Chapter II Circuit Analysis Fundamentals

Transcription

1 Chapter II Crcut nalyss Fundamentals Frm a desgn engneer s perspecte, t s mre releant t understand a crcut s peratn and lmtatns than t fnd eact mathematcal epressns r eact numercal slutns. Precse results can always be btaned thrugh prper crcut smulatns and thrugh numercal analyss wth sftware prgrams such as MTLB r Maple. Hweer, these tls cannt desgn the crcut fr yu, especally f yu are dealng wth analg crcuts. Een thugh desgn autmatn prgrams can synthesze cmplete dgtal crcuts, and prgress s beng made tward autmatng analg crcut desgn, engneers wll stll hae t mantan knwledge abut the functnalty f crcuts and systems. Hence, ths chapter wll reew and ntrduce sme basc crcut analyss methdlges that can be appled t gan nsghts nt perfrmance characterstcs and desgn trade-ffs. Generally, the apprach wll be t dentfy and utlze apprprate smplfcatns that enable apprmate analyss, whch s an essental skll when makng ntal cmpnent parameter selectns fr cmple crcuts and when perfrmng ptmzatns. Crcut analyss technques are f fundamental mprtance when determnng the man parameters f an electrnc system. Een thugh sftware smulatrs are etremely useful as cmplementary tls, they cannt cmpensate fr a lack f theretcal understandng n the part f the desgner. Therefre, an effecte system desgn prcedure begns wth selectng the mst sutable acte deces (transstrs, ddes, etc.) and passe deces (e.g. resstrs, capactrs, nductrs) alng wth pwer supples, and sgnal surces. Net, the prper cmpnent ntercnnectns hae t be defned, and the dece parameters hae t be chsen tgether wth the peratng pnt fr each f the acte deces n rder t realze the desred electrnc peratn. These steps requre theretcal knwledge as well as eperence. Fr ths reasn, fundamentals f crcut analyss are rested n ths chapter befre fcusng mre n practcal eamples thrughut the bk. Specal attentn s deted t the physcal nterpretatn f the results as well as the man prpertes f the basc crcut cmpnents and tplges that reccur frequently n cmple crcuts. It s mprtant t nestgate crcuts wth the prper analyss technques, but fr an engneer t s een mre sgnfcant t learn hw t apply the results durng the desgn f effcent and relable electrnc systems. Fnally, desgns shuld be erfed and cmpleted by ptmzng cmpnent alues thrugh accurate smulatns wth mre realstc and detaled mdels f the cmpnents. Befre we dscuss the prpertes f basc electrnc netwrks, t s necessary t ntrduce sme essental defntns fr the analyss f sgnal cmpnents, system transfer functns, decbel ntatns, and magntude/phase respnses. II.. DC and C sgnals typcal data acqustn system, such as the ne shwn n Fgure., cnssts f a sensr (transducer), a preamplfer, an analg flter, and the sgnal prcessr that typcally ncludes an analg-t-dgtal cnerter. The transducer detects the physcal quanttes t be measured and prcessed (temperature, pressure, glucse leel, frequency, wreless sgnal, etc.). Usually, the sensr s utput s a small sgnal (n the range f mcrlts t mlllts,.e., n the range f -6 t - lts) and t must be amplfed t ft wthn the lnear nput ampltude range f the analg-t-dgtal cnerter. Snce the desred sgnal s typcally accmpaned by undesred nfrmatn and randm nse, the sgnal may be cleaned thrugh a frequency-selecte flter that remes mst f the ut-fband cntents (at undesred frequences) befre the sensed sgnal s cnerted nt a dgtal frmat and further prcessed thrugh dedcated sftware. Blgcal System Blgcal Sgnal nalg Prcessng Blcks Dgtal Prcessng Blcks Transducer Preamp Flter (nse remal) nalg-t- Dgtal Cnerter Dgtal Sgnal Prcessr Electrcal Sgnal Fg... Frnt-end f a typcal b-electrncs system. - -

2 The electrcal sgnals at the utput f the transducer are nrmally cmpsed f tw cmpnents: the DC (drect current) and C (alternatng current) sgnals. s shwn n Fgure., the DC cmpnent s a tme-narant quantty, whle the C cmpnent s a tme-arant quantty and usually ths C cmpnent cntans the releant nfrmatn t be prcessed. Yu wll learn that the sgnals prcessed n the amplfers als cntan bth DC and C cmpnents. The general epressn fr sgnals such as the ne that appears at the utput f the transducer n Fgure. can be dented as: s C(t) = S DD + s ac(t) (.) where S DD and s ac dente the DC and C sgnal cmpnents, respectely. Befre we dscuss the manpulatn technques f these sgnals, let us defne sme f the nmenclature based n cmmn cnentns n electrncs lterature. The fllwng ntatns are used n ths tet t dentfy the nature f the sgnal cmpnents: CPITL CPITL labels represent nly the DC cmpnent; e.g. S DD = V; I X =. lwercase lwercase stands fr the C cmpnent nly; e.g. s ac(t) = sn(t + ) V; sgnal(t) = e j(t + ). lwercase CPITL stands fr the cmbnatn f DC and C cmpnents; e.g. s C = S DD + s ac(t); SIGNL = I X + sgnal. n eample f nddual sgnals and ther cmbnatn s llustrated n Fgure.. mpltude mpltude DC + C s C (t) = S DD +s ac (t) s ac (t) S DD DC cmpnent C cmpnent Tme Tme (a) (b) Fg... Tme dman sgnals: a) standalne DC and C sgnals, b) cmbnatn f C and DC sgnals. lthugh sgnals fund n mst real wrld scenars are nt perdc, the analyss and desgn f electrnc crcuts s cnducted based n perdc sgnals (snusdal waefrms, pulse tran, trangular waefrms, mdulated waefrms, etc.) because many waefrms can be apprmated wth summatns f perdc sgnals. The case f snusdal waefrms s especally nterestng because perdc waefrms can be represented by a Furer seres usng snusds as a bass f functns. It s well knwn that many practcal sgnals are cntnuus-tme, perdc wth perd T, and real functns f(t) are defned fr all t; whch allws representatn wth a smplfed Furer seres n the fllwng frm : f ( t ) Cn cs n t n, (.) T Ths s a smplfed frm f the Furer seres; the general frm s jn t f ( t ) Cne ; n T - -

