Chapter III The Operational Amplifier and Applications

Transcription

1 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba hapter III The Operatnal Amplfer and Applcatns The peratnal ltae amplfer (mre cmmnly referred t as peratnal amplfer) s ne f the mst useful buldn blcks fr the mplementatn f lw and medumfrequency anal snal prcessrs. The deal peratnal amplfer prcesses a dfferental nput snal (at ts nnnertn and nertn nputs) wth ery hh mpedance at each nput, ery hh ltae an, wde bandwdth, and ery lw utput mpedance. These prpertes are desrable because they make the peratnal amplfer ersatle, easy t nterface wth ther blcks, and rbust when cmbned wth passe and acte elements. Operatnal amplfers are usually csteffecte slutns fr the realzatn f anal snal prcessn such as amplfcatn, fltern, cmparsns f ltae, etc. A ast arety f peratnal amplfers are ffered by numerus nterated crcut endrs. Hence, the selectn f the ptmal peratnal amplfer fr a partcular applcatn s n many cases nt tral. Sme applcatns requre ery lw nse and mdest snal bandwdth whle thers may requre hh bandwdth but wth mderate accuracy and relaxed nse specfcatns. Pwer cnsumptn and cst are tw ther factrs that hae t be cnsdered when selectn an peratnal amplfer. Ths chapter deals wth the fundamental cncepts related t peratnal amplfers. Basc amplfer crcuts wll be studed and analyzed wth frst and secndrder system apprxmatns... Basc Operatnal Amplfer Mdeln. The OPeratnal AMPlfer (OPAMP, r pamp) s a key buldn blck n anal crcut desn. OPAMPs are cmpsed f seeral transstrs and passe elements (resstrs and capactrs) and are cnfured such that the lwfrequency ltae an s ery hh. Fr nstance, the D an f a μa7 OPAMP s arund 0 V/V, such that a 0 V nput ltae dfference leads t V at the utput. The desn f such a cmplex crcut s nt fully dscussed n ths bk, but the fundamentals f amplfers are dscussed n chapters V and VI. In ths chapter we wll use a smplfed lnear macrmdel t cney the prncples f basc OPAMP crcuts and ther peratn. Seeral crcuts are studed such as basc amplfers, frstrder flters, and secndrder flters. The ersatlty f the OPAMP wll be edent at the end f ths chapter. T btan macrmdel parameters f a snlenput ltae amplfer wth a snle utput, let us cnsder a lnear twprt system wth tw termnals runded, as shwn n F... Ths chapter presents fur arables (,,, and ) fr study. The nteractn between the fur arables can be defned n many dfferent ways. In realwrld applcatns, these defntns depend n the nput arable (current r ltae) and the mst releant utput arable. Usually, fr ltae amplfers, the nput snal s defned as whle the utput s. Z Electrnc rcut Z F... Electrnc crcut represented by a black bx. Snce we are assumn that the crcut s lnear, ne way t descrbe the electrnc crcut s by usn parameters n the fllwn matrx representatn:, (.a)

2 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba r. (.b) Ntce that we are mxn currents and ltaes n the matrx. Therefre, we call t a hybrd representatn f the crcut, and the resultn mdel s termed a hybrd mdel. In many textbks yu can fnd at least fur dfferent sets f parameters, but Eq.. s ne f the mst releant nes fr ltae amplfers. Bplar and metalxdesemcnductr (MOS) transstr mdels are frequently based n anther mdel called hybrd, t be dscussed n the fllwn chapters. In the abe equatns, the parameter s the nput cnductance, whch relates the nput current and the nput ltae f the crcut wthut cnsdern the effect f the utput current (.e., when = 0). The crcut s nput cnductance s frmally defned as fllws:. (.) Z 0 Ths parameter s measured by applyn a ltae ( ) at the nput and measurn the nput current ( ) whle the utput nde s left pen such that = 0. The unt f s amperes/lts, r /Ω. The parameter defnes the reerse current an f a tply, and t s defned as. (.) 0 The reerse current an s the current enerated at the nput due t the utput current. In the deal case ths parameter s zer fr ltae amplfers, whch we usually lke t be undrectnal, such that the nput snal enerates an utput snal. Hweer, the utput snals (current r ltae) d nt enerate any snal at the nput. T measure ths parameter, ne must shrtcrcut the nput prt such that = 0, t apply a current at the utput, and t measure the current enerated at the nput prt. In practcal crcuts, ths parameter s ery small and s ften nred. The frward ltae an s defned as the rat f the utput ltae and nput ltae wth an pencrcut at the utput: AV. (.) 0 Parameter = A V s certanly ne f the mst mprtant parameters f the twprt system. We als refer t AV as the pencrcut an (r penlp an when the feedback f a crcut s als remed). It represents the crcut s ltae an wthut any lad mpedance attached at the utput, resultn the n zer utput current. Anther mprtant parameter s the system s utput mpedance, whch relates the utput ltae and the utput current wthut the effects f the nput snal. It s defned as

3 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba Z. (.) 0 A twprt system can be represented by the fur afrementned parameters, whch are captured by the schematc f the macrmdel n F..a. Fr sake f clarfcatn we are usn mpedances nstead f admttances n ths representatn. Ntce that a current cntrlled current surce s used fr the emulatn f parameter snce t represents the nput current (nput prt) ben cntrlled by the utput current (utput prt). A resstr cannt represent ths parameter snce the current flws thruh the nput prt but the cntrlln current flws n the ther prt. Smlar cmments apply t the ltae cntrlled ltae surce represented by A V ( ). Z Z A V Z Z A V (a) (b) F... Lnear macrmdels usn hybrd parameters: (a) a typcal ltae amplfer and (b) an deal OPAMP. Mdel fr an OPAMP. The deal OPAMP s a dece that can be mdeled by usn the crcut f F..b, whch was btaned frm the ne n F..a wth A V =, = 0, Z =, and Z = 0. Input s cmmnly knwn as nnnertn nput, and the ther nput ( ) as nertn nput. Een thuh ths mdel s nt fully realstc, t s suffcent t understand the peratn f basc OPAMP crcuts. When cnnectn crcuts n cascade fr cmplex snal prcessn, bth nput and utput mpedances are mprtant parameters that affect the erall system perfrmance. Hweer, n the fllwn sectns the effects f these parameters are nt cnsdered unless nted therwse... Basc nfuratns: Inertn and Nnnertn Amplfers. 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, and a secnd mpedance that s cnnected frm the nertn termnal t the utput f the OPAMP. Z prdes a neate feedback (cnnectn the OPAMP s utput and the nertn termnal). Ths feedback s the man reasn fr the excellent prpertes f ths cnfuratn as ln as the system s stable by desn (based n ts phase marn). The smplfed lnear macrmdel f the OPAMP s used here fr the representatn f the nertn amplfer. The equalent crcut s shwn n F..b. By usn basc crcut analyss technques, t can be fund that and x x Z Z A 0, (.6) x. (.7) Sln these equatns as a functn f the nput and utput ltaes yelds

