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Iterative synthesis of a flat-staggered emitter-feedback transistor video amplifier Cameron, Frank Charles 1962

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ITERATIVE  SYNTHESIS  EMITTER-FEEDBACK  OP A  FLAT-STAGGERED  TRANSISTOR  VIDEO  AMPLIFIER  by FRANK C H A R L E S CAMERON B.S.,  A THESIS  Stanford University,  SUBMITTED  IN PARTIAL  1958  F U L F I L L M E N T OF  THE R E Q U I R E M E N T S FOR THE D E G R E E MASTER OF A P P L I E D  In  the  We a c c e p t  this  standards  required  degree  of  SCIENCE  Department  Electrical  OF  of  Engineering  thesis  as  from  Master.of  conforming candidates  Applied  to  the  for  the  Science  Members of the 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  AUGUST  British 1962  Columbia  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study.  I further agree that permission  for exte|^live copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  E l e c t r i c a l Engineering  The University of British Columbia, Vancouver 8, Canada. Date  ABSTRACT  B r u u n and G r i n i c h designs for  using  stages  alloy-junction  The  transistors  cascades  responses The  design  lation.  °  amplifier  drift-  Equivalent are  reviewed  feedback sistors  are  drift-  emitter  circuits  given  parameters problem of this  is  of  the  by  concerned  the  procedure  use  of  necessary  for  stages  Johnson— cases.  Butterworth-  w i t h an  types  of  alternative  to  obtain  pole-zero general  to  cancelto  in-  transistors.  Johrison-Giacoletto  newer  the  i n both  produced  sufficiently  the  lead  cancellation.  numerical  method i s  or d i f f u s i o n - t y p e s  and p o l e s  of  using  functions  the  spite  here  emitter  hybrid-n  treat  emitter  high-frequency  tran-  developed.  is  and the  pole—zero  amplifier  Diffusion-type  described  Grinich  and m o d i f i c a t i o n s  attention  Suitable  by  cascades without the  i n the  c i r c u i t , were used  an i t e r a t i v e  configuration  zeros  well  video  and d i f f u s i o n - t y p e - a l l o y - j u n c t i o n  amplifiers  Transfer either  reported  In addition,  both  :  type  research  feedback  which are  of  described  configuration.  described  t h r o u g h use  method u s i n g  broadband  clude  common-emitter  hybrid-n; e q u i v a l e n t  amplifier  type  previously  resistance-capacitance  i n the  Giacoletto  have  to  of  and cascades transistors  with emitter  feedback  the  of  of  the  for  property transfer  defining  synthesizing  dependence,  is  type  function  this for  this  are  described.  stages  i n the given. of  are  dependence  flat  of  using  commonSpecial  amplifier  that  interdependent. are  amplitude  developed,  response,  A numerical  in  iterative  method  of  solution is  A numerical using  2N384  a  response design  example  p-n-p  is  obtained  when u s i n g  the  of  delay,  and  t h a n by  the  small  dc  very  emitter  w i t h an i n p u t  results  iterative poses  pair  and t h a t  are  very  Suggestions  for  are  little  good  small  or very  or  design  to  a  is  is  linearity,  for  the  two  properties,  bandwidths,  or  resistances.  50-JTL l i n e ,  design,  obtained  output is  described.  using  circuit  and  construction  necessary. of  db,  49.6  pad and an  for  ob-  particular  theoretical  equivalent  an  no  designs  large  values  development  i i i  there  convergence  the  accurate  adjustment  given.  Butter-  shown t h a t  Phase  interstage  to  hybrid-it  further  the  comparable  shows  sufficiently  The  amplification  that  (300x0.  used  connection the  is  the  theoretical  impedance-matching  that  for  but  given.  of  It  amplitude  an a m p l i f i c a t i o n of  according  Johnson-Giacoletto method  The  currents  for  indicate  amplifier  method,  overshoot  for  Mc i s  6.5  G r i n i c h method  resistance  step-response  emitter-follower  modified  to  design  maximally-flat  g i v i n g maximum dc  method g i v e  The n u m e r i c a l m e t h o d  fitted  Test  cps  designs.  An a m p l i f i e r . b u i l t but  for  amplifier  G r i n i c h method.  iterative  Grinich  e x c e p t when d e s i g n i n g very  f r o m 40  resistance  interstage  designs.  for  a three—stage  transistor  the  iterative  db g r e a t e r  value  by  this  optimum f o r  t a i n e d by 7.6  of  compared w i t h an e q u i v a l e n t  optimum i n t e r s t a g e  true  drift  w i t h a passband  worth type  exists  proposed.  the  method  are  the the pur-  ACKNOWLEDGEMENT -—^  The Dr. of  author would l i k e  A . D . Moore, for this  his  afforded  many h e l p f u l in  the  by  like  the  supervising  and g u i d a n c e  throughout  professor, the  course  of  acknowledge the  the  programming  UBC C o m p u t i n g C e n t r e ,  given by h i s  fellow  graduate  assistand  the  students  work*  research  Grant  to  staff  suggestions  experimental This  Council  thank the  research,, He w o u l d a l s o  ance  help  to  (BT-68)  was  c a r r i e d out under  granted  to  ix  Dr« F *  a National  Noakes  0  Research  T A B L E OF CONTENTS  List  of  Illustrations  List  of  Tables  v i i i  Acknowledgement  .  1.  Introduction  2.  Transistor  .  .  .  ix 1  Equivalent  Circuits  Transistor  2.2  Physical  2.3  Admittance Parameters D i f f u s i o n Model The  Operation  .  .  .  .  .  .  .  of  the of  .  .  .  .  Junction Transistor a  Hybrid-ix  .  26  Transistor  2.7  Modified Equivalent Circuit for Parallel Capacitance Feedback i n the E m i t t e r Lead High-Level  Single-Stage Functions  35  Capacitance  .  43 .  Resistance. . 45  Injection  A m p l i f i e r Response  51  Functions Current-  .  and  .  .  .  .  3.5  M a x i m i z a t i o n of  .  .  54  Voltage-Amplification  for  a Stage  in  Stagger-Tuned  Cascade  the  .  DC C u r r e n t  iv  a .  Current-Amplification Representation Complex-Frequency Plane . . . The  .  54  Amplification Functions Cascade  3.4  8  Equivalent  The M i l l e r  3.3  .  11  2.6  3.2  6  One-Dimensional  The D r i f t  3.1  .  6  2.5  2.8  .  C i r c u i t Representation  Johnson-Giacoletto  Circuit  3.  .  2.1  2.4  Page vi  .  .  .  in  55  the 59 .  Amplification  .  61  .  62  4.  The 4.1 4.2  5.  6.  7.  Iterative  Method of  The S y n t h e s i s Realization  Problem . . .  Approximation Function .  . .  4.3  Realization  4.4  The  Numerical  —  Iterative  .  5.2  Grinich  Designs  5.3  Designs  U s i n g the  5.4  Characteristics  5.5  Pre-Design  . .  A p p r o x i m a t i o n and . . . . . .  .  64  Design  .  .  Using  .  .  .  Iterative  of  Construction Design  6.2  Low-Frequency  6.3  The E m i t t e r  6.4  Amplifier  Conclusions «  o  the  .  .  .  72  .  .  .  .  .  77  .  .  .  77  Transistor  .  .  .  79  .  .  .  .  Method  Physical  .  . .  88 .  Realizability.  94 96 98 98  .  101 103 106  . .  69  .  Results  o  .  .  and Performance  .  66  .  Follower  .  .  .  Method  Considerations  . .  Amplitude . • .  Transistor  Iterative of  .  Considerations  Test  .  .  Equations  .  a Drift  Prediction  General  «  .  RCA 2 N 3 8 4 G e r m a n i u m D r i f t  6.1  References  Page 64  Network  .  The  .  Synthesis  The M a x i m a l l y - F l a t . « . • . .  The  5.1  Amplifier  .  Method  Designs  .  Network  .  .  .  .  v  . .  . .  . .  . .  . .  . .  . .  115 117  LIST  OF I L L U S T R A T I O N S  Figure  "  1.  D i f f u s i o n model  2.  C o m p o s i t e t r a n s i s t o r : (a) common-base (b) common-emitter c o n f i g u r a t i o n . .  configuration, . . . . .  Generalized elements  extrinsic  3.  4.  5.  6.  of  Page  a junction transistor  Tt-equivalent  circuit  with  .  .  .  .  12  29  Johnson-Giacoletto equivalent c i r c u i t S i m p l i f i e d hybrid-TC circuit . . . .  hybrid-TC  common-emitter .  common-emitter .  .  32  . 3 3  equivalent  7.  The  circuit  8.  Single-stage  9.  S i m p l i f i e d hybrid-TC e q u i v a l e n t c i r c u i t for C ... Miller S i m p l i f i e d J o h n s o n - G i a c o l e t t o hybrid-TC c i r c u i t w i t h emitter feedback .  equivalent  circuit  .  equivalent  Common-emitter transistor  for  the  drift 37  amplifier  for  y^,  .  equivalent  .  .  circuit  .  .  .  .  41  .  .  .  .  44  modified 45  M  10.  28  11.  Equivalent  circuit  12.  F i n a l form of the common-emitter w i t h emitter feedback . . the  y-^-[  e  a n d y2i  •  e  equivalent  •  •  47  circuit 51  Equivalent  14.  Cascaded  15.  T h r e e - and f o u r - s t a g e B u t t e r w o r t h - t y p e d i s t r i b u t i o n s of p o l e s and z e r o s f o r a m a x i m a l l y - f l a t cascaded amplifier .  62  Amplifier pole locations c o m p l e x p o l e s ( Y} r e a l ) , ( imaginary)  70  17.  for  of  . 4 6  13.  16.  circuit  i n terms  equivalent  emitter-feedback  single-stage amplifier  .  amplifier .  .  .  .  . .  54 55  i n the s - p l a n e : (a) for and (b) f o r r e a l poles  Low-frequency a m p l i f i c a t i o n , methods, Rj v a r i a b l e . . . vi  i t e r a t i v e and . . . . .  Grinich . . .  84  Figure  Page  18. ^  T h r e e - s t a g e p o l e - z e r o c o n f i g u r a t i o n s f o r R-j- = 3 0 0 X 1 . : (a) G r i n i c h d e s i g n , and (b) d e s i g n b y t h e i t e r a t i v e me  19*  A m p l i t u d e r e s p o n s e c u r v e s by the G r i n i c h and i t e r a t i v e methods f o r R j = 300.n.: (a) o v e r a l l a m p l i f i e r r e s p o n s e , and (b) i n d i v i d u a l s t a g e r e s p o n s e s . . . 86  20.  Phase r e s p o n s e c u r v e s f o r R j = 3 0 0 H : (a) G r i n i c h d e s i g n , and (b) d e s i g n by t h e i t e r a t i v e method .  21.  22.  .  87  R e s p o n s e t o a u n i t - s t e p f o r R^ = 3 0 0 X L : ( a ) G r i n i c h d e s i g n , and (b) d e s i g n by the i t e r a t i v e method . .  87  Successive cycles a m p l i f i e r w i t h Rj  92  i n the i t e r a t i v e = 300A . .  diagram of  the  amplifier  .  an  23.  Circuit  24.  The t e s t  25.  (a)  T r a n s i s t o r c o m m o n - c o l l e c t o r a m p l i f i e r , and  (b)  low-frequency  Common-emitter output  27.  B l o c k diagram of generator . .  29.  .  amplifier .  .  T-equivalent  26.  28.  test  d e s i g n of  stage  to  .  99  .  .  circuit feed  .  .  .  .  .  .  .  .  50-J1. l i n e  test circuit using . . . . . . .  100  . .  103 .  sweep-frequency 107  Amplitude response using sweep-frequency generator, (a) e x a c t d e s i g n v a l u e s , (b) m o d i f i e d d e s i g n v a l u e s f o r m a x i m a l l y - f l a t response . . . . . . . . B l o c k diagram of phase  test  measurements  circuit for .  .  30.  T r i a n g u l a t i o n method of  31.  Amplitude response  32.  Phase  33.  B l o c k diagram of  response  response  test  .  phase  curve  curve  .  .  amplitude .  .  .  determination  for  the  the  test  amplifier  circuit for  transient  for  measurements  a m p l i f i e r response  . to  .  .  test  108  and .  . .  amplifier .  .  108  .  .  109  .  .  110  .  .  110  .  34.  Test  35.  Low-frequency amplitude response f o r amplifier . . . . . . . . . vii  105  112  a step-function input the t e s t . . . .  . .  113 114  LIST  OP T A B L E S  Table  Page  I.  Grinich  designs  II.  Designs  using  w i t h R-j- v a r i a b l e  the  iterative  variable  .  .  .  .  method w i t h  R  .  .  .  83  T  90  v i i i  ITERATIVE  S Y N T H E S I S OF A  FLAT-STAGGERED  EMITTER-FEEDBACK TRANSISTOR VIDEO  1. Video television, the  amplifiers radar,  transistor  posed  soon  to  after  transistor  is  pectancy.  band,  low-pass  early  transistor  acteristics not  necessary  made  to  this  network design  This  of  obtain  to  size,  been  to  low  the  proThe  life  ex-  to  only  broad-  fact  that  high-frequency  char-  circuit  representation.  "broad-band  amplifier  choice used  consideration  of  power  used  equivalent  The  of  1948.  high-gain,  due  broad-barid  be  of  was  and l o n g  has  suitable  to  design  application  in July  transistor  configuration  The  extremely  partly  stages.  i n the  amplifiers  transistor  a high-gain,  from a q u a l i t a t i v e  its  small  have  for  amplifier  circuit  device''"  is  the  advanced  systems.  synthesis  of  d i d not  importance  low-pass  m a i n l y due  state  cascade  transistor  c a n be  models  and the  to  of  amplifiers.  order  prime  ruggedness,  i n the  sufficiently In  lar  advent  physical  extent  of  these broad-band,  Modern a c t i v e  a limited  was  are  attractive  requirements*  INTRODUCTION  and communication  the  AMPLIFIER  of  the  i n the  of  the  it  is  particu-  cascade  following  2 3 three  configurations  emitter-input pedance, unity ciable The  configuration  a high  (very  initial  output  close  voltage  which give  to  is  amplification.' characterized  impedance,  unity  for  current  small  load  by  The a low  of  cascaded  video  input  amplification resistance),  a m p l i f i c a t i o n w i t h no p h a s e - r e v e r s a l  design  common-base  amplifiers  was  of  im-  less  than  and  appre-  the  signal.  based  on  this  configuration cutoff  to  frequencies  miniaturized the  due  the  of  the  output  pedance  was  necessary  gain  the  transistors.  necessary  to  bandwidths ple to  achieve  were  cutoffs  early  broad-band video  high collector  of  l o w common-base  of  to  the  transistor  interstage  impedance realize  high video  obtain low-frequency  models.  The u s e  transformers  the  low  to  emitter  of  match  input  im-  impedance-transformation  number o f  gains,  difficult  interstage  to  the  A large  extremely  current-amplification  and,  to  amplifier  stages  consequently,  achieve  transformers.  due  It  to  was  video  the  also  amplification with transformers,  was  multidifficult but  4 means  for  low—frequency  The extremely The  common-collector high input  current  power  is  at  less  gain is  mon-collector ing  impedance  the  least  connection or  output  connection.  gain is  close come  It  is  all  possible unity.  three if  common-emitter  of  an  lower  output  the  impedance. voltage  connections. for  am-  The  impedance  and  com-  match-  amplifier.  of  impedance  current of  characterinput  than the  im-  common-  and c u r r e n t  and g i v e s  Very large  advancement  is  magnitude)  phase-reversal,  common-base  an  phase-reversal,  of both voltage  connections.  can have  the  configuration  an o r d e r  capable  the  With  three  (by  with a signal  of  to  higher  stages  but  signal  primarily  common-emitter b a s e - i n p u t  plification,  have  used  The  and c o n s i d e r a b l y  is  the  input  pedance  cation  is  low output  appreciable, w i t h no  for  available.  configuration  a very  the  a somewhat  power  or  than u n i t y ,  i z e d by  base  base-input  a m p l i f i c a t i o n c a n be  plification the  c o m p e n s a t i o n became  am-  the  greatest  current  amplifi-  amplification,  the  transistor  current-amplification cutoff  a^,  art  frequencies  3 in  the  large  video-frequency power  interstage t h e n be  gain  available  transformers.  This  over  allows  a wide  Direct  use  t o he  band without  made the  o r RC c o u p l i n g b e t w e e n  of  the  need  for  stages  may  employed. In the  investigation  special  attention  as  configuration  this  for  range.  cascaded The  will  video  be  stages  is  of  i s o l a t i o n and  given  to  equivalent  common-emitter  offer  the  most  circuits,  representations  suitable  qualities  amplification.  video  of  transistor  seems t o  stagger-tuned  multiplicative  of  is  commonly u s e d  amplification.  generally so  cascade  dictated  simplify  A mismatch  i n order  the  to  achieving  interconnection  obtain  individual  in  stage  a good  degree  design.  The  5 analysis video ter pole  i n t r o d u c e d by Bruun  amplifiers  lead  has  been  ger-tuning  paper, by  extended  plane.  cancelled  by  sistor  i n the  the  cluding  the  The real  emitter  and G h a u s i , have  cascade  of  mally-flat  using  RC e m i t t e r  conjugate  pole lead.  alternate  common-emitter  feedback  give  i n the  a Butterworth  was  used  pole-pairs  by  Pederson  resistive,  to  of  pole-zero -cancellation.  real  produced  to  i n t r o d u c e d was  a  having  and h i s  methods  of  series-peaked,  stag-  complex-  zero  stage  a l l In  achieve  i n the  emit-  then  only  a  colleagues,  re-  Pepper  broadbanding,  in-  and s h u n t - p e a k e d i n 8 pair in a  common-collector-common-emitter  synchronously-tuned response.  broadbanding  design  feedback  extraneous  studied  the use of 7 t e r s t a g e s , and the  by Grinich:  response  introducing  frequency  the  with resistance-capacitance  maximally-flat  Grinich's  for  were n o t  Methods  (identical) of  considered  stages  to  give  multiple-stage  feedback  by  al  Pederson  et  due  to  maxifor their  4. overall  low gain-bandwidth It  cal)  is  method  to of  be  shown i n t h i s  network  amplifiers  tially  gain,  zero  cancellation The b a s i s  the  that  for  drift  to  junction well  as  added  circuit  a  on the  circuit  is  plane  of  the  is  essentially  (electric) alloy-  junctions,  and d i f f u s e d - a l l o y  lead  equivalent  much l e s s  manner  than the  network  outlined  hybrid-rc circuit  equivalent  is  valid  alpha-cutoff  transistor.  The v o l t a g e ^ - a n d c u r r e n t - a m p l i f i c a t i o n e x p r e s s i o n s rived  for  stage  i n a cascade.  best  a  single-stage  represents  design  It  emitter-feedback is  shown t h a t  a transistor  frequency-dependent the  as  (drift)  feedback  i n the  in  modified  parallel  This  which i s  pole-  lies  then v a l i d , f o r  then  a frequency  substan-  10 Johnson.. The  with a built-in  circuit  emitter  is  work of  modified Johnson-Giacoletto  o n l y up t o  a  of  small-signal  chosen  resistance-capacitance  simplified.  design  synthesis  suitable  circuit  micro-alloy,  w i t h the  the  (numeri-  t h a n does the  and the  is  frequency  by  equivalent  which exhibit  A parallel  i n series  by B r u u n ,  equivalent  transistors  surface-barrier,  to  method y i e l d s  amplifier  transistor  equivalent  transistors  transistors. is  The  for  an i t e r a t i v e  Grinich*  method of  hybrid-TC  account  field.  this  a given bandwidth*  the The  that  applicable  9 G i a c o l e t t o , based  by  Johnson-Giacoletto slightly  and t h a t  this  of  circuit*  developed  for  is  scheme u s e d b y  representation  equivalent  thesis  synthesis  common-emitter greater  product.  proceeds..  stage  the  amplifier current  i n a cascade,  amplification function For a p a r t i c u l a r  is  and f o r  de-  a  amplification and the  given,  transistor  are  from  complete which  operating-point,  5 a  c u t - a n d - t r y method y i e l d s  for  which the  low-frequency  A numerical design cascaded the  an optimum i n t e r s t a g e  amplifier,,  d e s i g n method  is  It  amplification is  is  is  greatest  interstage  resistance.  The  amplifier,  built  design  design  procedure.  to  the  for  the  large  performance  rate  emitter of  Rj,  a maximum^  carried through for  found that  resistance,  a  of  three-stage convergence  currents  a three-stage  specifications,  verifies  and  of large  cascaded the  6 2.: T R A N S I S T O R E Q U I V A L E N T C I R C U I T S 2.1  Transistor The  sidered  to  urately, for  Circuit  transistor,  for  be  active  a linear  a three-terminal  c i r c u i t design  eters  or  the  are  ated  of  and on the  the case  of  the  of  box'', t y p e  used  to  analysis  and c u r r e n t s ,  port  parameters  point at  out  which are  that  no  approximations  or  'exact'  d e s c r i p t i o n of  the  may v a r y In  this  even  for  of  each  is  design  for  partic-  depends to  be  types  both  of  is  only  a  amplifier  Although particular  device.  