ECEN 5014, Spig 013 Special Topics: Active Micowave Cicuits ad MMICs Zoya Popovic, Uivesity of Coloado, Boulde LECTURE 7 THERMAL NOISE L7.1. INTRODUCTION Electical oise is a adom voltage o cuet which is peset i a cicuit with o without the pesece of a sigal. Usually, oise is uwated. Noise should ot be cofused with itefeece, which is a sigal couplig fom aothe cicuit, o with fadig, which is adom vaiatios i the popagatio chaacteistics i a adio lik. Usually, the istataeous values of the oise cuets ad voltages caot be pedicted, so thei aveage values ae zeo. The aveage powe, howeve, is ot zeo, ad this is what oise is chaacteized by. Noise ca be caused by diffeet physical pheomea: themal (also efeed to Johso o white) oise associated with a esisto at some tempeatue, flicke o 1/f oise associated with low fequecy vaiatios, shot oise due to fluctuatios i paticle cuet i ay device with a dc cuet flow, ad diffusio oise poduced by caies i semicoductos which move due to diffusio. Flicke oise has a powe pe uit badwidth which vaies with fequecy as 1/ f, whee is close to uity. Typically, it is impotat i the fequecy age fom much less tha 1Hz to khz o MHz, depedig o the device. Fo example, bipola tasistos i geeal have lowe 1/f oise tha FETs, so fo low-phase oise oscillatos, oe should use bipola devices. As the fequecy iceases beyod a coe fequecy, themal oise stats domiatig. Thee also must be a fequecy i the low fequecy limit whee the oise becomes fequecy idepedet, othewise the itegated oise powe would be ifiite. Shot oise is due to dc cuet i devices. Ay cuet cosists of discete chaged paticles, with a umbe of paticles pe secod give by N I dc / e (e chage of electo), ad statistical stadad deviatio of N. Theefoe, the oise cuet magitude is i e N e I dc, (7.1) e ad sice N is give pe secod, the badwidth is 1Hz. The powe fom the oise cuet i a badwidth B is give by i ei dc B, (7.) whee the facto of appeas because oise is always measued i a fiite badwidth. Fo a moe detailed teatmet of shot oise, a good efeece is Micowave Semicoducto Devices by S. Ygvesso, Kluwe Academic Publishes, 1991. 1
The most sigificat oise at micowave fequecies is themal oise. Themal oise i a cicuit is closely elated to black body adiatio, ad ca be udestood statig fom themodyamic piciples. Below is a discussio o themal oise, which follows the pape Themal ad quatum oise by Olive, IEEE Poceedigs, May 1965. L7.. THERMAL NOISE SPECTRAL POWER DENSITY I the deivatio of themal oise powe pe uit badwidth (oise spectal powe desity), we stat fom the fist ad secod law of themodyamics ad Plak s law: i ay closed system, the total eegy is costat, ad the etopy is maximized; electomagetic eegy is adiated ad absobed i discete quata (photos) of eegy. Coside a lossless tasmissio lie of legth l, temiated i matched esistive loads, Fig.L7.1. If the two esistos ae at diffeet tempeatues, the hotte oe will lose eegy to the colde oe. We assume that the powe is oly o the tasmissio lie (o adiatio, covectio o coductio). What powe does the esisto give? Imagie the esistos ae shoted, tappig the themal eegy o the lie. The lie becomes a esoato that ca suppot may modes. These modes ae themally excited i ou sceaio. We eed to aswe the followig questios: - How ae the modes excited? - How may modes ae excited? - How much eegy does each mode cotai? R=Z 0 l, Z R=Z 0 0 Fig.L7.1. A lossless tasmissio lie l log, temiated i matched esistive loads. The esistos give ise to themal oise powe, which tavels o the tasmissio lie fom oe load to the othe. The tasmissio lie ad loads ae assumed to be a isolated system. To aswe how esoat modes ca be themally excited, we stat fom the Boltzma H-theoem, which says the followig. If a system ca assume ay of N discete modes, each havig a pobability p(), as the closed system appoaches equilibium, the quatity N H p( )log p( ) 1 (7.3) eve deceases, but teds to icease util the system eaches equilibium. The mathematical quatity H is diectly elated to physical etopy as S=kH, whee k is the Boltzma costat ( k 138. 10 3 J/K).