4 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba Z A V Z Z Z. (.8) The abe relatnshp s als knwn as the clsedlp an f the amplfer because the feedback resstr frms a clsed lp arund the OPAMP. If the penlp an f the OPAMP A V s ery lare, then the frst factr n Eq..8 can be apprxmated as unty and the clsedlp ltae an becmes Z Z. (.9) Ths result shws that f neate feedback s used and f the penlp an f the OPAMP s lare enuh, then the erall clsedlp ltae an f the amplfer depends n the rat f the feedback and nput mpedances. Unlke the penlp an f the OPAMP that can ary by mre than 0% due t transstr parameters aratns and temperature chanes, the clsedlp an s mre accurate, especally f same type f mpedances are used. Nrmally, the rat f mpedances s snfcantly mre precse than the abslute alues f cmpnents. Bth, rats f resstrs and rats f capactrs fabrcated n the same nterated crcut can hae msmatch errrs as lw as 0.0.% whle the abslute alues f the cmpnents may ary by mre than 0%. Thus, desnn amplfers wth feedback leads t rbust ans n the presence f manufacturn aratns. Z Z Z Z x A V x (a) (b) F... Inertn amplfer: a) schematc f the crcut and b) the lnear macrmdel assumn that the OPAMP nput mpedance s nfnty and that the utput mpedance s zer. Anther mprtant bseratn s that the dfferental ltae ( x n F. b) at the OPAMP nput s deally zer wth nfnte an. The reasnn behnd ths bseratn s as fllws: Accrdn t Eq..8, fr a fnte nput snal, the utput ltae s bunded (fnte) f Z s nnzer and Z s nt nfnte. Under these cndtns, and accrdn t Eq..7, the OPAMP nput ltae x s ery small f A s lare enuh. It fllws that the larer the penlp an f the OPAMP, the smaller the snal wll be at ts nput. Therefre, the nputs f the OPAMP can be cnsdered as a rtual shrt crcut. We use the wrd rtual because the ltae dfference between the tw nput termnals ( and ) s ery small but they are nt physcally cnnected. The rtual shrtcrcut prncple s extremely useful n practce because mst f the transfer functns can be easly btaned f ths smplfcatn s used durn the analyss. T llustrate ts use, let us aan cnsder the crcut f F..b. In ths crcut, the nnnertn termnal s runded, and the nertn termnal has the same ltae as the nnnertn termnal due t the rtual shrtcrcut when the OPAMP s penlp an s ery hh. Fr ths reasn, the nertn termnal can be referred t as a rtual rund. Snce x = 0 (rtual shrtcrcut apprxmatn), the nput current becmes = ( x) / Z = / Z. Snce the nput mpedance f the deal OPAMP s nfnte, flws thruhut Z leadn t an utput ltae

5 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba equal t = Z. After substtutn the expressn fr nt the equatn fr, the clsedlp ltae an s btaned as / = Z / Z, whch arees wth Eq..9. If the mpedances Z and Z are replaced by resstrs as shwn n F..a, we end up wth the basc resste nertn amplfer. The clsedlp an fr ths amplfer s. (.0) In the case f the nertn cnfuratn, the mpedance seen by the nput surce ( ) s Z = / =, whch s a result f the rtual rund present at the nertn termnal f the deal OPAMP. Hence, f seeral nertn amplfers are cnnected n cascade, we hae t be aware that the amplfer (nndeal OPAMP) must be able t dre the nput mpedance f the next stae. (a) (b) F... esste feedback amplfers: a) nertn cnfuratn and b) nnnertn cnfuratn. Nnnertn cnfuratn. If the nput snal s appled at the nnnertn termnal and s runded as shwn n F..b, then a cnfuratn wth nnnertn ltae an s btaned. Ntce that the feedback s stll neate. If was cnnected t the pste termnal, the crcut wuld becme unstable and useless fr lnear applcatns, whch wll be elabrated 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 equal. The current flwn thruh and (twards the real rund) s therefre equal t ( 0) / = /. Takn ths bseratn nt accunt, the utput ltae can be expressed as. (.) Accrdn t the abe equatn, the ltae an / s reater r equal than. An mprtant characterstc f the nnnertn cnfuratn s that ts nput mpedance s deally nfnty. Hence, seeral staes can be cnnected n cascade wthut ladn ssues. A specal case f the nnnertn cnfuratn s the buffer cnfuratn depcted n F... The ltae an s unty f = n Eq... In ths case the alue f s nt crtcal and can een be shrtcrcuted ( = 0). Ths tply s als knwn as untyan amplfer (r untyan buffer), and t s useful when drn small mpedances, e.., speakers, mtrs, etc. because the crcut has a hh nput mpedance (t ad ladn the preus stae) and a small utput mpedance. Keep n mnd that the mpedance lkn nt the utput f an OPAMP s deally zer, but actual OPAMPs hae fnte utput mpedances that are lsted n ther datasheets. F.. OPAMP n untyan buffer cnfuratn.

