Chapter IIIa The operational Amplifier and applications.

Transcription

1 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez hapter IIIa The peratnal Amplfer and applcatns. III.. Basc Mdel fr the Operatnal Amplfer. The OPeratnal AMPlfer (OPAMP) s a key buldn blck n anal nterated crcut desn. The OPAMP s cmpsed by seeral transstrs and passe elements (resstrs and capactrs) and arraned such that ts lw frequency ltae an s ery hh; the dc an f the OPAMP7 s arund 0 V/V (0 V at the nput e us V at the utput). The desn f such cmplex crcut s dscussed n chapter 6; here we wll use a smplfed lnear macrmdel t analyze the prncples f OPAMP based crcuts and ther peratn. Seeral crcuts are studed such as basc amplfers, frst rder and secnd rder flters and sme nncnentnal applcatns. The ersatlty f the OPAMP wll be edent at the end f ths chapter. T defne the fundamental parameters f a system, let us cnsder a lnear twprt system wth tw termnals runded, as the ne shwn n Fure.. There are arables,, and t be studed. The nteractn between the fur arables can be defned n many dfferent ways, dependn n the defntn f the dependent and ndependent arables; n real crcuts these defntns depend n the nput arable (current r ltae) and the mst releant utput arable. Usually n ltae amplfers the nput snal s defned as whle the utput s. 0 Electrnc 0 rcut 0 F... Electrnc crcut represented by a black bx. Snce we are assumn that the crcut s lnear, amn ther representatns, we can descrbe the electrnc crcut by usn the fllwn hybrd matrx representatn: parameters (.a) r (.b) Ntce that we are mxn currents and ltaes n the matrx, then we call t hybrd representatn f the crcut, and the resultn mdel s termed hybrd mdel. In many text bks yu can fnd at least dfferent set f parameters, but ths s ne f the mst releant nes fr ltae amplfers. Bplar and MOS transstrs mdels are based n anther mdel called hybrd, t be dscussed n the fllwn chapters. In equatns.b, the parameter defnes the nput cnductance, and t relates the nput current and the nput ltae needed by the crcut wthut cnsdern the effect f the utput current ( =0); the crcut s nput cnductance s frmally defned as fllws: (.) 0

2 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez Ths parameter s measured by applyn an nput ltae surce and measurn the nput current; the utput nde s left pen such that =0. Snce s the rat f the nput current t the nput ltae, whle the utput s leaed pen, ts unts are amps/lts r /Ω. The parameter defnes the reerse current an f the tply, and t s defned as (.) 0 Ths parameter represents the reerse current an: current enerated at the nput due t the utput current. In the deal case ths parameter s zer snce usually the peratnal amplfers are undrectnal; e.. the nput snal appled t the system enerates an utput snal, but the utput snals (current r ltae) shuld nt enerate any snal at the nput. Fr measurn ths parameter t s requred t shrt crcut the nput prt such that =0, then apply a current at the utput and measure the current enerated at the nput prt. In practcal crcuts ths parameter s ery small and usually t s nred. The Frward ltae an s defned as the rat f the utput ltae and nput ltae wthut any lad cnnected at the utput. AV (.) 0 Ths s certanly ne f the mst mprtant parameters f the twprt system; we als refer t A V as the penlp an f the OPAMP. It represents the crcut s ltae an wthut any lad mpedance attached (utput current equal zer). Anther mprtant parameter s the system s utput mpedance, whch relates the utput ltae and the utput current wthut takn the effects f the nput snal. It s defned by 0 (.) 0 Thus, the twprt system can be represented by the fur afrementned parameters; the resultn macrmdel s shwn n Fure.. Fr sake f clarfcatn we are usn mpedances nstead f admttances n ths representatn. Ntce that a current cntrlled current surce (IS) s used fr the emulatn f parameter snce t represents the nput current (nput prt) ben enerated by the utput current (utput prt). A resstr can nt represent ths parameter snce current s flwn n ne prt but the ltae at the ther prt cntrls t. Smlar cmments apply t the ltae cntrlled ltae surce represented by A V ( ). 0 0 A V 0 0 F... Lnear macrmdel f a typcal ltae amplfer usn hybrd parameters. Mdel fr the OPAMP. The deal OPAMP s a dece that can be mdeled by usn the crcut f F.. wth A V =, =0, =, and =0. Ths s f curse an unrealstc mdel, but t s enuh fr understandn the basc crcuts and ther peratn. We wll see that when yu cnnect seeral crcuts n cascade fr cmplex snal

3 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez prcessn, bth nput and utput mpedances are mprtant parameters that affect the erall perfrmance f the system. In ths sectn, hweer these effects are nt cnsdered. Dscuss here the man lmtatns such as nput mpedance, lmted D an and D ffsets and nput bas current. III.. Basc cnfuratns: Inertn and nnnertn amplfer. Inertn cnfuratn. The frst tply t be studed s the nertn amplfer shwn n f..a. It cnssts f an mpedance cnnected between the nput surce and the OPAMP s nertn termnal; the secnd mpedance s cnnected frm the nertn termnal t the utput f the OPAMP. prdes a neate feedback (cnnectn the OPAMP s utput and the neate nput); ths s the man reasn fr the excellent prpertes f ths cnfuratn. The smplfed lnear macrmdel f the OPAMP s used fr the representatn f the nertn amplfer, and the equalent crcut shwn n f..b. By usn basc crcut analyss technques t can be easly fnd that x x 0 (.) and A x (.6) Sln these equatns as functn f the nput and utput ltaes yelds; A V (.7) Ths relatnshp s als knwn as clsedlp amplfer s an snce the feedback resstr n cmbnatn wth the OPAMP frm a clsed lp. If the penlp an f the OPAMP A V s ery lare, then the frst factr can be apprxmated as unty and the clsedlp ltae an becmes (.8) Ths result shws that f neate feedback s used and f the penlp an f the OPAMP s lare enuh, the erall amplfer s ltae an depends n the rat f the tw mpedances. Unlke t the an f the penlp an f the OPAMP that can ary by mre than 0 % due t transstr s parameters aratns and temperature radents, as wll be explaned n the fllwn sectns, the clsed lp an s qute accurate, especally f same type f mpedances are used. Usually rat f mpedances s mre precse than the abslute alue f cmpnents; e.. rat f capactrs fabrcated n MOS technles can be as precse as 99. % whle the abslute alue f the capactance may chane by mre than 0%. x A V x 0 (a) (b)

