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Spectral distribution of noise Bain, William Arthur 1950

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SPECTRAL DISTRIBUTION 03? NOISE  William Arthur Bain  ABSTRACT of t h e s i s submitted i n p a r t i a l f u l f i l m e n t o f the requirements f o r the degree o f MASTER OF APPLIED SCIENCE i n the Department of ENGINEERING  PHYSICS  UNIVERSITY OF BRITISH COLUMBIA April,  1950  ABSTRACT An.apparatus f o r the measurement of n o i s e as a f u n c t i o n o f frequency i s d e s o r i b e d .  T h i s apparatus has been  used to determine the s p e c t r a l d i s t r i b u t i o n of the excess n o i s e oaused by the f l o w o f a d-o. o u r r e n t through a r e s i s t a n c e . -The samples used f o r the experiments were a z i n c oxide semiconductor and two.metal l a y e r r e s i s t o r s . r e g i o n i n v e s t i g a t e d was I t was  from 10 k c . to 400  The frequency ko.  found t h a t the excess n o i s e i n the ZnO  semi-  conductor obeyed a ^/*law a t room temperature w h i l e a t lower temperatures ( s o l i d C 0  2  and l i q u i d n i t r o g e n ) i t was  p r o p o r t i o n a l to '/ a t low f r e q u e n c i e s and £zat h i g h f r e q u e n c i e s . v  The excess n o i s e i n the metal l a y e r r e s i s t o r s  was  p r o p o r t i o n a l to '/^ a t room temperatures w h i l e a t 100°0 there was a marked d e v i a t i o n from t h i s law i n the d i r e c t i o n o f a z^z dependence f o r h i g h f r e q u e n c i e s . The measurements show t h a t the  law g r a d u a l l y  changes to a 'fa law a t h i g h f r e q u e n c i e s i n accordance w i t h the theory proposed r e c e n t l y by Dr. A. van der z i e l .  They  a l s o i n d i c a t e t h a t the c o r r e l a t i o n times i n v o l v e d are a f u n c t i o n o f temperature; the e x a c t nature o f t h i s dependence has y e t to be determined.  SPECTRAL DISTRIBUTION OP NOISE by William Arthur Bain  t h e s i s submitted i n p a r t i a l  fulfilment  the requirements f o r the degree o f MASTER OP APPLIED SCIENCE i n the Department of ENGINEERING PHYSICS  THE UNIVERSITY OP BRITISH COLUMBIA April,1950  TABLE OF  CONTENTS  Page Introduction  . . . . . . .  1  Apparatus, Samples  3 .  12  Theory  13  Results  . . . . .  22  Conclusions  . 35  Acknowledgements • •  38  Bibliography  39  .  Figure  I  B l o c k d i a g r a m "of a p p a r a t u s  • •  4  Figure  II  D e t a i l e d w i r i n g diagram o f p r e - a m p l i f i e r  . •  5  •Figure  III  Detailed wiring band  diagram  pass f i l t e r  Figure  IV  Circuit  of direct  Figure  V  Input c i r c u i t  Figure  VI  Equivalent  of o s c i l l a t o r ,  and  current  detector  mixer,  . . . . .  supply  7  of a m p l i f i e r . . . . . . . . .  input  circuit  of amplifier  6  . . .  9 9  SPEC THAI,  DISTRIBUTION  CHAPTER  OF  NOISE  I .  INTRODUCTION  The  random m o t i o n  small f l u c t u a t i n g p o t e n t i a l  of electrons differences  the t e r m i n a l s o f the conductor.  i n a conductor  causes  t o be d e v e l o p e d  across  T h i s i s , known a s  Thermal  N o i s e a n d i t c a n be d e s c r i b e d b y a n o i s e e.m.f. i n s e r i e s w i t h R whose mean square, v a l u e , f o r a f r e q u e n c y i n t e r v a l  A  i s given by the formula  where k i s B o l t z m a n n ' s c o n s t a n t , T i s t h e a b s o l u t e in  degrees K e l v i n ,  and S i s the r e s i s t a n c e .  temperature  The r e s i s t a n c e  R may depend on f r e q u e n c y i n w h i c h c a s e one must u s e t h e value  of R f o r the p a r t i c u l a r  vestigation. checked  etically  T H i s f o r m u l a , due t o l y q u i s t ,  and i s found  infra-red  r e g i o n o f the spectrum. from  under i n -  has been  carefully  t o be t r u e f o r f r e q u e n c i e s up t o t h e  thermodynamical  b e e n shown e x p e r i m e n t a l l y (3) of  frequency i n t e r v a l  I t has been proved  c o n s i d e r a t i o n s ( 1 ) , ( 2 ) and h a s that  this noise lis  the t y p e o f r e s i s t a n c e w h e t h e r i t be c a r b o n ,  wire-wound, t h i n m e t a l  theor-  independent composition,  layer,, or semi-conducting.  -The formula does not always hold i f a d-o current i s passing through the r e s i s t a n c e .  