a l o g Computer S i m u l a t i o n Diode S t a c k C i r c u i t  3.2  Modes o f 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 o f Output  3.4  V a r i a t i o n of C i r c u i t  3.4.1 V a r i a t i o n o f  Parameters  & I  3.4.2 S i m u l t a n e o u s V a r i a t i o n s o f 3.4.3 V a r i a t i o n o f I  B  3.4.4 S i m u l t a n e o u s V a r i a t i o n o f 3.4.5 V a r i a t i o n o f t  o f t h e Two  v  and  Tunnel  iv  3 . 4 . 6 V a r i a t i o n o f CQ, 3.5 4.  , and  .. .  42  Inductance E f f e c t s  45  ANALOG COMPUTER SIMULATION STUDIES OF A THREE B I T ANALOG-TO-DIGITAL CONVERSION CIRCUIT USING A TUNNEL DIODE STACK CONFIGURATION 4.1 4.2  5.  Page  S i m u l a t i o n o f a Three Circuit  Tunnel Diode  Stack 48  Interdiode Capacitance E f f e c t s  49  SUMMARY AND CONCLUSIONS  APPENDIX I  52  ANALOG COMPUTER CIRCUIT FOR SIMULATION OF TWO TUNNEL DIODE STACK CIRCUIT .  APPENDIX I I ANALOG COMPUTER CIRCUITS USED TO STUDY INDUCTANCE EFFECT I N A T¥0 TUNNEL DIODE CIRCUIT ..  A II.2 A II.3  Two T u n n e l D i o d e 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 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 n d u c t a n c e i n - S e r i e s w i t h CQ  ANALOG COMPUTER CIRCUIT FOR SIMULATION OF A THREE TUNNEL DIODE STACK CIRCUIT AND INTERDIODE CAPACITANCE EFFECTS ASSOCIATED WITH THIS CIRCUIT  REFERENCES  ....  58  with  Two T u n n e l D i o d e S t a c k C i r c u i t w i t h I n d u c t a n c e a n d Damping R e s i s t a n c e  APPENDIX I I I  55  58  :  A II.1  48  61 62  67 71  iv  V  L I S T OP ILLUSTRATIONS Figure  Page  1- 1  Tunnel Diode S t a c k C o n f i g u r a t i o n  2- 1  Proposed A n a l o g — t o - D i g i t a l Conversion C i r c u i t Utilizing  Large S i g n a l E q u i v a l e n t C i r c u i t  2-3  (a)  2—4 2-5 2—6  .»  a Tunnel Diode S t a c k  2-2  (b)  ......  1  4 o f a Tunnel Diode..  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 .. *. P l o t o f 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 o f Junction Voltage  7 8 8  C i r c u i t C o n t a i n i n g Two T u n n e l D i o d e S t a c k t o be Analysed  9  Equivalent Circuit t o be A n a l y s e d  9  Illustration  o f Two T u n n e l D i o d e S t a c k  o f A p p r o x i m a t i o n o f Tunnel Diode  Static Characteristic  Curve  2-7  P l o t I l l u s t r a t i n g Areas of V 2 V ^ Plane  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  12 18  P o i n t s o f 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  20  3- 1  Illustration  23  3-2  (a)  I l l u s t r a t i o n o f 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 Composite C h a r a c t e r i s t i c Curve Developed  24  when R e s i s t o r S h u n t s T u n n e l D i o d e  24  (c)  Circuit  24  (a)  I l l u s t r a t i o n o f Tunnel Diode C h a r a c t e r i s t i c Curve and Magnitude o f 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  25  Diode Shunted by B i a s Source I .  25  (c)  Circuit  25  (a)  I l l u s t r a t i o n o f Mode 1 O p e r a t i o n  27  (b)  Illustration  o f Mode 2 O p e r a t i o n  27  (c)  Illustration  o f Mode 3 O p e r a t i o n v  27  (b)  3-3  (b)  3-4  o f I n p u t P u l s e 1^  w h i c h P r o d u c e s C u r v e o f 3 . 2 ( b ) .......  w h i c h Produces Curve 3.3(b)  Figure  Page  3-5  F a m i l y o f Mode 1 O p e r a t i o n C u r v e s  28  3-6  F a m i l y o f Mode 2 O p e r a t i o n C u r v e s  29  3-7  F a m i l y o f Mode 3 O p e r a t i o n C u r v e s  30  3-8  I l l u s t r a t i o n o f t h e Manner i n w h i c h F o u r S t a b l e S t a t e s a r e O b t a i n e d i n t h e Time Domain f o r Mode 1 Operation  32  I l l u s t r a t i o n o f t h e Manner i n w h i c h F o u r S t a b l e S t a t e s a r e O b t a i n e d i n t h e Time Domain f o r Mode 2 Operation  33  3—9  3-10  I l l u s t r a t i o n o f t h e Manner i n w h i c h F o u r S t a b l e S t a t e s a r e O b t a i n e d i n t h e Time Domain f o r Mode 3 Operation  34  3-11  Variation  o f A V a l u e s as a F u n c t i o n o f  38  3-12  Variation  o f A V a l u e s when  and  & I  & I  I  .••  i s Decreased  i s Increased Simultaneously  39  3-13  Variation  o f A V a l u e s as a F u n c t i o n o f I g .......  41  3-14  Variation  o f A V a l u e s as a F u n c t i o n of  42  3-15  Variation  o f A V a l u e s as a F u n c t i o n o f  43  3-16  Variation  o f A V a l u e s ; as a F u n c t i o n o f C^(=C2) . •  43  3- 17 V a r i a t i o n o f A V a l u e s as a F u n c t i o n o f 4— 1 I l l u s t r a t i o n o f t h e Manner i n w h i c h E i g h t S t a b l e S t a t e s a r e O b t a i n e d i n t h e Time Domain f o r T h r e e Tunnel Diode C i r c u i t  44  A  57  I - I Analog C i r c u i t  A I I - l  o f Two T u n n e l D i o d e  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 n S e r i e s w i t h Tunnel Diodes  Stack C i r c u i t  50  w i t h Inductance 58  A I I - 2 A n a l o g C i r c u i t f o r Two T u n n e l D i o d e S t a c k C i r c u i t w i t h I n d u c t a n c e i n S e r i e s w i t h T u n n e l D i o d e s .. 60 A II-3  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 n S e r i e s w i t h CQ *  A I I - 4 Analog C i r c u i t Circuit A II—5  with  Inductance 61  f o r a Two T u n n e l D i o d e  Stack  w i t h I n d u c t a n c e i n S e r i e s w i t h CQ......  .Two T u n n e l D i o d e S t a c k C i r c u i t and Damping R e s i s t a n c e vi  with  Inductance ;  63 64  Figure  Page  A I I - 6 Analog C i r c u i t  f o r 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 w i t h I n d u c t a n c e and D a m p i n g R e s i s t a n c e A III-l  Three Tunnel Diode S t a c k C i r c u i t Additional Interdiode  A III-2  Containing  C a p a c i t o r , C^  .  66  One 67  A n a l o g C i r c u i t f o r a Three T u n n e l D i o d e 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 70 4  vii  L I S T OF TABLES Table 2-1  Page V a l u e s o f r , and r and f ( v )  0  f o r Approximation of f ( v j  A  i  2- 2  Description of S i n g u l a r i t i e s  18 ..  19  3— 1  V a l u e s o f 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  3-2  Mode 1, Mode 2, a n d Mode 3 C i r c u i t s ............... V a r i a t i o n of X** ^ 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  31  3- 3 4- 1  of C i r c u i t A n a l y s i s  a n t  V a r i a t i o n o f A V a l u e s w i t h Change o f C i r c u i t meters V a r i a t i o n o f T h r e s h o l d V a l u e s w i t h C^  viii  Para45 51  L I S T OF SPECIAL SYMBOLS AND TERMS Symbol I I V V  P v  F i r s t Defined i n Section  Diode Peak P o i n t C u r r e n t  2.2  Diode V a l l e y P o i n t Current  2.2  Diode Peak P o i n t V o l t a g e  2.2  Diode V a l l e y P o i n t V o l t a g e  2.2  Diode Forward V o l t a g e  2.2  J  P  6  v  V  f ( v ) Diode v v s . i S t a t i c  Characteristic  th I. mn State Threshold Current Value Amn th A Range o f mn State mn &  & I p  £ I v  2.2 3. 3.  D i f f e r e n c e i n t h e P e a k C u r r e n t s o f Two Diodes  3.4  Difference i n the V a l l e y Currents of Two D i o d e s  3.4  Term Tunnel Diode Stack  1.  Threshold Current  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  ix  X ACKNOWLEDGEMENT The  author would  like  to express h i s sincere  thanks  and a p p r e c i a t i o n t o D r . M.P. B e d d o e s , t h e 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 t h e c o u r s e o f t h i s r e s e a r c h . The to  author would  also l i k e  to express h i s indebtedness  D r . J . S . MacDonald o f t h e M a s s a c h u s e t t s  Institute  of Technology,  C a m b r i d g e , 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 a n d helpful private  communication.  The w o r k d e s c r i b e d i n t h i s t h e s i s was s u p p o r t e d b y t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada u n d e r G r a n t BT-68.  x  1. INTRODUCTION The  i n v e n t i o n of the tunnel d i o d e ^ ^  produced a switch-  i n g device with the f o l l o w i n g c h a r a c t e r i s t i c s : switching speed, low power d i s s i p a t i o n , and sions.  very high  small p h y s i c a l dimen  These c h a r a c t e r i s t i c s make the tunnel diode v e r y  tive for application in digital  circuits.  of c i r c u i t s i n which the tunnel diode  One  p a r t i c u l a r group  can be used i s the group  of c i r c u i t s used f o r a n a l o g - t o - d i g i t a l conversion. application, several a u t h o r s ^ ^ ' ^ ^ ' ^ ^ c o n t a i n i n g a number of tunnel diodes c i r c u i t i s known as a tunnel diode t i o n i s i l l u s t r a t e d i n Figure  D,  For  this  have proposed a c i r c u i t  i n s e r i e s ; t h i s type of  "stack".  The  stack c o n f i g u r a  1-1.  D  D.  Figure 1 - 1 .  attrac-  D  n  n  Tunnel Diode Stack C o n f i g u r a t i o n .  In the stack c o n f i g u r a t i o n a l l the tunnel diodes operated  as v o l t a g e dependent b i s t a b l e d e v i c e s .  s t a t e and the high v o l t a g e state of each,tunnel r e f e r r e d to as the Thus the steady  n  state and  the  low  diode  voltage  are  '1' s t a t e , r e s p e c t i v e l y .  s t a t e of the tunnel diode  in binary notation. i n f e r s that D  '0*  The  are  stack may  be  presented  For example, the b i n a r y n o t a t i o n 1 0 1 1 . . . ,  i s i n the high s t a t e , D , i s i n the low s t a t e , ' n-1 ' 6  2 etc.  When a tunnel diode  stack i s used i n an a n a l o g - t o — d i g i t a l  conversion c i r c u i t , the b i n a r y n o t a t i o n r e p r e s e n t i n g the steady state v o l t a g e s of the diodes magnitude of the analog  i n the stack a l s o represents the  s i g n a l d r i v i n g the c i r c u i t .  an a p p l i c a t i o n Diode 1 (D^) represents the lowest  For such  or " l e a s t  significant" b i t . (2) Renton and R a b i n o v i c i  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 v o l t a g e negative r e s i s t a n c e devices  (called  controlled  'S' type negative r e s i s t a n c e  devices) i n s e r i e s or n current c o n t r o l l e d negative r e s i s t a n c e devices  ('N' type) i n p a r a l l e l , when i t i s r e q u i r e d t h a t these  c o n f i g u r a t i o n s produce 2 these  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 .  c r i t e r i a are a p p l i e d to the tunnel diode  When  stack c o n f i g u r a -  t i o n i t i s seen t h a t 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 f a c t t h a t one of the c r i t e r i a r e q u i r e s that each successive diode  of the stack must have a forward  v o l t a g e equal to twice the forward v o l t a g e of the preceeding (3) Salama  i n v e s t i g a t e d 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 tunnel diode sampling  stack when the input  speed was i n the range of the switching speed of the  tunnel diodes. capacitances determining  This study p o i n t e d out the f a c t t h a t ' t h e j u n c t i o n  shunting the tunnel diodes p l a y a major r o l e i n the o p e r a t i o n a l c h a r a c t e r i s t i c s of t h i s  This t h e s i s proposes a means of o b t a i n i n g digital  diode.  