FLUCTUATIONS IN A CONDUCTOR by AMARNATH.R. KSHATRIYA B.Sc.(Hons) Uni v e r s i t y of Bombay, 1953-M.Sc. Un i v e r s i t y of Bombay, 1956. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE In the Department of PHYSICS We accept t h i s t h e s i s as.conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September., 1964 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f • B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y * I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . . D e p a r t m e n t o f PHYSICS The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r 8, C a n a d a D a t e September h, 196k. ABSTRACT Applying purely thermodynamic arguments i t has been shown that temperature f l u c t u a t i o n s i n a sample can be represented by introducing appropriate'series temperature generator 1 or .'shunt heat current generator'. The temperature f l u c t u a t i o n s lead to resistance f l u c t u a t i o n s i n the sample. These resistance f l u c t u a t i o n s due -to temperature f l u c t u a t i o n s can be detected as voltage-fluctuations (temperature noise") by using a sensing d i r e c t current through the sample. Sta t i s t i c a l - m e c h a n i c a l arguments are used to obtain t h e o r e t i c a l expressions f o r spectral density of heat current f l u c t u a t i o n s i n a m e t a l l i c conductor i n terms of the macroscopic properties of the conductor. 'Since the electrons are c a r r i e r s of heat and e l e c t r i c currents i n a metal., heat and e l e c t r i c current f l u c t u a t i o n s are correlated. Spectral density of c r o s s - c o r r e l a t i o n between e l e c t r i c and heat current i s derived. S t a t i s t i c a l considerations are extended to the c a l c u l a t i o n of the steady state spectral density of heat current f l u c t u a t i o n s between two black bodies i n r a d i a t i v e contact. Temperature noise i n a system i n which there i s only a p a r t i a l c o r r e l a t i o n between temperatures at d i f f e r e n t points along the length at any time (isothermal System) i s compared with a system i n which there i s a complete c o r r e l a t i o n between temperatures at a l l points along the length at -any time (Single Temperature System). Experimental r e s u l t s indicate that f o r the metal filament used Nyquist theorem can be applied-at the operating temperature. For frequencies close to the c h a r a c t e r i s t i c frequency of the system i t i s ' , observed that there i s an increase i n noise temperature of the filament due to temperature noise. v i . ACKNOWLEDGEMENTS I am indebted to .Professor-R.E. "Burgess f o r h i s guidance and •assistance during the preparation of the material f o r t h i s d i s s e r t a t i o n . I should also l i k e to thank the Defenpe Research Board of Canada f o r summer support. i i i . TABLE OF CONTENTS Page Chapter I. Introduction 1.1 Voltage f l u c t u a t i o n s i n thermodynamic equilibrium ... .... l 1.2 Temperature f l u c t u a t i o n s -in thermodynamic equilibrium . . . 2 1- 3 Object and scope of the t h e s i s .. ...... ... ... ....... k Chapter II. Spectral density of heat current f l u c t u a t i o n s from s t a t i s t i c a l considerations 2.1 Spectral density of e l e c t r i c current f l u c t u a t i o n s . . . . . 5 2.2 Heat and e l e c t r i c currents i n a conductor 5 2- 3 Spectral density of heat current f l u c t u a t i o n s . . 7 2- k Spectral density of radiant heat current f l u c t u a t i o n s between two black bodies . . . . .... . . . . . . ... 11 Chapter I I I . Temperature f l u c t u a t i o n s i n a conductor i n Thermodynamic Equilibrium. Isothermal Case. 3- 1 Equivalence between heat flow equations and transmission l i n e equations 15 3-2 Generalisation of Nyquist theorem .... . .... . . . . . 17 3-3 Spectral density of Resistance f l u c t u a t i o n s •. •• -. . . . . 17 Chapter TV Temperature f l u c t u a t i o n s i n a "'single temperature system' k.l Single temperature system . . . . . . . . . . . . 21 h.2 Temperature noise . . . . . . . 21 h.3 Discussion of temperature f l u c t u a t i o n s in• ' " ( l ) Isothermal System (Chapter III) and Single Temperature System 2k Chapter V Experimental Aspects 5.1 Selection of Sample ......... . ... . . . . . . . . . . . . . 27 5-2 Determination of parameters 28 5-3 Noise measurement . . . 29 i v . Page 5-4 Results . . . . . ... ... ... . ..... ... . 32 5.5 Noise Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I+ Chapter VI Conclusions and Recommendations . . . . . . 6.1 Conclusions . . . . . . . .... . 37 6.2 Recommendations f o r future project .... ... ... . . . . 37 Appendix l ( a ) Temperature f l u c t u a t i o n s i n thermodynamic equilibrium . ....... .... ... . . . . . . . . . . . . 39 Appendix 2(a) Relationship between Variance and Spectral density . . . .... . . . . . . 1+2 Appendix 3(a) D i s t r i b u t e d f l u c t u a t i o n generator approach fo r a transmission l i n e i n thermodynamic equilibrium . . . . . . . . ... . 1+5 Appendix 3(b) Properties of f(M) . . . . . . . . . . . . . . . . . . . 51 LIST OF FIGURES Figure Opposite Page 5.1+.1 I-V C h a r a c t e r i s t i c of the filament . . . . . . . . 32 5-4.2 A p l o t of R s(^) Vi iXs^ 'H/w at the operating point . ..... . .. . . . .32 5.4.3 Power dissipated-as a function of filament temperature . . . . . . . . . . . . . . 33 5.5.I Noise temperature T n of the filament as a function of frequency . . . .... . . .. .. .34 1. • CHAPTER I Introduction 1.1 Voltage f l u c t u a t i o n s i n thermodynamic equilibrium. By purely thermodynamic reasoning Nyquist (1928) proved that a r e s i s t o r R i n thermodynamic equilibrium with i t s surrounding at temperature T 0 e x h i b i t s spontaneous voltage fluctuations. Spectral density.of voltage f l u c t u a t i o n s i s given by Where k = Boltzmann constant. E i n s t e i n (191.0) has derived the following entropy p r o b a b i l i t y law f o r computing mean square f l u c t u a t i o n s ,of a macroscopic va r i a b l e x. P r o b a b i l i t y P(x) that the var i a b l e x l i e s between x and x+dx i s where A >S(x) = change i n entropy of the system as a r e s u l t of f l u c t u a t i o n i n x. Consider the following e l e c t r i c a l system i n equilibrium at a temperature T0.' Using (1.1.2) mean square voltage f l u c t u a t i o n i s obtained. Langevin equation f o r current i n the system gives voltage c o r r e l a t i o n function. (1.1.1) ASM fa (1.1.2) -0 F i g . (1.1.1) As f a r as the f l u c t u a t i o n s in,voltage V are concerned these are representable by a seri e s e-m.f generator or shunt current generator as i n F i g . ( l . l . l ) . The generators i n F i g . - ( l . l . l ) have following properties v 1 = kz/c 4 k r 0 fa (1-1.-3) 1.2 Temperature f l u c t u a t i o n s i n thermodynamic equilibrium. . Consider two bodies C-^ and C2 i n thermal contact at temperature "To. C-p C2 are the heat c a p a c i t i e s 1-^ and"T2 are instantaneous temperatures res p e c t i v e l y . T T 2 To -- < T > - <ir > Fig.(1.2.1) ^ H g = thermal conductance between C-j_ and C2 = "^ "rp where H = average heat flow between C i , C2 when there is.-a temperature difference T. Using E i n s t e i n Entropy-probability law i n Appendix l ( a ) following r e s u l t s f o r mean square temperature f l u c t u a t i o n s are derived. 