In general i t can he  s a i d that i f a d-o current i s passing through R there i s an excess noise generated that increases w i t h i n c r e a s i n g ourrent. U s u a l l y t h i s Increase i s as the square of the c u r r e n t .  One  can describe t h i s e f f e o t by i n t r o d u c i n g an a d d i t i o n a l noise e.m.f.  i n s e r i e s w i t h R so that f o r a small  frequency  interval  e«* where the f u n c t i o n of t h i s excess noise.  I fat)  A)J  z  describes the frequency dependence I t i s u s u a l l y of the form I! '  y  f o r a f a i r l y wide frequency range.  For a wire-wound r e s i s t o r  ;we have  fcv)«  o.  The object of t h i s research i s v/to measure  as  a f u n c t i o n of the frequency V and to measure <%* as a f u n c t i o n of current f o r various m a t e r i a l s a t d i f f e r e n t temperatures. i n order to measure. frequency band of width  the noise i n a o e r t a i n  small i n comparison to the f r e -  quency \f ) i s a m p l i f i e d and detected by a quadratic detector. If  i s the d e f l e c t i o n of the meter f o r 1 = 0 and.Dg the  d e f l e c t i o n f o r a f i x e d ourrent I then:  - 3 -  D,  4A  '  y-R.  so t h a t :  As soon as  and  Dg nave "been measured f o r d i f f e r e n t f r e -  quencies ^tO/Joan be c a l c u l a t e d as a f u n c t i o n of f r e q u e n c y . I f Dg  i s much l a r g e r than D i one might use a c a l i b r a t e d  attenuator  to b r i n g the r e a d i n g  a reasonable v a l u e . has  of the output meter down to  I f A i s the a t t e n u a t i o n  to s u b s t i t u t e ADg  f o r Bg  CHAPTER  i n the above  f a c t o r then  one  equation.  II  APPARATUS A b l o c k diagram of the apparatus used i s shown i n Figure I.  The  d e t a i l e d w i r i n g diagram of the  i s shown i n F i g u r e I I . ourrent  The h i g h t e n s i o n and  pre-amplifier the  are s u p p l i e d by a r e g u l a t e d power supply  t i o n a l design. i t s e l f was  The  filament o f conven-  frequency response.of the , p r e - a m p l i f i e r  found to be e s s e n t i a l l y f l a t between 1 ko.  400  ko.,  the half-power f r e q u e n c i e s  500  ko.  When the apparatus was  r e s i s t a n c e was  b e i n g at 500 v.  f i r s t t e s t e d the  found to be 3000 ohms.  and  cycles  and  noise  Tb4s has been reduced  - 4 -  FIGURE  I  to 800 ohms by u s i n g wire-wound r e s i s t o r s c i r c u i t s o f the f i r s t  two stages.  i n the p l a t e  The maximum v o l t a g e g a i n  of the p r e - a m p l i f i e r i s 40,000. The d e t a i l e d w i r i n g diagram of the l o c a l o s c i l l a t o r and mixer, c r y s t a l hand-pass in Figure I I I .  filter,  and a m p l i f i e r i s a2a.own  The frequenoy range o f the l o o a l  i s from 465 k c . to 1300 ko.  osoillator  The c r y s t a l band-pass  tuned a t 456 ko. and has a w i d t h of 200 c y c l e s .  filter i s  This allows  us to examine the n o i s e i n a narrow hand f o r a l l f r e q u e n c i e s from about 10 ko. up t o the p o i n t where the response o f the q i r c u i t as a whole f a l l s  to a v e r y low v a l u e .  T h i s upper  l i m i t of frequency i s i n the neighborhood o f 400 k c . A detector.  1U-23 c r y s t a l diode i s used as a q u a d r a t i c I t s q u a d r a t i c p r o p e r t i e s have been checked s e v e r a l  —  7 -  time during the course of the r e s e a r c h . The output meter i s a Rubicon galvanometer a t e n - c e n t i m e t e r s c a l e and a s e n s i t i v i t y  with  o f 0.0056 m i c r o -  amperes per m i l l i m e t e r . (0.56 microamperes f u l l  scale)  A diagram of the c i r c u i t used f o r s u p p l y i n g the d-c. c u r r e n t to be passed through the sample i s shown i n F i g u r e IV.  FIGURE  IV  Seven 4 5 - v o l t dry c e l l s a r e u s e i to supply the n e c e s s a r y d-o v o l t a g e .  These b a t t e r i e s and the ammeter are e n c l o s e d  i n a grounded metal box.  From t h i s b a t t e r y box the d-o.  c u r r e n t i s f e d through a s h i e l d e d c a b l e to another  grounded  metal box where i t i s f i l t e r e d and then a l l o w e d to pass through the sample under i n v e s t i g a t i o n . were taken to e l i m i n a t e the p o s s i b i l i t y the  These p r e c a u t i o n s o f feed-back through  power supply and to minimize the e f f e c t o f any  electrical  d i s t u r b a n c e s i n the v i c i n i t y .  