conversion u s i n g a tunnel diode  circuit. analog—to-  stack c o n f i g u r a t i o n i n  which the tunnel diodes are c o n s t r u c t e d from the same type of  s e m i c o n d u c t o r m a t e r i a l and using  'n' t u n n e l d i o d e s .  circuit  i n which the  suitably  i n which 2  s t a t e s can be  This e f f e c t i s achieved  capacitances  obtained  with a  t h a t shunt the diodes  novel must  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 t h e dynamic c h a r a c t e r i s t i c s a circuit  c o n t a i n i n g a two  C h a p t e r 2. technique the  teristics  be  and  a g r a p h i c a l technique d e s c r i b i n g the  of the  By PACE 231R  singularities  o p e r a t i o n of the  i n the  s i m u l a t i n g the  circuit  c o m p u t e r , i t was  r e s u l t s of t h i s  proximity  the  switching  to  character-  s t u d i e d i n C h a p t e r 2 on  p o s s i b l e t o obse'rve and  as w e l l as a s t u d y  of the  i n C h a p t e r 2.  e f f e c t s of p a r a s i t i c  o p e r a t i o n are presented  A three  t u n n e l diode  on t h e a n a l o g  record  circuit  i n Chapter  These  The  on  3.  s t a c k c o n f i g u r a t i o n was  computer.  results  inductance  also  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 i n C h a p t e r 4.  a  i n v e s t i g a t i o n were c o m p l e m e n t e d  a n a l y t i c a l t o o l s presented  simulated  charac-  circuit.  The  circuit  of  This a n a l y s i s proves  e f f e c t s o f d i f f e r e n t p a r a m e t e r v a r i a t i o n s on t h e  by t h e  in  c i r c u i t , the  system s w i t c h i n g t r a j e c t o r i e s  analog  operation.  the  stack i s presented  to 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 .  of the  of  of a n o n l i n e a r a n a l y s i s  a very valuable t o o l i n understanding  istics  the  tunnel diode  By a p p l y i n g a c o m b i n a t i o n  equations  of the  be  Also included i n t h i s chapter  are  presented  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 o f 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  i n the  circuit.  capacitance  4  2.  ANALYSIS OF AN ANALOG-TO-DIGITAL CONVERSION CIRCUIT U T I L I Z I N G A TWO  TUNNEL DIODE STACK CONFIGURATION R e n t o n and R a b i n o v i c i have i l l u s t r a t e d , b y means o f  Equation 2  n  ( 2 - l ) , t h e 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  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 f r o m n t u n n e l d i o d e s  to obtain connected  i n a stack c o n f i g u r a t i o n ,  V-f,m  = '.2^ fV-, l, f o r m = 2 t o n  (2-1)  m_1  where V„ i s the forward voltage f,m e  of the m  the forward v o l t a g e o f a t u n n e l diode  th  tunnel diode.  Since  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 o f t h e s e m i c o n d u c t o r m a t e r i a l f r o m w h i c h t h e device  i s c o n s t r u c t e d , t h e number o f d i f f e r e n t t y p e s  r e q u i r e d f o r the stack i s equal At present  t o t h e number o f d i g i t s  the c o n s t r i c t i n g forward voltage  Figure  available.  t h i s number i s l i m i t e d t o f o u r . This t h e s i s proposes a novel  vent  of diodes  2-1.  c i r c u i t which w i l l criterion;  circum-  see F i g u r e  Proposed A n a l o g - t o - D i g i t a l Conversion U t i l i z i n g a Tunnel Diode Stack.  Circuit  2-1.  5 1^  i s t h e 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 o f t h e  s i g n a l and  I g i s the b i a s c u r r e n t  analog  source.  By c o n t r o l l i n g t h e c h o i c e o f c a p a c i t o r s C^  C  Q  •-3  which 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  value  of CQ (the feedback c a p a c i t o r ) , i t i s p o s s i b l e to govern 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 Renton-Rabinovici  of the c i r c u i t  to the e x t e n t t h a t  f o r w a r d v o l t a g e c r i t e r i o n may  This chapter  presents  illustrated  be  i n F i g u r e 2-1 of the  where n = 2. circuit  the  relaxed.  a mathematical-graphical  a n a l y s i s aids the understanding  the  analysis This  operation  and  g i v e s i n s i g h t i n t o t h e manner i n w h i c h t h e o p e r a t i o n i s a f f e c t e d by t h e 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  parameters.  e f f e c t s of the parameter v a r i a t i o n s w i l l ter  The  be p r e s e n t e d  actual i n Chap-  3.  2.1 , R e a s o n 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 C i r c u i t C o n t a i n i n g a Two  Tunnel Diode  the  Stack.  T h e r e a r e t h r e e ways i n w h i c h t h e d u a l p r o b l e m o f ( 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 circuit is  parameters interdependency  an a n a l y t i c a l a p p r o a c h b u t  and  may  be  (2)  expressing  tackled*  The  the first  o f t e n t h i s approach i s too  tedious. The the  second approach i s a g r a p h i c a l technique  system s o l u t i o n curves  T h e s e t r a j e c t o r i e s may  illustrate  m e t e r as a f u n c t i o n o f t i m e o f one  called  i n which  " t r a j e c t o r i e s " are p l o t t e d . t h e v a r i a t i o n o f one  o r t h e y may  illustrate  p a r a m e t e r as a f u n c t i o n o f a n o t h e r  the  parameter.  paravariation The  main  6  disadvantage w i t h t h i s approach  i s that i t i s d i f f i c u l t to  o b t a i n a g e n e r a l i z e d c r i t e r i o n c o v e r i n g the system  behaviour  under a l l c o n d i t i o n s , although one p a r t i c u l a r case can be w e l l illustrated. The t h i r d approach  to the problem  an a n a l y t i c a l and g r a p h i c a l technique.  i s a combination of  I f the o p e r a t i o n of  a c i r c u i t can be d e s c r i b e d 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 s i n g u l a r p o i n t s of these  equations  i s very worthwhile,  s i n c e i t i s these s i n g u l a r p o i n t s which are  b a s i c i n determining the nature of the s o l u t i o n s of these equations.  By g r a p h i c a l l y r e p r e s e n t i n g the s o l u t i o n curves i n  the p r o x i m i t y of each s i n g u l a r p o i n t i t i s p o s s i b l e to s t r a t e 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  s o l u t i o n curves. illustrative  A s i n g u l a r i t y a n a l y s i s i s both r i g o r o u s and  since ( l ) the d e s c r i p t i o n of the  holds f o r a l l cases and curves may  illu-  singularities  (2) t h e i r i n f l u e n c e on the  be i l l u s t r a t e d f o r a l l cases at once.  solution The main  r e s t r i c t i o n on t h i s type of a n a l y s i s i s that only two  para-  meters can be i l l u s t r a t e d at once since the s o l u t i o n curve i s only p l o t t e d i n two dimensions. of t h i s approach  Because of these advantages  and because the r e s t r i c t i o n s t a t e d does not  apply f o r the case to be d i s c u s s e d , a s i n g u l a r i t y a n a l y s i s i s v e r y a p p l i c a b l e f o r the study of a two  tunnel diode stack  circuit. The complete mathematical  theory of the s i n g u l a r p o i n t (5)  a n a l y s i s i s d e s c r i b e d i n Cunningham not be repeated here.  (Chapter 5) and  shall  7 2.2  The E q u i v a l e n t C i r c u i t  o f a T u n n e l D i o d e t o be U s e d 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 tunnel  d i o d e c a n 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 dent j u n c t i o n c a p a c i t a n c e , a f u n c t i o n of the v o l t a g e  by i t s v o l t a g e  depen-  a n o n l i n e a r c u r r e n t source which i s across  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 t h e 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  inductor.  The c i r c u i t c o n f i g u r a t i o n u s e d f o r t h i s  sentation i s i l l u s t r a t e d i n Figure  2-2.  R  b u l k and c o n t a c t r e s i s t a n c e s  r  L  lead  s C(v)  junction f(v)  V  1-  Figure  2-2.  Large S i g n a l E q u i v a l e n t  Since  voltage  capacitance  nonlinear  Circuit  of a Tunnel Diode. s m a l l and  i s much l e s s d e p e n d e n t on j u n c t i o n  2-3(a) c a n a d e q u a t e l y r e p r e s e n t  diode used i n the design  current  source  than the j u n c t i o n current, the s i m p l i f i e d  shown i n F i g u r e  by  inductance  t h e magnitudes o f R , and L a r e v e r y s s  a l s o since the capacitance  repre-  t o be a n a l y s e d .  circuit  the tunnel  The c u r r e n t  produced  t h e n o n l i n e a r c u r r e n t s o u r c e i s p l o t t e d as a f u n c t i o n o f  voltage  i n Figure  2—3(b).  i=f(v)  (b) Figure  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 of a Tunnel Diode. (b).  P l o t of J u n c t i o n Current Junction Voltage.  The i v e r s u s t o as t h e " s t a t i c  v curve p l o t t e d i n Figure  Circuit  Function  of  2.3(b) i s r e f e r r e d  c h a r a c t e r i s t i c " of a tunnel diode.  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 a r e e x p l a i n e d I  =  Diode Peak P o i n t  =  Diode V a l l e y P o i n t  =  Diode Peak P o i n t  v  =  Diode V a l l e y P o i n t  V-,  =  Diode Forward V o l t a g e  I  P v  V V  2.3  below.  0urrent Current  Voltage Voltage at I  S i n g u l a r P o i n t A n a l y s i s o f t h e Two T u n n e l D i o d e  Stack  Circuit. A circuit  singular point analysis w i l l  shown i n F i g u r e  2-4.  be c a r r i e d o u t f o r t h e  9  Figure  2-4.  C i r c u i t Containing be S t u d i e d .  Using to  represent  t h e 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  D^ and B^  f  the c i r c u i t of Figure  i l l u s t r a t e d as i n F i g u r e will  2-5.  2-5.  o f t h e same t y p e .c^. V ~, V p,1 p,2' v, 1  and  2-4 c a n be  the v a l l e y  i n this  Stack  c i r c u i t are c o n s t r u c t e d  o f s e m i c o n d u c t o r m a t e r i a l and t h e r e f o r e Y.2'  and V„ , — V _ f , l f, 2  The p e a k ^  current  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  are g i v e n by the  that  diagram.  