Coside ow a electical esoato at themal equilibium. The esoato ca gai o lose eegy oly i discete amouts, so at ay momet thee will be eegy i the esoato. Hee is such that the pobability distibutio p() maximizes etopy. What is this pobability distibutio fuctio equal to? To aswe this questio, we will (1) fist fid p() fo a give S max ad (aveage umbe of modes), ad the () detemie (adjust the mea) to maximize system etopy. The etopy of the esoato is give by S kh k p ( )log p ( ), (7.4) 0 sice thee ae a ifiite umbe of modes i a esoato. Fo with the followig costaits: S, we set S / p( ) 0, max 0 0 p ( ) 1, whee is a itege p( ), whee does ot have to be a itege. (7.5) What happes fo small petubatios i p()? The two sums above do ot vay, fo a give aveage umbe of modes, ad whe S S max (aoud the extemum), the sum p ( )log p( ) will also ot vay with small vaiatios of p().this i tu meas that ay liea sum of these thee expessios will ot vay fo small deviatios p( ). Now we fabicate a quatity U that we kow will have a zeo vaiatio with p( ) : We set U / p( ) 0, ad the followig esults:. (7.6) U p( )log p( ) p( ) p( ) 0 0 0 0 U log p( ) 1 p( ) log p ( ) 1 0 p e e Ku Ke ue 1 1 ( ), whee ad. (7.7) Now, K ad ca be foud fom the costaits: 3
0 Ku 1 K 1u u p( ) (1 u) u (1 u) (1 ) u 0 0 (7.8) This meas that, sice u/(1 u), adjustig u adjusts the mea. The pobability distibutio fuctio fo the umbe of modes (that should maximize etopy) is p ( ) (1 uu ). (7.9) The ext questio we eed to aswe is: What is u equal to? Let us fist look at the evolved expessio fo the etopy, isetig the p() fom obtaied above: S k (1 u) u log (1 u) u 0 k (1 u) u log(1 u) (1 u) u log u 0 k(1 u) log(1 u) u k(1 u) log u u u k log(1 u) log u 1 u 0 0 (7.10) Imagie the esoato has o eegy (=0) at t=0 ad that the the etopy give i (7.10) is poduced by daiig a eegy fom the est of the system. If the system is at a tempeatue T, this poduces a etopy chage of u S (7.11) T 1 u T i the est of the system. I themal equilibium, u u S S klog(1 u) logu 1 u 1 u (7.1) the etopy is maximal, ad S / u 0. Settig the deivative with espect to u of (7.1), oe obtais u e log u 0, p( ) /, / / 1 e e. (7.13) 4
The actual eegy i the esoato is =W. The pobability of a state (mode) falls off as e W/ i the absece of quatizatio. Quatizatio of modes oly makes cetai discete modes possible. As fequecy iceases, the available levels become fewe ad have moe eegy. Whe ( / ) 1, the eegy levels become less umeous ad closely spaced, ad the a cotiuous p( ) q( W) ca be assumed. I the limit 0, the Boltzma distibutio is obtaied: 1 qw ( ) e W /. (7.14) The aveage eegy i the esoato is W uu 1 e / 1. (7.15) Now we eed to elate the eegy o the shoted tasmissio lie to themal oise powe poduced by the esistos that was tapped i the lie whe we shoted it. Thee will be some mode desity, i.e. umbe of modes pe uit legth of lie pe fequecy, deoted by m 1, whee the idex 1 stads fo oe dimesio. Each of these m 1 modes i themal equilibium has a aveage eegy W, so the themal eegy desity 1 is the poduct of the mode desity ad the aveage mode eegy: 1 mw 1 (i J/m pe uit badwidth). As the lie is made loge, the spectal desity of modes iceases i popotio to the legth, so the eegy desity alog the lie is ot chaged. Half of the eegy popagates i the +z, ad half i the z diectio. The spectal mode desity m 1 is foud as follows. The modes o a shoted lie log ae give by / ( c)/( f ), whee c is the popagatio velocity o the lie. Whe the lie gets vey log, the umbe of modes does ot deped o the fequecy. How may modes pe uit legth pe uit badwidth ae thee? The aswe is m 1 1 d lim. (7.16) df c If the shots ae ow eplaced by a matched load R Z 0, the powe desity must emai uaffected. Othewise, the load would eithe delive o absob eegy, but it caot do so i themal equilibium. Theefoe, a matched load both absobs ad adiates a powe W. The two loads ae exchagig eegy at this ate ad ow the system is ot esoat (the lie legth does ot play a ole). If oe of the loads is cooled to 0K, it will ot poduce oise powe, ad the eegy flow fom the hot load fo / 1 will be p lim e / 0 1 1 1 (7.17) To obtai the powe spectal desity i the limit / 1, the followig easoig applies: 5
the eegy spectal desity W is the equal to the mode spectal desity times the aveage mode eegy: W mw 1 / c. To obtai the powe o the lie, this eegy desity (i Joules pe mete pe Hetz) is multiplied by the velocity: p Wc (i watts pe hetz) The above eeds to be divided by fo the powe flowig i oly oe diectio dow the lie, so fially: p Powe spectal desity (watts/hz) of a esisto at tempeatue T fo / 1. (7.18) The impotat pactical implicatios of this esult ae: The powe spectal desity fom a esisto due to adom motio of electos is popotioal to the esisto tempeatue. p p( f ) fo the appoximatio i (7.18), which is why themal oise is ofte efeed to as white oise (at lowe fequecies). At oom tempeatue, T=90K, 410 1 W/Hz. The oise powe is give by B, whee B is the badwidth. As the badwidth of a system iceases, this powe does ot go up to ifiity, because (7.18) is oly valid fo a cetai fequecy age. Istead, the total themal oise powe available fom a esisto fom dc to ifiite fequecy is equal to, i watts, P R 0 df e 1 ( ) h 0 ( ) d e 1 h W. 6 (7.19) At T=90K, this powe is equal to P R 410 8 W, which is o the ode of 10W. This is the ate at which the esisto would cool though electomagetic adiatio ove a matched tasmissio lie to a matched load at T=0K. If both temiatios i Fig.L7.1 ae at tempeatue T, the total oise powe is P R, ad the total RMS oise voltage o the lie is V theal oise P R R mv at oom tempeatue fo a 50-ohm esisto. Fo a umatched load, with iput impedace Z i, i a cetai badwidth f, the oise powe deliveed to the umatched load ad the oise voltage ae give by V Rf P Ri R Z R Z R V fr. (7.0) themal oise i ad i i i Fo active etwoks, the above easoig is ot valid, sice the powe is supplied fom a exteal souce, violatig themal equilibium. The themal oise fom a esisto is the oe-dimesioal vesio of black body adiatio, ad the esult is that thee is o fequecy depedece fo may pactical fequecy ages. I thee dimesios, the mode spectal distibutio ca be show to be a fuctio of fequecy, so the spectal powe desity will also be a fuctio of fequecy. 6