representation  of  type  of  made  However, must be  transistor  is  two-port a  "black  with terminal  transistor  are  the  it  H e r e , we h a v e  exclusively  this  on  incorpor-  of  bandwidth.  advantage  the  of  used. two  frequency.  assumptions  a single  param-  wide—band r e p r e s e n t a t i o n , major  acc-  two m e t h o d s  is  con-  represented,  two-port  representation transistor  c a n be  o r more  useful  f o u n d by measurement  advantage  single-frequency  eters  of  and r e p r e s e n t s  foremost  that  a set  Although these  which deals  The  fact  the  a single  uency.  the  usually  in essentially  problem of  treatment  ages  this  is  d i s t i n g u i s h e d by t h e i r  general  of  choice  (two—port),  narrow-band representation  mentioned here method  either  i n t o which the  are  which are  problem of  by  It  p a r t i c u l a r method of  Transistors systems,  network.  independent,  The  circuit  four—pole  circuit.  not  applications. type  small-signal:applications,  purposes,  an e q u i v a l e n t  representation ular  Representation  at  by a set  of  a single  volttwo-  freq-  representation so  the  that  type,  we h a v e  'exactness'  examined  is  of  in light  certain  an  of  param-  considerably. thesis,  a broad-band  amplifier is  t o be  synthesized,  and  the  sistor  need is  for  a broad-band c i r c u i t representation  apparent.  approximated  over  single-frequency treatment  leads  transistor a  set  and  of  output,  erties. ing  range  t e r m i n a l parameter quite  n a t u r a l l y to  of  a set  parameters,  of  two  are  several  quadripole  features  of  a purely  internal  physical behaviour  of  the  separate  transistor will  describing familiar  Also,  have  to  be  the  several  the  approaches  band  equivalent  circuit.  band  quadripole parameters,  to  is  neglected  is  of  of  the  a change  o r TC-network as  the  of  the  most  possible  analysis  lead  an e q u i v a l e n t  c i r c u i t where  the  i n evidence.  The u s e  are  placed  is  for  quadripole  amplifier,  the  point. an of  set  form an  by  transistor.  deriving a  above  a  pa-  c i r c u i t represents  to  a  transistors  of  broadbroad-  equivalent  and G i a c o l e t t o .  a theoretical  transistor  that  13  tively, to  of  these the  common m e t h o d  problem of  shown b y P e t e r s o n  are  i n operating  12 T-  These  so  representation  F o l l o w i n g from the it  the  important of  familiar device,  the  of  empirical approach,  equivalent  less  of  lack-  synthesis  The  type  importance which are  and a l l  is  at  the  prop-  of  modified for  transistor.  of  be  transfer  device  type  representation  can  input  modified types  point,  a transfer-function  c i r c u i t elements  are  of  this  operating  equivalent-circuit  There  analysis  necessary.  For  This  tran-  the  The m o s t  is  rameters  an e x t e n s i o n  describing immittances  representation.  since  device  quadripole parameters^  this  fixed  that  the  a matrix representation  is  w o u l d be  by  the  measurements.  features  completely  of  a n d two d e s c r i b i n g f o r w a r d a n d r e v e r s e  There  i n the  terminal behaviour  a broad frequency  i n terms  four  The  of  a suitable  Alterna-  t r a n s i s t o r model  physical of  phenomena o f  an e q u i v a l e n t  will the cir-  8 cuit  representation  of  a transistor  advantages  over  the  quadripole  contact  is  kept  w i t h the  easy to  see  w h i c h are  transistor.  dependent  transistor  the  on the  circuit  advantage, an  elements though,  equivalent  circuit  frequency.  Since  ous  of  aspects  easily tage  the  imation  2.2  transistor  to  i n the  equivalent the  Physical  design,  'exact  for  the  point  circuit as  only  The m o s t a wide  elements may b e  it  is  choice  of  much  less  is one  or  is  frequency  are  it  range  identified with  is  only  of  of vari-  can  The m a i n  still  of  single  independent  circuit.  that  two  important  modification i n design  equivalent  circuit  closer  parameters  over  elements  a  device,  affected.  i n which the  Since  distinct  the  equivalent  validity  separate  incorporated  of  the  The  of  of  operating  may b e  is  operation  desirable  a high-frequency  a number  representation.  physical  the  t h e n has  be  disadvanan  approx-  device.  1  Operation  of  the  Junction  Transistor  i  A short transistor the  operation w i l l  equivalent  p-n-p  may b e holes  circuit.  discussed  in a  and e l e c t r o n s  the  physical  given  This  video-amplifier  as  leading  to  discussion w i l l this  design.  is  the  the be  involved  in  development  of  confined  polarity  chosen  The h ^ - p - n j u n c t i o n  s i m i l a r manner by  and by  processes  reversing  the  of  the  for  the  transistor  interchanging sense  to  all  the  terms  voltages  currents. Junction  and  be  junction transistor  present  and  d e s c r i p t i o n of  n-type  silicon.  transistors  semiconductor I n the  concentration  of  n-type donor  are  composed  materials,  usually  semiconductor atoms,  or  of  a c o m b i n a t i o n of  p-  either  or  material  an e x c e s s  of  germanium  there  is  electrons,  a  greater and  in  the  p-type  of  acceptor  trons). called  semiconductor atoms,  The  or  an excess  r e g i o n where  a p-n junction,  rectifying,  although  tion  i n that  rent  must be The  material  on b o t h  the  of  p-  differs  sides  of  is  holes  a greater  (i.e.,  and n - t y p e  w h i c h has  it  there  the  a deficit  materials  interesting  p-n-p  nority  carriers  is  composed  p-n junction which acts  (injector  of  holes)  into  as  the  n-type as  minority  carriers  the  base  emitter,  the  tion of  are  the  major is  concentration  into  tric  the  across  emitter  t h a n does  junccur-  r e g i o n has  the  base  region,  other  region  is  c o n t r o l l e d by  the  j u n c t i o n due  to  electron  into  by  flow the  i n the process  be  an e m p i r i c a l treatment  is  essentially  to  be  riers  m a k i n g many c o l l i s i o n s  times  during their  effect  transistor  discussed process  the  base  mi-  region, of At  much  way.  junc-  This  emission  voltage the  across  base.  with l i t t l e  or  no elec-  circuit  Diffusion individual  d i r e c t i o n of region.  n-type  a built-in  equivalent  later).  of  therefore  resistance  of  the  higher  emitter-base  w i t h the  and changing  across  low  diffusion, (the  i n the  a random t h e r m a l  travel  of  fields  included  by  relatively  of  composed  and  the  flow  is  a very  flow  base  base  junction  to  electrons  from e l e c t r i c  field will  the  forward-  region.  the  are  is  the  cross  the  material received  being  a collector  to  Hole-current  aid  flow  end of  a g r e a t many more h o l e s  emitter-base  base  opposite  current  because  than there  holes  at  a  an e m i t t e r  p-n junction which acts  there  of  and h o l e  of  and a r e v e r s e - b i a s e d  impurity  is  rectifying  junction electron  junction transistor  (conducting)  This  join  elec-  considered.  biased  holes.  of  property  from an o r d i n a r y  the  concentration  flow car-  m o t i o n many  On t h e  average,  10 it  acts  to  cause  throughout plied of  by  charge  of  region  is  very  trons,  and the  through  the  ing  base  transit  carrier  diffusion rived  The electron reach  the  lowed  to  from in the  the  will  is  on the  next  the  the  given  is  the  collector  The  effect  of  If  recombine  with  taken by h o l e s the  the  recom-  the  t r a n s m i s s i o n of  called  sup-:  nonuniformity  considered.  will  is  region,  i n the  that  composed  defined  flow by  bias  base  base  elec-  holes in  cross-  minority-car-  of  minority-  dependence and i s  to  of  the  be  de-  so  but  that are  the  n-type  is  base  The  very  :  which  immediately  the  i n the  base  region  current  al-  p-type  region.  are  impurity  the  prevented  concentration than that  flow  across  in the  holes.  through  the  * 8  t  X  base  region  from  multiplication factor,  by A f b = *  (in  holes  much g r e a t e r  the. m a j o r of  the  holes  from e n t e r i n g  a current  a  non-conducting  region  region.  region so  rendered  region,  prevented  collector  junction is  collector  to  representation  reverse  through  The h o l e - c u r r e n t to  due  junction potentials  a large  electrons  base  be  showing  junction is  are  the  also  for  uniform  section.  by  region  layer.  efficient  become  random m o t i o n i s  base  collector  a means  to  gradient  The m e a n t i m e  the  p-type*collector  collector  be  this  holes  An a d m i t t a n c e  enter  n-type  the  time.  collector  entering  of  to  collector  Similarly,  must  region  sense)  collector  i n the  region.  currents  i n the  few  result  diffusion  density  density  carriers  thin,  base  or  density  charge  charge  D i r e c t i v i t y to  a concentration  bination  rier  carrier  a region.  carrier  the  the  emitter a^,  11 where  ft  = i n j e c t i o n e f f i c i e n c y - the f r a c t i o n of t o t a l current c r o s s i n g the e m i t t e r j u n c t i o n t h a t i s c a r r i e d by m i n o r i t y c a r r i e r s i n t o the base,  6 = transport factor r i e r s that reach a n d oc  The  quency,  multiplication factor  but  quencies. a manner  may b e It  to  considered  decreases  be  hybrid-TC  the  principles of  the  fact  amplifier  available  2.3  circuit the  that  Admittance  of  real  car-  will  function  and c o n s t a n t  with  increasing  of  at  fre-  low  fre-  frequency  in  2.3.  the  modified  now b e  diffusion  the  synthesis  i n the  The  of  be  a complex  i n Section  derivation  equivalent  to  is  i n magnitude  discussed  A complete  view  injected  = the i n t r i n s i c - a = the r a t i o of t o t a l c u r r e n t c r o s s i n g the c o l l e c t o r j u n c t i o n to the m i n o r i t y c u r r e n t a r r i v i n g at i t .  current  basic  = the f r a c t i o n of these the c o l l e c t o r j u n c t i o n ,  made  process.  equivalent  Johnson-Giacoletto starting This  circuit  is  is  from  the  necessary  fundamental  and a complete): u n i f i e d d e r i v a t i o n  in to  is  not  literature.  Parameters  theoretical  of  a One—Dimensional D i f f u s i o n Model  analysis  of  the  junction transistor  for  14 small  signals  years  before  by  the  Shockley,  ction the  shortly  eters  of  device  the  physically  with  to  and f o l l o w i n g the  transistor.  T he  the  its  that, of  a  was  set  of  several  A second  theory  made  a number  one-dimensional following  paper  of  development.  widening  derivation  theoretical  classic  realized.  a n d Teal"'"'' e x t e n d e d  i n conjunction  thereafter,  for  was  i n Shockley's  space-charge-layer  contributed  junction  described  Sparks,  transistor  effects  have  was  of  by  the  authors  admittance  applies  jun-  A study of 16 Early 9 1 0 17  to  '  'is'  param-  d i f f u s i o n model  analysis  paper  of  the  alloy-  12 junction  as  well  Figure ity  carriers  (The  grown-junction  shows  across  symbols  on L e t t e r  1  as  the  the  and s i g n  Symbols  transistors  d i f f u s i o n model  base  region  of  convention used  for  Semiconductor  for  transport  a junction conform to  Devices)  of  transistor. the' I R E  Standards  19  E  V  v  0 Figure In  order  to  continuity carrier  1.  D i f f u s i o n model  satisfy  (hole)  density  generation  of  hole  flow  per  librium  hole  density  density  i n the  n-type  and q to  flow  to  i.e.,  the  from holes  the the per  of  holes  n-type  be  point  follows rate  at  used  per  i n the  to  effective  is  —^'J^/q.  expressed  -(p as  -  to  that  < 0  be  rate  to  P  the  net  the  then the  rate  equi-  actual  hole in  current  den-  rate  hole  of  recombination (natural  p r o p o r t i o n a l to  T h e n the  net  rate  of  P^)/"Cp'  Therefore,  of  rate  the  lifetime  hole  of  a  change  plus be  n  hole  the  of the  point  p be  characteristic  charge,  rate  Let  the  charge,  The n e t  recombination is  is  be  of  equal  material,  and ^  hole  time  point.  we d e f i n e  equilibrium density.  r e l a t i o n may b e  that  n-type  material, If  the  at  0  transistor  conservation  u n i t volume  the  u n i t volume  a junction  so t h a t  c  >  space charge regions  x  a n y p o i n t m u s t be  an e x p o n e n t i a l  of  of  c o n d i t i o n of  u n i t volume  material.  sity  carriers  the  r e l a t i o n may b e  of  the  Y  minor-  the  decay), deviation  generation the  of  of  continuity  This equation i s v a l i d i n the base r e g i o n and assumes that generation i s not stimulated through some e x t e r n a l Now i t field  is  is  agency«  assumed that the e f f e c t of an a p p l i e d e l e c t r i c  small so that the r e s u l t i n g d r i f t v e l o c i t i e s  r i e r s can be superimposed on the thermal v e l o c i t i e s riers;  hole  i n other words, the system i s l i n e a r *  then has two components, d i f f u s i o n component.  of the  car-  of the car—  The hole c u r r e n t  a d r i f t or conduction component, and a  The d i f f u s i o n component of h o l e - c u r r e n t den-  s i t y at any p o i n t i s p r o p o r t i o n a l to the g r a d i e n t of the hole dens i t y at that p o i n t , i . e . ,  it  i s due to a d i f f u s i o n p r o c e s s ,  and  may be w r i t t e n as  <Vdiff  =  where  v -v*> (  i s the hole d i f f u s i o n constant i n n—type m a t e r i a l , and  q ^ p i s the g r a d i e n t of the charge d e n s i t y .  The minus s i g n ap-  pears because hole motion must be such as to decrease The d r i f t  the  gradient.  component of h o l e - c u r r e n t d e n s i t y at any p o i n t i s  p o r t i o n a l to the a p p l i e d e l e c t r i c charge d e n s i t y at that p o i n t .  f i e l d s t r e n g t h and the  If a mobility  the d r i f t v e l o c i t y per u n i t f i e l d i n t e n s i t y ,  Adrift  pro-  hole  i s defined to be then  = iyipE  which gives f o r the t o t a l h o l e - c u r r e n t d e n s i t y  J  Since the d r i f t  p  =  ^p^P - ^ pV ° D  term i s n e g l i g i b l e we may w r i t e J  when put i n t o Equation ( 2 - l )  this  gives  ....(2-2)  p  = -  qD^yp;  14  a*  i V°(VP)(-<ID )  j i _  r  a  p  v  P - pn  V  "  / N  * p'  9  + D_ V  = -  ....(2-3)  P«  p  Since  the  diffusion region  a n d x = W, E q u a t i o n ( 2 - 3 )  This the  then i s holes  injected  It P , E  the basic  is  given  between  p a r a l l e l planes  may b e w r i t t e n  the  base  now t o  at  x = 0  as  r e l a t i o n which describes  into  convenient  lies  the  behaviour  of  region.  introduce  an excess h o l e  density,  by P  Substituting  =  i n Equation (2-4), ^P  ~  p  E  we P  E  P n  «.o (2-5)  *  0  have ^  E  2  P  E  + D  •p .2 ^ e I: 2 'x P  where  L_ = (D P P  Equation split  (2-4a)  into  a n ac  p  )  is  = the  2  . ^  ^ < P  p  hole  a linear  a n d a dc  _ J - , , L  diffusion length  differential  i n n-type  material.  e q u a t i o n w h i c h may  equation by assuming  a s o l u t i o n of  be the  form P (x,t) Q  = P  e  0  U )  + P  O L  (x)exp(jat).  ....(2-6.)  The d c e q u a t i o n t a k e s  the  form  <* eO ^__2  and t h e ac e q u a t i o n  - ^(1 L  x  ary  that  P ^ =  These  g  conditions  •..-.( 2-4b)  is  -v—\ d  noting  P = 0 2 eO n  T  + j « f J ? i p  equations  imposed by e m i t t e r  = 0  0 3 0 0  (2-4c)  x  must he s o l v e d w i t h b o u n d -  and c o l l e c t o r  voltages. 14  It for  c a n be shown b y t h e u s e o f B o l t z m a n n s t a t i s t i c s  an unbiased  and p - t y p e  j u n c t i o n the e q u i l i b r i u m hole  regions  are r e l a t e d P  where is  with mal  respect  3  reverse—bias  i n p-type  potential  (base),  m a t e r i a l , V^  of the p-type  a n d _A~^  region  ~ kT/q =  of the e l e c t r o n i c  and T = absolute  ).  temper a t u r e .  charge,  therk =  (At 300  K,  junction,  Note t h a t Vg i s  hand,  i f an e x t e r n a l  t h e n t h e new v a l u e  material i s , P (0) Q  s m a l l and n e g a t i v e ,  i.e., in  direction.  On t h e o t h e r  n-type  ....(2-7)  —1  A. - 1 0 / 2 6 v o l t "  the  region  w i t h q = magnitude  constant,  density  (barrier)  to the n-type  potential,  Boltzmann's  the  = P expCA.Vg).  n  i n the n -  by  P^ i s t h e e q u i l i b r i u m h o l e  the d i f f u s i o n j u n c t i o n  densities  that  of h o l e  (at the emitter  - P  expA(y + V B  de b i a s  is  applied  concentration  to  i n the  junction), =  P exp(AV ) n  E  ....(2-8a)  16 and,  (at the c o l l e c t o r P (¥)  = P expA(V  Q  Equations of  total  p  terms  The m o s t  e 0  (0)  P  direct  found by s o l v i n g  e Q  = P (expAV n  (¥)  n  parameters.  o f t h e more g e n e r a l  l/Lp  i s replaced by ( l + ja  origin  and t h a t  i s taken  then  i n the form  0  t h e same  i n terms  -  c  1) .  (2-9b)  density  o f t h e symmetry  ( x ) = A cosh(x  /L^  is just only  i s unaltered  solution will  about  - ¥/2)/L  p a r a m e t e r s may  f o r t h e ac c a s e .  equation  the simplest  i s t o be  and then o b t a i n i n g the i n c r e -  ac e q u a t i o n a n d d i f f e r s 2  are  (2-9a)  t h e dc e q u a t i o n  )  equations  1)  The ac a d m i t t a n c e  the d i f f e r e n t i a l  tage  P  conditions  s o l u t i o n f o r excess hole  case  that  -  E  = P (expAV  t h e n be f o u n d b y n o t i n g t h a t  in  (2-8b)  c  densities,  t h e dc e q u a t i o n  conductance  noted  n  t h e dc b o u n d a r y  of excess hole  and  be  c  densities.  P  mental  + V ) = P exp(Av )  B  (2-8) then represent hole  In  junction),  x = ¥/2.  result  a particular i n that It  should  by a change when  advan-  The dc s o l u t i o n i s  + B s i n h ( x - ¥/2)/L  . (2-10)  Using  P  (  x  the boundary  )  =  P  n(e  A  V  C  conditions,  + e ^  Y  2cosh(W/2L  Equations  E- 2) „. e  )  h (  x Jg2 f  (2-9),  )  this  ••  becomes  ....(2-11)  p  2sihh(¥/2L ) p  p  17 The h o l e is the  current  found from the form  (for  A is  procedure the (x  the  = 0)  P  similar  to  for  i n Equation  hole  Equation  (2-2)  ^p(x.t) °* P ^ x  cross-sectional  that  dc  using  ^  AT.  ^ p  and-(noting  current  area  (2-6),  across  of we  the  the  base.  can then  emitter  ( e  that  (x  =  write  junction  the  = W)  _  —  E  V  x  = 0  -  l)coth(¥/L  ) -  (e^ C  -  V  use  is  There  sense  that  ^ e O  of  for  U  l)csch(¥/L ) p  the  collector  dc h o l e  current  current across  is  i n the  the  a p-n-p If,  collector  F  x = V  (  n  e  A  V  E  -  l)csch(¥/L  p  of  also  transistor for  Np a s  neg-  ,-...(2-13b)  )  j  made is  )  as  ^ p A L  q  A  « • « « ( 2 1 3 cL  P  ) -  (e  A  V  C  -  l)coth(V/L  A  bility,  a  p  AD I„ = qAD. ^Cp ~ ^ ~ p  tion.  By  0  x-direction)  junction  in  in  . . ....(2-12)  ( x ) e,0  o  n  P  A L  where  base  field)  / \ ( x ) A = —qAD  P  x - d i r e c t i o n i n the  as  T  ative  / \ (x) = J  density  electric  effective  expression  positive  hole-current  negligible  i  where  i n the  the  relationship,  electron-current due  a p-type the  Einstein  to  the  region,  flow  voltage u  n  is  equilibrium electron  |x  p  -  from base  across  defined  as  density,  the the  P  )  14  AD^. to  emitter  emitter  junc-  electron as  the  mo-  decay  18 time  for  electrons,  then expressions derived Under  for  the  emitter that  L  that  and c o l l e c t o r the  the  primed  the  Then the  refer  total  to  dc  and  given  intrinsic-cx  large  I  E  c  A , A  V  J  V  = I  parameter,  can  and c o l l e c t o r  regions  junctions to  electron  L  n  be  to  and  currents  are  -  1)  ....