3 where (/T = f) s the fundamental frequency cmpnent n radans/sec used fr seres epansn. In practce, the f(t) has t be epressed as a lnear cmbnatn f sne and csne waefrms, but t smplfy ur dscussn let us gnre the sne functns. C n s the n th Furer ceffcent, and t s cmputed as fllws: C n T tt jn t t f t e dt. (.) In ths ntrductn, Equatn. s a smplfcatn f real sgnal representatn wth the ntentn t emphasze that the sgnal s spectrum ften has snusdal cmpnents at frequences whch are multples f. Furthermre, the analytcal epressns fr the sgnals prcessed by the electrnc crcuts durng the ealuatn f releant perfrmance characterstcs typcally hae the same frm as n Eq... Eamples n the use f Furer seres can be fund n a number f tetbks that deal wth sgnal prcessng, crcut analyss, and crcut realzatns. Ntce that C (n = ) n. represents the DC cmpnent (aerage) f f(t) btaned wth Dependng n the applcatns, ths DC cmpnent may r may nt cntan desred nfrmatn. The ceffcent C (n = ) s the fundamental (C) cmpnent f the sgnal, whch fr practcal purpses carres the mst releant nfrmatn t be prcessed n cmmuncatn applcatns. The ampltude f the ther terms, determned by C n (n > ) are knwn as the harmnc cmpnents and are rarely studed n ntrductry curses. Hweer, they are releant and an mprtant subject matter n adanced studes because they decde the qualty f the sgnal. Specfcally, the hgh-rder terms ndcate the amunt f sgnal dstrtn, and they generate undesred frequency cmpnents when prcessed by nnlnear crcuts, whch lmts perfrmance. Eqs. and. are benefcal because they allw ne t ad analyzng crcuts fr all pssble nput sgnals. Instead, t s preferred t analyze them fr the case f snusdal nputs and, frm thse results, nfer system behar fr any ther knd f nput sgnal. Ths s the s-called frequency dman analyss. In the frequency dman, we usually etend the analyss f the sgnals frm DC (= ) up t ery hgh frequences. Een thugh the frequency respnse f a system can be accurately cmputed and predcted, t s ften nt easy t fnd the eact transfer functn f a system, especally f the mathematcal representatn s cmple. In ther wrds, eact mathematcal deratns ften mply t much effrt fr ery lttle nsght snce a real electrnc system may cntan mllns f transstrs! In these cases, t s mprtant t hae a gd understandng f a crcut s behar frm a tp leel rather than beng cncerned wth mnr detals that can be btaned frm smulatns. Few bseratns abut the prpertes f frequency respnses and lgarthmc functns lead t useful tls fr ealuatng many cmple analg crcuts frm a tp-leel perspecte. Nte, n sme areas such as pwer electrncs, the tme-dman analyss s mre cmmnly used than the frequency dman analyss. Pulse and mpulse system respnses are emplyed n thse cases, and these tpcs wll als be partally cered n the fllwng subsectns. II.. Decbels and Bde Plts ) Magntude respnse. In electrncs, the magntude respnse s usually pltted n decbels (lgarthmc scale), makng t easer t cmpare strng and weak sgnals. In general, t s ften ery cnenent t use lgarthmc scales n cases where t s hard t sualze sgnal dfferences usng lnear scales. The magntude n decbels (db) f a cmple functn f() s defned as: lg ( f ) lg ( f ) f db C T t T t f t dt, (.) where f() stands fr the magntude f f(). The epressn lg ( f() ) s cmmnly used n cmmuncatn systems where the pwer f the sgnal ( f() ) s emplyed, whle the secnd defntn lg ( f() ) s mre ppular n baseband electrnc applcatns (e.g. bmedcal, aud, and de frequences) where the man quanttes f nterest are ether ltage r current. In any case, the use f decbel ntatn (lgarthmc scale) s ery cnenent when dealng wth ntrcate transfer functns. Sme releant prpertes f the lgarthmc functn are lsted n Table.. s an engneerng student, yu mst lkely already hae a gd backgrund n lgarthm peratns, but be sure t master these prpertes as they wll be used etensely thrughut yur study f electrncs.. - -

4 Ntce that: Table. Fundamental prpertes f the lgarthmc functn. Functn Prpertes lg () lg () lg ( N ) N lg () lg (/ N ) -N lg () lg (/) = lg ( -/ ) -. lg () = lg ( / ) +. lg (f g) lg (f) + lg (g) lg (f/g) lg (f) - lg (g) lg ( N ) N lg (f X /Bg X ) lg () = db shws that db wth lgarthmc scale crrespnds t unty gan wth lnear scale. The magntude f -/ (=.77) crrespnds t - db. The frequency at whch the gan f a crcut s reduced by db s ften referred t as db-freuncy (f db), cutff frequency, r crner frequency. The multplcatn f functns n lnear scale s equalent t addtn f lgarthmc functns. In many cases, t s a lt easer t manpulate lgarthmc equatns as well as t get a better ntutn abut the characterstcs f a system thrugh plts f the transfer functn n lgarthmc scale. The rat f tw quanttes maps nt the subtractn f tw lgarthms. Epnents n lnear epressns crrespnd t scalng factrs n frnt f lgarthms. T apprecate the benefts f the lgarthmc scale, let us cnsder the fllwng cmple functn: f, (.) j j( / ) lg (f X ) - lg (Bg X ) = lg () + X lg (f) - lg (B) - X lg (g) where and are real numbers, and j s the magnary part f the denmnatr. Ntce that j = - and that s a real arable whch can take alues n the range f { < < }. n the abe equatn defnes the frequency f the ple, whch s a sgnfcant parameter because t crrespnds t the frequency at whch the gan ( / ) f ths frstrder transfer functn s reduced by db. The squared magntude (pwer) f ths functn can be epressed as: f ( ) f ( ) f *( ), (.5) where f*() s the cmple cnjugate f f(). The magntude respnse (plt f the magntude f f() ersus ) fr ths functn wth = = s depcted n Fgure.a usng lnear scale. Yu can bsere that t s ery dffcult t etract nfrmatn frm ths plt. When s small, the alue f f() s clse t unty, and t decreases rapdly as appraches. Fr eample, t s ery dffcult t accurately predct the alue f f() fr >> = based n ths plt. Of curse, yu can see that the functn has a tendency t g t zer, but t wuld be hard t quanttately estmate hw small the alues are when s large. The same transfer functn s pltted wth db-scale n the y-as n Fgure.b, where the -as s n lg scale. T btan ths db-lg plt, Eq..5 s substtuted nt Eq.. prr t plttng. Clearly, we can etract sgnfcantly mre nfrmatn by sual nspectn f ths plt, especally under cnsderatn f the prpertes n Table.. Fr nstance, the functn s apprmately flat frm ery small -alues up t = 5 [nte: f() = = db], and the -db alue ccurs at = = [nte: f() =.77]. The gan f the - -

