present, and at the output the signal and noise are So, the noise figure of the amplifier is defined as (Fig.L8.1): / /

Transcription

1 ECE 504, Spring 06 ctive Micrwve Circuits Zy Ppvic, University f Clrd, Bulder LECTURE 8 LOW-OISE MPLIFIERS; MPLIFIER OISE PRMETERS L8 OISE FIGURE D OISE TEMPERTURE The nise figure f n mplifier r mixer give infrmtin ut the nise tht the mplifier dds t signl while mplifying it, r the nise tht the mixer dds t the signl while perfrming the frequency cnversin If the input t n mplifier is signl S i with nise level i present, nd t the utput the signl nd nise re S nd, the nise figure f the mplifier is defined s (FigL8): F Si S / / i This quntity descries the degrdtin f the signl-t-nise rti frm input t utput nd is usully defined with the input nise eing white nise with spectrl pwer density f kt 0 90K perfect mplifier wuld hve nise figure f, r 0dB nise fctr f (3dB) mens tht the SR hs een reduced y fctr f tw typicl lw-nise mplifier (L) yu will mke in the l will hve nise figure rund db By including the gin f the mplifier frm FigL8, the fllwing cn e written: S GS nd G, i i where is the nise dded y the mplifier mesured t the utput f the mplifier The nise figure cn nw e written s / G F i i, where is the nise dded y the mplifier referred t the input S i G S i, T FigL8 ise figure nd temperture f n mplifier 70

2 In rdistrnmy nd stellite receivers, the genertr nise is nwhere ner the stndrd t rm temperture, ecuse the nise cmes frm n ntenn tht is lking int the sky with very lw effective temperture f nly 3K In such cses, it des nt mke sense t tlk ut nise figure sed n rm temperture resistr nise, ut rther ut mplifier nise temperture T given y T kg k w the nise figure cn e relted t the nise temperture f the mplifier s T F T The nise temperture f -db nise figure L is 70K, nd the est rdistrnmy receivers hve nise temperture f K 8 ise f Cscded mplifiers Cnsider cscde f mplifiers with respective gins G, G,, G, s in FigL8 Wht is the equivlent nise figure f this cscde? With ll the mplifier nises referred t the inputs f the individul mplifiers, the fllwing fllws frm the definitin f nise figure: This is usully written in the fllwing frm 3 F i G GG GG G F F F F3 F G G G G G G In the previus frmul, the G i s re ville pwer gins f the individul mplifiers Therefre, the first mplifier in the chin hs the mst effect n the verll nise figure, nd the cscde design shuld minimize the first mplifier s nise figure nd mximize its gin Usully stility is n issue with such cscde nd limits the gin If the chin is cnnected t receiving ntenn, the ve frmul tells us tht the first L shuld e s clse t the ntenn feed pints s pssile ny length f lssy cle will hve negtive gin (lss) nd its resistnce will e surce f nise, s the nise figure f the cscde will e gverned y the cle ccrding t the cscde equtin 7

3 S i G S G G S i F F F FigL8 ise figure nd temperture f n mplifier cscde L83 ise Wves frm Pssive Multiprt etwrk Bsm s Therem nise wve cn e defined fr ne-prt, FigL83, s fllws The mplitude f nise wve c is rndm quntity with zer verge vlue, nd the expected vlue c is nise pwer, mesurle quntity When the nise surce impednce is equl t the nrmlizing impednce, the nise pwer is equl t the pwer delivered y the surce, kt When the surce impednce is nt the sme s the nrmlizing impednce, the reflectin cefficient f the surce is nt zer, nd in this cse the nise wve pwer per Hz ndwidth is equl t ( ) c kt s This fllws frm therml equilirium When the surce nd ld re t the sme tempertures, if they re mtched, the pwer flw n the line needs t e lnced If they re nt mtched, it still needs t remin lnced, therwise mre pwer wuld flw in ne directin, vilting therml equilirium c s FigL83 ise wve f ne-prt nise surce The S prmeter f the ne-prt is nrmlized t the impednce f the ld The ve discussin fr ne-prt netwrk cn e generlized t multiprt netwrks s fllws In the ne-prt in FigL83, ne cn write the fllwing expressin fr the nise wves in terms f the ne-prt scttering prmeter: s c s c 7