(2-14a)  C -  l)  ...,(2-14b)  V  to  at  ""-Ep  (2-13)  compared  dc  the  collector  =  electrons,  R  E  A  currents  I  The  are  refer  the  for  from the  and c o l l e c t o r  unprimed q u a n t i t i e s  quantities  emitter  distances  E  n  length  i n Equations  i n the  qAa N ' — — ( e A L ' n  •=  p t  where  those  qAu N = — p - £ ( e  I™  I  diffusion  boundaries  emitter  and  to  currents  assumptions  n  the  analogous  electron  a>"£ <<^l,  as  n  emitter  + I  and  the  and c o l l e c t o r  ""-En  +  a  region  region.  the  c  emitter  C n  are  ....(2-15a)  «  ....(2-15b)  , mentioned  in Section  2.2  is  by  a  = 1  ...,(2-16a)  H—— T Cp X  and  the  emitter  injection  efficiency,  Y = 1/(1  # > is  + — ) • I  Ep  given  by  ....(2-l6b)  19 For  practical  junctions, quency  junction transistors  Ig <<^Ig n  independent  c l o s e l y , ^foc  *=  lQ ^C  and  p  plane  ^ at  and a  ^-Qp'  n  a n d may b e  w i t h abrupt s  taken  o  to  I  v  Equations  = ^ i ( e  I  p  The may b e that  A  v  E  _ 1)  + ^ 2 ( e  G^^ =  G  ( e  A  V  E  -  G + -^(e  l)  A  21  l^n P. £  2  G  ~  _  A _ ^P!£  12  incremental  -^coth(¥/L  conductances slope  form  A  V  C  -  l)  ,...( -17a)  A  V  C  _ i)  ....( -17b)  2  2  of  the  c  s  c  at dc  h  (  ¥  /  (2-l8a)  )  L  . ..  a g i v e n dc  .(2-18b)  operating  characteristic  curves  point at  i.e.,  g  of  i n the  )  = G e E l l VQ = c o n s t  ....(2-19a)  E  E  = G e E 21 = const  (2-19b)  A  because  more  A  G^  found from the  point,  2  fre-  A  G = —  •  and  u n i t y " ^ ' ^ or  (2-15)  A  where  be  are  1.  ¥ e may now r e w r i t e  and  fl"  h  parallel  G ^  a n d G^  Vj,; and,  taking  21  ~  (which are into  account  u  A  U  V  e  functions the  V  e  of  ¥)  dependence  are of  independent ¥ on  16 ,  20  '  " o^c  1 2  =  const  "e  A  E - 1  V  /  1  L  Av  e  /  —-csch(¥/L E  V)• £v  c  AV 1 C  AL  A E V  -coth(W/L  )^  n  F  - +  csch(¥/L ) p  G  ^expCAVg)  ....(2-20a)  and g 22  A  X  C =  <lA|J.pP e n  L  e  A  V  A  const  E f" A E _ V  e  V  e  P C  - 1  _  _  _  E  G  22  e  L  -csch(¥/L  E  V  c  )•  ^¥  AVr  1 o J  Av [I  A  ±  t  h  (  ¥  /  L  p  ) _  s  e  c  h  (  ¥  /  L  )  p  coth(¥/L )  +  p  p  Av C  - i  !  +  (1  K  e  A  v  E  A  L  - s e c h W/L )c o t h ( V / L )  P (2-20b)  where K-l  e  =  A  •  E - 1 1 , , • c s c h ( ¥ / L )• A L , e 'A*V- RE V  A  C - l  V !  —r-n:  v  e  e  Av A  v  !  E  A L  c'  Av E For V first  n  < -0.5 volt,  and V g ^ O . l  term i n Equation  (2-2l)  volt  ^¥  7-—coth(¥/L ) ^ — P  P >  .. . .(2-21)  (a t y p i c a l value), the  predominates,  andthe K  1  term i n  C  Equation  (2-20b)  If  we  predominates•„  let  O  0  and  =  ¥  /  p  L  G =  ....(2-22a)  —exp(AV™)  ,.....(2-23)  ¥ then  the  conductance  parameters  g  and  = G  0 rcoth0 cj  written  . . . . (2-24a)  o  g  2  1  = -  G  0  O  csch0  g  1  2  = -  Gr -  0  O  csch0  g  The p a r a m e t e r  l l  may b e  G  2  0Q  2  = - 0  c a n be  „«,..( 2-24b)  Q  j  ....(2-24c)  Q  ... , . ( 2 - 2 4 d )  coth0 .  Q  Q  related  to  the  low-frequency  com-  A  mon-base noting  small-signal forward current  that  Jfoc  =1,  through the  ratio,  =  * a  o  =  ^ o 8  a  '  relation  I p C  ~ 21^ ll VQ .= c o n s t g  To o b t a i n t h e  g  ~  =  more g e n e r a l  G  2l/  G  ll  admittance  =  s e c h  ^o°  A2-25&)  parameters  we now  1 replace  l/L  by  (l  + j»'T") /L 2  0' £ 0 ( 1 + O  In  the  general  form,  0  replaces  so  that  j « 7p)'-  0Q  and y ^ .  ...,(2-22b)  replaces  g^ . ,  and  22 Equations  (2-24)  become  y  l l  y  21  y  0* coth0'  G  = ~  G  = -  2  K  (2-26a)  $  csch0  „...(2-26b)  0  csc.h0  . ,..(2-26c)  ,y = - 0 coth0  and  „...(2-26d)  00  The p a r a m e t e r signal  1  =  c a n be  0  forward current  related  r a t i o : i n the  to  the  common-base  same m a n n e r a s  small-  i n Equation  (2-25a) ' a  fb  -  —  8  i x  1 ~ 21 ll = const y  V  ep  r  C  / / y  =  s  e  c  h  ^  =  ' cos*0 {l o  +  >  T  p  ~T ) 2  9 a • o ( 2""~ 2'79') For  0  the  first  O  <3Cl>  = ¥/L two  "t  terms  n e  of  hyperbolic  cosine  its  expansion to  series  i  3  1/(1 + = ••• :•• '  —± 1  +  1 0 ( l + j«>r ) 2  1  p  o  +  may b e  that  mately  the  the  • ••.  r  3<oT l0 p  low-frequency value  j3 *  B /(l. + j « ^ 1 0 ) 2  0  p  0  of  *1/(1  8,  (2-27b)  2  2 o  is  approxi-  i.e.,  + i0  2 Q  = 1  )  = 8 / ( l + j«/« ) 0  ....  o  numerator term i n E q u a t i o n (2-27b)  J3  then  give  f0Q)  1 + i0  Noting  approximated by  0  ....(2-25b)  ....(2-27c)  23 where,  using  the  relation L  2  = D  P <°0 =  Here  is  Equation  the  frequency  (2-27c)  is  (  L  y  at  V  )  2  2  /  P  ^ , .P  /  L  ]  T n  which the  down t o  \  of  its  2D / ¥ .  ....(2-28)  2  =  squared magnitude low-frequency  of  value;  8  in  it  is  14 the is  a-cutoff here  with  a  frequency  observed single The  that  time  error  reduced by  o r i g i n a l l y defined 8 behaves  i n the  a refinement  To f i n d  cutoff  where  f  nority the  =  D  i t  ^ 2(L /W) p  Pritchard.  is  apparent  o  f  frequency is  = /%  0Q  2  i  s  through the  at  which the  down t o  \ 'of  its  t  h  e  f  0  diffusion  base  region*  squared  for  ¥/L  8 then behaves  = 8 sech(  p  ) '  D  The  p  (2-28)  <  0.3,  ....(2-27d) time  dc v a l u e of  value  of is  mi8Q  and  8 i n Equation is  called  t h e oc-  21 cutoff  frequency,  then takes 8  which  is  a^.  the  form  =  sech0*  found to  be  This  ^  requires  that  B sech(j2.43.tt/a> )2 0  a good  a  <* T'-Q = 2 . 4 3 , >  0i  A  8 sech0 o  approximation to  8 for  and 8  . ,,.(-2-27e)  BQ ^> 0 . 9 .  Also  0'/sinh0' *=" 0/sinh0  for  0  <  1.  is  as  (t r a n s i t )  magnitude  low-frequency  the  from Equation  and t h e r e f o r e  2  8 sech(ja7r 0 )^  carriers  (2-27d)  to  2  =  i.e.,  due  Prom E q u a t i o n ( 2 - 2 7 a ) , 8  system,  It  (2-27b)  a> , M . ^  20.  al,'.  i n Equation  that fi)>  a first-order  et  approximation used 21  more p r e c i s e l y , Q  Shockley  constant.  frequency near  as  by  ....(2-29)  24 The v a l u e the  of  is  o>  a  1.22  commonly measured  more  accurate  our  purposes. ¥e  are  times  value. 2  the  (2-26),  y  and  Referring  to  0Q  = W-/L  frequency  short-circuit and  too  complex  csch0  = -(G/K) 0  y  2  2  = ( G / B K ) 0 coth0,  parameters  .. ,.(2-30b)  csch0  ....(2-30c)  .,..(2-30d)  Q  L  for  ,.(2-30a)  2  J  be  as  1  Equation  is  w h i c h may  admittance  (2-29)  and  (2-17a),  we  see  that  for  V Q <^  -0,5  v V  G -7^(1  E  E  G  NI  A  1 9  + ^ ) l l  A  ....(2-31)  G  ( 2 - 2 5 a ) , w i t h 6Q )> 0 , 9 (2-3l)  is  and V g ^ O . l  dominant,  volt,  and from E q u a t i o n  (2-24a),  g  For  are  y  term i n Equation  and  but  4  = - G 0  using Equation  (2-19a)  2  al,  G  A  first  '  3  1  E  the  2  2  L  Then,  '  et  approximations  y  G ^ -^4  F  2  Shockley  ( /Po^ ^ coth0  =  volt, I  2  (2-27e),  l l  <0Q o f  Other  available "^ '  c a n now w r i t e  from Equations  the  n  = G 0 O coth0 Q = G  <[0.3,  which  is  easily  junction transistor,  g  l l  =  G / | 3  0  l  i  e  A  v  A  I  E  ^  attained  0QCoth0Q =  =  E  -«ee  A  I  E  .  ....(2-32)  i n a good a  n  d  high-  therefore  °* ' 0  ( " 2  3  3  a  )  25 and,  from Equation (2-24d),  g  Then,  from the  fact  22  =  / 0  Q  8  that  =  K  V  A  =  K  g  cc*  ....(2-33d)  = C*Q "= 8Q,  g  and  21  g  T h e n we may r e w r i t e  1  2  = "  = -  a  O  oc  g  g  Q  Equations  &  c o t h  c  ....(2-33b)  c  ....(2-33c)  #  (2-30)  as  follows:  0  y  l l  =  y  21  =  ~  a  0  g  ee  ^  c s c n  0  .,.,(2-34b)  y  12  =  "  a  0  g  cc  &  c s c h  0  . ...(2-34c)  y  22  =  g  g  ee  ee  cc  ^  c o t h !  ....(2-34a)  ^  (2-34d)  l where  0 = (j2.43co/a> ) a  = a-cutoff a  fbO g  =  a  Q  = A l  e e  IE and  These  g  c  c  2  ,  frequency  low-frequency factor,  =  for  the  d i f f u s i o n model,  common-base  current amplification  = q.Ig/kT = t h e l o w - f r e q u e n c y e m i t t e r a d m i t t a n c e , as g i v e n b y S h o c k l e y e t a l ,  E  10 /26 3  mhos a t . . 3 0 0 °  K,  = the low-frequency c o l l e c t o r p r a c t i c a l v a l u e s of K ) .  then are  one-dimensional  the  admittance  (<C!g  s h o r t - c i r c u i t admittance parameters  d i f f u s i o n model of  the  for e  e  for  junction transistor,  the valid  26 under V  c  <  2.4  the  assumptions  -0.5  The  ^  W / L  The as  racy  for  (XQ^>0«9,  V g ^ O . l  volt,  and  volt.  Johnson-Giacoletto  have  0.3,  transistor  few  frequency  equivalent  elements,  linear  of  video-frequency be  developed  with  corresponding or  mum d e p e n d e n c e measurement  so  that  are  the  also  which is  i n the  various  i n the  important  that  case  is  the cir-  associated  some  physical  phenomena,  operating  i n equivalent  the  equivalent,  have  point  circuit  or accu-  over  elements  transistor  on the  simple,  valid  the  transistor  as  sufficient  present  circuit  particular  elements  retain  s h o u l d be  desirable  the  regions  It  Circuit  s h o u l d be  and y e t  design.  It  represent of  circuit  possible  interest,  range.  cuit  significance  as  amplifier  range  Hybrid—TC E q u i v a l e n t  mini-  and ease  of  considerat-  ions. The into  two  analysis  sections,  one-dimensional the in  modified Section  resenting arrived sistor  at  the  first  theory  These  the  of  equivalent  which deals  effect  of  sistor  are  theory. extrinsic ite  have  added  The  been  for  the  elements  transistor.  gives  is  published.  of  '  defined  y.-parameters  were  by given  parameters,  rep-  first  intrinsic  Johnson"^ but  To t h i s not  intrinsic  a complete  broken  tran-  similar  18  elements the  by  is  idealized  well  one-dimensional  configuration  "extrinsic"  combination  which  an  minority-carrier diffusion,  common-base  since  with  admittance  IT analyses  circuit  and c o r r e s p o n d i n g  short-circuit  theoretically  i n the  to  "intrinsic" transistor,  Shockley  2.3. the  leading  "intrinsic"  covered  by  transistor  representation  of  the with the  tran-  Shockley the compos-  27 Practically, circuit  takes  mittance  minal  parameters  in  added  of to  are  are  are  the the  = the  2(a).  shown i n F i g u r e lead  (b'  junction  space-charge-layer)  transition itance,  both  tion  charge  across ances  the are  neglected  so  which are  the  equivalent  admittance  The  the  are  g^ = t h e  elements  difference  configuare:  r^^  of  the  and the  ac-  collector-junction /  transition  (depletion-layer  a n d C, = C = "tc c the  w i t h the  not  ter-  terminal  resistance  commonly c a l l e d  collector  collector  capac-  equilibrium condi-  potential series  measurable  set  lead  and hence  up resistare  circuit.  parameters  for  configuration  admittance  found.  connection  and e l e c t r o s t a t i c  they  the  range.  E m i t t e r and c o l l e c t o r  small that  common-emitter  for  ad-  these  common-emitter  to  base  associated  separation  i n the  common-base  metallic  (more  junctions.  The the  of  due  capacitance,  capacitance  are  The . e x t r i n s i c . e l e m e n t s  A = emitter  C,  and from  Extrinsic circuit account  = internal base);  conductance;  of  and compared w i t h the  i n the  resistance the  short-circuit  d i f f u s i o n model  frequency  is  tive  of  the  equivalent  and compared w i t h the  model.  circuit  between  or  then measured  over  four  the  theoretically  i n t r i n s i c model to  base  The  c i r c u i t elements  theoretical  semiconductor  leakage  obtained  Such an e q u i v a l e n t  series  d e f i n i t i o n of  then measured  t e r m i n a l parameters  ration  of  a one-dimensional  equivalent  parameters  parameters are  of  transistor  corresponding  terminal  method  following pattern.  parameters  intrinsic the  the  the  the  c a n be  parameters.  The  intrinsic transistor easily  found from  transformation  the  equations  are y  lle  =  y  l l  +  y  12  +  y  21  +  y  22  in  .-..(2-35a)  28  y  and  which  are  matrix  12e  = "  ( y  12  +  y  22  }  21e  = "  ( y  21  +  y  22  }  y  related  to  2  2  e  the  =  y  2  ....(2—35b)  ...(2-35c)  ...o(2-35d)  2  terminal voltages  and c u r r e n t s  l l e  y  12e  b e 1  ...  X  21e  y  Figure circuit  of  the  istor,  2(b)  the  y  shows  22e  the  transistor.  g  }  » ^^ >  n  e  a n <  composite  common-emitter  The y - p a r a m e t e r s  i n the  g  !  ^  c  are  of  the  the added  l l  y  12  y  21  y  22  'te  c  ° t c  +  i  b  Iv!  be  be C  (a) Figure  The  as  the  2.  transiselements,  C c  y  l l e  12e  y  21e  y  •"•c  22e  x  c  V. ce  te (b)  C o m p o s i t e t r a n s i s t o r : (a) common-base c o n f i g u r a t i o n , (b) common-emitter c o n f i g u r a t i o n  admittance  sum o f  repre-  AAA y  bb  box  extrinsic  e  y  equivalent  intrinsic  C. 1 tc  o-  (2-36a)  c iVe  v  m i n o r i t y - c a r r i e r admittances  and r ^  c  the  equation  y  sent  by  matrix i n Equation  a symmetrical matrix,  (2-36a)  representing  may b e the  written  passive  29 elements, trolled  and a n o n - s y m m e t r i c a l m a t r i x ,  representing  the  source,  l l e  y  y  0  12e  0  b 'e (2-36b)  + y  This  12e  then leads  y  to  22e  the  Lumped—element ted  section  priate  sums  common-base power  of  Figure  ( y  21e  ~  ^-equivalent  y  12e  of  o b t a i n e d by  the  elements  Equations  phase  slope  introduced  by u s i n g  parameters  of  correct is the  Equations  for  correct  the at  (2-26)  exact so  that  squared-magnitude dc.  approximate  Excessive  form of  the  Using Equations that,  half-  function  common-base  22e +y- L2e  (  |21e"  y  I I  tr Figure  the  error would  I  o-  appro-  form of the  dot-  be y-  (2-34).  y  e  3.  i n the  approximating the  y-parameters is  , c 'e  shown i n F i g u r e  obtained from the  of  V  0  circuit  representations 3 are  }  and d i f f e r e n c e s  frequency  and the  con-  12e  ) V  be  V  ce —o e  3.  Generalized TX-equivalent extrinsic elements  (2-35),  from Equation  (2-26),  (2-2l),  (2-33a),  circuit  (2-33d),  with  and the  fact  30  K-l * J -  the  element  values  0 0  of Figure  „  l l _ * J _ J^L<^!  3 may b e w r i t t e n  0 l l e  y  "  y  +  12e  y  22e  y  21e  +  12e  y  -  = 12 y  y  l l  y  +  y  +  y  21  ~ Weesi'nitff  = Mcc—^  22  12e  = " 12 =  12e  = 12  -(cosh0  -  l)  ^ ~l}  ....(2-37a)  --.(2-37b)  0 sinh0'  cc  y  cosh  as  (2-37c)  and "  y  Using cy,  and t h a t  y  "  y  the facts tt a  21  e e  that  0' sihh0''  (2-37d)  ^$>1 n e a r  <c  = 2 . 4 3 , the magnitude  the half-power  frequen-  0'/sinh0'  of the term  c a n be a p p r o x i m a t e d b y 1  0'  slnh0  1  +  ....(2—38a)  j0.263»/tt  5 This  is  s i m i l a r to the approximation by Bruun  except  that  0> /0.263 r e p l a c e s h i s tt /0.256. E q u a t i o n (2-38a) i s than r e c t e d t o y i e l d t h e c o r r e c t p h a s e s l o p e a t dc so t h a t a  a  0' , sinh0'  e  -j0.142«./co  a  (2-38b)  + j0.263Wto a *  l  cor-  0  The  term  —r(cosh0 sinh0  -  l)  =  0 tanh(0/2)  i s then  expanded  i n a  31 series,  and n o t i n g  omitting  terms  expansion,  from Equation  (2-25b)  that  3Q = 1 - 0 ^ / 2 ,  of degree  greater  than  two i n t h e  with  0  '  ,| (cosh0'  -  l) * ( l -  a  0CQ = 8Q =  We m a y t h e n w r i t e  l l e  - y  1  +  2  y  e  12e  =  (  1  = (1 - a  0  "  O  a  ) g  c  ) g  c  +  22e  +  y  12e  = V c c ,  e  a  n  where at  d  y  21e " <j±e  the l a s t  analysis  circuit, cuit  a  O ee u ee  5  2 1 5 g  -j0.142«/«  +  j  0  <  2  ee<°  ./«  c c  .263«/.  j 0  g  1  *  1  jl.21 g  two a p p r o x i m a t i o n s  equivalent is  shown  / f i )  cx  ••••(2-40a)  ....(2-40b)  a  a  *  a  ^  ; -  (  2  -  4  0  ^  )  a  T~ 63a/a  "  a  O ee  a  ••••(2-40d)  g  U  e  e  are permissible  c i r c u i t which results  called  for  operation  with  from t h e above  the Johnson-Giacoletto  i n Figure  4.  The e l e m e n t s  are simply the i n t r i n s i c elements  equations, The  12e " l^e  I  +  as  w^tt . The  of  y  ee  (2—37)  -j0.142«/a  +  (2-39)  a  1.  Equations  e  y  ) + jl.215a/a>  u  the s u b s t i t u t i o n  y  series  we may w r i t e  smh0 with  and  r  the added  i n t r i n s i c elements  extrinsic  hybrid-n;  equivalent  of the equivalent  derived  elements  from the  described  are  g^,  e  = emitter  diffusion  conductance  = ( l - o^Jg  C^,  e  = emitter  diffusion  capacitance  = 1.215g  type  /tt  cir-  diffusion previously.  32 g  b'c b 'c  g  and It  lent  diffusion  conductance  = (l  collector  diffusion  capacitance  =.1.215g  = i n t r i n s i c transconductance  m  g °ce  is  collector  = collector-emitter  important circuit b  Y  *>  +  to  are  note  that  independent  -  «o)g  cc  /«  = oc„g O ee e  diffusion  conductance  all  elements  of  of  frequency.  the  = oc-.g 0°cc  above  equiva-  b  v  v  +  ce  Figure  4.  Johnson-Giacoletto hybrid-ic equivalent c i r c u i t  Bruun  has  s i m p l i f i e d the  cation  to  video  amplifiers,  tances  of  g^, ,  g^,  to  that  c  of  and C ^ ,  as  C , which accounts  lateral  nature  ration.  The  of  the  equivalent  is  are  c  for  g ce  is  circuit,  shown i n F i g u r e assumed  transistor  conductance  common-emitter  the  i n the  5.  negligible  major  for The with  p o r t i o n of  common-emitter  omitted because  it  appliadmitrespect  the  b i -  configu-  shunts  the  6  small  load  amplifier •C^  ,  is  resistance  used  considered  assumed  capacitance.  here.  negligible  Equations ylie  12e  y  i n the The  optimum d e s i g n  emitter  with respect  (2-40)  transition to  may t h e n b e  = g_A* 'ee  +  of  the  video  capacitance,  emitter  expressed  jl.215«/« a  the  diffusion  as ....(2-41a)  33 -  22e  y  and  where  it  is  y  of  the the  a  y  ~  12e g  a  O  g  which l i m i t  Figure  the  Q  diffusion  high-frequency as  to  process  o b t a i n an  performance  a guide  to  the  be  6  ee  persion but  Simplified equivalent  fundamental  in transit  finite,  base  time  quency,  oc^ b e g i n s  to  an a—cutoff  have  of  layer. to  of  a^g V, ' 0 e e be  1.215g  The m o s t  the  c -o  be  5.  of  transistor.  +  Figure  idea  choice  tt  +  (2-41d)  a ).  5 i n order  T h i s may s e r v e  high-frequency  = l / ( l -  q u a l i t a t i v e l y the of  oa o  ..  ee  0 and a '  cc  (2-41c)  o  0  =  circuit  transistor.  suitable  12e  y  "  ....(2-41b)  consider  equivalent  junction  = 0  that  us  effects  +  21e  noted  Now l e t and  v 12e  y  V ee  ce  oc  hybrid-TC c o m m o n - e m i t t e r circuit  cause the  of  cutoff  minority carriers  T h i s means  decrease  frequency  beyond  across  the the  a certain  disthin,  fre-  that  the  t r a n s i s t o r may b e  oa^c  It  has  been  frequency  is  inversely proportional  frequency,  so  that  is  shown b y  said  Shockley  15 et the  al  that  square  of  the the  a-cutoff base  w i d t h and hence  increases  r a p i d l y as  to  the  34 base  is  effect C  made  thinner.  of  is  I n the  represented  by  the  emitter  circuit,  diffusion  the  capacitance,  b 'e ' The  other  capacitance  transition  capacitance.  across  p - n j u n c t i o n due  any  up a c r o s s region ance  the  of  charge. "where  into  The  the  density  and donor  atoms  are  i n the  and n - t y p e  the  electric  to widen or  increase  of  the  exists  the  C^ , e  much s m a l l e r  of  is  but  into  carriers by mobile  across  layer,  is  serve  the  and the than the  low.  Accepas  to  terminate tends  decrease  the  emitter emitter  diffusion  collector  they  capacitance.  than the  collector  unbal-  "depletion-  in a  transition  much g r e a t e r  n-  junction  resulting  set  the  charge  junction potentials, is  capacitance  a thin  instead  or  a  junction  potential  holes  across  neutralized  g  there  a  with a resultant  mobile  regions,  is  £  capaci-  transition  ca-  b c ' 1  pacitance,  C.^.  and e m i t t e r  The  effective  i n Figure  diffusion  admittance,  tional  C.j.  to  circuit  c  = C ,  resistance giving 1  5 is y^,  to  c  then behaves  > c, , , bb' b e'  i~Tl  p-region  , ,  ,  tance,  the  applied  tance,  lead  diffusion  of  C ,  electrostatic  depletion-layer  capacitance,  '  the  depletion  transition  is  circuit,  the  not  capacitance,  C, ,  circuit,  An a p p l i e d b i a s the  i n the  Because diffusion  are  field.  