ECE594I Notes set 6: Thermal Noise

Transcription

1 C594I ots, M. odwll, copyrightd C594I Nots st 6: Thrmal Nois Mark odwll Uivrsity of Califoria, ata Barbara , fax

2 frcs ad Citatios: C594I ots, M. odwll, copyrightd ourcs / Citatios : Kittl ad Kromr : Thrmal Physics Va dr Zil : Nois i olid - tat Dvics Papoulis : Probabil ity ad adom Variabls hard, comprhs iv Pyto Z. Pbls : Probabili ty, adom Variabls, adom igal Pricipl s itroduct ory Wozcraft & Jacobs : Pricipl s of Commuicat ios giri g. Motchbak r : Low Nois lctroi c Dsig Iformatio thory lctur ots : Thomas Covr, taford, circa 98 Probabilit y lctur ots : Marti Hllma, taford, circa 98 Natioal micoductor Liar Applicatio s Nots : Nois i circuits. uggstd rfrcs for study. td Va dr Zil, Wozcraft & Jacobs, Pbls, Kittl ad Kromr Paprs by Fukui dvic ois, mith & Prsoik optical rcivr dsig Natioal mi. App. Nots! Covr ad Williams : lmts of Iformatio Thory

3 Boltzma Law C594I ots, M. odwll, copyrightd But k l g g xp { / k} P ε P ε g g 0 0 ε ε xp xp { 0 ε / k} 0 ε 0 ε xp { ε / k} k 0 But If 0 ε 0 ε 0 ε O ε T th rsrvoir is big, th driviativs ar small, ad P ε ε ε xp P ε st - ordr is ough : Importat : this is th probabilty of a sub - systm with dgr of frdom g big i a particular stat, ot th probability distributio of rgy. Ths diffr bcaus stats ar ot uiformly distribut d i rgy.

4 Partitio Fuctio C594I ots, M. odwll, copyrightd If th systm has allowd stats s, th P ε i whr xp ε / Z Z partitio fuctio i xp s / s

5 Backgroud: Harmoic Oscillator C594I ots, M. odwll, copyrightd Photos i a mod at frqucy f harmoic oscillator at f

6 rgy of Photos i om Mod C594I ots, M. odwll, copyrightd lctromagtic mod with frqucy. Allowd rgis of th stat : / s h / s hf whr s is a itgr. Partitio fuctio : { s } Z xp / stats

7 rgy of Photos i om Mod C594I ots, M. odwll, copyrightd Z s / h h s h xp xp xp s 0 stats h xp xp h / { } Probabilit y of P s occupacy of xp s / Z h xp s This is th probabilty of a stat xp s s : h xp{ h / } havig s xp h h xp h xp / photos i mod h.

8 rgy of Photos i om Mod C594I ots, M. odwll, copyrightd h P s xp s xp{ h / } xpctd valu of s : [ s] s P s skip stps xp h s 0 But h s /, so h h h [ s / ] for << h h xp This is th avrg mod rgy at frqucy h.

9 Nyquist's Nois Drivatio from Va dr Zil C594I ots, M. odwll, copyrightd Trasmissi o - li with matchd rsistors. Tmpratur T. ach rsistor has a thrmal ois voltag. This causs wav to flow lft- > right V ad right - lft V. a voltag Dfi P av powr availabl from ach rsistor i frqucy itrval Δ f. From this, V V f V V f 4 Pav / Δ f. Powr o li PLi Pav l / v

10 Nyquist's Nois Drivatio from Va dr Zil C594I ots, M. odwll, copyrightd Trap this propagati g radiatio by closig switchs. With switchs closd, allowd frquci s ar v / l # allowd frquci s withi cosidrd badwidth Δf : # Δ f l / v

11 Nyquist's Nois Drivatio from Va dr Zil C594I ots, M. odwll, copyrightd rgy i li rgy i li o Powr from rsistors l / v P AV # mods i Δf Δ f l h h / v xp h / propagati o tim rgy pr mod h P AV Δf xp Ad sic V f f 4 P / Δf V V V av hf hf V f 4 xp hf / V V h hf hf Δf h / xp hf /

12 Nyquist's Nois Drivatio from Va dr Zil C594I ots, M. odwll, copyrightd P AV Δf hf xp hf hf / dp AV df hf xp hf hf / hf hf f 4 xp / hf V V k J/K, h J s, h J s

13 Commt about Nois Drivatio C594I ots, M. odwll, copyrightd Aothr drivatio uss a LC rsoator. Th physics is simplr, th math mor complx. I both cass w hav th sam udrlyig difficulty. I couplig th rsoator to rsistors, th rsoator liwidth bcoms o - zro, ad allowd frquci s xtd ovr som small badwidth, rathr tha big rstrictd to th sigl frqucy of a quatum harmoic oscillator.

14 Availabl Thrmal Nois Powr C594I ots, M. odwll, copyrightd Maximum powr trasfr : load matchd With matchd load, voltag across load is With matchd load, currt th rough load is I N to grator / Giv that 4 dpload f 4 or I I f df N /. I P load dp is th maximum th availabl ois powr, hc availabl, ois df All rsistors hav qual availabl ois powr. Ay compoot udr thr mal quilibriu m o bias follows this law.

15 Thrmal Nois C594I ots, M. odwll, copyrightd hf hf f 4 xp hf / 4 hf hf f * xp hf / * I I I For hf f << I f I 4 ths 4 bcom Availabl Powr hf Frqucy

16 C594I ots, M. odwll, copyrightd Nois from ay impdac udr thrmal quilibrium For ay compot or complx twork udr thr mal o rgy supply quilibriu m dpavailabl, ois df f 4 Z or I I f 4 Y This This This follows from th d law of thrmodyamics. allows quick ois calculatio of complx passiv allows quick ois calculatio of atas. tworks Biasd smicoduc tor dvics ar NOT i thrmal quilibriu m.

17 C594I ots, M. odwll, copyrightd Nois from ay impdac: xampl ois to calculat First mthod C C C C C C Z whr X Z C C C C C 4 whr f C X Z V V 4 C

18 C594I ots, M. odwll, copyrightd Nois from ay impdac: xampl ois to calculat mthod cod * *, C total * * * *,, C C total total idpd t ar procsss th 0 bcaus But * * f C C C 4 4,,,,,, C f f f f total total π aswr....sam 4 C f π

19 Nois from a Ata C594I ots, M. odwll, copyrightd dp availabl, ois df f 4 Z Th ata has both Ohmic ad radiatio rsistac s. Th Ohmic rsistac has a ois voltag of spctral dsity 4 ambit Ohmic, whr T ambit is th physical ata t mpratur rad,rad By th dsity 4 of th d law, th radiatio rsistac has a ois voltag of fild rgio rad, whr T fild is th avrag tmpratu r from which h th ata rcivs sigal powr spctral loss,loss Itr - galactic spac is at 3.8 Klvi...

20 Nois o a capacitor C594I ots, M. odwll, copyrightd From f W fid V V c c f 4 that or πfc I I f 4 πfc f 4 f C π o th ma stord Capacitor rgy is / C V V * f df 0 4 f C π f [ ] / c c C Vc - This also follows dirctly from th Boltzma law.