stray  As a f u r t h e r  - 8 p r e c a u t i o n a g a i n s t these s t r a y e l e c t r i c f i e l d s  (due to  fluorescent lights,  done i n s i d e  a cage made of two wooden frame.  etc.) a l l the r e s e a r c h was  l a y e r s o f chicken-wire mounted on a  Both these l a y e r s are s e c u r e l y grounded and  power i s brought i n t o the cage by means o f an transformer.  Even w i t h these p r e c a u t i o n s the apparatus  a tendency to be u n s t a b l e a t times; t h i s i s due  isolating has  I t i s not known whether  to s t r a y f i e l d s or to some f a u l t y component i n  the apparatus  itself.  These p e r i o d s of i n s t a b i l i t y , however,  are v e r y i n f r e q u e n t and  short-lived.  During normal o p e r a t i n g c o n d i t i o n s the  apparatus  i s q u i t e steady but shows a slow v a r i a t i o n of g a i n so t h a t over a p e r i o d of say 30 minutes the r e a d i n g of the g a l v a n ometer may  change by as much as 10$.-  the apparatus  was  I t was  found t h a t when  l e f t untouched f o r . a p e r i o d of many hours  t h i s d r i f t never exceeded 10$ and from day to day w i t h i n t h i s  data c o u l d be  reproduced  limit.  The galvanometer has a r a t e d time constant of three seconds.  T h i s was  i n c r e a s e d to approximately  30  seoonds^by:vlhsertding three stagescof- f i l t e r i n g i n f r o n t o f :  the galvanometer.  Each stage c o n s i s t s of a 5000  ohm  r e s i s t o r i n s e r i e s w i t h a 2000 m i c r o f a r a d condenser. was  This  done to damp out some v e r y r a p i d f l u c t u a t i o n s of the  reading.  The cause o f these i s b e l i e v e d to be due  f a o t that- r e s i s t o r s g i v e out b u r s t s of e x t r a n o i s e and above those expected. b u r s t s may  to the over  I t has been s a i d t h a t these  be as l a r g e as 300$ of the average v a l u e .  In order to show that the apparatus was working properly the f o l l o w i n g t e s t was performed.  Nyquist's  formula:  or:  —  7  46 ~r z\\J  7?. shows that the mean square value of the noise e;m.f• increases l i n e a r l y w i t h r e s i s t a n c e . However, i n any p r a c t i c a l c i r c u i t t h i s r e s i s t a n c e i s i n p a r a l l e l w i t h both the input c a p a c i t y and the g r i d r e s i s t a n c e of the f i r s t stage o f the a m p l i f i e r . This g r i d r e s i s t o r a l s o produces a noise v o l t a g e . Consider the c i r c u i t s shown i n Figure V and Figure VI where C i s the input c a p a c i t y , R^ i s the g r i d r e s i s t a n c e and R  a  i s the r e s i s t a n c e of the sample under  consideration.  IT, 7  ©  FIGURE V  FIGURE Y l  -  F i g u r e  V  s h o w s  F i g u r e  7 1  s h o w s  T h e  i s  g i v e n  a n d  t h e  t h e  b y  m e a n  -  a c t u a l ; c i r c u i t  t h e  e q u i v a l e n t ,  v o l t a g e  t h e  1 0  i  n  t h e  a m p l i f i e r  a n d  c i r c u i t .  d e v e l o p e d  b e t w e e n  t h e  g r i d  a n d  g r o u n d  f o r m u l a :  s q u a r e  v a l u e  o f  t h e  v o l t a g e  i  s  g i v e n  b y :  w h e r e  3.  ft. T h i s  t h e r e  i  s  i  t h e  s  a  '  e q u a t i o n  m a x i m u m  o  f  a  f a m i l y  w h e n :  /  7 T h i s  m a x i m u m  v a l u e  i  s  :  ^s * 1  o  f  c u r v e s  a n d ,  i  f  -w<?^j >  / J  - 11 -  /0O  •  "•  so  <2  s o o *  / o c a *  A/o J I f we l e t x approaoh i n f i n i t y H  ( i . e . l e a v e the i n p u t open and  becomes i n f i n i t e ) we have  2  —  By p l o t t i n g a graph o f mean square v o l t a g e v e r s u s  i t is  p o s s i b l e to o b t a i n the maximum value o f the curve and we oan then s o l v e equations 5 and 6 f o r the two u n k n o w n s ^ and 0. T h i s has been done and the r e s u l t s a r e shown on Graph U o . l f o r a frequency o f 50,000 c y c l e s .  The R^ has a value o f  460,000 ohms and maximum value o f ^ I s when R, ohms.  =  190,000  U s i n g t h i s data we g e t a value o f 24 micro-micro ^-farads  f o r C which i s q u i t e reasonable f o r a c i r c u i t  of t h i s  type.  - 12 S u b s t i t u t i n g the known v a l u e s f o r C andJ@ i n t o equation a l l o w s us to c a l c u l a t e a ourve. a l s o p l o t t e d on Graph U o . l and  2  T h i s t h e o r e t i c a l curve i s i t s olose resemblance to the  a c t u a l curve i s a good i n d i o a t i o n t h a t the apparatus i s working p r o p e r l y .  