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Spectral distribution of noise 1950

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SPECTRAL DISTRIBUTION 03? NOISE William Arthur Bain ABSTRACT of thesis submitted i n p a r t i a l f u l f i l m e n t of the requirements f o r the degree of MASTER OF APPLIED SCIENCE i n the Department of ENGINEERING PHYSICS UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1950 ABSTRACT An.apparatus f o r the measurement of noise as a function of frequency i s desoribed. This apparatus has been used to determine the spectral d i s t r i b u t i o n of the excess noise oaused by the flow of a d-o. ourrent through a resistance. -The samples used for the experiments were a zinc oxide semi- conductor and two.metal layer r e s i s t o r s . The frequency region investigated was from 10 kc. to 400 ko. I t was found that the excess noise i n the ZnO semi- conductor obeyed a ^/*law at room temperature while at lower temperatures ( s o l i d C0 2 and l i q u i d nitrogen) i t was proportional to '/v at low frequencies and £zat high frequencies. The excess noise i n the metal layer r e s i s t o r s was proportional to '/^ at room temperatures while at 100°0 there was a marked deviation from t h i s law i n the d i r e c t i o n of a z^z dependence f o r high frequencies. The measurements show that the law gradually changes to a 'fa law at high frequencies i n accordance with the theory proposed recently by Dr. A. van der z i e l . They also indicate that the c o r r e l a t i o n times involved are a function of temperature; the exact nature of th i s dependence has yet to be determined. SPECTRAL DISTRIBUTION OP NOISE by William Arthur Bain thesis submitted i n p a r t i a l f u l f i l m e n t the requirements f o r the degree of MASTER OP APPLIED SCIENCE in the Department of ENGINEERING PHYSICS THE UNIVERSITY OP BRITISH COLUMBIA April,1950 TABLE OF CONTENTS Page I n t r o d u c t i o n . . . . . . . 1 Apparatus, 3 Samples . 12 Theory 13 R e s u l t s . . . . . 22 C o n c l u s i o n s . 35 Acknowledgements • • 38 B i b l i o g r a p h y . 39 F i g u r e I B l o c k diagram "of apparatus • • 4 F i g u r e I I 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 I I I D e t a i l e d w i r i n g diagram of o s c i l l a t o r , mixer, band pass f i l t e r and d e t e c t o r . . . . . 6 F i g u r e IV C i r c u i t o f d i r e c t c u r r e n t supply 7 F i g u r e V Input c i r c u i t o f a m p l i f i e r . . . . . . . . . 9 F i g u r e VI E q u i v a l e n t i n p u t c i r c u i t o f a m p l i f i e r . . . 9 SPEC THAI, DISTRIBUTION OF NOISE CHAPTER I . INTRODUCTION The random motion o f e l e c t r o n s i n a conductor causes s m a l l f l u c t u a t i n g p o t e n t i a l d i f f e r e n c e s to be developed a c r o s s the t e r m i n a l s o f the conductor. T h i s is, known as Thermal Noise and i t can be d e s c r i b e d by 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 frequency i n t e r v a l A i s g i v e n by the formula where k i s Boltzmann's constant, T i s the a b s o l u t e temperature i n degrees K e l v i n , and S i s the r e s i s t a n c e . The r e s i s t a n c e R may depend on frequency i n which case one must use the value of R f o r the p a r t i c u l a r frequency i n t e r v a l under i n - v e s t i g a t i o n . THis formula, due t o l y q u i s t , has been c a r e f u l l y checked and i s found to be t r u e f o r f r e q u e n c i e s up to the i n f r a - r e d r e g i o n of the spectrum. I t has been proved t h e o r - e t i c a l l y from thermodynamical c o n s i d e r a t i o n s (1 ) , ( 2 ) and has been shown e x p e r i m e n t a l l y (3) t h a t t h i s n o i s e l i s independent of the type of r e s i s t a n c e whether i t be carbon, composition, wire-wound, t h i n metal layer,, or semi-conducting. -The formula does not always hold i f a d-o current i s passing through the resistance. In general i t can he said that i f a d-o current i s passing through R there i s an excess noise generated that increases with increasing ourrent. Usually t h i s Increase i s as the square of the current. One can describe t h i s effeot by introducing an add i t i o n a l noise e.m.f. i n series