Solving for the constant no, we obtain n,= Pww( B)/2B, which we put into Eq (73. 20)to get the spectral density as a function of temperature and resistance using Nyquist's result above Swu(w)=n,= Pww(B)/4B= 4kTR2T B/4TB= 2kTR watts/(rad/s)(73.26) Some Examples The parasitic capacitance in the terminals of a resistor may cause a roll-off of about 20 dB/octave in actual resistors [Brown, 1983, P. 139]. At 290K (room temperature), we have 2kT=2X 1.38 X 10-2X 290=0.8 X 10-20 W/Hz due to each ohm [see Ott, 1988. For R= 1 MQ2(1022), Swwdw)=0.8x 10-4. Over a band of10°Hz, we have Pwn(B)=Swww)B=0.8×10-×10=0.8×106w=0.8 uw by Eqs.(73.24)and (73. 26). In practice, parasitic capacitance causes thermal noise to be bandlimited(pink noise). Now consider Fig. 73. 6(b)and let the temperature be 300k,R=10632, C=I pf (1 picofarad = 10-12 farads), and assume L is OH. By Eq (73.26), the thermal noise power is (w)=2kTR=2×1.38×10-23×300×106=828×10-17W/Hz The power across a bandwidth B=10% is Pww(B)= Sww(w)B=8280 X 10-l2W, so the rms voltage is wms [Pw(B)]=91μV. Now let Y(t) be the output voltage across the capacitor. The transfer function can be seen to be H(w) I(w)(1/fwOH{r(w)R+(1/jiO=(1/mO/[R+1/jnO=l/1+jwRCl(wherew)istherouriertransfor of the current). The output psdf [see Eq(73. 22)is Srr(w)=H(w)/(w)=(1/[1+w2R"C21)Sww(w) Integrating Snx(w)=(1/[1+w2R-CD)Swwfw)over all radian frequencies w=2f [see Eq.(73. 21)], we obtain the antiderivative(828 X 10-7)(1/RC)atan(RCw)/2. Upon substituting the limits w= too, this becomes 828 X 1017[m2+m/21/2RC=414×10-(1/2RC)=207×1017×106=2070×10- W/Hz. Then o2=Y()= y(-∞,∞)=2070×10-w,soYm(t)=y=[P(∞,)2=45.5μV. The half-power(cut- off) radian frequency is w=1/RC=10rad/s, orf= w/2T=159.2 kHz. Approximating Snx(w) by the rectangular spectrum Srrw)=no-10< w<10 rad/s(0 elsewhere), we have that RnT)=(w/sinc(wr), which has the first zeros at:|=T, that is =1/(2f)[see Fig. 73. 4(b)]. We approximate the autocorrelation by Rrr(t)=0 for 521/2) Measuring Thermal noise In Fig. 73.7, the thermal noise from a noisy resistor R is to be measured, where R, is the measurement load. The incremental noise power in R over an incremental frequency band of width dfis Pwu(df)=4kTRdf w by Eg(73. 24). Pn(df) is the integral of Sn(w)over df by Egs.(73. 21), where Sn(w)=H(w)PSwn(w), by Eq (73. 22). In this case, the transfer function H(w)is nonreactive and does not depend upon the radian frequency we can factor it out of the integral). Thus, Rr(df)= H()I(2KTR)df=(R/(R+Ri)14KTRdf To maximize the power measured, let RL=R. The incremental available power measured is then Prdf) 4kTR-df/(4R2)=kTdf [see Ott, 1988, P. 201; Gardner, 1990, P. 288; or Peebles, 1987, P. 227]. Thus, we have the result that incremental available power over bandwidth df depends only on the temperature T. Prrdf=kTf (output power over df) Albert Einstein used statistical mechanics in 1906 to postulate that the mean kinetic energy per degree freedom of a particle,(1/2)mE[v(o)) is equal to(1/2)kT, where m is the mass of the particle, yt)is© 2000 by CRC Press LLC Solving for the constant no , we obtain no = PWW(B)/2B, which we put into Eq. (73.20) to get the spectral density as a function of temperature and resistance using Nyquist’s