CHAPTER  III  SAMPLES The metal l a y e r r e s i s t o r s used were obtained C o n t i n e n t a l Carbon i n c o r p o r a t e d . ohm  from  They are o r d i n a r y 50,000-  p r e c i s i o n r e s i s t o r s and c o n s i s t o f a t h i n m e t a l i z e d  l a y e r deposited on a oeramio m a t e r i a l . The z i n c oxide semi-conductor was  made i n the  l a b o r a t o r y as f o l l o w s : z i n c oxide powder was mould and  subjected to a pressure of 10,000 pounds per  square inqh. was  placed i n a  The r e s u l t i n g b l o c k o f m a t e r i a l (2" < /  there f o r twelve hours, p e r i o d o f twenty-four  sealed.  i t was  hours.  were p a i n t e d on the ends,  tube and  l  e  heated to 1200°C a t a r a t e o f 100°C per hour and  these c o n t a c t s and  * / )  ?  allowed  &  held  to c o o l over a  After cooling silver  contacts  p l a t i n u m l e a d s were j o i n e d to  the whole assembly was  placed i n a glass  CHAPTER  IV  THEORY A d-o.  current flowing  through a r e s i s t o r causes  an e x t r a n o i s e whioh, f o r a s m a l l frequency i n t e r v a l i s given by the  ^1^,  formula:  where I i s the d-o.  o u r r e n t and  ftv)  i s a function describing  the frequency dependence o f t h i s induced n o i s e . We we  oan  e x p l a i n the 1^  dependence of the n o i s e i f  assume that the r e s i s t a n c e shows spontaneous f l u c t u a t i o n s  i n value, extra  i f the f l u c t u a t i o n i n r e s i s t a n c e i s ^R  then an  noise =  i s developed a c r o s s  T<d  the r e s i s t a n c e R.  *  The mean square  value  of t h i s noise i s  To e x p l a i n these f l u c t u a t i o n s i n r e s i s t a n c e we to the d e f i n i t i o n of c o n d u c t i v i t y as g i v e n by the theory  of matter.  The  turn  electron  conductivity cTis:  cr  = — — —  C/tr  where n i s the number of f r e e e l e c t r o n s per c u b i c  centimeter,  - 14e the e l e o t r o n i o charge, L the f r e e path l e n g t h b f the e l e c t r o n s , and v the v e l o c i t y o f the e l e c t r o n s . uations  The f l u c t -  o f r e s i s t a n c e may he due to a f l u c t u a t i o n i n the  number o f f r e e e l e c t r o n s p e r c u b i c c e n t i m e t e r .  There i s a l s o  the - p o s s i b i l i t y that the f r e e path l e n g t h I» or the v e l o c i t y v of the e l e c t r o n s show f l u c t u a t i o n s .  To take a l l these  p o s s i b i l i t i e s i n t o account we i n t r o d u c e  a f l u c t u a t i o n i n the  c o n d u c t i v i t y as f o l l o w s : /\ This  =  C7~ - C 7 7  ACT-  =O  Si  gives  to a good approximation.  We now have f o r the value o f  We must o a r r y out a F o u r i e r a n a l y s i s o f  :  and c a l c u l a t e  C o n s i d e r a f l u c t u a t i n g q u a n t i t y X ( t ) which i s known f o r the i n t e r v a l 0<-t<t where t o  i s large but f i n i t e .  I f we  assume t h a t Z ( t ) i s a continuous f u n c t i o n i n the i n t e r v a l we may develop i t i n t o a complex F o u r i e r s e r i e s as f o l l o w s :  x  # J =  ^  =  «j  fa  *  *  15  9.  ~  tf -0 o  7  because  X&)  = O.  I f a ^ i s the complex conjugate o f a ^ e see t h a t a _ = a^*" n  The F o u r i e r  component x  n  o f frequency £ ^ i s  and i t s mean square v a l u e i s  because a l l terms c o n t a i n i n g the time to drop out when the average  i s taken.  2 a  .• From equation 9 we o b t a i n :  n  We now have to c a l c u l a t e the v a l u e o f  XU)X(«r)  e-'^*^*****  /z.  - 16 We now change to a new v a r i a b l e w.  where x(u) X(u+w) i s independent only,  o f u and i s a f u n c t i o n o f w  we a l s o know that Z(u) X(.u+w) i s symmetrical  equals zero f o r /.^/-/ > 8 c o r r e l a t i o n time.  where  Therefore i f %  S  i s a measure o f the  >> S  equation 14 may be  rewritten as:  °4  And, i f we put Y  4 -  = & \J  to  So  i n w and  , equation 15 becomes:  -17  -  sinoo the imaginary term c o n t r i b u t e s n o t h i n g to the v a l u e o f the i n t e g r a l .  E q u a t i o n 16 may  be more c o n v e n i e n t l y w r i t t e n as:  /7  where: /a  and i s c a l l e d the "normalized" c o r r e l a t i o n c o e f f i c i e n t .  