4 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez F... Inertn amplfer: a) the crcut and b) the lnear macrmdel assumn that OPAMP nput s nfnty and utput mpedance s zer. Anther mprtant bseratn s that the dfferental ltae at the OPAMP nput x (see F. b) s deally zer. The reasnn behnd ths bseratn s as fllws: accrdn t.8, the utput ltae s bunded (nt nfnty) f s nt zer r s nt nfnte, hence s als bunded. Under these cndtns and accrdn t expressn.6 the OPAMP nput ltae x s ery small f A s lare enuh; the larer the OPAMP pen lp an the smaller snal at the nput f the OPAMP. Therefre the nputs f the OPAMP can be cnsdered as a rtual shrt; the ltae dfference between the tw nput termnals ( ) s ery small but they are nt physcally cnnected. The rtual shrt prncple s extremely useful n practce; mst f the transfer functns can be easly btaned f ths prperty s used. T llustrate ts use, let us cnsder aan the crcut f f..b. Due t the rtual rund at the nput f the OPAMP, x =0 (rtual shrt) and the nput current s determned by /. Snce the nput mpedance f the OPAMP s nfnte, flws thruhut, leadn t an utput ltae en by. As a result, the clsedlp ltae an becmes equal t /, as stated n equatn.8. If the mpedances and are replaced by resstrs as shwn n F..a, we end up wth the basc resste nertn amplfer. The clsedlp an s then btaned as (a) (b) F... esste amplfers: a) nertn cnfuratn and b) nnnertn cnfuratn. (.9) Ntce that n the case f the nertn cnfuratn, the amplfer s nput mpedance s determned by ; ths s a result f the rtual rund present at the nertn termnal f the OPAMP. Hence, f seeral nertn amplfers are cnnected n cascade we hae t be aware that the amplfer must be able t dre the nput mpedance f the next stae. Nnnertn ltae an cnfuratn. If the nput snal s appled at the nnnertn termnal and s runded, as shwn n f..b, the nnnertn cnfuratn s btaned. Ntce that the feedback s stll neate. If s cnnected t the pste termnal, the crcut becmes unstable and useless fr lnear applcatns; ths wll be edent n the fllwn sectns. The clsedlp ltae an f the nnnertn cnfuratn can be easly btaned f the rtual shrt prncple s used. Due t the hh an f the OPAMP, the ltae dfference between the nertn and nnnertn termnals s ery small; hence the ltae at the nnnertn termnal f the OPAMP s als. The current flwn thruh and s then en by / ; therefre the utput ltae s cmputed as (.0) The ltae an s therefre reater r equal than. An mprtant characterstc f ths cnfuratn s that deally ts nput mpedance s nfnty; hence seeral staes can be easly cnnected n cascade. A specal case f the nnnertn cnfuratn s the buffer cnfuratn shwn n fure.. If =, accrdn t equatn.0, the ltae an s unty; n ths case the alue f s nt crtcal and can een be shrtcrcuted ( =0). Ths tply s als knwn as unty an amplfer r buffer, and t s ery ppular fr drn small mpedances; e.. speakers, mtrs, etc.

5 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez f Emphass n ladn effects; dscuss these ssues n class! F.. OPAMP n unty an buffer cnfuratn III.. Amplfer wth multple nputs and superpstn. Input snals can be appled t the tw nputs f the OPAMP, as shwn n F..6. and are a ltae dder; the nput ltae at the nnnertn ( ) termnal s then F..6. Amplfer cnfuratn wth tw nput snals appled t the nnnertn and nertn termnals. (.) If a rtual shrt at the OPAMP nputs s assumed, the ltae at the nertn and nnnertn termnals s the same, and determned by. Usn KL at the nertn termnal f the crcut leads t (.) Usn equatns. and., the utput ltae s btaned, yeldn (.) Ntce that the utput ltae s a lnear cmbnatn f the tw nput snals: the frst cmpnent s determned by the and the ther ne determned by. Ths analyss can always be used, but we can als take adantae f the prpertes f lnear systems. Superpstn prncple fr a lnear system. If the OPAMP s cnsdered as a lnear dece and nly lnear mpedances are used, then the utput ltae s a lnear cmbnatn f all the nputs appled. If seeral nputs are appled t the lnear crcut, then the utput can be btaned cnsdern each nput snal at a tme; e.. rundn all ther nput snals and applyn the nput snal under study. Therefre, the fllwn prperty hlds: f

6 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez N,,..,N k jj (.a) k k j 0 j then,,..,,0..0 0,,..0 0,0,.. N N (.b) Ths prperty s knwn as the superpstn prncple. Let us apply ths prncple t the tply shwn n F..6, where tw nputs are appled t the amplfer. The crcut s analyzed by applyn ne nput snal at a tme: f s cnsdered, s made equal zer. The equalent crcut s shwn n F..7a. (a) (b) F..7 Equalent crcuts fr the cmputatn f the utput ltae: a) fr and b) fr. Snce the nput mpedance f the OPAMP s nfnte, the current flwn thruh s zer, and =0. The resultn crcut s the typcal nertn amplfer where the utput ltae s en by ( / ) ; ntce that ths utput crrespnds t the frst term n equatn.. If the frst nput s runded and snal s cnsdered, the resultn equalent crcut s depcted n f..7b. The ltae at the nnnertn termnal s en by equatn., and the utput ltae s equal t =( / ) ; the fnal result leads t the secnd term f equatn.. eneralzatn f basc cnfuratns. The cncepts f nfnte nput mpedance, zer utput mpedance, rtual shrt f the OPAMP nputs and lnear superpstn are especally useful when cmplex crcuts are desned. An anal nertn adder s shwn n f..8a; the utput ltae can be easly fund f we apply the superpstn prncple t each nput. The equalent crcut fr the jthnput s depcted n f..8b. Snce nly j s cnnected t j, the resstrs cnnected t the ther nputs must be runded. These resstrs are cnnected between the physcal rund and the rtual rund enerated by the neate feedback and the lare penlp ltae an f the OPAMP, as a result f ths the current flwn thruh all runded resstrs s zer, then j, enerated by j, j and the rtual rund nde x (=0) flws thruhut f. The utput ltae s then btaned as ( f / j ) j. By usn the same cncept t all nputs, the amplfer s utput ltae s fund as f f j j j N J j J x =0 N N F..8. a) Anal adder and b) equalent crcut fr analyzn the utput ltae due t j. N 6