5 transfer functn at = s - db, whch crrespnds t a magntude equal t.. Yu can check ths alue usng yur calculatr and Equatn.. Magntude E+ E+ E+ 6E+ 8E+ E+ (a) Magntude (db) E+ E+ E+ E+ Fgure.. Magntude respnse f the functn /( + j) usng: a) lnear-lnear scale, b) db-lg scale. (b) T get mre nsght, let us mre fully eamne the preus transfer functn. Takng adantage f the prpertes f the lgarthmc functn, the magntude f the transfer functn gen n Eq..5 can be epressed n db: lg ( ) *lg ( ) f db (.6) Ntce that: Fr << ths equatn can be apprmated as lg ( ) - lg ( ) = lg ( / ), whch s equal t db fr the eample case wth = =. Fr = =, the magntude f f() becmes f() = lg ( ) - lg ( ) = lg () - [lg ( ) + lg ()] = lg () - lg () - lg () = - lg () = -.db. Fr >> equatn.6 can be apprmated as lg ( ) - lg ( ) = lg() - lg() db. Ntce that the epressn cnssts f a cnstant term and a negate term that decreases prprtnal t the - lg functn f. If we use a lg scale n the -as, ths crrespnds t a straght lne wth a slpe f -, whch agrees well wth the plt shwn n Fgure.b fr >> =. If =, then f() - lg ( ) = - db, whch agan fts well wth the sual nspectn that can be made n Fgure.b. The smple analyss abe can be perfrmed fr any alue. The transfer functns can be easly pltted n lg -db scale after sme addtnal bseratns. Fr frequences beynd the ple frequency ( n the preus eample) f() s apprmately equal t lg ( ) - lg () db; sme alues fr the preus case are prded n Table.. Beynd the frequency f the ple, a ne-decade ncrease n the -as crrespnds t a gan change f - db (equalently, ths can be referred t as db attenuatn). Hence, the slpe f the functn wth lg -scale n the X-as s -db/decade. It can als be shwn that an ncrease by a factr f ( ctae) n the -as crrespnds t an attenuatn f 6dB fr f() wth >>, leadng t a slpe f 6dB/ctae. Scales n ctaes are frequently used n musc and medcal equpment, and they prde mre reslutn (change by a factr f per dsn n the -as) cmpared t lg -scale n the - as. In bth cases, the scales f the aes make t easer t plt cmple transfer functns and t dentfy releant prpertes thrugh sual nspectn f ther magntude respnses

6 Table. Eample alues fr the magntude f the lg() ( < ) functn n db ) The phase respnse f the frst-rder transfer functn (sld lne) dsplayed n Fgure. requres a bt mre detaled analyss than fr the magntude respnse. bref analyss s presented n ppend (t be added). It s mprtant t recgnze that a cmple epressn can always be wrtten n plar frm: f() = f() e j, where s the phase f f() defned as = tan - [Imagnary(f)/Real(f)]. Due t the prpertes f the epnental functn, t s edent that the phase prpertes lsted n the Table. hld; where R and Im represent the real and magnary parts, respectely. Table. Releant characterstcs f cmple numbers. f g f Functn Ealuatn R j Im f f f R f tan Im/ j f g f g e f f j f e g g f... fn f... fn g... gm g g... g e j f Im R f n m j f f gk k m g e g T plt the phase f a transfer functn by hand, yu must frst ealuate the phase f each nddual term and add the phases f the functns n the numeratr whle subtractng the phase f the functns n the denmnatr. Fr nstance, the phase respnse f Eq.. s depcted (sld lne) n Fgure. ( = = ). Crrespndngly, the phase shft f the cmple functn can be assessed frm the fllwng epressn: Phase [ f ( )] radans Phase [ Numeratr ] Phase[ Denmnatr ] Phase [ ] Phase[ j] tan ( / ). (.7a) Ths equatn s gng the phase n radans. T cnert the phase t degrees, yu must multply the alue n radans by 6/ 57., yeldng Phase[ f ( )] 57. Phase[ f (. (.7b) degrees )] radans The phase plt shwn n Fgure. s a nnlnear functn. Sme bseratns wll allw us t apprmate ths functn wth smpler but qute useful pecewse lnear functns: The phase at = s degrees because the functn s real and pste fr ths alue f. Negate real functns hae an ntal phase f radans (r 8 degrees) at =

7 The phase shft f f() at the lcatn f the ple = s -5 degrees. The phase at = s - radans (r -9 degrees). If we ealuate the derate f the phase respnse at the ple s lcatn ( = ), t takes n the larger alue f mre than -5 degrees/decade. Fgure.. Phase respnse fr the transfer functn n Eq.. wth = =. s a specfc eample, let us apprmate the phase respnse (Eq..7a) f the transfer functn by a pecewse lnear functn such that belw =. = where the phase s apprmated as degrees, and abe = =, t s apprmated by -9 degrees. The phase arund = = (the frequency f the ple) s apprmated t hae a slpe f -5 degrees per decade. Bth the eact phase respnse (sld trace) and the pecewse lnear apprmatn (dashed trace) are shwn n Fgure.. The largest errr ntrduced by the apprmatn s +/-5.7 degrees, whch s acceptable fr mst hand calculatns. The preusly dscussed prpertes are als ald fr the general case. Fr nstance, cnsder the fllwng cmple functn: Ths equatn can be rearranged t: f ( ). (.8) j f ( ). (.8b) j Therefre, ths equatn becmes smlar t Eq..5., and can be epressed n db as: f ( ) lg lg. (.9) db Nw, the transfer functn can be easly sketched by eamnng each part separately fr the dfferent ranges that arable can be n whle ntng that: - 7 -

8 f ( ) db lg lg lg lg lg lg lg lg lg lg f f f (.) It s straghtfrward t plt the apprmate transfer functn, as demnstrated n Fgure.5. Usng a pecewse lnear apprmatn s a ppular methd that s used t get an dea f a crcut s frequency respnse and ts mprtant characterstcs. In ths eample, the magntude f the transfer functn can be apprmated as cnstant [= lg ( ) - lg ( ) = lg ( / )] up t = /. fter ths frequency, the magntude f the transfer functn decreases wth a slpe f db/decade. mamum errr f - db ccurs at = / wth the lnear apprmatn. Fr mst f ur practcal applcatns, we wll use the pecewse lnear apprmatn unless t s crtcal t make mre accurate calculatns. It shuld be edent that the parameter / plays an mprtant rle n shapng the transfer functn, snce ths parameter s n fact the lcatn f the ple fr whch further mplcatns are dscussed n the fllwng sectns. ppmatn lg ( / ) eact db - db/decade / (lg) Fg..5. Plt f the transfer functn descrbed by Eq..9 wth ts pecewse lnear apprmatn. (The mamum errr f - db ccurs at the frequency ( / ) f the ple.) Snce f() s cmple, we hae t cnsder ts phase respnse as well. Cnsderng f() gen by Eq..8, the phase respnse s Phase[ f ( )] Phase[ ] tan tan (.). Ths functn s smlar t the ne pltted n Fgure.. Fr =, the phase shft presented by the crcut s degrees. It s edent frm Eq.. that the phase shft s radans (= -5 degrees) fr = / ; the phase shft when appraches nfnty s -.57 radans (-9 degrees). Ntce that the phase shft s clse t the lmts f the pecewse lnear apprmatn (arund º when. /, and arund -9º when / ). The transtn frm t -9 degrees takes place wthn apprmately tw decades arund the system s ple = /. The slpe f the phase arund a ple s apprmately 5 degrees/decade, and the mamum errr s +/-5.7 degrees. ddtnally, the phase shft s -5 degrees at the ple ( = / ). ) Magntude and phase f a general functn. The cmple transfer functn mght cnsst f a cmbnatn f ples and zers; hence, anther beneft f the lgarthmc scale s that t helps t dstngush between the tw cases. In the preus eamples, we fund that a ple reduces the magntude f the transfer functn wth a rll-ff f

