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A unilateral tunnel-diode frequency converter Little, Warren David 1963

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A UNILATERAL  TUNNEL-DIODE FREQUENCY CONVERTER  by WARREN DAVID L I T T L E B.AoSc,  The U n i v e r s i t y  of B r i t i s h  C o l u m b i a , 1961  A THESIS SUBMITTED IN PARTIAL FULFILMENT  OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED  SCIENCE  i n the Department o f Electrical We  accept  this  thesis  Engineering as c o n f o r m i n g  required  THE UNIVERSITY  to the  standard  OF BRITISH COLUMBIA  J u n e , 1963  •r  In  presenting t h i s  thesis in p a r t i a l fulfilment  of  the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree a v a i l a b l e for reference  that the L i b r a r y s h a l l make i t  and study.  I f u r t h e r agree  that  freely per-'  mission for extensive copying of t h i s t h e s i s f o r . s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representativeso  It  i s understood that copying, or p u b l i -  c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n .  Department of E l e c t r i c a l  Engineering  The U n i v e r s i t y of B r i t i s h Columbia,. Vancouver 8, Canada. Date  July  25, 1963  ;  ABSTRACT  Frequency converters and  pumped n o n l i n e a r  flow The  capacitance  i n one d i r e c t i o n unilateral  u s i n g pumped n o n l i n e a r with  only are the subject of t h i s  property  c a n be o b t a i n e d  t e r m i n a t i o n a t t h e image f r e q u e n c y the  of s i g n a l thesis.  e i t h e r by a s u i t a b l e  o r by q u a d r a t u r e  pumping  c o n d u c t a n c e and c a p a c i t a n c e . Of  the  the property  conductance  great  output  importance  frequency  i s a frequency  i s lower than  converter  i n which  t h a t of the i n p u t .  Such a  d o w n — c o n v e r t e r b a s e d upon a p r o p o s e d image t e r m i n a t i o n method is are in are  examined b o t h  analytically  g i v e n w h i c h t h e c o n d u c t a n c e and c a p a c i t a n c e order  t h a t t h e image t e r m i n a t i o n be p a s s i v e .  fulfilled  forward-  by a s i n g l e  tunnel—diode.  to reverse-gain r a t i o  bandwidth i s f e a s i b l e ; The  quadrature  image t e r m i n a t e d the  and e x p e r i m e n t a l l y .  former  pumped c o n v e r t e r  converter  must  satisfy  The c o n d i t i o n s  I t i s found  of at l e a s t  the estimated  Conditions  20 db o v e r  noise figure  that a a 5%  i s 3.4 db.  i s compared w i t h t h e  and i t i s shown i n p a r t i c u l a r  c a n be u n i l a t e r a l  o n l y a t one  frequency.  that  ACKNOWLEDGEMENT  The Dr«  author wishes t o thank t h e s u p e r v i s i n g  M, P. B e d d o e s i  course of t h i s  f o r his help  professor*  and encouragement d u r i n g t h e  study.  Grateful  acknowledgement i s g i v e n  to the B r i t i s h  C o l u m b i a T e l e p h o n e Company f o r a s c h o l a r s h i p awarded i n 1961 j and  to the National The  the  National  Research Council  work d e s c r i b e d  i n this  f o ra Studentship  t h e s i s was s u p p o r t e d by  R e s e a r c h C o u n c i l u n d e r G r a n t BT-68.  vi  i n 1962*  TABLE OP CONTENTS Page L I S T OP ILLUSTRATIONS  ..  v  ACKNOWLEDGEMENT  v i  L I S T OP SYMBOLS  v i i  1.  INTRODUCTION  1  2.  MIXING-ELEMENT ANALYSIS  5  2.1 2.2 3.  6  The M i x i n g - E l e m e n t  9  Matrix  FREQUENCY CONVERTER ANALYSIS 3.1 3.2  4.  D i f f e r e n t i a l Equation of the M i x i n g Element ••«..••.... «..  14  Network R e p r e s e n t a t i o n o f t h e M i x i n g E l e m e n t • ••••  14  Unilateral  16  Frequency  Conversion  UNILATERAL DOWN-CONVERSION BY THE IMAGE TERMINATION METHOD  22  4.1  Approximations  22  4.2  Optimum T e r m i n a t i o n s  4.3  Converter  4.4  G a i n w i t h Optimum T e r m i n a t i o n s  4.5  Frequency C h a r a c t e r i s t i c s o f a Computer A n a l y z e d Model w i t h " P r a c t i c a l " T e r m i n a t i o n s ...  ......................  Stability  23 ........•••..*... ......  27 30  32  5..  NOISE  39  6.  EXPERIMENTAL  40  6.1  7.  Experimental Converter Elements .....  and I t s 40  6.2  Experiment  6.3  Suggestions f o r Further Experimental Study .«•«.««....»..•............*...  CONCLUSIONS  Techniques  Circuit  and R e s u l t s  ...  45  47 48  Page APPENDIX I  P o s i t i v e Real D r i v i n g - P o i n t  Admittance  •  49  APPENDIX I I T e r m i n a t i n g Components f o r a " P r a c t i c a l " C o n v e r t e r • • • . . » • • • • • . » •••• •  51  REFERENCES  53  LIST  OP ILLUSTRATIONS  Figure 1- 1  Page B l o c k Diagram I l l u s t r a t i n g  Frequency-  Converting Process  3  2— 1  Mixing—Element  2- 2  S m a l l - S i g n a l Frequency  3- 1  Mixing—Element  4- 1  B o u n d a r i e s o f a P a s s i v e Image T e r m i n a t i o n .............. I n p u t and Output C o n d u c t a n c e v s . N o r m a l i z e d Input Frequency  29  F o r w a r d G a i n w i t h Optimum T e r m i n a t i o n s v s . Normalized Input Frequency  31  Equal Gain Contours of Converter Optimum T e r m i n a t i o n s  33  4-2 4-3 4-4 4-5 4-6  Circuit  ..... Spectrum  11  with Terminations  .......  and R e v e r s e  with"Practical"  with  Terminations  ...........  Experimental Converter C i r c u i t  6-2  E q u i v a l e n t C i r c u i t and C h a r a c t e r i s t i c s of a Pumped 1N2939 T u n n e l - D i o d e B l o c k D i a g r a m o f C o n v e r t e r and M e a s u r i n g Equipment ..............................  v  34  Gain of a Converter  6-1  6-3  15  25  Mixing—Element w i t h '^Practical" Termina t i o n s ••*...».............. Forward  7  .........  36 41  42 45  List  o f Symbols  V  (t)  F i r s t defined in Section t o t a l voltage across nonlinear conductance  2 . 1  c o n d u c t a n c e pump v o l t a g e  2 . 1  v(t)  small-signal voltage  2 . 1  I  t o t a l current through conductance  ^ V  (t)  (t) ^  g(t) I  S P  i  instantaneous ance (t)  nonlinear 2 . 1  incremental  conduct2 . 1 2 . 1  c o n d u c t a n c e pump c u r r e n t s m a l l - s i g n a l conductance  I p(t)  c a p a c i t a n c e pump c u r r e n t  2 . 1  q V  c h a r g e on n o n l i n e a r c a p a c i t a n c e c a p a c i t a n c e pump v o l t a g e  2 . 1 2 . 1  capacitance bias voltage  2 . 1  conductance b i a s v o l t a g e  2 . 1  C  cp  (t)  V ^  i (t)  small-signal capacitance  C(t)  instantaneous ance "fc h  g  n  c  n  current  2 . 1  (t)  current  incremental c a p a c i t -  2 . 1 2 . 1  conductance harmonic  (real)  2 . 2  n  c a p a c i t a n c e harmonic  (complex)  2 . 2  th n  c a p a c i t a n c e harmonic  (real)  2 . 2  i ^  v o l t a g e harmonic  (small-signal)  2 . 2  c u r r e n t harmonic  (small-signal)  2 . 2  th C C  '  n  V\  n  "fch i WQ  pump a n g u l a r f r e q u e n c y  2 . 2  to^  r.f.  angular frequency  2 . 2  fi>2  i.f.  angular frequency  2 . 2  G>2  image a n g u l a r f r e q u e n c y admittance at  across  f r e q u e n c y a. vii  2 . 2  mixing-element 3 . 1  List  o f Symbols  F i r s t defined in Section  Y^  Y ^ minus  Y.. ^  admittance parameters mixing—element matrix  1  forward  admittance  of f i l t e r s  3.1  of t h e 3.1  transducer  gain  3.2  G^2^  reverse transducer  gain  3.2  a  nonlinear factor  4.1  P  loading factor  4.1  normalized  4.1  frequency  Y. in  converter  i n p u t admittance  4.2  Y ^.  converter  output  4.2  F  converter noise  Q U  admittance figure  5  A UNILATERAL TUNNEL—DIODE FREQUENCY CONVERTER  1.  INTRODUCTION  Most modern h i g h f r e q u e n c y the  heterodyne p r i n c i p l e , ^ ^  frequency  of the  intermediate c a t i o n and i n the  incoming  frequency  detection.  c o m m u n i c a t i o n r e c e i v e r s use  i n w h i c h power a t the  signal  gives r i s e  down-converter stage  of the  the  receiver.  advent of the  best  noise (6)  Chang, e t a l  figures  amplifiis carried  The  out  noise p e r f o r -  importance  since i t  (2) of t h e r e c e i v e r . ' Until (3) i n 1957, c r y s t a l - d i o d e s were (4)  as t h e  frequency  changing  elements.  c r y s t a l - d i o d e c o n v e r t e r s have c o n v e r s i o n l o s s e s of 3  converters with  all  tunnel-diode  exclusively  t o 4 d e c i b e l s and 1960  lower  the u l t i m a t e s e n s i t i v i t y  used almost The  for  conversion  mance of t h e d o w n - c o n v e r t e r i s o f prime determines  t o power a t a  w h i c h i s more s u i t a b l e This frequency  carrier  figures  of 4 to  ' showed t h a t t u n n e l - d i o d e  22  reciprocal.  