with R so that for a small frequency i n t e r v a l e«* - Izfat) A)J where the function describes the frequency dependence of t h i s excess noise. I t i s usually of the form I! ' y for 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 function of the frequency V and to measure <%* as a function of current f or various materials at d i f f e r e n t temperatures. i n order to measure. the noise i n a oertain frequency band of width small i n comparison to the f r e - quency \f ) i s amplified and detected by a quadratic detector. I f i s the deflection of the meter for 1 = 0 and.Dg the deflection for a f i x e d ourrent I then: - 3 - D, ' 4A y-R. so that: As soon as and Dg nave "been measured fo r d i f f e r e n t f r e - quencies ^tO/Joan be calculated as a function of frequency. I f Dg i s much larger than Di one might use a c a l i b r a t e d attenuator to bring the reading of the output meter down to a reasonable value. I f A i s the attenuation factor then one has to substitute ADg f o r Bg i n the above equation. CHAPTER II APPARATUS A block diagram of the apparatus used i s shown i n Figure I. The detailed wiring diagram of the pre-amplifier i s shown i n Figure I I . The high tension and the filament ourrent are supplied by a regulated power supply of conven- t i o n a l design. The frequency response.of the ,pre-amplifier i t s e l f was found to be e s s e n t i a l l y f l a t between 1 ko. and 400 ko., the half-power frequencies being at 500 cycles and v. 500 ko. When the apparatus was f i r s t tested the noise resistance was found to be 3000 ohms. Tb4s has been reduced - 4 - FIGURE I to 800 ohms by using wire-wound r e s i s t o r s i n the pla t e c i r c u i t s of the f i r s t two stages. The maximum voltage gain of the pre-amplifier i s 40,000. The detailed wiring 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 f i l t e r , and amplifier i s a2a.own i n Figure I I I . The frequenoy range of the l o o a l o s o i l l a t o r i s from 465 kc. to 1300 ko. The c r y s t a l band-pass f i l t e r i s tuned at 456 ko. and has a width of 200 cycles. This allows us to examine the noise i n a narrow hand f o r a l l frequencies from about 10 ko. up to the point where the response of the q i r c u i t as a whole f a l l s to a very low value. This upper l i m i t of frequency i s i n the neighborhood of 400 kc. A 1U-23 c r y s t a l diode i s used as a quadratic detector. I ts quadratic properties have been checked several   — 7 - time during the course of the research. The output meter i s a Rubicon galvanometer with a ten-centimeter scale and a s e n s i t i v i t y of 0.0056 micro- amperes per millimeter. (0.56 microamperes f u l l scale) A diagram of the c i r c u i t used f o r supplying the d-c. current to be passed through the sample i s shown i n Figure IV. FIGURE IV Seven 45-volt dry c e l l s are usei to supply the necessary d-o voltage. These ba t t e r i e s and the ammeter are enclosed i n a grounded metal box. From thi s battery box the d-o. current i s fed through a shielded cable to another grounded metal box where i t i s f i l t e r e d and then allowed to pass through the sample under investigation. These precautions were taken to eliminate the p o s s i b i l i t y of feed-back through the power supply and to minimize the e f f e c t of any stray e l e c t r i c a l disturbances i n the v i c i n i t y . As a further - 8 - precaution against these stray e l e c t r i c f i e l d s (due to fluorescent l i g h t s , etc.) a l l the research was done inside a cage made of two layers of chicken-wire mounted on a wooden frame. Both these layers are securely grounded and power i s brought into the cage by means of an i s o l a t i n g transformer. Even with these precautions the apparatus has a tendency to be unstable at times; I t i s not known whether th i s i s due to stray f i e l d s or to some f a u l t y component i n the apparatus i t s e l f . These periods of i n s t a b i l i t y , however, are very infrequent and sh o r t - l i v e d . During normal operating conditions the apparatus i s quite steady but shows a slow v a r i a t i o n of gain so that over a period of say 30 minutes the reading of the galvan- ometer may change by as much as 10$.