2 ( i ) A T ; 2 = •/?Xcl/c,{c,+cx) (1.2.1) ( m ) A T A T Q = ~ ^ /fo^ j ^T-j_ and A T2 are f l u c t u a t i o n s i n temperatures of the bodies characterised by the heat c a p a c i t i e s C-j_ and C2 • I f C-j_/'c^ /V*/ ^' e* ^°0^ with heat cap-a c i t y C-j_ i s i n contact with an i n f i n i t e heat bath then AT, 1 - ti%l/c, (1.2.2) This i s consistent with Gibbs'1 result that i f i s the energy of a system i n thermodynamic equilibrium then energy fluctuations are: This result i s v a l i d no matter what s t a t i s t i c s apply to the process of heat conduction between and the i n f i n i t e heat bath. Equation (1/2.3) i s the well known energy fluctuation theorem. Temperature fluctuations i n C-^ when i n contact with an i n f i n i t e heat bath are derived i n Appendix l ( a ) . I t i s seen that as far as temperature fluctuations are concerned these are representable by a series'"temperature generator" or a shunt heat current generator analogous to the generators i n (1.-2-.-3) T Fig.(1.2.3) We have shown that ( i ) Spectral density of series "temperature generator'" Thermal impedance of the body C, to i t s surroundings ( i i ) Swt^ 13 + j ^ l ) 1.3 Object and scope of the t h e s i s . G i l l (1958) claims that temperature f l u c t u a t i o n s and voltage f l u c t u a t i o n s i n a conductor are manifestations of the same underlying phenomena. One ,of the objects of t h i s i n v e s t i g a t i o n i s to show that such a claim i s erroneous. Temperature f l u c t u a t i o n i s a macroscopic e f f e c t and temperature noise can be detected i n the following way. A H - > A T - > A R - > A ^ { 1 , 3 m X ) where A^ H = f l u c t u a t i o n i n heat current C-j_ .= heat capacity of the system ^ T-^ = temperature f l u c t u a t i o n = i- ^ - 5 - = temperature c o e f f i c i e n t of resistance R dT I = b i a s i n g current V = voltage f l u c t u a t i o n due to temperature f l u c t u a t i o n , o This i n v e s t i g a t i o n i s concerned with t h e o r e t i c a l and experimental aspects of temperature f l u c t u a t i o n s . Theoretical aspects of temperature f l u c t u a t i o n deal with the foll o w i n g : ( i ) Thermodynamic Equilibrium General s t a t i s t i c a l considerations are used f o r obtaining the spectral d e n s i t i e s of heat current f l u c t u a t i o n s and c r o s s - c o r r e l a t i o n s between the heat and e l e c t r i c currents. A d i s t r i b u t e d system i n which the temperature at d i f f e r e n t points are p a r t i a l l y c o r r e l a t e d (isothermal System) i s investigated and compared with a system i n which there i s a complete c o r r e l a t i o n between temperatures at d i f f e r e n t points (Single Temperature System). ( i i ) Steady State Temperature noise i n the steady state f o r a single temperature system and spectral d e n s i t i e s of heat current f l u c t u a t i o n s between two black bodies are obtained. 5-CHAPTER I I Spectral density of heat current f l u c t u a t i o n s from s t a t i s t i c a l considerations 2.1 Spectral density of e l e c t r i c current f l u c t u a t i o n s . Employing general s t a t i s t i c a l considerations Bakker and H e l l e r (1939) have derived the s p e c t r a l density of e l e c t r i c current f l u c t u a t i o n s i n a conductor at temperature T5. . Applying Lorentz theory of the electron and assuming that e l e c t r i c currents T ( t ) and l ( t + ^ ) are corr e l a t e d only when \t!T\ i s small they e s t a b l i s h the following r e s u l t . temperature Ty. 2.2 Heat and e l e c t r i c currents i n a conductor. In a conductor the motion of electrons leads to a simultaneous flow of e l e c t r i c current and heat -energy. To the f i r s t approximation e l e c t r i c and heat current densities are dependent l i n e a r l y on both the 'forces' - e l e c t r i c and thermal. Accordingly i f J i s the e l e c t r i c current density and i s the heat current density i n the conductor when there i s an e l e c t r i c f i e l d F and (2.1.1) where QfJ = $e Yi|l"f) = Real part of the conductance of the conductor at frequency w and temperature gradient then quite generally, 1 = ^ ,U I + ^M, (2.2.1) where (a*t) = 1,2) are s p e c i f i c conductances and i s the ' e l e c t r i c force' ^2 i s the 'thermal force'. Using Boltzmann transport equatipn e x p l i c i t forms for U, and U t and the admittances M^. i s evaluated. For the sake of generality we s h a l l assume an 6. a l t e r n a t i n g e l e c t r i c f i e l d i s applied. f(|<) = Fermi Dirac D i s t r i b u t i o n function f o r the electrons i n the presence of applied f i e l d s (steady state conditions). In the steady state f ( K ) varies with the same frequency as the applied f i e l d To'. T 0 ax (2.2.2) where = Fermi Dirac D i s t r i b u t i o n function f o r the electrons i n thermodynamic equilibrium. si-- N F ex|={(E(^)-^)/r?T0\ -H ECk) = energy of an electron with wave vector K = Fermi energy relaxation time of-electrons with wave vec t o r s - i n the i n t e r v a l K and K+dK ^ = angular frequency of applied f i e l d 0 dl<x X-component of v e l o c i t y of energy transport of an electron. , Y\ - n/zTT h is phncts Ca»S^nt Following Mott and Jones (1936) pages 305-306 1 = ^ F c f t U x f c f t ^ K (2.2.3) 3 d°K t ^ an element of volume i n K space - £ = charge on an electron. Substituting f o r f(if) from (2.2.2) i n (2.2-3) leads to (2.2.1+) 470 7" x where K = - J_ j«a g f o ^ f 0 Comparing (2.2.4) with (2.2..1) leads to the following e q u a l i t i e s \ x ' ^ (2.2.5) 1 lo when i-j^ = o 0 - i<i <0> (2.2.6) where = = thermal conductivity of the specimen conductor when Ljj_ = o. 2.-3 Spectral density of heat current f l u c t u a t i o n s i n a conductor. . In t h i s section Bakker and H e l l e r type approach i s extended to the c a l c u l a t i o n of sp e c t r a l density of heat current f l u c t u a t i o n s i n a.conductor at temperature TQ.. Consider a conductor of length L and area of cross section A. 'N-£» (t) i s the number of electrons i n the conductor at a c e r t a i n time t with t h e i r wave vectors l y i n g between K and K+dK. P ( T;K) i s the p r o b a b i l i t y that an electron which has not suffered a c o l l i s i o n at t continues to move f r e e l y i n the time i n t e r v a l t and t + T . I t i s assumed that'P i s independent of t, i . e . the c o l l i s i o n process i s a Poisson Process. P(ttF) = e,Xp>(- Vfetft) ) (2.3-1) dH(t) = f l u c t u a t i o n of heat current along x d i r e c t i o n due to N^(t). = jEi !? l> x A>-^ (2-3-2) dH(t+7) = M ^ ^ f t + T j (2-3-3) (2.3-4) I f i s the occupancy number of the electrons then Mj? = ALYi^dJL (2.3.5) and VV/^ = M W O g A (2.3-6) ^ H t ) ^ ) , hi' Ety \£?(<j:m<r^A (2.3-7) Auto c o r r e l a t i o n function f o r the heat currents at (t) and (t-Hj) due to a l l e lectrons: Hit;v\(t+<r) - £ k ( E \ P J V £ ?(ie )Vn/ K ( 2- 3- 8 ) Using Wiener-Khinchin theorem, s p e c t r a l density of heat current f l u c t u a t j i s rx> Htfe) H(t-wr) Cos *r a<rp 0 (2.3.9) Substituting i n (2.3.9) from (2.3.8) and i n t e g r a t i n g w.r.t leads to whe whe re %Y ftg - _ fa iff V • = AL"' W 121 C7aa (Talman R.C. (19U6)) (2.3.II) Motion of electrons i n the conductor leads to a simultaneous flow of e l e c t r i c and heat currents. The spectral density of cross c o r r e l a t i o n between the e l e c t r i c and heat current i s c a l c u l a t e d as f o l l o w s : d Hit) = eg?> rvg,w (2.3.12) Here on following the procedures leading to equations (2.3.k) to (2.3.9) leads to where ^ © - A L r ' ^ t O = - K, 6 AL"' (2.3.I3) The negative sign i s expected as the heat current and e l e c t r i c current c a r r i e d by the electron are i n opposite d i r e c t i o n s . Equations (2.