To  the  '  In  frequency  d e c i b e l s o f c o n v e r s i o n g a i n and  o f 3 d e c i b e l s were p o s s i b l e .  tunnel—diode  5 decibels*  noise  author's  knowledge  d o w n - c o n v e r t e r s t h a t have been s t u d i e d  T h i s p r o p e r t y i s u n d e s i r a b l e b e c a u s e the  i n no  way  isolated  from the  output  and  as a r e s u l t  noise  i s i n c r e a s e d by n o i s e  i n the  output  the  are input i s input  (o) * For a r e c i p r o c a l the reverse g a i n .  transducer  the  forward  circuit.-  '  g a i n i s equal  to  2 The  main t o p i c  tunnel—diode  of t h i s  frequency  i s the  development  of a  unilateral  down—converter*  Previous considered  thesis  s t u d i e s of tunnel-diode  the n o n l i n e a r c o n d u c t a n c e c o n v e r t i n g element*  converters  of the d i o d e s  have as t h e  sole  In a d d i t i o n to a n o n l i n e a r  c o n d u c t a n c e however, t u n n e l — d i o d e s  have a n o n l i n e a r j u n c t i o n  (3) capacitance I important  w h i c h the  equations  The two  The  as w i l l  frequency  be  i s the  concept  signal  and  The  output  examples  * the  capacitance  1,  frequency f, - f .  1—1.  2 and  The  3.  The mixing-element*  i s a sinewave of f r e q u e n c y fQ. t o be  The  of t h e s e  In g e n e r a l , the  For a u n i l a t e r a l t r a n s d u c e r the reverse gain i s zero*  f ^ and  developed  at f ^  forward  and  frequencies*  frequencies —  converter  the  nonlinearities  d i f f e r e n c e s of t h e h a r m o n i c f 2 i s one  frequency  converter.  sinewave of f r e q u e n c y  a t sums and  in  in parallel*  2 i s d e f i n e d as the  cause v o l t a g e o r c u r r e n t h a r m o n i c s and  having  conversion i s introduced  b l o c k diagram of F i g u r e  p o r t i o n of the  input  For  are then a p p l i e d to t u n n e l - d i o d e s  of f r e q u e n c y  o f b l o c k s 3.  pump i s a n o t h e r  f^y  nonlinear  c o n s i s t s of blocks  t  active  The  f o r a converter  n o n l i n e a r elements are d i r e c t l y  c o n s i d e r i n g the  combination  shown, can p l a y a v e r y  converting process.  are developed  equations  converter,proper  It  this*  n o n l i n e a r c o n d u c t a n c e and  elements*  by  and  p a r t i n the  generality, separate  '  for  exhibits  gain i s f i n i t e  and  ** C o r r e s p o n d i n g to o p e n — c i r c u i t - t e r m i n a t i o n s , v o l t a g e h a r m o n i c s w i l l be p r o d u c e d : s h o r t — c i r c u i t t e r m i n a t i o n s w i l l l e a d to current harmonics.  3 harmonic load (1)  r nonlinear  1  signal  "1  elements  source  (2)  -Mixing-element  _J Figure  bilateral that  but  a special  1-1.  not  with  * an  reciprocal gain.  termination  (image) p r o d u c e s The  B l o c k Diagram I l l u s t r a t i n g Converting Process.  zero  theory  experimental  a t one  g a i n i n one  of u n i l a t e r a l techniques  and  In t h i s  normally  Frequency-  thesis,  unused  i t i s shown  harmonic  direction. frequency  conversion,  together  f i n d i n g s , are d i s c u s s e d  i n the  Gain i s l o o s e l y b i l a t e r a l . An i n p u t s i g n a l a t f , p r o d u c e s o u t p u t r e s p o n s e a t f ; and v i c e - v e r s a an i n p u t s i g n a l a t ±  p r o d u c e s an  2  output  a t f^t  the  i n p u t and  output  signals  2  are  s e p a r a t e d by f r e q u e n c y r a t h e r t h a n d i s t a n c e b u t usage has i t t h a t the g a i n from f ^ to f i s d e s c r i b e d i n terms o f d i r e c t i o n * 2  The c o n v e r t e r , bilateral•  having  g a i n i n b o t h such d i r e c t i o n s , i s thus  4 chapters of  that follow.  the mixing-element  form and  the mixing—element  I t i s derived, using Fourier Series  admittance  matrix  t o a two—port.  of the reduced  important  i s found  but  that with  realized  simple  Chapter  4.  termination of  This study 2 and C h a p t e r  i s based 3 and  r e a l i z a b i l i t y * s t a b i l i t y and g a i n c o n d i t i o n s * optimum t e r m i n a t i o n s  frequency  o n l y over  the s h o r t - c i r c u i t  by p r o p e r  e s t a b l i s h e d i n Chapter  that with  band u n i l a t e r a l  The  are o u t l i n e d .  image c i r c u i t i s s t u d i e d i n C h a p t e r  includes  study  n e t w o r k , two methods by w h i c h  A c o n v e r t e r made u n i l a t e r a l  It  Using  c o n v e r t e r may be made u n i l a t e r a l  upon t h e m a t r i c e s  under  i s e q u i v a l e n t to a t h r e e — p o r t network.  i s reduced  an  mixing-element.  3 i t i s shown t h a t f o r t h e case  three—port  the  i s i n the  small-signal voltage  the d i f f e r e n t i a l e q u a t i o n o f the  Chapter  description-  This d e s c r i p t i o n  matrix r e l a t i n g  current harmonics.  In  2 a mathematical  i s developed.  o f an a d m i t t a n c e  methods, f r o m  I n Chapter  ( S e c t i o n 4.2) w i d e -  c o n v e r s i o n w i t h g a i n c a n be o b t a i n e d  terminations the u n i l a t e r a l  a narrow f r e q u e n c y  property i s  band.  5 contains a very b r i e f  noise analysis  of the  converter. In The with for  results  Chapter  6 an e x p e r i m e n t a l  obtained with t h i s  i t , are presented further  converter i s discussed.  c o n v e r t e r , and p r o b l e m s a s s o c i a t e d  and compared w i t h t h e t h e o r y .  e x p e r i m e n t a l work a r e a l s o  given.  Suggestions  MIXING-ELEMENT  2.  Essential suitable element  ANALYSIS  to the a n a l y s i s of a frequency  converter  mathematical d e s c r i p t i o n of the n o n l i n e a r (Figure 2-1).  compared t o t h o s e signal voltages differential  mixing-  Assuming t h a t s i g n a l v o l t a g e s  o f t h e pump, i t i s shown i n t h i s  and c u r r e n t s a r e r e l a t e d  equation.  i sa  are small  chapter  that  through a l i n e a r  By c o n s i d e r i n g t h e h a r m o n i c  content  o f t h e s m a l l — s i g n a l power t h a t i s made a v a i l a b l e by t h e m i x i n g (7)  a c t i o n , and by a p p l y i n g the P r i n c i p l e a s o l u t i o n of the d i f f e r e n t i a l  o f Harmonic B a l a n c e ,  equation  i s later  '  7  obtained.  I n g e n e r a l , t h e s o l u t i o n i s i n t h e f o r m o f a n by n m a t r i x relating  the n voltage  harmonics t o the n c u r r e n t  I n t h i s work 3 h a r m o n i c s a r e c o n s i d e r e d description  of the mixing—element  harmonics.  so t h a t t h e m a t h e m a t i c a l  i s i n t h e f o r m o f a 3 by 3  matrix. The nonlinear  mixing-element  conductance g ( v ) i n p a r a l l e l  capacitance  incremental  with  C ( v ) . These n o n l i n e a r i t i e s  associated with  semiconductor diodes  a nonlinear  a r e assumed t o be  lead inductances  on t h e u n d e r s t a n d i n g  strictly valid  those  where g ( v ) i s t h e  conductance and C(v) i s the j u n c t i o n  Ohmic l o s s e s and s e r i e s neglected  t o be i n v e s t i g a t e d c o n s i s t s o f a  capacitance.  of the diodes are  t h a t t h e a n a l y s i s w i l l be  only f o r r e l a t i v e l y  low f r e q u e n c i e s  (less  than  a b o u t 150 m c . ) .  These  series  components c a n be a c c o u n t e d f o r by c o n s i d e r i n g  6 2.1  Differential  Equation  A differential  of the Mixing-Element  equation  relating  the s i g n a l  current  -I  i(t) be  to the s i g n a l voltage v ( t ) of the mixing-element  derived  (equation  2-10)s  this  S e c t i o n 2.2 t o r e l a t e v o l t a g e mixing—element elements w i t h  their  interconnected sinusoidal  circuit  of Figure  d-c,  and  d-c, but block  fg»  b u t pass a l l other a l l other  on  and pump v o l t a g e  frequencies.  the nonlinear  frequency f ^  Filters  frequencies.  F Q pass fQ  With t h i s  v ( t ) and s i g n a l  i n f l u e n c e d b y t h e pump s o u r c e s  sources  The pump v o l t a g e s a r e  F Q block  Filters  arrangement the s i g n a l v o l t a g e  The  2-1 c o n s i s t s o f two n o n l i n e a r  respective bias  at frequency  i s used i n  and c u r r e n t h a r m o n i c s .  i n a system o f f i l t e r s .  and  are  equation  i s to  circuit  current i ( t )  only through t h e i r  action  elements.  From F i g u r e  2-1 t h e v o l t a g e  across  the nonlinear  conductance i s  V  The V  gP  signal voltage  ( t ) = -V ( t ) + v ( t ) gP  v ( t ) i s small  ( t ) and t h e r e f o r e  conductance  g  the t o t a l  •..(2-1)  compared t o t h e pump v o l t a g e  current  I (t) through the g 5  c a n be expanded i n a two-term T a y l o r  I  g  (t) = I V ( t )  V (t) L gP J v  Series.  + g(t)v(t)  ,.*(2^2)  where  •ct)  =  &  I[TJ  ...(2-3) V  g  = V  (t) gP  7 a ii(t) i  (t) F  7  i (t g  i (t)  +  (t)  V„(t)£  gp  v  v(t) g(v)  g p  (t)  P  v  /\  v  I  i (t) c '  p  (t)  v  (t) cp ' v  C(v)  cos(2Tif ij)  |V cos(27xf t+^  0  c p  0  cb Conductance I pump and b i a s | I circuit Ismall-signal I I  Figure  2-1*  I Capacitance j I ' pump and b i a s | circuit I I  circuit  Mixing-Element  Circuit.  also,  y*>  = igptt) + i  g  (t)  ...(2-4)  therefore  I gP  (t) + i (t) = I V (t) g . gp :  T h i s e q u a t i o n c a n be s e p a r a t e d i n t o  + g(t)v(t)  large  ...(2-5)  and s m a l l - s i g n a l  components,  V*> =  1  v  (t)  . gp  .  ...'(2-6)  8  and  i  The  (t) = g(t)v(t)  small-signal  pumped n o n l i n e a r  current—voltage  capacitor  expanding t h e charge  .  i s derived  r e l a t i o n s h i p f o r the  i n a s i m i l a r manner by-  on t h e c a p a c i t o r  a b o u t t h e pump v o l t a g e *  o . . ( 2—7 )  i n a two^-term T a y l o r  The r e s u l t i n g l a r g e  and  Series  small-signal  equations are  I  cp  (t) ' x  (t)  V  IT  *  3  C(t)v(t)  • • • (2—8 ).  cp  and  i  The and  at  differential  from e q u a t i o n s  (2-7) and ( 2 - 9 ) .  Since  c a n be combined t o g i v e  C  i(t)  = g(t)v(t) +  s h o u l d be n o t e d t h a t  w i t h time v a r y i n g  coefficients*  a l i n e a r 'device.  | ST  C(t)v(t)  this i s a linear differential Since  equation i s l i n e a r j the s m a l l — s i g n a l be  ..(2-9)  equation r e l a t i n g v ( t )  = i ( t ) + i ( t ) , these equations 6  It  (t) '  v  small-signal  i ( t ) follow  i(t)  c  this  ....(.2-10)  equation  differential  frequency  converter  will  2.2  The  Mixing-Element A solution  of d i f f e r e n t i a l  c e r t a i n assumptions evident  from  Matrix  and  equation  equation  restrictions  (2—3)  t h a t the  i s required.  It is  instantaneous  incremental  conductance g ( t ) i s a f u n c t i o n of the V  gP  (t).  s i n u s o i d a l pump v o l t a g e  Therefore, g(t) i s a periodic  fundamental frequency  fQ.  S i n c e the  f u n c t i o n of time  instantaneous  capacitance C(t) i s s i m i l a r l y p e r i o d i c , g ( t ) and  C ( t ) can be  (2-10) u n d e r  with  incremental  i t follows that  both  expanded i n F o u r i e r S e r i e s ,  2  g„  e  jntt t n  ..,(2-11)  U  n z=~^-cO  and  3 n^n't C(t) =  >  C ' n  e  ...(2-12)  U  n = — OO  <o„  where  =  2uf 0  0  Consider angular  frequency  the  effect  tt^r  being a p p l i e d to the  (between t e r m i n a l s a-b w i t h the angular  o f an i n p u t s i g n a l  of F i g u r e 2-1).  components o f g ( t ) and  i2(t),  at  mixing—element  This s i g n a l w i l l  C ( t ) to produce harmonics  frequencies  o»  n  = n  <D  0  ±  <D  2  n = l ,  2,  .•  "mix" at  By t e r m i n a t i n g t h e m i x i n g — e l e m e n t frequencies  c a n be i s o l a t e d .  p r o p e r l y , power a t t h e v a r i o u s  I f terminations  t h a t power c a n f l o w  only at frequencies  resulting  converter  frequency  i s said  are provided  such  g i v e n by n = 1, t h e  t o be o f t h e f u n d a m e n t a l -  (4) mode t y p e *  '  A 3 by 3 m a t r i x  converter w i l l Let can  be  of such a  developed.  the angular  e x i s t be w^, to^  a> = w  a)  f o r the mixing-element  1  a n <  frequencies  ^ 3  at which s m a l l - s i g n a l  power  where  w  + tt.  Q  and  ...(2-13) b)  This  fl>  3  frequency The  = <o  -  0  «  2  s p e c t r u m i s shown i n F i g u r e  small-signal voltage  2-2.  and c u r r e n t c a n be  expressed  (o) by  the f o l l o w i n g  v(t)  3 >  = i  series} V. e  + V.r e  + high-order harmonics  voltage  = 1 ...(2-14)  jtt.t i(t)  I.  = i  e  1  -3»,t + I.* e  + high-order harmonics  1  current  = 1 ...(2-15)  Since  power  either current  i s r e s t r i c t e d to angular  the h i g h - o r d e r  f r e q u e n c i e s (o^  v o l t a g e harmonics, or the  h a r m o n i c s must be a b s e n t .  a  t  n  d  to^t  high-order  Which i s a b s e n t  depends upon  11  A  Power  6)  (0,  Figure  2-2*  (0  0  S m a l l - S i g n a l Frequency  >  1  frequency-  Spectrum,  the t e r m i n a t i o n s of the mixing—element,  S u b s t i t u t i o n of  equations  (2-10) and  the  ( 2 - l l ) to  coefficients  (2-15)  of e  the e q u a t i o n fox the  *0  +  «-l  I  3  +  into  equation  ) i = 1,' 2, 3, y i e l d s ,  equating  i n matrix  form,  mixing—element•  ^lV  «i  ^2°-!*  «0  * > i V  g  «l  +  *  2  + D^CJJ*  +  J 2l ft)  C 1  V  3  *  ...(2-16)  or  tt  • H  W  ...(2-17)  12 In general matrix  Y are  the  complex.  shown i n F i g u r e  However, w i t h  of  c o n d u c t a n c e pump v o l t a g e  as  2-1;  V  g ( t ) i s an  off-diagonal Fourier Coefficients  gP  ( t ) = V . + V . cos gb gp  e v e n f u n c t i o n of time w i t h  (« t), 0 ' n  g n  real  and  6  6  g n  = g , -n 6  Also for  V  C(t)  cp  l e a d s g ( t ) by  C . n  where C  n  = V , + V cb cp  ( t )7  and  M a t r i x Y c a n now ance c o e f f i c i e n t s CQ>  C ^ and  a  g^,  g^  =  g  + 3«2 1 C  x  (<D_t  0  C  = C  n  —n  +  can be  ©),  '  replaced  .  w r i t t e n i n terms of r e a l  and  a n (  ^  by  real  capacitance  conductcoefficients  follows!  s  g  1- 0" Y  be  v  so t h a t C  0 radians  is real  cos  6  l  g  0  3  +  +  w C  i  3« C 2  e  3©  g  8l  0  + 3« C  2  l  0  1  e  2  J' 2 1  +  W  "  C  3*> C 3  e  J20  3©  0  ...(2-18)  This mixing-element matrix frequency  3 by  used to analyze  c o n v e r t i n g p r o p e r t i e s of a n o n l i n e a r  a nonlinear The  can be  capacitance,  3 matrix  c a n be  or a p a r a l l e l reduced  conductance,  combination  t o a 2 by  the  2 matrix  of the by  two.  13  eliminating so t h a t no  row  column i i f t h e  converter  parameters g  s u c h are d e t e r m i n e d  Q  and  C  n  is  terminated  co^.  power e x i s t s a t f r e q u e n c y  The as  i and  are F o u r i e r C o e f f i c i e n t s  and  fromi  2TZ  -jn» t n  gn  g(t) e  2%  d(» t) = g (V 0  n  ,V ) gb' gp'  ...(2-19)  and r  C  n  "  2%  -3n(tt t C(t) e ' n  27  U  J  +0) d ( t t t ) = C (V , ,V 0 n cb' ft  v  ) cp  0 ...(2-20)  C u r v e s o f g (V , ,V ) for tunnel-diodes n gb* g p 6  v  7  f o r v a r a c t o r - d i o d e s are The  parameters are  diode  of C  o b t a i n a b l e f r o m the  a l l positive  which are n e g a t i v e  and  over  except  a limited  v o l i a g e s V , and V respectively. S SP  g^  and  n  (Y . ,V ) c b ' cp  v  literature•*(l^) g^  of t h e  tunnel-  range of b i a s and  pump  3.  The be  used  of t h i s  FREQUENCY CONVERTER ANALYSIS  mixing-element  i n a unilateral chapter  d e s c r i b e d by Y - m a t r i x  down—converter *  a 3-port  the Y-matrix  the  second  The 3 - p o r t  of the l a t t e r  section,  3.1  two s p e c i a l  cases  i s reduced  t o a 2-port In  of the 2-port are case may be made  uni-  are o u t l i n e d .  Network R e p r e s e n t a t i o n o f t h e Equations  Mixing-Element  (2-16) t o (2—18) d e s c r i b e t h e  when power a t t h e t h r e e f r e q u e n c i e s <o^, exist.  section  network i s determined.  c o n s i d e r e d , and methods b y w h i c h each lateral  In the f i r s t  network e q u i v a l e n t of the m i x i n g -  element m a t r i x i s d e f i n e d * and  (2-18) i s t o  The f o r m  equations  o f these  of a l i n e a r  equations  3—port  co^ and OJ^ i s a l l o w e d t o  i s t h e same as t h e n o d a l  network.  The m i x i n g - e l e m e n t  t h e r e f o r e be r e p r e s e n t e d as i n F i g u r e 3-1. frequencies  at ports 1,2  mixing-element  and 3 o f t h i s  can  The s i g n a l  n e t w o r k a r e Oi^, (a^ and  ft>2 r e s p e c t i v e l y . Using  the 3-port  e q u i v a l e n t of the mixing-element  in a  (9) n o n i n v e r t i n g down-converter,  ' p o r t s 1, 2 and 3 a r e d e s i g n a t e d  respectivelys (1)  r . f . input port  (2)  i.f.  (3)  image p o r t  In subsequent  output  analysis*  port  subscripts  associated with the r . f . , i . f *  1, 2 and 3 w i l l  be  and image p o r t s , r e s p e c t i v e l y .  15  Image Circuit  n  '3  M  l-=  Two-Port  F 3 Source u s e d reverse gain calculations  in signal  •3V  source  signal  circuit  _ 3-port  1  Mixing-Element  output  circuit  t — Z Z L T j B  mixingf l  3  S  l  F  l  element  'out  in  Figure  3-1*  Mixing-Element  The 3 - p o r t m i x i n g - e l e m e n t reduced  with  ( F i g u r e 3-1) c a n be  t o t h e 2 - p o r t by t e r m i n a t i n g t h e image p o r t w i t h an  admittance  X^.  