or narrow  to  potential  tor  p-  i n the  On o p e n  j u n c t i o n by  and e l e c t r o n s  layer,"  r  simplified equivalent  and C c  to all  composed  of  the  discussed  later.  like  a lowpass  filter  effective  an e m i t t e r lead  between  internal  sum o f  the  to  The  i n t e r n a l base cutoff  effect.  an i n c r e a s e  emitter  composed  i n the  to  of  base  emitter  , and a M i l l e r - a d m i t t a n c e term  be  and the  rise  admittance  proporinput  the  emitter  baseadmit-  A reduction emitter  of  cutoff  frequency.  2.5  The D r i f t It  has  transistor short the  Transistor been  depends  transit—time  collector,  resistance. the  (b)  junction  high-frequency  fulfillment  of  injected  the  to  collector the  demands  an upper  the  on the  low  Prior  conflicting  placed cy  shown t h a t  capacitance,  and  of  the  these  on t h e  of  a  requirements:  from the  development  limit  three  carriers  imposed by  frequency  of  operation  three  emitter  (c)  drift  low  to  base  transistor,  requirements  operation  (a)  of  had  high-frequen-  transistors. 25  Kroemer in  the  base  to  move  realized  c o u l d be  reduced  i n an e l e c t r i c  a comparatively  that  slow  field  process  the  minority-carrier  considerably rather  t h a n by  because  of  emitter very  low v a l u e  the  base  and d e c r e a s e d  constant in  i n the  at  electric  base The  drift the  field  of  a p-n-p  transistor  net  N(x)  = 0 is  field  that  carriers  were  which  random n a t u r e . control  the  The  impurity  it  was  very  high  through  the  base  region  This  parallel  is  at  the to  a  d i s t r i b u t i o n introduces  a  to  the  diffusion  direction  region.  to  x  collector.  drift  related  N(x)  so  exponentially  the  F,  where  region  to  the  diffusion,  its  new p r i n c i p l e i n t r o d u c e d b y K r o e m e r w a s concentration  if  transit-time  — net  the  is  by  the  drift  impurity concentration  field  i n the  intensity, base  region  by  = N(0)exp(-AFx)  impurity  emitter  defined  side  (electron) of  the  base  (2-42) concentration region,  and  i n the A ^ -  base,  = kT/q =  36 thermal  potential. The  fined  (normalized)  N(¥) Inj^y  A  where  F = drift  and used  free for  to  ¥ = base  compare  transistor. the  drift  the  which  y,  . = AF¥  field  is  de-  .....(2-43)  intensity  width,  drift  transistor  The p a r a m e t e r  field^free  transistor,  to  the  generally and  ip = 8 ,  (electric)  lies  for  a  field-  between  = 0,  "maximum-field"  transistor. The  gion  parameter,  by  ^ -  is  base-field  drift  field  and i n c r e a s e s  reduces  the  the  oc-cutof f  transit-time  frequency  of  i n the  the  base  re-  transistor.  a g i v e n base w i d t h i n a germanium t r a n s i s t o r ,  an improvement  a factor  high  sity  of  near  eight  the  emitter  the  low  tor  capacitance  tion  c a n be  attained.  leads  impurity density  layer.  performance  produces  because  limit  tion  transistor  the  is,  advantage  sistor,  it  is  the  of  quite the  necessary  An e q u i v a l e n t  collector  due  to  a further  base  high-frequency  To t a k e  mon-emitter  near  the  a low base-lead  and conductance  This  also  to  Also,  the  leads wider  operation.  This  naturally,  called  modify  circuit  configuration,  for  the the  Figure  6,  the  drift  the  collec-  of  the  deple-  transiscapacitance  drift  of  junc-  transistor. drift  tran-  circuit.  transistor  been  den-  collector of  by  and  low  improved type  equivalent  has  to  and c o l l e c t o r  improved c a p a b i l i t i e s to  resistance,  improvement  resistance  impurity  For  i n the  developed  com-  by  26 te¥inkel,  i n the  same  f o r m as  the  Johnson—Giacoletto  hybrid-n  37 circuit  commonly u s e d  The v a l u e s number  of  of  the  for  alloy-junction  circuit  elements  multiplying factors  that  diffusion  transistors  c a n be  f o u n d b y means o f  depend  on the  drift  a  field  only. b  b  i  b  r  cc  S-AA/V 'be  'te  be  be  o-  e  Figure  Using (2-38), fusion  6.  Common-emitter e q u i v a l e n t f o r the d r i f t transistor  a method  identical  an a p p r o x i m a t i o n transistor  c a n be  a fb  The  numerical  excess drift  phase  value (at  transistor  of  2  written „  a  0  ^ '  2  that  as  -j0.215«/« +  in deriving + j<o"Jp)  l 2  Equation  for  a  dif-  27  e  I  used  = sech0Q(l  ja/tt  a  (2-27f)  a  in this  A similar  a  s  oc^  0.215  w )« i  for  to  circuit  expression  expression  for  is  called  alpha  of  the the  ^ ^-jOco/c^  a fb  where ljf,  of  0 = the oc  f b  excess  ^  a  0  1  phase  e  (2-27g) +  at  ja/a  a^,  a  related  to  the  phase  angle,  by 0 = -  \fj -  TC/4.  (2-44)  38 For  the  field-free  transistor,  transistor,  0 . 2 1 5 ^ 6 ^ 1  6 = 0.215  radian,  radian;  depending  for  on t h e  the  drift  drift-field  parameter. In the istic  of  oc^  mon-emitter  common-emitter is  extremely  small-signal  the of  important current  oc„  By  configuration,  =  may b e  seen  a  f  (2-27g)  exponential  t e r m c a n be  approximated by the  where  one—half  of  at  its  a  f  e  + ja> (1  squared  low-frequency  frequency  is  f  that,  for  first  two  a<^fi> , a  terms  ~  29 parameter, a ,  this given  a  0>  Q  magnitude is  given  n) 0  at  T  a  Q  "  l  of  oc^  +  is  down  to  by  • 1  point by  ....(2-45b)  a e)  +  a ).  value  e  convenient  (  Q  which the  a  a  = oc /(l -  Q  <°,V-P~ = ( !  It  and n o t i n g  expansion,  a  frequency  com-  ....(2-45a)  i n Equation  series  character-  from the  .  b  substituting  its  phase  ratio  —  1  The  as  the  ....(2-46)  a^9 to  introduce  a new  cutoff  39 which  c a n be  suitable effects  considered  for of  to  common-emitter  excess is  most u s e f u l  frequency  at  which the  «/w  ^>(l  a  -  an e f f e c t i v e  representation,  a )  to  note  that  magnitude  fe  <o(l  0  the  compensated  is  is  0<°a +  a a> at  also  frequency for  the  approximately  the  equal  to  unity,  i.e.,  then  Q  a  is  this  of  a  This  alpha-cutoff  phase.  It  for  be  a =  frequency  at  1  +  =  a 9)  1  0  a  =  «„. T  aQ Q  which the  real  part  of  oc^  is  equal  30 to  one-half The  ter  a  i n the  low-frequency parameter,  value. co^,  because  it  is  applicable  to  equivalent  expressions  requires phase,  its  frequency  t h a n (o  directly «  of  to  be  for  much more  the  specified  w h e r e a s tt^ a l o n e  is  is  a more  easily  measured  c i r c u i t work;  elements  to  attractive  include  of  the  the  parame-  and i s  i.e.,  the  equivalent  effects  of  more use  of  circuit  excess  sufficient.  26 teWinkel the (for  excess-phase  7^9),  has  given  parameter,  an e m p i r i c a l e x p r e s s i o n w h i c h 9,  to  the  base-field  relates  parameter,  as 9 = 0.221  A second  expression  effective  a-cutoff  relates  + 0.098 ^ the  frequency,  radian.  a-cutoff  a^,,  by  frequency,  (2-48) (o^,  to  the  40 (0 = ^  P  =  Equation 2.43  used  be  transistor,  + a„9  (2-49)  i n the  transistor  1  *  1.21  suggests  d e r i v a t i o n of  r e p l a c e d by  o r more  w a  + 0 . 0 9 7.  then that  the  r e l a t i o n (6^  E q u a t i o n (2-27e)  Tj)  2.43  =  + 0.18  a>^^C^  s i m p l y by  o..a(2-49)  for  the  diffusion  for  the  drift  2 . . The { . . l a t t e r  =F  =  expression  31 was  used by  authors  of  Stephenson the  for  preceding  the  four  diffusion transistor  references  for  the  and by  drift  the  transis-  tor .  Equations  l l e  y  "  y  +  (2—40)  12e  y  12e  =  (  =  ~  1  then  (  1  a  0  )  "  a  0  g  cc  become  ee  ) g  5s «/»  +  ee  J cc  +  g  a / a )  T  ....(2-50b)  -j0.117«/a y  22e  +  y *21e 0 1  Using of lent  *12e = O « c c i V j b T 2 p £ 7 « ^ " ^ c c  -  y, 12e  = oc„g — 0 e  0  J  e  an e x a c t  \j , t e W i n k e l h a s  where  J  g  e e  C, . b'e  e  i  -j0.117a/a ;—• = a g jp.216a/«  .  /a'  for  for  e  e  ^  T  y^ ,  g  e  =  + y^2e  expressed  = g  e e  /a'  . . . . (2-50d)  e  low frequencies  may b e  = g /«m. ee T b  0  T  y^,  + J g  .  A  +  shown t h a t of  ""(2-50c)  T  expression  c i r c u i t elements  'b'e  T  a  e  and  ....(2-50a)  T  the  """  n  ^  e  r  m  equiva-  as  + j«C  b t e  ....(2-51)  ....(2-52)  s  41 By p l o t t i n g y^, has  found  i n the  g  that, that  on t h e  axis.  good to  pass  through  This  of  a series  is  accurate  factor  P accounts  to  parameter  the  to  J  ^  P/g  the  i n the  frequencies. 7.  part The  is  bination  of  a'/g  assumed  higher  their  b t e  centres  the  a  impedance  i.e.,  ,  a/a^ - ^ 1 .  field  on  frequencies  considering  +l/3-C  for  <  to  The  and i s  multiplying  related  graphically  0.51  a n d C, ,  be  determined  circuit  may b e  'b'e  ,  for  by  y^,  is  s i m p l i f i e d to  just  as  i n the  the  , shown  at  lower  in  Figure  parallel  com-  Johnson-Giacoletto  D-.e  ee hybrid-u  by  at  nearly  0 < Y ^ < 9 .  equivalent  F o r «/«_, ^ 0 . 2 ,  very  and c a p a c i t a n c e ,  ^ P  for imaginary  lie  te¥inkel  range  0.167  The  variable,  curves  that  e e  drift  with  o r i g i n and have  obtained  w i t h i n 1$  for ;  the  resistance  b'e  which  the  suggests  a p p r o x i m a t i o n c a n be  consist  plane  u p t o tt/to^, = 2 ,  semicircles real  complex  equivalent  circuit.  It  has  been  shown t h a t  g  this  change  g  e  de-  32 creases  slightly  with  increasing  but  1  neglected.  ybe  Figure  =  7.  y i l e  T he  +  y  a'/g  B a  for  y^,  'ee  12(  equivalent  small  circuit  >P/g  ee  is  42 The te¥inkel  expression  for  y  ~  v  a  N is  related  m = !  +  08ee j W  12e  &  ^  g i  s 0  v  e  by  n  graphically  ....(2-53)  V  to  Vp i n t h e  0.333 for This  y  as  y  where  = 21e  m  representation  for  is  m  a n d w i t h i n Vfo i n m a g n i t u d e w/fi>  crude,  but  is  The  commonly u s e d  T  = 0.5.  The  used  for  ^0.87  0 ^V? ^ 9 . y  6% f o r  range  w i t h i n ifo i n p h a s e  for  a/tt^  0.25,  approximation y the  sake  of  for  w i t h an e r r o r  % a g  is  n  overall  tt/coVp  0.5,  of  again  only  quite  simplicity.  lumped r e p r e s e n t a t i o n  of  the  distributed  33 base  resistance  ment,  to  be  practical  been  shown,  a reasonable  applications  Because emitter  has  of  the  diffusion  approximation; of  the  drift  increase  capacitance  is  a diffusion  junction  t r a n s i t i o n capacitance,  the  ter,  is  diffusion not  conductance  is  for  the  even  fusion  capacitance  is  collector  pacitance  to  drift  transistor,  the  due  at  also  high  the  which  the  drift  that is  for  most  larger  The  transistor  for is  the  drift  here  the  at  emitter than the  that emit-  collector  than for  The  the  transistor  the  high doping  frequencies.  smaller  measure-  frequency,  drift  transistors.  t r a n s i t i o n capacitance  anyway.  adequate  cutoff  T h i s means e  for  fusion  for  C^ ,  negligible smaller  is  of  transistor.  less  transistor.  transistor  it  methods  in effective  than for  of  by v a r i o u s  the  dif-  collector  dif-  transistor,  but  dominating  ca-  43 It  is  seen  hybridan  equivalent  with  a d d i t i o n of  the  f o r a/a^, < ^ 1 ,  then that  circuit is  valid  C, , t h e ty e •  for  the the  Johnson-Giacoletto drift  transistor  emitter—junction transition  32  34  '  capaci-  tance. The accounted  effect for  of  the  collector  lead—resistance,  i n a manner g i v e n by G r i n i c h ,  but  it  r is  '.may cc "  be  usually  neglected. O t h e r more e l a b o r a t e e q u i v a l e n t c i r c u i t s h a v e b e e n p r o 35 posed, but they are u n n e c e s s a r i l y complex f o r our p u r p o s e s .  2.6  The M i l l e r  Capacitance  The m a i n e f f e c t analogous been the  that  of the 5  shown b y B r u u n . collector—base  mately by  to  to  of  1  Prom F i g u r e  tion  is  it  is  =  -  ( y  21e  effect  reference  base-emitter  to  8,  showing the  seen  "  y  that  12e  O ee L g  the  s  intrinsic C  the  ) R  equivalent  low-frequency  , leads  +  L  r  has  transistor, approxi-  '^fjiller*  & i  bb  +  a  ' / g ee  v  e  n  . . (2-54a)  c i r c u i t of voltage  as  an  ampli-  amplifica-  a V g ee F~ + ' r , \ ! + a ' / g bb ee  R  R  i n a manner  i n vacuum t u b e s ,  capacitance,  7 g ee l  36  for  low-frequency base-collector voltage amplification  +  V, v~  accounted  t r a n s i t i o n capacitance,  Stiller  stage,  c a n be  c  Miller  With  an e q u i v a l e n t  fier  C  ....(2-55a)  \  44 and f o r  a'/g  S  6 6  + r,^ this  becomes  0 0  v ^  -  "  ( y  21e  -  y  12e  L  ) R  "  =  a  .(2-55b)  O ee I/ g  R  (y i--V e be ) V  9  1 2  *L  Figure  8.  Single-stage  The e f f e c t ternal  base  the voltage  ty,  C  can therefore  c  and e m i t t e r  C  If  of  the M i l l e r  Miller  amplifier equivalent  =  C  c  (  a  +  capacitance  e e  ~  a  ^L»  parallel  with  the  cuit  a better  this  capacitance  is  g  R  in-  the  (2-54c)  effect  plus  uni-  by  t h e n m o d i f i e d as  current generator  c a n be  than  C  approximately equal  approximation, but  much g r e a t e r  O ee L c*  and t r a n s i t i o n c a p a c i t a n c e s , A capacitance  between  (2-54b)  represented  represents  e  represented  g  may b e  circuit is  ^ ff>  circuit  O eeV  a  The c a p a c i t a n c e , fusion  l  a m p l i f i c a t i o n , o&  equivalent  o  by  "Miller The  be  V  of  the to  i n Figure  the  Miller C  c  t o make t h e  9.  emitter  dif-  capacitance.  s h o u l d be  added  equivalent  i n a p p l i c a t i o n s where  ciris  low  neglected.  The m o d i f i e d J o h n s o n - G i a c o l e t t o h y b r i d - i t e q u i v a l e n t  in  cir-  45 cuit are  fits well  transistor described  alloy-type  drift  transistors  are  transistors  r  -  with plane-parallel  i n one-dimensional  well  •u o  types  A  x/ /  well  as  by  this  represented  bb A  as  V  -  form.  [rile*  y  12e  equivalent  ^  +  2.7  9.  As was the is  choice the  of  ( y  cuit  with  Circuit for  Feedback  mentioned circuit  transistor  The  t e  C  diffusion  circuit.  M ] }  i n the  Emitter  earlier,  to  be  the  used.  common-emitter feedback  feedback,  Z , g  is  b'e  Resistance-  of  circuit  configuration in  ) V  Lead  method  Th e  yi2e  circuit  Parallel  series  simplified Johnson-Giacoletto emitter  +  21e-  S i m p l i f i e d hybrid-TC e q u i v a l e n t modified for CL,.,, Miller  resistance-capacitance lead.  C  and  B  Modified Equivalent Capacitance  that  Surface-barrier  alloy-junction  be  Figure  junctions  shown  design to  be  with  w i t h the  depends used  on  here  parallel common  emitter  hybrid-u equivalent  cir-  i n Figure  cir-  10.  This  5 cuit  will  be  analyzed  From F i g u r e  10 H  using  the  it  seen  is  method  = Ve^lle  of  Bruun.  that + y  12e  +  ^  C  ± e  )  ..-.(2-56a)  46 X  so  2 = V e ^ l e  that V, , b'g  = V, , b'e  =  e^lle (  = V e t  and  I  =  1  1  bb  +  ,  b  +  ( y  e  +  2  y  1  y2ie Z  e  (  -ll  y  +  e  Z  e  (  1  l l e  + Z (y  = V  2  y  lle  ( y  I,  i i j z  2' i  q  I 1  or  i (I + •1  +  (  l  -+ y  e  -  e  y  -  y i  l l e  +  +  ^  +  1  2  2  e  2  1  e  )  )  v  t e  C  }  +  e  b '  +  }  >  C  te ]  V  (2-58a)  j«C .J te +  . (2-58b)  g  J" te>''  +  C  (yno+  y i  c  -o 2  e  Z  l_l  Figure  T h e n we  can  10.  ••••(2-57)  }  ft  b  + +  a C  21e  t e  C  J te'  +  ^  y  2 1 e  y  21e  +  lle  12e l  J  12e  y  y  .(2-56b)  ^12^  "  2=<y2ier -  a  y  12e>Ve  O ee be g  V  S i m p l i f i e d Johnson-Giacoletto hybrid-Tt equivalent c i r c u i t with emitter feedback  define i l T  b'g  (  1  +  y  l l e  +  y  12e  V^lle  +  +  y  J'  21e  a C  +  te  )  ^ t e *  . . . . (2-59a)  47 and  The  y  2  1  V g  '  -  ^21e1  +  circuit  Z  e  (  l l e  y  c a n be  y  +  y  12e  )  21e  ...(2-59b) ^ t e  +  5  redrawn i n terms  of  the  parameters  i  l l e  a  n  d  y  21e  It back the  _  equivalent  i y  2  1  =  e  is  a  s  s  seen  impedance, transistor  21e  b°'  i  w n  that  Z , g  Figure  n  the  is  to  11.  effect modify  equivalent  of  the  the  c i r c u i t by  external  emitter  feed-  admittance  parameters  of  the  | l + Z (y-^  e  factor  e  +  + ^ te>]' C  L  bb  o-wv  + 21e bg  bg  Figure  11.  E q u i v a l e n t c i r c u i t i n terms y  Further Miller-effect Equation  v  lie  a n d  y 2  ie  s i m p l i f i c a t i o n , c a n be admittance  (2-54a),  term,  w i t h the  of  achieved  g i v e n by  by  lumping  the  a relation similar  modified emitter  d i f f u s i o n and  to tran-  i  sition  admittance The  stage  . p -•y le"L 2  which  replaces  term,  V  voltage  +  Z  Equation  e  amplification is ( Y 2o1l eo ~  = i  -Q «  e  ( y  l  l  e  +  1 2y^oJ e " L^ y  now  (for  coR^C<<C[l) c  R  J  / X  2  1  e  +  (2-60)  jrt^)  (2-55b).  The M i l l e r - e f f e c t a d m i t t a n c e  may t h e n b e  written  as  48  lie  y  = ^  C  ( y  1 +  c  1  w h i c h may be s i m p l i f i e d , to  "  y  12e  + Z (y, e lle -i  J  +  ) R  L  ..(2-61a)  21e  y  ^°-te\  +  u s i n g t h e same argument  as t h a t  this  case  plification,  lle  ^ - !  J^(y le z (y  -  2  +  e  l  l  y  e  +  12e y 2  1  c L * J-C  ) C  R  e  t o  (2-61b)  )  the approximation i s rather  crude  but i s used  simplicity.  f o r the sake  of  f o r l o w s t a g e am-  C o m b i n i n g t h e m o d i f i e d sum o f t h e e m i t t e r transition  admittances  y y  l l e .  +  In  y  lle  w i t h the M i l l e r - e f f e c t  +  lle  12e +  +  Z  ^  +  e  (  y  te  t o C  ile  (y-Q  e  +  y  + y  with 0  21e ^  +  (2-62a) *  n  ^  C  d i f f u s i o n and  admittance '  t e  y  12e  e  s  a  L c C  )  ....(2-62a)  we m u s t n  ) R  gives  m  e  o b t a i n an a p -  manner  as  used  (2-37),  y  21e  t e r m 0 coth0  terms  t t ( y  21e  21e^  =  y  l l  +  y  12  = S g o  = P  The  J  +  to expand E q u a t i o n  to the term  Equations  lie  y  1  order  proximation for  leading  Equation (2-54c), to  y  In  21e  is  e e  g o  0 ' ( c o t h 0 ' - |csch0')  ee  0  whe r e K~  <3C[ 1.  expanded  i n a power  of degree  greater  c o t h  .(2-37e)  0  series,  t h a n t w o , we  have  and o m i t t i n g  49 •rile  +  y  21e  g  -  -  where  use  is  made  of  e  + J O . S W ^ )  d  e  g (l 'ee Q  0Q <^0-  a  n  + ;j0.667to/to )  O  a  a  (2-50e)  rp  o  ~  Q  8  ~  ¥ e may now e x p a n d E q u a t i o n ( 2 ^ 6 2 a ) ,  using Equations  (2-50),  as  n y ln i e* +^ l y  1  j i a ' [l •+ < A ( C / g  +  T  T  le  ^'/See^  1  +  Z  1 (a'/g  ee  )[l L  e  (  t e  l l e  y  W c  + e e  +  y  21e  !  1  J^te*]  +  + . jft/ttj,  + . Z (y e lie  + y  + jtoC  21e  )1 te J (2-62b)  i  where  l/«  = ^ [ l  r  and  Here,  the  frequency c a n be  T  1  A  frequency of  seen  effective  H  + <> (C  the by  eff  T  t  e  + a  e e  /g  e  e  R C )] L  -  G  a'HA>  ....(2-63)  T  + oc R C ). L  0  p a r a m e t e r to^ i s  transistor  ( )  the  .,..(2-64)  c  effective  half-power  internal-base-emitter (2-52)  and  circuit.  (2-54c)  to  This  give  capacitance  =  =  shown i n F i g u r e  + ^ (C  /g  combining Equations  input  C  t e  b'e  C  +  C  te  +  C  M =  g  ee  / t t  T  +  C  te  +  a  0 ee L c g  R  C  *ee /*T H  9.  Combining t h i s  to r  = g /a'c ee eff b  w i t h E q u a t i o n (2-63)  = l / r , , C „„. b'e eff  gives  an  50 If  Z  i s defined  g  as the p a r a l l e l  combination of R  and C ,  g  e  then R  R 1 + jfi>R C  e  e  where  <o, 1  The d e n o m i n a t o r  1 + Z (y e  + y  n e  l/RC . e e  A  (2-66)  of Equation  (2-62)  + 3«C ) = 1 + Z [ g  2 1 e  e  t e  1 (1  i  + g R ) ee e  + jw K  g  See^  +  U  1  e  1 (1  + g  + jft)/a  and M  0.667  A  Finally,  + «  T  Equation  lle  Thus, equivalent ure  12.  +  (l + J0.667*-)  R (0.667 e  V  1  1  +  +  y  l l  e  the f i n a l  * C T  .