6 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba.. Amplfer wth Multple Inputs and Superpstn. Input snals can be appled t the tw nput termnals f the OPAMP as n F..6 fr example. and mplement a ltae dde such that the nput ltae at the nnnertn ( ) termnal s. (.) If a rtual shrt crcut at the OPAMP nputs s assumed, the ltaes at the nertn and nnnertn termnals are the same. Usn KL at the nertn termnal f the crcut (wth = ) leads t. (.) F..6. Amplfer cnfuratn wth tw nput snals appled t the nnnertn and nertn termnals. The utput ltae can be determned usn Eqs.. and., whch after alebrac rearranement es. (.) Ntce that the utput ltae s a lnear cmbnatn f the tw nput snals. The precedn ndal analyss apprach that ncludes all the nput surces can always be used, but yu can als take adantae f lnear system prpertes such as superpstn t arre at the same result. Applcatn f the superpstn prncple t OPAMP crcuts. If the OPAMP s cnsdered as a lnear dece and nly lnear elements are used fr ts external netwrk, then the utput ltae s a lnear cmbnatn f all nput snals. If seeral nputs are appled t the lnear OPAMP crcut, then the utput can be btaned cnsdern each nddual nput snal ne at a tme (by replacn ltae surces wth a shrt crcut t rund and current surces wth an pen crcut). Mathematcally, the utput s lnear cmbnatn can be wrtten as N K K K,.., K NN (.a) where K K N are the ltae ans frm each nput t the utput. Based n the superpstn prncple, the utput ltae can be btaned by determnn the utput ltae cmpnent enerated by each surce at a tme whle elmnatn all ther nput surces as captured n the fllwn equatn: 6

7 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba,,..,,0..0 0,,..0 0,0,.. N N. (.b) Ths prperty falls under the superpstn prncple that was ntrduced n hapter I. Let us apply ths prncple t the crcut wth tw nputs n F..6. The crcut can be analyzed by applyn ne nput snal at a tme: If s cnsdered, s set t zer as n the equalent crcut f F..7a. Snce the nput mpedance f the OPAMP s nfnte, the current flwn thruh s zer, entaln that = 0. The resultn crcut s the typcal nertn amplfer fr whch the utput ltae s en by ( / ). Ths utput crrespnds t the frst term n Eq... If the frst nput s runded and snal s cnsdered, then the resultn equalent crcut wll respnd as shwn n F..7b. The ltae at the nnnertn termnal s dctated by the ltae dder n Eq... Furthermre, the utput ltae s equal t = ( / ), whch s the an frm the nnnertn nput t the utput f the nnnertn cnfuratn. mbnn ths an wth the ltae dder n Eq.. leads t the secnd term f Eq... (a) (b) F..7 Equalent crcuts fr the cmputatn f the utput ltae usn superpstn: a) fr and b) fr. eneralzatn f basc cnfuratns. The cncepts f nfnte nput mpedance, zer utput mpedance, rtual shrtcrcut at the OPAMP nputs, and lnear superpstn are especally useful when cmplex crcuts are desned. Fr example, an anal nertn adder s shwn n F..8a, fr whch the utput ltae can be fund f we apply the superpstn prncple. The equalent crcut fr the th nput s dsplayed n F..8b. Only s cnnected t, and the resstrs cnnected t the ther nputs are runded snce the ther nput ltaes are replaced wth shrt crcuts t rund. These resstrs are cnnected between the physcal rund and the rtual rund enerated by the neate feedback wth the lare penlp ltae an f the OPAMP. nsequently, the current flwn thruh all runded resstrs n the crcut s zer, such that the nly current flwn thruh f s the current ( = / ) enerated by the ltae drp acrss (frm t the rtual rund nde x = 0). Therefre, the utput ltae s equal t f = ( f / ). By usn the same arument t all nputs, the amplfer s utput ltae can be fund as N f. (.6) An nterestn beneft f the nertn adder s that each f the nput resstrs ndependently cntrls the ltae an fr the crrespndn nput. Ths s an mprtant crcut prperty used fr the desn f analtdtal cnerters, whch s the case when the addtn f bnary wehted snals s requred. 7

8 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba f f N x =0 N N N (a) (b) F..8. a) Anal nertn adder and b) ts equalent crcut fr analyzn the utput ltae due t. An anal nnnertn adder s depcted n F..9a. Smlarly t the preus case, the utput ltae can be fund usn superpstn. F..9b shws the equalent crcut fr the th nput snal wth all ther nputs set t zer. The ltae at the nnnertn termnal s the result f the ltae dder determned by the resstrs lumped t nde as fllws: ( N N X X ). (.7a) f f X X N N (a) (b) F..9. a) Nnnertn adder wth multple nputs and b) ts equalent crcut fr the th nput. N Snce many cmpnents f the nnnertn adder are n parallel, t s ften mre cnenent t use admttances nstead f mpedances fr the analyss f ths type f netwrk. Fr ths example, the preus equatn can als be wrtten as (.7b) N X N X 8

9 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba 9 where = /. The numeratr ( ) s dentfed as the admttance f the element cnnected between the nput snal and. The denmnatr represents the parallel cmbnatn f all elements attached t. Equatn.7b can be expressed wth a shrter equatn: N X (.8) Once s btaned, the utput ltae enerated by can be fund snce s the ltae at the nnnertn termnal; hence, t fnd the utput ltae fr ths nput s strahtfrward as = ( f / ). 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: N N X f N X N f N J f. (.9) In the nnnertn adder cnfuratn, each nput snal has a cntrbutn t the utput ltae that depends n all resstrs unlke the case where nertn tply ccurs. The nput mpedance fr each nput depends n the array f resstrs. Fr nstance, the nput mpedance seen by the th nput snal s X N Z. (.0) Smlar expressns can be btaned fr all nput resstances at the ther surces. Desn Example. Let us cnstruct a crcut that mplements the fllwn equatn: ) ( 0 ) ( 0 t t t Assume that nly supply ltaes f / V are aalable f needed. The crcut n F..0 can be used t realze the abe equatn, but keep n mnd that ths s nt the nly slutn. Fr nstance, an alternate crcut culd be bult usn a cmbnatn f nertn and nnnertn OPAMP crcuts. When usn the crcut n F..0, the desn prcedure cnssts f fndn the prper resstance alues. Thus, accrdn t Eq..9, the fllwn equatns must be sled: 0 0

10 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba Snce the desn space cnssts f three equatns and fe unknwns, tw cndtns can be added t sle the set f equatns. Fr nstance, desn cnsderatns n nse and pwer cnsumptn may requre specfc alues fr sme resstrs. Wthut such restrctns, yu hae mre freedm t chse the resstance alues. F..0. Nnnertn adder example crcut t sum up tw nput snals and a D ltae... Amplfers wth Very Hh an/attenuatn Factrs. Hhan amplfers requre lare spreads f cmpnent alues. In anal nterated crcuts desn, t s dffcult t accurately cntrl such lare rats, and typcally passe cmpnents wth hh alues ccupy lare areas n chps, thereby ncreasn the prduct cst. Frtunately, sme amplfer cnfuratns facltate a reductn n the spread f cmpnent alues. Hhan nertn amplfer usn a resste Tnetwrk. The amplfer n F.. wth a feedback resstr netwrk can be used t ncrease the effecte feedback resstance. The crcut s transfer functn can busly be btaned by usn cnentnal crcut analyss technques such as KL at ndes x and the OPAMP s nertn termnal. But, t s nshtful t analyze the crcut based n the fllwn bseratns:. 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. Hence, at nde x, resstr s n parallel wth when yu lk nt the feedback netwrk frm the utput. Therefre, x depends n the utput ltae based n the fllwn ltae dder: x (.). Als, as a result f the rtual rund at the OPAMP s nput, the current flwn thruh s equal t x / = = /. Nte that the current flwn thruh s equal t the ne flwn thruh because the OPAMP s nfnte nput mpedance preents current flw nt the nertn (and nnnertn) termnal. Substtutn the abe relatns nt the expressn f the current prdes an equatn that relates the utput ltae and nput ltae t each ther n terms f the resstrs: After rearrann Eq.., the ltae an becmes x. (.) 0