The  2-port  i s shown w i t h c u r r e n t s o u r c e s  admittances  as w e l l as t h e r e q u i r e d f i l t e r s .  the f i l t e r s  a r e shown t o be o f t h e t y p e  h i g h - o r d e r v o l t a g e harmonics* voltage  That  a t f r e q u e n c y (o^ t o d e v e l o p The e q u a t i o n s  equation  Terminations.  of the 2—port  (2-16) and m a t r i x  that  and t e r m i n a l  To be  specific,  short-circuit  i s ,filter  F^ a l l o w s o n l y a.  across i t s t e r m i n a l s . are obtained  from  (2-18) by e l i m i n a t i n g V_* and I _ *  16 t h r o u g h the r e l a t i o n put  i n the  I ^ * = -Y^V^**  ^ e n  e q u a t i o n s c a n be  form;  Y x  Y  L  l l  12 ...(3-1)  Y  Y 22  21  x  where  (g X  T  1 2  =  i 2 ,=  g  21  =  4- ^ C  2  + j  *  eJ  Si  >>1 1  +  C  e  g  j©  gl  2  =  g  e-3  2  o  + g  - j». C 3  2  n  e  j 2  0  C  +  g l  "  °) (  3«2 i "  +  -  2  V  2 0  Ja  J<o C 3  *1  g l  +  > 3  C  e"^  1  0  + j« C ^2^1 0  e  1  j e  e  g 2 2  )(g -  ^ eJ ^) (  V  Y  2 Q  + j(0-,C 1~0  n  g T  2  l  + g  + jO) C 2  j» C 2"0  1  e  j  0  "  J'  ) (g  9  n  tt  C 3  0  - ja^C, '3 1  e ^  V  0  V  + g  0  "  3« C 3  0  (3-2) These 3.2  parameters are the b a s i s Unilateral The  of f u r t h e r a n a l y s i s .  Frequency Conversion  frequency converting  p r o p e r t i e s of a  c a n be d e t e r m i n e d f r o m a knowledge  converter  o f the g a i n o f t h e  device*  (13) The r  transducer  gain  1  defined  _ power d e l i v e r e d t o l o a d T ~~ a v a i l a b l e power f r o m s o u r c e  purpose.  From t h i s  definition  as, ,  will  be u s e d f o r t h i s  t h e f o r w a r d g a i n G,j,^  2  ^  s  g i  v  e  n  17 by,  4 G  T12  |T  1  T  +  1 2  '21  G  )II + Y  n  2  - Y  2 2  I  1 2  i a  4G,  I  2  = 0 ...(3-3)  and  the r e v e r s e  g a i n G ^ i by  2  G,  4 G  'T21 Y  l  +  1 2 | 12l G  - 11 T  T  T  2  +  Y  22 " 12 21 Y  Y  4G,  I, = 0 ...(3-4)  For is  finite.  a unilateral This  down—converter G ^ l i s zero  i m p l i e s t h a t Y^  From Y - p a r a m e t e r s (3—2) down-conversion are placed Method  I.  Quadrature  2  = 0 and Y ^  two methods  proper  choice  G^  2  ^ 0.  of o b t a i n i n g  unilateral  i n evidence.  Pumping  T h i s method, a p p l i c a b l e i n p a r t i c u l a r a short-circuit  and  image t e r m i n a t i o n  to converters  (Y^ = OO), depends  o f t h e pump phase a n g l e  9.  upon  with  18 With I  3  = 0 0 the Y m a t r i x  12  11  g  0  +  (3-1)  of equation  O^O  g  l  +  j  <  0  l l C  becomes.  j©  e  ...(3-5) 21  22  For u n i l a t e r a l  g  l  + j * ^  e  + d« c 2  0  0  gain  T  12  =  g  i  +  1.  9 = +2"  2.  gj^ +' tt C = 0  converter w i l l  be u n i l a t e r a l .  Therefore, i f  g  e"J'  ^ i  0  3©  !  6  = 0,  radians  and  the  1  by w h i c h t h e c a p a c i t a n c e  1  Since  pump v o l t a g e  0 i s t h e phase  leads the conductance  pump v o l t a g e , t h e two n o n l i n e a r e l e m e n t s must be pumped quadrature. nonlinear directly and  This  c o n d i t i o n c a n o n l y be s a t i s f i e d  elements are p h y s i c a l l y together  as i s t h e c a s e  the nonlinear capacitance The  present  unilateral  separate,  of a single  i f t h e two  conductance  tunnel-diode.  from t h i s  frequency  i n time  i . e . , not j o i n e d  f o r the n o n l i n e a r  feature obtained  only a t the band—centre  angle  method i s  since the balance  * The t h e o r y o f u n i l a t e r a l f r e q u e n c y c o n v e r s i o n b y t h e q u a d r a t u r e pumping method, f o r a c r y s t a l - d i o d e and a v a r a c t o r d i o d e u s e d as an u p - c o n v e r t e r j h a s b e e n o u t l i n e d i n t h e literature*(H)»(12) The method c a n e q u a l l y w e l l be a p p l i e d t o a t u n n e l - d i o d e and a v a r a c t o r — d i o d e f o r w h i c h u n i l a t e r a l down— conversion with gain i spossible*  r e q u i r e d by c o n d i t i o n 2 above  c a n o n l y be  s a t i s f i e d at  one  frequency. Method I I .  Image T e r m i n a t i o n  T h i s method f o r o b t a i n i n g u n i l a t e r a l depends  upon p r o p e r c h o i c e o f t h e image t e r m i n a t i o n T^.  case i n w h i c h 0 = 0 case the n o n l i n e a r can  down-conversion  will  be d i s c u s s e d  i n detail  since  c o n d u c t a n c e and the n o n l i n e a r  be t h a t o f a s i n g l e  The  for this  capacitance  tunnel—diode.  The T - p a r a m e t e r s *  f o r © = 0,  from e q u a t i o n s ( 3 - 2 )  become:  J l 2)fs  +  (*2  W  -  C  V  2  i» c ) 3  g "-  +  2  3« C 3  0  0  fg + J > c )(g - ^ 3 l ) T* + g • " J' 3 0 fl  1  2  T  i2  =  Si  i°>i i c  +  2  C  1  0  3  ( 2 " J 3 W  g  T  T  21  22  =  =  Si  S  0  J"2 1  +  C 2  ) (si  W  +  C  J*2 l) C  C  + J« C 2  V  -  0  S  +  V  +  •" J 3 0 to  0  s "0  C  J 3 0 tt  C  ...(3-6)  The  down-converter  will  be u n i l a t e r a l  (g .+ ^ 2  T  i2  = s i + d-iCj. -  v  provideds  3« C )(g 1  2  1  - j^Cj) =  ?* 3  + g  Q  -  j« C 3  0  o  20 The  image t e r m i n a t i o n Y^  *  = -g  3  - 3" C  0  3  r e q u i r e d to s a t i s f y t h i s  (g ^  ++  0  2  ~ 3« C )(g +' j ^ C j ) '' ' ^ g - 3« C 1  2  1  •••O-?)  n  x  This p a r t i c u l a r  identity i s :  1  1  image t e r m i n a t i o n g i v e s u n i l a t e r a l  down-  conversion. can be case where the vanish  realized,  capacitance  zero:  Capacitance  essential  g and  satisfied, for  but  both  Y.^  terms as w e l l  for unilateral  a  n  (  the  i  as  conversion  C p a r a m e t e r s t h a t make up  pumped s e m i c o n d u c t o r d i o d e s positive  g p a r a m e t e r s the  ( S e c t i o n 4.2b)  so t h a t Y^  elements.  Since  unilateral  g a i n by  converter,  can  parallel  close  the  5fo b a n d w i d t h It  real  c a n be  admittance  o n l y be  using  combination  (Section  t o Y^  t o the  capacitance  and  to give  these  and  positive  linear  quadrature  passive  pumped  frequency.  a capacitor i s a  20 db.  inherent  of r e j e c t i o n  narrow-band  image t e r m i n a t i o n  methods depend upon the  For  this  a  approximated,  methods. reverse  c o n d u c t a n c e e l e m e n t t o c a n c e l t h a t due element.  of  With  4.5).  quadrature  Roughly speaking,  o n l y be  a t the b a n d - c e n t r e  i s p o s s i b l e t o e x p l a i n the  of the  (see A p p e n d i x I ) .  approximated with  of a r e s i s t o r  those  c a n n o t be  p a r t of Y-j can be  can  obtained  are  shown t h a t  t h i s method, as f o r the  enough a p p r o x i m a t i o n  feature  i t can be  r e a l d r i v i n g - p o i n t admittance,  tunnel-diode  due  (3-7)  method. When the  The  equation  terms are  simultaneously.  c o n d u c t a n c e terms a r e this  and  c o n d i t i o n , a net  to  gain  the  forward  gain  over  a  still  exists  because the  forward  and  r e v e r s e g a i n s due  to  the  (14) capacitance  are not  equal*  The  g a i n p r o v i d e d by  the  (4) conductance  i s frequency  independent (o)  whereas t h a t o f  capacitance  i s frequency  dependent*  Hence the  required  f o r the u n i l a t e r a l  f e a t u r e can  o n l y be  F o r the q u a d r a t u r e  by  the t e r m i n a t i o n s * overcome t h i s b a s i c  whereas, f o r the c a n be  obtained  equation  (3-7)  image-terminated over  c a n be  one  at  one  cannot,  limitation;  c o n v e r t e r , zero r e v e r s e  as l a r g e a b a n d w i d t h as t h e synthesized.  balance  obtained  frequency. altering  pumped c o n v e r t e r  the  of  gain  22 4.  UNILATERAL DOWN-CONVERSION BY THE IMAGE TERMINATION METHOD  The  first  p o r t i o n of t h i s  study  of a converter with  4.4).  Such t e r m i n a t i o n s  chapter  i s devoted  optimum t e r m i n a t i o n s c a n be r e a l i z e d  number o f e l e m e n t s , b u t t h e y  enable  to the  ( S e c t i o n s 4.2 t o  only with  the t h e o r e t i c a l  an i n f i n i t e study  of the  c o n v e r t e r t o be made u n d e r t h e b e s t p o s s i b l e c o n d i t i o n s .  Such  topics  part  as g a i n and s t a b i l i t y  of t h i s  chapter,  terminations  are considered.  a more p r a c t i c a l  i s discussed.  converter with  A model o f t h i s  2—port  Y-parameters  an image t e r m i n a t e d  frequency  capacitance  range b e i n g  curves.  3-6) a r e a p p l i c a b l e  i n which the n o n l i n e a r  are d i r e c t l y  conduct-  i nparallel.  For  c o n s i d e r e d and t h e n o n l i n e a r  t h a t of a semiconductor  can be n e g l e c t e d and  (equations  converter  ance and n o n l i n e a r c a p a c i t a n c e the  response  Approximations The  to  simple  c o n v e r t e r was a n a l y z e d  on t h e computer i n o r d e r t o o b t a i n f r e q u e n c y 4.1  In the l a s t  ( S e c t i o n 6.1).  