«ee i B  t  e  jacj  +  /g  e  e  )  1  J  )  +j«/«  (2-67)  R M/cc j/(l  e e  c  e  t  e  ( l  +  T  /g  e  e  (2-62a)  g y  e e  9  = [ l / ^ + g  2  e e  may t h e n b e w r i t t e n  —r-^  R )  w h e r e l /«  (2-65)  1 + jtt/aj  e  +  e e  g  e e  (2-68)  e  (2-69) •• « •  -  takes  the form  W<* )(l + r  a ' ( l + g  R )  e  e  R )(l e  +  W^) j«A> )2  form of the common-emitter  c i r c u i t with emitter  feedback  i s that  ....(2-62c)  transistor shown i n F i g -  51  b  b  b  h  +  lie  2  J  bg  bg  = y ie bg V  2  g Figure  2.8  12.  F i n a l form of the common-emitter c i r c u i t with emitter feedback  High-Level The  jection quency  Injection  preceding  theory. for  Equation  For  small  (2-28)  development  has  a diffusion  transistor,  emitter i n the w  By  extending  emitter almost  the  currents  = 2.43D  a  should a  which for is  the  actual  increases  with  high-frequency an advantage Such  is  (o  a  a-cutoff  obtained by  in-*  fre-  developing  P  is  (2-70)  high-level  shown t h a t  I ) E  less  this  (large  frequency  = 4 . 8 6 D /V .  than t h i s emitter  by J  is  for  the  (2-71)  a correction  current.  operation with diffusion  case  injection  give  increasing  the  the  on l o w - l e v e l  /W .  i n using high-level not  based  2  has  <o (high  However, '  c a n be  to include 38  Rittner  which  been  form  theory  currents), doubled,  equivalent  factor  Nevertheless,  transistors,  there  injection. drift  transistor.  It  has  52 been  pointed  drift  out  by  transistor  Kroemer  tends  to  that  at  operate  high  as  levels  a diffusion  of  injection,  the  transistor.  The  32 minority-carrier to  increase  level, rocal  transit  quite  rapidly  and t h e r e f o r e relation  higher  to  current  creasing ed f r o m  the  the  at  i n the  base  emitter  a-cutoff  transit  levels.  emitter  time  currents  frequency,  time,  increase  is  much more  has  been  above  a  quite  in transit  shown  certain  which bears  decreases  This  current  region  a  recip-  rapidly  time  with  r a p i d t h a n w o u l d be  at in-  expect-  theory.  39 It a ^  might  increases  creasing  be  to  here  that  a maximum a n d t h e n  emitter  and d r i f t  mentioned  current.  This  Webster  decreases  effect  is  has  shown  steadily  noted  with  i n both  that in-  diffusion  transistors.  The  low-frequency  emitter-base  conductance,  g  e e  >  has  been  32 shown  to  creasing rents.  vary  to The  capacitance For  a  significant  increasing  above  effects  to  quite  markedly  effects  approximately The  with  a minimum and t h e n  high-level  non-linear  amount  the  emitter  with  cause  simple  vary  the  increasing  operation,  which  like  tend  there  the  injection for  large  level,  emitter.cur-  emitter-base emitter  diffusion  current.  appear  to  be  transistor  to  behave  theory  de-  a variety not  of even  predicts.  t r a n s i t i o n capacitance  is  quite  independent  33 of  emitter  sistance, tion  r^^,  level, It  the  current,  is  but  is  and Das  very  tends  obvious  nearly to  shown t h a t  constant,  decrease  then  hybrid-TC e q u i v a l e n t  has  that  circuit  very  some are  of not  the  base-lead  independent slightly the  at  circuit  correct  for  of  re-  the  high  injec-  levels.  elements high  of  injec-  53 tion  levels.  equivalent design ter  Until  circuit  procedure  currents  currents  is  an a n a l y t i c a l for  based not  which are  equivalent  circuit  transistor  and i s  high  or  injection levels  on t h e  equivalent  considered  well  Figure suited  12 i s for  of  is  circuit  worthwhile.  w i t h i n the" r a n g e of  e m p i r i c a l r e l a t i o n to established, for  However,  low-level  large for  a emit-  emitter  injection,  a valid representation use  the  i n c i r c u i t design  of  the the  problems.  54 3.  3.1  A M P L I F I E R RESPONSE FUNCTIONS  Single-Stage The  equivalent  now e n a b l e functions for  us  to  for  cascades  circuit  Current-  for  and V o l t a g e - A m p l i f i c a t i o n  circuits  calculate  derived  current-  a common-emitter of  a  such  single  s ^>  and  stage  previous  with  emitter  The N o r t o n f o r m o f  stage  shown i n F i g u r e  is  chapter  voltage-amplification  stages.  r  ,  i n the  Functions  feedback the  and  equivalent  13.  ' bb  i  bg v ' V ?' 2 1 e b g 1  v  o  g Figure  It the  = - I  Equivalent circuit stage a m p l i f i e r  useful  single-stage  cation  G. i  is  13.  to  define  amplifier.  function  two The  -  [1+  the  single-  amplification functions short-circuit  current-amplifi-  Z  e  ( y  n  e  +  y  2  1  e  ( y  +  21e  "  yi2e  J«P. )][l+(Z t e  ) Z  S  .+ r  a  b  b  ) (  y  |  i  e  +  .... with  for  is  " s  for  a low-frequency  y ^ ) ] (3-la)  value -oc^a'R  G  .(0) 1  R^ + r  A voltage-amplification 1  s  2 s  = b  b  + a ' / g' e_e  function,  +  a*R  i n terms  .  (3-lb)  Q  of  measured  voltages,  55  V G  = V  m  with  =  -  = i  V  t  1  =  M  +  l l e  y  a low-frequency  +  y le  In  vm  amplification system. the  It  overall  individual  3.2  is  R  Ig,  R  all  are  , + a /g ee b bk  a single-stage  meet  the  stage  to  Functions for  + a R  +  y  lie)]  (3-2b) Q  a m p l i f i e r has of  insufficient  a video  cascade  (broadband)  amplifier  just  interstage  a Stage  a cascaded  resistances,  the  stages  product  of  so  that  the  in a  Cascade  emitter-feedback Rj,  and a l l  amplifier,  emitter  currents,  identical.  i  r-AAH  +E  -in(k-l)  1  "Rn  Je  el  Figure  The the  y  amplification functions.  14 s h o w s  assumed  ob( ne  r  L  requirements  then necessary  Amplification  where  =  amplification function is  Figure  V  to  +  • o••(3~•2a)  b  general,  1  C  value  Q  (0)  ^ te)] I  +  2  -a a G  ( y21e  R  (k-l)  1  14. C a s c a d e d  l o a d impedance  interstage  resistance,  T  'R-,  _lc e (  ink  >1)B  T  E  1  J c ek  emitter-feedback  of  a stage  R , T  inn  Rn  R  ek  en  1 en  amplifier  i n a cascade  i n p a r a l l e l with  consists  the  of  i n p u t imped-  ance  of the f o l l o w i n g The  script  input  stage,  impedance  "k", is  Z^^.^)*  of a t y p i c a l  seen from F i g u r e  f o r low frequencies,  impedance  Lk  =  I  B  Z  in(k+1)  (2-26c),  of a t y p i c a l  1 Z  =  B  I  1  sub-  +  +  . . . . (3-3a)  this  becomes (3-3b)  + a'R , . ek  b  load  by the  + y, ' ' ) ^lle/k  from E q u a t i o n  R- i = r ' + a ' / g ink bb ee The  denoted  13 t o b e  Z- , = r , ' + 1/(y-, ' ink bb lie and  stage,  r  stage  bb yij (  (Rj  r  +  b  b  is  yii^k-n  +  e  ) (  y  therefore  i  {  e  +  y  i  ; i  )  e  k  +  1  ....(3-4a) which  reduces  R  The stage  at low frequencies  T 1  Lk  = 'R  The stage  I R I  equivalent  i n a cascaded  ' bb +  + a'/g„„ + a R ^ { > ' + a'/g + a'R bb ee e (k+1) !  e  r  circuit amplifier,  I  1  +  Z  e yile (  +  noting  13 a p p l i e s  that  Z  g  to a  single  = R j a n d Z-^ = Z ^ . function  for a  is ~  k  of Figure  current-amplification  G l  (3-4b)  k + 1  e  short-circuit  i n a cascade  to  ^ l e  +  ( y  21e "  ^ t e f l  1  yi2e +  < I R  ) R  +  I ' b ^ l i e  +  ^li^k] ....(3-5a)  The  symbol  "  "  signifies  " i n parallel  with"  57 with  a low-frequency  value -a a'R , / n  G  (0) l  =  ,  •  k  I  R  The m e a s u r a b l e in  a cascade  +  r  bb  1  /See  a  +  a  R  •  ....(3-5b)  ek  a  stage  is  " L  ;  voltage-amplification function for  Q  vmk  +  T  +  Z  e  (  l l e  y  +  y  ( y  21e  21e  "  yi2e  ^ t e ^  +  ) Z  1  Lk  +  r  bb  (  lie  y  +  y  lie>k] ....(3-6a)  reducing  at  low frequencies  to  -a a'R , , ,> ^ . r bb + a ' / g ee + a ' R e k ' 0  G  It  is  function  is  function  by  v m k  seen  (0)  =  then that  related  to  the  the  L k  measurable  ....  (3-6b)  voltage-amplification  short-circuit current-amplification  Lk i k 7 ~ — ^L(k-l) Z  G  The  optimum i n t e r s t a g e  low  that  we may n e g l e c t  put  impedance The  resistance  vmk  of  the  the  resistance  following stage,  a n d we may  G  (see  •  ••*  Section  v a r i a t i o n with frequency  correction factor ratio  =  G  i.e.,  Z^  i n E q u a t i o n (3-7a)  n  is  (  ~  3  3.5) of  7  a  )  is  the  in-  R^ « n  then a  constant  write  vmk  =  G  i k p  ^ k i ' L(k-l)  so  ....(3-7b)  58 A similar analysis lent R , T  of the current  c a n b e made u s i n g  source,  and the i n t e r s t a g e  1^'»  with which i t i s i n p a r a l l e l ,  voltage-amplification  the Thevenin  giving  the  f u n c t i o n f o r a stage  equiva-  resistance,  open-circuit  i n a cascade  as  (V ) Q  vk  _  ~  V  Q  P C  s  -  I  "  1  V^lle  +  +  ^ l e  ^21. -  ^te'H  +  *12e I )R  1  *  (  E  I +  » b  T  )  (  ' l i .  +  . • . • (3—8) This caded  expression  short-circuit  is identical  to the isolated-stage  current-amplification function  of  cas-  Equation  (3-5a). The for  overall  a cascaded  stage  short-circuit  amplifier  functions,  is just  as mentioned  G  Likewise,  the overall  i  current-amplification the product  earlier,  = TT  measurable  G  i  of the i n d i v i d u a l  and i s given  k  function  by  ."  ....(3-9)  voltage-amplification  function  is n G  and  is related  vm  =  TT  G  vmk  t o G^ b y R G = G . -4 vm I  + Z. , i S i .  T  vm  It  is  obvious  (3-10)  that  the overall  ....(3-11)  Z  m l  open-circuit  voltage-amplification  59 function  is  identical  fication  f u n c t i o n by  Because amplifier  the  examine 3.3  the  the  overall  short-circuit  comparing Equations  overall  depends  amplification  to  on the  functions  (3-5a)  and  amplification function  position of  the  of  the  poles  for  Current-Amplification Representation  (3-8).  of  the  cascaded  and z e r o s  individual stages,  amplification expressions  current-ampli-  we m u s t  a stage i n the  of  in a  the  now  cascade.  Complex-Frequency  Plane To e x t e n d  the  complex-frequency The v a r i a b l e cy v a r i a b l e  be  be  be  (3-5a)  i n the  cascaded  This  (2-62c),  expression  and -  (2-67),  a a'u A  R  to  the  p =  for  general  case  shall  replace  a normalized  of  the jco b y  p.  complex-frequen-  later.  is  the  current-amplification function  amplifier c a n be  to  more  + jco, we  reserved  defined  Equation used  form.  variable,  s will to  analysis  design,  expanded  in its  most  to  general  using Equations  (2-50d),  give  T  (3-12) (  R  l  +  r  bb  }  p + .co. co. + co + 1 r  <°2 ee g  The  subscript This  p,  "k" w i l l  rational  may b e w r i t t e n  1  r ( R  I  be  +  r  bb  }  +  a g  dropped while  function  r  1  of  the  discussing  (1 ee  (  + R  I  !ee e) R  +  r  a single  complex-frequency  bb  }  stage.  variable,  as N(p) G  1  (  P  )  =  K  D ( P ) '  (3-13)  60 Here the  the zero  numerator  of t r a n s m i s s i o n of the stage  factor = p + <o  N(p)  which depends o n l y By  proper  at  any p o i n t  . , . .(3-14)  1  on t h e e x t e r n a l f e e d b a c k  adjustment  The  i s d e f i n e d by  of these  on t h e n e g a t i v e  denominator  elements,  R  t w o e l e m e n t s we may p l a c e real  and C „  g  G  the zero  axis-.  polynomial  to-.to a ( l + g R ) 1 r e e e. to, + to + . ^ .2g „e„e( R I + r b b') 1  D(p)  P  A  +  2  v  1  r  b  T  + to,to  1  defines  the poles  e o o o( 3 - 1 5 )  5  r  1  R ) a '(1 + g ee e g ( R + r, J ) ee I bb'.  +  6  6  v  T  of t r a n s m i s s i o n , or p o i n t s of i n f i n i t e  amplifi-  cation The transistor operating ing  ing  are seen t o depend upon the  some o f w h i c h  only  transistor  parameters  depend  on t h e t r a n s i s t o r parameters,  includ-  and t o a l i m i t e d  at a particular  operating point,  a r e to-^, to > to^, a n d R . to-^ a n d to d e -  on t h e f e e d b a c k extent  2  elements,  x  R  g  on t h e v a l u e  2  a n d C , a n d to^ d e p e n d s o n G  of R  e  used  i n the  follow-  stage. Alternatively,  are  quadratic  elements.  a given  important  pend I  parameters,  the feedback  the  T  of t h i s  p o i n t , and upon t h e e x t e r n a l c i r c u i t  For  R  zeros  R , o  C , R , q  T  f o r a given  and I g (which  transistor,  determines  g  the four  parameters  ) . The i n t e r s t a g e ee • Q  Q  61 resistance,  R ,  is  T  chosen  current-amplification. large  as  and C ,  maximize  As w i l l  be  shown l a t e r ,  Thus  noted  poles  and zeros  are  once  The  that  the  we may f i n d  form of  of  a stagger-tuned  Stagger-Tuned order  to  the  as  multiple  the  achieve  That  poles  cascade,  A stagger-tuned  readily  to  thereby  t h a n does Grinich  has  The  pattern  passband  as  more  lo-  (3-12)  design.  like  amplifier,  cancellation  maximum a t t a i n a b l e  the  ammust  amplifica-  bandcentre  than  appearing  case  of  a single  lends  i n the  video  amplifier  about  i n the  response  a better  for  of  rather  syn-  amplifier  itself  functions  a  quite by  the  complex-frequency  amplification-bandwidth cascade.  a maximally-flat  cascaded  pole-zero  l i m i t s the  and z e r o s  synchronously-tuned  designed  stagger-tuned  g  of  zero  amplifier  cascaded  amplifier  prescribed  generally  the  the  and z e r o s  which acts  pole-zero  giving  poles  i n the  cascaded  approximating its  of  function desired,  (identical)  of  cascaded  the  Equation  bandwidth necessary  the  distributed  stage.  adjustment  the  is,  response  chronously-tuned  R  Cascade  i n d i v i d u a l stages  are  are  positions  the  achieve  which  point  of  use  as  positions.  An a m p l i f i c a t i o n f u n c t i o n the  s h o u l d be  parameters  pole-pair.  to  lation.  this  Ig  the  function  worth)  zero  low-frequency  of  stagger-tuned.  product  at  and  design  interdependent;  we k n o w t h e  plification,  plane,  effective  the  location  In  tion  the pole  s h o u l d be  suggests  be  to  It  cation  3.4  as  c o n t r o l l i n g both  g  the  practical.  so  all-pole  using  method  (Butter-  pole-zero  imposes  amplification.  a  cancel-  restriction As  an  example,  62 typical  three-  and f o u r - s t a g e  pole-zero  patterns  of  the  type  40 used by G r i n i c h are  to  obtain a Buttervorth  shown i n F i g u r e  yield  is  of  type and erly  of  discussed  no p o l e - z e r o  this  method of  t r a n s m i s s i o n to  all-pole step  poles  i n the  cancellation. than that  following  chapter  T h i s method  should  o b t a i n e d by  the  Grinich.  Because poles  be  current amplification greater  method  of  15.  A d e s i g n method w i l l i n which there  distribution  design,  responses  compare  the  as  it  design allows  exist, is  well  two d e s i g n  in contrast  necessary as  zeros  the  to  to  as w e l l  as  the B u t t e r w o r t h -  investigate  the  a m p l i f i c a t i o n response  phase to  prop-  methods. s-plane  Figure  3.5  15.  Maximization In  T h r e e - and f o u r - s t a g e B u t t e r w o r t h - t y p e d i s t r i b u t i o n s of p o l e s and zeros f o r a m a x i m a l l y f l a t cascaded a m p l i f i e r  of  choice  for  of  terstage  DC C u r r e n t  e x i s t i n g designs"''^  maximized by the  the  case  of  as  dc  current  an optimum i n t e r s t a g e  no e m i t t e r  resistance  the  Amplification  feedback.  g i v e n by B r u u n ,  This  amplification is resistance,  optimum v a l u e  but modified to  Rj p-k» 0  of  in-  include  63 the  effect  of  excess  phase,  is -1  R  Iopt  where  R  for  I°°V  a»/g  e e  r  ee + ' / g ee  bb  ^>B  +  T  R  These  expressions  feedback  are  the  of  design,  ing  solution. is  serves to  find  either tion.  but  iterative  R  and C  g  It  a good  Rj p-j.» 0  side  +  a  ' / g ee  ....(3-16b)  ao  s h o u l d be  is  not v a l i d  as  bb  bb  interstage  the  Rlopt  T  r  m o d i f i e d when t h e  effects  of  emitter  included.  Using mum v a l u e  .(3-16a)  a  to  method  resistance g  are  evident  when u s i n g  starting  not that the  point.  of  network  must be  the  value  for  chosen  found u n t i l the  above  iterative  method  maximum dc  is of  method,  (3-16),  the  before  there  A cut-and-try  starting with Equations find  synthesis,  a  starting converg-  choosing  although  method  is  it  used  and d e v i a t i n g current  opti-  to  amplifica-  64 .4... THE I T E R A T I V E - METHOD OF. NETWORK 4.1  The  Synthesis Methods  two  problems The  Problem  of  of  approximation  function,,  called  specified  frequency  the  for  the  commonly  called  the  number  freedom  For  the  of  numerator  just  or  single  stage  of  four  degrees  are to  available be  due  to  the  of  more  of  finding to  most are:  methods  by  the  the  a  suitable  approximate  a  commonly u s e d  ap-  maximally-fHat  am-  delay,  sum o f less  From E q u a t i o n cascade  number  of  on t h e  number  transfer  number  placed of  more  degrees  the  number  should  imposed by  3.3; the  on the  this  it  interdependence  system.  of of  the imposed  seen  that  a maximum  However, may b e  requirements  less  degrees  is  afford  ra-  of  function  the  (3-12)  a  available  available  approximation.  i n Section  the  is,  i n the  than the  for  explained  i.e.,  that  restrictions,  amplifier  a constraint  on t h e  polynomials,  freedom  amplification Formal  solve  a p p r o x i m a t i o n when u s i n g  depends  function,  one  the  of  as  realizability, of  or  restrictions.  a  of  coefficients  and denominator  conditions  to  and m a x i m a l l y - f l a t  function,  conditions,  is  three  amplifier  function  amplification  freedom  that  function,  The  accuracy  i n the  variable of  attempt  realization. is  amplitude,  Realization  linear-phase.  approximation  independent  problem  low-pass,  attainable  degrees, of  and  response.  equal-ripple  tional  synthesis  approximation  plitude,  The  A p p r o x i m a t i o n and  modern network  approximation  proximations  -  SYNTHESIS  only  two  considered for  physical  of  the  zeros  and  are  not  generally  poles  function. of  network  synthesis  ap-  65 plicable  to  because  of  the  synthesis  restrictions  acterization  of  lent  composed  circuit It  cuit as  is  also  external  possible  the  to  so  obtained  by  This  formal  ence  of of  pole-zero  dition)  the  under  polynomial  be  of  for  ization,  two  specified the  as  by  elements  the  useful  char-  transistor,  by  an  equiva-  matter,  the  cir-  c i r c u i t be  as  is  adjusted  easily  networks  simple  w h i c h may  has  the  been to  be  active  net-  i n the  pres41,42  described  work  is  previously.  that  the  upon the  numer-  denominator  function. method  involves  determining  amplification function  function  poles  response  directly  design  that  (the  i n designing  prescribed  the of  transistor  difficult  method  solution.  of  the  simultaneous  the  excludes  transfer  the  to  a practical  method  response  to  is  due  amplifier  depend  restriction  related  w o u l d be  iterative  must  configuration  circuit  The  the  often  case,  the  This  methods.  this  the  present  a way p r e s c r i b e d  external  at  (zeros)  (poles) the  proximate  the  definitely  method  amplifier  equivalent  finished  condition for  polynomial  In  in  that,  synthesis  circuits.  elements.  realizability restrictions  polynomial  the  lumped  known c o n f i g u r a t i o n  A necessary ator  on t h e  transistor the  An i t e r a t i v e works  of  amplifier  element,  important  that  linear  placed  active  the  and e c o n o m i c a l .  of  (the  zeros the  realizability  ap-  approximation  of  the  con-  numerator  amplification  circuit  to  parameters  function and  the  condition).  requirements,  a p p r o x i m a t i o n and  real-  to  analytical  but  provides  satisfy  by  a numerical  procedure  means, for  arriving  66 4.2 Approximation The mise  quency  in  range  of  percent.  