9 - - 9 db/decade beynd the ple s lcatn. Furthermre, the ple ntrduces negate phase shft. zer, n the ther hand, ncreases the transfer functn wth a slpe f + db/decade and ntrduces pste phase shft. Let us cnsder a frst-rder transfer functn wth a ple and a zer: j j j j f ) (. (.) / and / are termed the zer and ple f the cmple functn, respectely. The zer crrespnds t the pste alue f that sles X z + / = n the numeratr, where X z = j z. Ths yelds X z = - /. Smlarly, the ple crrespnds t the alue f X p such that X p + / = n the denmnatr; hence, X p = - /. The magntude (n db) f f() can be epressed as fllws: lg lg lg ) ( f (db). (.) Smlar t the case f a sngle ple functn, the three terms f ths epressn can be apprmated fr dfferent ranges f. s an eample, let us cnsder / < /. In ths case, the frequency f the zer s lwer than that f the ple, resultng n a zer lcatn n the -as that s t the left f the ple lcatn. Frm Eq.. t fllws that. lg, lg lg lg, lg lg lg, lg, lg lg lg lg ) ( ) ( f f f f f f db. (.) These results can be pltted by usng the pecewse lnear apprmatn shwn n Fgure.6. db / (lg) lg ( / ) db/decade / db lg ( / ) Fg..6. Magntude respnse f a functn wth a ple-zer par where zer lcatn < ple lcatn.

10 T plt phase respnse f transfer functns, we can use the afrementned prpertes. Fr the transfer functn frm the preus eample, the phase n radans can be btaned thrugh the fllwng rearrangements: Phase f ( ) Phase tan Phase j j tan j Phase. j (.5) s edent abe, the phase cntrbutns f the zer and ple can be bsered ndependently, and the erall phase respnse s the superpstn f the nddual respnses, whch s pltted n Fgure.7 fr the case n whch the zer s lcated at a sgnfcantly lwer frequency than the ple ( / << / ). Snce the zer s lcated t the left f the ple, the phase starts at degrees and ncreases due t the mpact f the zer, reachng a phase shft apprmately equal t +5 degrees at the lcatn f the zer. The phase ges up t +9 degrees at = / << /, and remans cnstant untl the ple begns t affect the phase respnse by ntrducng negate ecess phase that s ntceable at -alues arund the ple lcatn /. t the ple alue ( = / ) the phase shfts due t the zer s +9 degrees whle the ple cntrbutes wth -5 degrees, leadng t an erall phase equal t +5 degrees. t ery hgh frequences. The phase shft s zer because the +9 degrees ntrduced by the zer s cancelled by the -9 degrees frm the ple. In the net subsectn, we wll cnsder the effects that ccur when ples and zers are lcated clse t each ther. 9 degrees 5 degrees degrees / / (lg) Fg..7. Phase respnse f the frst-rder transfer functn under eamnatn. ) Bde plts: General case. Bde plt s a well-knwn graphcal representatn f magntude and phase plts s. frequency n lgarthmc scale. In mst practcal cases, the characterstc equatn f an electrnc system cnssts f seeral terms that mdel nddual sectns cnnected n a cascade frmng a mult-stage system. lthugh the sectns may need a mre cmple representatn, let us assume that each stage can be descrbed by a frst-rder equatn s that the electrnc system can be mathematcally represented as: f ( ) N k M k C jb k jd N k M B k k Bk C D j D j, (.6) - -

11 where the symbl dentes the multplcatn f the factrs. s wll be edent sn, ths factrzatn s ery cnenent, especally fr lw-pass transfer functns. Usng the prpertes f the lgarthmc functn, the magntude (n db) f the generalzed transfer functn yelds: N k k f ( ) lg db, ( ) M C (.7a) N Bk N M k k C f ( ) lg db lg M lg. ( ) k Bk D D (.7b) Nte that.7a s nly ald at DC ( = ), whle.7b hlds at any frequency. Generally, the cmplete epressn must be cnsdered because we d nt hae nfrmatn abut the lcatn f the ples and/r zers. Hweer, smplfcatns can be made when relate ple/zer lcatns are knwn. ls ntce that we nrmalzed the functns wth respect t the zers and ples t make an analyss by nspectn easer. Sme nsghtful characterstcs nlng the abe equatns fllw:. By settng = n Eq..7b, the DC gan f the functn (magntude f the transfer functn at = ) can be btaned. Ths s the startng pnt f the erall magntude respnse n the y-as, and t depends n the ceffcents k and C snce ther terms wll cancel each ther.. The zers and ples must be dentfed. fter frequency =, each zer f the transfer functn wll ncrease the magntude at a rate f db/decade. Smlarly, each ple wll decrease the magntude f the functn by -db/decade after the ple s lcatn n the -as. Snce a db-scale s used n the y-as tgether wth lgarthmc scale n the -as, the afrementned effects f ples and zers n the magntude can be added algebracally.. t a gen -alue, the rll-ff (slpe) f the magntude can be easly btaned by the fllwng equatn: # f zersbelw # f plesbelw db slpe f f (.8). Fnally, yu can ealuate the functn fr t see that the fnal alue depends n seeral factrs: a. If the number f zers s greater than the number f ples, the transfer functn ges t nfnty (pste nfnte n db) as. b. If the number f ples s greater than the zers, whch s the typcal case n electrncs, the fnal alue s zer (negate nfnte n db) when. c. In case the number f ples and zers are equal, then the magntude s fnal alue s a functn f the ceffcents B k and D. Smlar rules can be establshed fr the phase respnse. The general phase equatn can be wrtten as fllws: Phase [ f ( )] Phase N k N k tan B k k j Phase B k k M tan M D C C D j. (.9) - -