7 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez N j f j j (.) An nterestn adantae f ths crcut s that each ne f the nput resstrs ndependently cntrls the ltae an fr each nput. Ths s an mprtant crcut prperty used fr the desn f analdtal cnerters fr nstance where the addtn f bnary wehted snals are requred. An anal nnnertn adder s depcted n f..9a. Smlarly t the preus case, the utput ltae can be cmputed by usn superpstn. Shwn n f..9b s the equalent crcut fr the jthnput snal, and makn all ther nputs equal zer. The ltae at the nnnertn termnal s the result f the ltae dder determned by the resstrs lumped t nde j as fllws: f f j j j J X J X j j N N a) b) F..9. a) Nnnertn amplfer wth multple nputs and b) equalent crcut fr the jth nput. J J J J N X J N X N (.6) Snce many cmpnents are n parallel, fr the analyss f ths type f netwrks t s ery ften mre cnenent t use admttances nstead f mpedances. Fr ths example, the preus equatn can als be expressed as j J (.6b) j J J J N X The numeratr (/ j ) s dentfed as the admttance f the element cnnected between the nput snal and j. The denmnatr represents the parallel f all elements attached t. The parallel cnnectn f mpedances s als equalent t the recprcal f the addtn f ther admttances; then.6 can be expressed by the fllwn equalent expressn j j N J X (.7) where =/. Once j s btaned, the utput ltae enerated by j can be btaned. Snce j s the ltae at the nnnertn termnal, t fnd the utput ltae fr ths nput s strahtfrward as j =( f / ) j. Takn nt accunt all the nput snals and applyn the superpstn prncple, t can be shwn that the erall utput ltae s a lnear cmbnatn f all nputs; the result f ths analyss yelds, 7

8 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez 8 N j j J N X f N j X N j J f N J j f (.8) Each nput snal has a cntrbutn t the utput ltae that depends f all resstrs, unlke the case f the nertn tply. The nput mpedance fr each nput depends n the array f resstrs; fr nstance the nput mpedance seen by the jthnput snal s X N J J J j (.9) Smlar expressns can be btaned fr all ther nput surces. Example: Desn a crcut that mplements the fllwn equatn: 0 0 t If needed use the supply ltaes / V. The crcut shwn belw can be used; ths s nt the nly slutn, cmbnatns f nertn and nnnertn crcuts mht be used as well. The desn prcess cnssts f fndn the resstance alues. Fr that purpse, the fllwn equatns must be sled: 0 0 Sle ths set f equatns, and fnd the numercal alues; t s edent that = and that =. III.. Amplfers wth ery lare an/attenuatn factrs. Very lare an amplfers requre lare spread f the cmpnents. In anal nterated crcuts desn, t s dffcult t cntrl prperly such lare rats, and ery ften they are ery demandn f slcn area. Sme cnfuratns allw us t reduce ths spread. Lare an nertn amplfer usn resstrs n a Tarray. Lare ltae an factrs requre ery lare resstrs; the array f resstrs shwn n fure.0can be used t ncrease the effecte feedback resstr. The crcut s transfer functn can busly be btaned by usn cnentnal crcut analyss technques such as KL. It s hweer t useful t analyze the crcut based n the fllwn bseratns:

9 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez. Snce the nertn termnal s at the rund ptental due t the rtual shrt at the OPAMP nput, the resstr s cnnected between nde x and (rtual) rund; then t s n parallel wth f yu analyze the crcut frm the utput. Therefre, the ltae x can be cmputed as x 0 (.0) Ntce that x s an attenuated ersn f the utput ltae.. Als, as a result f the rtual rund at the OPAMP s nput, the current flwn thruh s en by x /. If the OPAMP s deal, the current flwn thruh s equal t the ne flwn thruh, and then the utput ltae can be cmputed frm the fllwn expressn: x 0 (.) Therefre, the ltae an s 0 (.) The ltae an s therefre equalent t the an f a twstae amplfer; larer an factrs are btaned wth reduced resste spread. x F..0. esste ltae amplfer fr hhan applcatns. If further reductn n the resste spread s needed, tw Tnetwrks can be used as shwn n F... Smlarly t the preus case, the ltae x s equal t ( / ) ; ntce that x s an attenuated ersn f y and ths last ltae s an attenuated ersn f. These ltaes are related by the fllwn expressns: x y (.) 9

10 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez y 0 6 (.) Then the clsed lp ltae an s btaned as y y x x n 6 (.) Ths ltae an can be ery lare; t s n fact determned by multplcatn f terms, whch s equalent t a stae amplfer. Ntce that the nput mpedance f the nnnertn crcut s determned by. x y 6 F... esste amplfer usn a Tcnfuratn fr lare an factrs Very lare attenuatn factrs. The Tnetwrk can als be used fr the desn f lare attenuatn factrs, fr nstance n applcatns related t pwer amplfers where the nput snal mht be n the rane f 00 V r mre but the OPAMP can handle nly V r less. The resultn crcut usn a Tnetwrk at the nput s shwn n fure.. The ltae x s an attenuated ersn f the ncmn snal, and the ltae an s adjusted t the prper leel by the typcal nertn cnfuratn ( and ). The attenuatn factr, cnsdern aan a rtual rund at the nput, s determned by and the parallel f and ; the ltae an between x and s determned by the rat f resstrs and. Therefre, the utput ltae s en by: x 0 0 x x F... esste amplfer usn a Tcnfuratn fr lare attenuatn factrs (.6) The frst factr s the result f the ltae dder at the nput f the structure, whle the secnd factr s the result f the nnnertn amplfcatn f x. Yu can als use the duble Tcell structure; t s left t the student t fnd ut the ltae an n that case; DO IT, ths can be a mdterm questn. III.. rcuts: Interatrs and dfferentatrs. Basc nteratrs. If the feedback resstr s replaced by a capactr, we btan a lssless nteratr; the crcut s shwn belw. In ths crcut, the nput ltae s cnerted nt a current by and the rtual rund present at the nput f the OPAMP, smlarly t the case f the resste amplfer. The resultn current s njected nt the feedback capactr where t s nterated; the resultn utput ltae s the nteral f the njected current. The 0