diode,  i t i s found  t h a t 0^  Under t h e f o l l o w i n g a s s u m p t i o n  definitions,  c  2  a  = 0 A  ¥ l  Si A <°0 0  nonlinearity factor)  C  loading factor)  '0  1  «0  -2C-2 2  3  <o a>  00  normalized  frequency)  11  1 1 )  11  1 1 )  ...(4-l)  23 p a r a m e t e r s of  the Y  simplified  can be  g 2 2  + ji^p)  Q  Y *  + g ( i - jrijP)  3  g (i  "12  g g ( 1  +• j r ^ a )  1  0  2 1  =  g l  + jn a)  (l  2  1  Y * 3  These p a r a m e t e r s w i l l image t e r m i n a t e d 4.2  Optimum  be  2  u s e d as  ...(4-2)  jOjP)  0  x  22  jil a)  + g (l -  g 2(l  2  3  +  2  Y *  + jn P) -  0  3  0  g g (l  -  J^ a)  + g ( i - ja p)  Y *  3  g (i  "  1  2  3  Y  w r i t t e n i n the f o l l o w i n g  forms  g (l  11  (3-6)  + jf^aMl  - jOjOc)  + g ( l - jA P) Q  3  the b a s i s f o r a n a l y s i s o f  the  converter.  Terminations  A unilateral properties w i l l  be  d o w n — c o n v e r t e r t h a t has  considered  the f o l l o w i n g  t o have optimum  a)  No  b)  Wide-band u n i l a t e r a l  terminations.  r . f . , i . f . or image a t t e n u a t i o n due behaviour  with  to f i l t e r s .  a passive  image  termination. c) a)  Input  Optimum It  give  zero  = Y^  optimum. infinite  output  admittance  real  over  a wide  band,  Filters i s assumed t h a t the  a t t e n u a t i o n and  centredaround Y^  and  over  their  F  phase  over  respective centre  a wide b a n d .  In a d d i t i o n , i t admittance  zero  f i l t e r s F^, shift  2  and  F , 3  wide  (Figure pass-bands  frequencies, that i s ,  These f i l t e r s w i l l  be  termed  i s assumed t h a t f i l t e r F^  to s i g n a l s at f r e q u e n c i e s  other  presents than  f..  3-1)  24 This  latter  a s s u m p t i o n has  d e r i v a t i o n of the  been i m p l i c i t l y  mixing-element matrix  assumed i n  equations  the  (2-16) t o  (2-18). b)  Optimum Image  Termination  Consideration  of equations  (3-7)  and  (4-1)  reveals  that  if (1 + T_  the Y^  = -g  0  down-converter w i l l i s not  fore  c a n n o t be  t o Y^  realized  0.  t o be  f  e q u a l i t y of a as The  0 — 2  exactly.  For  a converter  termination  e x p a n s i o n of Y^  and with  thereoptimum  approximation  band. i t i s necessary  i t follows  that  that  for  3  _fl^ with  d r i v i n g - p o i n t admittance  a passive  From the  g  The  ...(4-3)  for a l l frequencies.  c l o s e o v e r a wide f r e q u e n c y  F o r Y^  (Y ) ^ 0  unilateral  jfUcx) —  i t i s assumed, however, t h a t the  i s very  Re ^Y^J-  be  a positive real  terminations  Re  (1 + j f u 0 ) + g  Re  20. a  0  2  (4-4)  *  1  -  2  —± -i . rt 2 2 1 +iL a  i s plotted i n Figure  ...(4-4)  4-1  as a f u n c t i o n o f  a parameter. (T<J)  will  be  p o s i t i v e f o r a given a only i f  g  g  is less  t h a n the v a l u e  g i v e n by  the  graph.  For  a  crystal-  25  g  2  0  oc  1.0-  a = -0.3  0.5-  a = -1.0 —i  0  1.2  1.1  1.0  1.4  1.3  1.5  fr»  n.  a = -5.0 -0.5"  -1.0-  F i g u r e 4-1.  Boundaries of a Passive (Re ( Y ) = 0 ) .  Image  Termination  3  (l) 0 i t c a n be shown ' that — 2 g  diode  x  \ ^ 1.  T h i s r u l e s out the  g  possible with  use o f s u c h a d i o d e  a passive  to achieve  image t e r m i n a t i o n .  unilateral  A tunnel-diode,  satisfy  c o n d i t i o n 4-4  over  Further  analysis w i l l  t h e r e f o r e be c a r r i e d  standing diode.  conversion however, c a n  a r e g i o n of i t s c h a r a c t e r i s t i c  curve.  o u t on t h e u n d e r -  t h a t the r e q u i r e d g parameters are those  of a t u n n e l -  (10)  26 c)  Optimum  Input  and Output  Terminations  The i n p u t and o u t p u t suitably follows  choosing  and  in  =  T  a r e made r e a l by  respectively.  t h a t w i t h optimum  X  admittances  From F i g u r e 3-1 i t  filters;  l l "  ^-  Yj  L  Y 22 x  +  + Y 2  + j B,  ...(4-5)  and  Y  out  = 22 * T  l  1 2 l  11  1  Since Y^ will  = 2  +  f  + Y  X  l  3  B  2  —  ®t by c h o i c e o f Y^, t h e i n p u t and o u t p u t  <-> 4  6  admittances  be r e a l i f  and  2  *2 = ^ m  ( 2 ) T  2  =  —  a  «h  F o r t h e optimum t e r m i n a t i o n s and f i l t e r s above, t h e c o n v e r t e r e q u a t i o n s *  from  V « 0 —  ( 4  ~  d e f i n e d i n a, b and o  equations  (4-2),  become;  8 )  27 1 -XL-jfLjOc' I, 1  G  2  + g  - g  0  0  2  i  +n  2 3  a  2  / 1 + a-n- \  -  a  2  0  •j 2  g  l  a  ( \1  - j/1-a/  G  2  ^  +  >0 ...(4-9)  It impossible The  should  be n o t e d  istics  of a converter with  with  optimum t e r m i n a t i o n s .  4.3  Converter  are a l l  isstill  c a n be s y n t h e s i z e d  degree o f a c c u r a c y .  2  number o f p a s s i v e  optimum t e r m i n a t i o n s  the t e r m i n a t i o n s  required  and B  f  to r e a l i z e with a f i n i t e  converter with  since  t h a t Y^  of i n t e r e s t  theoretically  A l s o , the band-centre  simple  elements.  terminations  t o any  character-  can equal  those  (Section4.5).  Stability  I t has b e e n e s t a b l i s h e d ( S e c t i o n 4.2b) t h a t t h e g parameters o f a t u n n e l - d i o d e ,  i n conjunction with  image t e r m i n a t i o n , a r e s u i t a b l e f o r u n i l a t e r a l Since  spurious  oscillations  are d i f f i c u l t  a passive  down-conversion.  to avoid i n c i r c u i t s  (15) employing are  tunnel-diodes  o f prime  ' the converter  stability  conditions  importance.  V i t h reference  t o the 2-port  necessary conditions f o r s t a b i l i t y (1) G +• Be ( l ) > 0 ±  i  n e t w o r k o f F i g u r e 3—1, (16) ares  n  ...(4-10) ( 2 )  G  2  +  R  e  ( ou^° Y  For a converter with  optimum t e r m i n a t i o n s  and f i l t e r s ,  28 I. and T , a r e r e a l , and have v a l u e s xn out ' equations  (4-5)»  G. and G , g i v e n by in out 6  J  (4-6) and ( 4 — 9 ) ,  2(2 - r ^ V g  0  - 2 g  1  ~  1 + (2 - ^ ) a 2  2  1  j  (4-11)  G  o u t  >1_  =  g 0  - r-  >2  Si  f  =  1  g  o  1 - (r^-i) ^ a ' ..(4-12)  These f u n c t i o n s a r e p l o t t e d particular  i n F i g u r e 4-2 f o r a  s e t o f g and a p a r a m e t e r s .  predominantly  negative.  From  (4-10),  Both G^  n  and G  ^ are  t h e c o n v e r t e r w i l l be  stable i f G, •+- G, > 1 i n ^  0  and  .*(4-13) G  for * and  a l l values  0  of frequency.  The g p a r a m e t e r s g^ = 2 mrT  G ,> out  2  1  c h o s e n , namely gQ = -4 mrr-,  are approximately  those  v.-^  -2 m r f  1  o f a 1 ma. germanium  t u n n e l - d i o d e b i a s e d a t t h e p o i n t where g ( v ) i s a minimum and pumped w i t h a 100 mv. peak t o p e a k s o u r c e . T h i s mode o f o p e r a t i o n s a t i s f i e s the c o n d i t i o n g^ ^ ( e q u a t i o n 4-4) and p r o v i d e s  £ 1 c o n v e r s i o n g a i n w i t h s o u r c e and l o a d r e s i s t a n c e s w i t h i n t h e common range o f 50 t o 300 ohms. These g p a r a m e t e r s a r e u s e d i n f u r t h e r i l l u s t r a t i v e graphs i n t h i s c h a p t e r . For these o p e r a t i n g c o n d i t i o n s the n o n l i n e a r capacitance o f t h e t u n n e l d i o d e i s s u f f i c i e n t t o g i v e v a l u e s o f a between a b o u t 0 and -1.0 f o r pump f r e q u e n c i e s l e s s t h a n about 150 mc. V a l u e s o f a l e s s t h a n —1.0 c o r r e s p o n d t o pumped e l e m e n t s c o n s i s t i n g o f a t u n n e l — d i o d e i n p a r a l l e l w i t h a varactor-diode„  F i g u r e 4-2.  I n p u t and Output  Conductance  v s . Normalized Input  Frequency  30 4.4  G a i n W i t h Optimum  Terminations  From e q u a t i o n s converter with  and  (4-9)  optimum t e r m i n a t i o n s  16 G G 1  G  (3—3)  g-^oc  2  the  forward  g a i n of  the  becomesj  i + o-n-a  2  2  1 -  j-0- a 3  T12 1 -ajTLjOC  i -. G  +  e  o  G  1  JJO  2  + g0  2  g  2  I-  2  \1 +fLy a / ...(4-14)  The  denominator of t h i s  ( l S  e x p r e s s i o n i s of the  °in)( 2  +  G  +  w h i c h upon c o m p a r i s o n w i t h t h e 4—13) is,  form  °out)  stability  conditions  i s s e e n t o a p p r o a c h z e r o when i n s t a b i l i t y the  condition for infinite  conditions for i n s t a b i l i t y . value  of g a i n can be  gain i s obtained Variation p a r t i c u l a r values respectively,  gain  d e f i n e d t o be A  With such  r e a l i z e d by  at the  of forward of source  a device  the  any  and  G^.  but  1 2  with frequency  for  1  Grj i 2  °^  a  reasonable  l o a d c o n d u c t a n c e s G-^ and  gain significantly  frequency  c o n v e r t e r w i t h these  equal—gain  high  contour  G  dependent.  