-2.1), (2.I.I), (2-3.ll) and (2.3.1k) suggest that f o r a conductor i n thermodynamic equilibrium at a temperature T Q, following k terminal network f u l l y describes i t s response to external forces and i t s properties noise-wise. Fig.(2.3.l) 10. ( i ) £ = 0 i . e . open c i r c u i t (2,2). Corresponding to t h i s s i t u a t i o n and the spectral density of current f l u c t u a t i o n s i s given by (Bakker and H e l l e r ) where \{{) « A L' ' ^ ( - 0 ( i i ) Open C i r c u i t ( l , l ) i . e . J=0 then from (2.2..1) 2. - ^ , M 2 Substitution f o r W Is made from (2.2-5). ^ ( k a . " * U ) (2.3-16) K = Conventional Thermal Conductivity of the Conductor I f H 0 i s the heat current flow when 1=0. Spectral density of H Q f l u c t u a t i o n s i s 5 0 1 , (2.3-17) where := Admittance between the terminals (2,2) when (l , . l ) are ' open c i r c u i t e d ' ^ = ( ^ £ ) = ^ ( , 3 . 1 8 , hence; r ,r\ it u-^z n a (2-3-19) ^ = AL 'K = Thermal Conductance of the Conductor at. T^. Equation (2.3-14) g i v i n g the spectral density of cross c o r r e l a t i o n between the currents H and I i s an example of.the generalised Nyquist theorem (Takahasi, 194l) i n which the elements of the •admittance matrix do not a l l have the same dimension. Spectral density of radiant heat current f l u c t u a t i o n s between -]_i n i two black bodies i n r a d i a t i v e contact 2 . 4 ;" .". '. . . „ — — — : : , — — — . . . .. . :... ;:. . . . . . . Two black bodies with surface areas A-j_ and A2 are maintained at temperatures T-j_ and T2 r e s p e c t i v e l y T-j_.^ T2 • There i s a net flow of radiant energy from A-^ to A2 which i s supplied by a power source i n A-j_ . Fig.(2 .4.1) F i r s t consider the radiant energy emitted by A^ . Let J^j/^i) be the occupancy of a c e l l i n bos.on phase-space. The number of photons i n the frequency range y and emitted from Aj to A2 i n time t i s (Planck, M. •1913')-where A = *S&. t y>} t r a n s i t time of photons from A^ to Ag C = speed of l i g h t V * * N y * ^ t (2.4.2) I f i s the energy flow from the surface of A^ i n time t due to photons i n the frequency rangeV and H+d'i) and 2, By {h) lAv/V, -(2.^-3) Using (2.U.2), (2.4.3) becomes w 2 (2 . 4 . 4 ) V«y = 2 Ah fe^Hj^W^ 12. spectral density of heat current f l u c t u a t i o n s f o r the radiant energy due to photons of frequency andV'+fl/v^ then °lfy " (Appendix 2a) (2,4..5) Substituting f o r Var Ey from(2.4.4) since the r a d i a t i o n of d i f f e r e n t frequencies are s t a t i s t i c a l l y independent, the spectral density of radiant heat current f l u c t u a t i o n s from A^ to Ag due to photons of a l l frequencies i s obtained by i n t e g r a t i n g (2.4.5). ^- 0 Any photon emitted by Aj and reabsorbed by i t w i l l not give r i s e to any f l u c t u a t i o n s . Hon '= heat current composed of r a d i a t i o n of frequency between)) and })+^}) radiated from A-j_ to Ag T o t a l heat current from A-j_ to Ag i s the i n t e g r a l o v e r a l l frequencies of the r a d i a t i o n i n equation (2.4.8) H z s *Lhk Uv (2.4.9) Average heat flow from A-j_ to Ag i s 0 2 • £i> f £ ( j ^ T/*^ * (2.4.10) 13-It i s assumed that the d i s t r i b u t i o n function f o r photons emitted by A-|_ i s one appropriate to i t s steady state temperature (Tolman R.C, 19k6, page 630). - r -! 4^ VW W» (h»lkT?J (2.4.11) AT, Substituting i n (2.4.10) from ( 2 - 4 . l l ) Comparing (2.4.7) with (2.4.12) one can e a s i l y write s H I,T ^ ' l (2.^.12) Since A i s symmetric i n su f f i x e s 1 and 2 introducing an appropriate geometrical f a c t o r which determines the f r a c t i o n of photons emitted by Ag and absorbed by Aj_ and following arguments s i m i l a r to the ones outlines above, one can show that J > "the spectral density of heat current f l u c t u a t i o n s from Ag to A-]_ i s given by <3S„ - W r i 4 J & (2.k:l3) 1. Spectral density of t o t a l heat'current f l u c t u a t i o n s , Is C, - 'K. 4-C Radiations from A-^ and Ag being independent (no r e f l e c t i o n s ) . < T J - P = * fe (TMBI* 4. Tv i ^ O (2 . 4 . 1 4 ) • I f the two bodies A]_ and A2 are i n thermodynamic equilibrium with each other then ( X ? * O i > * and C^-f) =" ^ f c T . ^ (2.4.15) where g = conventional thermal conductance between the two bodies "A-^ and Ag $ £ JllLJlil i's ^ l ^ - _ ? Hxj appendix (2b). T ~ 7 ^ c>T, ^ T a -Ik. Steady state c a l c u l a t i o n of heat current f l u c t u a t i o n s "becomes extremely complicated when the bodies and Ag are not black. R e f l e c t i o n from the bodies w i l l destroy the s t a t i s t i c a l independence of the radiant heat currents. 15-CHAPTER I I I Temperature f l u c t u a t i o n s i n a conductor i n Thermodynamic Equilibrium,Isothermal Case 3-1 Equivalence between heat flow equations and transmission-line equations. Consider a one dimensional non-inductive transmission l i n e . Let Re = Resistance per unit length Gic = Conductance per unit length Ce = Capacity per un i t length then shunt admittance per unit length and . = series Impedance per un i t length = Rc I f l ( x ) and V(x) are current and voltage re s p e c t i v e l y at some point x then and (3-1-1) Consider a thermal conductor of length L and area of cross-section A. The specimen i s i n thermodynamic equilibrium at temperature T Q . Microscopic temperature f l u c t u a t i o n s w i l l be present along the length of the specimen, except at the two ends. Two ends of the specimen are held constant at T Q . 16. Heat balance equation, neglecting l a t e r a l temperature gradient i n the specimen compared to the l o n g i t u d i n a l temperature gradient, i s given by: - A A 0 -_CAM (3.1.-2) where A = l a t e r a l conductivity K = l o n g i t u d i n a l conductivity C = heat capacity per u n i t volume Response of (3..I.2) to a sinusoidal s i g n a l at frequency i s where 9g = l a t e r a l heat conductance per unit length = AA R9 = l o n g i t u d i n a l thermal resistance per unit length =(I<A) CQ = heat capacity per unit length = C A If h i s heat current, Comparing (3.L3) and (3-1-4) with (3.I-I) i t i s seen that equations of voltage and current d i s t r i b u t i o n s i n a RCG transmission l i n e and those f o r temperature and heat current d i s t r i b u t i o n s i n a specimen with l a t e r a l and l o n g i t u d i n a l heat current flows are s i m i l a r , i . e . (3.1-3) and''(3.1-4) can be obtained by replacing ( i ) M b y e ( i i ) L by h ( i i i ) by R& ( i v ) G\t by 9e (v) C e by Ce i n (3-1.1). •17-Thus a conductor with heat flow described by (3.I.2) has an e l e c t r i c a l transmission l i n e as i t s analogue. 3-2 Generalisation of Nyquist theorem. Consider a h terminal network i n thermodynamic equilibrium at temperature T Q . Fig.(3-2.1) Impedance matrix i s defined as follows: (3.2.I) Reciprocal of the Impedance Matrix defines the Admittance Matrix: I , ^ Y„ + ^ V,x i r - \i,yw + vtv« (3.2.2) Twiss, R.-Q (1955) has shown that the s p e c t r a l density of cross c o r r e l a t i o n s between the voltages \|\ and and the currents I,.