11 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba ) (. (.) The abe ltae an s equalent t the an f a twstae amplfer cnsstn f an nertn stae and a nnnertn stae. But the adantae f the crcut n F.. s that the cmbnatn f resstrs permts acqurn lare an factrs wth a smaller dfference between the hhest and lwest resstr alues. x F... esste ltae amplfer fr hhan applcatns. If further reductn n the resstance alue spread s needed, then tw Tnetwrks can be cmbned as shwn n F... Smlar t the preus case, the relatnshp between and x can be dered frm the current flw at the nput: x = ( / ). On the ther end, the ltae, x s als determned by ltae dsn but t s an attenuated ersn f y. Furthermre, ltae y s an attenuated ersn f. Mre specfcally, these ltaes are related by the fllwn expressns: ) ( y x, (.) 6 ] ) [( ) ) (( y. (.) The clsedlp ltae an can be btaned based n the abe relatnshps between the ltaes: 6 x x y y. (.6) Ths ltae an can be ery hh because t s determned by the multplcatn f three terms, makn t equalent t a stae amplfer. Ntce that the nput mpedance f ths nertn crcut s equal t.

12 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba x 6 y F... esste amplfer wth a mre cmplex Tcnfuratn fr lare an factrs. Very lare attenuatn factrs. A Tnetwrk can als be used fr the desn f crcuts wth lare attenuatn factrs, such as 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. As an example, the crcut n F.. uses a Tnetwrk at the nput fr snal attenuatn. The ltae x s an attenuated ersn f the ncmn snal, and the ltae an s adusted t the prper leel by the typcal nertn cnfuratn wth dependence n the resstances at the nertn termnal. Assumn aan a rtual rund at the nertn termnal f the OPAMP, the nput attenuatn factr s determned by and the parallel cmbnatn f and. The ltae an frm x t depends n the rat f resstrs and. Therefre, the utput ltae s en by: x x ( ). (.7) The frst factr n Eq..7 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 Tnetwrk frm F.. fr lare nput attenuatn factrs. It s left t yu t fnd the ltae an f the crcut n F.. wth a duble Tnetwrk at the nput. x F... esste amplfer usn a Tcnfuratn fr lare attenuatn factrs... rcuts: Interatrs and Dfferentatrs. Basc nteratrs. If the feedback resstr f the standard nertn amplfer s replaced by a capactr, then we btan the lssless nteratr shwn n F... In ths crcut, the nput ltae s cnerted nt a current by. As n the case f the resste amplfer, the rtual rund s present at the nertn nput f the OPAMP. Thus, the current thruh s nected nt the feedback capactr where t s nterated. The resultn utput ltae s the nteral f the nected current. The analyss f ths crcut can be perfrmed n the frequency dman wth an mpedance f the capactr equal t /( ). The transfer functn f the nertn cnfuratn s, as n the preus cases, determned by the rat f the mpedance n feedback and the mpedance at the nput, leadn t the fllwn result:

13 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba H( s ) (.8) s where s =. Ths crcut has a ple at the rn (when = 0). The mantude respnse has extremely hh alues at lw frequences, and t decreases wth a frequency at the rate f 0 db/decade. The phase s 90 (70) derees at all frequences. Ntce that the mantude f the ltae an s unty at = /( ). Usually ths cnfuratn s nt used as a standalne crcut, but t s a key buldn blck fr hhrder flters and analtdtal cnerters. F... Lssless nertn nteratr. In eneral, the cmbnatn f resstrs and capactrs leads t the eneratn f ples and zers. Fr example, the crcut mplementatn f a frstrder flter s dsplayed n F..a. Ths crcut s als knwn as a lssy nteratr because the resstr dschares the capactr (.e., ntrduces lsses) whle nects chare nt. The an f ths crcut s determned by the rat f the equalent mpedance n feedback and the mpedance at the nput. In ths crcut, the equalent feedback mpedance s cmpsed f the parallel cmbnatn f the capactr s mpedance (/ ) and. At frequences at whch the mpedance f the capactr can be nred, the lwfrequency an ( / ) depends nly n the rat f the tw resstrs. At ery hh frequences, the mpedance f dmnates the feedback and the crcut behaes as the lssless nteratr shwn n F.. wth a hhfrequency an defned by /s. The erall ltae an f the crcut n F..a at any frequency s en by: H( s ) s. (.9) 0l0 / > 0 db P l u (a) (b) F... Frstrder lwpass flter: a) crcut schematc and b) sketch f ts mantude respnse.

14 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba Equatn.9 cnfrms that the lwfrequency an s determned by the rat f the resstrs. The ple s lcated at = /( ), and fr frequences beynd ths frequency, the ltae an decreases wth a rllff f 0 db/decade. The man dfferences between ths crcut and the passe lwpass flter (ltae dder wth a resstr and a capactr) are twfld: a) n the acte realzatn (wth OPAMP) the lwfrequency an can be reater than unty by adustn the rat f the resstrs, whle n the passe flter the an s always less than ne r equal t unty; b) the OPAMP allws us t cnnect the crcut t the next stae wthut affectn the transfer functn, whch s thankfully a result f the lw utput mpedance f the OPAMP. A typcal mantude respnse btaned wth ths crcut s pltted n F..b. If the untyan frequency s anther mprtant parameter f the crcut n F..a. It can be btaned by takn the mantude (r squared mantude) f Eq..9, and equatn t t. The resultn equatn can be sled fr the untyan frequency, leadn t the fllwn result that yu shuld erfy fr yurself: u. (.0) Fr >>, ths frequency s apprxmately en by u = /. Ntce that fr < the slutn s manary, meann that the untyan frequency des nt exst. In fact, yu cannt fnd any frequency where the an s unty fr an attenuatr (wth D an less than ). Fr <, the lwfrequency an s less than unty, and cnsequently there s n untyan frequency. A eneral frstrder transfer functn can be mplemented by usn the tply n F..6. Snce the elements are cnnected n parallel, t s ery cnenent t fnd the ltae an as the rat f the equalent nput admttance and the equalent feedback admttance as fllws: s s H s. (.) s s Wth ths transfer functn fr the crcut n F..6, yu can desn the fllwn flters: (a) Lwpass flters f s remed. The ple s frequency s en by /( ). (b) Amplfer f and are remed. an = /. (c) Amplfer f the resstrs are remed. an = / (nt ery practcal because ery hh resstance alues are needed, especally fr lwfrequency applcatns) (d) Hhpass f s remed. The ple s frequency s at /( ), and the hh frequency s equal t /. F..6. eneral frstrder flter.