11 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez analyss f ths crcut can easly be dne n the frequency dman where the characterstc mpedance f the capactr s en by /(j ). The transfer functn f the nertn cnfuratn s, as n the preus cases, determned by the rat f the mpedance n feedback and the nput mpedance, leadn t the fllwn result: F... Lssless nertn Interatr H( s ) (.7) s where s=j. Ths crcut has a ple at the rn and a phantm zer at = (why?). The mantude respnse s extremely lare at lw frequences, and decreases at hher frequency wth a rllff f 0 db/decade. The phase respnse s 90 (70) derees and ndependent f the frequency. Ntce that at =/, the mantude f the ltae an s unty. Usually ths crcut s nt used as a standalne dece, but s the key buldn blck fr hhrder flters and analdtal cnerters. The cmbnatn f resstrs and capactrs lead t the eneratn f pes and zers; fr example, the crcut mplementatn f a frst rder flter s shwn n fure.. Ths crcut s als knwn as a lssy nteratr because whle njects chare nt, the resstr leaks (ntrduces lsses) the chare stred n the capactr. The an f ths crcut s als determned by the rat f the equalent mpedance n feedback and the nput mpedance. In ths crcut, the equalent feedback mpedance s cmpsed by the parallel f the mpedance f the capactr (/j ) and. The lw frequency an, where the mpedance f the capactr can be nred, s determned by the rat f the tw resstrs; t s edent that the ltae an s en by /. At ery hh frequences, the mpedance f dmnates the feedback and the crcut behaes as the lssless nteratr shwn n fure.9 (an defned by /s). The erall ltae an s en by: H( s ) F... Frst rder lwpass flter (.8) s The lwfrequency an s, as expected, determned by the rat f the resstrs. The ple s lcated at =/, and fr frequences beynd ths frequency the ltae an decresases wth a rllff f 0 db/decade. The man dfferences between ths crcut and the passe lwpass flter (ltae dder) are twfld: a) n the acte realzatn (wth OPAMP) the lw frequency an can be reater than by adjustn the rat f the resstrs whle n the passe flter the an s always less than ; b) the OPAMP allws us t cnnect the crcut t the next

12 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez stae wthut affectn the transfer functn; ths s due t the small utput mpedance f the OPAMP. A typcal transfer functn btaned wth ths crcut s depcted n the fllwn plt; 0l0 /> 0 db l P u F... Mantude respnse fr a frst rder flter. The unty an frequency s anther mprtant parameter; t can be btaned by takn the mantude (r square mantude) f equatn.8, and equatn t t. The resultn equatn can be sled fr the frequency, leadn t the fllwn result (fnd t yurself) u (.9) Fr >>, ths frequency s apprxmately en by u =/. Ntce that fr < the slutn s manary, meann that the unty an frequency des nt exst; n fact yu can nt fnd any frequency where the an s unty fr an attenuatr (dc an less trhan ). Fr <, the lw frequency an s less than ; hence the transfer functn des nt hae any ntersectn wth the 0 db cure. The eneral frst rder transfer functn can be mplemented by usn the tply shwn n f..6. Snce the elements are n parallel, t s ery cnenent fnd the ltae an as the rat f the equalent nput addmttance and the equalent feedback addmtance as fllws: 0 F..6. eneral frst rder flter s0 s 0 H s (.0) s s Wth ths crcut, yu can desn the fllwn flters: (a) Lwpass flters f 0 s remed. The ple s frequency s en by / (b) Amplfer f 0 and are remed. an = / (c) Amplfer f the resstrs are remed. an = 0/ (nt ery practcal, especally fr lw frequency applcatns)

13 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez (d) Hhpass f s remed. Ple s frequency at /, and hh frequency an = 0/. The hhpass transfer functn can als be realzed f a seres f a capactr and resstr s used, as shwn n the fure belw. Lw frequency cmpnents are blcked by the capactr due t ts hh mpedance at lw frequences. At hh frequences the capactr behaes as a shrt crcut, and the an s en by the rat f the resstrs. Usn typcal crcut analyss technques, the erall transfer functn can be btaned as F..7. Frst rder Hhpass flter usn a seres capactr. s H s s (.) s A D zer and a ple lcated at =/ can be bsered. After the ple s frequency the ltae an s manly determned by the rat f the resstrs /. Nnnertn nteratr. A nnnertn amplfer s mplemented by usn the fllwn crcut. It s usually an expense mplementatn, snce the tply requres matched elements. The transfer functn can be easly btaned by ntn that the ltae at the nnnertn termnal s the result f a ltae dder between and (/n=/(s)). The ltae at the nnnertn termnal s then amplfed by a factr plus the rat f the feedback mpedance /s and the resstr. The resultn transfer functn yelds; ' ' n F..8. Nnnertn lssless nteratr s'' H (s) (.) s'' s That crrespnds t a nnnetn nteratr f = ; n ths case, the ltae an decreases when the frequency ncreases wth a rllff f 0 db/decade and ts phase s 90 derees. The phase at D s 90 derees. III.6. Instrumentatn amplfers. In many practcal applcatns, t s desrable t use amplfers wth ery lare nput mpedance and ery lw utput mpedance. Ths s the case when the sensrs hae lare utput mpedance and r lmted current drn capabltes. Fr thse applcatns, the nertn amplfer based n tw resstrs can nt be used snce ts nput mpedance s fnte; e.. defned by the nput resstance. The nly ptn s t use nnnertn amplfers, as the nes shwn n fure.9. In case the ncmn snal s dfferental carred ut by and, therefre tw nn