We  see t h a t o(w) = 1 and where  for  w = o  o(-w) •= o(w) o(w) = 0 i f  |w|  >>  i s the c o r r e l a t i o n time o f the f l u c t u a t i o n s . -  When we  c o n s i d e r the case of a f l u c t u a t i n g q u a n t i t y which i s oaused by a l a r g e number o f independent and random events 2 " i s the d u r a t i o n o f the event (e.g. the t r a n s i t time of an e l e c t r o n i n a radio tube).  i n the case of f l u c t u a t i o n s i n v o l v i n g  problems 2"is a measure o f the average l i f e  decay  o f the decaying  quantity. We now  have f o r the mean square v o l t a g e i n a s m a l l  frequenoy i n t e r v a l  ^  V  - 18 Tne  spectral  mined by  the  applying a Fourier  It  f u n o t i o n ^ / ^ J i s completely  distribution correlation  -  f u n c t i o n o(w).  f o r we  find  deterby  transform:  i s .evident from t h i s  equation  that i t i s not p o s s i b l e f o r  j&6J) t o be o f t h e f o r m  &<  for  integral also  AJ <  i f we  substitute E l into  that i s divergent at  divergent at ^ J '  3 1 0 0  &J-0±QT  f o r w = o.  o f 20 f o r a l l f r e q u e n c i e s we  To  20 we  a l l values ensure the  must impose the  o b t a i n an o f w and  is  convergence  following re-  strictions: must v a r y  slower  than  /£/ f o r v e r y  low  must v a r y f a s t e r  than  /fa f o r v e r y  high  frequencies. b) ^Xu) frequencies. Of  course  the ^  law  i s satisfactory f o r intermediate  fre-  quencies. I t has  u s u a l l y b e e n assumed t h a t t h e  f u n c t i o n r e p r e s e n t s an  correlation  e x p o n e n t i a l decay o f h a l f - l i f e  i.e.  and  hence, from equation  £l>)  -  19 we o b t a i n :  C0/13-/<?s?/- /  €  2-  COS AJ^US-  <#C<SS-  23.  T h i s s t a t e s t h a t j ^ t v ) i s independent o f frequency, a t low frequencies relationship  and v a r i e s a s ^  at high  i s i n marked c o n t r a s t  frequencies.  to the experimental  where one u s u a l l y f i n d s t h a t t h e n o i s e large frequency  This results  v a r i e s a s />/ i n a  range.  A s o l u t i o n t o t h i s problem h a s been proposed r e c e n t l y b y p r . A. v a n d e r z i e l .  (4)  instead o f using a  s i n g l e c o r r e l a t i o n t i m e "2" we i n t r o d u c e of c o r r e l a t i o n  a wide  distribution  times,  let  be  t h e p r o b a b i l i t y o f a c o r r e l a t i o n t i m e between  and  H^f*  T h i s g i v e s f o r jkv) i n s t e a d o f 23:  Of  course by a proper o h o i o e . o f ^ ^ J o n e  a g r e e m e n t between t h e o r y with  c a n always  a n d e x p e r i m e n t e v e n i f one s t a r t s  t h e wrong c o r r e l a t i o n f u n c t i o n .  Therefore  we c a n a t t r i -  bute p h y s i c a l meaning t o t h e whole p r o c e d u r e o n l y are  obtain  sound a r g u m e n t s i n f a v o u r  o f the d i s t r i b u t i o n  i f there function  -  o h o s e n .  i  f u n c t i o n  t  w i l l  I f  w e  w e  b e  l a t e r  i n t r o d u c e  o b t a i n  f t ' ? ' ) * ' ?  a n d  s h o w n  t h e  •£/  i  t  t h a t  f o l l o w i n g  l a w  e x a c t l y .  f o r  i n t o  t h i s  t h e  •= 0  s u b s t i t u t e  2 0  2 5 .  s  a  c  t  u  a  l  n o r m a l i z e d  l  y  t h e  c a s e .  d i s t r i b u t i o n  L e t  ?*  e q u a t i o n  i  ?>  >V  T h i s  Z  g i v e s  27  = ear?*/**?/ //r?  — ) 2  Thi3 l o w  g i v e s  a s  .  W e  d i t i o n s  n o w  a )  i  s  a s  h a v e  a n d  b )  i  a n  o n  a s  '/j  h i g h  s  l o s s e s  (5))the  eJ.  t h a t  i  f o r  n e c e s s a r y  e x p r e s s i o n  P a g e  w h e n  s  i n d e p e n d e n t  i n t e r m e d i a t e  f r e q u e n c i e s .  b y  a  O n e  ^  f o r  v e r y  e x t e n d  c h o i c e  t h a t  •  f r e q u e n c i e s ,  o a n  p r o p e r  for/c)))  o f  .  o f  s a t i s f i e s  a n d  t h e  27  a n d  c o n -  1 8 .  i n t r o d u c t i o n  r e a s o n a b l e  d i e l e c t r i c  p h y s i c s  v e r y  a s f a r  T h e  t i m e s  v a r i e s  f o r  r e g i o n  offtv)  v a l u e  f r e q u e n c i e s ,  v a r i e s  ?2  a  *J n - *"-cA?« *J  1  o f  w e  ( w h i c h  i  d i s t r i b u t i o n  c o n s i d e r  s  c o r r e l a t i o n  T--  a  a l s o  a  t i m e  i  t h a t  p r o b l e m  s  g i v e n  i  n  o f  o f  t h e  c o r r e l a t i o n  t h e o r y  s o l i d  o f  s t a t e  b y  za  -  21  where E i s the a c t i v a t i o n energy.  