Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: January 02, 2011) Johnson noise Johnson-Nyquist theorem Boltzmann constant Parseval relation Correlation function Spectral density Wiener-Khinchin (or Khintchine) Flicker noise Shot noise Poisson distribution Brownian motion Fluctuation-dissipation theorem Langevin function ___________________________________________________________________________ John Bertrand "Bert" Johnson (October 2, 1887–November 27, 1970) was a Swedish-born American electrical engineer and physicist. He first explained in detail a fundamental source of random interference with information traveling on wires. In 1928, while at Bell Telephone Laboratories he published the journal paper "Thermal Agitation of Electricity in Conductors". In telecommunication or other systems, thermal noise (or Johnson noise) is the noise generated by thermal agitation of electrons in a conductor. Johnson's papers showed a statistical fluctuation of electric charge occur in all electrical conductors, producing random variation of potential between the conductor ends (such as in vacuum tube amplifiers and thermocouples). Thermal noise power, per hertz, is equal throughout the frequency spectrum. Johnson deduced that thermal noise is intrinsic to all resistors and is not a sign of poor design or manufacture, although resistors may also have excess noise. http://en.wikipedia.org/wiki/John_B._Johnson 1 ____________________________________________________________________________ Harry Nyquist (February 7, 1889 – April 4, 1976) was an important contributor to information theory. http://en.wikipedia.org/wiki/Harry_Nyquist ___________________________________________________________________________ 10S.1 Histrory In 1926, experimental physicist John Johnson working in the physics division at Bell Labs was researching noise in electronic circuits. He discovered random fluctuations in the voltages across electrical resistors, whose power was proportional to temperature. Harry Nyquist, a theorist in that division, got interested in the phenomenon and developed an elegant explanation based on fundamental physics. hence this type of noise is called Johnson noise, Nyquist noise, or thermal noise. 10S.2 Spectral density Suppose we measure a physical quantity x(t) as a function of time t. We define the Fourier transform of x(t) as 1 X () dteit x(t) 2 Note that 2 1 X * () dteit x(t) X () . 2 since x(t) is real. The inverse Fourier transform is given by 1 x(t) deit X () . 2 (a) Parseval relation. 1 1 x(t)x* (t)dt dt deit X () d'ei't X *(') 2 2 1 dX () d' X *(') dtei(')t 2 1 2 dX () d' X * (') ( ') 2 d X () 2 2 d X () 2 0 since X () 2 X ()X * () X ()X () X () 2 . (b) We use a truncated signal. It is sometimes necessary, in order to avoid convergence problem, to approach the definition of the integral as a limiting process. Using the Parseval relation, we get T / 2 1 2 2 x2 (t) lim x2 (t)dt lim d X () . T T T T / 2 T 0 We define the spectral density as 3 2 2 lim X () G() , (1) T T which is the ensemble average of X () 2 . Then we have x2 (t) dG() . (2) 0 ((Note)) We assume that the average of process parameter over time and the average over the statistical ensemble are the same (ergodicity). 10S.3 Correlation function The limiting process required to calculate G() is sometimes awkward to use in practice. Here we show that correlation function can be closely related to the spectral density. The correlation function associated with a stationary random function of time t, x(t), is defined by x(t)x(t . (3) which means that Eq.(3) is independent of t. The Fourier transform of x(t ) is given by 1 1 dteit x(t ) ei dteit'x(t') ei X () . 2 2 Correspondingly, the inverse Fourier transform is given by 1 x(t ) dei(t ) X () . 2 We need to calculate the product given by 1 x(t)x(t ) d d'ei(t )ei' X ()X (') . 2 It is sometimes necessary, in order to avoid convergence problem, to approach the definition of Eq.