-and l a i s : M,,\l, ^ _ / w (3-2-3) The r e s u l t s (3-2-3) were f i r s t derived by Takahasi•(l9^l). •3-3 Spectral density of Resistance fluctuations,Isothermal System. For c a l c u l a t i n g the spectral density of resistance f l u c t u a t i o n s i n the specimen equivalent "temperature generators" i n series with thermal resistance and'"heat current generators" i n shunt with the thermal conductance are introduced. These generators are assumed to be uniformly d i s t r i b u t e d along 18. the length of the specimen, the generators are not c o r r e l a t e d and t h e i r s p ectral d e n s i t i e s of f l u c t u a t i o n s i n temperature and heat current are given hy (1.2.U). The generalisation of Nyquist theorem provides an elegant t o o l f o r computing temperature c o r r e l a t i o n s i n the specimen. I 1 r0 • "1 1 • I I 1 X= X, Fig.(3-3-l) Analogous to (3-2.3) s p e c t r a l density of cross c o r r e l a t i o n s i n temperature %ft " ^ e ' i x i j = Z e ' " ' 2 ( 3 " 3 ' i ) Appendix 3(a), equation (3-a.l3) from appendix "3a f o r the ends of the conductor clamped at T Q. , - z ^ ^ ( L - j ^ ) (3.3.2) 9 " &'C ^ K V „ L X K < / Z where H f t c C h a r a c t e r i s t i c thermal impedance' of the specimen (3-3-3) i f thermal propagation constant of the specimen D.C. e l e c t r i a c l resistance of the specimen •g- —rzz = temperature c o e f f i c i e n t of Resistance at T t0 dT0 * o e l e c t r i c a l r e s i s t i v i t y of the specimen at 'T . (3-3-h) 19-spectral density of resistance f l u c t u a t i o n s ^ ^ ^ r / J j /?« ^ . ^ x ^ x , (3.3.5) .Substituting f o r ^)Q^(U)^> f r o r a e ( l u a t i o n (3-3-1). F i n a l l y on inte g r a t i o n i n (3-3-5) 1 - _ / Putting SA(f) = ^ r 0 V > 0 V ^ ( 3 . 3 . 6 ) If .1 'is a sensing current through the specimen then the sp e c t r a l density of voltage f l u c t u a t i o n s due to temperature f l u c t u a t i o n s : vSjH~9 is given by 6 ( / ) ^ 1 '2.. 2 (3.3.7) where \|0 .= D.C. voltage drop across the specimen Real part of fa) = [Co£$- y^ ')r £ ^ - f c J/l)} j (3-3-8) where o j (I 20. As shown i n Appendix 3(b) -f(A) has following p r o p e r t i e s : ( i ) i f \M « W ( T « | Re fa) 3 % ( i i ) i f \M\ » l •4 1 u- 1 , j U0«T #9 C3-3..2; 21. CHAPTER IV Temperature f l u c t u a t i o n s i n a "s i n g l e temperature" system 4.1 Single temperature system. In Chapter I I I an expression f o r spectral density of voltage f l u c t u a t i o n s due to temperature f l u c t u a t i o n s was obtained. In the model considered there Macroscopic and Microscopic temperatures were not the same. This resulted i n temperature c o r r e l a t i o n s , spectral density of which i s given by (3-3-l)-In the 'single temperature' system i t i s assumed that both the temperatures, Microscopic and Macroscopic, of the specimen are the same.. 4-2 Temperature noise. Let V and I be the instantaneous values of current -and voltage applied to the system characterised by a single temperature. I f the rate of loss of heat i s determined only by the excess temperature, then heat balance equation i s c , | | + 3 e = p = i v CU.-2.-l) where (j = instantaneous excess of temperature ..= T-T c C ( = heat capacity of the system 3 -- 9-y~ = ' l o c a l ' thermal conductance ( i . e . at the operating point.) A.C. Impedance of the system under operating temperature T Q i s given by; Burgess, R.'E. (1955) ' ^+ 0 ( ? O -r JU)C,' (4.2.2) where i \ 0 = -r = D.C- resistance at the operating point. ^Q = \l0To = Steady state power di s s i p a t e d i n the system. o( = J_ = Temperature c o e f f i c i e n t of resistance at T Q. (4.2.3) • 1 - 9 - - ? . •22. The-non l i n e a r i t y of I-V c h a r a c t e r i s t i c of the specimen i s purely a thermal e f f e c t . I f o(.= 0 then ^ = K 0 . For frequencies ^ ^ £ M = R G ; due to i t s thermal i n e r t i a the specimen i s not able to follow rapid v a r i a t i o n s i n temperatures. A.C Resistance of the specimen at .angular frequency ^ i s P r s cxPo 1 RM = K«[l+(9-*P.)f l4Wx«r.*;J (a) (4.2.4) and the A.C. reactance i s given by (b) (4.2.4) where .(c) (4.2.4) Eliminating (14 W <07 / from (4.2.4) (a) and'(b), (Burgess, R.E.) (4.2.5) Thus f o r a system characterized by a single time constant a p l o t of R(w) and linear-. Intercept on the XW/w = O axis gives the D.C. resistance and slope of the l i n e a r p l o t gives the r e c i p r o c a l of time constant ^ . A.C. Responsivlty of the single temperature system: Consider the following set up: 4^ A V Fig.(4.2.1) 23-R i s the instantaneous resistance of the single temperature system. V, I are the voltage and current respectively. There i s a f l u c t u a t i o n i n heat current from the system to the ambient and vice-versa. R-^ = Load resistance. (?f^ and 6|_ .are the equivalent Johnson noise e.m.f. generators i n series with the specimen resistance R and the load resistance R^ • To a f i r s t approximation the A.G". components of the equation (4.2.l) .satisfy the following equation of heat balance. * e = v, i o - u ( V 2 - 7 ) further, from K i r c h o f f ' s law Substituting f o r 0 i n (4 . 2 .6) from (k. 2.7) and evaluating ^l/r,' from (4.2.8) leads to o<V0 H, f^-t^u) + ^c,-v<3-o<Pe (4.2.9) equation (4.2.9) states that the current f l u c t u a t i o n Xt i s the r e s u l t of three noise e.m.fs .acting i n the c i r c u i t . The e.m.f. generator due to temperature f l u c t u a t i o n s of the system has the value &\\ given by ek - - <Vo\\\ (u.2.10) Current f l u c t u a t i o n due to heat current f l u c t u a t i o n s only i s 1\Q where o(\l0 Hi (4.2.II) I, 24.. and the spectral density of voltage f l u c t u a t i o n s due to heat current f l u c t u a t i o n s S W JJL-(^-^P 8) l(KwVJ ( 1 1 . , 1 2 ) i? - Ru The term Yl r _ f i n (4.2.11) takes i n t o account the electro-thermal feedback. Electro-thermal i n t e r a c t i o n i s due to the f a c t that the power dis s i p a t e d i n the specimen i s a function of time because the resistance i s changing. The time constant °J^= ^ </($-+°<f%rlJ i s a function of operating condition. Under constant current conditions, i . e . ^ | ?(coJ| 4-. 3 Discussion of temperature f l u c t u a t i o n s i n ( l ) Isothermal System (Chapter III) and Single Temperature System For the sake of comparison we w i l l regard the Isothermal and the single temperature systems to be i n thermodynamic equilibrium at a temperature 'T . Isothermal System: Two ends of the specimen conductor, i . e . X = 0 and )( ..= L are •'clamped' at a temperature T Q. Temperature f l u c t u a t i o n ( s p e c t r a l density) at a given point on the conductor I s given by s e t t i n g X t - X| = X i n (3-3-2). At the given point and f o r f i x e d parameters of the system sp e c t r a l density of temperature f l u c t u a t i o n s i s a maximum when UJT^< I . With t h i s approximation 25-(4.3.1) 2 *2» where i -L96 = A L A . The expression (4.3.I) vanishes at the two ends y(. = 0 and X= 1— • This i s •expected "because the two ends are clamped at T Q . (4.-3.l) has a maximum at *--L/2-(4.3.2) .Single temperature system: The single temperature system i s characterised by a complete c o r r e l a t i o n between the temperatures at d i f f e r e n t points at any instant of time. i. This s i t u a t i o n represents a " t h e r m a l l y short" conduction. Spectral density of temperature f l u c t u a t i o n s at a l l points i s the same. In thermodynamic equilibrium ^ ^ / ^ - G . With t h i s value (4.2.10) becomes A p l o t of s p e c t r a l density of temperature f l u c t u a t i o n s i n the two systems as a function of distance looks l i k e Fig. ( 4.3 . 1 ) . T0 L F i g , ( 4 . 