4 The ltter is true ecuse the incident wve emntes frm the termintin nd is uncrrelted with the nise wve c Since therml equilirium requires kt, it fllws tht c ( ) kt s Fr tw-prt netwrk, FigL84, ne cn write the sme in mtrix frm: s s s s c c, nd the crrespnding crreltin mtrix f the nise wves is defined s c cc C S cc c S () S () FigL84 ise wves f tw-prt pssive netwrk () nd the sme tw-prt with the prts cmined in directinl cupler () The cupler is initilly ssumed t e 80 nd then 90 3-dB cupler Since the nise frm the tw termintins re nt crrelted, nd the tw-prt is in therml equilirium, the fllwing cn e written: kt nd 0, kt The nly quntity tht remins unknwn is the crreltin f the scttered (ut-ging) wves, The fundmentl prperties f nise wves llw methd y which it cn e determined, utlined elw In rder t clculte this crreltin, we mix the tw utging wves y dding directinl cupler t the utput f the tw-prt, FigL84 The primed quntities in the figure re the nes tht ptentilly chnged frm FigL84 y the dditin f the lssless directinl cupler w the tw nise wves re given y ( ) nd ( ), 73

5 since the hyrid in this cse is 3-dB 0 /80 directinl cupler The nise pwer f the dditin f tw rndm vriles cn e clculted s p where the tw lst terms re cmplex cnjugtes f ech ther nd their sum is therefre rel, s we tin j I R R w the ve expressins cn e cmpred in rder t tin the crreltin f the utput wves s fllws Fr the input wves t the tw-prt, similr expressins cn e written: j I R R nd since the nise wves emnting frm the tw termintins re uncrrelted nd their spectrl pwer densities re kt, it fllws tht I I R R j j kt kt kt kt This mens tht the directinl cupler did nt ffect the nise wves input t the pssive tw prt, nd therefre in therml equilirium, kt is still vlid Therefre, the scttered wves t the utput f the cupler re 74

6 kt R kt R { } { } Since the pwer f nise wves fr therml equilirium hs t e kt, then 0 R w the 0-80 hyrid cn e replced y 0-90 hyrid, nd the result is shwn t e similr, ut fr the imginry prt f the crreltin: I { } 0 Therefre, since th rel nd imginry prts re zer, it fllws tht 0 This discussin cn e generlized t multiprt netwrk, where cmplex cnjugtes ecme Hermetin vectrs (cmplex cnjugte trnspse) S, fr n n-prt netwrk, we cn write: kt I with the sign mening Hermetin vectr, nd I eing the identity mtrix The directinl cupler experiment cn nw e pplied t ny tw prts i nd j t time, nd ech time the fllwing is true: kt nd 0 i j i j Therefre, in mtrix frm, kt I This is surprising result! Input nise wves t the input (the s) prduce scttered wves which re nw crrelted (the s ij i s ), ut the net crreltin vnishes! This mens tht the pssive tw-prt prduces nise wves (the c s) tht cncel the crreltin It cn e prven (see pper y Wedge nd Rutledge) tht this result is cnsistent with Bsm s therem fr the fundmentl prperty f the nise crreltin mtrix The cnclusin f this therem is tht the nise wves cntriuted y the netwrk re simply thse necessry t cncel the effects f crreltin present in the scttered wves S in rder t mintin the lck f crreltin in utput wves nd the chieve lnce f pwer flw Bth f these cnditins re required fr thermdynmic equilirium L84 ise Figure nd ise Wves f MESFET mplifier Given trnsistr, hw des ne design lw-nise mplifier ptimized t hve the lwest pssile nise figure nd lrgest pssile gin? The specifictins given y the mnufcturer give three nise prmeters, usully fr numer f frequencies: the minimum nise figure ( F min ), the ptimum input refectin cefficient (usully written s Γ pt ) nd the nise resistnce ( R n ) These re mesured using surce-pull tuners (t the input) nd nise-figure meter, nd 75