14 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez nertn amplfers are used. In a dfferental system, the nfrmatn s determned by the ltae dfference between the tw nputs rather than by the ltae at each nde. The crcut shwn n fure.9a s cmpsed by snleended nnnertn amplfers. The crcut s a partcular case f the crcut shwn n F. 7b; =0, =, =, =0 that leads t a untyan amplfer (buffer) wth ery lare nput mpedance. Snce the OPAMP utput mpedance s ery small yu can easly cnnect nertn amplfers after ths structure f requred. The benefts f these buffers wll be edent n the fllwn chapters. f f (a) (b) F..9. Fullydfferental amplfers based n: buffers and b) nnnertn amplfers. The tply shwn n fure.9b s mre useful and can prde ltae amplfcatn reater than ; the nput mpedance s ery lare as well and determned by the nput mpedance f the OPAMP. The analyss f ths crcut s strahtfrward f we take adantae f the rtual shrt prncple. As shwn n fure.0, the ltaes at the nertn termnals are en by and, respectely. The current flwn thruh s then en by ( )/; ths current flws thruhut the resstrs f, eneratn a ltae drp due t F en by ( ) f /. The utput ltae s then equal t () f / whle the utput ltae s en by () f /. The dfferental ltae an s then cmputed as d d f (.) f f F..0. Practcal fullydfferental nstrumentatn amplfer. Ntce n ths expressn that the utput ltae s als dfferental d =. Snce the an f the preus expressn s defned as the rat f the dfferental utput ltae and the dfferental nput d t s knwn as dfferental ltae an. Dscuss here the prpertes f dfferental and cmmnmde snals. mmnmde nput mpedance and dfferental mde nput mpedance.

15 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez The mst ppular snleended nstrumentatn amplfer s depcted n F..; ths tply s ery useful fr nstrumentatn applcatns. There are tw nputs, and the mprtant nfrmatn s n dfferental frmat d=. A majr adantae f fullydfferental crcuts s that they are lttle senste t nse and snal nterferences that affect bth nputs; e.. cmmnmde snals. The snle ended utput then shuld be prprtnal t d, and snals that are present n bth nputs wth same ampltude and same phase are cancelled by the dfferental nature f the amplfer. Applyn the superpstn prncple t the crcut shwn n f.., t can be shwn that the utput ltae s en by a lnear cmbnatn f and as fllws: 6 F... Practcal snleended nstrumentatn amplfer. 6 6 (.) Usn the preus equatns. and., the utput ltae can be btaned as 6 6 (.) After sme alebra we et the fllwn result: (.6) If we ntrduce the cndtns = and =6, ths equatn smplfes t the requred dfferental utput (.7) The mprtant prpertes f ths amplfer are:. The nput mpedance s extremely lare and determned by the OPAMP. Therefre, t can be easly cnnected t a number f sensrs reardless the type f sensr s utput mpedance.. The utput ltae s senste t dfferental nput ltae d=.. mmnmde nse present at bth termnals s rejected by the dfferental nature f the tply. The ablty t reject cmmnmde nse (electrmanetc nterference present at bth amplfer s nputs fr

16 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez nstance) by an amplfer s measured by the cmmnmde rejectn rat (M) parameter, t be dscussed n next sectns. III.6. Practcal OPAMP lmtatns. 6

17 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez ELEN. Part III.b 7. Desn examples. Wrkut sme addtnal examples: e.. f. P.9, SedraSmth st edtn. 8. Flters and ther nncnentnal crcuts. Flters are used n electrncs fr the selectn f nfrmatn that s lcated n a specfc frequency band. A ppular structure s the scalled multple feedback tply shwn n fure.b. Tw feedback paths can be bsered n ths crcut, the frst ne due t Y and the secnd ne due t Y. Bth feedback trajectres prde neate feedback, that makes the crcut s stable. The ltae an f ths structure can be fund by sln the ndal equatns at nde x and the OPAMP s nertn termnal. Thse equatns can be wrtten as (fnd t!): Vn Y Y Y x Y Y V F..b. Multple feedback secndrder flter. Y Y 0 x Y Y Y Y Y n Y Y Y Y (.b) Please wrte the equatns and check the matrx (dn t belee n prfessr s result!). Sln the system f equatns we shuld be able t fnd the utput ltae f the crcut. Ntce that we are nt wrtn any equatn fr the utput nde f the OPAMP, and we shuld nt! The OPAMP utput ltae s enerated by a ltae cntrlled ltae surce, where the current demanded by the elements cnnected t the OPAMP s prded by that element. Fr deal OPAMPs, s cntrlled by the elements cnnected n feedback and the nput ltae, and nt by the elements cnnected at the OPAMP utput! Fr the slutn f the equatns.b make _=0 snce we hae a rtual rund at the nput f the OPAMP. We mht wrte these equatns by nspectn ntcn that:. Fr the frst rw we cnsder the ndal equatn at nde x. Fr the element f the admttance matrx we hae t cnsder all the admttances cnnected t x; snce x s the frst nde we are cnsdern. In f..b, the elements cnnected t x are Y, Y, Y, and Y.. The element f the matrx s determned as the neate f the admttance between and, snce the secnd nde cnsdered n the admttance matrx s.. The element s the neate f the element(s) cnnected between x and, n ths case Y.. Fr the secnd rw we cnsdered the ndal equatn at nde. Matrx term s the neate f the admttance cnnected between and x.. The element s cmpsed by all admttances cnnected t (the nde under cnsderatn). 6. Fnally the matrx term s the neate f the admttances cnnected between and. 7. Fr the rht hand sde f.b, we cnsdered the nput ltae and the admttance cnnected between n and x. 8. Snce we d nt hae any element cnnected between and the secnd term f the rht hand sde s zer. Anther apprach fr fndn the transfer functn s usn the prpertes f lnear crcuts. Applyn superpstn, x s a lnear cmbnatn f n and. Snce the OPAMP nertn termnal s a rtual rund, cmputatn f x yelds (check t!) 7