I t i s evident that a rather  narrow d i s t r i b u t i o n of E w i l l g i v e q u i t e a l a r g e d i s t r i b u t i o n i n T because  kT i s a s m a l l q u a n t i t y a t room temperature.  Of  course, i t i s not to be construed from these remarks t h a t t h i s i s the s o l u t i o n to the problem of the s p e c t r a l  distribution  of n o i s e - equation 28 i s i n t r o d u c e d f o r the s o l e purpose of g i v i n g a p h y s i c a l meaning to a d i s t r i b u t i o n of c o r r e l a t i o n times. We might a l s o t u r n to the f i e l d of N u c l e a r where we have t h a t the h a l f - l i f e by an equation of the  physios  of an e x c i t e d s t a t e i s g i v e n  form  where E i s the e x c i t a t i o n energy.  Again, a narrow v a r i a t i o n of  E w i l l g i v e r i s e to a wide d i s t r i b u t i o n i n the v a l u e s of Equations 28 and (and thus the form of ftv))  f  29 both show t h a t the v a l u e s o f ~Z might be dependent on  temperature.  This v a r i a t i o n of the frequency dependence of the n o i s e w i t h temperature  will,  .  of course, be the governing f a c t o r i n the  choice of a s u i t a b l e expression f o r  - 22 CHAPTER  V  RESULTS Excess  noise as a f u n c t i o n of current a t a constant  frequency.  •Graph Ho.2 shows, on l o g a r i t h m i c c o o r d i n a t e s , the, excess the  noise  i n the metal  layer r e s i s t o r s as a f u n c t i o n o f  d-c. c u r r e n t f l o w i n g .  The r e a d i n g s were t a k e n  temperature.  F o r low v a l u e s  as  o f the ourrent while a t higher  the square  increases less rapidly, with  o f the c u r r e n t the n o i s e  ourrent.  these  i n the frequency  c u r r e n t s i>t  a r e independent o f  range examined.  Bemamont  same r e s u l t s f o r a n e v e n w i d e r f r e q u e n c y All  plotted period  the readings  on t h e g r a p h .  f o r t h e n o i s e a t 15 k c . a r e  F o r the sake o f c l a r i t y  o f the r e a d i n g s  (6) f o u n d  range.  These r e a d i n g s were o b t a i n e d  of several houls.  average v a l u e s  taken  increases  From t h e g r a p h i t c a n  a l s o be s e e n t h a t t h e s h a p e s o f t h e c u r v e s frequency  a t room  are plotted  over  a  only the  f o r 31 k c .  The  n o i s e v a l u e s on t h i s and a l l s u b s e q u e n t g r a p h s a r e p l o t t e d in arbitrary  units.  A l l curves  to t h e same s c a l e b u t t h e r e nitudes  o f the n o i s e  on t h e same g r a p h a r e p l o t t e d  i s no r e l a t i o n between  on eaoh s e p a r a t e  graph.  t h e mag-  The a c t u a l u n i t s  O  e  of noise are (volts)  p e r cy&le  measured i n m i l l a m e t e r s multiples since  thereof.  deflection  Since  but, o f course,  the n o i s e  i s simply  o f the g a l v a n o m e t e r o r  the apparatus  the band-pass i s c o n s t a n t  correct  but here  this  i s q u a d r a t i c and  unit  o f measurement i s  the a b s o l u t e magnitude  i s dependent  - 23 -  - 24 -  /OOO &/?&/?//  A/Q, 3-  - 25 -  - 26 on the g a i n of the apparatus which oan be v a r i e d over a wide range. Graph Ho. 3 shows the excess noise i n the same metal l a y e r r e s i s t o r s f o r a temperature of about -75°0 ( s o l i d carbon d i o x i d e ) .  Only the average p o i n t s have been p l o t t e d .  The., J r e s i s t a n c e at t h i s temperature i s almost the same as a t room temperature.  Again we n o t i c e that the shapes of the  curves are independent of, frequency and that the slopes a t low currents are approximately 2.  We also see that the slopes  decrease as the current r i s e s . Graph Ho. 4 i s a p l o t of the excess noise i n the semi-conductor as a f u n c t i o n of d-o. current a t room temperature.  The r e s i s t a n c e at t h i s temperature i s 600,000 ohms.  The semi-conductor was connected i n s e r i e s w i t h a one-megohm wire-wound r e s i s t o r .  This wire-wound r e s i s t o r c o n t r i b u t e d  nothing to the excess noise (since^/-cv) O. s  ) butf i t s presence  was necessary to keep a l l the semi-conductor noise from being grounded through the f i l t e r .  The slope of t h i s curve i s  approximately 1.4 over the whole range of c u r r e n t .  