7 the ptiml input impednce is fund when the nise figure is mesured t e miniml Then the input f the L cn e designed y designing mtching circuit fr tht impednce, while cnjugte-mtching the utput t mximize gin These prmeters, hwever, d nt give us ny insight int the surce f nise in the trnsistr, r why there is n ptiml input impednce Furthermre, fr very lw nise mplifiers, the surce-pull methd is difficult, since the tuners hve t e very stle nd clirted crefully ll the time The nise emnting frm trnsistr cn e descried lterntively y input nd utput nise wve quntities nd their crreltin, s we will see elw c s g c S c s g S () () FigL84 Equivlent tw-prt nise wves fr n mplifier () nd the sme nise wves referred t the genertr side () In the cse f n mplifier presented s tw-prt netwrk (usully, cmmn surce), the nise equivlent circuit is given in FigL84 The trnsistr is prducing nise wves t th input nd utput, c nd c The genertr t the input is shwn s ne-prt with reflectin cefficient s g The nise wves cn th e referred t the genertr in such wy tht the ttl nise t the utput f the mplifier remins the sme The new cmined nise wve is equl t c c s c g T find the mplifier nise temperture nd nise figure, we slve cs g c c kt( sg ) T k ( sg ) Frm the reltinship etween nise figure nd nise temperture, the mplifier nise figure is given y F cs c kt 0 g ( sg ) 76

8 The nise figure cn e written s sme minimum chievle nise figure, fr n ptiml input nise mtch, plus sme dditinl term tht depends n the genertr reflectin cefficient nd n ptimum reflectin cefficient f the genertr ( s ): 0 cs c f s s kt g g F min ( sg ) ( sg ) If c nd c, s well s their crreltin c c, re nrmlized t kt 0, this fctr disppers in the denmintr f the lefthnd side, nd the previus equtin cn e written s c { s c c } ( F ) ( s ) f s f s f { s s } s g c R g min g g R g If this is t hld fr ny genertr reflectin cefficient, the cefficients f ll the pwers f s g must e equl, resulting in the fllwing three equtins: c F f s min g c ( F ) f c c These equtins cn nw e slved t find fs min f s c ( Γ pt c cc ) f c c c c F min f c f is fund y slving qudrtic equtin, nd ne rt f the equtin is chsen The criterin fr chsing the rt with the plus sign is tht the ptimum reflectin cefficient needs t e smller thn unity In the prcess f this derivtin, we use the fct tht the expected vlue f the nise pwer which is sum f tw nise wves is p c c c c c c c c The first tw terms re f the nise wves nd the lst tw terms re crreltin terms These tw terms re cmplex cnjugtes f ech ther, s their sum is rel The crreltin terms re zer if 77

9 the tw nise wves cme frm physiclly different nise surces, such s tw resistrs If the nises re crrelted, such s in MESFET, ne tries t chse the reflectin cefficient f the genertr t cncel s much f the nise s pssile t the utput If the tw nise wves cme frm the sme surce, they re prprtinl t ech ther nd they re sid t e cmpletely crrelted In this cse, the crss-crreltin ecmes c c c c By serving the expressins fr the ptiml reflectin cefficient, it is seen tht it is prprtinl t the crreltin f the tw nise wves If the tw nise surces re uncrrelted, then s 0, ut in MESFET usully the mgnitude f the ptimum genertr scttering cefficient is clse t unity, nd this indictes tht the tw nise surces re clsely crrelted nd tht they physiclly cme frm ne surce inside the trnsistr In trnsistr specifictin sheets, usully minimum nise figure nd ptiml scttering prmeter re given fr numer f frequencies Insted f f, relted prmeter referred t s the nrmlized nise resistnce is given: f 4Rn k 90K Z s 0 Wht re the surces f nise in MESFET? With reference t very simplified unilterl mde, the resistnce etween the gte nd surce prduces nise vltge tht ecmes prt f the input vltge, nd is cupled t the utput thrugh the trnscnductnce It cn e shwn (see the pper y Wedge nd Rutledge) tht the input nd utput nise pwer spectrl densities nd the input nd utput nise wve crreltin fr trnsistr (MESFET r HEMT) cn e writing in terms f the device s-prmeters nd the effective nise tempertures f the gte nd drin which derived frm Bsm s therem Since the trnsistr is nt pssive, therml equilirium cn e chieved nly if the dc input is mdeled y n effective increse in nise temperture In prctice, the drin effective nise temperture is the min effect L84 ise Mesurements Fr MESFET r HEMT, s mentined, minimum nise figure nd crrespnding ptiml input reflectin cefficient re usully given These re mesured using tuner t the input f the device (referred t s surce tuner ), while simultneusly mesuring the nise figure Hw is the nise figure mesured? There re severl methds fr determining the nise figure f n mplifier methd esy t understnd, ut nt s esy t implement ccurtely in prctice, is the Y-prmeter mesurement, r ht-nd-cld mesurement, FigL85 First resistr R cnnected s the nise surce t the input f n mplifier is heted t temperture T H nd then cled t temperture T C The resulting mesured nise pwer densities in ndwidth f t the utput f the mplifier re PH GkTH f GkT f P GkT f GkT f C C 78