15 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba The hhpass transfer functn can als be realzed f a seres cmbnatn f a capactr and resstr s used at the nput, as shwn n the F..7. Lwfrequency snal 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 dctated by the rat f the resstrs as n the standard nertn cnfuratn. Usn cnentnal crcut analyss technques, the erall transfer functn can be btaned as s H s. (.) s s A D zer and a ple lcated at =/( ) can be bsered frm the abe transfer functn. After the ple s frequency the ltae an appraches /. F..7. Frst rder hhpass flter usn a seres capactr fr nput. Nnnertn nteratr. A nnnertn nteratr can be mplemented wth the crcut n F..8. It s usually an expense mplementatn because the tply requres matched elements (.e., hhprecsn cmpnents). The transfer functn can be btaned by ntn that the ltae at the nnnertn termnal s the result f a ltae dder between and, such that / n = /(s ). The ltae at the nnnertn termnal s then amplfed by a factr f plus the rat f the feedback mpedance /(s ) and the resstr. The resultn transfer functn yelds: s H ( s ). (.) s s Ths crcut behaes as a nnnertn nteratr f =, such that the ltae an decreases wth a rll ff f 0 db/decade when the frequency ncreases and ts phase shft s 90 derees at all frequences. n F..8. Nnnertn lssless nteratr.

16 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba.6. Instrumentatn Amplfers. In many practcal applcatns, t s desrable t use amplfers wth ery hh nput mpedance and ery lw utput mpedance. Ths s the case when the sensrs hae lare utput mpedance r lmted current delern capabltes. Fr thse applcatns, the nertn amplfer based n tw resstrs cannt be used snce ts nput mpedance s fnte;.e., defned by the nput resstr(s). The nly ptn s t use nnnertn amplfers, as the nes shwn n F..9. Here, the ncmn snal s assumed t be dfferental and delered by and ; therefre, tw nnnertn amplfers are used n ths case. 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 n F..9 s cmpsed f tw snleended nnnertn amplfers. Each amplfer s a partcular case f the crcut shwn n F. 7b where = 0, =, = and = 0, leadn t a untyan amplfer (buffer) wth ery hh nput mpedance. Snce the OPAMP utput mpedance s ery small yu can safely cnnect nertn and nnnertn amplfers at the utput f ths structure f requred. The benefts f such buffers wll be edent n the fllwn chapters. F..9. Fullydfferental amplfer based n tw buffers. The tply shwn n F..0 s mre useful and can prde ltae amplfcatn reater than. Its nput mpedance s als ery hh and entrely determned by the nput mpedance f the OPAMP. The analyss f ths crcut s strahtfrward f we take adantae f the rtual shrtcrcut prncple. As anntated n F..0, the ltaes at the tw nertn termnals are equal t and. The current flwn thruh s = ( )/. Ths current flws thruh the resstrs, eneratn a ltae drp f ( ) ( / ) acrss each resstr. The utput ltae s then equal t ( ) ( / ), whle s equal t ( ) ( / ). The dfferental ltae an s therefre en by d d. (.) F..0. Practcal fullydfferental nstrumentatn amplfer nput stae. 6

17 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba 7 Ntce n Eq.. that the utput ltae s als dfferental: d =. Snce the an n ths equatn s defned as the rat f the dfferental utput ltae and the dfferental nput ( d ), t s knwn as the dfferental ltae an. Fr cmmnmde snals ( c = = ) appled at the nput f the crcut f F..0, the ltae dfference acrss s zer, leadn t = 0. As a result, the ltae drp acrss resstrs s zer, and c = = = c. Hence, the dfferental utput ltae d = s zer, ndcatn that the cmmnmde nput snals d nt hae any effect n the crcut s dfferental utput snal. In ther wrds, ths nstrumentatn amplfer nput stae cmpletely suppresses any cmmnmde nput snal cmpnents when the OPAMPs are deal. In many measurement applcatns, cmmnmde nput snals nterfere wth the dfferental measurement, whch s why ths amplfer tply s ery helpful. Fr nstance, the crcut s tlerant t electrmanetc nterferences that equally affect bth nputs. Ntce that when = = c, the cmmnmde snals are present at each amplfer utput. But, cmmnmde snals that are present at the crcut s nput wll nt hae any effect n the crcut s utput after subtractn the tw dfferental utput snals f each amplfer. A ppular snleended nstrumentatn amplfer archtecture s depcted n F... There are tw nputs, and the nfrmatn f nterest s n dfferental frm: d =. A mar adantae f fullydfferental crcuts s that they hae lw senstty t nse and snal nterference that affect bth nputs; e.., snals present at amplfer nputs wth same phase and same mantude. The snleended utput shuld be prprtnal t d, and snals that are present at 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: (.) Usn Eqs.. and., the utput ltae can be btaned as (.6) After sme alebra we et the fllwn result: (.7) If we ntrduce the cndtns =, = 6 and = 7, ths equatn smplfes t the fllwn utput:. (.8) The mprtant prpertes f ths amplfer are:. The nput mpedance s extremely hh and depends n the selected OPAMP. Therefre, t can be easly cnnected t a number f sensrs reardless f the sensr s utput mpedance. Snce the amplfer s nput mpedance s hh, t des nt snfcantly affect the peratn f the sensr when cnnected;.e., ladn effects are aded.