to  It  plifier  is  of  almost the  the  is  also  the  The  finity,  number  i.e.,  at  familiar  function  is  i.e.,  one  N(p)  input  is  is  a  compro-  over  variety),  usually  and  less  for  the  relatively  approximation to  amplitude  of  the  origin for  fre-  linear overshoot  than  ten  handle  all  pole-zero starting  a useful  in  am-  i n which  the  for  flat  all-pole  not  necessarily  maximally-flat  of  equal  G(p)  to  trans-  are  unity. of  its  at  roots  poles  function is  function, to  r  the  3  but  amplifier  it  amplifier  is  designs.  mathematically  the  efficient  with finite  in-  It  is  unit  half-  of  stages  important  although  functions  at  ....(4-1)  in certain  most  plane  = -1.  usedf '^ our  left-half  at  amplifier.  maximally-flat  zeros  is  zero  Butterworth^ function  many o f  lead  the  (3-13)  become  r^ -order  cancellation point  of  one  low-pass  i n which  2  not  the  is  G(j<c)  (Butterworth)  (-s )  is  of  all-pole  b  b a n d w i d t h has  function  derivatives  i n Equation  known that, the  This  response  is  requirements  amplitude  minimum-phase  simplest  function  response  constant  phase  a step-function  maximally-flat  bandcentre,  power  has  interest,  maximum p o s s i b l e  well  and t r a n s i e n t  Function  design. The  fer  It  function  response  amplitude-response  steady-state  amplifier.  (when t h e  The M a x i m a l l y - F l a t A m p l i t u d e  maximally-flat  between  a video  -  as  The  a  maximally-  simpler,  network;  unless  does  i.e.,  zeros  may p o s s i b l y  zeros  perturb  have  44 higher tions  figures  of  from those  merit. given  The  finite  by E q u a t i o n  (4-1)  by  the  an amount  pole  posi  depending  67 on t h e p r o x i m i t y o f t h e z e r o s The g e n e r a l  transfer  to the poles.  function of a multistage  a m p l i f i e r may  be w r i t t e n a s , a l+ , •. • • • + a , s + a + , s = K ^ i " „_ ± 0 s + b s + . . . . + b s + b^ r-1 1 0 s  G(s)  m  m  n  m  1  r  where  m <^r.  i.e.,  N(s) i s given,  Equation a)  1  n  a l l finite  we w o u l d l i k e  zeros  to find  G(0)  The  amplifier cutoff  of G ( s ) are known,  a G(s) i n the form of  number o f p o l e s ,  be m a x i m a l l y - f l a t a t z e r o  b)  normalization,  r  with a fixed  G(s) w i l l  ....(4-2)  v  n  Assuming that  (4-2),  -  r,  such  that  frequency,  s = 0,  G(;jl)  so t h a t  frequency,  «  u  , i s used  the normalized frequency  for  frequency  variables  are  defined by s It  i s convenient  p/« = £ + j f t .  A  t o work w i t h the f u n c t i o n  G(s)G(-s)  On t h e r e a l - f r e q u e n c y the  equal  even  powers  to zero The  be  axis,  above  restated  as  of s thereby  at the  = K  this  amplification function,  only  (4-3)  u  2  N(s)N(-s)  is just  G(jft) having  ....(4-4)  D(s)D(-s)  the squared magnitude This  function  a l l odd-order  contains  derivatives  origin.  requirements  o n t h e a p p r o x i m a t i o n f u n c t i o n may  follows:  a)  G(s)G(-s)  must be m a x i m a l l y - f l a t a t s = 0 , a n d  b)  2 G(jl)  = G (0).  2  2  of  68 Prom E q u a t i o n ( 4 - 4 ) , written  as  G(s)G(-s)  a Taylor  = K (c 2  series  + c s  Q  2  the about  +  2  expression the  + c  +  c  where c  To make setting equal  the  to  cessive  of  these  of  the  that  zero  number as  c^,  etc.,  2  set are  of  to  it  s  (  2  r  (  r  "  l  )  +  l  )  +  c  *  +  2  s  r  2  at  of  zero  ••"( - )  o )  4  possible  successive  used  5  of  means  G(s)G(-s) the  It  the  specify  be  of  2 r - l , but of  suc-  can  derivatives  function,  to  of  zero.  s = 0 is  an even  are  frequency  derivatives  s h o u l d be  at  r  ....(4-6)  through choice  is  freedom  l )  2  be  only  r-1  coefficients  and  therefore  maximal-flatness  frequency. In  order  to  satisfy  and denominator of  conditions G(s)  Z  It available  s h o u l d be i n G(s)  noted  since  i n Equation (4-2).  is  f i x e d by  is  used  the  dc  in setting  that  N(s)  is  These  (a)  must be  / \ / \ / 2\r D(s)D(-s) = ( - s )  pear  +  s  )  many as  zero  zero  f u n c t i o n , because degrees  l  can  as  2  i.e.,  c ,  c a n be  2 ( r  -  2  maximum number o f  derivatives  r-1  merator  the  r  maximally-flat  s = 0,  coefficients  G(s)G(-s)  at  at  (  origin  G(s)G(-s)  = G (0)/K .  Q  maximum p o s s i b l e  zero  shown t h a t  only  2  G(s)  2  for  +  r  there  (b)  related  are  r+1  half-power  by  4  4  '  4  2  the  K,  r+1  b ^ , b-^,  •••  a n d one  frequency.  nu-  ' ^ 4  . . (4-7)  degrees  known, because are:  above,  N(s)N.(-s) . N(jl)N(-jl)  amplification level, the  and  of  freedom  unknowns b  r  _2 •  degree  of  ap-  Here K freedom  The r e m a i n i n g  r-1  69 degrees  of  freedom  are  available  the  amplification function,  4.3  Realization The  function  -  r e a l i z a b i l i t y requirement  upon u s i n g  the  given  positions,  transfer  itance  amplification  (3-15),  the  It  resulting  has  been  feedback,  zeros  are  stage  by  a  transfer  zeros  those  with  if,  from  of  the  the  that,  given  for  a  resistance-capac-  the  zero  i n the  found from the  pole  positions.  polynomial for  may b e w r i t t e n a f t e r  of  upon poles  compute  p o s i t i o n of  f u n c t i o n may b e  denominator  to  zeros  shown b y G r i n i c h ^  amplifier the  fulfilled  of  dependence  derivatives  Equations  is  dependence  common-emitter  emitter  The  is  prescribed  function.  transistor  adjusting  The N e t w o r k D e s i g n  i n which there  pole  for  a single  stage,  n o r m a l i z a t i o n i n the  current-  Equation  simplified  form D(s) The ure  16,  zeros  of  = s  Equation  + 2 tfs +  2  (4-8)  (4-8)  2  located,  as  shown i n  Fig-  at  s  k'*k  =  ^  where  The pole  are  ^ ,  pole-pair  positions  y  A  radius,  ~\Jf  2  -  ^V?  ±  3 ?  -  2  -  ft  is  the  2  tf'  ....(4-9)  \  geometric  mean o f  the  ? and the  average  pole-pair  v  =  s  k  0  p o s i t i o n , ~Y»  - X  = (  s  k  -*-  g i  s  v  e  n  0  o(4-10)  by •(4-11)  s )/2.  +  0  k  s-plane  Figure  If D(s),  of  Equation nomial  of  equations  16.  we the  A m p l i f i e r pole locations complex poles ( real), (YJ imaginary)  equate  to  Equation the  Equating  2 y = ^  and  equating  of  the  denominator  current-amplification function  (3-15))  for  coefficients  i n the s - p l a n e : (a) f o r and (b) f o r r e a l p o l e s  the  the  corresponding  (4-8), zero  the  we  positions  coefficients  + fi  r  +  constant  from the of  (corresponding  coefficients  can then f i n d  the  gives  +  r  of  the  required  pole-pair  s  polynomial, to poly-  design  positions.  ....(4-12a) ftog_(R  T  coefficients  b  b  )  gives  + g R ) ee e I (R, + r ') ee I bb .  l +  Substituting tion  (2-68)  the expression  1  +  n  Solving R  g  ^  r  Equations  yields  (4-13)  B  f o r te^ ( n o r m a l i z e d t o i " ^ )  ^;  r+  1  and  x  i n t o E q u a t i o n (4-12a)  a^Q  71  a'(1  gives  ^;^f  , ( T  g  e  (  e  (4-12b)  the design  °^ E q u a -  I  R  e M )  O  O  C  O  o  (4-12b)  + bb) r  and (4-13)  simultaneously  for  equations 1  l  A!  +  e e ('^1:  g  + r )J b b  ..(4-14a)  a . (R_ + r M ee I bb J and  «ee  R  I  +  r  These  expressions  normalized parameter,  ft  ."  2  ....(4-15a)  0  may b e s i m p l i f i e d s o m e w h a t  ELQ*  "ft:  a'H  resulting  f  bb>  g ee  e  a  ( B  such  1  define  that  a  + *ee  1  i f we  ( R  I  +  r  b b  }  a  + g  e  e  ( I R  J  ...(4-16) +  r  bb)  in  Q  ayflj  - M^  2  -  flgflp  ....(4-14b)  72 and  where  R  a  =  e  g  from Equation  +  (  R  I  bb  r  (2-64)  T  t e  /g  (2-69)  T  By e l i m i n a t i n g expressed  R  e  (4-3),  +  r  b b  may b e m o r e  conven-  and R  g  ..(4-15c)  )H-  compensating  using  the  capacitor,  C , g  defining Equations  may  (2-66)  be and  giving C  Equations equations zeros  ee  as  = (Bj  from  t e  E q u a t i o n (4-15b)  The h i g h - f r e q u e n c y found  W e )  + e e  M £ 0.667 + «> C /g  iently  o o o (o4 - 1 5 b )  }  ee  H = 1 +« (C  and from E q u a t i o n  +  of  (4-14b)  which state the  transfer  the  e  and  = l/tt.R . 1 e (4-15c)  co ( 4 - 1 7 )  are  then the  two  r e a l i z a b i l i t y requirement  f u n c t i o n to  the  poles  for  the  design  relating  the  known c i r c u i t  configuration. 4.4  The  Iterative The  to  solve  Method  iterative  a variety  of  method of  synthesis  network problems  has  p r e v i o u s l y been  concerned w i t h  used  vacuum-tube  73 41 amplifiers,  distributed  vacuum-tube  44 networks, and  interstage  4 6 47 stagger-tuned author's to  multicavity klystron amplifiers,  knowledge,  networks  this  is  the  involving solid-state  The m e t h o d a l t e r n a t e s and r e a l i z a t i o n ,  first  the  taking  and f i n d i n g from t h e s e  scribed  configuration; to  approximates from  the  known as network  find the  the  a new desired  corresponding the is  pole  of  poles  differ  must be  process, vergent.  the  from the  found.  If,  The  iterative  In the  of  positions  this,  plane  particular  the  are  poles  t h a n w o u l d be  is  a new  some  set  lie  closer  of  trans-  in a  pre-  network  an  of  satisfied set,  approximation  new s e t  so  are  cycles the  differs amount  of  poles  because  of  the  the  new  set  of  iterative  process the  is  con-  error  of  amount.  in this  paper,  real  axis."  to  real  axis  an a l l - p o l e  a  The  a new  on the the  zeros,  zeros.  again  prescribed  tend to  for  by  continued u n t i l  to  case  of  poles  set  decreases,  constrained  the  applied  Each pole  application treated  lie  the  along w i t h the  Prom t h i s  previous  than  been  from t h i s  previous  no l o n g e r  process  of  function.  the  approximation error  To  network  which,  on s u c c e e d i n g  a p p r o x i m a t i o n becomes l e s s  zero  poles  r e a l i z e d which gives is  set  obtained  approximation error.  again  zeros  of  has  requirements  a realizable  response  approximation requirement set  of  it  '  devices. two  zeros  set  time  an assumed  mission  then used  first  48  of  the  Because the  maximally-flat  s-  func-  tion. The erties. lour  iterative However,  45 may o c c u r ,  method g e n e r a l l y degenerative,  a n d no w a y  of  shows  good  oscillatory, anticipating  convergence  or divergent such behaviour  propbehavis  known.  74 The  iterative  method  may b e  restated  more  specifically  42 for  the  design 1.  in this  Start of  with  G(s)  plane  paper  as  an i n i t i a l  for  roots  n = number  worth  all-pole  )  2  s  of  amplifier  finding  the  zeros  is  2  r •= 2 n  as  the  poles left-half-  + 1 = 0  n  i n the  from the  given  ....(4-1)  This  maximally-flat  the  tion  the  response  stages.  Realize  ^  of  of  where  where  3.  definition  a maximally-flat  (_  2.  follows:  is  just  the  Butter-  d i s t r i b u t i o n of prescribed  poles  by E q u a t i o n  using  (4-10)  poles.  configuration, the  and  relation  ^ by  Equa-  (4-11).  Form the  denominator  isfy  approximation  the  polynomial  from the  c o n d i t i o n as  given  zeros in  to  sat-  Equation  (4-7) N(s)N(-s) D(s)D(-s)  = (-s ) 2  2  n  +  ....(4-7) D(s)D(-s)  and f i n d  its  (-s ) " 2  The  2  left-half  zeros  +  from  • t o  plane  f  c  —  roots  > —  2  of  = 0.  Equation  ....(4-18)  (4-18)  are  the  75 new  p o l e s w h i c h make t h e a m p l i f i c a t i o n  function, G(s),  maximally-flat. 4.  Compare t h e new  s e t o f p o l e s w i t h the p r e v i o u s s e t o f  poles: a) I f t h e y a r e d i f f e r e n t ,  the  s o l u t i o n has  v e r g e d ; u s i n g the p o l e s f o u n d Step b)  2 and  r e p e a t the  i n Step  not  3,  con-  r e t u r n to  cycle.  I f t h e y a r e t h e same ( w i t h i n t h e p r e s c r i b e d a p p r o x i mation  error),  the  s o l u t i o n has  then only necessary to f i n d elements  R^  and  converged,  and  the v a l u e s of the  from E q u a t i o n s  (4-15c)  i t is circuit  and  (4-17). The  iterative  method i s b a s i c a l l y v e r y s i m p l e , b u t  f a c t o r i z a t i o n of p o l y n o m i a l s i n Step 3 n e c e s s i t a t e s Use  was  the  t h e use  high-speed  digital  computer.  made o f an Alwac  electronic  digital  computer, a m e d i u m - s i z e machine w i t h an  of a  III-E 8,192-  word m a g n e t i c - d r u m memory. A p r o g r a m has video  amplifier  S t e p s 1-4  above.  been w r i t t e n f o r the d e s i g n of a  f o r up  able  stages i n cascade,  to a u n i t  and  phase-frequency  Autograph  programs were w r i t t e n u s i n g a 6-26 system  w h i c h has  digits  i n the m a n t i s s a .  response  output  curves  s t e p i n the t i m e domain, i n a f o r m  f o r r e c o r d i n g on t h e M o s e l e y  through  following  Programs have a l s o b e e n w r i t t e n w h i c h  the a m p l i t u d e - f r e q u e n c y the r e s p o n s e  to e i g h t  transistor  6 binary digits  X-Y  suit-  Recorder.  The  floating-point-arithmetic  i n t h e e x p o n e n t and  A s u b r o u t i n e p r o g r a m made  t h e Computing C e n t r e was  and  used  26 b i n a r y  available  f o r s o l v i n g the h i g h -  order  polynomials  floating-point decimal fraction unit  digits.  using  system  which  A second  expansions  step.  Bairstow's  was  is  method;  equivalent  subroutine  used  for  it to  used seven  forming  i n determining  an  8-24  significant  partial-  response  to  a  77 5o N U M E R I C A L D E S I G N S The m e t h o d cancellation  to  amplification  of  design  employed by G r i n i c h uses  obtain a stagger-tuned  function.  To e x t e n d  drift  transistor  clude  the  effects  using  the  2N384 g e r m a n i u m d r i f t  signs  are  possible  pole-zero that  for  tially  his  design  equations  of  excess  phase;  if  the  cancellation.  It  all-pole  maximally-flat  analysis  to  must be  several  will  be  method  a m p l i f i c a t i o n , at  modified to  a given  will  the in-  be  given  More e f f e c t i v e is  used  shown f o r  iterative  include  designs  transistor.  iterative  a g i v e n bandwidth the  greater  the  pole-zero  rather  a number  method g i v e s transistor  dethan  of  designs  substan-  operating  point.  5.1  The R C A 2 N 3 8 4 G e r m a n i u m D r i f t As  stated  transistor tor  previously,  performance  capacitance,  low  are:  alpha-cutoff  amplifier  d e s i g n m u s t be  requirements  low b a s e - l e a d  emitter  effective  Giacoletto  the  Transistor for  high-frequency  resistance,  t r a n s i t i o n capacitance,  frequency. well  The  transistor  represented  hybrid-n equivalent  by  the  c i r c u i t besides  low and  collechigh  chosen  for  the  modified Johnson-  f i t t i n g the  above  requirements. The RCA 2 N 3 8 4 a l l o y - t y p e  P - N - P germanium d r i f t  transistor 33 3 2 29 4 9  fits  the  above r e q u i r e m e n t s  have  s t u d i e d the  parameters  well.  RCA 2 N 3 8 4 d r i f t  and t y p i c a l  operating  A number transistor,  of  authors  and the  characteristics  are  '  '  '50'  following taken  from  51 their ratory  findings  as  well  measurements.  as One  from the of  the  manufacturer's  objectives  data  in this  and  design  labo-  pro-  78 cedure  was t o u s e t y p i c a l  rameters  to achieve  a low value order  of c o l l e c t o r  a low interstage  a converging, resistance,  due  to the effects  begins  to  of the parameter  designs:  = 50-S2.  common-base factor  a^  common-emitter factor  = emitter  7 'ee  + 0.7 pf  = 2 pf = 0.984  current= 60  t r a n s i t i o n capacitance  = 35 p f  = 5.5  low-frequency common—base conductance E  = 1.3  current-  base f i e l d parameter  A l  at an  = 2 u ( l 0 0 Mc)  oc^ = l o w - f r e q u e n c y amplification  C, 'te  deterioriate  values,  resistance  = low-frequency amplification  solution for  injection.  = collector capacitance plus glass header, metal case, and i n t e r lead capacitance  e  in  as p o s s i b l e , b u t  frequency  a  assures  be shown l a t e r ,  o f 25° C , t o be u s e d i n t h e  base—lead  for  physically-realizable  of h i g h - l e v e l  a> = a l p h a - c u t o f f  C  As w i l l  performance  The f o l l o w i n g i s a l i s t  bb  design.  I j , must be as l a r g e  I g ^>2 m a , h i g h - f r e q u e n c y  temperature  a general  The c h o s e n v a l u e  capacitance.  for  ambient  of the t r a n s i s t o r p a -  p o i n t f o r t h e t r a n s i s t o r was c h o s e n as  and I j , = 2 ma.  to achieve  values  as f a r as p o s s i b l e  The d c o p e r a t i n g = -12 v o l t s ,  (average)  emitter  = I (ma)/26  = 0 . 0 7 7 xx (I = 2ma).  E  E  The  effective  alpha-cutoff .= effective frequency  frequency  common-base  i s , using Equation  (2-49),  alpha-cutoff = 2u(58.5 M c ) .  79 5.2  G r i n i c h Designs  Using a Drift  Transistor  The d e s i g n m e t h o d u s e d b y G r i n i c h equivalent Bruun.  c i r c u i t with emitter  He u s e s  a cascaded  transfer  function,  pensated  by a capacitor  and a r e a l  zero  made  pairs  cle,  feedback  to coincide  o f t h e above  obtained.  form,  lost  w h i c h c o u l d be u s e d , of:the  real  cascading response  is cir-  i n the Butterworth  o f one s t a g e  say, to further  other  (normalized)  two d e g r e e s  includes  the effect  to cancel  of freedom  increase  of the emitter  f r e q u e n c y , to^, r e l a t e d  b'e  From E q u a t i o n  0.09y^),  i n the p a i r ,  pole-pair  the  the  are  amplifi-  transition ca-  by defining  to the emitter  a modified  diffusion ca-  by  C  drift  pole  G^g, f o r a d i f f u s i o n t r a n s i s t o r  alpha-cutoff pacitance  on the u n i t  com-  a single  By  stagger—tuned  two-pole  pair.  Grinich pacitance,  and the  the zero.  t o be m a x i m a l l y - f l a t  By c o n s t r a i n i n g the s i n g l e stage  conjugate  plane,  a  resistor  o n l y to produce  By p l a c i n g a l l the poles  of the other  the  resistor  hybrid-TC  to give  feedback  a complex  an a l l - p o l e  zero  cation  an e m i t t e r  w i t h and cancel  t h e f u n c t i o n i s made  sense.  of t r a n s i s t o r s  t ° produce  the  i n the form given by  i n the complex—frequency  having an emitter pole  feedback  pair  one u s i n g  employs  = > *eeK 1  215  (2-49),  +  =  1  a  g  ee  / a  T  21  the effects  ....(5-la) of excess  phase  in  i s r e p l a c e d b y to^ - to /(1.215 +  « /l.215  and Equation (5-la)  b»e  t e = ' ^eeK'  to include  transistor,  C  C  becomes  +  C  te =  g  e e  /  a  r  ....(5-lb)  ,  8 0  Using the parameters given i n the preceding s e c t i o n f o r the 2N384  drift transistor,  fied cutoff  frequency  at the s p e c i f i e d  operating p o i n t ,  the modi-  is  to*  ='2TC-(50.2 M C ) .  The r e s u l t a n t modified expression  1  f o r the  amplification-  bandwidth product^'^ GB, f o r a stage i n a cascaded a m p l i f i e r without emitter feedback  is  <4/2 71  GB = 1  +  (1  +  r  bb  R-r  +  a  a'Tg  ,  /  g  v ee  . . . . ( 5 - 2 )  ee  By modifying the c u r r e n t - a m p l i f i c a t i o n expression used by G r i n i c h we o b t a i n (s +  G . ( s ) = Ks  2 +  [ f i d 0  +*  B  e  \ & )  Q) ±  + g R ) e e e'  +  6  » o9 o(5 3) —  where  Q"-l . = a,/V 1' u = l• / u eC ( a  B  J  2TI(GB) 0  u  « g R u ee I  . . . . ( 5 - 4 )  T  e  and  (6 = a m p l i f i e r c u t o f f u normalization.  By equating the c o e f f i c i e n t s  frequency used f o r  of the denominator polynomial  81 of  Equation  (5-3)  to  the  corresponding  coefficients  of  the  poly-  nomial  D(s) the in  resulting design the  manner  = s  equations  + 2 2Ts +  ^  Grinich's  method are  2  for  indicated i n Section  2  obtained  4.3  1 ? = - — ( s\ ^ 2  and  R e  For the  single  located  the  g  case  real  ee  of  pole  L  l±  h  a stage  of  the  having  the  choosing  zero,  the of  A special  •  a n odd number e  Equation  case of  of  P  B  for  R ,  an e m i t t e r  resistor, is  P  e  e  B ),  it  g  ....(5-7)  e  c a n be  used  to  cancel  i.e.,  "  1  )*  (5-6b)  g  (5-6b)  i n order  e  low-frequency  + g  = ^ - ^ 7 ^ ee  Equation  stages  •  ( l  (5-5),  • t  The  0  proper value  e  1  only  at  R  of  . ...