3 . i ) ' 26. Supposing a sensing current I Q i s passed through the two systems, spectral density of the voltage f l u c t u a t i o n s due to temperature f l u c t u a t i o n s i n each case i s given by the follow i n g : Isothermal model 8«) (3-3-7) Single temperature model where »j" - S / ^ * Ce/^ c^d «P0/^«\ f o r LO <TJX< | (4.2.13) (4-3-4) For-a specimen of f i x e d length L increasing the l o n g i t u d i n a l conductivity K and or decreasing the l a t e r a l conductivity ^ reduces the voltage f l u c t u a t i o n s due to temperature f l u c t u a t i o n s as compared to that i n a single temperature system. When W<T»I 9 t t f l ) ~ Temperature f l u c t u a t i o n s i n the Isothermal system are equivalent to fl u c t u a t i o n s i n temperature of a ' s t r i n g ' of single temperature systems. Temperature f l u c t u a t i o n s at points f a r t h e r from the thermally clamped ends of the system are not sens i t i v e to the boundary condition. Fig.(4-3-2) shows the spe c t r a l density of temperature f l u c t u a t i o n s i n the two systems. To Stogie "u.v i^j.tYftjMv'e-^S te.vw Ic Fig.(4.3-2) 27-CHAPTER V Experimental Aspects 5-1 S e l e c t i o n of Sample . In a metal current noise i s not detectable because the number of electrons i s fi x e d . Only two sources of noise are'present. They, are ( i ) Johnson Noise and ( i i ) Temperature Noise. The experimental i n v e s t i g a t i o n outlined below i s therefore concerned with -a m e t a l l i c conductor. The choice of sample i s dictated by the following requirements, ( i ) Material of the sample be known. ( i i ) Uniformly high temperature be attainable. This w i l l increase the heat current f l u c t u a t i o n s . ( i i i ) S u f f i c i e n t l y high d-c. resistance at the operating point. This helps i n improving the noise f i g u r e of the noise detecting c i r c u i t . ( i v ) Tod± should be s i g n i f i c a n t l y l a rger than unity so that a check can be made f o r the appropriate expression f o r Johnson noise. (v) S u f f i c i e n t l y small thermal time constant at the operating point. It i s the r e c i p r o c a l of the time constant that determines the frequency below which temperature noise increases. P i h l i t e s A x i a l leaded s t y l e (Type 13-7) manufactured by Kay E l e c t r i c Co. were found, acceptable. The lead material of the p i n l i t e i s platinum and the filament material i s tungsten. The lamps were examined with a microscope f o r uniformity of temperature at a suitable value of current. It was observed, i n case of some of the lamps, that at a c e r t a i n bias current the resistance dropped. This on v i s u a l check l e d to the observation that decrease i n resistance of the •lamp was due to shorting out of adjacent c o i l s of the filament.. This was the case with a non-uniformly c o i l e d lamp, Fig.(5-1-1). Fig.(5.1.1) 28. %2 Determination of Parameters In the measurements outlined below the sample was immersed i n an o i l -bath contained i n a m e t a l l i c container. This was done to assure that the ends of the filament were thermally clamped to the bath temperature and the metallic container provided an e l e c t r i c s h i e l d . ( i ) , Measurement of R 0, P Q, T Q and . D.'C. resistance R Q of the sample was measured using a standard wheatstone's bridge c i r c u i t f o r d i f f e r e n t currents through i t . Power dis s i p a t e d i n the lamp was noted. Knowing the r e s i s t i v i t y of tungsten as a function of temperature Smlthels..'. (1955) the temperature at the operating points was determined. Temperature c o e f f i c i e n t of resistance at the operating temperature was then derived from these tables f o r tungsten at the operating temperature. ( i i ) Measurement of A.C. Impedance Following bridge c u r c u i t was used f o r measuring the a . c resistance and reactance of the lamp at the operating point. A l l the bridge components were c a r e f u l l y shielded against extraneous pickup. 1-1 Fig.(5.2.1) E]_ i s a model 202C Hewlett-Packard low frequency o s c i l l a t o r C]_, C2 are f i x e d capacitors 29-R3 are standard capacitance box and General R'adio resistance box respectively. They constitute the variable arm of the bridge L-j_ i s a bridge transformer (?v is- the p i n l i t e lamp under I n v e s t i g a t i o n D i s the detecting c i r c u i t c o n s i s t i n g of Keithl e y Low Noise pre-amplifier connected to a General Radio Wave Analyser (Type No..736-A) B i s a variable output d.c. supply A i s a c a l i b r a t e d ammeter. Balance condition corresponds to An advantage of using the c i r c u i t of F i g . (5-2.-1) i s that no d.c. passes through the other arms of the bridge excepting,^ . Use of Wave -Analyser as the n u l l detector has the advantage of eliminating any possible harmonics at n u l l point. Band width of the Wave Analyser was Performance of the bridge was checked by replacing ^ by a f i x e d carbon r e s i s t o r -at room temperature and i t s a.c. impedance measured over the frequency range ^ =Zoc/$ to (• = 5"Kc/S . The a.c. bridge readings' agreed within 5$ with the value of the r e s i s t o r determined by a d.c. VJheatstone bridge. 5•3 Noi&e- Measurement 1-, ( i ) Assembly A pre a m p l i f i e r using E810F P h i l i p s tube was assembled. Choice of the tubes was d i c t a t e d by i t s high value of trans-conductance, which implies low noise tubes. Careful sh i e l d i n g of the input c i r c u i t from the 30. r e s t o f t h e c i r c u i t was n e c e s s a r y t o p r e v e n t o s c i l l a t i o n s . F o l l o w i n g i s a b l o c k d i a g r a m o f t h e n o i s e d e t e c t i n g c i r c u i t u sed. vi-S P(?E. A>v\|?. Amp; Fig.(5.3.1) W-A CR.O 1-1 i s t h e i n p u t t o n o i s e m e a s u r i n g c i r c u i t . W-S i s a w a f e r s w i t c h . T h i s c o n n e c t s t h e i n p u t o f t h e p r e - a m p l i f i e r t o t h e lamp ^ o r ^54^ • ' i s a G e n e r a l R a d i o Decade R e s i s t a n c e box k e p t a t room t e m p e r a t u r e . W-A i s a narrow b a n d (Af = 4 $ ) G e n e r a l R a d i o Wave A n a l y s e r (Type 736-A). Wirewound r e s i s t o r s a r e u s e d as s o u r c e r e s i s t a n c e t o en s u r e t h a t J o h nson n o i s e i s t h e o n l y . s o u r c e o f n o i s e i n t h e power s u p p l y . Source r e s i s t a n c e i s v e r y much g r e a t e r t h a n . T h i s i s done t o o b t a i n a c o n s t a n t c u r r e n t s o u r c e . ( i i ) Measurement o f E q u i v a l e n t N o i s e R e s i s t a n c e (Req). E q u i v a l e n t n o i s e r e s i s t a n c e o f t h e n o i s e m e a s u r i n g c i r c u i t a t a f r e q u e n c y i s o b t a i n e d as f o l l o w s . The i n p u t ( l , l ) i s s h o r t e d . The mean r.m.s. o u t p u t i s n o t e d i n W-A a t t h e chosen f r e q u e n c y . Then a v a r i a b l e R e s i s t a n c e S t a n d a r d , ^ s t d * room t e m p e r a t u r e i s i n t r o d u c e d a t t h e i n p u t ( l , l ) and i t s v a l u e a d j u s t e d t i l l t h e ave r a g e i n d i c a t i o n o f t h e r.m.s. o u t p u t V,,.^ i n W-A i s such t h a t V s t d / ^ s h = \J~2~. The v a l u e o f R s t a c o r r e s p o n d i n g t o t h i s s i t u a t i o n i s R g ^ ( f ) , o f t h e n o i s e m e a s u r i n g c i r c u i t . 31-Rg^ ( f ) of the noise detecting c i r c u i t was measured at f=40c/s, f=100e/s, f=1000c/s and f=5000c/s. The procedure used was to p l o t V g t d as a function of Rg-^ jL at each of these frequencies. It was observed that the p l o t was l i n e a r and therefore W-A readings were r.m.s. readings f o r the values of R s t d used-Values obtained f o r R e q ( f ) are as follo w s : R e q ( f ) = 35OO ohms -f=40c/s. R e q ( f ) = 85O ohms -f=UOc/s. R e q ( f ) .