terminations i s  curves  2  Only f o r  zero.  s e t of t h r e e  That  stability.  gain Grp and  same as  c h o i c e o f G^  expense o f  occurs.  ( b o t h 8 mri- ) i s shown i n F i g u r e 4-3.  |oc| l a r g e i s t h e reverse  g a i n i s the  (equations  i n the  The  31  -5-  Figure  4-3.  F o r w a r d G a i n w i t h Optimum T e r m i n a t i o n s v s . Normalized Input Frequency.  32 G-^ - G^ p l a n e £L^  corresponding  to three values  = 1«25 i s g i v e n i n F i g u r e 4-4.  of a w i t h  F o r a = -.3 and -1.0 b o t h  G. and G . a r e n e g a t i v e , b u t f o r a = -5.0 G. i s negative in out • • o T in and  Cr ^. p o s i t i v e :  consequently  Qu  a different  form.  S i n c e Grj,^  1  the curves practically  S  j f l ^ f o r a = -.3 and -1.0 t h e c o n t o u r s values 4.5  are approximately  valid  independent of  corresponding  to these  of -TL^.  f o r a l l values  F r e q u e n c y C h a r a c t e r i s t i c s o f a Computer A n a l y z e d with " P r a c t i c a l Terminations" The  optimum f i l t e r s  and t e r m i n a t i o n s  S e c t i o n 4.2 c a n n o t be r e a l i z e d towards t h e p r a c t i c a l element t e r m i n a t i o n s bars)  case  practically.  suppress  by the l a t t e r  closer  passive-  superscript  (L^£» ^ i f * * 7  ^  e  components i s assumed adequate t o  unwanted h a r m o n i c s i n t h e s e n s e d e s c r i b e d f o r  optimum f i l t e r s violated  (Section 4*2a).  This assumption  when L ^ / C ^ i s a v e r y h i g h The  equations  f o r this  mixing-element parameters components f o r t h i s  (4—2).  "practical"  ( S e c t i o n 4 . 2 ) . By c h o o s i n g characteristics  component v a l u e s  ratio.  The t e r m i n a t i n g and f i l t e r converter  are chosen t o g i v e  as optimum  terminations  the components i n t h i s manner, the  of the converter  obtained with  a t band-centre  optimum t e r m i n a t i o n s .  a r e t h e same as  The t e r m i n a t i n g  are g i v e n i n Appendix I I .  A c o n v e r t e r w i t h t e r m i n a t i n g and f i l t e r described  i s only  c o n v e r t e r a r e b a s e d upon t h e  same l o a d i n g , a t b a n d - c e n t r e *  those  One s t e p  ( F i g u r e 4-5 e l e m e n t s w i t h filtering  Model  specified i n  i s p r o v i d e d by s i m p l e  and a n t i r e s o n a n t f r e q u e n c y  filtering  the  f o r a = -5.0 a r e o f  ( F i g u r e 4-5) was a n a l y z e d  components as  on t h e d i g i t a l  computer.  F i g u r e 4-4.  Equal-Gain  C o n t o u r s o f C o n v e r t e r w i t h Optimum Terminations  u> U>  34  r-VWV-i  G\, image c ircuit  signal  circuit  output  t  Figure  ^if/C^f For  ratios  small  ratio t o the  (3-3)  and  (3-4)  and  L^/C^  the  w i d t h c o u l d be L^/C\£  violates  the  section.  , The  independent  The  the  gain L  if/C  f  for various  shown i n F i g u r e s narrow.  the  using  of F i g u r e  4-6  upper  As  4-6. the  limit  a d d i t i o n a l frequency correspond first  ratio,  indicating  the  f o r l a r g e s t bandfiltering.  i s l a r g e , and paragraph of  " b a n d w i d t h " seems t o be i  gain  n e c e s s i t y t o c h a n n e l power a t  a s s u m p t i o n s t a t e d i n the  of the  reverse  bandwidth i s v e r y  to which they  reverse  and  These a r e  curves  r e a l i z e d by  ratio  forward  bandwidth i n c r e a s e s :  b a n d w i d t h i s s e t by  desired frequencies.  The  of a.  ratios the  the  Practical  r e s p e c t i v e l y , were o b t a i n e d  values  i s increased  T?2f.  Mixing-Element with Terminations.  F r e q u e n c y r e s p o n s e s b a s e d on equations  1  mixingelement  4-5.  circuit  thus this  practically  that this  "band-  35 width"  i s limited  narrow r e v e r s e  by  gain  the  critical  tuning  The  response i s not  the  approximate  image t e r m i n a t i o n .  "bandwidth" p r o b a b l y of t h e  experimental  g r e a t l y a f f e c t e d by  has  repercussions  converter, a  gained  at these  capacitance  capacitance.  Thus, t h e r e  f r e q u e n c i e s by  of a tunnel-diode  6.2).  the  property  is little  i n c r e a s i n g the  in  (Section  ( a measure of  n o n l i n e a r c a p a c i t a n c e ) , even though the u n i l a t e r a l depends upon t h i s  The  to  effective  be nonlinear  by b r i d g i n g i t , f o r example, w i t h  a  varactor-diode. In a d d i t i o n to the g a i n contours to  the  gains be  o f F i g u r e 4-4  converter with o f F i g u r e 4-6,  obtained  response are  practical  4-4.  of F i g u r e  also a p p l i c a b l e at terminations.  f o r w h i c h G^  from F i g u r e  curves  = 8m.Hr and 1  4-6,  band-centre  Thus, t h e 5^  =  the  mid-band  8m.nl , can 1  also  •G (db)  G (db)  T  a --- -5.0  T  L  i f  =  8000  'if  10 4 -<:G  T21  -10 t.  -20  -30  4  -30  (5  X  = 8 mil" , 5  Figure 4-6(a).  2  = 8  F o r w a r d and  , p  =  Reverse  -.75,  Gain  g  Q  = -4  mXV , 1  g ; L  = -2  m-TiT , g 1  of a C o n v e r t e r w i t h P r a c t i c a l  2  = 2  m^" ) 1  Terminations. UJ  0^  f (db)  •& (db)  T  T  •  r  10+  H*l  T12  G„(db) T  10f  1.2 \  G  T21  -10t  -10  T21  /  / /  -20  -20T  t  -20'  I  I  I  \  l  -30  -30T  -30" a = L  -40t  -40  (S  = 8 mn." , G  C  1  1  Figure 4-6(b).  mA . _ 1  2  =  if.  =  8  F o r w a r d and  P = --75, Reverse  g  if  a =  -1.0  'if  -40"  = -4 m-TL" , g  -5.0  = OO  'it.  OO  1  Q  '  I I I  - -2 m n " , 1  x  g  G a i n of a C o n v e r t e r w i t h P r a c t i c a l  ?  = 2  mil ) 1  Terminations,  u>  Figure 4 - 6 ( c ) »  F o r w a r d a,nd R e v e r s e  G a i n of a C o n v e r t e r w i t h P r a c t i c a l  Terminations.  00  39  5.  The v e r y  NOISE  important  topic  been s t u d i e d e x p e r i m e n t a l l y * two s o u r c e s !  the s i g n a l  o f n o i s e p e r f o r m a n c e has n o t  The n o i s e a t t h e o u t p u t  source  i s due t o  and t h e e q u i v a l e n t d i o d e  noise;  (8) and  t h e image c i r c u i t  the  first  source  noise*  of noise  0 ^7-  From Kim,  a limiting  figure f o r  i s g i v e n by  G  F = 1 +  where G  n  = e q u i v a l e n t shot n o i s e  Q  = 10 mxC  conductance*  °i For the experimental F = 3 db.  G  by a c o n d u c t a n c e a c r o s s  of t h i s  Kim s noise t  f i g u r e now  G  F = 1 +  0  1  3 + _  indicate  1  db.  poorer  than the b i l a t e r a l  figure  converter.  c a n be r e v e r s e d i f t h e c o n v e r t e r f e e d s  I n such  converter  gain  i s G^ ( 2 m . r L ) .  b a s e d on Kim's l i m i t i n g n o i s e  n o i s e power a c r o s s  noise figure  circuits,  power  t h a t the n o i s e performance of the u n i l a t e r a l  converter i s s l i g h t l y  transistor).  terminals.  and has a v a l u e F = 3.4  expression,  This r e s u l t  the output  c a n be  G  calculations,  converter.  = lOmrT ;  becomes;  These n o i s e  biLateral  and G  c o n d u c t a n c e * b a s e d on an e s t i m a t e d  o f u n i t y between image and o u t p u t  producing  1  A d d i t i o n a l n o i s e due t o t h e image c i r c u i t  represented The v a l u e  diode,  i t s input terminals;  a case, the n o i s e  into  a  circuit  (e.g. a  a t the i n p u t of the  i s i n c r e a s e d by f e e d b a c k ;  c a n be g r e a t e r t h a n t h a t w i t h  and t h e o v e r a l l  the u n i l a t e r a l  40 6.  An  EXPERIMENTAL  experimental  the u n i l a t e r a l  s t u d y was u n d e r t a k e n i n o r d e r t o v e r i f y  property  s e t f o r t h i n the theory.  m i n d j a 127 mc, t o 27 mc, c o n v e r t e r u t i l i z i n g c o n d u c t a n c e and t h e n o n l i n e a r c a p a c i t a n c e diode 6,1  With t h i s i n  the n o n l i n e a r  of a single  tunnel-  was c o n s t r u c t e d , Experimental  Figure  The  circuit  6-1,  Before  worthwhile circuit  Converter  Circuit  and I t s E l e m e n t s  chosen f o r experimental considering this  circuit  t o examine t h e c h a r a c t e r i s t i c s  o f t h e GE IN 2939 t u n n e l — d i o d e  work i s shown i n i ndetail,  i t is  and e q u i v a l e n t  used.  These a r e shown  i n F i g u r e 6-2. In theory t h e minimum v a l u e  i t was f o u n d  t h a t a b i a s p o i n t a t or near  of the conductance g ( v ) i s d e s i r a b l e . t  From t h e m a n u f a c t u r e r s  specifications  and f r o m  experimental  measurement, t h e c o n d u c t a n c e and t h e c a p a c i t a n c e point  are approximately  at this  bias  -6,6 mn"''' and 12 p . f , r e s p e c t i v e l y .  