(5-6a)  current-amplification function  s = - Q  and by  l ) .  to  is  place  i -i = Z—(rY~ ~ e e S^O  l  )  -  used  i n the  a pole  at  design s =  ....(5-6c)  g  current-  and  -1,  voltage-amplification  82 expressions  are  the  same  as  those  i n S e c t i o n 3.2$  from Equation  (3^5b) G. ( 0 )  R  and from E q u a t i o n  -ocQa'Rj , bb /See  = — I  +  r  +  —  a  +  j e  a  R  (3-6b)  i  G  v  r  ("0 )  r  An upper h a l f - p o w e r all  designs.  stage  F o r ease  a m p l i f i e r was  normalized  Mc), Q j  b  e  the  poles  given for  able*  the  pears  I gives  curves the  r e l a t i o n s , w i t h R^, These that  (Equation relatively  results  the  6.5  Mc w a s  specified  For t h i s  a Grinich  design  for  three-  case  the  is  shown  in  t r a n s i s t o r , w i t h te^: = 2TXX  optimum i n t e r s t a g e  value  (3-16)),  I o 0  Iopt  =  4  the  plotted  for  Bj p^.  meaningless  6  of  a  '  17.  a number o f  designs  resistance,  i n Figure  a stage without  when e m i t t e r  feedback  configuration for  as  using the  is  a Grinich  the  vari-  17-.•from w h i c h i t  computed from B r u u n ' s to  is  dc-amplification versus  interstage  are  0  4  Figure  results  applicable  The p o l e - z e r o  of  resistance  = 282x5.  comparison w i t h the  interstage-resistance  above  e  7.74.  R  Table  •+ a ' R *  1  example. for  R  is  of  2N384 d r i f t  From E q u a t i o n ( 3 - 1 6 ) ,  which  T  e  c a l c u l a t i o n and c o n s t r u c t i o n , a  c h o s e n as  U s i n g the =  0  b  frequency  d i s t r i b u t i o n of  F i g u r e - . 1 5 (a) v r (6.5  of  -oc„a R + a 7 g  ap-  expression  feedback,  is  used. three-stage  Rj (a)  220  240  270  300  330  390  470  560  680  820  1000  1200  1500  0.314  0.292  0.265  0.242  0.224  0.193  0.163  0.139  0.116  0.098  0.081  0.068  C.055  0.580  0.601  0.626  0.648  0.666  0.695  0.723  0.745  0.765  0.784  0.795  0.808  0.820  R  el  11.0  13.8  17.7  21.8  25.8  33.8  44.8  56.6  72.6  91.5  115  142  182  R  e2  28.4  31.4  36.0  40.6  45.1  54.3  66.5  80.5  98.6  120  148  179  224  B  e3  58.3  61.0  65.2  69.7  74.3  83.8  97.0  157  189  224  276  G  il  7.70  7.60  7.49  7.40  7.30  7.20  G  i2  4.80  4.88  4.97  5.05  5.12  G  i3  2.90  3.04  3.23  3.38  G (0)  107.2  113.0  120.0  G (0)  40.60  41.06  41.58  ±  ±  (db)  112  133  7.08  7.02  6.95  6.90  6.88  6.83  6.80  5.23  5.34  5.40  5.50  5.55  5.60  5.65  5.69  3.52  3.74  3.96  4.13  4.30  4.45  4.55  4.66  4.75  126.2  132.0  141.0  150.0  157.0  164.5  170.5  175.5  180.0  184.0  42.02  42.41  42.9c*  43.52  43.92  44.32  44.63  44.87  45.11  45.30 i  00  Table  I  Grinich  designs  with R  T  variable  51 •• 50  I t e r a t i v e Method Designs  -  49  t  48 47  4 6 •• G (0) ±  \ 45 " :  ( d b )I 4 4 ••  Grinich  43  Three-Stage  4 2 "  f  u  Designs  Amplifier 2N384:  = 6 . 5 Mc  2 ma  41  te= P <o = 2TT(58O5 C  4 0 •-  3  5  f  T  /)  MC)  =5.5,  3 9 •• 200  Figure  17.  300  400  Low-frequency  500  600  700  amplification, iterative  800  900  and G r i n i c h  1000  methods,  1100  1200  Rj  variable  85 amplifier  is  figuration ative  shown i n F i g u r e  of  the  method.  a value stage  equivalent  In both  The  amplitude  and phase  shown i n F i g u r e s  shows  its  response  cutoff  that at  6.5  frequency the with  time  the  above domain,  to  The  the the  for  responses  the  iter-  of  300Si.,  frequency-dependent  input.  response  phase  cutoff  is  response  frequency  overshoot  is  product  a bandwidth  of  for  the  design  and 20 r e s p e c t i v e l y .  a unit-step  a risetime-bandwidth  nanoseconds  using  resistance  appreciable  19(a)  amplitude  Mc.  designed  conr-  used.  are  seen  comparison w i t h the  amplifier  avoids  18(a)  is  for  cases, an i n t e r s t a g e  which d e f i n i t e l y  l o a d i n g $ was  18(a)  6.5  is  flat  19(a)  with  relatively seen  i n Figure  about  8.1$  for  0.370  Mc.  The  or  21 it  a 3 db  linear  as  of  Figure  Figure  From F i g u r e  indeed  of  to  20.  In  a unit-step  a risetime  50$ time  of  delay  a  input, 57  is  53  nanoseconds.  s—plane  Figure  18.  Three-stage (a) G r i n i c h method  p o l e - z e r o c o n f i g u r a t i o n s f o r R-j- = 3 0 0 - n . : d e s i g n , and (b) d e s i g n by t h e iterative  0  86 Grinich Design ( G . ( 0 ) = 4 2 . 0 2 db)  -3  rl o  -  Design by the Iterative Method( G ( 0 ) = 49.61. db)  -6--  •H  -P  o  i  •H <HH •H  - 9 "  PH Ci)  -12 a>  OS  (a)  1.5  4-  2.5  3 f  -+-  4  Iterative Grinich  (b) Figure  1.5 19.  2.5  3  6.5  8  10  6.5  8  10  (Mc)  Design Design  A m p l i t u d e response c u r v e s by the G r i n i c h and i t e r a t i v e methods f o r Rj = 300XL : (a) o v e r a l l a m p l i f i e r response, and (b) i n d i v i d u a l s t a g e r e s p o n s e s  240  T  D e s i g n by the I t e r a t i v e Method  200 +  Grinich  Design  150 +  8 6, .7 10 11 12 13 14 (Mc) 2 0 . Phase r e s p o n s e c u r v e s f o r R = 300.rx: (a) G r i n i c h d e s i g n , and (b) d e s i g n by the: i t e r a t i v e method 5  Figure 1.1  f  T  r  6  7  (sec. R e s p o n s e t o a u n i t - s t e p f o r R-j- =^300jn_: ( a ) G r i n i c h d e s i g n , and (b) d e s i g n b y t h e : . i t e r a t i v e : method L  Figure  21.  15  88 5.3  Designs  U s i n g the  A number o f using  the  outlined  Iterative  three-stage  computer  programs  i n Section  Method a m p l i f i e r designs  written for  (4-16), depends  stages.  This  is  does not  restrict  In the  This  fier  the  where  noise  because  stage. have  of  or  first  the  following  product  somewhat,  pedance  of  greater  stages, but  effect,  the  This  gives  =  R  j  r  the  stage  first  most  increase  been  order  to  i d e n t i c a l values  of  the  the  the  noise  stages least ampli-  amplifier figure  a r b i t r a r i l y arranged  of  so  for  convenience  by r e a r r a n g i n g  the  tuning  the  loading  started bb  +  a  '/gee  effect  of  assuming ^  for  1 +  of  the  identical i  o r  the  QQ^  V te/gee C  as  to  in  amplification-bandwidth input  g a i n e d w o u l d p r o b a b l y be  by  a  pattern  current-amplification function is advantage  A  design  purely  the  i d e n t i c a l values H  of  has  the  stage  following  whose  n <  ii idual v  stages  +  from E q u a t i o n  Wc* (4-16)  load  ima small re-  stages.  from Equation  (2-64)  and  method,  a high-gain low-level  increase  since  on the  or  feedback,  to  on t h e  placement  for  to  possible,  a stage  made  a typical  by G r i n i c h ,  contributes  have  The d e s i g n may b e sistances,  the  tends  stages  may b e  second-order  stage  feedback  Suceeding  It  neglected  specify  for  a small extent  advantageous  progressively  design.  to  apparently  is  iterative  given  QQJ^?  following designs,  feedback.  been  4.4.  The n o r m a l i z e d p a r a m e t e r , by E q u a t i o n  the  have  89 The  iterative  convergence  method  i s applied for several  of the poles,  and a p p r o x i m a t e  values  s  k' k' s  of R  g  a  n  d  z  e  r  o  s  >  cycles  (until  ^ i k ' ''"  are then found using  k  s  a  PP  r  a  o  a  c  n  rough ed) ,  Equation  (4-15c)  n +  ?  2 0  R  where  ek^  (  I  R  from Equations  * Q  +  (2-69),  (4-10),  M  ^  and  As  an a l t e r n a t i v e  made  for values  of  2  based  of R  the load resistance  g  ^ The v a l u e s the  of Q g k  iterations  (4-15c)  iterative  plotted  1  a  r  e  r  B  I  method,  i n Figure  T  +  k  M <>  2  t e  /g 'ee t  s )/2. k  +  r  stage  +  |  bb  '  a  +  /  g  initial  e  to give  guess  e  +  a  of R j .  improved  estimates  (3-4b)  R  + ' e(k+l) " a  E  (usually  the c i r c u i t  be  ' e(k+l)  to a converging  the r e s u l t s  can often  using Equation  aVgee  ^^en m o d i f i e d  then give  I I gives  T  upon t h e known v a l u e  bb  are continued  and (4-17) Table  the  k  + « C  a good  f o r each  = R  0  and (4-11),  are then used  k  -  ft Q  0  = s,_*s w  to the above, e  The v a l u e s  0.667  A  - ft = ( s  o f & ^f  _  T  - 2 ^ Q  2  only  solution.  element  o f a number  w i t h R j as t h e v a r i a b l e .  17 a l o n g w i t h t h e r e s u l t s  s l i g h t l y ) , and Equations  values.  of designs  using  The r e s u l t s  from the  Grinich  are  _  A>i  _ ^-02 _^03  _n _ii  12 1 3  i  s  _ s  2' 2 s  _  "  R  el  R  e2  R  e3  G  i l  G  i2  G  i3  "  G (0) ±  " G.(0) ' * (db)  300  330  390  470  560  680  820  1000  1200  0.358  0.334  0.296  0.258  0.226  0.196  0.169  0.146  0.127  0.352  0.328  0.288  0.251  0.220  0.189  0.163  0.140  0.122  0.349  0.325  0.286  0.249  0.217  0.187  0.161  0.138  0.120  0.443  0.626  0.751  0.840  0.909  0.970  1.019  1.063  1.096  1.174  1.129  1.145  1.173  1.202  1.228  1.250  1.271  1.286  0.360  0.364  0.385  0.406  0.427  0.449  0.466  0.483  0.497  0.360  0.364  0.385  0.407  0.428  0.449  0.467  0.484  0.498  0.441  0.598  0.665  0.697  0.716  0.729  0.737  0.743  0.747  0.769  0.735  0.726  0.724  0.724  0.724  0.724  0.724  0.724  +J0.308  +J0.336  +J0.369  +J0.392  +J0.409  +j 0 . 4 2 3  +J0.433  ±3 0 . 4 4 1  +j 0 . 4 4 8  0.387  0.377  0.368  0.362  0.356  0.352  0.349  0.346  0.344  +j 0 . 9 0 0  +J0.902  +J0.906  +j0.910  +J0.912  ±j0.915  ±j0.916  +J0.918  +J0.919  0.05  0.70  3.11  6.81  11.30  18.30  27.20  40.50  56.70  12.78  15.23  20.92  28.80  38.00  51.46  69.00  93.30  125.0  130.4  142.4  162.3  189.0  222.0  272.0  333.0  420.0  527.0  15.88  16.48  16.65  16.50  16.23  15.62  15.00  14.10  13.25  9.50  9.55  9.46  9.31  9.15  8.90  8.50  8.07  7.55  2.01  2.04  2.14  2.23  2.28-  2.29  2.27  2.22  2.14  302.5  321.0  336.5  343.0  339.0  318.0  290.0  252.0  214.0  49.61  50.13  50.50  50.69  50.60  50.05  49.25  48.03  46.60  Table  II.  Designs  using  the  iterative  method w i t h R  T  variable  1  91 designs ative  of  Table  I.  method becomes  levels,  which y i e l d Grinich there  is  The  method  ratio  levels,  levels  of  though is  Figure  the  of  The  get be  iterations,  solve  four the It  to  response  the  less  at  iter-  impedance  negligible  chosen,  fre-  designs the  R-j- = 3 0 0 X 1 . ,  amplification using  corresponds  to  a voltage  gain ratio  higher  significant;  at  of  interstage  very  high  or  6.2:1. imped- ~  impedance  l o a d i n g p r o b l e m , where  neither  to  be  about  next  shown,  close fifteen  More on t h e section.  w i t h the  major  successive  to  that  zeros  the  the  closest  The  w i t h the  convergence  iterations matter  p o r t i o n of  to  real the  i n d i v i d u a l stages  in  final  dotted  lines  of  after  were  that  needed  to  origin.  will  approxi-  time Used  i n each  poles  are  only  convergence  Each i t e r a t i o n took  two  steps  pole-pair.  polynomial ^hich arises  noted  two  for  are  d i s t r i b u t i o n of  R^ = 3 0 0 x i . .  w h i c h goes w i t h each  sixth-degree  the  and the  problem for  accuracy.  s h o u l d be  (Butterworth)  plane  and z e r o s  minutes,  curves  of  here  a power  initial  design  though  i n the  cancelling  level  This  capacitive  s o l u t i o n seems  discussed  mately  the  zero  three-place  interstage  a m p l i f i c a t i o n t h a n do  or  complex-frequency  the  the  applicable.  both poles  indicating  three  the  22 s h o w s  s o l u t i o n of  values  2.5:1,  of  method produces  db i n c u r r e n t  design.  quite  lower  iterative  somewhat  still  we r u n i n t o  i n the  of  advantage  requirement  greater  7.6  the  the  impedance  of  almost  d e s i g n method i s  poles  the  an advantage  advantage,  ance  At  At  the  the  substantially  iterative  current  satisfy  loading,  designs.  figure,  apparent*  which e a s i l y  quency-dependent  the  Prom t h i s  cycle.  come v e r y The  to  close  amplitude-  shown i n F i g u r e  19(b)  s-plane  * 0  -1.532  x =. p o l e  position  0 = zero  position  = p o l e and zero paths to convergence •= f i n a l p o l e - p a i r a n d associated zero  Figure  22.  Successive cycles i n the i t e r a t i v e a n a m p l i f i e r w i t h R-j- = 300-CL.  -1.0 design  of  93 for  comparison w i t h the  19(a),  for  the  The be  first  discussed  have  for  same  and  e  from the  has  next  essentially  section.  and  C  R - = 130.4A,  C  low-frequency  voltage  =--15.88  G. (0)  =  -9.50  G  =  -2.01  i 3  retical seen  19(b)  design.  At  initially Grinich time which  the  at  is  of  frequencies a rate  the  somewhat  The  -302.5 ~  the  is  pf.  which i s phase  overshoot higher  the  stages  are  Figure  cutoff^  the  the  than that is  is  9.9% f o r  than the  21  shows  the  19(a)  comparable  for  relatively  figure  and  it  of  the  8.1%  for  the  is  off  comparable  linear.  a unit-step  theo-  Grinich  amplification falls  response about  amplitude-  From F i g u r e  than for  greater  db.  theoretical  input.  flatter  above  49.61  amplifier.  a unit-step  amplitude  design.  domain,  to  stages  is  a n d 20 show t h e  curves  response  that  (0)  amplification  i  phase-response  to  pf,  = 579  G (0)  G (0).=  Figures  and t h i r d  = 1811  2  amplification for  2  overall  second  a result  e3  ±1  and the  Figure  values  ei The  no f e e d b a c k ,  The  = 12.78.n-,  e 2  design,  I^.  feedback-element R  Grinich  a  stage  i n the  their  R  curves  In  the  input, Grinich  94 design. to  The  risetime-bandwidth  a risetime  essentially delay 5.4  is  of  the  58  58 n a n o s e c o n d s same  Although  tion are  may b e  for two  ling.  of  the  properties,  haviour  for  the  the  as  mentioned  0.378,  corresponding  Mc b a n d w i d t h .  design.  The  This  is  50% t i m e  encountered.  Figure  designed  real  pole-zero  pairs  a result,  the  infinite.  is  4.4,  18(b)  which are  shows some  shows  the  of  the  compensating  an example  of  the  good  conver-  abnormal final  Here,  essentially  resistor  and i t s  This  generally  w i t h R^ = 300-fl..  feedback  zero,  Method  in Section  amplifier  become  Grinich  method  the  As  a 6.5  Iterative  iterative  comes a p p r o x i m a t e l y to  as  for  is  nanoseconds.  Characteristics  gence  product  be-  soluthere  self-cancel-  first  stage  capacitor  be-  tends  degenerative  ef-  44 feet  mentioned by Another  Chang.  characteristic  tremely  rapid for  in  five  or  six  iterations.  closest  to  the  o r i g i n was  zero  this  of  and the  pair  large  cancelling  pole  was  made  was  the  tendency  come first  stage to  for  as the  to  to  of  of  case  last  first  a second zero  the  stage  stage. came  real  cancelling pole.  The  R . T  arrived  critical  exat  pole-zero  pair The  of  the  amplifier,  As  the  interstage  more  slowly.  pole-zero pair  most  real  was  self-cancelling.  this  the  of  convergence  solution being  convergence  o r i g i n was  low values  the  the  The  d i d the  that  approximately  lower,  self-cancelling.  closest verge  to  w i t h the In this  belonged  resistance due  R^,  n o t e d was  pair  belonged  This to  to  be-  the  pole-zero  pair  and s l o w e s t  to  con-  95 A s R j was first  stage  made  became  and e v e n t u a l l y  even  smaller,  closest  crossing  led  to  oscillatory  was  o b t a i n e d by u s i n g  to  over  the  the  pair  approaching  right  network  this  of  the  the slowly  half-plane,  process,  because  of  it  Although a converging  an a v e r a g i n g  a physically-realizable  pole-zero  origin  into  behaviour.  the  which  solution  d i d not  yield  right-half-plane  zero. The in  the  critical  design  is  parameter  QQ»  Q U  This the  is  given  Q [l  T  L  critical  pole  designs  converge  faster  fixed  zeros  transistor  externally  to  parameters  C ,  be  small to  make  small,  in  the  requirements  a given  c  g e  but  crease  i n interstage  resistance  values  of  and f o r  tions of  current  large  negligible  = 0. is  (lj;)  less  a  n  <  these  C ,  and C ^ ,  i  •  The  right-  by  the the  transistor  large,  and  tt^  requirements  are  disregarded  they  i n emitter  are  we  should  in conflict  with  performance.  current  make Q Q  resistance  frequency-dependent  for  and by  e  and/or  smaller.  we r u n i n t o h i g h - l e v e l  interstage  small,  determined  transistor  will  For  tendency  is  s h o u l d be  high-frequency an i n c r e a s e  lems,  e e  )]  0 L c J  c  because  transistor,  emitter  b b  + a R C )] *  r ^ ,  and C^. ,  a transistor for  ee  r  parameter  a',  parameters  a',  the  +  and t h e r e This  parameters,  of  te  realizability  (4-16)  position for  occur.  variable  choice  by E q u a t i o n  a ' f l + <*JC / g  °  physical  + a ' / g ^  T  the  half-plane  governing  an i n -  For  injection  exceed the  loading.  For  For t h i s  large prob-  restricpar-  96 ticular QQ  design,  <C 0 . 3 6 0 .  physically  This  I-g, = 2 m a , R-j- = 3 0 0 r i - ,  is  very  nonrealizable  The reasons  where  amplifier  of  physical  will  design  realizability,  shown i n t h e  5.5  P r e - D e s i g n P r e d i c t i o n of possible  a particular the  the  c r i t i c a l pole  and/or ly  design.  no p r o b l e m s  bility poles  (move  will will  be  upper  occur.  necessarily  real  lie  extremely  are  large.  within are  not  the  fast  and the  pole  critical  and t h e r e  the  not startis  QQ  are  of  1^,  general-  than the  real  r e g i o n where  amplifier  realizable  problems  the  closest  on the  approach  realizable  bandwidths, case,  or  values  of  the  is  critical  pole  physical  to  some  not  i n d i v i d u a l stage  the  realiza-  tend  to  have  to  possible.  bandwidths greater  o r i g i n may a g a i n  physically  real  c r i t i c a l distance  solution is  poles  cut-  solution  small bandwidths,  closer  For  In this  lim-  before  section,  For large  bounds  poles  extremely  o r i g i n and a p h y s i c a l l y  separation,  previous  is  origin),  the  large  within prescribed  realizable  = 0.  and lower  the  for  realizability.  the  For  a  restricted,  p r e d i c t i o n whether  w i t h i n which a p h y s i c a l l y  toward  which y i e l d s  Realizability  physically  convergence  of  Mc,  section.  a rough  i n the  is  u  lie  Physical  give  physical  Yhen any  position  to  position for  exist  frequency  possible.  of  value  <o ,  following  As m e n t i o n e d  R-j- ( s m a l l ^ Q ) ,  There off  to  solution will  ing  the  bandwidth,  as  is  to  = 6.5  u  solution.  its  It  be  close  a n d f*  realizable  lie  solutions  possible. As  an example,  approximate  upper  cutoff  frequency  bounds  97 were  found  drift  quite  upper  300-n.,  the  realizable 5.5  two  transistor  is for  for  three-stage  w i t h 1^,-2  small,  cutoff  ma.  physically  frequencies  bounds  amplifier  were  solutions  very  existed  designs  the  F o r R-j- = 5 6 0 . n - , a c a s e  realizable  between  solutions  were  4 Mc a n d 6 0 M c .  much c l o s e r for  using  upper  together, cutoff  and  2N384  where possible  For Rj  =  physically  frequencies  between  Mc a n d 12 M c . For  stages,  define poles  cascaded  physical  amplifiers  containing greater  r e a l i z a b i l i t y becomes  more  Through f u r t h e r  investigation,  a region  complex-frequency  must  lie  of to  the  assure  it  of  a  numbers problem.  s h o u l d be plane  no r i g h t - h a l f - p l a n e  of  possible  to  w i t h i n w h i c h the zeros.  98 6.  The for  transistor  a three—stage  300x1.. put  It  was  = -  2.67  interstage give  ues  cascade  of  50-rx.  db)  to  a low  output  is  on the  line  50-Si.  and a t w o - s t a g e  impedance of  the  The  side  to  to  of  the  compensate  (pad  was  300-XL  used  output  amplifier.  23, w i t h t h e o r e t i c a l  nominal r e s i s t o r  out-  300-xu input  l o a d i n g the  stage  of  designed  the  is  and  emitter-follower  without  third  5  and l o a d r e s i s t a n c e s  an a m p l i f i e r w i t h i n p u t  shown i n F i g u r e  low  i n Chapter  A m i n i m u m - l o s s p a d was  shown i n b r a c k e t s .  