result above. SWW(w) = no = PWW(B)/4pB = 4kTR2pB/4pB = 2kTR watts/(rad/s) (73.26) Some Examples The parasitic capacitance in the terminals of a resistor may cause a roll-off of about 20 dB/octave in actual resistors [Brown, 1983, p. 139]. At 290K (room temperature), we have 2kT = 2 3 1.38 3 10–23 3 290 = 0.8 3 10–20 W/Hz due to each ohm [see Ott, 1988]. For R = 1 MW (106 W), SWW(w) = 0.8 3 10–14. Over a band of 108 Hz, we have PW W(B) = SW W(w)B = 0.8 3 10–14 3 108 = 0.8 3 10–6 W = 0.8 mW by Eqs. (73.24) and (73.26). In practice, parasitic capacitance causes thermal noise to be bandlimited (pink noise). Now consider Fig. 73.6(b) and let the temperature be 300K, R = 106 W, C = 1 pf (1 picofarad = 10–12 farads), and assume L is 0H. By Eq. (73.26), the thermal noise power is SWW(w) = 2kTR = 2 3 1.38 3 10–23 3 300 3 106 = 828 3 10–17 W/Hz The power across a bandwidth B = 106 is PWW(B) = SWW(w)B = 8280 3 10–12 W, so the rms voltage is Wrms = [PWW(B)]1/2 = 91 mV. Now let Y(t) be the output voltage across the capacitor. The transfer function can be seen to be H(w) = {I(w)(1/jwC)}/{I(w)[R + (1/jwC)]} = (1/jwC)/[R + 1/jwC] = 1/[1 + jwRC] (where I(w) is the Fourier transform of the current). The output psdf [see Eq. (73.22)] is SYY(w) = *H(w)* 2SWW(w) = (1/[1 + w2R2C2 ])SWW(w) Integrating SYY(w) = (1/[1 + w2 R2C2 ])SWW(w) over all radian frequencies w = 2pf [see Eq. (73.21)], we obtain the antiderivative (828 3 10–17)(1/RC)atan(RCw)/2p. Upon substituting the limits w = ±`, this becomes 828 3 10–17[p/2 + p/2]/2pRC = 414 3 10–17(1/2RC) = 207 3 10–17 3 106 = 2070 3 10–12 W/Hz. Then sY 2 = E[Y(t)2 ] = PYY(–`,`) = 2070 3 10–12 W, so Yrms(t) = sY = [PYY(–`,`)]1/2 = 45.5 mV. The half-power (cut-off) radian frequency is wc = 1/RC = 106 rad/s, or fc = wc /2p = 159.2 kHz.Approximating SYY(w) by the rectangular spectrum SYY(w) = no, –106 < w < 106 rad/s (0 elsewhere), we have that RYY(t) = (wc /p)sinc(wct), which has the first zeros at uwctu = p, that is utu = 1/(2fc) [see Fig. 73.4(b)].We approximate the autocorrelation by RYY(t) = 0 for usu ³ 1/2fc . Measuring Thermal Noise In Fig. 73.7, the thermal noise from a noisy resistor R is to be measured, where RL is the measurement load. The incremental noise power in R over an incremental frequency band of width df is PWW(d f ) = 4kTRdf W, by Eq. (73.24). PYY(d f ) is the integral of SYY(w) over df by Eqs. (73.21), where SYY(w) = *H(w)* 2 SWW(w), by Eq. (73.22). In this case, the transfer function H(w) is nonreactive and does not depend upon the radian frequency (we can factor it out of the integral). Thus, To maximize the power measured, let RL = R. The incremental available power measured is then PYY(d f ) = 4kTR2 df /(4R2 ) = kTdf [see Ott, 1988, p. 201; Gardner, 1990, p. 288; or Peebles, 1987, p. 227]. Thus, we have the result that incremental available power over bandwidth df depends only on the temperature T. PY Y(df) = kTdf (output power over df) (73.27) Albert Einstein used statistical mechanics in 1906 to postulate that the mean kinetic energy per degree of freedom of a particle, (1/2)mE[v2 (t)], is equal to (1/2)kT, where m is the mass of the particle, v(t) is its P df H f kTR df R R R kTRdf YY L L df df ( ) = ( ) ( ) = { + ) }( ) -Ú * *2 2 2 /( 4