18 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba. The utput ltae s senste t dfferental nput snals d = nly.. mmnmde snals present at bth nput termnals are reected by the dfferental nature f the tply prded that the resstrs are matched. Hence, accurate resstrs wth lw tlerance specfcatns shuld be selected durn the desn f ths type f nstrumentatn amplfer. Wth prper desn, the ablty t reect cmmnmde nse (electrmanetc nterference at bth amplfer nputs fr nstance) s a mar adantae f ths archtecture. 6 7 F... Practcal snleended nstrumentatn amplfer. Desn examples. The fllwn subsectns dscuss seeral crcuts that shw hw OPAMPs can be used fr the mplementatn f anal snal prcessrs..6.. Multple Feedback ndrder Lwpass Flter. 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 flter tply dsplayed n F... Tw feedback paths can be bsered n ths crcut: The frst ne due t the admttance Y, and the secnd ne due t Y. Bth elements prde neate feedback, makn the crcut stable. The ltae an can be fund by sln the ndal equatns wth KL at nde x and the OPAMP s nertn termnal. These equatns can be summarzed n matrx frm: Y Y Y x Y Y Y Y Y Y Y Y. (.9) 0 The slutn f the system f equatns allws us t fnd the utput ltae f the crcut. Ntce that we are nt wrtn any equatn fr the utput nde f the OPAMP. When fndn the slutn f Eq..9, we can set = 0 snce there s a rtual rund at the nput f the OPAMP. Frm nspectn, we can bsere that:. Fr the frst rw, we cnsder the ndal equatn (KL) at nde x. Fr the element, (frst rw and frst clumn) f the admttance matrx we hae t cnsder all the admttances cnnected t x n F.., whch are Y, Y, Y, and Y.. The element, f the matrx asscated wth the frst rw secnd clumn s determned as the neate f the admttance between x and because the secnd nde cnsdered n the admttance matrx s. Hence, the element s lsted as Y.. The element n the frst rw and thrd clumn s the neate f the admttance(s) cnnected between x and, whch s Y n ths case. 8

19 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba. Fr the rht hand sde term, we cnsder the nput ltae and the admttance cnnected between and x, whch s Y.. Fr the secnd rw we cnsdered the ndal equatn (KL) at nde. The matrx term n secnd rw and frst clumn s the neate f the admttance cnnected between and x. 6. The element n secnd rw, secnd clumn s cmpsed f all admttances cnnected t (the nde under cnsderatn fr ths equatn). 7. Fnally, the matrx term n secnd rw and thrd clumn s the neate f the admttances cnnected between and. 8. The secnd rw f the rht hand sde clumn s zer because we d nt hae any elements cnnected between and. Y Y Y x Y Y F... Multple feedback secndrder flter. Anther apprach t analyzn the crcut n F.. s t fnd the transfer functn usn the prpertes f lnear crcuts. Applyn superpstn, x s a lnear cmbnatn f and. Snce the OPAMP s nertn termnal s a rtual rund, x can be expressed as the sum f tw ltae dders: x Y Y. (.0) Y Y Y Y Y Y Y Y Als, ntce that depends n Y, Y and x as fllws: Y x. (.) Y x The ltae an can be fund by sln Eq.. fr x, substtutn the result fr x n Eq..0, and rearrann the expressn: H ( s). (.) Y Y Y Y Y Y Y YY F.. shws a specfc aspect f the crcut called the secndrder lwpass transfer functn. By prperly selectn the admttances, the crcut shwn n F.. behaes as a secndrder lwpass flter. Frm Eq.. t can be dered that ts transfer functn s 9

20 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba H( s ) (.) s s where the admttances Y, Y and Y were replaced by the cnductances f the resstrs, and the substtutns Y = s and Y = s were made. =/ =/ x =/ F... Multple feedback lwpass flter. The basc prpertes f the crcut n F.. becme edent frm ust nterpretn the lcatns f the ples and zers f Eq... There are tw ples defned by the cmpnents n the netwrk external t the OPAMP. If the ples are real, then the mantude respnse wll reman relately flat dependn n the frequency f the dmnant ple. 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 as prtrayed n F... 0 l 0 ( H(s) ) 0 db/decade P P 0 db/decade P P (l) F... Mantude respnse fr a secndrder transfer functn wth tw real ples. Fr better reectn f hhfrequency cmpnents the ples are ften lcated clse t each ther as sualzed by the dashed lne n F... In ths case, the hhfrequency rllff f the mantude respnse s 0 db/decade. Fr the desn f a secndrder lwpass flter t s mre cnenent t express Eq.. as fllws: 0

21 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba s s s s ) s H(, (.a) r s s ) s H(, (.b) The ples f the abe transfer functn can be determned by fndn the rts f the denmnatr, whch are:, P (.) Dependn n the alues f the cmpnents, the ples can be real r cmplex cnuates. The cndtns fr these cases are: f cnuated mplex f eal, P (.6) The phase respnse f Eq.. 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 by ealuatn the transfer functn n Eq.. at s = 0. Each ple ntrduces a phase shft f derees arund ts ple frequency as dscussed n hapter II. If the ples are far away frm each ther, the phase respnse lks lke the ne depcted by the sld lne n F... If the system has the tw ples clse t P, then the system s phase respnse has a rllff f 90 derees/decade arund P due t phase cntrbutn f the tw ples, as exemplfed by the dashed plt n the fure.

22 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba Phase(H(s)) P P derees/decade derees/decade 60 P P P P (l) F... Phase respnse f an nertn secndrder transfer functn: bde apprxmatn..6.. Lwpass Flter Desn Example: D an = 0 db, p = p = 00 Krad/sec. The key desn equatn s the desred flter transfer functn n a smlar frm as Eq..b: H ( s) 0 P 0 P s P s P s Ps P. (.7) Nte that the numeratr 0 P s needed t btan the desred D ltae an f 0 db. The terms f ths equatn are equated ne by ne wth the terms n Eq..b used t btan the fllwn desn cnstrants: / = 0, ω P = (0 ) = /( ), ω P = (0 ) = (/ / / ) /. Let us desn the flter based n pwer cnsumptn cnsderatns. T ad the use f ery small resstrs, whch mples ery lare currents and hh pwer cnsumptn, let us fx the smaller resstr t = 0 k. Furthermre, we can use = fr smplcty. Hence, = = 0 = 00 k, = / (ω P ) = / ( ) = 0 0 F, = (/ / / ) / ( ω P ) = (. 0 ) / ( 0 ) = F. It fllws that = / = (0 0 F ) / ( F) = F. Ths desn s dsplayed n F..6a. The crcut was smulated n PSPIE t ealuate the mantude and phase respnses n F..6b and F..6c. Yu can bsere that the lwfrequency an s 0 db and that the phase shft s 90 at the frequency f the tw ples ( p = p = 00 Krad/sec f p = f p =,9 Hz).