= 275 ohms f=lKc/s and 5Kc/s. These fi g u r e s show that l / f noise becomes apparent i n the system at frequencies below about 300c/s. ( i i i ) Measurement of noise i n the sample. (a) For checking the accuracy of noise measurements, the lamp i n F i g . ( 5 - 3 - l ) was replaced by a f i x e d r e s i s t o r at room temperature. W-S was switched to 3 and the mean output meter reading V n s i n W-A was noted. Then W-S was switched to 2 and the value of Rg^a adjusted t i l l the mean output meter reading Vs^.^ was equal to V n s . I t was noted that at f=40c/s the random error i n resistance measurement was about 20$> while at f=5Kc/s the error i n measurement was l e s s than 10$. 'This i s because the equivalent noise resistance of the noise detecting c i r c u i t Increases with decreasing frequency. (b) .Lamp current was adjusted to 6 .3 m.-a. The choice of b i a s i n g current i s dic t a t e d by the need to keep the evaporation rate small enough while obtaining a high enough uniform temperature. As outlined i n (a) above W-A was set to kOc/s. W-S was switched to 3 and the mean output meter reading V n was noted. Then W-S was switched to 2 and the value of R sta adjusted t i l l the mean out put meter reading V was equal to V n . Keeping the current constant the procedure was repeated f o r various frequencies up to 5Kc/s. ft$(U)) \'i |A^ wyw ,at the operating point 32. 5.k Results (i) D.C. and A.C Data R a = Resistance of the lamp at the ambient temperature . 296°K = 30 ohms. T = Current through the lamp at the operating point = 6 .3 m.-a. RQ = D.C. resistance at the operating = 206 ohm. _3 P q = D.C Power dissipated in the steady state •= 8.2 x 10 watt. RD/Ra = 6 . 8 Using Smithels (1955) page 638, from the resistivity-temperature relationship for tungsten the ratio RQ/Ra = 6 . 8 corresponds to the operating temperature T = 1440°K o From the I-V characteristic Fig . ( 5-4.l) 4-¥ at the operating point dl i s equal to kOO ohm. A.C •Impedance'measurements at low frequency f = 20c/s gives Re (Z) = 395 + 15 ohm. (jt\j Thus ^ obtained from I-V characteristic (at the operating point and the values of Re Z(f) for f=20c/s agree within the experimental error in measurement. A plot of Rg(^) and | X s ^ f / 6 0 " is linear, Fig. (5 .4.2). The slope of the graph gives the value of characteristic time •= 5xlO~3sec. This implies that the lamp at the operating point can be treated as a system having a single characteristic time-. Power d i s s i p a t e d as a f u n c t i o n of Temperature 33-( i i ) Discussion of D.C. and A.C Data For a single temperature system (4.2.3) S u b s t i t u t i n g the measured values of and R„ gives = 0.-32 • As a check the value of ^'0/^ c a n b e independently c a l c u l a t e d by determining the i n d i v i d u a l parameters involved. i s found from Smitheljs ( l o c c i t ) at the operating point. = 2-1 X lo4o/c '' Knowing P Q as a function of T Q Fig.(5 .4.3) Q - at T Q i s c a l c u l a t e d Using the i n d i v i d u a l values of <?(. , -P0 and ^ gives e<tfy = 0.37 The two values of obtained from independent means show f a i r l y close agreement providing a confirmation of the lamp at the operating point as a Single Temperature system. Thus by two d i f f e r e n t c r i t e r i a (plot of R^(^) ,vs I Yi^fur and the °{?O/Q comparison) we have corroborated t h e - e s s e n t i a l feature of the model used i n the theory. It i s also noted from Fig.(5 .-4.3) that the 4 power di s s i p a t e d tends to be proportional to T f o r temperatures i n the neighbourhood of T Q. Noise Temperature of filament as a fu n c t i o n of frequency Experimental Curve (Dotted l i n e ) T h e o r e t i c a l Curve ( s o l i d l i n e ) . - 2Z0OK N $4s 34. 5-5 Noise Data We define a noise temperature T n ( f ) of the lamp at the operating point as a function of frequency as follows: (5.5,2) where Q t o - t a l ( f ) i s mean square voltage f l u c t u a t i o n i n the lamp at frequency f.. k = Boltzmann constant o R Q - D.C. resistance of the lamp at T Q K, Experimental values of T n ( f ) as a function of frequency are p l o t t e d i n Fig.(5-5-l). Discussion of noise data-We have shown that the lamp at the operating point acts as a Single Temperature System. T o t a l noise i n the system i s composed of two components ( i ) Johnson noise, ( i i ) Temperature noise. We assume that as: f a r as temperature f l u c t u a t i o n s due to r a d i a l heat losses are concerned the two components are independent. In the t h e o r e t i c a l expression f o r noise temperature T n ( f ) we add the mean square voltage f l u c t u a t i o n s due to each CUB = Or*0 + ^ c " - * ) We adopt the view that f o r a metal (^(f) = Johnson noise of the conductor = 4kT 0R Q (5-5-3) Since the filament of the lamp i s i n r a d i a t i v e contact with the ambient at 296°K and t o t a l e m i s s i v i t y of tungsten at the operating temperature Is.0.-2 Smithels (1955) page 666, we assume that equation (2.4.14) i s v a l i d . At the operating temperature (T Q/ip )^ J .' Hence the dominating term i n S H ( f ) i n (2.4.14) i s S H *0 = 2kT.X% (5-5-4) 35-Temperature noise from (U.2.13) and su b s t i t u t i n g f o r S j f f ) from (5-5.4) i s (5-5-5) ve as before,' (5.5.6) mean value of Substituting f o r the parameters i n (5.5-6) at the operating point y i e l d s frequency. In passing, we note that Brophy (1963) has reported measurements of noise i n a si m i l a r sample of p i n l i t e operating under s i m i l a r conditions. He has used expressions f o r temperature f l u c t u a t i o n s and heat current f l u c t u a t i o n s that are not appropriate to the experimental conditions to which they are presumed to apply. For example, at the high temperature (l600°K) at which h i s noise measurements are made, contrary to h i s assumption, Newton's law of cooling does not apply. Electro-thermal i n t e r a c t i o n , our fa c t o r ( (|— ), at the operating point i n h i s experiments does not appear i n h i s expression f o r temperature noise. His expression f o r Nyquist noise i s erroneous f o r he has assumed that Nyquist noise i s given by ^~IAT> i n the metal. It i s worth while to r e c a l l at t h i s point that i t i s purely the thermal i n e r t i a , of the system at the operating point that gives r i s e to Z ( w ) d i f f e r e n t from RQ. His experimental r e s u l t In Fig.(5.5.I) t h e o r e t i c a l value of T n ( f ) i s p l o t t e d as a function of 36.. " ^ / f ( 0 ) = i S p r o l : ' a b l y d u e t o a component of l / f noise.. It i s noted that the product of mean square noise voltage and frequency i n the range f=0 to f=30c/s i s appreciably constant. Wow reverting to the discussion of our own data, i t i s observed that f o r frequencies much l e s s than the c h a r a c t e r i s t i c frequency ^ D - ^IJ3]^ , the upper l i m i t to the t h e o r e t i c a l T n i s ( Q = 2100°K and T n - T Q for frequencies i n the range 800 c/s to 5Kc/s. Comparing the experimental value of T n ( f ) with the t h e o r e t i c a l value i n F i g . ( 5 - 5 ' l ) w e observe that the experimental curve agrees with the t h e o r e t i c a l curve f o r frequencies i n the range 800c/s to 5Kc/s. In t h i s frequency range the experimental noise temperature T n of the filament i s observed to be 1550°K. This agrees with the conventional temperature (Thermodynamic) of the filament "at the operating point within the experimental error. This implies that the temperature noise i s absent at higher frequencies and the conventional Nyquist theorem ( ljkT0&o ) i s applicable to the filament at L450°K. There i s a d e f i n i t e increase i n the experimental noise temperature f o r frequencies l e s s than 800c/s. The r i s e i s of the type expected from the appearance of temperature f l u c t u a t i o n s at low frequencies. However, the frequency at which t h i s appears i s higher by a f a c t o r of about 5 than the c h a r a c t e r i s t i c frequency ( ^ f f j ^ ) derived from impedance data. This discrepancy remains unexplained. 