- I t was found that when the apparatus was l e f t untouched for.a period of many hours t h i s d r i f t never exceeded 10$ and data could be reproduced from day to day within t h i s l i m i t . The galvanometer has a rated time constant of three seconds. This was increased to approximately 30 seoonds^by:vlhsertding: three stagescof- f i l t e r i n g i n front of the galvanometer. Each stage consists of a 5000 ohm r e s i s t o r i n series with a 2000 microfarad condenser. This was done to damp out some very rapid fluctuations of the reading. The cause of these i s believed to be due to the faot that- r e s i s t o r s give out bursts of extra noise over and above those expected. I t has been said that these bursts may be as large as 300$ of the average value. In order to show that the apparatus was working properly the following 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 with resistance. However, i n any p r a c t i c a l c i r c u i t t h i s resistance i s i n p a r a l l e l with both the input capacity and the g r i d resistance of the f i r s t stage of the am p l i f i e r . This g r i d r e s i s t o r also produces a noise voltage. Consider the c i r c u i t s shown i n Figure V and Figure VI where C i s the input capacity, R̂  i s the g r i d resistance and Ra i s the resistance of the sample under consideration. IT, 7 © FIGURE V FIGURE Y l - 1 0 - F i g u r e V s h o w s t h e a c t u a l ; c i r c u i t i n t h e a m p l i f i e r a n d F i g u r e 7 1 s h o w s t h e e q u i v a l e n t , c i r c u i t . T h e v o l t a g e 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 i s g i v e n b y t h e f o r m u l a : a n d t h e m e a n 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 ft. ' 3. T h i s i s t h e e q u a t i o n o f a f a m i l y o f c u r v e s a n d , i f -w<?^j >/J t h e r e i s a m a x i m u m 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 * /0O - 11 - 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 ( i . e . leave the input open and H becomes i n f i n i t e ) we have 2 — By p l o t t i n g a graph of mean square voltage versus i t i s possible to obtain the maximum value of the curve and we oan then solve equations 5 and 6 f o r the two unknowns^ and 0. This has been done and the r e s u l t s are shown on Graph Uo.l for a frequency of 50,000 cycles. The R̂  has a value of 460,000 ohms and maximum value of ̂ I s when R, = 190,000 ohms. Using t h i s data we get a value of 24 micro-micro ^-farads for C which i s quite reasonable f o r a c i r c u i t of t h i s type. - 12 - Substituting the known values f o r C andJ@ into equation 2 allows us to calculate a ourve. This theore t i c a l curve i s also plotted on Graph Uo.l and i t s olose resemblance to the actual curve i s a good indioation that the apparatus i s working properly. CHAPTER III SAMPLES The metal layer r e s i s t o r s used were obtained from Continental Carbon incorporated. They are ordinary 50,000- ohm p r e c i s i o n r e s i s t o r s and consist of a t h i n metalized layer deposited on a oeramio material. The zinc oxide semi-conductor was made i n the laboratory as follows: zinc oxide powder was placed i n a mould and subjected to a pressure of 10,000 pounds per square inqh. The r e s u l t i n g block of material (2" < ?/e * l/& ) was heated to 1200°C at a rate of 100°C per hour and held there f o r twelve hours, i t was allowed to cool over a period of twenty-four hours. Afte r cooling s i l v e r contacts were painted on the ends, platinum leads were joined to these contacts and the whole assembly was placed i n a glass tube and sealed. CHAPTER IV THEORY A d-o. current flowing through a r e s i s t o r causes an extra noise whioh, f o r a small frequency i n t e r v a l ^ 1 ^ , i s given by the formula: where I i s the d-o. ourrent and ftv) i s a function describing the frequency dependence of t h i s induced noise. We oan explain the 1̂  dependence of the noise i f we assume that the resistance shows spontaneous fluctuations i n value, i f the f l u c t u a t i o n i n resistance i s ^R then an extra noise = T<d * i s developed across the resistance R. The mean square value of t h i s noise i s To explain these fluctuations i n resistance we turn to