The n o i s e  f o r lower currents could not be determined w i t h any degree of aocuracy because of a large v a r i a t i o n i n the readings. I t was found that the lower the currents used the more e r r a t i c were the readings.  This f a c t was also n o t i c e d by Bernamont  (6). Excess noise as a f u n c t i o n of frequency f o r a r e s i s t o r c a r r y i n g a constant current.  - 27 Graph No.  -  5 shows, on l o g a r i t h m i c c o o r d i n a t e s ,  the  excess n o i s e i n the metal l a y e r r e s i s t o r s as a f u n c t i o n of frequenoy f o r three d i f f e r e n t v a l u e s of d-c. c u r r e n t . readings are f o r room temperature.  The curve f o r  m i l l i a m p e r s o n l y snows the average v a l u e s . all  three curves a r e -1 w i t h i n the experimental  o±fcv)±a given  i n d i c a t e s t h a t the form  f<v)  in  The  t h i s frequency  -  ,The  0.25  s l o p e s of error.  This  by:  p  range.  Graph No.  6 shows t h a t the excess n o i s e a t -75°C i s  a l s o i n v e r s l y p r o p o r t i o n a l , to frequency  i n the range inves-?  tigated. However, Graph No. d i s t i n c t departure to  a frequency  frequency  7, taken a t 100°C, shows a  from t h i s law a t h i g h e r f r e q u e n c i e s ,  up  o f about 40 ko. the slope i s -1 and above t h i s  the slope a t t a i n s a value of  -1.6.  A f t e r the readings a t 100°C had been taken r e a d i n g s a t room temperature were repeated and were the same as f o r the f i r s t  trial.  the  the results  This i n d i c a t e s that  h e a t i n g and c o o l i n g the r e s i s t o r s had no permanent e f f e c t the s p e c t r a l d i s t r i b u t i o n of the n o i s e . a l l three temperatures was being l e s s than 10$. p l o t t e d and found All  on  The r e s i s t a n c e a t  almost constant - the v a r i a t i o n  The response of  the apparatus  to be, the same a t a l l t h r e e  was  temperatures.  r e a d i n g s have been oorreoted f o r the decrease  of gain at  - 28 -  -  29 -  J _____—,  S>/it  \©  N.  £>A-  \ G  S'sTf* -  /o  /  O  G  /ooo  - 31 higher  frequencies.. Graph l b . 8 shows the excess n o i s e  i n the semii-  conductor as a f u n c t i o n of f r e q u e n c y a t a constant and  a t room temperature.  current  The slope o f t h i s curve i s -2  over  the whole frequency range whioh i n d i c a t e s t h a t the form o f /CV) i s :  Costs 6&<n f  y*.  /cv)  Graph Ho. 9,  f o r a temperature o f r-75°0,  the noise,at, low f r e q u e n c i e s  shows that  i s p r o p o r t i o n a l to V'  the n o i s e a t h i g h f r e q u e n c i e s  and t h a t  i s p r o p o r t i o n a l to V ~  Z  . By  extending the two p o r t i o n s o f t h i s we see that, they meet a t a frequency o f about 70 ko.  T h i s " t r a n s i t i o n frequency", as  i t might be c a l l e d , cannot be a c c u r a t e l y determined but i t i s the most convenient q u a n t i t y t o use when comparing graphs. Graph No. 10,  f o r a temperature o f - 1 8 6 ° C ,  also  shows t h i s t r a n s i t i o n o f the n o i s e dependence from low  frequencies  to V~  Z  a t high  frequencies.  at  I t i s a l s o t o be  noted t h a t the t r a n s i t i o n frequency, a t -186?0 i s almost the same as t h a t f o r The  -75°C  r e s i s t a n c e of the semi-conduotor changes v e r y  r a p i d l y w i t h temperature, a t u r e , 15 megohms a t - 7 5 ° C ,  i t i s 0.6  megohms a t room temper-  and 60 megohms a t - 1 8 6 ° C .  The  frequency response of the a m p l i f i e r i s g r e a t l y i n f l u e n c e d by the i n p u t r e s i s t a n c e and a response curve was p l o t t e d f o r eaoh temperature.  A l l readings  on the graphs have been  - 33 -  /OO  O  /o  05  /O  V?  /<DO (s^&^>/s A/o.  /&O0 9.  - 34 -  /GO  \  o  f a r\\ \\  0  /O  °  \ \ \ \ \\  C  0 \ \ \ \  \A V  /  / o  / a o  / a o o  - 35 -  corrected using these response ourves. CHAPTER  VI  CONCLUSIONS  (Hie measurements i n d i c a t e that the s p e c t r a l d i s t r i b u t i o n of noise shows a marked d e v i a t i o n from a /V law a t high frequencies. This seems to i n d i c a t e that i t i s permissable to introduce a d i s t r i b u t i o n of c o r r e l a t i o n times as was done i n the preceding theory (Chapter I V ) . The dependence of the t r a n s i t i o n frequency upon temperature seems to i n d i c a t e that the c o r r e l a t i o n times depend on temperature.  