E q u i v a l e n t C i r c u i t o f Two T u n n e l D i o d e 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  2-3(a)  The c u r r e n t s and v o l t a g e s  be u s e d i n t h e a n a l y s i s a r e shown i n t h i s  Figure  V  Two T u n n e l D i o d e S t a c k t o  inequalities;  diodes  10 I  <  -1  p,l  I  p,2 0  (2.2) ^ , 1 ^ , 2 In rise  time  reality  and f a l l  1^ i s a c u r r e n t p u l s e w h i c h h a s a d e f i n i t e time, but f o r t h i s analysis i t w i l l  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  be  quiescence,  s i n c e I . and I., a r e z e r o , o n l y I_, f l o w s i n t o t h e s e c t i o n o f the  circuit  c o n t a i n i n g the tunnel diodes f(v )  = f(v )  x  Consider this  at  2  the equations  and t h e r e f o r e ,  t  =  0  (2.3)  d e s c r i b i n g the o p e r a t i o n of  circuit* dv 1 ~d¥  =  < l  =  ( I ,+ I ) - f ( v )  X  V  +  "  f  < V  (2.4)  dv 2  \ dt  B  C i ^ l + ^ l ' °0 dt dt,  1  =  +  A  =  I  0  +  2  I 0  (2.6)  (2.7)  l  I  (2.5)  By r e a r r a n g e m e n t and s u b s t i t u t i o n o f t h e p r e v i o u s four equations  t h e 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 J  A~  a I  B  +  C  2  C f  < V  C T ^ V  +  where; a  =  0  and  6  =  1 + a  (2.8)  11 From E q u a t i o n s  (2.4) a n d ( 2 . 8 ) ,  dv,  q  dt where  K]  .  [ 2 C  ( I  A  ~ <0  V  +  C  + Cfa  = Similarly  +  l  C  }  f  V  (  C  by m a n i p u l a t i n g Equations  1  (  A  I  +  V  +  the f o l l o w i n g K7[ 2 C  dv  jL- [  t  2  C  l  ( I  +  + I )  A  B  C  f  V "  (  0  ( C  + C  1>  simultaneously. ( i  +  A  i  B  f ( v  (2.10)  2>  (2.9) and E q u a t i o n  0  f  0  (  v  2  f(  -  )  V  l  ( C  0  +  ) - (C  2  C  0 +  C l  the s i n g u l a r i t i e s  )  f  )  (  v  l  }  ]  (2.10)  (2.11)  f(v ) ] 2  of Equation  o c c u r when v ^ a n d v ^ a r e s u c h t h a t d v ^ / d t  2  (2.5) and (2.8)  , a , and |3, t h e e x p r e s s i o n  of Equation  C  +  By d e f i n i t i o n ,  c  }  expression:  A  ( l  0  C  Taking the r a t i o  dv  (2.9)  2 _  becomes  dv _2 dt  yields  f ( v  +  and u s i n g t h e same e x p r e s s i o n s f o r f o r dv^/dt  °0  +  (2.1l)  = 0 and d v / d t 2  — 0  I n other words, ) +  c  0  f(v ) 2  -  (C  +  Q  c )  f (  2  V  l  )  =  0  (2.12) C  at  1  ( I  A  +  V  +  C  0  f  (  V  "  ( C  0  +  C  l>  t h e same t i m e . S o l v i n g E q u a t i o n s  yields  f  (  v  2  ='  }  0  (2.12) s i m u l t a n e o u s l y  t h e f o l l o w i n g v a l u e s o f f ( v ^ ) and f ( v ) a t a s i n g u l a r 2  point: f(  V l  )  =  f(v ) = 2  I  A  + I  B  (2.13)  The  e q u i l i b r i u m p o i n t s of the system or of one p a r t i c u -  l a r element of the system are g r a p h i c a l l y i l l u s t r a t e d by  the  i n t e r s e c t i o n s of the system's or the elements*s c h a r a c t e r i s t i c curve with the appropriate  load l i n e .  In t h i s case the system  l o a d l i n e i s an i n f i n i t e r e s i s t a n c e l o a d l i n e since the i s being f e d from c u r r e n t  circuit  sources.  For the purpose of t h i s a n a l y s i s i t i s acceptable 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  a f i v e l i n e a r segment curve. Figure  by  This approximation i s shown i n  2-6.  F i g u r e 2-6.  I l l u s t r a t i o n of Approximation of Tunnel Diode S t a t i c C h a r a c t e r i s t i c Curve.  Let v  be the value  of v  be the value  of v  at a s i n g u l a r i t y , l e t v_ J_  1S  S  •at a s i n g u l a r i t y , and also f ( v . ) and f ( v _ )  be the r e s p e c t i v e tunnel diode c o n s i d e r a p o i n t on the 2 1 V  point.  to  V  c u r r e n t s at a s i n g u l a r i t y *  P-*-  ane  Next  "that i s very near a s i n g u l a r  Since the f ( v ^ ) and the ^(v^) f u n c t i o n s have been  approximated by l i n e a r segments, the values of f ( v ^ ) and at t h i s p o i n t near the following  way:  s i n g u l a r i t y may  be expressed  i n the  f(v^)  13 f(v.) 1'  =  f(v. Is  ) +  v  d f(v-) . dv. 1  a  Is  (2.14)  df ( v j f (v )  =  2  f(v  2 s  )  + 2s  df(v ) —z l x  Defining  — 1  d  1  and  — 2  v  =  Equation  dv„  r  ( 2 . 1 4 ) becomes  (2.14a) f(v )  =  2  f(v  2 g  )  b  +  r  For t h i s p a r t i c u l a r  p r o b l e m t h e m a g n i t u d e s o f r ^ and r  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 e a c h  will  2  linear  segment. Now c o n s i d e r E q u a t i o n at d(T  (2.9) and E q u a t i o n  a p o i n t near a s i n g u l a r i t y .  ls> .  d  V  (  dt  dt  1  S^A  Equation  + V  + C  (2*10)  ( 2 . 9 ) becomes  0 < 2s> f  V  C  Q b V  -< 6 C  ( C  0  +  +  C  2  C  2>  f ( v  ls>  }  a d  (  v  l s  1  )  C  =  2^A  V  +  to the f o l l o w i n g dv  a 1 d t ~ K,  C  0  F  ^ 2 8  )  -  ( C  0  + C  2  )  f  ^1.)  o  by d e f i n i t i o n a t t h e s i n g u l a r i t y . reduced  °0  +  V  Therefore, the equation i s  form: b  (  C  0  +  C  2  )  a  (2.15)  14 By s i m i l a r manipulation of Equation (2.10) f o r a p o i n t close to a s i n g u l a r i t y , the equation becomes (c  0 a  dt  0 + C l  (2.16)  2  l  x  )  Taking the r a t i o of Equation (2.15) and Equation (2.16) y i e l d s the expression (c  V b  dv  j a dv.  L  r  2  in  r  c ) 2  V r  (c  [ V a _  0 +  l  0 + C l  l  r  a  (2.17)  )  2  V  b  t h e same f o r m a s t h e  namely, dv du  cu au  + dv + bv  (2.18)  Therefore, f o r t h i s problem the values of the constants a, b, c, and d are given below: K,  *  r ')  '0  b = K  l  r  l  (2.19) '0 c = Kf *l 2  _I ( o+ C  d  i  r  The values of the r o o t s  and \^ of the  r  %  i  solution  curves i n the p r o x i m i t y of a s i n 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 Therefore, r  l( 0 C  +  C  l) 2  K  +  r  2  (  C  0  +  C  2  l^ 0  r  )  C  +  C  l l 2 r  r  l)  +  r  2  (  C  0  +  C  2  )  l l 2  K  r  r  (2.21) K  Since the tunnel diode both  l l 2 r  r  static  characteristic  contains  r e g i o n s o f 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 o f n e g a t i v e  r e s i s t a n c e , i t i s important  t o i n v e s t i g a t e the type  r ^ a n d x^ a r e b o t h  t h a t would e x i s t f o r the f o l l o w i n g cases:  n e g a t i v e , r ^ a n d x^ a r e o f o p p o s i t e p o l a r i t y , are both  of s i n g u l a r i t y  and r ^ a n d x^  positive. By s u b s t i t u t i n g  i n the values  o f a, b , c , a n d 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  - a d ) •=  K  If  e i t h e r r ^ o r x^ i s n e g a t i v e  t h a t the term 4 ( b c - ad) w i l l  unstable  s i n g u l a r i t y formed w i l l  (a  be r e a l and o p p o s i t e  next  r ^ and ^  (2.22) s t a t e s  This being the i n s i g n and t h e  be a s a d d l e  point.  the s i n g u l a r i t i e s that l i e i n regions  are negative. V  + d) '=  C  From t h e e x p r e s s i o n , 1  ^2 it  r  be p o s i t i v e .  and \^ w i l l  where b o t h  r  then Equations  case,  Consider  (2.22)  l l 2  , 0 C  A  + C  2  i s s e e n t h a t when b o t h r ^ and r ^ a r e n e g a t i v e  expression i s positive  s i n c e n e i t h e r CQ, C ^ , n o r C  Since both  (2.23)  l the whole 2  c a n be a  negative  quantity.  ( a +. d) and 4 ( b c - ad) a r e  negative  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 f o c u s .  Therefore, i t f o l l o w s  from the conclusions f o l l o w i n g Equations  (2.22) and  (2.23)  t h a t , i f a s i n g u l a r i t y f a l l s i n a r e g i o n where e i t h e r r ^ or x^ i s negative, or where both r ^ arid x^ are n e g a t i v e , then the s o l u t i o n curves i n the p r o x i m i t y of t h i s s i n g u l a r i t y w i l l  be  unstable. The requirements (a + d) = 0  and  f o r a vortex to be formed are ( l )  (2) 4(bc - a d ) < 0 simultaneously.  If either  r ^ or x^ i s n e g a t i v e , then c o n d i t i o n (2) cannot be met. e i t h e r r ^ and r ^ are p o s i t i v e , condition  ( l ) cannot be met.  or i f both r ^ and x^ are negativ Therefore, f o r t h i s c i r c u i t ,  v o r t e x cannot e x i s t anywhere on the "^v^ The and r  2  a  plane.  s i n g u l a r i t i e s t h a t occur i n regions where both r-^  are p o s i t i v e w i l l be s t a b l e since (a + d ) < 0  4(bc - a d ) < 0 . - ( a + d)  If  and  These s i n g u l a r i t i e s w i l l be s t a b l e f o c i i f  >4(bc — ad) or they w i l l be s t a b l e nodes i f  -(a + d ) < 4 ( b c - ad). 2  For t h i s 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 s t a t e c o n d i tions;  i . e . 1^ = 1 ^ = 0 .  Since there are two  there are nine s i n g u l a r i t i e s .  Four of these  tunnel diodes, singularities  wil^L occur i n regions where r ^ and x^ are p o s i t i v e and therefore stable.  are  These s i n g u l a r i t i e s couid be e i t h e r s t a b l e  nodes or s t a b l e 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 e i t h e r r ^ or x^ i s negative and t h e r e f o r e these s i n g u l a r i t i e s are u n s t a b l e .  The  these regions are saddle p o i n t s . when both r, and x  0  s i n g u l a r i t i e s occurring i n One  singularity w i l l  occur  are negative and thus i t i s u n s t a b l e :  it will  be  e i t h e r an u n s t a b l e node o r an u n s t a b l e  S i n c e t h e r e i s no no  oscillations  c a n be  inductance  c a n o c c u r and  formed.  focus.  i n the c i r c u i t  t h e r e f o r e , no f o c i o r  expected  t o o c c u r n e a r some o f t h e  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  if  the magnitude of the o s c i l l a t i o n s of the u n s t a b l e  analysis.  one  c o n s i d e r a two parameters:  d>2  diodes  state.  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 to a P h y s i c a l System.  graphical technique  v  For i n s t a n c e ,  of the t u n n e l  For the purpose of i l l u s t r a t i n g t h i s  I  c i r c u i t would  b u i l t up t o a l a r g e v a l u e  singularities,  m i g h t end up a t an i n c o r r e c t  oscilla-  singularities.  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  2.4  vortexes  I f an i n d u c t i v e e l e m e n t were t o e x i s t ,  t i o n s w o u l d be  n e a r one  considered,  2  =  will  = -0.23 1*1 be  of s t u d y i n g the behaviour  t u n n e l diode  I  = 1,1  p,l  mA,,  P-^*  approximated  v a l u e s of r ^ , r > 2  The  I  mA.,  , = 0.40 v,l C  Q  2  v^  linear  plane  V  P^  be  a n e  known as A r e a  (1-2).  V 2  0  = 1.2  and  Each of the nine  mA.,  = 2pF., for  to the f i v e Each of the  identified  60 mV.  p,2  the 2—1.  