(a) Noisy resistor(b)Noiseless resistor (c)Equivalent R W FIGURE 73.6 Thermal noise in a resistor The voltage N(o) generated thermally between two points in an open circuit conductor is the sum of an extremely large number of superimposed, independent electronically and ionically induced microvoltages at all frequencies up to f = 6,000 GHz at room temperature [see Gardner 1990, P. 235], near infrared. The mean relaxation time of free electrons is 1/f=0.5 X 10-10/Ts, so at room temperature of T= 290K, it is 0. 17 ps (1 picosecond= 10-2s). The values of N(t) at different times are uncorrelated for time differences(offsets) greater than T.=1/fe. The expected value of N() is zero. The power is fairly constant across a broad spectrum and we cannot sample signals at picosecond periods, so we model Johnson noise N(t) with Gaussian white noise W(t). Although u= E[W(t)]=0, the average power is positive at temperatures above OK, and is o Rww(O)[see the right side of Eq. (73.21). A disadvantage of the white noise model is its infinite power,i.e Rww0)=0x=oo, but it is valid over a limited bandwidth of B Hz, in which case its power is finite. In 1927, Nyquist [1928] theoretically derived thermal noise power in a resistor to be Pww(B)= 4kTRB(watts) 73.24) where R is resistance(ohms), B is the frequency bandwidth of measurement in Hz(all emf fluctuations outside of B are ignored), Pww(B)is the mean power over B(see Eq. 73.21), and Boltzmanns constant is k= 1. 38 x 023 J/K [see Ott, 1988; Gardner, 1990, P. 288; or Peebles, 1987, P. 227. Under external emf, the thermally induced collisions are the main source of resistance in conductors (electrons pulled into motion by an external emf at OK meet no resistance). The rms voltage is Wms=ow=[(4kTRB)]V over a bandwidth of B Hz Plancks radiation law is SN(w)=(2h fD/exp(hlf)/kT)-1], where h=6.63 X 10-34 J/s is Plancks constant, and fis the frequency [see Gardner, 1990, P. 234]. For If much smaller than kT/h=6.04 X 102 Hz=6,000 GHz, the exponential above can be approximated by exp(hlf//kT)=1+hlf kT. The denominator of SNN(w)becomes hlfVkT, So SNN w)=(2hl//(Hf/kT)=2kTW/Hz in a 1-Q2 resistor Over a resistance of RS2 and a bandwidth of B Hz(positive frequencies), this yields the total power Pww(B)=2BRSNN w)=4kTRB W over the two-sided Thermal noise is the same in a 1000-2 carbon resistor as it is in a 1000-Q2 tantalum thin-film resistor [see Ott, 1988]. While the intrinsic noise may never be less, it may be higher because of other superimposed noise(described in later sections). We model the thermal noise in a resistor by an internal source(generator ), as shown in Fig. 73.6 Capacitance cannot be ignored at high f but pure reactance(C or L)cannot dissipate energy, and so cannot generate thermal noise. The white noise model W(n) for thermal noise N(n) has a constant psdf Swuw)=n W/(rad/s)for -o0< w<oo. By Eq. 73.21, the white noise mean power over the frequency bandwidth B is Pww (b)= w(whw=n2(4兀B/2π)=2n (73.25) e 2000 by CRC Press LLC© 2000 by CRC Press LLC The voltage N(t) generated thermally between two points in an open circuit conductor is the sum of an extremely large number of superimposed, independent electronically and ionically induced microvoltages at all frequencies up to fc = 6,000 GHz at room temperature [see Gardner 1990, p. 235], near infrared. The mean relaxation time of free electrons is 1/fc = 0.5 ¥ 10–10/T s, so at room temperature of T = 290K, it is 0.17 ps (1 picosecond = 10–12 s). The values of N(t) at different times are uncorrelated for time differences (offsets) greater than tc = 1/fc . The expected value of N(t) is zero. The power is fairly constant across a broad spectrum, and we cannot sample signals at picosecond periods, so we model Johnson noise N(t) with Gaussian white noise W(t). Although m = E[W(t)] = 0, the average power is positive at temperatures above 0K, and is sW 2 = RWW (0) [see the right side of Eq. (73.21)]. A disadvantage of the white noise model is its infinite power, i.e., RWW(0) = sW 2 = `, but it is valid over a limited bandwidth of B Hz, in which case its power is finite. In 1927, Nyquist [1928] theoretically derived thermal noise power in a resistor to be PWW(B) = 4kTRB (watts) (73.24) where R is resistance (ohms), B is the frequency bandwidth of measurement in Hz (all emf fluctuations outside of B are ignored), PWW (B) is the mean power over B (see Eq. 73.21), and Boltzmann’s constant is k = 1.38 3 10–23 J/K [see Ott, 1988; Gardner, 1990, p. 288; or Peebles, 1987, p. 227]. Under external emf, the thermally induced collisions are the main source of resistance in conductors (electrons pulled into motion by an external emf at 0K meet no resistance). The rms voltage is Wrms = sW = [(4kTRB)]1/2 V over a bandwidth of B Hz. Planck’s radiation law is SNN(w) = (2hu f u)/[exp(hu f u/kT) – 1], where h = 6.63 3 10–34 J/s is Planck’s constant, and f is the frequency [see Gardner, 1990, p. 234]. For u f u much smaller than kT/h = 6.04 3 1012 Hz ª 6,000 GHz, the exponential above can be approximated by exp(hu f u/kT) = 1 + hu f u/kT. The denominator of SNN(w) becomes hu f u/kT, so SNN(w) = (2hu f u)/(hu f u/kT) = 2kT W/Hz in a 1-W resistor. Over a resistance of R W and a bandwidth of B Hz (positive frequencies), this yields the total power PWW(B) = 2BRSNN(w) = 4kTRB W over the two-sided frequency spectrum. This is Nyquist’s result. Thermal noise is the same in a 1000-W carbon resistor as it is in a 1000-W tantalum thin-film resistor [see Ott, 1988].While the intrinsic noise may never be less, it may be higher because of other superimposed noise (described in later sections).We model the thermal noise in a resistor by an internal source (generator), as shown in Fig. 73.6. Capacitance cannot be ignored at high f, but pure reactance (C or L) cannot dissipate energy, and so cannot generate thermal noise. The white noise model W(t) for thermal noise N(t) has a constant psdf SWW(w) = no W/(rad/s) for –` < w < `. By Eq. 73.21, the white noise mean power over the frequency bandwidth B is (73.25) FIGURE 73.6 Thermal noise in a resistor. P B WW SWW w dw no B noB B B ( ) = ( ) = ( ) = -Ú 1 2 4 2 2 2 2 p p p p p /