the d e f i n i t i o n of conductivity as given by the electron theory of matter. The conductivity c T i s : cr = — — — C/tr where n i s the number of free electrons per cubic centimeter, - 1 4 - e the eleotronio charge, L the free path length b f the electrons, and v the v e l o c i t y of the electrons. The f l u c t - uations of resistance may he due to a f l u c t u a t i o n i n the number of free electrons per cubic centimeter. There i s also the - p o s s i b i l i t y that the free path length I» or the v e l o c i t y v of the electrons show flu 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 into account we introduce a f l u c t u a t i o n i n the conductivity as follows: / \ = C7~ - C 7 7 ACT- = O Si This gives to a good approximation. We now have f o r the value of : We must oarry out a Fourier analysis of and calculate Consider a f l u c t u a t i n g quantity X(t) which i s known for the i n t e r v a l 0<-t<to where t i s large but f i n i t e . I f we assume that Z(t) i s a continuous function i n the i n t e r v a l we may develop i t into a complex Fourier series as follows: x # J = ^ = «j fa * * 15 ~ 7 9. tfo-0 because X&) = O. I f a ^ i s the complex conjugate of a ^ e see that a _ n = a^*" The Fourier component x n of frequency £ ^ i s and i t s mean square value i s because a l l terms containing the time to drop out when the average i s taken. We now have to calculate the value of 2 a n .• From equation 9 we obtain: XU)X(«r) e-'^*^***** /z. - 16 - We now change to a new variable w. where x(u) X(u+w) i s independent of u and i s a function of w only, we also know that Z(u) X(.u+w) i s symmetrical i n w and equals zero f o r /.̂ /-/ > 8 where S i s a measure of the cor r e l a t i o n time. Therefore i f % >> S equation 14 may be rewritten as: ° 4 4 - And, i f we put Y = & \J , equation 15 becomes: t o So -17 - sinoo the imaginary term contributes nothing to the value of the i n t e g r a l . Equation 16 may be more conveniently written 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 that o(w) = 1 f o r w = o o(-w) •= o(w) and o(w) = 0 i f |w| >> where i s the c o r r e l a t i o n time of the fluctuations.- When we consider the case of a f l u c t u a t i n g quantity which i s oaused by a large number of independent and random events 2"is the duration of the event (e.g. the t r a n s i t time of an electron i n a radio tube). i n the case of fluctuations involving decay problems 2"is a measure of the average l i f e of the decaying quantity. We now have f o r the mean square voltage i n a small frequenoy i n t e r v a l ^ V - 18 - Tne s p e c t r a l d i s t r i b u t i o n f u n o t i o n ^ / ^ J i s completely d e t e r - mined by the c o r r e l a t i o n f u n c t i o n o(w). f o r we f i n d by a p p l y i n g a F o u r i e r transform: I t i s .evident from t h i s equation t h a t i t i s not p o s s i b l e f o r j&6J) to be of the form f o r &< AJ < i f we s u b s t i t u t e E l i n t o 20 we o b t a i n an i n t e g r a l t h a t i s d i v e r g e n t a t &J-0±QT a l l v a l u e s of w and i s a l s o d i v e r g e n t a t ^ J ' 3 1 0 0 f o r w = o. To ensure the convergence of 20 f o r a l l f r e q u e n c i e s we must impose the f o l l o w i n g r e - s t r i c t i o n s : must va r y slower than /£/ f o r v e r y low f r e q u e n c i e s . b) ^Xu) must v a r y f a s t e r than /fa f o r v e r y h i g h f r e q u e n c i e s . Of course the ^ law i s s a t i s f a c t o r y f o r i n t e r m e d i a t e f r e - quencies. I t has u s u a l l y been assumed t h a t t h e c o r r e l a t i o n f u n c t i o n r e p r e s e n t s an 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 19 we o b t a i n : £l>) - 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 of frequency, a t low fr e q u e n c i e s and v a r i e s as ̂  a t h i g h f r e q u e n c i e s . This r e l a t i o n s h i p i s i n marked c o n t r a s t to the experimental r e s u l t s where one u s u a l l y f i n d s t h a t the n o i s e v a r i e s as />/ i n a l a r g e frequency range. A s o l u t i o n to t h i s problem has been proposed r e c e n t l y by pr . A. van der z i e l . (4) i n s t e a d o f u s i n g a s i n g l e c o r r e l a t i o n time "2" we i