into  linear twenty-five  by q u o t i n g t h e For  and  and  given i n Table  r e g i o n s o f t h e a x e s w h i c h bound i t ( F i g u r e 2 - 7 ) . the a r e a bounded by 0 <  I  are each d i v i d e d  regions which correspond  on t h e ^2 1  mA.,  following  segments h a v i n g  segments o f t h e a p p r o x i m a t e f ( v ) c u r v e s . areas  the  curves  and v o l t a g e i n c r e m e n t s v  of a c i r c u i t ,  = 12 pF.,  characteristic  with five  axes of the  five different  mA.,  I g = 0.45 static  c i r c u i t w h i c h has  analytical-  two  example,  60 mV. *^ v ^ <^ 200 mV. singularities  will  be  i d e n t i f i e d by t h e same number as t h e a r e a i n w h i c h i t l i e s .  is  18  Area  V o l t a g e Range  0 —  1  2  Diode  Diode 1  (mV*)  (n)  V o l t a g e Range  0 —  +54.3  60  (mV.)  r (A) 2  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 —  T a b l e 2-1.  V a l u e s o f r . and r and f ( v ) . ^  +35.7  f o r Approximation of f ( v )  1  1  2  1-5  Z-5  3— 5  4-5  5-5  1-4  z- 4  3-4-  4-4  5-4-  1-3  Z-3  3—3  4-3  5-3  1-2.  2 — 2  3—  e  4-2  5-2  I-I  2^1  3-  I  4-1  5-1  I  4  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 2 i  T a b l e 2—2 g i v e s a d e s c r i p t i o n of t h i s  system.  3  V  v  of the nine  •*  >lane  *  singularities  19  Coordinates  Area  Table  (v  2  —  Type o f  v . ^ mV.  Stability  Singularity  1-1  15 —  15  node  stable  1-3  15 —  240  saddle  unstable  1-4  15 —  330  node  stable  2-1  175 —  15  saddle  unstable  2-3  175 —  240  node  unstable  2-4  175 —  330  saddle  unstable  5-1 5-3  435 —  15  node  stable  435 —  240  saddle  unstable  5-4  435 —  330  node  stable  2-2.  Description of S i n g u l a r i t i e s  The s i n g u l a r i t i e s p r o x i m i t y of these  of C i r c u i t A n a l y s i s .  and t h e s o l u t i o n c u r v e s  singularities  i n the  a r e 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 o u t e a r l i e r i n t h i s  chapter,  s o l u t i o n curves  p o s i t i o n s only f o r steady  exist  i n these  c o n d i t i o n s , i . e . when 1 ^ = 1^ = 0. to  singularities  curves  plete  i t w o u l d n o t be p r a c t i c a l  set of s o l u t i o n In  the d i f f e r e n t  state  are a l t e r e d .  are nonlinear, the  p o s i t i o n s o f t h e s i n g u l a r i t i e s move i n a v e r y Therefore,  s i n g u l a r i t i e s and  When.an i n p u t i s a p p l i e d  the c i r c u i t the p o s i t i o n s of these  Because t h e f ( v ) c h a r a c t e r i s t i c  these  intricate  manner.  t o a t t e m p t t o p l o t a com-  curves.  order to describe, q u a l i t a t i v e l y , singularities  the i n f l u e n c e of  on t h e shape o f a s o l u t i o n c u r v e 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 t h e p a t h o f a point i s constructed after  the input i s a p p l i e d .  t h e c i r c u i t i s i n t h e 00 s t a t e a n d t h e r e f o r e t h e v r e m a i n s a t t h e s t a b l e node i n A r e a ( l - l ) . i s a p p l i e d to the c i r c u i t ,  v  2 i  Consider  v  that  point  When an i n p u t  pulse  t h e v - v . p o i n t l e a v e s t h e node i n  F i g u r e 2-8.  S o l u t i o n C u r v e s i n t h e P r o x i m i t y o f t h e S i n g u l a r P o i n t s o f a Two Diode S t a c k C i r c u i t .  Tunnel  21 A r e a ( l - l ) s i n c e t h e e q u i l i b r i u m p o i n t o f t h e system has been moved.  The f i r s t  s e c t i o n of the path along which v^v^ point  moves i s p r i m a r i l y d e t e r m i n e d b y t h e r a t i o s o f CQ, C^, a n d C ^ w i t h each other  s i n c e i t i s t h e s e r a t i o s t h a t d e t e r m i n e how t h e  charge, and thus the v o l t a g e , circuit.  i s d i s t r i b u t e d throughout the  I f t h e l o a d l i n e p o s i t i o n becomes s u c h t h a t i t does  i n t e r s e c t Segment 1 o f 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 ,  not  t h e n i t i s s a i d t h a t t h e A r e a ( l - l ) s i n g u l a r i t y becomes a "virtual  singularity".  Area(l-2) V  2 1 Y  This  singularity will  o r Area(2—2) d e p e n d i n g upon t h e v a l u e  P -*- ^ moves o u t o f one a r e a 0  t h e n be f o u n d i n  11  o f 1^.  As t h e  i n t o another, i t continues  to seek an e q u i l i b r i u m p o i n t and t h u s t h e p a t h i t t r a c e s o u t i s p r i m a r i l y determined by whether o r n o t the e q u i l i b r i u m p o i n t by  2-8  i s s t a b l e . - As F i g u r e  unstable  shows, t h e s o l u t i o n c u r v e s n e a r an  point are forced t o avoid  t h a t p o i n t a n d so t h e ^2 l Y  p o i n t i s f o r c e d t o e n d up e v e n t u a l l y on one o f t h e f o u r points.  I fthe  P ^ '*' a p p r o a c h e s a s a d d l e 0  11  node n e a r a t —*> + oo a s y m p t o t e i t w i l l if  i t approaches t h e t  in  another d i r e c t i o n .  o  near-  stable  o r an u n s t a b l e  be f o r c e d i n one d i r e c t i o n ;  o asymptote i t w i l l  be f o r c e d t o go  The manner in w h i c h a n u n s t a b l e  approached i s governed by the choice  point i s  o f CQ, C^, C ^ a n d t h e  r e l a t i v e magnitudes o f f ( v ) and f ( v ) . 0  22 3.  ANALOG COMPUTER STUDIES OF AN ANALOG-TO-DIGITAL CONVERSION CIRCUIT U T I L I Z I N G A TWO  TUNNEL DIODE STACK  F o r 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 , t h e change o f i n p u t q u a n t i t y ( v o l t a g e o r c u r r e n t o r c h a r g e ) n e e d e d t o change the output  b y t h e l e a s t s i g n i f i c a n t b i t must be t h e same  out t h e e n t i r e i n p u t range. chapter in  to evaluate  This c r i t e r i o n w i l l  through-  be u s e d i n t h i s  ( l ) the f e a s i b i l i t y of the c i r c u i t  analysed  C h a p t e r 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) t h e  e f f e c t s on t h e o p e r a t i o n o f t h e 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 . Since  a two t u n n e l d i o d e  stack c i r c u i t  studied i n this^ chapter, there w i l l to  i s t o be  be o n l y f o u r s t a b l e s t a t e s  d i s c u s s , n a m e l y , t h e 00, 01, 10, and 11 s t a t e s .  value mn^  o f 1^ t h a t i s r e q u i r e d t o make t h e c i r c u i t state w i l l  be d e s i g n a t e d °  I. . Amn  I. will Amn  The l e a s t  end up i n t h e  a l s o be known  t h as t h e 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 r a n g e o f i n p u t th v a l u e s w h i c h w i l l p r o d u c e t h e mn s t a t e w i l l be d e s i g n a t e d b y A . mn In  some c a s e s  the ranges e q u a l to  be u s e d .  00 r a n g e tal  i t may n o t be p o s s i b l e t o make a l l o f  and thus  some f o r m o f c o m p e n s a t i o n w i l l  One r a n g e w h i c h c a n e a s i l y be c o m p e n s a t e d i s t h e  (AQQ).  T h i s may be done b y a d d i n g  to the input p u l s e s .  an a p p r o p r i a t e  I n order to avoid a d d i t i o n a l  t i o n c i r c u i t r y which would complicate necessary  t o ensure t h a t the remaining  of t h e 2  - 1 r a n g e , be e q u a l .  n  have  pedes-  compensa-  the o v e r a l l device, i t i s ranges,  with the exception  23 3.1  A n a l o g Computer S i m u l a t i o n o f t h e  Two  Tunnel Diode  Stack  Circuit. The  c i r c u i t p r o p o s e d i n C h a p t e r 2 and  Figure  2-5  was  s i m u l a t e d on t h e PACE 231R  analog  circuit for this In order  illustrated  analog  computer.  The  s i m u l a t i o n i s given i n Appendix I .  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 0.5  n.se.o. and  -9 10  t h a t the  a fall  by  simulated  t i m e o f 1.0  n s e e , was  used,  of  ( l n s e c . .• =  \ second).  These v a l u e s  T e k t r o n i x 111 p u l s e , 1^,  pulse  and  correspond  generator.  to the  Figure  time d e s i g n a t i o n s  0 Ofi i.o  3-1  o u t p u t of  illustrates  a the  associated with i t .  n  (n«-i.o)  time (Lnsec.) Figure  3-1  The  Illustration  actual values  of the  of the  c h o s e n i n a h e u r i s t i c manner. w o r k s o f R e n t o n and  Input  Pulse  I  c i r c u i t p a r a m e t e r s were  However, i t was  R a b i n o v i c i , and  known f r o m  the  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 o b e y e d : I < I and I i > I „• ! p,l P,2 v,l v,2 The v a l u e s o f CQ* C^ and were c h o s e n t o y i e l d A Q ^ = A^Q f o r . n  0  J  f  set values  of I  ,> p*l'  I  * I ,, I , and p,2' v , l ' v,2' n  0  t~. 2  Having found e  24 suitable  c o m b i n a t i o n s o f C Q , C^> and C ^ , a l l t h e c i r c u i t  para-  m e t e r s were v a r i e d i n e i t h e r d i r e c t i o n a b o u t t h e s e v a l u e s a n d the  corresponding effects  on t h e c i r c u i t  studied*.  o p e r a t i o n were "\  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 which w o u l d y i e l d A Q ^ - k-^Qt i t & s w  I  , » and I  o t  found t h a t the magnitudes of  as quoted i n t h e 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 t o produce the d e s i r e d results*  I n order t o increase the e f f e c t i v e value of I  , a  V fl  r e s i s t o r was s h u n t e d a c r o s s D^. t i v e v a l u e of I w i t h Dg*.  0  In order to decrease the e f f e c -  , a D.C* b i a s s o u r c e was p l a c e d i n p a r a l l e l  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  these e f f e c t s ,  respectively.  v  (a)  v  (b)  (c)  F i g u r e 3 — 2 ( a ) * I l l u s t r a t i o n o f T u n n e l D i o d e and R e s i s t o r C h a r a c t e r i s t i c Curves. (b) .  C o m p o s i t e C h a r a c t e r i s t i c C u r v e D e v e l o p e d when R e s i s t o r Shunts Tunnel Diode.  (c) .  C i r c u i t w h i c h Produces Curve  3-2(b)*  25  (c)  (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 .  F i g u r e 3-3 ( a ) (b)  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 h i c h P r o d u c e s C u r v e 3-3(b)«  3.2 Modes o f O p e r a t i o n * During the course  of the experiments  obtain d i f f e r e n t combinations y i e l d AQ^  =  ^IO  ^  0 T  "^  ^  ne  of capacitance v a l u e s t h a t would u n n e  i  diode  stack c i r c u i t y  d i f f e r e n t modes o f o p e r a t i o n were d i s c o v e r e d . c a l l e d Mode 1, Mode 2  r  designed to  a n d Mode 3.  three  These modes a r e  The t h r e e modes a r e s i m i l a r  i n t h e manner i n w h i c h t h e 01 s t a t e i s o b t a i n e d b u t d i f f e r e a c h o t h e r i n t h e manner i n w h i c h t h e 11 s t a t e i s o b t a i n e d .  from In  a d d i t i o n , Mode 1 a n d Mode 2 d i f f e r f r o m Mode 3 i n t h e manner i n w h i c h t h e 10 s t a t e i s o b t a i n e d .  With  t h e a i d o f F i g u r e 3.-^4 ( a ) ,  '(b)* a n d ( c ) t h e manner i n w h i c h t h e c i r c u i t o f t h e t h r e e modes w i l l  s w i t c h e s f o r each  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 t h e p o s i t i o n o f the  26 00  For I^QJ <  s t a b l e node.  1^ <  I^Q.'  v  ^ increases t o the high  voltage value while the value  of v ^ remains i n the low range.  