18 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez x Y Y n (.b) Y Y Y Y Y Y Y Y Als, ntce that s enerated by Y, Y and x as Y x (.b) Y Sln thse equatns, the ltae an can be fund as: H( s ) (.b) n Y Bth methds are useful. Y Y Y Y Y Y YY A Partcular case f ths crcut s the Secnd rder LOWPASS transfer functn. eplacn sme f the admttances n F..b by the resstrs and capactrs shwn belw, the crcut behaes as a secnd rder lwpass flter. Frm equatn.b t can be fund that Vn V H( s ) F..b. Multple feedback Lwpass flter. (.b) s s The prpertes f ths crcut are edent just analyzn the lcatn f ples and zers. There are tw ples defned by the dfferent cmpnents, and tw zeres at = (why?). Therefre, the mantude respnse wll reman relately flat untl the frequency f the dmnant ples f real ples are mplemented. Fr frequences abe the frst (dmnant) ple, the mantude respnse decreases mntncally wth a rllff f 0 db/decade. Beynd the frequency f the secnd ple the rllff becmes 0 db/decade due t the effect f the secnd ple, as shwn n the fure belw. 8

19 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez 9 (l) 0*l0( H(s) ) 0 db/decade P P 0 db/decade P P F..b. Mantude respnse fr a secnd rder transfer functn wth tw real ples. Fr better rejectn f hh frequency cmpnents usually the ples are lcated clse t each ther as shwn n the dashed cure; n ths case the hhfrequency rllff f the mantude respnse s 0 db/decade. Fr the desn f a lwpass secnd rder transfer t s mre cnenent t express equatn.b as s s s ) ( H (.6b) The ples f the system are determned by the denmnatr f the preus equatn and can be btaned as, P. (.7b) Selectn the prper alues fr the cmpnents, the ples can be real r cmplex cnjuate; the cndtns fr these cases are the fllwn: f cnjuated mplex f eal, P (.8b) The phase respnse s als mprtant fr the full characterzatn f the lwpass flter. The nertn flter cnfuratn has a phase shft f 80 derees at ery lw frequences; ths can be erfed n the rnal transfer functn, equatn.6b, ealuatn t at s=0. Each ple ntrduces a phase shft f derees arund ts ple

20 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez frequency, as dscussed n preus sectns. If the ples are far away frm each ther, the phase respnse lks lke the ne depcted by the sld lne n the fllwn phase plt. Phase(H(s)) P P derees/dec derees/dec 60 (l) P P P P F..b. Phase respnse fr an nertn secndrder transfer functn. If the system hae the tw ples clse t P, then the system s phase respnse presents a rllff f 90 derees/decade arund P, due t phase cntrbutn f tw ples, as shwn n the dashed plt. The fllwn desn example wll be dscusssed n class: Lwpass flter desn: D AIN = 0 db, p = p =00 Krad/sec. The desn equatns are btaned frm the desred flter s transfer functn as fllws: H (s) 0 0 P s P s P s Ps P Nte that the numeratr 0 P s needed t btan the desred D ltae an. Equatn the terms f ths equatn wth the nes f equatn.6b, the desn cndtns are btaned / =0 (0 ) =/( ) (0 )=(/ / / )/ Lets desn the flter based n pwer cnsumptn cnsderatns. T ad the use f ery small resstrs, whch mples ery lare currents and mre pwer cnsumptn, lets fx the smaller resstr t =0 k and t use =. Hence: = =00 k =/(0 0 x0 0 )= 0 0 =(.x0 )/ (x0 )=0.6x0 9 Then =0 0 /0.6x0 9 =6.6x0. The fnal desn s shwn belw. The crcut has been smulated n PSPIE, the mantude and phase respnse are als shwn. 0

21 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez Vn 0k 0.6 nf 00k 00k 6.6 pf V mpnent alues fr the LOWPASS flter 0 0 Vltae an (db) E E E E E E6 Frequency (Hz) Spce esults: Mantude respnse 80 Phase (Derees) 90 0 E E E E E E6 Frequency (Hz) Spce esults: Phase respnse elatnshp between frequency dman and tme dman. In many cases we are mre nterested nt seee the respnse f the crcut n tme dman; e.. mpulse and/r pulse respnse. An apprach fr the anaylss f a crcut

22 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez n tme dman s t wrte the ndal r mesh equatns n tme dman usn the nterdfferental equatns fr capactrs and nductrs. Anther apprach s t btan the transfer functn n the frequency dman as n the preus examples and t cnert t nt a dfferental equatn by usn the prpertes f the laplace transfrm. Amn many ther prpertes f the laplace transfrm, ne f the fundamental nes s the fllwn: t N N d x a s x a (.9b) 0 0 dt Ths prperty f the laplace transfrm s used fr the cnersn f ratnale lnear functns n the sdman t the dfferental equatn n the tdman. T llustrate ts use, let us cnsder the fllwn sdman (frequency dman) lwpass transfer functn: n s a0 s s bs b0 (.0b) It can als be rewrteen as s s s bs b0 a0n (.b) f the laplace transfrm s appled t bth sdes f ths equatn, the tmedman equalent s batned leadn t the fllwn secndrder dfferental equatn d dt d 0 n (.b) dt t b t b t a t 0 The next step s t sle ths equatn takn nt accunt the type f nput snal; e.. mpulse respnse, pulse respnse r een snusdal nput respnse. It s nt the purpse f ths chapter t dscuss the tme dman analyss, and the student shuld be refereed t mre specalzed bks fr detaled analyss and mre examples. Bandpass transfer functn. Very ften the nfrmatn t b prcessed s wthn a en pass band, hence lwpass and hh pass fltern mht nt be the mst effcent apprach fr snal detectn. A bandpass flter s mre sutable fr ths purpse; t can be btaned f n addtn t the tw ples f the lwpas transfer functn, ne f the zers s lcated at lw frequency and the ther ne at ery hh frequences. These zers can be easly mplemented f they are lcated at =0 and =, respectely. If the eneral multple feedback transfer functn, equatn.b, s cnsdered the zer at lw frequences s enerated f ne f the tw elements Y r Y s replaced by a capactr whle the ther ne s a cnductance, respectely. A sutable ptn fr flter realzatn s shwn n the fllwn fure. The analyss f ths crcut s smlar t the ne used fr the lwpass flter; the transfer functn s Vn V F..b. Multple feedback bandpass flter.