v a l u e s , t h e s e r i e s r e s i s t a n c e R = In. and t h e s e r i e s ' s i n d u c t a n c e L = 5 n.h, a r e n e g l i g i b l e a t f r e q u e n c i e s below s  For these  the  r . f . frequency  o f 127 mc*  This f a c t  had been p r e v i o u s l y  assumed i n t h e t h e o r y . A 100 mc. pump s o u r c e conductance  and c a p a c i t a n c e .  peak t o peak a c r o s s t h e d i o d e ^ fundamental, the f i r s t  For  effects  the v a r i a t i o n  F o r a pump v o l t a g e t h e o r e t i c a l values  of the  o f 100 mv. of the  and s e c o n d h a r m o n i c s o f t h e time  example, f o o t n o t e  o f S e c t i o n 4.3.  varying  41  Pump  0. Image L  3  r-Ofin-, 5  1N2939 T . D  3  P  g  6  •AAAA-  0  ^  4f  signal  Output  '0  'b  Pump f r e q u e n c y  100 mc  —  "5  _zz  I  Signal  frequency  Image f r e q u e n c y G  o  5  3  L  2  C  l  = 20mrL~' j 5  = lOrnn" ,  = 2ml\~ , G  = 2 0 0 m r L , 1^ *  1  1  C  0  1  —  =7-45 p . f . , C  2  =  b  = .1 j i f ,  .2  = 7-45 p . f . , C  Figure  6-1.  .15  f J - . h . ,  2  LQ  = lOrn-fL ,  -  Experimental  4  (i.h.,  .05  f x . h . ,  .1  200-250p.f ., C G  73 mc  -1  _ 1  6  ( J , » h » ,  —  G  127 mc  3  = 7-45  = 100  Converter  mn" . 1  Circuit  •1  42  Figure  6—2.  E q u i v a l e n t C i r c u i t and IN 2939 T u n n e l - D i o d e *  Characteristics  o f a Pumped  c o n d u c t a n c e and  c a p a c i t a n c e wave forms a r e  g  Q  = -4.0  mil"  g  1  =-2.0  mrT  g  2  = 2.0  mn."  These v a l u e s  =  C  1  °°0 —  =  t o an a and  2% x• 100 x 1 0  0  6  ap =  -2.0  a  -4.0  parameter v a l u e s  (6i6  theory*  The if  mn.  2  =  0.5  p.f.  a ^ of:  ,= —u.o^o  1 2  n  Q  1 2  =  ,-1.9 n  J  above, t h e  theory p r e d i c t s that  than ) .  l a t t e r  t h e real  ( 6 — l ) be  conductance  the a b s o l u t e The  G^f  s t a b l e at the  c o n d u c t a n c e r e g i o n * a d-c stability  ( e q u a t i o n s 4—13)  i f the  1  p.f.  J  x 10  c o n d i t i o n , t o g e t h e r w i t h the  greater  2.0  conversion with gain i s p o s s i b l e .  i n the n e g a t i v e  satisfied  =  -3  xIO  In order that c i r c u i t  i n the  ±  6  unilateral  point  p.f.  2TX x 100 x 1 0 x 12 x 10 _  u  With the  = 12.0  x 2 x 10"  g  1  Q  C  1  give r i s e  "O ! ——— l  a =  C  1  approximately?  value  c o n d i t i o n s are p a r t of the  obeyed.  satisfied  admittance  The  former i s  a c r o s s the d i o d e  of the diode  G^ c o m b i n a t i o n  stability  conditions outlined  must be  ( a t d—c)  bias  provides with  is  conductance a suitable  value,  a margin of s a f e t y ,  a c r o s s the  diode  i s greater  than the absolute value  of the diode  f r e q u e n c i e s below t h e d i o d e  resistive  In order t h a t the admittance wiring  inductance The  tuned  a h i g h admittance  be k e p t  circuit be  cut-off  ata l l  frequency.  be l a r g e , i t i s e s s e n t i a l  (15)  that  t o a minimum.  circuits  of the converter  are designed  a t harmonic f r e q u e n c i e s o t h e r than  centre frequencies. equivalent  conductance  t o have  their  With such a d e s i g n the c o n v e r t e r i s  t o t h e computer a n a l y z e d model o f S e c t i o n 4.5.  elements w i t h  determined  the e x c e p t i o n o f  from equations  and G^ c a n t h e r e f o r e  AII*-2 t h r o u g h  are chosen t o g i v e the d e s i r e d s t a b i l i t y  The  AII-6.  and g a i n .  G^ and G^ Since  lumped e l e m e n t s a t V.H.F. f r e q u e n c i e s have a s s o c i a t e d p a r a s i t i c s the v a l u e s the  g i v e n by t h e above e q u a t i o n s  components a r e mounted i n t h e a c t u a l In the experimental  were u s e d  i n an e f f o r t  t o minimize p a r a s i t i c  o f #14 t i n n e d c o p p e r w i r e .  compared t o t h e l o a d c o n d u c t a n c e s * disc  components  effects.  The f o u r  were wound w i t h \ t o 3 t u r n s  T h e i r Q a t 100 mc. was above 50.  a Q the equivalent c o i l  were o f t h e c e r a m i c  after  circuit.  model, h i g h q u a l i t y  i n d u c t o r s , e a c h 3/4" i n d i a m e t e r *  With such  must be a d j u s t e d  type.  conductances are small The a d j u s t a b l e c a p a c i t o r s  The components were  on a 3/16" s h i e l d e d b r a s s c h a s s i s f i t t e d  with  placed  s t a n d a r d B.N.C.  * I f the r e a l p a r t of the admittance i s n o t s u f f i c i e n t l y l a r g e * s e l f — o s c i l l a t i o n s w i l l occur* By a r r a n g i n g t h e s e o s c i l l a t i o n s t o t a k e p l a c e a t t h e pump f r e q u e n c y , i t i s p o s s i b l e to e l i m i n a t e t h e need f o r an e x t e r n a l pump s o u r c e * Converters of t h i s s e l f o s c i l l a t i n g t y p e ( 8 ) were e x p e r i m e n t e d w i t h , b u t g r e a t d i f f i c u l t y was f o u n d i n c o n t r o l l i n g t h e a m p l i t u d e o f t h e o s c i l l a t i o n s . Since the adjustments of the converter f o r d i r e c t i o n a l g a i n are c r i t i c a l t o b e g i n w i t h ( S e c t i o n 6*2), t h e i n v e s t i g a t i o n o f s e l f — o s c i l l a t i n g c o n v e r t e r s was n o t p u r s u e d beyond t h e p r e l i m i n a r y stage*  connectors  i n s u c h a manner t h a t no w i r i n g between t h e  components was 6*2  required.  Experimental Techniques The  and R e s u l t s  equipment a v a i l a b l e f o r a l i g n m e n t and c h e c k - o u t  was a l l o f t h e 50X1 c h a r a c t e r i s t i c a match, i n d e p e n d e n t  impedance t y p e .  o f f r e q u e n c y * between t h e equipment and  t h e c o n v e r t e r , i t was n e c e s s a r y t o u s e r e s i s t i v e works a t a l l t i m e s * noise  These  networks  on t h e v a r i o u s s i g n a l s  measure t h e n o i s e f i g u r e A sweep—frequency  so t h a t  additional  of the c o n v e r t e r . s i g n a l g e n e r a t o r and a d e t e c t o r  0  circuits.  Oscillator  = 100 mc resistive  matching  networks Detector (RCA, AR-88LP)  808D)  Radar D e t e c t o r (AN/APR-4)  Hewlett-Packard S i g n a l Generator (M  Figure  6-3.  net-  i t was n o t p o s s i b l e t o  -Crystal  (M  matching  superimposed  were u s e d t o a l i g n r o u g h l y t h e f o u r t u n e d  Hewlett-Packard S i g n a l Generator  To p r o v i d e  808D)  B l o c k D i a g r a m o f C o n v e r t e r and M e a s u r i n g  Equipment*  46 Figure  6-3  i s a b l o c k d i a g r a m o f the  tuning  and  measurement o f t h e  The realized  very  sequence o f t u n i n g , i n order  db.  difficulty o f any  was  tuned  realized  completely  trusted. the  i s contained  of the  c o u l d be  circuits  (25 db.)«  It is likely  e f f e c t i v e value  t h a t the  reverse  the  for this  into  slight  g a i n has  detuning  c o n d i t i o n the  tuning  and  gain converter be  substantially alters  (4-3)  a very  of  the  a high forward  and  Computer a n a l y z e d  the  gain  f a r as measurements c o u l d detuning  not  completely properties.  model  (Figures  4-6)  narrow b a n d w i d t h ! other  adjustments  difficult. I n s h o r t , the  be  but  way,  was  adjustments,  insight  required for unilateral of t h e  gains.  a reverse  that with  i n one  t h a t the  t h i s f u r t h e r e x p l a i n s why m i g h t be  fact  of 8 i n e q u a t i o n  of  response curves  indicate  i n the  An  fine  expended i n a  bias point  g a i n and  frequency.  r e c i p r o c a l as  changes t h e v a l u e The  of forward  spot  reverse  e f f o r t was  l o a d m a t c h i n g and  a t one  and  gain possible i n theory  Considerable  t o o b t a i n 1 db.  afeout -10  forward  large forward  in practice.  equipment u s e d f o r  obtained  e x p e r i m e n t s show t h a t d i r e c t i o n a l  as p r e d i c t e d f o r the m a t h e m a t i c a l models  adjustments to achieve  extremely  such g a i n  ( i n the  gain  but  c i r c u i t used)  can  that are  critical.  * P r e c i s e measurement o f the r e v e r s e g a i n was n o t p o s s i b l e b e c a u s e o f the v e r y low s i g n a l — t o - n o i s e r a t i o o f the s i g n a l t r a n s m i t t e d t o t h e 127 mc* d e t e c t o r . Measurements were f u r t h e r hampered by t h e n e c e s s i t y t o have t h e magnitude of a l l f o r c i n g s i g n a l s s m a l l compared t o t h e pump s i g n a l (100 mv. p . t o p . ) .  47 6,3  Suggestions  effects  f o r Further Experimental  The  experimental  and  tuning problems.  overcome i n two at a s t i l l  study vas  ways; one* use  higher frequency);  high frequency  practical  distributed o r two,  conversion.  use  elements  the  o f t h e two  methods.  