In  o f v ^ and v ^ a r e d e t e r m i n e d b y t h e  t h i s case t h e f i n a l v a l u e s  p o s i t i o n o f t h e 01 s t a b l e n o d e .  For  <  1^^  v  a h i g h v a l u e w h i l e v ^ s t a y s i n t h e low v o l t a g e range*  2 attains The end  p o i n t v a l u e s o f v ^ and v ^ a r e d e t e r m i n e d by t h e p o s i t i o n o f t h e 10 s t a b l e n o d e * until to  v  2  For  V  h a s assumed a h i g h v a l u e  a high value.  The f i n a l v a l u e s  I remains i n t h e low range  a f t e r which time v^ r i s e s o f v ^ and v  2  are described  by t h e p o s i t i o n o f t h e 11 s t a b l e node ( F i g u r e 3-4(a))» Mode 2 F o r 1^<=Z I ^ Q > t h e 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 a r e s i m i l a r t o t h o s e ^<  ^"All*  ^  d e s c r i b e d f o r Mode 1 o p e r a t i o n .  For  t r a j e c t o r i e s are s i m i l a r t o those of  e  Mode 1 b u t i n t h i s c a s e t h e m a g n i t u d e o f 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 higher than F o r 1 ^ ^j^]_]_» >  eously  (Figure  v  i  a  n  d  v  i n t h e p r e v i o u s mode.  2 a p p r o a c h t h e i r end s t a t e v a l u e s  simultan-  3—4(b))*  Mode 3 As  stated earlier, f o rI  <  ^XIO ^  e  p i t c h i n g trajec-  t o r i e s f o r t h i s mode o f o p e r a t i o n a r e v e r y s i m i l a r t o t h o s e developed  i n Mode 1 a n d Mode 2 o p e r a t i o n s . *  remains a t a low value u n t i l t h i s has o c c u r e d , v decreases  2  R  v ^ has o b t a i n e d a h i g h v a l u e .  After  increases to the high state value while v^  t o i t s low s t a t e v a l u e .  a low value u n t i l v  2  When I . . . < I . < I . , , , v _ A10 A A l l 2  F o r 1^ >  v  2  r  e  m  a  i n s at  v ^ has i n c r e a s e d t o a l a r g e magnitude and then  increases to i t s high state value  (Figure 3-4(c)).  27  100  200  300  400  500  6  100  200  300  400  500  100  v Uv.)  V (mV) 2  (b)  (a) F i g u r e 3-4 ( a ) .  200  300  400  500  VJmV)  2  (c)  I l l u s t r a t i o n o f Mode 1 O p e r a t i o n .  (b) .  Illustration  o f Mode 2 O p e r a t i o n .  (c) .  I l l u s t r a t i o n o f Mode 3 O p e r a t i o n .  F a m i l i e s o f c u r v e s f o r Mode 1, Mode 2, a n d Mode 3 t y p e s o f o p e r a t i o n h a v e b e e n 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 t h e p o s i t i o n s a n d t h e e f f e c t s of t h e 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  t h e v a l u e o f t h e b i a s c u r r e n t I g , arid t h e v a l u e o f for  each of the three cases.  v a r i e d i n these  a  r  e  diodes, "^  ne  The o n l y p a r a m e t e r s t h a t have b e e n  e x a m p l e s a r e t h e v a l u e s o f CQ, C^, C^  and t h e  y  m a g n i t u d e o f I^»  The m a g n i t u d e o f 1^ was v a r i e d o v e r a r a n g e  to y i e l d  states.  a l l four  F o r t h e t h r e e examples chosen I I  = 1.2 mA.,  0  Table for to  I  0  same  = -6.23 mA.,  I  D  , = 1.1 mA*.,  = 0.45 mA.,  and t  0  I r Vt  =. 40 mA.  = 3nsec.  3-1 g i v e s t h e v a l u e s o f 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  each of the three c i r c u i t s .  Although  t h e example u s e d h e r e  i l l u s t r a t e Mode 3 o p e r a t i o n d o e s n o t y i e l d  trajectories  are characteristic  of t h i s  AQ^ = A ^ Q , t h e  t y p e o f mode.  500  VOLTAGE  Figure 3-5.  \fc GTMDUB)  Family of Mode 1 Operation Curves.  31  Mode  C (pF.) C^pF.) C (pF.) I Q  2  A 0 1  (mA.) I  A 1 0  (mA.) I  A 1 1  (mA.)  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 o f 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 2 a n d Mode 3 C i r c u i t s . t  The all  nine  combined s e t o f examples d i s p l a y t h e e f f e c t s o f  s i n g u l a r i t i e s on t h e c i r c u i t  d i s c u s s e d i n S e c t i o n 2.4.  operation:  were  The i n f l u e n c e o f t h e c a p a c i t a n c e  e s p e c i a l l y C^/C , u p o n t h e manner i n w h i c h 2  is  these  the c i r c u i t  ratios,  operates  l u c i d l y d i s p l a y e d b y c o m p a r i n g F i g u r e s 3-5, 3-6, a n d 3—7. To i l l u s t r a t e  t h e v a r i a t i o n o f t h e c u r r e n t s and  v o l t a g e s w i t h i n t h e c i r c u i t as a f u n c t i o n o f t i m e , I I Q , V ^ , and v of o p e r a t i o n .  2  A  , 1^ + I g ,  w e r e p l o t t e d v e r s u s t i m e f o r e a c h o f t h e t h r e e modes F i g u r e s 3—8, 3-9, a n d 3-10 d e m o n s t r a t e  these  plots  f o r Mode 1, Mode 2, a n d Mode 3 t y p e o f o p e r a t i o n , r e s p e c t i v e l y . 3.3 A d v a n c e D e t e r m i n a t i o n o f O u t p u t . When t h e v a l u e o f I  A  became v e r y c l o s e t o t h e v a l u e  o f I^o^» i t was f o u n d t h a t f o r a l l modes t h e t i m e 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 : h o l d v a l u e s a n d o t h e r modes t h e same was t r u e . increase the computation technique used  time  enormously.  i s d e s c r i b e d here which  enables  f o r other This e f f e c t  threswill  However, a c o m p a r a t o r a s i m p l e c i r c u i t t o be  t o g i v e advance i n f o r m a t i o n o f t h e f i n a l d i g i t a l  output.  32  F i g u r e 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 f o r Mode 1 Operation,  33  54-  'A (mA)  32 I  -I-I—i—r—i—r  T (Idiv.<= 2-5n S e c . )  Figure  3-9.  I l l u s t r a t i o n o f t h e Manner i n w h i c h t h e F o u r S t a b l e S t a t e s a r e O b t a i n e d i n t h e Time Domain f o r Mode 2 Operation.  34  Figure  3-10.  I l l u s t r a t i o n o f t h e Manner i n w h i c h t h e Four S t a b l e S t a t e s a r e O b t a i n e d i n t h e Time Domain f o r Mode 3 Operation.  35  A c l o s e look at the c h a r a c t e r i s t i c s of a saddle p o i n t s i n g u l a r i t y i l l u s t r a t e s t h a t there are only two a c t u a l l y terminate  on the s i n g u l a r p o i n t ,  t r a j e c t o r i e s which  (See F i g u r e  These t r a j e c t o r i e s are the asymptotes f o r t + In other words, i t would take an i n f i n i t e  oo and  2-8). t—•-oo.  l e n g t h of time to  reach a saddle p o i n t i f the t r a j e c t o r y c o i n c i d e d w i t h the t -> + oo asymptote.  S i m i l a r l y , i f the t r a j e c t o r y c o i n c i d e d w i t h  the t-*-+oo asymptote of an unstable node i t would take  infinite  amount of time to move away from t h a t unstable p o i n t . Consider the saddle p o i n t i n A r e a ( l - 3 ) of F i g u r e 3 — 5 .  -^oi'  For  V  2 1 V  P '- ^ 01  11  -*-  s n  e  v  e  r  placed i n a p o s i t i o n  near  t h i s saddle p o i n t such t h a t i t w i l l be f o r c e d to go toward the stable point i n A r e a ( l — 4 ) . approaching  However, as 1 ^ takes on v a l u e s  the I^Q-^ v a l u e , the v^v^ p o i n t t r a v e l s on a t r a j e c t o r y  nearer to the two  asymptotes which w i l l f o r c e i t to r e t u r n to  the 00 s t a b l e node.  In other words, as 1 ^ tends  towards I^Q-^  the time r e q u i r e d f o r the c i r c u i t to reach the steady s t a t e condition i s increased.  T h e o r e t i c a l l y , i t should take an  infinite  amount of time f o r steady s t a t e c o n d i t i o n s to be reached when 1 ^ = I^Q-^ • it will node.  S i m i l a r i l y , when 1 ^ = I^Q-^ + £> where e i s v e r y small,  take the c i r c u i t a very long time In both the case where 1 ^ = I ^ Q ^  -  to reach the 01 s t a b l e e  a n <  i where 1 ^ = IJ^Q^+E  the value of v^ only v a r i e s a small amount from the 00 s t a t e value.  In the former  case, v^ r i s e s to a value n e a r i n g the v^  c o o r d i n a t e value of the saddle p o i n t i n A r e a ( l - 3 ) and then r e ceeds slowly to i t s 00 s t a t e v a l u e .  In the l a t t e r case v^  rises  s l i g h t l y above the v^ coordinate value f o r the A r e a ( l - 3 ) s i n g u l a r ity,  then i t r i s e s slowly to i t s h i g h s t a t e v a l u e .  From t h i s  36 d i s c u s s i o n i t can be seen that there i s a t h r e s h o l d value of v o l t a g e over which v^ must pass i n order to go to i t s high state value or c o n v e r s e l y , the t h r e s h o l d which v^ i s not to exceed i n order to r e t u r n to i t s low state v a l u e . Therefore, i n order t h a t the end state of the diodes be known before the c i r c u i t has reached  i t s steady  state c o n d i -  t i o n , a l l t h a t i s r e q u i r e d i s t h a t the value of v^ be compared to the known t h r e s h o l d Value from the comparator.  and an appropriate output be given  The comparing of v o l t a g e s could be c a r r i e d  out only a few nanoseconds a f t e r 1^ has been reduced  to z e r o .  This system of comparing diode voltages, to known t h r e s h o l d v o l t a g e s would decrease  the o v e r a l l time r e q u i r e d f o r readout  since i t i s not necessary reached  to wait u n t i l the stack c i r c u i t has  quiescence. For the two tunnel diode  case considered i t was found  that f o r Mode 1 o p e r a t i o n when 1^ <  IJQQ> ^  w  o  u  ld  state v o l t a g e l e v e l by the time 1^ had been reduced Thus i t would not be necessary used f o r t h i s p a r t i c u l a r diode.  reach i t s high to z e r o .  f o r a comparator c i r c u i t to be However, f o r other modes and  f o r c i r c u i t s c o n t a i n i n g more diodes a comparator might be necessary f o r B^' 3.4  V a r i a t i o n of C i r c u i t Parameters. To gather more i n s i g h t i n t o the c h a r a c t e r i s t i c s of the  two  tunnel diode  stack c i r c u i t , a l l of the parameters of t h i s i  c i r c u i t were v a r i e d 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  s t a b l e s t a t e s ( S e c t i o n 3.2).  The i n f l u e n c e  of these v a r i a t i o n s on the c i r c u i t performance were s t u d i e d and those quoted below are r e p r e s e n t a t i v e  f o r a l l modes of  operation. The  tunnel diodes used i n these experiments had the  f o l l o w i n g c h a r a c t e r i s t i c s : I , = 1.1 mA., I = 1.2 mA., P >1 P> I •• , = 0.40 mA. , I _ = —0.23 mA. The basic values f o r some v,l v,2 0  of the other parameters were C  Q  = 3 n s e c , Ig = 0.45 mA., and  = 12 pF.  3.4.1  V a r i a t i o n of  Si  . v  Let value I  61 = (I , — I „ ) . In t h i s experiment the v v,1 v,2' *  of I , was h e l d constant v,l  _ was v a r i e d so t h a t  0.63 mA. , 0.40 mA., decreasing  the value  Si  at 0.40 mA. and the value of  assumed the f o l l o w i n g  0.20 mA. , and 0.10 mA.  values:  As a r e s u l t of  of S I j AQQ remained e s s e n t i a l l y constant, y  A.