12 Each zer ntrduces +5 degrees at the lcatn f the zer and rughly +9 degrees after ne decade beynd the zer s lcatn, whle each ple ntrduces -5 degrees at ts lcatn and -9 degrees after abut a decade. ccrdng t Eq..9, the phase cntrbutn f the ples and zers shuld be algebracally added. Eample: Fr a better understandng f the use f these rules, let us cnsder the fllwng eemplary equatn wth tw ples and tw zers: 5 j j5 f ( ) (.) j j Basc algebra allws us t put ths equatn n a mre cnenent frm: 5 5 j j f ( ). (.) j j The DC gan can be dentfed as f() =.5 (-db). Further nspectn reeals that the zeres f the system are lcated at Z = and Z = 5, whle the ples are placed at P = and P =. In decbels, ths functn yelds lg 5 lg lg f ( ) db db lg ( ). (.) Fgure.8 llustrates the magntude respnse that s sketched usng a pecewse lnear apprmatn based n the fllwng bseratns f the abe epressn: Fr <, the magntude s rughly cnstant and equal t - db (snce zers and ples belw = ge us a slpe f accrdng t Eq..8). Wthn the nteral < <, the gan decreases wth a slpe f - db/decade due t the ple lcated at = ( ple belw = ). t =, we hae the effect f the ple (- db/decade) and the effect f the zer at =, whch cntrbutes t a slpe f + db/decade after ths frequency. Thus, n the nteral > > 5, the effect f the ple and zer cancel each ther and the gan remans cnstant, snce zer and ple lead t a slpe f n ths nteral. The zer lcated at = 5 cmes nt the pcture fr > 5, addng an etra ncrement f + db/decade. Fnally at >, all ples and zers determne the shape f the erall functn. Snce the number f ples () and zers () s the same n ths case, the slpe f the transfer functn fr > s zer. The fnal alue f the transfer functn s btaned by ealuatng the rgnal functn at =, whch fr ths eample results n 5/( ) =.5 (- db). Magntude (db) - (lg) - - f() (db) - Fg..8. Magntude respnse f the cmple transfer functn gen by Eq... Let us nw cnsder the phase respnse f the preus eample (Eq..). Yu are strngly encuraged t check yur wrk s as t make sure that the phase f ths functn can be wrtten as - -

13 Phase f tan tan tan tan. (.) 5 The pecewse lnear apprmatn fr ths phase respnse s shwn n Fgure.9. The DC phase shft s btaned by ealuatng t frm the rgnal transfer functn at =, yeldng degrees n ths case. fter ths frequency, we hae t cnsder the phase cntrbutns f the ples and zers and add them tgether. The ple lcated at = ntrduces phase shft that ges frm zer degrees at -alues belw, t -5 degrees at =, and reachng -9 degrees at >. Its effect can be apprmated by a lnear phase aratn wth a rll-ff f -5 degrees/decade wthn the range < <. Smlarly, the zer lcated at = ntrduces degrees fr <, +5 degrees at =, and +9 degrees at >. The zer lcated at = 5 rases the phase at a rate f +5 degrees/decade startng at = 5 and endng at = 5. Fnally, the last ple adds a rll-ff f -5 degrees/decade n the range < <. Fgure.9 als ncludes the superpstn f the nddual phase cntrbutns that frm the erall phase respnse. Snce pecewse lnear apprmatns f phase respnses are smetmes mre bscure than thse f magntude respnses, yu wll hae t plt seeral eample n rder t master bde plts. 9 st zer nd zer 5 Phase (f()) -5-9 st ple 5 nd ple (lg) Fg..9. Phase respnse apprmatn fr the cmple functn gen by Eq... II.. Frequency Respnse f Frst-Order Systems. Steady-state analyss: resste, capacte, and nducte mpedances. When applyng fundamental crcut thery, the ltage-current relatnshp fr a resstr s dctated by Ohm s law. Frtunately, mst f the resstrs n mcr-electrnc crcuts can be apprmated as lnear deces een thugh they are nt perfectly lnear. Hence, t s easy t descrbe ther electrcal characterstcs wth smple algebra: (.) R R R where R s the ltage drp acrss a resstr whle current R flws thrugh t. The ltage plarty and current drectn are defned n Fgure.a. The mpedance f a dece s defned as the rat f the ltage appled t ts termnals dded by the resultng current flw, whch n case f a resstr ges: Z R R R. (.5) R resstr s mpedance s real and pste; therefre, the ltage and current are n phase wth each ther. Snce resstance and mpedance are equalent fr resstrs, the terms are cmmnly used nterchangeably. But when dealng wth capactrs r nductrs, be aware that the mpedance s frequency-dependent and nt equal t the resstance. Current and ltage arables asscated wth a capactr (Fgure.b) are related by the fllwng fundamental relatnshp: - -

14 + + R V R - C + (a) (b) (c) Fg... Passe cmpnent mdels: a) resstr, b) capactr, and c) nductr. c V C - V L - dc C. (.6) dt In ths epressn, C s the capactance, usually measured n Farads (F). Sme capactrs ehbt nnlnear capactance characterstcs; hweer n ths curse, we wll assume that capactrs reman cnstant (tme, ltage, and current narant) under all peratng cndtns. When epressng the C ltage appled at the capactr termnals n the general cmple frm c = V C e jt, the epressn fr the current flwng thrugh t can be dered frm Eq..6 as j C. (.7) c c Thus, cntrary t a resstr n whch ltage and current are n phase, the ltage acrss the capactr s plates and the current flwng thrugh the capactr are 9 degrees ut f phase. nther remarkable prperty edent frm.7 s that the current flwng between the capactr plates ncreases prprtnally wth the frequency f peratn (sgnal frequency), leadng t ery large alues at hgh frequences een f the ltage aratns are small. Based n Eq..7, t s bus that the mpedance f an deal capactr C s a functn f frequency, and t s gen by L Z c c c j C sc (.8) where (=f) s the sgnal frequency n radans/sec. It fllws that the mpedance f a capactr s negate and magnary: Z C = /(jc) = -j (/C). T smplfy the algebra, usually the frequency arable s defned as s = j. Ntce that s s an magnary arable. t =, the magntude f the capacte mpedance s nfnte (pen crcut). Cnsequentally, any DC ltage appled at the termnals f the capactr des nt generate any current n steady state. Fr ths reasn, capactrs are ften utlzed t cuple electrnc crcuts tgether whle mantanng dfferent DC leels at the ntercnnected nputs and utputs. In such practcal applcatns, the capactrs are ften referred t as DC blckng capactrs. Sme transent current s generated f a capactr s ntal cndtn s dfferent than the appled ltage, but ths current anshes as sn as the capactr s charged t the DC ltage acrss ts termnals. It s als mprtant t recgnze that the capacte mpedance decreases wth frequency, and ths lw mpedance affects the behar f electrnc systems at ery hgh frequences. s an eample, let us cnsder a pf capactr ( - Farads). The magntude f ts mpedance s 7 at Krad/sec but drps dwn t at Mrad/secand the capactr nly has an equalent mpedance f at Grad/sec. In mst practcal lnear crcuts, wth the eceptn f sme specal technques such as swtched-capactr crcuts, the effects f the capactrs n the range f a few pcfarads can be neglected at lw frequences (< MHz) because ther mpedance s usually ery hgh cmpared t resstrs. Hweer, yu shuld check any assumptns befre smplfyng yur crcut analyss by mttng capactrs n equatns. Befre clsng ths sectn, t s mprtant t nte that the ltage and current related t the same capactr are phase-shfted because the phase f a purely capacte mpedance s -9 degrees. Fllwng a smlar prcedure as fr the capactr, the mpedance f the nductr under steady state cndtns can als be fund easly. The relatnshps between the quanttes anntated n the deal nductr mdel f Fgure.c are gen belw. - -