23 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba 00 kω 6.6 pf 0 kω 0.6 nf 00 kω (a) 0 0 Vltae an (db) E E E E E E6 Frequency (Hz) (b) 80 Phase (Derees) 90 0 E E E E E E6 Frequency (Hz) F..6. Secndrder lwpass flter example: a) schematc wth cmpnent alues, b) smulated mantude respnse, and c) smulated phase respnse. The bde apprxmatns are the dashed cures. (c) elatnshp between frequency dman and tme dman. In many cases we are mre nterested n seen the respnse f the crcut n tme dman; e.., mpulse and/r step respnse. An apprach fr the analyss f a crcut n the tme dman s t wrte the ndal r mesh equatns n the tme dman usn the nterdfferental equatns fr capactrs and nductrs. Anther apprach s t btan the transfer functn n the frequency dman, as shwn n the preus examples, and t cnert t nt a dfferental equatn by usn the prpertes f the Laplace transfrm. Amn the many ther prpertes f the Laplace transfrm, ne f the fundamental nes s the fllwn:

24 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba N a s 0 N x a 0 t d x. (.8) Ths prperty f the Laplace transfrm s used fr the cnersn f ratnal lnear functns n the sdman t dfferental equatns n the tme dman. T llustrate ts use, let us cnsder the fllwn sdman (frequency dman) lwpass transfer functn: The abe transfer functn can be rewrtten as n dt s a0 s s bs b0. (.9) s b s b s a s. (.0) 0 0 If the Laplace transfrm prperty frm Eq..8 s appled t bth sdes f ths equatn, the tmedman equalent s btaned leadn t the fllwn secndrder dfferental equatn: n d dt d 0 n. (.) dt t b t b t a t 0 The next step s t sle ths equatn whle takn the type f nput snal nt accunt, whch culd be an mpulse, a pulse r a snusdal nput. It s nt a fcal pnt f ths chapter t dscuss the tme dman analyss f lnear systems, but yu can refer t mre specalzed bks fr detaled analyss methds and examples..6.. Bandpass Transfer Functn Implementatn. Often the nfrmatn t be prcessed s wthn a en pass band; hence, lwpass r hhpass fltern mht nt be the mst effcent apprach fr snal detectn. A bandpass flter s mre sutable fr ths purpse, whch can be btaned f a zer s placed at a lw frequency n addtn t the tw ples f the lwpass transfer functn. The zer can be easly mplemented wth a crcut f t s lcated at = 0. A d example f ths s shwn n Eq.. where the multple feedback transfer functn enerates a lwfrequency zer f ne f the tw elements Y r Y s a capactr and the ther ne s a cnductance. A sutable ptn fr such a bandpass flter realzatn s shwn n F..7. The analyss f the crcut s smlar t the ne used fr the preus lwpass flter, and the transfer functn f ths bandpass flter s H( s ) s s s s s s. (.) The abe transfer functn has the desred zer at D. If the ples are at the same frequency, the mantude and phase respnses can be apprxmated by pecewse lnear functns as depcted n F..8.

25 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba =/ =/ x =/ F..7. Multple feedback bandpass flter. 0 l 0 ( H(s) ) Phase(H(s)) 0 db/decade 0 db/decade 90 derees/decade 80 P P P P derees/decade P P (l) 70 P P P P (l) (a) (b) F..8. nd rder bandpass flter transfer functn: a) mantude respnse and b) phase respnse..7. rcuts wth Partal Pste Feedback..7.. esste Amplfers wth Partal Pste Feedback. Partal pste feedback can als be used fr the mplementatn f hhperfrmance crcuts n applcatns wth demandn specfcatns. Fr nstance, neate resstrs hae t be used fr the desn f ltaecntrlled scllatrs t cancel the effects f resstances asscated wth nductrs and capactrs (due t resste lsses). In crcuts wth partal pste feedback, bth termnals (nertn and nnnertn) are part f feedback lps, whch s exemplfed by the crcut n F..9 where the ltae at the pste termnal s an example f the utput ltae based n the fllwn ltae dder: x. (.) The utput ltae s nfluenced by the cntrbutn f (as n an nertn amplfer wth a ltae an = / ) and x (as n a nnnertn amplfer wth a an f ( / )). Thus, the utput ltae can be expressed as:. (.) x earrann the abe equatn t relate the nput ltae t the utput ltae yelds:

26 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba. (.) The pste feedback f the crcut n F..9 s reflected n the neate term f the denmnatr n the abe equatn. The ltae an can be ery hh f ( ) / ( ( )) s slhtly less than unty. Ntce that the an can ptentally be nfnte, whch n a practcal crcut wuld cause the utput t be stuck at the pste r neate supply ltae leel. The stuatn wth a denmnatr n Eq.. han a alue clse t zer s undesrable because a small aratn n any f the cmpnents has a ery hh mpact n the erall ltae an. Such aratns culd be due t cmpnent manufacturn tlerances, temperature chanes, r cmpnent an effects. Thus, f pste feedback s used, t s d practce t ensure that neate feedback s dmnant and that cmpnent aratns d nt drastcally affect the crcut s perfrmance. x F..9. esste amplfer wth neate and pste feedback..7.. ealzatn f Neate Impedances. The crcut n F..0 uses partal pste feedback snce the resstr 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 the superpstn prncple, x s cmpsed f cntrbutns frm and. The frst cmpnent can be btaned by cnsdern and rundn n the analyss, whch 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. The cmbnatn f these tw cmpnents s: x Z Z. (.6) Z Z The abe expressn fr x can be substtuted nt the nnnertn an relatnshp between x and : Z Z x. (.7) Z Z Wth sme alebra, yu can dere the erall transfer functn as 6