23 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez s s H (s) (.b) s s s s The resultn transfer functn has tw zers, ne at D and the ther ne (phantm) at =. If the ples are at the same frequency, the mantude and phase respnses can be apprxmated by pece wse lnear functns as depcted n the fllwn plts. 0*l0( H(s) ) Phase(H(s)) 0 db/decade 0 db/decade 90 derees/dec P P 80 P P derees/dec (l) 70 (l) P P P P P P F..6b. Mantude and phase respnse f a secnd rder bandpass transfer functn. Exercse: Desn a Bandpass flter wth bth ples at 0 MHz and peak an f 0 db. D t! 0. Partal pste feedback. 0. esste amplfers wth partal pste feedback. Partal pste feedback can als be used fr the mplementatn f demandn applcatns. Fr nstance, neate resstrs hae t be used fr the desn f ltae cntrlled scllatrs t cancel the effects f resstrs lumped t nductrs and capactrs (resste lsses). In partal pste feedback crcuts, bth termnals nertn and nnnertn are embedded n feedback lps; an example s the crcut shwn n F..7b. The ltae at the pste termnal s a sample f the utput ltae, and then t can be fund that x F..7b. esste amplfer wth neate and pste feedback x (.b) The utput ltae s cmpsed by the cntrbutn f (nertn amplfer wth a ltae an =/) and x(nnnertn amplfer en by ( /)x. Hence, the utput ltae can be btaned as:

24 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez x (.b) The slutn f ths expressn fr utput ltae yelds; (.6b) The pste feedback s reflected n the neate term f the denmnatr. The ltae an can be further ncreased f ()/(()) s clse t unty. Ntce that the an can ptentally be nfnte. Ths stuatn s undesrable because a small aratn n any f the cmpnents has a hue mpact n the erall ltae an; these aratns culd be due t temperature aratns r an f the cmpnents. Thus, f pste feedback s used, be sure that neate feedback s dmnant and that aratns n cmpnents d nt drastcally affects crcut s perfrmances. ealzatn f neate mpedances. The crcut shwn belw uses partal pste feedback as well snce the lnks the utput ltae and the nnnertn termnal. T understand the peratn f the crcut, let us fnd the ltae at the nnnertn termnal. Applyn superpstn, x s cmpsed by the cntrbutn f and. The frst cmpnent can be btaned by cnsdern, and rundn ; ths can be dne because the utput f the OPAMP s a lwmpedance nde, and s defned by the ltaes appled at the OPAMP nputs. The secnd cmpnent s btaned by cnsdern and rundn, then x.8b. Amplfer wth partal pste feedback. x (.7b) Once ths ltae s btaned, the utput ltae can be easly determned, because

25 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez x (.8b) After sme alebra we can fnd the erall transfer functn as (.9b) Once aan, the pste feedback s reflected n the neate term f the denmnatr. An mprtant specal case s when = and =; the preus equatns can be smplfed as fllws: (.0b) Therefre, the crcut behaes as a nnnertn amplfer. The mst nterestn prpertes f ths crcut are asscated wth the nput mpedance. Frm.7b, t s btaned fr the case = and = that x (.b) Therefre the current flwn thruhut can nw be btaned as x (.b) As can be ntced n ths result, the current flwn thruh s determned by ; ths current s ndependent f. Therefre, ths crcut can be cnsdered as a ltae cntrlled current surce; the current s cntrlled by the nput ltae and the resstrs =, and ths current s frced t flw thruh. On the ther hand, the mpedance at the nput prt s determned as fllws (please wrk ut the fllwn expressn). (.b) Fr < the nput mpedance s pste, and neate fr >. A useful crcut ften used n flter s desn s the neate mpedance cnerter. The crcut, shwn n F..9b, s a aratn f the schematc depcted n F..8b. The nput ltae s appled t the nnnertn termnal, and the utput ltae s en by (/). The nput current s cmputed as ()/, leadn t the result en n expressn.b.

26 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez F..9b. Neate mpedance cnerter. (.b) Therefre, the equalent mpedance s neate. A neate mpedance means that, cntrary t the case f a pste mpedance, the crcut delers current when pste snals are appled. The reasn fr ths behar s that the OPAMP altether wth and amplfy the nput and the utput ltae s reater r equal than. Hence pste enerates > ; snce s cnnected between them t enerates a current that flws frm t. Thnk abut the fllwn questns: What s the meann f neate mpedance? Dfference between pste and neate mpedance? 0.. Sallen & Key Flter. Pste feedback has been use fr the desn f flters fr ln tme. The flter based n fnte an amplfer uses admttances and an amplfer wth fnte an K. Snce the amplfer s a nnnertn structure, the feedback prduced by Y s pste leadn t a flter wth partal pste feedback. By fllwn the analyss prcedure used fr the multple feedback flters, the transfer functn can be btaned by wrtten the admttance matrx as fllws: y n y x y V y K y y F..0b. Sallen and Key secndorder flter. y y y y y y y 0 K y 0 0 y 0 0 x n y y (.b) The frst tw rws crrespnd t the ndal equatns f ndes x and y, respectely. The thrd rw crrespnds t the fnte amplfer an en by =Ky. The slutn f ths system leads t the fllwn flter s transfer functn: 6

27 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez H( s ) y y y y y y y y y y K K (.6b) Selectn the prper elements lwpass, bandpass and hhpass flters can be desned. These especal cases are: Selectn Y and Y as cnductances, and Y and Y as capacte admttances, the resultn transfer functn leads t a lwpass transfer functn. Een mre, Y mht be remed n ths case; the resultn flter s shwn n the fure belw. Equatn.7b shw the transfer functn. ` Vn K V H( s ) F..b. Secnd rder Sallen and Key Flter. Ntce that tths flter uses partal pste feedback (.7b) s s Smlarly t can be shwn that K K Y and Y cnductances, and Y and Y capactrs lead t a bandpass transfer functn. Y and Y cnductances, and Y and Y capactrs lead t a hhpass transfer functn. Fnd yurself the resultn crcuts! Yu may fnd ne f these tples n the frst mdterm.. Practcal Lmtatns f the Operatnal Amplfers. Frst at all, we must recnze that practcal OPAMP are nt een clse t the deal mdel: nfnte nput mpedance, nfnte an, nfnte bandwdth, and unlmted utput current capablty. Thse parameters depends n the tply used (array f transstrs, resstrs and capactrs, technly used and pwer cnsumptn); there are tns f dfferent OPAMPs ffered by dfferent endrs; e. Texas Instruments, Farchld, Natnal Semcnductr, etc. Althuh the rn f thse lmtatns are nt dscussed n ths curse, the effects f n the erall transfer functn f these parameters are brefly dscussed n ths sectn. A mre realstc OPAMP macrmdel s depcted n the fllwn schematc. n n r (A)n Fure.b. Macrmdel fr the OPAMP. Sme alues fr cmercally aalable OPAMPs are: = M, r 0 =0, and A=0. These parameters ntrduce errrs n the transfer functn. Usually t s cumbersme t btan the fnal results, and t s dffcult t ealuate 7