Future  be  (and  a tunnel-diode  work in  frequency. to  f r e q u e n c i e s w o u l d promote  a t low  frequencies, i t should  c o u l d be  g a i n s i n c e the  closely  experimental  study  study would a l s o p r o v i d e  experimenting  f o r noise f i g u r e  optimum  of a quadrature  pumped  a converter i s  a realistic  s h o u l d a l s o be  be  approximated.  i s r e q u i r e d to i n d i c a t e whether such  practical;  could  work w o u l d l e a d d i r e c t l y  t o o b t a i n wide-ba.nd u n i l a t e r a l  converter  high-frequency  of t h e phenomena i n v o l v e d i n u n i l a t e r a l  Furthermore*  I n a d d i t i o n , an  by  work a t a l o w e r  experimental  t e r m i n a t i o n s ( S e c t i o n 4,2)  suitable  and  a p p l i c a t i o n s , work a t low  a b e t t e r understanding  possible  hindered  These d i f f i c u l t i e s  p a r a l l e l with a varactor—diode Although  Study  comparison  done w i t h  circuits  measurements.  The v a r a c t o r — d i o d e i s r e q u i r e d as the n o n l i n e a r c a p a c i t a n c e o f a t u n n e l — d i o d e i s i n s u f f i c i e n t a t low f r e q u e n c i e s .  7.  CONCLUSIONS  c This passive with  study  has  shown t h a t a f r e q u e n c y  image t e r m i n a t i o n  gain.  For t h i s type  have b o t h n o n l i n e a r s u c h as  t h a t of  terminations,  can p r o v i d e  a tunnel-diode  the  narrow f r e q u e n c y  and  ( S e c t i o n 4.5),  n o i s e mechanism o f the  complex, b u t  approximate  i s only  (Chapter  this  With  but w i t h  gain  can  must  conductance,  only  more  be  simple over  a  complex  achieved*  converter  i s extremely  calculations indicate that  i t s noise  s l i g h t l y worse t h a n t h a t of a b i l a t e r a l  converter  5). Unilateral  obtained  nonlinear  a  down-conversion  mixing-element  u n i l a t e r a l feature i s present band  with  4.4)• The  figure  the  ( S e c t i o n 4.2).  t e r m i n a t i o n s , wide-band u n i l a t e r a l (Section  unilateral  of conversion  capacitance  converter  by  down—conversion w i t h  g a i n can  the q u a d r a t u r e pumping method.  Converters  method r e q u i r e f e w e r p a r a m e t e r s t h a n image  converters*  but  o v e r a narrow  the  unilateral  bandwidth.  property  also  be using  terminated  i s possible  only  49 APPENDIX I  Positive  Real This  terminating  Driving-Point appendix deals  The  =  the c o n d i t i o n s  gain  , i .0 " 0 -J*3°0 g  admittance  admittance  variable  Y  3=  g  ( 2 ~ g  + +  (*z-*o)  j  W  l  2 )  C  (l ~ °1 1 S  3 <  2  +  required  to give  s = jco^  +  4  a  n  <  i  C  C  also noting  ( i - o i) -o) o) V g  « o \  «l l(«2 C  c  g  3  2  C + s  2 x  3  (g 0, I  2  *1  Q  C  2g  +  (c  1  2  o)  C  ( 2 " C  - C )  2  0  o)  C C  i ^o ^ 2  l C l  C  (2 -  ~ 2 l)f l +  C  2+  +  ( 2 "  2  + 3 »o( l*2 g  that  2  - g )  2 2  frequency  c  C  +  wide-  >3 l)  +  g  + s  t o be a p o s i t i v e  c a n be w r i t t e n i n terms o f t h e complex  s by l e t t i n g  r  the image  (3-7)i  i s , from equation  «1 -  This  i n order  that  admittance.  image t e r m i n a t i n g  band u n i l a t e r a l  X  with  a d m i t t a n c e must s a t i s f y  real driving-point  Y 3  Admittance  2+s  2g  C  ii c  g  +s 2 G  g  i  s  C  l)  2  ..•(AI-1) For  this  least  t o be a p o s i t i v e r e a l d r i v i n g - p o i n t a d m i t t a n c e  i t i s at  n e c e s s a r y t h a t t h e f o l l o w i n g c o n d i t i o n s be s a t i s f i e d ;  (17) '  50  Condition  (1)  c  (2)  C  (3)  gC_ 1  l g 2  *  2  ]  - c  C  2 g l  = 0  0  ^ 0  (2) c a n n o t be  s a t i s f i e d w i t h s e m i c o n d u c t o r d i o d e s or  (9) o t h e r known n o n l i n e a r to  prove t h a t  admittance.  capacitance elements.  c a n n o t be  a positive  real  ' This  is sufficient  drivin -point g  51  APPENDIX I I  Terminating  Components  for a "Practical"  The t e r m i n a t i n g l o a d i n g as optimum  components  terminations  The e q u a t i o n s  Converter  are chosen t o g i v e  ( S e c t i o n 4.2) a t b a n d - c e n t r e .  d e s c r i b i n g the converter  can be w r i t t e n i n t h e f o l l o w i n g s i m p l i f i e d  Y  l  +  Y  t h e same  l l  T  ( e q u a t i o n s 4-2)  form;  12 ...(AII-1)  Y  "21  Let  the normalized  angular  the  converter  22  r . f . * i . f * and image  f r e q u e n c i e s be d e n o t e d b y -H-^O* "^"20 The r e q u i r e d l o a d i n g w i l l  + Y  2  a n < 1  be r e a l i z e d  "^"30  r e s  P  band-centre e c  "ki  v e  ly *  i f t h e components o f  ( F i g u r e 4—5) a r e t h e f o l l o w i n g v a l u e s s  L  if  ^10  C  c  if  i  i  =  = 1,  2, 3  ...(AII-2)  -n. io 2  =-^11) Q  i  = x l  io g a 2  = - ioP«o a  +  2  1 +-nr oc 30  ...(AII-3) 2  52 " n^22y  - 20 2 f L  C  I  n  =XL  2  2f 2.  •^20^0  G  3  + a^ c 30  =• T  3  ^10  +  go  from  3  =  n  3  =  " 0 g  +  g  (AII-4)  20  which  3 0 /  G  + r i  2 \  2 1  +Xl  2 1 0  a  2  2g « 0  30 3  rL  C  =  "So^O  .....(AII-6)  +  i +r\ a 2  2  0  The equations  inductance  and c a p a c i t a n c e v a l u e s g i v e n by t h e s e  a r e n o r m a l i z e d w i t h r e s p e c t t o t h e pump  frequency*  S h o u l d .TL^QC^ be n e g a t i v e , an i n d u c t i v e t e r m i n a t i o n is  r e q u i r e d , and ^ ^ Q C ^ i s r e p l a c e d b y  -1  xO These e q u a t i o n s d e t e r m i n e  I  a l l but the f i l t e r  components  uniquely.  I f the L ^ / C ^ r a t i o $ which i s a s s o c i a t e d w i t h the  bandwidth^  i s also  specified*  a l l components  c a n be  determined*  53 REFERENCES  1.  T o r r e y , H.C. and Whitmer, C.A.* C r y s t a l R e c t i f i e r s , McGrawH i l l Book Compaay* I n c . * New Y o r k and L o n d o n , (1948).  2.  van  3.  E s a k i , L.,  4.  E d w a r d s , C.F., " F r e q u e n c y C o n v e r s i o n by Means o f a N o n l i n e a r A d m i t t a n c e " * B e l l S y s * T e c h . J . . V o l . 35, pp. 140316, (November* 1 9 5 6 ) .  5*  S t e r z e r , F . and P r e s s e r * A** " S t a b l e Low-Noise T u n n e l - D i o d e F r e q u e n c y C o n v e r t e r s " * RCA Rev,. V o l . 23, p p . 3-28, (Mar* 1 9 6 2 ) .  6*  Chang* K*K*N** H e i l m e i e r * G*H* and P r a g e r , H . J « , "Low-Noise T u n n e l - D i o d e Down C o n v e r t e r s H a v i n g C o n v e r s i o n G a i n " * P r o c * IRE* V o l * 48, pp* 854-58, (May, I 9 6 0 ) *  7*  Cunningham* W*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 , McGraw^ H i l l Book Company* I n c . , New Y o r k and L o n d o n , ( 1 9 9 8 ) *  8*  Kim* C.S.*  9*  B l a c k w e l l * L*A. and K o t z e b u e * K*L.* Semiconductor-Diode Parametric A m p l i f i e r s , P r e n t i c e - H a l l , Inc., Englewood C l i f f s * N*J.* (1961)*  10*  P u c e l , R.A., "Theory of the E s a k i Diode Frequency C o n v e r t e r " * S o l i d - S t a t e E l e c t r o n i c s , V o l . 3, pp. 167-208, (Nov.-Dec. 1 9 6 1 ) *  11*  der Z i e l , A», Noiseg P r e n t i c e - H a l l , C l i f f s , N.J.* (1956)*  I n c . , Englewood  "New Phenomenon i n Narrow 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).  " T u n n e l - D i o d e C o n v e r t e r A n a l y s i s " * IRE T r a n s * PGED* V o l . ED-8* pp* 394-404, ( S e p t . 1 9 6 1 ) .  E n g l e b r e c h t , R.S.* " P a r a m e t r i c E n e r g y C o n v e r s i o n by N o n l i n e a r A d m i t t a n c e s ' * P r o c * I R E , V o l . 50, pp. 312-21, (Mar. 1 9 6 2 ) * 1  12*  13*  R o s s * P.W*  Linville,  and S k a l n i k * J * F * * ''Parametric F r e q u e n c y C o n v e r t e r s V i t h A r b i t r a r y Pumping A n g l e s " , P r o c . IRE, V o l . 51* p . 239, ( J a n * 1 9 6 3 ) * J.G. and G i b b o n s * J . F . , T r a n s i s t o r s and A c t i v e C i r c u i t s , M c G r a w - H i l l Book Company, I n c . * New York* T o r o n t o and London* ( l 9 6 l ) .  54 14*  Manley, J.M., and Rowe* H.E.* "Some G e n e r a l P r o p e r t i e s o f N o n l i n e a r Elements - P a r t I , G e n e r a l Energy R e l a t i o n s " * P r o c . IRE., V o l . 44, pp. 904-913, ( J u l y 1956).  15*  Sommers, H.S., "Tunnel-Diodes as High Frequency D e v i c e s " * P r o c . IRE. V o l . 47, pp. 1201-06, ( J u l y 1959).  16*  G a r t n e r , W.W.* T r a n s i s t o r s ; P r i n c i p l e s , D e s i g n and A p p l i c a t i o n s * D. Van N o s t r a n d Company, I n c . , P r i n c e t o n , N.J., ( i 9 6 0 ) .  17.  G u i l l emin* E.A.* S y n t h e s i s o f P a s s i v e Networks. John W i l e y and Sons, I n c . * New Y o r k , (1959)•  •.  

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