10 () () (c) FigL85 () Ht nd cld (Y-fctr) nise figure mesurement, () spectrum nlyzer nise figure mesurement nd (c) utmted nise figure meter lck digrm The Y-fctr is defined s the rti f these tw nise pwers nd cn e written s T Y T T T H C, TH YTC s tht T Y By mesuring the Y fctr, the nise temperture f the mplifier, nd therefre the nise figure, re determined The ccurcy f the mesurement depends n the tw temperture eing s fr prt s pssile, nd requires tht the reflectin cefficient f the ld resistr R des nt vry with temperture Since we strt with tw equtins with tw unknwns (the gin is ls unknwn), the gin f the mplifier cn ls e determined Since the gin cn e mesured 79

11 independently with netwrk nlyzer, this cn help check r imprve the ccurcy f the mesurement nther wy t mesure nise figure is directly with spectrum nlyzer, FigL85 The methd is nt very ccurte fr nise figures elw 3dB knwn pre-mplifier, with gin G pre-mp nd utput nise spectrl pwer density pre-mp is used in ne rnch f the system Fr limited frequency ndwidth, the spectrum nlyzer IF ndwidth is set t B, nd the resulted mesured DUT nise figure is F ktg pre-mp pre-mp G B, where G is the unknwn mplifier s (DUT s) gin The nise pwer mesured y the spectrum nlyzer integrted ver the IF ndwidth will hve crrectin cnstnt f C-dB which includes different detectin effects, fr exmple the vide filter used fr the pst-detectin disply, which is usully set t ne hundredth f the IF ndwidth In db, the nise figure cn e written in terms f the mesured pwer y the spectrum nlyzer: F( db) P - G - G - 0lg B 74 C n pre-mp (Where des the 74 numer cme frm?) In the utmtic nise figure meter, nise surce is used t prvide different levels f nise This is usully pn dide ised in reverse vlnche rekdwn, nd the ville nise pwer is inversely prprtinl t the dide current simplified lck digrm f gilent nise figure meter is shwn in FigL85c ise figure cn ls e mesured indirectly y mesuring nise wves This is especilly useful fr chrcterizing trnsistrs fr the ptiml input nise mtch The mesurement des nt require surce tuner, nd elimintes errrs due t tuner limited rnge nd tuner clirtin Cnsider the system in FigL86 The nise wves emnting frm the device re c nd c They re dded t nise wves generted y tw nise surces tht cn ech e t tw equivlent tempertures, T nd T The resulting wve pwer spectrl densities t the input f the switched fur-prt re d c kt s kt s d c kt s kt s d d c c kt s s kt s s These re mesurle quntities, nd in rder t tin frm them the unknwn c, c nd cc, first thru is inserted in the plce f the switched fur-prt In this cse, d e nd d e, nise pwer mesurement is perfrmed t ech temperture f the nise surce, resulting in fur nise pwer mesurements T mesure the crreltin, first 0-80 degree 3-dB cupler, nd then 0-90 degree cupler re inserted in the plce f the fur-prt 80

12 The first set f mesurements t the tw tempertures f the nise surces gives the rel prt f the crreltin, nd the secnd set gives the imginry prt Fr exmple, fr the 0-80 degree hyrid cse, the fllwing cn e written: e e ( d d ) ( d d ) e e R ( d d ) [ R( c c ) kt ( s s ) kt ( s s )] The resulting numer f mesurements prvide n verdetermined system f equtins, nd in dditin t tining the s-prmeters f the device, sme sttisticl nlysis f these mesurements cn e perfrmed FigL86 ise wve mesurement system lck digrm (frm Wedge nd Rutledge) 8