37-CHAPTER VI Conclusions and Recommendations 6.1 Conclusions Experimental r e s u l t s indicate that f o r frequencies much greater than the c h a r a c t e r i s t i c frequency.the noise temperature of the p i n l i t e filament i s i n close agreement with the conventional (thermodynamic) temperature. This would imply that the electrons i n the conductor -are e s s e n t i a l l y i n equilibrium with the l a t t i c e and that temperature f l u c t u a t i o n s are n e g l i g i b l e at frequencies greater than the c h a r a c t e r i s t i c frequency. Results f o r noise temperature f o r frequencies of the order of c h a r a c t e r i s t i c frequency seem to indicate' a d e f i n i t e increase i n noise temperature. The r e s u l t s though not conclusive make a strong case f o r the existence of temperature fluctuations.. However, there remains an unexplained discrepancy between the frequency at which these f l u c t u a t i o n s ' begin to appear and the t h e o r e t i c a l c h a r a c t e r i s t i c frequency. 6.2 Recommendations f o r future p r o j e c t s P i n l i t e s being coiled.filaments with e s s e n t i a l non-uniformities are not n e c e s s a r i l y the most elegant smaples f o r measurement of temperature f l u c t u a t i o n s at frequencies f o r which small enough noise fi g u r e of the noise detecting c i r c u i t i s attainable. The only parameter an experimenter can hope to adjust i s the temperature of the filament. This again i s l i m i t e d by the need to keep the rate of evaporation small. Following experimental projects are expected to lead to conclusive experiments regarding temperature fluctuations.. ( i ) A.thin s t r a i g h t wire of a metal, with diameter of a few microns and length a few cms. To increase the c h a r a c t e r i s t i c frequency ^ w i l l have to be Increased. This can be done by choosing a material of high emissivity. •38. I t i s advisable to work under conditions of r a d i a t i v e cooling only. This provides the p o s s i b i l i t y of operating point at which -£s <^L i s much greater than unity. ( i i ) Using a se n s i t i v e i n f r a - r e d detector, spectral density of heat current f l u c t u a t i o n s from a black body can be measured. This experiment w i l l provide a check f o r the expression f o r non-thermodynamic equilibrium fluctuations i n heat current between two black bodies derived i n (2.k). Following t h e o r e t i c a l problems suggest themselves.. ( i ) Extension of general s t a t i s t i c a l considerations f o r obtaining an expression f o r heat current f l u c t u a t i o n s i n a conductor.with a temperature gradient and also when both electrons and phonons are the c a r r i e r s of heat energy. ( i i ) Extension of the arguments outlinec(in (2.U) to sp e c t r a l density of heat current f l u c t u a t i o n s between two non-black bodies i n r a d i a t i v e contact. APPENDIX 1(a) Temperature f l u c t u a t i o n s i n Thermodynamic Equilibrium 39-Two bodies with heat c a p a c i t i e s Cj_ and Cg and instantaneous .temperatures T-j_ and Tg r e s p e c t i v e l y are i n thermal contact. ^ is.the thermal conductance. T 0 i s the thermodynamic equilibrium temperature of the system. j c aJf H = nett heat flow between the two bodies 0 - difference i n temperature between the bodies leading a flow of heat H. 1> ^ F i g . ( l . a . l ) T o t a l change i n entropy of the system due to a spontaneous flow of heat from Tg to T-^ i s c a l c u l a t e d as follows X = T ° + ^ ( l . a . l ) Spontaneous flow of heat from Ag to A^ r e s u l t s i n no nett change i n energy AS= C ^ + ( ^ = t o t a l change i n entropy = - _L £*AJi (c^) (l.a.2) P r o b a b i l i t y of a f l u c t u a t i o n i n temperature ^ T-^ i n C-^ i s oC e 2 * T » c* (l-a-3) .Covariance (l.a.K) ko. Negative sign of the covariance implies that there Is a n t i c o r r e l a t i o n between the temperatures of the two bodies. This i s a oorsequenee of the f a c t that the -spontaneous flow of heat i s not accompanied by any nett -change i n energy of the system. Spectral density of temperature f l u c t u a t i o n s Consider a body of heat capacity C^ i n thermal contact with an i n f i n i t e heat bath at temperature T Q. Qft) i s an instantaneous temperature difference between the body CQ_ and the ambient. Heat balance equation i s C\$ + f6 - « W ( 1 . A . 5 ) r\(i) = f l u c t u a t i n g heat current with a sharp c o r r e l a t i o n function and = O R f Solution of (l.a.5) i s Q (t) = Q (o) £ + ^ 6 J e ' hi^ij^t, - £ 0*0 ( i . a . 6 ) Spectral density of temperature f l u c t u a t i o n s i s (Wiener-Khinchin theorem) W i (l.a-7J where £ f e •= /<! f rom = A T( and =. 0« i n (l.a-3) As f a r as f l u c t u a t i o n s i n temperature i n are concerned analogous to e l e c t r i c a l case ( l . - l ) , we introduce series temperature generators and shunt heat current generators to give (l.a-7) I ' " —(3 T r i ^ Fig.(1.a.2) :kl. •Spectral d e n s i t i e s of the series temperature generator and shunt heat corrent generator are given by *lu> = ^ + > C ' J" ' (l.a.8) ,Thus i n thermodynamic equilibrium we have -an analogue of Wyquist theorem f o r heat current and temperature f l u c t u a t i o n s . k2. APPENDIX 2(a) Relationship between variance and spe c t r a l density Consider the following system. It has per f e c t memory up to a c e r t a i n time i n t e r v a l T and no memory l a t e r . T i s the sampling time. Xft) A(0 l ( t ) i s the frequency response of the system to ,a un i t d e l t a function. 0 e (2.a.l) Let ^ ( t ; - J t - T Then = s P e G t r a l density of f l u c t u a t i o n s i n ^ = l A C f ; | * ^ ( ^ (2.a,2.) where ^(f).= s p e c t r a l density of f l u c t u a t i o n s i n X • (Var^) = Variance of ^ , as measured i n the sampling time T. = W^/^-f from(2.a.2). o For sampling time T ^ longest c h a r a c t e r i s t i c time J n ^ y of the system ( i . e . i n the case treated i n (2.h) • .T max t r a n s i t time of photons between A-j_.• and :Ag) S^(f) i s uniform (y^jd) - S p ( ^ ) i ^ - f - (2,.a..3) This s i m p l i f i e s to . r ( O - d ^ ) T ( 2.a.,) 43-(2..a. 3) i s quite a general statement. I f has the following form ••Fig-. (2,a , 1) then f o r (.2. a. 3) to be v a l i d sampling time T should s a t i s f y the following i n e q u a l i t y : APPENDIX 2(b) Thermal Conductance i n Equilibrium . Consider the two bodies A-|_ and Ag in. thermodynamic equilibrium. H 0 i s the heat flow i n each d i r e c t i o n . Temperature of Aj_ i s ra i s e d so that ^ = ^ . ( - ^ > T > (2.b.l) S i m i l a r l y i f "17"""X r K i _ ~ Hi-i (thermodynamic equilibrium) ^ = ^ ( 2 . b . 2 ) from (2.b..l) and (2.b.2) r 1 s the conventional thermal conductance. 