Though f u r t h e r experimental data are  needed we oan a t l e a s t draw some negative c o n c l u s i o n s . L e t us assume that the shape of the d i s t r i b u t i o n f u n c t i o n f o r 2" does not depend upon temperature." The t r a n s i t i o n frequency a t whioh the'/^dependence changes to a '/yz dependence then determines the value of ^< .. The f a c t that the t r a n s i t i o n from '/^ to j j  2  occurs a t a lower frequency f o r higher temp-  eratures then i n d i c a t e s that the c o r r e l a t i o n time ^ must increase w i t h i n c r e a s i n g temperature.  This means that  equation 28 of Chapter IV  does not e x p l a i n the experimental r e s u l t since i t / g i v e s values f o r "2^ that deorease w i t h i n c r e a s i n g temperature.  On the  - 36 o t h e r  h a n d  e q u a t i o n  2 9  o f  C h a p t e r  I V  -__T  d o e s  g i v e  v a l u e s  i n c r e a s i n g  t h e  r i g h t  g i v e  o f  "  t h e  ^  a  t  f o r  t h e  c o r r e l a t i o n  t e m p e r a t u r e .  t r e n d  r i g h t  a s  a  f u n c t i o n  s h a p e .  d i f f e r e n t ,  H o w e v e r ,  T a b l e  o f  I  t i m e s  w h i l e  f  t h i s  o  r  t h e  E  i n c r e a s e  r e l a t i o n  t e m p e r a t u r e ,  g i v e s  t e m p e r a t u r e s  t h a t  i  t  = l . O e V ,  g i v e s  d o e s  p r e d i c t e d  w l t h  n o t  v a l u e s  O . l e V ,  a n d  O . O l e V .  -2T _ r - 0.0/  & = O. / eV  /O  90 ZOO  /O**? ? /o' r  Z  0  /o'  /o- ? 06  a  0  0  T A B 1 - B  o f  a t  o f  F o r  i s  f  a  r  t o o  o b t a i n e d  c a s e  T .  v a l u e s  A  2  _  t o  a  o f  t  3?  i n d i c a t e  t  l o w  "21  t h e  i  s  d i f f e r e n t  d e p e n d e n c e  o f  w o u l d  o f  o f  T  H o w e v e r ,  e v e n  i  d e p e n d e n c e  o f  d e c r e a s e  a n d  t e m p e r a t u r e s  i n d e p e n d e n t  u p o n  u p o n  E .  r i g h t  v a l u e s  2 ) .  d e p e n d e n c e  t e m p e r a t u r e s  h i g h e r  t h a t  t h e  v a l u e s  2  a n d  e q u a t i o n  s l o w  g i v e  e q u a t i o n  a  E  r a t h e r  n o t  ¥ e r y  t o  o f  s m a l l  d o e s  I  t e m p e r a t u r e s  ( a c c o r d i n g  a s s u m i n g  A c c o r d i n g  t o  E  s t r o n g .  e q u a t i o n  i n d e p e n d e n t  d i f f e r e n t  l a r g e  b y  d e c r e a s i n g  s e e m  ?~  /6  0  600  2-*  /# ~ *2-  0  30O  V a l u e s  e v  r a p i d l y  w o u l d  w h e r e a s  o f  T  a t  b e  o u r  l o w  Tr  T  i  n  s  t h a t  u p o n  w i t h  p r a c t i c a l l y  e x p e r i m e n t s  t e m p e r a t u r e s  -  37  and i n c r e a s e s with. I n c r e a s i n g T a t h i g h e r  temperatures*  T h i s means that n e i t h e r equation 1 nor equation 2 r e p r e s e n t s the r i g h t dependence of T> upon  temperature.  More experimental data i n a wider frequency range and f o r h i g h e r temperatures  are needed b e f o r e any d e f i n i t e c o n c l u s i o n s  oan be made about t h i s p a r t of the problem.  --38 -  AO KPT 0 WliB DGEHEH T S I should p a r t i c u l a r l y l i k e to express my s i n c e r e a p p r e c i a t i o n f o r the guidanoe and h e l p g i v e n by Dr. A. van der Z i e l who s u p e r v i s e d the r e s e a r c h .  I am indebted to  Mr. R.H,. O a r l y l e who spent many hours c o n s t r u c t i n g the equipment and.preparing-the  semi-conductor.  could not.have been undertaken without  This p r o j e c t  the use o f the  f a c i l i t i e s o f the p h y s i o s Department and a grant f o r equipment g i v e n by the Defense Research Board. conducted  The researoh was  w i t h the a s s i s t a n c e o f a S c h o l a r s h i p donated by the  B r i t i s h Columbia Telephone Company L i m i t e d .  39 -  BIBLIOGRAPHY  6.  Bemamont,  J.  Annales de Phys.  2.  Bridgeman, P.W.  7.  Campbell, R.H. and Chapman, R.A.  Phys. Rev.  7^, 71  31, 101  (1936). (1928),.  Proo. I.R.E.  37, 938  (1949). 5,  Gervers, M.  Thesis;  Delft, Holland  (1947)  A l s o r e p r i n t e d i n three s e c t i o n s : P h i l l i p s Research Reports  1, 197  (1947),  P h i l l i p s Research Reports  1, 279  (1947).  P h i l l i p s Researoh Reports  1, 361  (1947).  3.  Johnson,  J.B.  1.  H y q u i s t , H.  4.  van der Z i e l  Phys. Rev.  Phys. Rev. Physioa  32, 97  32, 110  16, Ho. 3  (1928). (1928). (March  1950).  

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