n t r o d u c e a wide 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, l e t be the 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 time 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 can always o b t a i n agreement between theory and experiment even i f one s t a r t s w i t h the wrong c o r r e l a t i o n f u n c t i o n . Therefore we can a t t r i - bute p h y s i c a l meaning to the whole procedure o n l y i f there are sound arguments i n favour o f the d i s t r i b u t i o n f u n c t i o n - 2 0 o h o s e n . i t w i l l b e s h o w n l a t e r t h a t t h i s i s a c t u a l l y t h e c a s e . I f w e i n t r o d u c e t h e f o l l o w i n g n o r m a l i z e d d i s t r i b u t i o n f u n c t i o n w e o b t a i n t h e • £ / l a w e x a c t l y . L e t f t ' ? ' ) * ' ? •= 0 f o r ?* ?> >VZ a n d s u b s t i t u t e i t i n t o e q u a t i o n 2 5 . T h i s g i v e s 27 = ear?*/**?/ //r? — ) *J n - *"-cA?« *J 2 1 eJ. . • Thi3 g i v e s a v a l u e o f f t v ) t h a t i s i n d e p e n d e n t o f ^ f o r v e r y l o w f r e q u e n c i e s , v a r i e s a s '/j f o r i n t e r m e d i a t e f r e q u e n c i e s , a n d v a r i e s a s f o r v e r y h i g h f r e q u e n c i e s . O n e o a n e x t e n d t h e r e g i o n a s f a r a s i s n e c e s s a r y b y a p r o p e r c h o i c e o f 2 7 a n d ?2 . W e n o w h a v e a n e x p r e s s i o n f o r / c ) ) ) t h a t s a t i s f i e s c o n - d i t i o n s a ) a n d b ) o n P a g e 1 8 . T h e i n t r o d u c t i o n o f a d i s t r i b u t i o n o f c o r r e l a t i o n t i m e s i s r e a s o n a b l e w h e n w e c o n s i d e r t h a t i n t h e t h e o r y o f d i e l e c t r i c l o s s e s ( w h i c h i s a l s o a p r o b l e m o f s o l i d s t a t e p h y s i c s ( 5 ) ) t h e c o r r e l a t i o n t i m e i s g i v e n b y T-- 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 give quite a large d i s t r i b u t i o n i n T because kT i s a small quantity at room temperature. Of course, i t i s not to be construed from these remarks that this i s the solution to the problem of the spectral d i s t r i b u t i o n of noise - equation 28 i s introduced f o r the sole purpose of giving a physical 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 also turn to the f i e l d of Nuclear physios where we have that the h a l f - l i f e of an excited state i s given by an equation of the 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 give r i s e to a wide d i s t r i b u t i o n i n the values of f . Equations 28 and 29 both show that the values of ~Z (and thus the form of ftv)) might be dependent on temperature. This v a r i a t i o n of the frequency dependence of the noise with temperature w i l l , of course, be the governing factor i n the choice of a suitable expression f o r - 22 - CHAPTER V RESULTS Excess n o i s e as a f u n c t i o n of c u r r e n t 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 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 o f the d-c. c u r r e n t f l o w i n g . The re a d i n g s were taken a t room temperature. For low va l u e s of the c u r r e n t the n o i s e i n c r e a s e s as the square o f the our r e n t while a t h i g h e r c u r r e n t s i>t i n c r e a s e s l e s s r a p i d l y , w i t h o u r r e n t . From the graph i t can a l s o be seen t h a t the shapes of the curves are independent o f frequency i n the frequency range examined. Bemamont (6) found these same r e s u l t s f o r an even wider frequency range. A l l the re a d i n g s taken f o r the n o i s e a t 15 kc. are p l o t t e d on the graph. These readings were obtained over a p e r i o d o f s e v e r a l h o u l s . For the sake of c l a r i t y o n l y the average v a l u e s of the readings are p l o t t e d f o r 31 kc. The n o i s e v a l u e s on t h i s and a l l subsequent graphs are p l o t t e d i n a r b i t r a r y u n i t s . A l l curves on the same graph are p l o t t e d to the same s c a l e but there i s no r e l a t i o n between the mag- n i t u d e s of the n o i s e on eaoh separate graph. The a c t u a l u n i t s e O of n o i s e are ( v o l t s ) per cy&le but here the n o i s e i s simply measured i n m i l l a m e t e r s d e f l e c t i o n o f the galvanometer or m u l t i p l e s t h e r e o f . Since the apparatus i s q u a d r a t i c and s i n c e the band-pass i s constant t h i s u n i t o f measurement i s c o r r e c t but, o f course, the a b s o l u t e magnitude i s dependent - 23 - - 24 - /OOO &/?