~, i n c r e a s e d , A . « decreased. 01 '10  SI  I t was found that when  A , ~ was reduced to zero and t h e r e f o r e f o r t h i s value 10 c i r c u i t would not y i e l d four s t a t e s  (Figure  v  =0.10  mA., '  of 61 the v  3-ll).  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 a f f e c t e d .  When I  0  i s increased  for V < T < Y „ i s altered: P— — f  the s e c t i o n of the f ( v ~ )  the s e c t i o n 0 < v -< V P  i s unaltered.  This l a t t e r c o n d i t i o n i m p l i e s that AQQ i s not changed. a l 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 the s i n g u l a r i t i e s on the v , ^ P l 1  X-^ and X » e t c . 2  a l t e r e d values  a n e  »  a  n  The  of r , the p o s i t i o n s 2  d i n the values of  These changes w i l l be r e f l e c t e d , f i n a l l y , i n of AQ^ and A ^ Q .  38  C «I2 pP C, = &OpE 0  C = | | F. e  P  <fl (mA) y  Figure 3.4.2  3-11»  Simultaneous Let  and 61  Variation  61  as a F u n c t i o n o f o"l .  V a r i a t i o n o f o ' I and 0*1^. y  = ( l „ - I -, ) . P,2 p , l  P  7  For this  experiment *  I ~ were v a r i e d w h i l e I , and I - r e m a i n e d v,2 p,l v , l was v a r i e d  f r o m 0*1 mA. up t o 0.5 mA. w h i l e  v a r i e d f r o m 0.63 mA* I  of A Values  decreased;  t o 0.23 mA.  AQQ remained  and A ^ Q d e c r e a s e d  (Figure  The v a r i a t i o n s  ¥hen  & I  essentially  0  constant* o I r  v  was  was i n c r e a s e d and  constant, AQ^ increased,  3-12). i n t h e A v a l u e s c a n be i n t e r p r e t e d i n  the  same manner as f o r t h e p r e v i o u s c a s e .  and  A ^ Q i n ,this case d i f f e r  F i g u r e 3—11 b e c a u s e  I p,2  from  The c u r v e s f o r AQ-^  t h o s e f o r A Q ^ and A-^Q i n  t h e mode o f o p e r a t i o n o f t h e c i r c u i t  was  changed f r o m Mode 1 t o Mode 3 i n t h e i n t e r v a l between o ' l ^ = 0.63 and  = 0.43 mA*  By c o m p a r i n g  the e f f e c t s  o f v a r y i n g (f I  T  alone w i t h the e f f e c t s of v a r y i n g  Si  i s seen t h a t t h e v a r i a t i o n o f  circuit  &I  Si  and v  it  39 simultaneously,  p h a s much more e f f e c t on t h e  operation than the v a r i a t i o n of  Si  .  This  statement  P assumes t h a t t h e r e the  i s no change i n t h e mode o f o p e r a t i o n f o r  circuit.  C = 12 pF. C,= 2 F. C = I.I pF0  P  1-5-  (mA)  2  l-OMODE CHANGE  0-5-  A IQ,  0  •5  -6  -7  T~ Figure  3-12.  V a r i a t i o n o f A V a l u e s when £ I Increases.  3.4.3  V a r i a t i o n of I g . The  value  ^Iy ( A-) m  A" I  Si  P  (mA-)  D e c r e a s e s and  o f t h e 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. , t o 1.00 mA a n d t h e f o l l o w i n g A values c h a n g e d i n t h e f o l l o w i n g manner: (Figure  3-13).  AQQ d e c r e a s e d , A ^ i n c r e a s e d , and A decreased >10 Nn  40 I t was a l s o f o u n d t h a t f o r h i g h e r v a l u e s o f I g t h e s w i t c h i n g time of the system decreased near the t h r e s h o l d current values.  This c h a r a c t e r i s t i c  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 is  increased the s i n g u l a r i t i e s  may be e x p l a i n e d b y a and t h e X v a l u e s .  When I g  o f t h e s y s t e m a r e moved s i n c e t h e  q u i e s c e n t l o a d l i n e now i n t e r s e c t s t h e f ( y ^ ) and t h e f ( v g ) c u r v e s a t h i g h e r p o i n t s on t h e c u r r e n t s c a l e . the s i n g u l a r i t i e s w i l l approximate  fall  F o r some v a l u e s o f I g  on d i f f e r e n t  segments o f t h e  f ( v ) curve and thus the v a l u e o f r w i l l  T h i s change w i l l X^ and X »  cause  a c o r r e s p o n d i n g change i n t h e v a l u e o f  I n the p a r t i c u l a r  2  case  s t u d i e d , the s i n g u l a r i t i e s  change segments ( a n d t h u s A r e a s ) f o r o n l y D^. c h a n g e s i n X-^ a n d X  2  change.  The  resulting  when I g was i n c r e a s e d f r o m 0.45 mA. t o  0.65 mA. a r e g i v e n i n T a b l e 3-2.  (AjX Area  Ig=  10~ )  (X x  9  .45 mA. Ig=  2  .65 mA. Ig=  10~ )  .45 mA.  Switching  9  Ig=  .65 mA.  Time  1-1  -0.7490  -0.7490  1-3,2  +0.0975  +0.3418  -6.772  -6.086  decreased  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  5-3,2  +0.0961  +0.3267  5-4,5  -0.1857  -0*5061  T a b l e 3-2.  -12.48  -15.22 -9.617 -10.70  V a r i a t i o n o f X-^, X , a n d C i r c u i t Change i n ! „ . 2  -12.48  -15.2.2 -8.917 -12.38  no change  no change decrease decrease  S w i t c h i n g Time w i t h  41 1-5 C  0  = I 2  P  R  C, = * F . P  Ca = 1-lpF  1-0-  A  0 5  — i —  •35  Figure 3.4.4  i  •55  •45  3-13.  Variation  Simultaneous ¥ith I  0.63  mA.,  <^I »  of A Values  Variation  o f 61  = 0*65 mA.,  SI  11 was i n t e r s p e r s e d  thus the output  assumed t h e f o l l o w i n g v a l u e s : F o r each  Variation  of t  following:  For t  2  from  decreased,  Mode 3.  Q  f o r state  10 and  I t was f o u n d t h a t I g  0.45 mA.  and t h e c o r r e c t  = 0.40  mA.  *  experiment  t h e v a l u e s o f c a p a c i t a n c e s were  = 20 p F . , C  = 1 n s e c . t h e 01 s t a t e  increased A^Q  C  2  of these v a l u e s of  i n t h e range  sequence was i n c o r r e c t .  For t h i s the  as a F u n c t i o n o f I g .  sequence w o u l d be p r e s e r v e d f o r (Si  3.4.5.  B  and I - .  c o u l d be i n c r e a s e d t o 0*625 niA. f r o m output  ±(mA.')  1.05  •95  0.40 mA. , arid 0.20 mA.  state  V  B  —i—  •85  •75  •65  ;L  = 4. 5 p F . , and C" = 4.5 pF* 2  d i d not appear.  3 nsec. to 5 n s e c ,  When t  A Q Q decreased,  2  was  A Q ^ decreased,  and t h e mode o f o p e r a t i o n changed f r o m Mode 2 t o  The d e c r e a s e  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 e x p e c t e d  s i n c e t h e 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 area of the input pulse. 3.4.6  V a r i a t i o n o f CQ*  a n d C^-  To i n v e s t i g a t e t h e e f f e c t s o f c h a n g i n g 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 ,  the values of  one c a p a c i t a n c e was  v a r i e d w h i l e t h e 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 ^ a n d C ^ w e r e e q u a l a n d were s i m u l t a n e o u s l y v a r i e d . When CQ a n d C ^ were h e l d c o n s t a n t a n d C ^ v a r i e d , i t was f o u n d and C  t h a t AQQ r e m a i n e d e s s e n t i a l l y c o n s t a n t , A Q ^ i n c r e a s e d ,  A^Q decreased.  For this  s t u d y CQ = 1 2 p F . , C-^ = 2 p F . , and  was v a r i e d f r o m 1 p F . t o 4.5 p F .  2  S i n c e t h e mode o f o p e r a t i o n  c h a n g e d f r o m Mode 1 t o Mode 3 w h i l e C ^ v a r i e d f r o m 1.2 p F . t o 2.0 p F . t h e r e was a n a l t e r a t i o n i n t h e r a t e o f change o f A Q ^ and A  1  Q  (Figure 3-14). n  C =I2 F; 0  P  C, = 2?F. C is varied t  I  2.  3  Figure 3-14. V a r i a t i o n of A Values  4  C (pF) £  as a F u n c t i o n o f C  43  For CQ and C^ constant and i n c r e a s i n g the value of C^, AQQ  arid  i n c r e a s e d and A Q ^ decreased  A^Q  (Figure  3-15).  A  15-1  "  "  c =ie F e  P  C = 4.5 F. 2  P  C| is varied  10-  A (mA)  —i  -  4  — i —  C,( F)  4,5  P  F i g u r e 3 - 1 5 . V a r i a t i o n of A Values as a F u n c t i o n of C ^ When CQ was constant and C^ and  were equal and  v a r i e d simultaneously", A Q Q and A ^ Q i n c r e a s e d and A Q ^ decreased i n magnitude  (Figure  3—16).  L . — ©  —  C = |£ F. 0  p  C,=C2  A (mA.) •o-  -r  -  4-  — i — 4v5  C,= C . ( F ) P  F i g u r e 3 - 1 6 . V a r i a t i o n of A Values as a F u n c t i o n of C-^  varied  44 Figure 3 — 1 7  i l l u s t r a t e s the f a c t t h a t a l l A values  i n c r e a s e i n magnitude f o r an increase i n C Q when C ^ and C  2  are  constant.  i£ Figure 3 — 1 7 .  .  C ( F:)  20  0  V a r i a t i o n i n A Values  P  as a F u n c t i o n of C Q .  These r e s u l t s i n d i c a t e t h a t a v a r i a t i o n i n the value of one or more capacitances i n t h i s c i r c u i t has a decided on the magnitude of the d i f f e r e n t A ' s .  effect  A change i n the value  of C Q has much l e s s e f f e c t on these A values than a corresponding change i n e i t h e r 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 e i t h e r C ^ or C opposite d i r e c t i o n s by about 40%.  2  changes A Q ^ and A ^ Q i n  Therefore, i t would be a d v i s -  able to use tunnel diodes with low j u n c t i o n capacitance and then shunt the diodes with l a r g e r f i x e d c a p a c i t o r s .  45  Table 3 - 3 summarizes a l l t h e r e s u l t s of i n d i v i d u a l parameter  variations*  Parameter V a r i e d by  i c\oL  B a t e o f Change o f A ' s ( m A . / u n i t )  l\j/0  oo  A  Ll o*I  v  0  /mA.  +1.5  +1.8  /mA.  +0.8  increased  -0.7  /nsec.  -0.25  /nsec. - 0 * 6  increased  +0.17  /pF.  -0.73  /pP.  increased  +0.06  /pF.  increased  (C^=C^) i n c r e a s e d increased  Table 3 - 3 .  Inductance  inductance  /mA.  +0.10  /pP.  +0.07  /pP.  /mA.  -0*63  /mA.  /mA.  +0.34  /pP.  /pF.  -0.25 +0*05.  /pP.  -0.6  /mA. /nsec.  +0.78  /pP.  -0.49  /pP.  +0.26  /pF.  +0*08/pF.  Effects*  a means o f i n v e s t i g a t i n g t h e e f f e c t s o f p a r a s i t i c  on t h e o p e r a t i o n o f t h e c i r c u i t b e i n g  d i f f e r e n t analog  circuits and  -0*75  V a r i a t i o n s o f A V a l u e s w i t h Change o f D i f f e r e n t C i r c u i t Parameters.  As  two  10  decreased  Lg  3.5  /mA.  +1  increased  CQ  A  /mA*  olp  tg  01  0  decreased V  A  contained  circuits  Were e m p l o y e d .  t h e two t u n n e l d i o d e s  discussed,  Each of these  a s shown i n F i g u r e 2 - 5  i n a d d i t i o n e a c h h a d a n i n d u c t o r p l a c e d i n one s e c t i o n o f t h e  circuit*  The f i r s t  circuit  s i m u l a t e d had an i n d u c t o r  i n s e r i e s w i t h the tunnel diodes  and t h e second c i r c u i t had an  inductor placed i n s e r i e s w i t h CQ. given i n Appendix I I *  placed  These a n a l o g  c i r c u i t s are  46 F o 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 o f o p e r a t i o n were s i m u l a t e d a n d t h r e e v a l u e s o f i n d u c t a n c e , n a m e l y , 1 nH. , ;  10 nH., a n d 30 n H were u s e d w i t h e a c h mode.  I n a d d i t i o n , two  v a l u e s o f CQ were u s e d f o r e a c h mode a n d i n d u c t a n c e  valuej  t h e s e v a l u e s were 12 p F . , a n d 20 p F . I t was f o u n d  that f o r small values of inductance  ( l n H . ) , damped o s c i l l a t i o n s w o u l d o c c u r  i n a l l modes o f o p e r a -  t i o n a n d t h e r a n g e o f 1^ o v e r w h i c h t h e y o c c u r r e d d e p e n d e d u p o n t h e mode.  