28 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez system deradatn especally fr cmplex crcuts. Here we btan sme results fr a snle nertn amplfer but mst f the cnclusns fr ths crcut are als ald fr cmplex crcuts. Let us cnsder the crcut shwn n Fure.b; we are cnsdern the fnte nput mpedance f the OPAMP and the nput mpedance f the stae dren by the OPAMP. Usn the macrdel f F..b wth r 0 =0, the equalent crcut can be btaned as depcted n Fure. (b). The transfer functn can be btaned f the ndal equatn at nde s wrtten; ntce that ndal equatn at the OPAMP utput can nt be wrtten because s cntrlled by the ltae dependent ltae surce ( =A ( ). The current demanded by F and L s prded by the deal ltae surce. Sln the fundamental equatn ( = 0 ) the crcut s transfer funcn yelds, F 0 F L /A V A V ( ) L F..b. a) Inertn amplfer wth OPAMP nput mpedance and lad mpedance L, and b) equalent crcut wth r 0 =0. F H( s ) F A F (.9b) The effect f the fnte D an and fnte nput mpedance n the nertn amplfer can be better apprecated f the errr functn s cnsdered; frm equatn.9b t fllws that F F H ( s ) (.0b) where the errr functn ξ s defned as ( s ) F A F A F (.b) If the OPAMP D an s lmted, the assumptn f rtual rund s nt lner ald snce any utput ltae aratn enerates a fnte aratn n the dfferental nput snal en by /A. The smaller the OPAMP an the larer the ltae aratns at the OPAMP nput are; hence the errr shuld be nersely prprtnal t A. The ltae aratns n the nnnertn termnal lead t current errrs: frstly the nput current s en by ( )/ hence an errr prprtnal t ( errr = / ) s ntrduced; Secndly, the OPAMP nput mpedance takes part f the current enerated by, leadn t a secnd current errr en by errr = / I. These current errrs are cnerted nt ltae errrs by the feedback resstr F, and are edent n equatn.b. Ntce that een f the OPAMP nput mpedance s nfnte, a an errr due prprtnal t the deal an F / s present. Fr a en OPAMP penlp an A, the larer the clsedlp amplfer s an the larer the errr s. The errr that can be tlerated depends n the applcatns; ntce that t kept the errr belw % fr nstance t s requred t satsfy 8

29 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez F ( s ) 0. 0 (.b) A Fr nstance f = M, and ltae an f 0, the ltae an needed depends n the abslute alues f the resstrs used, as shwn n table belw A V F / 00/0 > 000 0k/k > 00 M/00k >00 0M/M >000 Ntce that the errrs are mprtant when the nput mpedance s cmparable wth the OPAMP nput mpedance. Anther mprtant lmtn factr s f curse the desred amplfer an F/. Ntce that.b can als be rewrtten as ( s ) A F (.b) Effects f the OPAMP Fnte bandwdth. Unfrtunately the OPAMP bandwdth s ery lmted; belee r nt, the bandwdth f the 7 s arund 6 Hz nly whle he penlp D an s arund x0 V/V. The prduct f the penlp D an and bandwdth s defned as OPAMP s anbandwdth prduct BW. Fr the OPAMP 7, BW~. MHz. These parameters are llustrated n the fllwn plt 0*l0( A V (s) ) 06 db 0 db/decade f (Hz) F..b. Typcal penlp mantude respnse f an OPAMP The OPAMP penlp ltae an can be mdeled by a fnte D an and a lwfrequency ple as A V s AD s P (.b) Ntce that BW=A D x P ; t s als knwn as the unty an frequency u. Ths frequency s mprtant because t defnes the upper lmt fr the peratn f the OPAMP: beynd ths frequency the OPAMP s nt lner an amplfer but an attenuatr. That des nt mean that yu can always use the OPAMP untl u! Usually the useful frequency rane s well belw ths lmt. We learn n the preus dscussn that an errr s ntrduced f the pen 9

30 ELEN. Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez lp an f the OPAMP s fnte. Fr the nertn cnfuratn, the errr determned by equatn.b. If A s frequency dependent, then usn.b we can btan a eneral frm fr the errr functn ncludn the effects f the fnte OPAMP bandwdth as shwn n the fllwn expressn: F s ( s ) AD P (.b) Therefre, the ple f the OPAMP leads t a zer fr the errr transfer functn as shwn belw. Therefre, the hher the bandwdth the larer the errr s. 0*l0( A V (s) ) 06 db 0 db/decade Errr (db) 0 db 0 db/decade BW f (Hz) F..b. OPAMP penlp mantude respnse and errr respnse fr an nertn amplfer. Fr the example dscussed n the preus sectn we hae: AD=0 /, F / =0, =M then t flws: / / P errr ~ ~.x ~ ~ ~ ~ ~0 00 ~x0 Ntce that the errr ncreases further fr hh frequency applcatns. Ths s a result f the lmted bandwdth f the OPAMP; we hae t remember that the penlp ltae an reduces prprtnal t the frequency. The errr s less than % f and nly f the frequency f the appled snal s belw 00 tmes the OPAMP ple s frequency. Fr the OPAMP 7; f P =6 Hz; hence the snal s frequency has t be less than 600 Hz fr an deal amplfcatn factr f 0. The effects f the OPAMP fnte nput mpedance are nt ery releant f the used mpedances and F are lesser than /0. Slewrate lmtatns: wll be dscussed n class. mmnmde snals: wll be dscussed n class. 0