&/?// A/Q, 3- - 25 - - 26 - on the gain of the apparatus which oan be varied over a wide range. Graph Ho. 3 shows the excess noise i n the same metal layer r e s i s t o r s for a temperature of about -75°0 ( s o l i d carbon dioxide). Only the average points have been plotted. The., J resistance at t h i s temperature i s almost the same as at room temperature. Again we notice that the shapes of the curves are independent of, frequency and that the slopes at 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 plot of the excess noise i n the semi-conductor as a function of d-o. current at room temper- ature. The resistance at t h i s temperature i s 600,000 ohms. The semi-conductor was connected i n series with a one-megohm wire-wound r e s i s t o r . This wire-wound r e s i s t o r contributed nothing to the excess noise (since^/-cv) sO. ) 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 current. The noise fo r lower currents could not be determined with 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 fact was also noticed by Bernamont (6). Excess noise as a function of frequency for a r e s i s t o r carrying a constant current. - 27 - Graph No. 5 shows, on logarithmic coordinates, the excess noise i n the metal layer r e s i s t o r s as a function of frequenoy f o r three d i f f e r e n t values of d-c. current. ,The readings are for room temperature. The curve f o r 0.25 milliampers only snows the average values. The slopes of a l l three curves are -1 within the experimental error. This indicates that the form o±fcv)±a given by: f<v) - p i n t h i s frequency range. Graph No. 6 shows that the excess noise at -75°C i s also i n v e r s l y proportional, to frequency i n the range inves-? tigated. However, Graph No. 7, taken at 100°C, shows a d i s t i n c t departure from t h i s law at higher frequencies, up to a frequency of about 40 ko. the slope i s -1 and above th i s frequency the slope attains a value of -1.6. After the readings at 100°C had been taken the readings at room temperature were repeated and the r e s u l t s were the same as for the f i r s t t r i a l . This indicates that heating and cooling the r e s i s t o r s had no permanent e f f e c t on the spectral d i s t r i b u t i o n of the noise. The resistance at a l l three temperatures was almost constant - the v a r i a t i o n being less than 10$. The response of the apparatus was plotted and found to be, the same at a l l three temperatures. A l l readings have been oorreoted f o r the decrease of gain at - 28 - - 29 - J _____—, S>/it £>A- \© N. \ G S'sTf* - /o / O G /ooo  - 31 - higher frequencies.. Graph l b . 8 shows the excess noise i n the semii- conductor as a function of frequency at a constant current and at room temperature. The slope of this curve i s -2 over the whole frequency range whioh indicates that the form of /CV) i s : /cv) Costs 6&<n f y*. Graph Ho. 9, for a temperature of r-75°0, shows that the noise,at, low frequencies i s proportional to V' and that the noise at high frequencies i s proportional to V ~Z . By extending the two portions of t h i s we see that, they meet at a frequency of about 70 ko. This " t r a n s i t i o n frequency", as i t might be c a l l e d , cannot be accurately determined but i t i s the most convenient quantity to use when comparing graphs. Graph No. 10, f o r a temperature of - 186°C , also shows t h i s t r a n s i t i o n of the noise dependence from at low frequencies to V~Z at high frequencies. I t i s also to be noted that the t r a n s i t i o n frequency, at -186?0 i s almost the same as that for - 7 5 ° C The resistance of the semi-conduotor changes very r a p i d l y with temperature, i t i s 0.6 megohms at room temper- ature, 15 megohms at - 7 5 ° C , and 60 megohms at - 1 8 6 ° C . The frequency response of the a m p l i f i e r i s greatly influenced by the input resistance and a response curve was plotted f o r eaoh temperature. A l l readings