15 L dl L (.9) dt where L s the nductance n henry (H). If the nductr s current has the general cmple frm L = I L e jt, then the ltage acrss ts termnals s j L. (.) L L Ths ltage generated acrss the nductr and the current flwng thrugh t are +9 degrees ut f phase. The mpedance f the nductr can be dered frm Eq.. as Z L L j L sl. (.) L Cntrary t the case f the capactr, the nductr presents ery small mpedance acrss ts termnals at lw frequency (Z L = at DC) and ery hgh mpedance (pen crcut) at ery hgh frequences. Vltage Dder. The ltage dder cnssts f tw mpedances cnnected n seres. Its utput ltage s defned as the ltage drp acrss ne f the elements, whch s a fractn f the nput ltage. typcal ltage dder s shwn n Fgure., n whch mpedances Z and Z are emplyed t reduce the nput ltage t a lwer utput ltage. By usng basc crcut thery (KVL) t can be easly shwn that the ltage acrss the grunded mpedance Z s gen by the fllwng epressn: Z Z Z. (.) If the nnzer mpedances Z and Z are f the same type (ether real r magnary), the ampltude f the utput ltage s smaller than the ampltude f the appled sgnal. Therefre, ths tplgy causes sgnal lss, and t can be referred t as an attenuatr. In the mst general case, Z and Z can be cmpsed by seeral elements, leadng t a arety f pssble transfer functns wth dfferent prpertes. The frequency respnse f sme f these functns s analyzed net. Z R Z R (a) (b) Fg... Vltage dder: a) general scheme and b) resste ltage dder. Let us frst cnsder the case f a smple resste dder shwn n Fgure.b. The ltage gan n ths case s R R R. (.) Snce the resstrs are deally ltage and frequency ndependent elements, the ltage gan f the resste ltage dder s a real number wth a magntude less than unty and a phase shft f zer degrees fr all frequences. Thus, the utput sgnal s an attenuated replca f the ncmng sgnal. Fgure. shws the magntude respnse f the resste ltage dder fr dfferent eample resstr alues. In thery the crcut des nt hae any frequency- - 5-

16 dependent lmtatns, but parastc capacte and nducte elements are unadable n practce, whch wll affect the magntude and phase respnses. Fr these shwn smulatn results, R = k and R = {,. k, and.k}. The ltage attenuatn factrs are / (-.8 db), / (-5.6 db), and / (-. db) fr the respecte alues f R. Vltage Gan (db) R = k, R =.k R = k, R =.k R = k, R = -5 E+ E+ E+ E+ E+5 Frequency (Hz) Fg... Magntude respnse f the resste ltage dder. It s mprtant t emphasze that the attenuatn factr reduces f the seres resstance R s smaller than the grunded resstance R because the nput ltage s splt between the tw mpedances, and the larger ne absrbs mst f the sgnal. In fact, as R appraches zer, the ltage gan becmes clse t unty (.e., the utput sgnal s almst equal t the nput sgnal). The utput ltage (acrss Z ) s the dfference between the nput ltage and the ltage drp acrss the seres mpedance Z. Hence, the larger the seres resstance Z s, the larger the ltage drp acrss t, and the smaller the utput ltage wll be. Ntce that f a ltage dder s frmed by resstrs, then. (.b) R R Frst-rder lw-pass transfer functn. T btan mre nsghts nt the prpertes f electrnc netwrks cntanng capactrs, we wll analyze the ltage dder wth a capactr and resstr. If Z s replaced by a capactr (C ) n the ltage dder f Fgure. whle keepng Z = R, then the crcut becmes a lw-pass flter (Fgure.). Lw-pass flters are frequently used t suppress undesred hgh-frequency sgnal cmpnents and nse. In aud applcatns, fr eample, the sgnal bandwdth apprmately s khz, and receed sgnals abe that frequency shuld be suppressed befre the nfrmatn s cnerted nt dgtal frmat fr further prcessng and recrdng. + R - R + C C - Fg... Frst-rder lw-pass flter. Usng the epressn fr the capactr s mpedance (Eq..8) and the ltage dder relatnshp (Eq..), the ltage gan f the resultng frst-rder system becmes - 6-

17 Z Z Z jc R jc RC j R C RC s R C. (.) It shuld be edent that the ltage gan has becme a cmple functn f the frequency arable due t the cmbnatn f the resstr (real mpedance) and the capactr (magnary mpedance). Fr ery lw frequences ( ), the transfer functn s pste and real (unty n ths case). t the frequency f the ple (= p = /[R C ]), the transfer functn becmes equal t p/( p + j p) =/( + j), and the magntude f the transfer functn s equal t.77 (- db). On the ther hand, the phase shft between utput and nput s clse t zer at ery lw frequences and -5 degrees at = p. Fr frequences beynd p = /(R C ), the transfer functn s dmnated by the factr /jr C ), causng the magntude f the transfer functn t decrease as the sgnal frequency ncreases. The magntude f the transfer functn reduces at hgh frequences because the capacte mpedance s lwer at hgh frequency. Ths s eplaned by the fact that the alue f the resste mpedance s greater than that f the capacte mpedance beynd p. Cnsequently, mst f the nput sgnal appears acrss the larger mpedance, whch s the resstr when >> p. The regns f peratn fr ths eample flter can be summarzed as fllws: lg db db lg RC lg f, RC f, RC f. R C (.5) The tw regns f peratn (pass-band wth flat respnse and stp-band wth rll-ff respnse) can be dentfed by bserng the lg-lg plt depcted n Fgure.a fr tw eample cases: p = /(R C ) = rad/sec (f p = Hz) and p = /(R C ) = rad/sec (f p = khz). Recall that the ple s frequency p s determned by the R C prduct. It s nterestng t rest the physcal eplanatn fr the behar f ths frst-rder lw-pass crcut. Fr lw frequences (<< /[R C ]), the seres resstance R s much smaller than the magntude f the grunded capactr mpedance gen by /C ). Snce the nput ltage s splt between the tw elements ( n = R + C) and ne f the mpedances s much greater than the ther ne, mst f the ltage wll appear acrss the hgher mpedance. Mst f the nput ltage s absrbed by the capactr at lw frequences, leadng t C = n at DC (unty gan and zer phase shft). t = /(R C ), the magntudes f bth mpedances are equal, and the nput ltage s equally splt between the tw elements, resultng n R = C =.7 n snce the current flwng thrugh bth elements s the same. But, the resstance s real and the capacte mpedance s magnary. t hgh frequences, the mpedance f the capactr reduces further, and mst f the nput sgnal s then absrbed by the seres resstr. The hgher the frequency s, the smaller the capacte mpedance and the larger the attenuatn factr wll be. t ery hgh frequences, the capactr s mpedance becmes etremely small (practcally zer) and effectely shrtens the utput nde t grund (.e. the capactr frms a shrt crcut at = ). - 7-