F o r Mode 1 a n d Mode 2 t h e damped o s c i l l a t i o n s  occurred  n e a r t h e I ^ Q p o i n t a n d f o r Mode 3 o p e r a t i o n t h e damped o s c i l l a t i o n s occurred near tance  IJ^QI  a  n  d  "*"A11*  ^  o r  l  A  R  S  E  R  v  a  (10 nH. a n d 30 n H . ) , s u s t a i n e d o s c i l l a t i o n s  over ranges centered a t these  l  u  e  s  of induc-  occurred  t h r e s h o l d s a n d damped o s c i l l a t i o n s  occurred 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 points.  B o t h t h e damped o s c i l l a t i o n s a n d t h e s u s t a i n e d  t i o n s were s u p e r i m p o s e d on t h e v ^ and for  an i n d u c t a n c e  free c i r c u i t ,  oscilla-  curves p r e v i o u s l y obtained  c . f . F i g u r e s 3-8, 3-9, 3-10.  Comparison of t h e r e s u l t s from the c i r c u i t i n which the inductance  was i n s e r i e s w i t h t h e t u n n e l d i o d e s w i t h t h e  r e s u l t s o f t h e c i r c u i t i n w h i c h t h e 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  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 .  slightly  I t was f o u n d  that f o r  i n c r e a s i n g values of inductance, a l l the threshold current v a l u e s f o r Mode 1 a n d Mode 2 d e c r e a s e d A10  decreased  while I  a n d t h a t f o r Mode 3>  , and I . , , i n c r e a s e d . A01 All A r k  F o r an i n c r e a s e  i n t h e v a l u e .of C Q * t h e r a n g e s o v e r w h i c h t h e damped o s c i l l a t i o n s and  t h e 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  order 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 The  was p l a c e d i n t h e c i r c u i t i n s e r i e s w i t h CQ.  analog c i r c u i t used t o simulate t h i s  Appendix I I .  The v a l u e  c i r c u i t i s given i n  of t h e damping r e s i s t o r  u s e d was t w i c e  the v a l u e r e q u i r e d t o stop s u s t a i n e d o s c i l l a t i o n s . value, the t r a n s i e n t o s c i l l a t i o n s  For this  were r e d u c e d t o t h e p o i n t w h i c h  w o u l d make a d v a n c e d e t e r m i n a t i o n o f t h e o u t p u t f e a s i b l e . the Mode 1 c i r c u i t  ( C = 12 pF. ,  w i t h L = 10 nH. a v a l u e factory results. C  2  =2pP., C  Q  F o r Mode 2 c i r c u i t  ( C = 12 p F . , C Q  1  satis-  ( C = 12 pF. , C-^ =4.5 Q  o f R o f 8X1 was  = 1.1 p F . , C  w i t h L = 10 nH. R was c h o s e n t o be 15-TL . of inductance  = 1.1 p F . ) ,  o f r e s i s t a n c e ( R ) o f 1 5 i T gave  = 4.5 p F . ) , w i t h L = 10 nH. a v a l u e  F o r Mode 3 c i r c u i t  2  For  2  =4.5  pF. ,  sufficient. pF.),  For increasing values  l a r g e r values of R are r e q u i r e d .  48 4.  ANALOG COMPUTER STUDIES OF A THREE B I T ANALOG-TO-DIGITAL CONVERSION CIRCUIT USING A TUNNEL DIODE STACK CONFIGURATION A three b i t three tunnel diode  stack c i r c u i t ,  i n a s i m i l a r f a s h i o n t o t h e two b i t two t u n n e l d i o d e 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 d i o d e  operating  stack  circuit,  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 c o m p u t e r and t h e r e s u l t s o f t h e b r i e f studies c a r r i e d out are described i n t h i s  chapter.  4.1 S i m u l a t i o n o f a T h r e e T u n n e l D i o d e S t a c k The diode  Circuit.  analog c i r c u i t used to simulate the three  stack c i r c u i t i s given i s Appendix I I I .  T h i s c i r c u i t was  also used f o r the study of i n t e r d i o d e capacitance purpose of the experiment  effects.  states: I C  ±  s e t of c i r c u i t parameters y e i l d e d these I  , = 1.1 mA., p,l  T  „ = 1.2 mA., p,2  , = 0.40 mA.,  I  = 2.8 p F . , C  =' 2.5 p F . , C  2  = -0.23 mA.,  0  c i r c u i t d i d not y i e l d would undoubtedly  The  was t o a s c e r t a i n w h e t h e r 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 d i o d e following  tunnel  3  I  I  stack.  The  eight stable  _ = 1.4 P»3  = -1.05 mA.,  mA., ' C  n  = 12 p F . ,  = .5 pF. , and t = 3 n s e c . 2  This  equal A values but f u r t h e r experimentation  y i e l d the d e s i r e d equal A value  result.  F i g u r e 4-1 i l l u s t r a t e s t h e 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 b y t h i s c i r c u i t , b y d e p i c t i n g p l o t s o f v ^ , v ,•• and v ^ v e r s u s d i f f e r e n t values of i n p u t . are as f o l l o w s : A011 I  A110 n  i  n  -"-^001  The v a l u e o f t h e t h r e s h o l d c u r r e n t s  ^*681 mA. , I ^ Q ^ Q  =  = 1.357 mA.,  I  = 1.551 mA.,  and I . = 2.193 mA. Alll  A  1  0  0  = 1.407 mA., i n n  time f o x  I  A  = 1.203 mA. , 1  0  1  = 1.490  mA.,  49 I t was f o u n d d u r i n g t h e s e r i e s o f e x p e r i m e n t s w i t h t h e three tunnel diode and  c i r c u i t t h a t i f t h e v a l u e s o f C^, C^,  f ( v ) t h a t y i e l d e d A Q ^ = A ^ Q f o r t h e two t u n n e l d i o d e  case  2  were u s e d , t h e n t h e s e  equal  increments  w o u l d be  preserved.  Because o f 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 to g i v e t h e d e s i r e d r e s u l t by w o r k i n g g r o u p s o f two r a t h e r t h a n w o r k i n g one  f(v^),  c a n be b u i l t  w i t h tunnel diodes i n  w i t h the complete s t a c k a t  time. Since the behaviour  of the three tunnel diode  i s an e x t e n s i o n o f t h e two t u n n e l d i o d e possible to u t i l i z e  circuit,  the comparator technique,  c r i b e d , a s a means o f a d v a n c e o u t p u t 4-1 i n d i c a t e s t h e s w i t c h i n g t i m e  l o n g e r t h a n t h a t o f t h e two t u n n e l d i o d e expected  s i n c e t h e r e a r e more s t o r a g e  circuit.  T h i s f a c t does n o t p r e c l u d e  i t will  circuit case.  devices  be  p r e v i o u s l y des-  determination.  of t h i s  circuit  As F i g u r e  i s somewhat T h i s i s t o be  i n the former  the c i r c u i t from  being  u s e d a s 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  Interdiode Capacitance  Effects.  By p l a c i n g a f i x e d c a p a c i t o r f r o m t h e t o p o f the  top of  an e x t r a c i r c u i t p a r a m e t e r i s c r e a t e d .  to This  p a r a m e t e r , c a l l e d C^, c a n be u s e d t o a i d i n a c q u i r i n g a which w i l l  yield  equal A values using diodes  and  circuit  capacitances  t h a t would not otherwise  give t h i s r e s u l t .  u s e d t o d e s i g n an a n a l o g  c i r c u i t to simulate t h i s e f f e c t are  given i n Appendix I I I .  The c i r c u i t  equations  50  F i g u r e 4—1.  I l l u s t r a t i o n o f t h e Manner i n w h i c h E i g h t S t a b l e S t a t e s a r e O b t a i n e d i n t h e Time Domain f o r Three Tunnel Diode C i r c u i t .  51 The p r e s e n c e  of  i n the c i r c u i t has the e f f e c t o f  slowing,down the s w i t c h i n g of ing  of  and  relative  to the s w i t c h -  As an example o f t h e e f f e c t i v e n e s s o f  i n the  c i r c u i t , consider the v a r i a t i o n i n t h r e s h o l d current values for D  2  a c i r c u i t where I. = 1.40 mA., p,3 a  a r e t h e same as b e f o r e , C = 0.4 p F . , and  Q  I  - = -.96 mA., D, and v,3 ' 1  = 12 p F . , C.^ = 2.7 p F . , C  i s v a r i e d f r o m 0 p F . t o 0.8 p F .  = 2.5 p F . ,  2  See T a b l e  4-1. Threshold  I  A001  I  A010  •'"AO 11 I  c  4  = 0 pF.  C, = 0 .8 p F . 4  0.696 mA.  0.659 mA.  1.274 mA.  1.296 mA.  1.371 mA.  1.390 mA.  A100  1.454 mA.  "^101 ^110 :I  Table 4-1.  A111  1.507 mA. 2.230 mA.  1.658 mA.  2.289 mA.  2.246 mA.  V a r i a t i o n o f T h r e s h o l d V a l u e s w i t h C^  As T a b l e 4-1 i n d i c a t e s ,  does n o t a f f e c t a l l o f  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 a d d i t i o n a l parameter proves ing  this  t o be a v e r y u s e f u l t o o l i n a t t e m p t -  to design a suitable c i r c u i t  sion.  that  for analog-to-digital  conver-  52 5.  SUMMARY AND CONCLUSIONS  The main purpose of t h i s study has been to i n v e s t i g a t e the c h a r a c t e r i s t i c s of a novel tunnel diode  circuit  i n order to  a s c e r t a i n whether or not i t could f u n c t i o n as an analog-to— digital  conversion c i r c u i t .  circuit  i s the f a c t t h a t 2  tunnel diodes  n  The n o v e l t y a s s o c i a t e d with  this  s t a t e s can be obtained u s i n g n  c o n s t r u c t e d from the same type of semiconductor  material. Prom the s i n g u l a r i t y a n a l y s i s of the two tunnel circuit two  diode  case and the analog computer s i m u l a t i o n s t u d i e s f o r the  tunnel diode  circuit  and the  three tunnel diode  circuit,  the f o l l o w i n g summary of r e s u l t s and c o n c l u s i o n s can be made: (1) . Three modes of operation e x i s t f o r the two tunnel diode  case and these modes are determined by the r a t i o of  the capacitances which shunt the tunnel d i o d e s . circuits  For  c o n t a i n i n g more than two tunnel diodes, more modes  would be  developed*  (2) . The switching time f o r t h i s c i r c u i t v a r i e s .  The longest  switching occurs a t the current t h r e s h o l d values f o r different states. approximately wait u n t i l  For the c i r c u i t s  40 nsec.  explored t h i s time was  A means of a v o i d i n g the need to  the c i r c u i t has reached  i t s steady s t a t e c o n d i t i o n  i n order to o b t a i n the output has been proposed.  By  u s i n g t h i s scheme i t i s conceivable that i t would be p o s s i b l e to o b t a i n the output input pulse had been a p p l i e d .  i n 10 nsec. a f t e r a 4 nsec.  53 (3) . f o r  circuit*  containing mora than two  additional' interdiode capacitances can tha  dosign of a c i r c u i t  suitable for  be  tunnel diode*, used to aid  in  analog-to-digital  conversion* (4) . P a r a s i t i c inductance can the c i r c u i t .  cause incorrect operation of  H o w e v e r , t h i s p r o b l e m 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 t h e c i r c u i t t o dampen these u n d e s i r a b l e  effects.  (5) . Compared t o o t h e r p a r a m e t e r v a r i a t i o n s , a v a r i a t i o n in  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  causes the circuit*  greatest  the  To r e d u c e t h i s p r o b l e m l o w j u n c t i o n c a p a c i t a n c e  tunnel diodes to  e f f e c t on t h e o p e r a t i o n o f  s h o u l d be u s e d and  shunt these  f i x e d capacitances  used  diodes.  ( 6 ) . A summary o f t h e e f f e c t s o f a v a r i a t i o n i n t h e m e t e r s o f a two Table  diodes  tunnel diode  para-  stack c i r c u i t i s given i n  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 which w i l l  t h e a p p r o p r i a t e A v a l u e s so t h a t t h e c i r c u i t w i l l  yield  produce  the proper 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  time*  T h i s a r i s e s from the f a c t t h a t  s t a c k i n g s u c c e s s i v e diodes, the A r e l a t i o n s h i p of  by  one  diode w i t h the n e x t , f o r a g i v e n s e t of 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  t h e b a s i s t h a t r e c o m m e n d a t i o n s ( 2 ) , ( 4 ) , and  (5)  \  54 are implemented, t h i s study has i n d i c a t e d that the proposed c i r c u i t would be a f e a s i b l e means of producing h i g h speed analog-to-digital  conversion.  55  APPENDIX I A I.  A n a l o g Computer Stack C i r c u i t . The  circuit  t h e PACE 231R  C i r c u i t f o r S i m u l a t i o n o f Two  Tunnel Diode  illustrated  simulated  i n Figure  c i r c u i t may  be w r i t t e n as  The  three major e q u a t i o n s  follows:  dv  1 dt  cf^l  dv _2 dt  1  h  +  -  f ( v  [ i + i - f(v j]  l  i  -  A  a i  a =  f ( v ^ ) and f (  v 2  C  (2.17)  2  B  B  C,  where  (2.16)  l>]  r  x  x  on  a n a l o g c o m p u t e r and t h e a n a l o g c i r c u i t u s e d f o r  t h i s p u r p o s e i s shown i n F i g u r e A 1-1. for this  2-5 was  +'^f(v )  + ^ f ( v  1  and  l  C  1 + a  2  ) are the e f f e c t i v e  by t u n n e l d i o d e s D^and D , 2  (2.21)  )  2  characteristic  respectively,  curves produced  as o u t l i n e d  i n Section  3.1. The p a r a m e t e r s o f t h e a c t u a l a p p r o p r i a t e l y to f a c i l i t a t e designate real the computer  simulation.  c i r c u i t were Lower  case  scaled letters  time parameters w h i l e upper case l e t t e r s  scaled parameters.  The  follows:  " =  1 0  <\olts>  i = .2(1  m V  -  r ) mA. volts' t = 10~ (T ) sec sec v  9  n  designate  s c a l i n g f a c t o r s a r e as  56 C  n  —12 i s i n p i c o f a r a d s ( l pF. = 10~  By s c a l i n g Equations f o l l o w i n g analog c i r c u i t  (2.16), (2.17), and  equations  Farad) (2.2l) the  result:  dV  (1.3)  Since the maximium value of t ^ was  chosen to be  20  nsec. and since the a m p l i f i e r s of the analog computer were r a t e d f o r a maximium i n p u t or output of 100 v o l t s , then the slope f a c t o r K, was In  chosen to be  the analog c i r c u i t F(V^) and P ( V ) were set up 2  f u n c t i o n generators w i t h I  . = 4.4 P»l  and the appropiate s c a l i n g was marked  4.  &and  S2*  mA.,  and I  „ = 4.8 P»2  on  mA.  employed by u s i n g potentiometers  respectively.  F i g u r e AI--1  A n a l o g C i r c u i t o f Two  Tunnel Diode S t a c k .  58  APPENDIX I I A II.  A n a l o g Computer C i r c u i t s U s e d t o S i m u l a t e I n d u c t a n c e E f f e c t s i n 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 .  A I I . 1 Two T u n n e l D i o d e S t a c k C i r c u i t w i t h I n d u c t a n c e 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 AI I - l .  X  X  A  x  0  *  B  l  c  7  1  L  =  Af( ) V l  "B c  a )  2 +  f  K  }  I Figure A I I - l .  Two T u n n e l D i o d e S t a c k C i r c u i t w i t h i n S e r i e s w i t h Tunnel Diodes.  Inductance  The e q u a t i o n s d e s c r i b i n g t h e o p e r a t i o n o f t h i s are  circuit  as f o l l o w s :  f ld tl  = of  [  * i  +  A  B  -  f  (  V ]  (II.1)  59 dv _2 dt  +  d (i dv _3 = L dt  + i  2  x  1  B  (II.2)  f(v )  B  2  ) (II.3)  dt' (II.4)  i. = d.(v  = c0  + v  1  2  + v ) 3  dt  (II.5)  By s u b s t i t u t i o n and r e a r r a n g e m e n t o f t h e above equations the f o l l o w i n g  equation r e s u l t s :  ,2/. ( i  l  +  i  B  c  1  )  dt'  C  0  i  L  five  _  A  a  i  _ ( i  B  +  a  )  i  i  + 5  c  fi'f(v )  + ^ f ( v  1  2  )  (II.6) where *  °1  =  Equations  °2 (Il.l),  ( I I . 2 ) , and ( I I . 6 ) a r e t h e e q u a t i o n s  used t o d e s i g n the analog c i r c u i t . factors  as were s t a t e d  U s i n g t h e same  i n Appendix I , these three  scaling equations  become: dV, dT dV, dT d  2  ^  + I ) f i  dT  10 C  0  20  r j  f  [  (II.7)  I  + I, -  ±  (II.8)  F(V )] 2  c  3 I  A  _ a I  B  -  (1 + a ) l . 1  +  s  c  ap(V ) 1  +7^P(V ) 2  L  (II.9)  F i g u r e AII-2  Analog C i r c u i t f o r Two Tunnel Diode Stack C i r c u i t with Inductance i n S e r i e s with the Tunnel Diodes.  61 where C" i s i n p i c o f a r a d s a n d L i s i n n a n o h e n r i e s .  Figure  A lt-2 illustrates  simulation,  n  A II. 2  the analog  c i r c u i t used f o r t h i s  Two t u n n e l D i o d e S t a c k C i r c u i t 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  w i t h Inductance i n  is illustrated  i n Figure  A II-3.  Figure A  II-3.  The are  Two T u n n e l D i o d e S t a c k i n S e r i e s w i t h CQ  equations  Circuit  with  Inductance  describing the operation of this  circuit  as f o l l o w s : dv  1 dt  57  [  h + i  B  (II.1)  -f (  dv  2 dt  f(v )] 2  dv _2 = L dt  0 dt  k  (II.2)  (11.10)  dv  0 0 dt " C 0 i .  =  *1  (II..11) +  *(>  (II.4)  62 d(v  L  + v )  d(v + v )  Q  1  2  (11.12)  dt  dt  By s u b s t i t u t i o n arid rearrangement  of the above s i x  equations the f o l l o w i n g equation r e s u l t s :  * <JA - *1> 2  dt  1 L  ( p  i  (i, • i ) .l!vV.±il  +  B  ^0 f(v ) 2  Equations  (ll,l),  (11.13)  ( I I . 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 F i g u r e A I I - 3 .  Using 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 three equations become:  f  dV  1 dT  20 C  (II.7)  dV,  (II.8)  dT  dT  103 L  2  i ^ (i, + i ). ifigiiil F ^ )  °1  +  °2  B  °0 F(V )  (11.14)  2  where C i s i n p i c o f a r a d s 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 f o r this simulation. A II.3 Two Tunnel Diode Stack C i r c u i t w i t h Inductance and Damping R e s i s t o r .  The  c i r c u i t to be simulated i s shown i n F i g u r e A II-5  F i g u r e AII-4  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 i n S e r i e s with C . n  64 "1  B  —  R  >  V  R  J-C  '2  (f  0  V  2  '=2 '0  ©  f  U  1  B  2 >  .0  F i g u r e A I I - 5 . Two Tunnel Diode Stack C i r c u i t with and Damping Resistance.  Inductance  The equations 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  circuit  are as f o l l o w s :  •7  dt  [ h  +  i  B -  f  (  v  i ) ]  (II.1)  (II.2)  dt d i 2  dt  = L  0  (11.10)  dt'  dv  9. - l£L dt " c  (11.11)  dv di 0 _R = R dt dt  (11.15)  0  d ( v  R  +  V  L V dt +  _  d  (  ~  v  l  +  v  2  }  dt  (11.16)  By s u b s t i t u t i o n and rearrangement of tfie above s i x equations y i e l d s the f o l l o w i n g equation:  65  a (i 2  -  V  1 L  ( 1  ^ A "  1) ( i  +  ! *  1  d  (  which  are used  circuit  (II.2),  of Figure A I I - 5 .  three equations  I>are  l dT " av v  =  2  dT "  =  10 L  (11.17)  and ( I I . 1 7 ) a r e t h e e q u a t i o n s to simulate the  The same s c a l i n g f a c t o r s t h a t were  used  for this  circuit  and t h u s  20  these  (II.7)  °1 20 C  2  [  h  h  +  -  F ( v  (11.8)  2>  F(V )  3  2  <•  " A " '0  , R L  d  (  l  Figure A II—6 i l l u s t r a t e s this  )  become: d  for  l  to design the analog c i r c u i t  s t a t e d i n Appendix  dT  i  dt  (H.l),  1 W 1  B  A -  i  E  '0 Equations  i ) .  x+  simulation*  A dT  I  l  )  the analog c i r c u i t  (11.18)  used  Figure AII-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 .  67 APPENDIX I I I  A I I I Analog Computer C i r c u i t f o r Simulation of a Three Tunnel Diode Stack C i r c u i t and Interdiode Capacitance E f f e c t s A s s o c i a t e d with t h i s C i r c u i t .  The If  c i r c u i t to be  simulated i s shown i n Figure A  C^ i s removed from t h i s f i g u r e ,  three tunnel diode  stack c i r c u i t .  the c i r c u i t becomes a  three tunnel diode  which r e l a y the e f f e c t s  The  of C. to other s e c t i o n s of the c i r c u i t .  ,  ,  Three Tunnel Diode Stack C i r c u i t Containing One A d d i t i o n a l Interdiode C a p a c i t o r , C^.  equations  are as f o l l o w s :  simulate  c i r c u i t by removing the paths  _p  Figure A I I I - l .  simple  Because of t h i s f a c t , the  analog c i r c u i t f o r t h i s s i m u l a t i o n can a l s o be used to the simple  III-l.  d e s c r i b i n g the operation of t h i s c i r c u i t  68  d  v  d v  l dt ~  1 C  2  3  (  (IH.I)  V  ^2  B "  2  C  -  f  + v )  V l  ^ l V  (III.2)  )  v  2  +  V  dt  B  (III.3)  - f (v ) 3  *2  2  +  -2 + i  *1  3  dt d  f  x  dt d(  -B ~  i—.  1  dt - c dv  _ *1  1  (IH.4)  3^  10 3  (III.5)  0 (III.6)  By s u b s t i t u t i o n and rearrangement equations y i e l d s the f o l l o w i n g  *2 = °4  ' l  equations:  (III.7)  ' 2  C  C  f(v ) 1  1 + a  X  A  ~  of the above s i x  a l  B  f(v ) 2  6  +  +  '0  c;  f  (  v  3  }  (III.8)  where C  and  Equations  0°4  (ill.l),  (III.2),  (ill.3),  (ill.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 . u s i n g the same s c a l i n g f a c t o r s o u t l i n e d  i n Appendix I these  By  69 equations become: d  v  l 20 a T • l1 ;  = C  20 dT " C 2 dV 20 ~df ~ 33 =  L  h h  4  +  l  "  B  B ~  1  F  (III.9)  V  (  F ( V  2  (III.10)  }  3  =  I  1  +  I  2  +  I  B  P ( V  -  3  (III.11)  )  C  I„ =  C  4  << C  ' l  C  2 (III.12) F^)  1  1 + a  X  l  "  a I  B  +  P ( V  2  )  e  +  ^ P ( V  3  )  (III.13)  Figure A I I I . 2 i l l u s t r a t e s for  this simulation.  the analog c i r c u i t used  Figure AIII-2  Analog C i r c u i t f o r a Three Tunnel Diode C i r c u i t with an A d d i t i o n a l Interdiode C a p a c i t o r , C..  Stack  71 REFERENCES  1.  E s a k i , L.,  "New Phenomenon i n N a r r o w P-N J u n c t i o n s " , P h y s . 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. , " C o m p o s i t e C h a r a c t e r i s t i c s o f 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.  S a l a m a , 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 o f S e r i e s C o n n e c t e d T u n n e l D i o d e s 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 o f B r i t i s h C o l u m b i a , December, 1962.  4.  Kiyono,  5.  Cunningham, ¥. J . , 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 Book Company, I n c . , New Y o r k and L o n d o n , 1958.  6.  Gibson, J.E., N o n l i n e a r Automatic Book Company, I n c . , New  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 d i o d e t h e o r y " , P r o c . I.R.E.. V o l 4 9 , pp 1268-78, A u g u s t 1961.  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 C o m p u t e r s . V o l . E C - 1 1 , No. 16, pp. 7 9 1 - 2 , December 1962.  C o n t r o l , McGraw-Hill Y o r k and L o n d o n , 1963.  

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