on the graphs have been  - 33 - /OO /o 05 O V? /O /<DO /&O0 (s^&^>/s A/o. 9. - 34 - /GO /O / \ o \ \ \ \ 0 \ \ \ \ \ \ 0 \ \ \ \ f a r- ° C \A V / o / a o / a o o - 35 - corrected using these response ourves. CHAPTER VI CONCLUSIONS (Hie measurements indicate that the spectral d i s - t r i b u t i o n of noise shows a marked deviation from a /V law at high frequencies. This seems to indicate that i t i s per- missable to introduce a d i s t r i b u t i o n of co r r e l a t i o n times as was done i n the preceding theory (Chapter IV). The dependence of the t r a n s i t i o n frequency upon temperature seems to indicate that the co r r e l a t i o n times depend on temperature. Though further experimental data are needed we oan at least draw some negative conclusions. Let us assume that the shape of the d i s t r i b u t i o n function for 2" does not depend upon temperature." The t r a n s i t i o n frequency at 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 at a lower frequency for higher temp- eratures then indicates that the co r r e l a t i o n time ^ must increase with increasing temperature. This means that equation 28 of Chapter IV does not explain the experimental r e s u l t since i t / g i v e s values for "2̂  that deorease with increasing 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 f o r t h e c o r r e l a t i o n t i m e s t h a t i n c r e a s e w l t h i n c r e a s i n g t e m p e r a t u r e . H o w e v e r , w h i l e t h i s r e l a t i o n g i v e s t h e r i g h t t r e n d a s a f u n c t i o n o f t e m p e r a t u r e , i t d o e s n o t g i v e t h e r i g h t s h a p e . T a b l e I g i v e s t h e p r e d i c t e d v a l u e s o f " ^ a t d i f f e r e n t , t e m p e r a t u r e s f o r E = l . O e V , O . l e V , a n d O . O l e V . -2T & = O. / eV _ r - 0.0/ e v 90 ZOO 30O 600 /O**?0 ?0 /o'a r0 /o'/6 ?~ /O 2-* /# ~Z*2-0 /o-06?0 T A B 1 - B I V a l u e s o f a t d i f f e r e n t t e m p e r a t u r e s a n d d i f f e r e n t v a l u e s o f E ( a c c o r d i n g t o e q u a t i o n 2 ) . F o r l a r g e v a l u e s o f E t h e d e p e n d e n c e o f u p o n T i s f a r t o o s t r o n g . A r a t h e r s l o w d e p e n d e n c e o f u p o n T i s o b t a i n e d b y a s s u m i n g s m a l l v a l u e s o f E . H o w e v e r , e v e n i n t h a t c a s e e q u a t i o n 2 d o e s n o t g i v e t h e r i g h t d e p e n d e n c e o f Tr u p o n T . A c c o r d i n g t o e q u a t i o n 2 " 2 1 w o u l d d e c r e a s e r a p i d l y w i t h d e c r e a s i n g _ a t ¥ e r y l o w t e m p e r a t u r e s a n d w o u l d b e p r a c t i c a l l y i n d e p e n d e n t o f 3? a t h i g h e r t e m p e r a t u r e s w h e r e a s o u r e x p e r i m e n t s s e e m t o i n d i c a t e t h a t i s i n d e p e n d e n t o f T a t l o w t e m p e r a t u r e s - 37 and increases with. Increasing T at higher temperatures* This means that neither equation 1 nor equation 2 represents the r i g h t dependence of T> upon temperature. More experimental data i n a wider frequency range and f o r higher temperatures are needed before any d e f i n i t e conclusions oan be made about t h i s part 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 sincere appreciation for the guidanoe and help given by Dr. A. van der Z i e l who supervised the research. I am indebted to Mr. R.H,. Oarlyle who spent many hours constructing the equipment and.preparing-the semi-conductor. This project could not.have been undertaken without the use of the f a c i l i t i e s of the physios Department and a grant f o r equip- ment given by the Defense Research Board. The researoh was conducted with the assistance of a Scholarship donated by the B r i t i s h Columbia Telephone Company Limited. 39 - BIBLIOGRAPHY 6. Bemamont, J. Annales de Phys. 7̂, 71 (1936). 2. Bridgeman, P.W. Phys. Rev. 31, 101 (1928),. 7. Campbell, R.H. and Chapman, R.A. Proo. I.R.E. 37, 938 (1949). 5, Gervers, M. Thesis; Del f t , Holland (1947) Also reprinted i n three sections: 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. Phys. Rev. 32, 97 (1928). 1. Hyquist, H. Phys. Rev. 32, 110 (1928). 4. van der Z i e l Physioa 16, Ho. 3 (March 1950).

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