chosen  build  match the  resistance  used  a m p l i f i e r designed  w i t h source  to  resistance,  interstage circuit  video  decided  impedances  loss  to  A M P L I F I E R C O N S T R U C T I O N AND PERFORMANCE  design  values  p a r t i a l l y for  The  used  val-  were  emitter-lead  resistance.  6.1  General  Design  Considerations  Standard high-frequency stage the  layout  complete  eight were  inches  cury battery Figure The 5/l6 The  employed.  a m p l i f i e r was l o n g by  provided at  P r o v i s i o n was  in  were  both  also  square.  BNC c o a x i a l dc  i n t e r n a l mounting of The  test  inline  purposes,  about  connectors  power  terminals.  a 15-volt  amplifier  is  mershown  24. individual transistor  2N384 h a s base  i n an aluminum box  along with external  for  with  s h i e l d i n g and g r o u n d i n g  enclosed  inches  ends  made  For  w i t h a n ON-OFF s w i t c h .  inch i n length  tween  two  construction techniques  and were  a fourth  lead  and c o l l e c t o r  to  leads  soldered connected  were  cut  directly to  minimize the  to  approximately  into  the  an i n t e r n a l feedback  circuit.  shield  be-  capacitancej  designed  Figure  23.  emitter-follower output  amplifier  Circuit  diagram  of  the  test  amplifier  100  101 this  lead  tion  was  is  grounded.  that  o c ^ be  The  only  criterion for  w i t h i n 25$ of  its  transistor  nominal value  of  selec-  60  for  l g .= 2 m a .  6.2  Low-Frequency For  to  design  biasing the  low-frequency for  Considerations  video  and s t a b i l i z a t i o n r e a s o n s ,  extending  cutoff,  40 c p s ,  f^,  made  of  13.8  R-j- = 3 0 0 - n .  \if f o r  coupling  as  capacitors,  capacitors  age  which causes  current,  by u s i n g heavy capacitors to  The made  use  to  practical  dc .  The  a typical  sensitive  to  to  feedback  ensure  temperature  resistance  not used  bypassed.  The  for  emitter  that  change.  value  i n multistage  currents,  for  single  to high  of  use leak-  biasing.  and it  electrowas  found  coupling-.  resistance dc  a  a value  their  leakage  emitter  i n each  the  compensation  p r o b l e m has  stage  currents  The p o r t i o n o f  high-frequency bypassing  of  feedback  the  of  desirable  emitter  electrolytics  emitter  not  c o u p l i n g because  l o w dc  basis  s h o u l d have  is  difficulties  with very  total large  for  low-frequency  15-^if  C-j-,  Normally, i t  electrolytic  been  chosen  R C - c o u p l i n g , and on the  the  possible  was  not  amplifiers.  interstage,  lytic  is  a m p l i f i e r w i t h a passband  Use was  But,  it  are  has in-  feedback is  been w e l l  heavily discussed  52 by M u r r a y , reactance  using of  the  the  low-frequency  emitter  bypass  T-equivalent  capacitor  is  circuit.  given  by  The  102 where  R  s  p  = source G . (R , ei  _ "  R  and  resistance  G  i< el R  present) -  °)  '  „ = emitter feedback broadbanding  resistor  used  for  high-frequency  R -. = e m i t t e r f e e d b a c k el stabilization  resistor  used  for  dc  n  where  the  elements  shown i n F i g u r e  25(b),  and  of  the  are  low-frequency  (for  the  r^  = base  r  = collector  r  c e  = emitter  For  the  current  frequency,  resistance  = 12A  set  at  R  total  emitter  520-fi. t o  2  are  are  (= 4  MXL)  for  Ij, = 2 ma.  (6-1)  is  e l . be  d o w n 3 db a t  = 1/2  a  partic-  feedback  give  good  shown i n F i g u r e  s h u n t e d w i t h 0«2-p,f  ....(6-2)  C , ^ .= - l / f l ^ X , , - , . de 1 Ode  ceramic  23.  .,..(6-3)  resistance,  current  The r e s u l t i n g d e s i g n v a l u e s capacitors  transistor)?  then  and  been  «  a m p l i f i c a t i o n to  P  The  circuit,  ( = 200-fiu)  The d o m i n a n t r e s t r i c t i o n o n E q u a t i o n  Cde  current  resistance  E  X  emitter  T-equivalent  2N384 d r i f t  resistance  = 25/l (ma)  ular  :  for  R . ' e l  + R , , eh'  has  stability. the  emitter  A l l electrolytic  capacitors  to  ensure  bypassing capacitors good  high-  103 frequency  6.3  performance  The E m i t t e r In  the  last  order stage  ter-follower its  Follower to of  in  r  b  The  4.  , = r + out e  ( 1  +  is  (  output  impedance  without  it  necessary  to  transistor  T-equivalent  !  +  a low  amplifier,  53 Stansel. It  given by  R  the  pair.  low-frequency  R  achieve  f  b  common—collector  circuit  evident  _ a  1  was  )(r  are  then  c  r  emit-  amplifier  and  25,  as  that  e  R )  ....(6-4a)  L  " " f b ^ ^ C —— : -— 1 + / r, + R c' b s +  an  shown i n F i g u r e  + r )/(r +  b  use  loading  }  ....(6-5a)  1  r  v  and G (0) vm  Figure  Here, then  a^ the  b  =  b  25.  c  (a) (b)  (1  -  a  f  b  )r  b  .«..(6—6a)  + RL  T r a n s i s t o r common—collector a m p l i f i e r , low-frequency T-equivalent c i r c u i t  = (r^ + r ) / ( r b  above  +  relations  + * " ) , and i f b  become  we  assume  that  and  r <^r , b  c  104 R. in  = r,  b +  - b r  R  +  r  The v o l t a g e off with  fo)  der  to  e  ( r  .  +  -  input  r  < b. r  R  a  s  )r  b  c^  +  R  > a  '  ( r  e  (6-4b)  V '  +  ....(6-5b)  s>]  )  - , f  ,..  + RL)  E  f o r  c  +  •  ( 1  V '  +  b  U  quite  ....(6-6b)  +  b  constant  impedance,  to beyond  however,  which i s  the  falls  reason  for  the  quite using  a-cutrapidly two  cascade.  and the  used by the drawing  r  ^  +  frequency,  The f i r s t plifier,  -  The  increasing in  ( r  amplification is  frequency.  stages  ' e  , = r + —rlr out e a l  - e and G  a  a  emitter-follower output  stage  emitter-follower  1^, = 1 m a , a n d t h e give  adequate  R  The v o l t a g e  is pair  = 32.4A/  R.. m4  = 48.8 k a , '  a m p l i f i c a t i o n of  the  vm4  fourth  fifth.  is  t o be  stage  voltage.  out4  and G  the  the  output  output  G  is  o  u  t  =  5  are  =  °*  ,(0)  =  0*815.  9  6  6  the  12*., k n ,  am-  current first  = 3 ma i n  have  R. _ = 3 . 6 8 m5 stages  of  4 ma, w i t h the  drawing Ig  ( 0 )  vmD  The t o t a l  We t h e n  B  stage  or-  105 For  the  emitter-follower R .  n  - 4 9 k ^ ,  and G  ¥e  t h e n have  although  stage  of  the  quite  is  output  another,  c a n be  made  common-emitter  Figure The v a l u e  slight  -  of  the  equal  loss  to  i n the  does not  overlooked, is  it  impedance,  l o a d the  last  passband.  by the  is  method of  use  possible  of to  case  for  stage w i l l  have  to  close  emitter  db.  in this  to  output  2.06  and a h i g h i n p u t  a high level, to u n i t y .  achieving  a heavily design the  be  but  for  an  50--n_ l i n e .  included  the  Figure  loaded  to  overall  26 s h o w s  a  ampos-  stage.  26. Common-emitter output  amplification the  buffer  i n p u t impedance  plification sible  matchj  - 1 2 ^ ,  t  dependent,  often  Here,  impedance  u  impedance  and t h a t  stage.  An e m i t t e r - f o l l o w e r the  o  = 0,79 ~  frequency  impedance,  common-emitter  raise  (0)  R  a m p l i f i e r w i t h i n the  There  exact  y m  a low output  which  low output  pair,  resistor  or greater  is  stage  to  chosen  feed to  than u n i t y ,  common-collector buffer  50-.fi- l i n e  make to  the  dc  compensate  stage.  The  voltage for low-  ,  106 frequency  voltage  amplification  G  .(0)  is  = l/G  vm5  (0)  =  A  1.045  vm4  w h e n R _ = 14.4x1.  e5  Although be u s e d in  the  as test By  output peak  the  the  output  stage,  the  common—collector  cascade  p a i r was  could used  amplifier.  operating  voltage  is  stage,  into  a 50-xz. l i n e  limited  output voltage  output  common-collector-common—emitter  is  to  about  and about  0.15  at  low  current  levels,  a low l e v e l .  The m a x i m u m  0.30 v o l t  the  volt  for  for the  the  peak-to-  emitter-follower  common-emitter  output  stage. 6.4  Amplifier  Test  Results  A number of  different  amplitude  response  and phase  responses  low-frequency input,  stage  necessary  it  was  i n order  sweep-frequency  a sweep-frequency  amplifications, amplitude  expected to  generator  displayed  on an o s c i l l o s c o p e  of  bypassing,  for  500 k c  flat  coupling,  to  a  some  measurements, step-function  900-B)  20 M c .  The  detected  and was  used  and a l s o  and g r o u n d i n g .  the  test  circuit  used.  would  amplitude was  response  i n adjusting i n checking  Figure  27  be  response, used  the  the  shows s  diagram of  made:  amplitude  small adjustment  Model  response  been  response.  maximally-flat  to  have  generator,  response  (Jerrold  band from  elements  that  achieve  sweep t h e  back  measurements  using point-by-point frequency  and low—frequency Since  a  using  response  to was feed-  adequacy the  block  107  SweepFrequency Generator Jerrold Model 900-B  Figure  The  element  c a n be  27.  values  after  compared w i t h the were  a m p l i f i e r w i t h the  modified  design values  shown f o r are  adjustment  1 Mc  different  design values t a k e n of  exact for  comparing design  compensating  capacitors  the  second  and 12$ f o r  the  accuracy In  resistors  stage  are  to which the  Order to  phase  in  ranges  they  display both  and the  i n Figure  23;  brackets.  oscilloscope  find  response  very are  to  the  for  amplifier with These  28; the  are  markers  somewhat  the  third  lower,  stage.  upper half-power  the  signals  The l o w - f r e q u e n c y measured; 29.  the  were  The m o s t  seen  These  measurements  used  value  the of  that  are  within  an  and  meas-  generator  amplifier.  test  for  known.  frequency  each  but  4$  are  a m p l i f i e r , a VHF s i g n a l  a m p l i f i c a t i o n of  desirable  is  by about  d i s p l a y e d and measured:on  voltage  above  it  design values,  t r a n s i s t o r parameters  the of  close  values  along w i t h a wideband d i f f e r e n t i a l  and o u t p u t  Figure  given  and a d j u s t e d  the  used  using  shown on F i g u r e  design values  frequency  feedback  was  Display  maximally-flat response.  the  the  the  CRO  apart.  In  ure  are  o  Detector Jerrold Model 50-D  B l o c k diagram of t e s t c i r c u i t sweep-frequency generator  Photographs the  Video Amplifier under t e s t 50-n G = 50 db  Fixed Atten. 4 0 db  The  input  oscilloscope.  stage circuit  was  also  shown  attenuation would  be  in  108  (a) Figure  equal of  to  fixed  output,  (b)  28. Amplitude response u s i n g sweep-frequency generator, (a) e x a c t d e s i g n v a l u e s , (b) m o d i f i e d d e s i g n v a l u e s for maximally-flat response  the  a m p l i f i c a t i o n of  attenuation  was  and d i f f e r e n c e  RF S i g n a l Generator Radiometer Type MSllh  Figure  29.  50-n.  the  test  available.  v o l t a g e s were  Fixed Atten. 4 0 db  amplifier,  but  The m a g n i t u d e s  of  o n l y 4 0 db the  input,  measured.  Video Amplifier under t e s t  B l o c k diagram of t e s t c i r c u i t and phase measurements  u Wideband Differential Input to CRO for  amplitude  109 The  phase  shown i n F i g u r e  was  determined by  30 f o r  each  the  measured  method of  triangulation  frequency.  (V  Figure The  - V.) o 1 3 0 , T r i a n g u l a t i o n method of resulting  amplitude  Figures  31  but  upper half-power  the  response  curve.  The h i g h e r  that  to  gradual  the  fact  sistor  actually  Figure  7)  frequencies input  the  because  cies  Miller  frequency  cutoff  base-emitter  Miller the of  element  collector the  to  thought  to  a conductive broaden the  stage;  partly  w i t h C^-i  (see  has  been at  neghigher  shunted  by  introduces  been n e g l e c t e d .  response.  values.  tran-  is  at  high-  (the  decrease  this  component  the  be  of  fact  the  this  l o a d impedance  following  the  design to  circuit  i n series  Mc, design  stages  than the  tends  to  in  flat,  i n the  due  two  response;  capacitance  l o a d i n g which has  tends  input  is  response,  last  is  shown  than 6.5  as  is  flat  smaller  to broaden the  term contains  which also  above  sharply  the  are  response  7 Mc r a t h e r as  of  chosen  a resistive  capacitance  frequency-dependent the  the  which tends  lected. ~ Ttlso,  the  has  were  responses  achieve  capacitors  falloff  that  fall  to  determination  amplitude  is  half-power  elements  stages)  The  frequency  upper the  The  does not  compensating  high-frequency  due  curve  i n adjusting  frequency  and phase  a n d 32 r e s p e c t i v e l y .  and the  phase  In addition,  higher  On t h e  frequen-  other  hand,  110  I  1  1  1  1.5  2  Figure  31.  Figure  1  1  2.5 3  ,  1 — I  4  5  f Amplitude response  32. Phase  response  1  6.5 (Mc) curve  I  1  8 for  f (Mc) curve f o r  1  II  10 12  15  20  the  test  the  1  test  > -  25  amplifier  amplifier  Ill the  assumption  Miller for is  admittance  the l a s t taken  (OCQRJ^^^I) i n t h e  justified  see E q u a t i o n  which has an a m p l i f i c a t i o n  account,  the amplifier  amplification  is hardly  the t e s t  than predicted.  accounting the  stage  term  stage  into  bandwidth for  of large  It  amplifier  f o r the M i l l e r  difficult  the errors  admittance  If  s h o u l d have  i s therefore  cascade whether  of two.  (2-54) this  a  smaller  to  predict  i n approximation  should  increase  or  in  decrease  bandwidth. The m e t h o d  small  angles,  phase  shift  This  b u t the phase  at the higher  i s also  compensating than  probably  curve  is relatively  frequencies  accurate  linear  for  with  than  i n the design  that  the  less  curve.  high-frequency stages  are  smaller  values. stage  G  G  vml  (0)  v m l  2  were  found  t o be  =-12.7  -  is  (0) = -310 ~ 49.83 d b ,  s h o u l d be compared  G  amplifications  vm (°>  amplification  G  which  i s not very  of the high-frequency  low-frequency  and the t o t a l  phase  due t o t h e f a c t  capacitors  the design The  f o r measuring  w i t h the design  (  0  )  +  G  pad= -  (  1  8  '  2  values  "  2  '  6  of  7  )  =-15.53  G  vm2<°>  -  -  vm3  =  -  G  and the  total  first  which  is  sistance The  2  '  9  0  8  '  amplification G  The  ( 0 )  1 0  stage  (0)  due  to  the  — 3X1, w o u l d  e  following  = -352 - ^ 5 0 . 9 3  amplification  probably (i" g  v m  stages  are  is  lower  presence  account i n close  than of  for  db.  the  design  value,  some e m i t t e r - l e a d  the  lower  re-  amplification).,  agreement w i t h the  design  values. The using  the  response test  response  is of  the  spaced  of  the  leading  test  Figure  of  signal  was  measured  33.  Tektronix T y p e 581 CRO & Type N Sampling Unit  Video Amplifier under t e s t  the  amplifier.  step-function edge  input  B l o c k diagram of t e s t c i r c u i t f o r t r a n s i e n t response measurements  a photograph  80-nanosecond is  33.  step-function  Fixed Atten. 4 0 db  50xx  Figure  34  a  circuit  Tektronix Type 111 Pre-Trigger Pulse Generator  Figure  to  of  1-nanosecond per  the  step-function  input  The u p p e r m o s t  display  input waveform. input waveform;  centimeter.  The  and is  The m i d d l e the  lower  grid  the  display  lines  display  output  is  are the  Figure  amplifier  response  centimeter. is  faster  ment  of  The  the  source plug-in  with grid lines  was  low-frequency s i m i l a r to Wave  is  was  step-  10-nanoseconds  design value.  No v a l i d  amplitude  response  was  that  of  Figure  was  545A o s c i l l o s c o p e  about  was  44 cps  used as  for  which measure-  available.  measured u s i n g  29 e x c e p t  (Model AG-10)  per  50 n a n o s e c o n d s ,  equipment  preamplifier)  frequency  by  a  w i t h the  a n d a T e k t r o n i x Type  half-power  to  possible  Generator  dual-trace  spaced  approximately  58-nanosecond  overshoot  circuit  Sine-Square  Test a m p l i f i e r response function input  risetime  than the  The test  34.  used  that for  a  a Heathkit the  signal  ( w i t h a Type display.  shown i n F i g u r e  CA  The 35.  lower  114  0  f  /o  -3  o/  (design)  /  (measured)  /  G  (0)  50.93  db  G(0) vm  49.83  db  y m  7  /  -6 +  -10 10  20  Figure  35.  -f-  40  50  100 f  200 (cps)  Low-frequency amplitude f o r the t e s t a m p l i f i e r  500  response  •4-  1000  115 7 . CONCLUSIONS Two p r i n c i p a l c o n c l u s i o n s this  work.  The  applicable  to  amplifiers,  iterative  the  design  leading  to  method of of  designs  o b t a i n e d by the  Giacoletto  hybrid-n equivalent  in  of  the  the test For  up t o  with a fixed  7.6  db w a s  corresponding  was  less  linear  the  dc  and t h e r e  varying  the  interstage  of  meaningless  small,  the For  of  dc  when u s i n g  indeed video  i n some  a useful  respects  representa-  frequency  range  used  using  2N384  drift  delay  more I n the  emitter  stages.  but  for  amplifier phase  the  was  used  over  response in  designs  low-frequency  a cut-and-try  It  advantage  observed  method, that  the  by B r u u n and G r i n i c h  optiis  feedback. one  Since  tuning patterns  amplifier,  the  an  and overshoot  design,  optimized by  the  2 ma,  three-stage  was  resistance  successive  of  However,  t u n i n g p a t t e r n u s e d was  different  improve  the  resistance.  interstage  for  three-stage  for  current  design.  method.  a m p l i f i c a t i o n was  feedback  the  frequency,  emitter  achieved  current  The  of  The m o d i f i e d J o h n s o n -  c i r c u i t gives  throughout  is  transistor  performing better  cutoff  Grinich  iterative  mum v a l u e  synthesis  G r i n i c h method.  transistor  Mc u p p e r  the  using  network  results  amplifier. a 6.5  transistor of  drift  drawn from the  emitter-feedback  than those  tion  c a n be  were  more  of  progressively  the not  loading  effect  investigated  stages  this  greater is  for  m i g h t be  the  used  to  performance. low v a l u e s  emitter  current,  of  interstage  convergence  resistance was  slow  and/or  low  and problems  values of \  116 physical  r e a l i z a b i l i t y appeared.  realizability Further  has  been  w o r k c a n be As a u s e f u l  has  suggested  done  designs  byproduct  w i t h up t o  Several w o u l d be  desirable  realizability operating  currents, of  to  the  being used  eight  stages.  areas  for  for  future  impedance  point.  It  would also  be  and l e t  interstage  them be  and to  possibly  problems.  quite  far  further  as  are  program ampli-  apparent.  ensuring or  transistor  useful  to  remove  and e q u a l to  allow  impedance  maximize  the  dc  It  physical  level  resistances  as  a computer  study  design variables, stages  physical  maximally-flat  a means o f  interstage  equal  these  research,  any  of  predicting  direction*  investigate  i n p u t and output  concerned,  of  of  of  avoiding  at  restrictions  ling  possible  for  in this  been w r i t t e n capable  tude  A method  the  emitter  easy  hand-  matching current  is  am-  plification. A study circuit stage  of  high-level  c o u l d be v e r y  signal-handling  useful,  injection effects especially  on the  equivalent  i n r a i s i n g the  output  capabilities.  7 It peaked with  has  of  worthwhile peaked  shown b y P e p p e r  transistor video  pole-zero  product  been  to  a m p l i f i e r , w h i c h has  dependence,  any of  the  apply  amplifiers  to  and P e d e r s o n  gives  the  the  iterative  o b t a i n the  greatest  shunt-  function  gain-bandwidth  techniques.  method to  the  a transfer  greatest  common b r o a d b a n d i n g  that  the  design  possible  It of  may  be  shunt-  amplification.  117 REFERENCES 1.  Bardeen,  J . , and B r a t t a i n , W . H . , "The T r a n s i s t o r , A S e m i c o n ductor T r i o d e , " P h y s i c a l Review, 74:230-1, ( J u l y 15, 1948)  2.  Ryder,  3.  Wallace,  R . L . , J r . , a n d P i e t e n p o l , W . 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