27 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba Z Z. (.8) Z Z Once aan, the pste feedback s reflected n the neate term f the denmnatr. An mprtant specal case ccurs when = and =, such that the preus equatn smplfes t Z Z Z. (.9) The abe transfer functn allws nnnertn amplfcatn. The mst nterestn prperty f the crcut n F..0 s asscated wth the nput mpedance. Frm Eqs..6 and.9, the nput mpedance s btaned fr the case where = and = as shwn n Eq..60. x Z Z Z Z Z Z. (.60) Therefre, the current flwn thruh Z s: Z x. (.6) Z x Z Z Z.0. Amplfer wth partal pste feedback. It can be ntced frm Eq..6 that the current flwn thruh Z depends n but s ndependent f Z. Hence, ths crcut can be cnsdered as a ltaecntrlled current surce: The current s cntrlled by the nput ltae and the resstrs =, and ths current s frced t flw thruh Z. On the ther hand, yu can dere the expressn fr the mpedance at the nput prt yurself and cmpare t wth ths result: 7

28 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba Z Z. (.6) The nput mpedance s pste fr Z <, and neate fr Z >. Thus, f desred, the crcut n F..0 can be desned wth a neate nput mpedance. A useful crcut that s ften emplyed n the desn f flters s the neate mpedance cnerter shwn n F.., whch s a arant f the crcut depcted n F..0. The nput ltae s appled t the nnnertn termnal f the neate mpedance cnerter, and the utput ltae s = ( / ). The nput current s = ( )/Z, leadn t the fllwn expressn f the nput mpedance: Z Z Z. (.6) Ntce that the equalent mpedance at the nput s neate. A neate mpedance means that, cntrary t the case f a pste mpedance, the crcut delers current when pste ltae snals are appled. The reasn fr ths behar s that the OPAMP crcut wth and amplfes the nput snal (wthut nersn) and the utput ltae s reater than r equal t. Hence pste enerates >, and snce the element Z s cnnected between the utput and nput termnals, t enerates a current that flws frm t. Z Z F... Neate mpedance cnerter..7.. SallenKey Flter. Pste feedback has been used fr the desn f flters fr a ln tme. The flter n F.. cnssts f fe admttances and an amplfer wth fnte an K. Snce the amplfer s nnnertn, the feedback prduced by Y s pste. By fllwn the analyss prcedure dscussed earler n ths chapter fr the multple feedback flters, the transfer functn can be btaned by wrtn the admttance matrx as fllws: y y y y y y y y 0 K y 0 0 y 0 0 x n y (.6) 8

29 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba Y Y x Y y K Y Y F... Secndrder SallenKey flter. The frst tw rws n Eq..6 crrespnd t the ndal equatns f ndes x and y, respectely. The thrd rw crrespnds t the amplfer an en by = K y. The slutn f ths system leads t the fllwn transfer functn fr the flter: H( s ) y y y y y y y y y y K K. (.6) Lwpass, bandpass and hhpass flters can be desned based n the abe transfer functn by selectn the prper elements and cmpnent alues. The specal cases are: ) Selectn Y and Y as cnductances, and Y and Y as capacte admttances, whch leads t a lwpass transfer functn, Y can be remed n ths case, resultn n the flter dsplayed n F.. wth the fllwn transfer functn. H ( s) K K (.66) s s Smlarly, t can be shwn that the cndtns belw lead t bandpass and hhpass transfer functns. ) Y and Y shuld be selected as cnductances and Y and Y as capactrs t realze a bandpass flter wth the transfer functn n Eq..6. ) If Y and Y are selected as cnductances, and Y and Y are capactrs, then a hhpass transfer functn s btaned. T practce, yu shuld wrte the transfer functns fr cases ) and ) abe, and draw the schematcs f the asscated crcut mplementatns. T sualze the results, yu can substtute s = ω nt the transfer functns and plt H(ω) s. ω t bsere the mantude respnses. 9

30 Intrductn t Electrnc rcuts: A Desn Apprach Jse SlaMartnez and Marn Onaba K F... Secnd rder SallenKey Flter wth pste feedback..8. Practcal Lmtatns f Operatnal Amplfers. Frst at all, we must recnze that practcal OPAMPs are nt een clse t the deal mdel wth nfnte nput mpedance, nfnte an, nfnte bandwdth, and unlmted utput current capablty and utput ltae rane. The actual parameters and lmtatns depend n the OPAMP tply (arranement and parameters f transstrs, resstrs and capactrs as well as technly used and pwer cnsumptn). There are many dfferent OPAMPs ffered by endrs such as Texas Instruments, Farchld, Natnal Semcnductr, etc. Althuh the specfc rns f OPAMP desn lmtatns are utsde the scpe f ths bk, the effects f these parameters n the erall transfer functn are brefly dscussed n ths sectn. 8.. Amplfer Mdel wth Fnte D an, Fnte Input Impedance and Nnzer Output Impedance. A smewhat mre realstc OPAMP macrmdel s depcted n F... Example ranes fr sme parameter alues f cmmercally aalable OPAMPs are: = MΩ Ω, = 00Ω, and A = V/V (60 0 db). The an usually decreases at a rate f 0 db/decade abe the cutff frequency n the 0 Hz khz rane. These lmtatns ntrduce errrs n the transfer functn. Nrmally, t s cumbersme t ealuate system deradatns wth analytcal equatns, especally fr cmplex crcuts. Here, we wll btan sme results fr a snle nertn amplfer stae, but mst f the cnclusns frm the analyss f ths crcut are als ald fr cmplex crcuts. A V F... An peratnal ltae amplfer wth fnte nput resstance, fnte ltae an, and nnzer utput resstance. Let us cnsder the crcut shwn n F.a and nclude the effects f bth the OPAMP fnte nput mpedance (Z ) and the OPAMP fnte an. Nte, Z s mre eneral than because the nput mpedance s typcally dctated by bth a resste and a capacte part. It s assumed that penlp amplfer an (A ) s fnte but wth nfnte bandwdth. Please keep n mnd that ths s nt a realstc case. The effect f the fnte OPAMP bandwdth s cnsder later n ths chapter. Usn the macrmdel f F.. where = 0 (assumn that << Z F, L ), the equalent crcut can be drawn as shwn n F..b. The transfer functn can be btaned f the ndal equatn at the nertn termnal s wrtten. Snce s cntrlled by the ltaedependent ltae surce [ =A ( )], the utput ltae s entrely defned by the ltae acrss the OPAMP nput termnals and the external mpedance elements. The current demanded by Z F and Z L s prded by the deal ltaecntrlled ltae surce, when can take n any necessary alue t sle the equatns. Hweer, a real OPAMP has a specfed maxmum utput current (that s lsted n the datasheet), and as a cnsequence yu shuld be careful when selectn external resstrs. 0