(3) as a limiting process. The integral of this product is 4 T / 2 1 T / 2 x(t)x(t )dt d d' dtei(')tei' X ()X (') T / 2 2 T / 2 1 2 d d'ei' ( ')X ()X (') 2 dei X ()X () dei X () 2 2 d cos( ) X () 2 0 Note that T / 2 dtei(')t 2 ('). T / 2 Then we have 1 T / 2 x(t)x(t lim x(t)x(t )dt T T T / 2 2 2 lim d cos( ) X () T T 0 The correlation function C() is obtained as the ensemble average of x(t)x(t , C( ) x(t)x(t 2 2 lim d cos( ) X () T T 0 , (4) d cos( )G() 0 1 dei G() 2 where we use the relation(Eq.(1)). The inverse Fourier transform of G() is given by 5 1 2C( ) dei G() . 2 2 The corresponding Fourier transform is 1 2C( ) G() dei 2 2 2 dei C( ) . (5) 2 4 d cos( )C( ) 2 0 Equation (5) is the Wiener-Khinchin (WK) theorem. G() and C() are a Fourier transform pair; G(): spectral density C(): correlation function 10S.4 Properties of the correlation function The correlation function C( ) x(t)x(t is an even function of . C( ) x(t)x(t x(t )x(t) x(t)x(t ) C( ) . We also have [x(t) x(t )]2 0 , or x 2 (t) x 2 (t ) 2 x(t)x(t ) 0 , or C( 0) x2 (t) x(t)x(t ) C( ) . 10S.5 Johnson-Nyquist theorem 6 At any non-zero temperature we can think of the moving charges as a sort of electron gas trapped inside the resistor box. The electrons move about in a randomised way — similar to Brownian motion — bouncing and scattering off one another and the atoms. At any particular instant there may be more electrons near one end of the box than the other. We consider a resistance R which has a length l, a cross section A. The relaxation time of the carriers (electrons) is e. There are n electrons per unit volume (n is called the electron density). The total number of electrons is N; N nAl . From the Ohm's law, the open circuit voltage is V IR RJA RA(neu), where V is a voltage, I is a current, J is the current density, A is the conductor cross section area, e is the absolute value of the charge of electron (e>0), and u is the average velocity of electrons. Since 1 N u ui , N i1 we have N 1 N Re N N V RA( e) ui ui Vi , Al N i1 l i1 i1 where Re V u . i l i Suppose that the correlation function is expressed by 2 2 0 Re 2 0 C( ) Vi (t)Vi (t Vi e ui e . l The spectral density is given by 7 4 G() d cos( )C( ) 2 0 2 4 Re 2 u d cos( )e 0 i 2 l 0 2 4 Re 2 0 ui 2 2 l 1 ( 0 ) (Lorentz-type spectral density) In general, the spectral density G() has a Lorentz-type when the correlation function C() is described by a exponential decay; exp(-/t0). For the voltage signals, the spectral density G() is customary to use units of [V2 s] = [V2 Hz−1]. We note that 1 1 mu 2 k T , (equipartition theorem) 2 i 2 B where m is the mass of electron and kB is the Boltzmann constant. In metal at room temperature, -12 e is very small; e<10 s. For the DC and microwave regions, 2f c 1. Then we get 2 4 Re kBT G() 0 , (white noise) 2 l m which is independent of . We get V 2 dG() G() i 0 2 f G() , 2 Re kBT 4f 0 l m since = 2f. We have Re2 NA V 2 N V 2 4k TRf ( 0 ) 4k TRf . i B ml 2 B Here we note that the resistance is define by 8 1 l l ml 2 R , A N e2 NAe2 0 A 0 Al m and 2 2 2 Re 0 NA ml e 0 NA 2 2 2 1. ml NAe 0 ml Experimentally it is predicted that the plot of V 2 vs R may exhibits a straight line. The slope will give some estimation for the value of the Boltzmann constant kB. 10S.6 Evaluation of V 2 The Boltzmann constant is given by -23 kB = 1.380650410 x 10 J/K. 2 V 4kBTRf , where T is the temperature (K), R is the resistance, and f is the frequency range. (i) T = 293.15 K, f = 106 Hz, R = 1 M V 2 127.238V . (ii) T = 293.15 K, f = 1.9980 x 104 Hz, R = 200 V 2 254.349nV 0.254V . (iii) T = 293.15 K, f = 1.0 x 104 Hz, R = 1000 V 2 402.362nV 0.402V . 10S.7 Equivalent circuit (Thevinin theorem and Norton theorem) 9 Noise volatge source (the left of the above figure) and noise current source (the right of the above figure); 1 G R 2 V 4kBTRf V 2 2 4kBTRf I 2 2 4kBTG(f ) . R R In Fig. we show the equivalent circuits for the noise voltage source and noise current source. These circuits are equivalent according to the Thevinin's theorem and Norton's theorem. ________________________________________________________________________ 10S.8 Measurement of Johnson noise 10 The above figure shows an equivalent circuit for the Johnson noise produced by a resistor. n is an invisible random voltage generator, connected in series with an ideal (noise free) resistor R.