18 Vltage Gan (db) E+ E+ E+ E+ E+5 Frequency (Hz) Fg..a. Magntude respnse f the frst-rder lw-pass flter fr tw cases: /(R C ) = and /(R C ) = Phase (Degrees) E+ E+ E+ E+ E+5 Frequency (Hz) Fg..b. Phase respnse f the frst-rder lw-pass flter fr the tw eample cases. The phase respnse (n radans) f the lw-pass flter crcut s transfer functn s btaned frm. as Phase tan RC j R C tan R C. (.5) The crrespndng phase plt s depcted n Fgure.b. s dscussed n the preus subsectn, the phase aratn s manly allcated wthn tw decades arund the ple s frequency (. P < < P). Fr the case P = /(R C ) = rad/sec (f P = khz), the phase shft transtns frm t -9 degrees wthn the frequency range f khz - khz, beng eactly -5 degrees at f P = khz. Of curse, the magntude and phase plts are crrelated wth the tme respnse f the crcut. If a snusdal sgnal s appled at the nput f the lw-pass flter, whch s a lnear system, then the utput wll als be a snusdal functn but wth a dfferent ampltude and dfferent phase. In case the frequency f the nput sgnal s belw the frequency f the ple, the ampltude f the utput sgnal s ery clse t the ampltude f the nput sgnal, and the phase - 8-

19 dfference s small as well, as shwn n Fgure.5a. t the ple s frequency, the ampltude f the utput s.7 n and the phase lag s -5 degrees (Fgure.5b). Fr frequences beynd tmes the ple s frequency, the phase lag s clse t -9 degrees and the magntude f the utput sgnal s ery small as sualzed n the Fgure.5c. Input sgnal Output sgnal (a) (b) (c) Fg..5. Sgnals present at the nput (cntnuus cure) and utput (dashed cure) f the frst-rder lw-pass flter: a) < p, b) = p (the magntude f the utput sgnal s.7 Vn and the phase shft s 5 degrees), and c) > p (the utput sgnal s small and the phase shft s clse t 9 degrees) Befre we cntnue plttng the transfer functns wth Bde plt apprmatns, t s desrable t ntrduce anther useful parameter: the crcut s tme cnstant. The frst-rder lw-pass flter s descrbed by Eq.., and f the s arable s used nstead f j, the ple s lcated at S p. (.6) R C S p s a left-hand plane ple n the cmple s-plane. It wll be edent sn that t s cnenent t use p = /(R C ). The crcut s tme cnstant s defned as the prduct R C, and the unt f ths RC prduct s ndeed a tme-dman quantty epressed n secnds. In general, f the crcuts are cmpsed by acte elements, capactrs, and resstrs, then the frequency f bth ples and zers can be determned by crrespndng RC prducts (tme cnstants). Hgh-pass transfer functn. nther nterestng crcut s the s-called frst-rder hgh-pass flter. It can be btaned by cnnectng a capactr n seres wth a resstr as shwn n Fgure.6. Ths cnfguratn s fund at the nput f many C-cupled amplfers because t blcks the DC leel frm the preus stage (at the termnal). Usng basc crcut analyss, the transfer functn can be btaned as:, (.7) R R sc s s R C where s = j. zer s lcated at z = and a ple ccurs at p = /R C. s mentned befre, a zer ncreases the magntude at a rate f +db/decade, whle a ple decreases t at the same rate. s a result, three regns can be dentfed when plttng the magntude respnse f the frst-rder hgh-pass flter: a) the behar fr << p; b) the magntude at = p; and c) at the trend fr >> p. The fundamental equatns fr the square f the magntude n thse regns are as fllws: - 9-

20 C R Fg..6. Frst-rder hgh-pass flter. ( RC ).5 f f f, RC, RC ; R C (.8a) r, n decbels: db lg R C lg f f f, RC, RC. R C (.8b) The transfer functn starts at zer (- n db) and ncreases at a rate f +db/decade when the frequency s swept up t the ple frequency, at whch the gan s -db. t hgher frequences, the utput ltage s almst n phase wth the nput sgnal and the ampltudes f bth sgnals becme nearly dentcal. It s mprtant t ntce that the capacte mpedance s etremely large at DC and lw frequences, hence mst f the nput sgnal s absrbed by ths element when t s n the drect sgnal path frm the nput t the utput. Therefre, the ltage drp acrss the resstr s ery small at lw frequences. t ery hgh frequences, the capacte mpedance decreases and can een be cnsdered as zer fr quck estmates at frequences much larger than the ple frequency. In ths case, the nput sgnal s mstly absrbed by the resstr, whch leads t a ltage gan that s clse t unty (db). The magntude and phase respnses f the flter depcted n Fgure.7 are fr tw cases: p = /R C = rad/sec and p = /R C = rad/sec, where the -db frequences ccur when the plts crss the hrzntal dashed lnes. - -

21 Vltage Gan (db) E+ E+ E+ E+ E+5 Frequency (Hz) Fg..7a. Magntude respnse f the frst-rder hgh-pass flter fr tw cases: f p = Hz and f p = khz Phase (Degrees) E+ E+ E+ E+ E+5 Frequency (Hz) Fg..7b. Phase respnse f the frst-rder hgh-pass flter fr tw cases: f p = Hz and f p = khz. General frst-rder transfer functn. The mst general frst-rder flter shwn n Fgure.8 realzes a zer and a ple. Regardless f where the zer and ple are lcated, the crcut can be analyzed by nspectn:. t ery lw frequences the capactrs can be cnsdered as pen crcuts, leadng t a lw-frequency gan gen by R /(R +R ) and a phase shft f zer degrees.. Fr ery hgh frequences, the capactrs dmnate the ltage dsn because ther mpedances becme sgnfcantly smaller than the resstances. Fr ths reasn, the hgh-frequency gan s determned by C /(C +C ). Under ths cndtn, the phase shft s als degrees.. If the lw-frequency gan s smaller than the hgh-frequency gan, then the zer s at a lwer frequency than the ple, allwng the magntude respnse t ncrease.. If the lw-frequency gan s greater than the hgh-frequency gan, then the frequency f the ple s lwer than the zer lcatn t cause a decrease f the magntude. - -

22 C R R C Fg..8. General frst-rder flter. T btan the crcut s transfer functn, the equatn shuld be wrtten as: C C C s s RC RR C R R C. (.9) The DC gan and apprmate lw-frequency gan can be btaned remng the capactrs under the pen crcut assumptn, and slng the resultng crcut wth ω = ; whch leads t R R R. (.) t ntermedate frequences, the gan depends n the lcatns f the zer and ple, whch are gen by z R C p RR R R,. C C (.) Fnally, the magntude f the transfer functn at hgh frequences can be btaned by ealuatng the transfer functn fr =, whch results n C C C. (.) Pecewse lnear apprmatns can be easly pltted fr bth the magntude and the phase by fllwng the rules dscussed n preus sectns. The magntude respnses n Fgure.9 and the phase respnses n Fgure. represent the fllwng eample cases: ) C =.59F, C =.59F, R = R =, ) C =.59F, C =.59F, R = R =.k. - -