45. . APPENDIX 3(a) D i s t r i b u t e d f l u c t u a t i o n generator approach f o r a transmission l i n e i n thermodynamic equilibrium. Method: E.m.f. generators i n series with e l e c t r i c a l resistance and current generators i n p a r a l l e l with e l e c t r i c a l conductance are introduced uniformly along the l i n e . Spectral density of these generators are given by 1.1.5- Voltage and current components due to the two types of generators at given points are obtained. Replacing voltage by temperature current by heat current equivalent expressions f o r temperature c o r r e l a t i o n s i s written down. Appropriate value f o r sp e c t r a l density of the.series temperature generators and shunt heat current generators, proved i n 1.2.3, i s used. Outline of procedure. Fig.(3.a.l) * *' Assumptions: ( i ) The g e n e r a t o r s ^ / , ) , , 6(xV) and are completely uncorrelated. ( i i ) Response of the system i s l i n e a r . 1(X) = amplitude of shunt ( e l e c t r i c ) current generator \ = t o t a l ( e l e c t r i c ) current at a point X\ 1% = t o t a l ( e l e c t r i c ) current at a point Xa. V-j_ and Vg are voltages at the points %\ and X-Lrespectively £lXj= amplitude of series e.m.f generator YjtP) = admittance per unit length 'i^i = impedance per un i t length ^ = propagation constant = Y??e = c h a r a c t e r i s t i c impedance = e^/Y^ 46. Series e.m.f generators and shunt current generators contribute to current and voltage at any given point on the l i n e . These contributions are c a l c u l a t e d from the following' formulae. -4^ Y e ^ V -V- l_(x Boundary ; condition i s = 0 (3-a.l) Part I: Current and voltage.at two points due to an e.m.f generator. 'Switch o f f the shunt current generators. Superscript .'g. ' r e f e r s to component due to s e r i e s e.m.f generator and 's' r e f e r s to the somponent due to shunt current generators. '6-g- V'i^) = current at a point / = due to a shunt current source at x' With t h i s notation: tlx'; <•*«•»' , (3-a.2) ;Part I I Currents and voltages at X, and Xi- due to a shunt current generator. t K^Ki X-U Fig,(3.a.3) T o t a l voltage at y and /T,. i s obtained by adding the components due to shunt current generators and serie s e.m.f generators and then integr a t i n g over )(' . Thus taking into account a l l the generators. (3-a.U) *• S i m i l a r l y an expression f o r • (complex conjugate of VJX) ) i s obtained. Integral expressions f o r V|QO and V^)- are m u l t i p l i e d and bearing i n mind the assumption ( i ) i s a d e l t a function. (a) (3-a <A/|\jt^ -is obtained. (3,. 2?I2 f o r a RCG transmission l i n e shorted at y T^ B and y = L where 2 l 2 - ( 2 U - 2I6C2)H; ^ fat, Sfyvh y^ .v-:|r() Hi] -(3-« 2;a v * c (3, k9. A c a r e f u l algebra then leads to the following < V , V * > - S£ = 4*7.fc:2J . (3.1 .8) Spectral density of current c o r r e l a t i o n s . < Total current at X\ and yC^_ > as i - n the preceeding case i s obtained by adding the components due to serie s e.m.f. generators and shunt current generators and then i n t e g r a t i n g over From (3.a.2) and (3-a.3) ; L * -f I (3-a.9) S i m i l a r l y an i n t e g r a l expression f o r ^2 ^ i s obtained. As before, bearing i n mind the r e l a t i o n s (3-a.5) an expression f o r l | L i / "is obtained. f o r a shorted transmission l i n e . Admittance matrix i s defined by: ' ( 3.a,10) 50. Zc^svWvjrL Xi>y, (3-a.ll) Careful algebra leads to the following < T , r * > - - * * T . & ^ ( 3 . a . l 2 ) Temperature Correlations . For calculating the spectral density of temperature correlations following replacements are made i n the preceeding analysis, hy © 0 0 TV) by H60 v, *y 0 , L, by H i by Zevw) The heat and current generator and temperature generator at any given point are not correlated; also the heat current generators and temperature generators at different points are not correlated with themselves:;Jnor with each other; (3-a.13) For spectral density of temperature correlations where 2^ * **c ^ 51-APPENDIX 3(b) Properties of f(p) \/U\« i CO 01« \ expanding £, and £.. <7 3 • up to,terms i n 1 0 fefw - ^ - f ; [ ( i - s*- . )+ u V( J s| ' - . - ' ) f o r W J has.an upper bound.= ly H » tQ fa) ^ J u \ (3-b.l) (3-V2) 52. BIBLIOGRAPHY Bakker, C. and H e l l e r , G. -Physica 6, 262 (1939). Brophy, J . J. Appl. Phys. 3_4, 1890 (1963). Burgess, R. Proc. Phys. Soc. B. 68, 766 (.1955). Einstein,A. Ann. Physik. 33, -1275 (l910). G i l l , T. Nature 182, 239; (1958). Mott, N. and Jones, H. The Theory of the properties of Metals and Al l o y s . Dover Publications Inc., New York. Nyquist,, H. Phys. Rev 32, ; 110 (1928). Planck., M. Waermestrahlung (1913).• Tr a n s l a t i o n : The Theory.of Heat Radiation. Dover Publications Inc., New York. Smithels, C. . Metal Reference Book, Vol.2,-(Second E d i t i o n ) . Butterworths S c i e n t i f i c P u b l i c a t i o n s , London (1955)-Tak-ahasi,-H. Proc. The Physico-Mathematical Society of Japan, ser. 3, 23, 5^ 8 (1941). Tdlman, R. ;The P r i n c i p l e s of S t a t i s t i c a l Mechanics. ( F i r s t E d i t i o n ) Oxford U n i v e r s i t y Press (1946). Twiss, R. J . Appl. Phys. 26, 599 (1955).
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Fluctuations in a conductor
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Fluctuations in a conductor Kshatriya, Amarnath R. 1964
pdf
Page Metadata
Item Metadata
Title | Fluctuations in a conductor |
Creator |
Kshatriya, Amarnath R. |
Publisher | University of British Columbia |
Date Issued | 1964 |
Description | Applying purely thermodynamic arguments it has been shown that temperature fluctuations in a sample can be represented by introducing appropriate 'series temperature generator’ or 'shunt heat current generator'. The temperature fluctuations lead to resistance fluctuations in the sample. These resistance fluctuations due to temperature fluctuations can be detected as voltage-fluctuations (temperature noise) by using a sensing direct current through the sample. Statistical-mechanical arguments are used to obtain theoretical expressions for spectral density of heat current fluctuations in a metallic conductor in terms of the macroscopic properties of the conductor. Since the electrons are carriers of heat and electric currents in a metal, heat and electric current fluctuations are correlated. Spectral density of cross-correlation between electric and heat current is derived. Statistical considerations are extended to the calculation of the steady state spectral density of heat current fluctuations between two black bodies in radiative contact. Temperature noise in a system in which there is only a partial correlation between temperatures at different points along the length at any time (isothermal System) is compared with a system in which there is a complete correlation between temperatures at all points along the length at any time (Single Temperature System). Experimental results indicate that for the metal filament used Nyquist theorem can be applied at the operating temperature. For frequencies close to the characteristic frequency of the system it is observed that there is an increase in noise temperature of the filament due to temperature noise. |
Subject |
Electric conductors |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-09-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0085298 |
URI | http://hdl.handle.net/2429/37653 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
Download
- Media
- 831-UBC_1964_A6_7 K7.pdf [ 2.41MB ]
- Metadata
- JSON: 831-1.0085298.json
- JSON-LD: 831-1.0085298-ld.json
- RDF/XML (Pretty): 831-1.0085298-rdf.xml
- RDF/JSON: 831-1.0085298-rdf.json
- Turtle: 831-1.0085298-turtle.txt
- N-Triples: 831-1.0085298-rdf-ntriples.txt
- Original Record: 831-1.0085298-source.json
- Full Text
- 831-1.0085298-fulltext.txt
- Citation
- 831-1.0085298.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085298/manifest