23 In the frst case, the DC gan s / (-6dB) and the hgh-frequency gan s.99 (-.8dB). The frequency f the zer s z = 6.8 krad/sec (f z =khz) and the ple s lcated at p =.5 krad/sec (f p =.8kHz). Smlarly, the student can easly erfy the plts that crrespnd t the secnd case. -5 Pece-wse lnear apprmatn Vltage Gan (db) E+ E+ E+ E+ E+5 Frequency (Hz) Fg..9. Magntude respnses f the generalzed frst-rder flter fr cases ) and ). Phase (Degrees) E+ E+ E+ E+ E+5 Frequency (Hz) Fg... Phase respnses f the generalzed frst-rder flter fr cases ) and ). The current dder. Many mprtant ntegrated crcut realzatns are based n the relatnshps between nput and utput currents because the sgnals at the dece termnals are currents r are easly epressed as such (e.g., at the cllectr f a bplar junctn transstr). The basc structure f the current dder s sualzed n Fgure.. Z Z Fg... Current dder. - -

24 Here, the prblem s t fnd the utput current as functn f the nput current sgnal and the asscated mpedances. Certanly, the utput current depends n the relatnshp between the mpedances Z and Z. Cnentnal crcut analyss technques allw us t fnd the current gan: Z Z Z. (.a) The current flwng thrugh Z ncreases wth ncrement f Z r decrement f Z. Here, t s mprtant t nte that mst f the current flws thrugh the smaller mpedance. Remember that the relatn = + must hld at the summng nde; hence, f mre current flws thrugh ne f the mpedances, then less current flws thrugh the ther ne such that the addtn remans equal t the nput current. smlarty wth the ltage dder s that the current gan s a functn f the relate alues f the mpedances rather than the abslute alues f the mpedances, as clarfed n the fllwng rearranged epressn:. (.b) Z Z Fr a resste current dder (Z = R, Z = R ), t can be bsered that the utput current s sgnfcantly reduced f R >> R, snce mst f the ncmng current under ths cndtn flws thrugh the smaller mpedance (R ), and ery lttle current s cllected at the utput. On the ther hand, mst f the current flws thrugh R f R << R. Smlar t the ltage dder, the current transfer functn f the current dder becmes a frequency-dependent cmple functn n the case where ts mpedance elements are cmbnatns f resstrs and capactrs. lw-pass transfer functn s btaned f the cnfguratn shwn n Fgure. s cnstructed. Usng Eq.., t fllws that jc R jc RC s R C. (.) t lw frequences, the capacte mpedance s much hgher than R, and mst f the current flws thrugh the resstr. Cnsequently, the current gan s clse t unty. t = /R C (when R = /jc ) the magntude f the current gan s equal t / /, and fr hgher frequences the current gan reduces nersely prprtnal t. Ths structure s useful fr the analyss f multstage current-mde amplfers. The magntude and phase plts f the current gan are smlar t the nes btaned fr the ltage dder and are nt shwn here, but we encurage yu t sketch them wth pecewse lnear apprmatns as well as plt them wth a sftware prgram. C R Fg... smple resstr-capactr (RC) current dder. It s nterestng t nte that the current C flwng thrugh the capactr C ehbts hgh-pass behar, but ths can be easly understd by bserng that the addtn f C and must be equal t. Snce reduces at hgh frequences, the current flwng thrugh C must ncrease t mantan the sum f the currents equal t the nput current. The current gan relatng C t the nput current s gen by the fllwng epressn: - -

25 C R jr C R jrc jc. (.5) The tme cnstant cncept. Frm the preus eamples, t s clear that the resstr-capactr (RC) prducts defne the frequences f ples and zers f the transfer functn. The RC prduct s als termed as tme cnstant, and frequences f ples (r zers) (n radans/secnd) are usually defned by /. Furthermre, fndng the tme cnstants lumped t the ndes n a crcut allws us t dentfy ts ples and zers. Yu wll learn mre abut crcut analyss thrugh such nspectns n the fllwng sectn. - 5-

26 II.. Frst rder system: Tme respnse. If the frequency respnse f a crcut s knwn, the tme respnse can als be btaned by usng the prpertes f the Laplace transfrm whch relates the tme and frequency dmans. Tme dman analyss wll be cnsdered n the fllwng chapters, but t s suffcent t say nw that frequency dman analyss s smpler t apply, especally fr cmple systems under steady state cndtns. Bth technques allw us t get nsghts nt the behar f the crcut. Snusdal sgnals are ften assumed n frequency analyss f lnear tme-narant crcuts because they make t easy t etraplate the results t cases n whch the nput sgnal s smth (n jumps, spkes, r square waes). If f(t) s a functn defned fr t >, the Laplace transfrm s dented as F(s) and defned as F s st f t e f t dt, (.6) where s = j The Laplace transfrm f f(t) ests f and nly f Eq..6 cnerges t a fnte alue fr all alues f s. Sme mprtant prpertes f the Laplace transfrm are lsted n Table.. Prperty Epressn Lnearty f t C f t C f t C f t C F s C F s C Frequency shftng e at f t Fs a s a Scalng f at F a Dfferentatn f ' t sfs F Secnd dfferentatn Integratn Frequency dfferentatn f '' t s Fs sf F' t f d Fs s n n n d t f t f s ds n Table.. Prpertes f the Laplace transfrm. mng ther prpertes f Laplace transfrms that are nt dscussed n ths bk, we hae t cnsder the ntal and fnal alue therems, mpulse and step transfrmatns, and system respnse fr perdc functns. Please refer t a tet bk n lnear systems t learn mre abut these tpcs. - 6-

27 The Laplace transfrm f sme mprtant functns are gen n Table. as fllws: Table.. Laplace transfrms f sme releant functns. Functn F(t) Laplace transfrm F(s) (t) (t-a) u(t-a), t (unt step) t n- /(n-)! -as e e -as t n- e at /(n-)! n e bt e b a at bt b e a e b a at s s s n s a s as b s s as b, a b, a b T llustrate the relatnshp between the tme dman analyss and the s-dman transfer functn, let us cnsder the fllwng s-dman transfer functn: m m s s am s... as a n n s s bn s... bs b. (.7) s llustrated n Fgure. fr a frst-rder system, any system can be characterzed n the frequency dman by ntng the magntude and phase f the nput and utput sgnals at dfferent frequences t btan ts transfer functn H(). lternately, the characterzatn can be perfrmed by determnng the transent mpulse respnse f the crcut;.e., by applyng (t) = (t) and fndng (t) = h(t) (t), where h(t) = l - H(). In general, the system respnse t any nput sgnal can be btaned by calculatng the cnlutn f the mpulse respnse h(t) f the crcut wth the nput sgnal. nther reasn why the mpulse respnse s frequently used fr the tme-dman characterzatn s that t can be easly btaned frm the s-dman transfer functn, especally because there are plenty f tables aalable n the lterature. - 7-