The Operational Amplifier and Application

Transcription

1 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj The Operatnal Amplfer and Applcatn The peratnal ltage amplfer (mre cmmnly referred t as peratnal amplfer) s ne f the mst useful buldng blcks fr the mplementatn f lw- and medum-frequency analg sgnal prcessrs. The deal peratnal amplfer prcesses a dfferental nput sgnal (at ts nnnertng and nertng nputs) wth ery hgh mpedance at each nput, ery hgh ltage gan, 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 cst-effecte slutns fr the realzatn f analg sgnal prcessng such as amplfcatn, flterng, cmparsns f ltage, etc. A ast arety f peratnal amplfers are ffered by numerus ntegrated crcut endrs. Hence, the selectn f the ptmal peratnal amplfer fr a partcular applcatn s n many cases nt tral. Ths chapter deals wth the fundamental cncepts related t peratnal amplfers. Basc amplfer crcuts wll be studed and analyzed wth frst- and secnd-rder system apprxmatns. - -

2 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj.. Basc Operatnal Amplfer Mdelng. T btan macrmdel parameters f a sngle-nput ltage amplfer wth a sngle utput, let us cnsder a lnear tw-prt system wth tw termnals grunded, as shwn n Fg... Ths chapter presents fur arables (,,, and ) fr study. The nteractn between the fur arables can be defned n many dfferent ways. In real-wrld applcatns, these defntns depend n the nput arable (current r ltage) and the mst releant utput arable. Usually, fr ltage amplfers, the nput sgnal s defned as whle the utput s. Electrnc Crcut Fg... Electrnc crcut represented by a black bx. - -

3 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Snce we are assumng that the crcut s lnear, ne way t descrbe the electrnc crcut s by usng g-parameters n the fllwng matrx representatn: r g g g g g g g g, (.a). (.b) In the abe equatns, the parameter g s the nput cnductance, whch relates the nput current and the nput ltage f the crcut wthut cnsderng the effect f the utput current (.e., when = 0). The crcut s nput cnductance s frmally defned as fllws: g 0. (.) - -

4 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Ths parameter s measured by applyng a ltage ( ) at the nput and measurng the nput current ( ) whle the utput nde s left pen such that = 0. The parameter g defnes the reerse current gan f a tplgy, and t s defned as g 0. (.) The reerse current gan g s the current generated at the nput due t the utput current. In the deal case ths parameter s zer fr ltage amplfers, whch we usually lke t be undrectnal, such that the nput sgnal generates an utput sgnal. T measure ths parameter, ne must shrt-crcut the nput prt such that = 0, t apply a current at the utput, and t measure the current generated at the nput prt. In practcal crcuts, ths parameter s ery small and s ften gnred. - -

5 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj The frward ltage gan s defned as the rat f the utput ltage and nput ltage wth an pen-crcut at the utput: g A V 0. (.) Parameter g = A V s certanly ne f the mst mprtant parameters f the tw-prt system. We als refer t A V as the pen-crcut gan (r pen-lp gan when the feedback f a crcut s als remed). It represents the crcut s ltage gan wthut any lad mpedance attached at the utput, resultng the n zer utput current. Anther mprtant parameter s the system s utput mpedance, whch relates the utput ltage and the utput current wthut the effects f the nput sgnal. It s defned as g 0. (.5) - 5 -

6 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj A tw-prt system can be represented by the fur afrementned parameters, whch are captured by the schematc f the macrmdel n Fg..a. Fg... Lnear macrmdels usng hybrd parameters: a typcal ltage amplfer and Mdel fr an OPAMP. The deal OPAMP s a dece that can be mdeled by usng the crcut f Fg..b, whch was btaned frm the ne n Fg..a wth A V =, g = 0, =, and = 0. Input + s cmmnly knwn as nn-nertng nput, and the ther nput ( - ) as nertng nput

7 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Basc Cnfguratns: Inertng and Nn-nertng Amplfers. Inertng cnfguratn. The smplfed lnear macrmdel f the OPAMP s used here fr the representatn f the nertng amplfer. The equalent crcut s shwn n Fg..b. By usng basc crcut analyss technques, t can be fund that 0 x x, (.6) x 0 A. (.7) Slng these equatns as a functn f the nput and utput ltages yelds A V. (.8) If the pen-lp gan f the OPAMP A V s ery large, then

8 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj. (.9) Ths result shws that f negate feedback s used and f the pen-lp gan f the OPAMP s large enugh, then the erall clsed-lp ltage gan f the amplfer depends n the rat f the feedback and nput mpedances. Unlke the pen-lp gan f the OPAMP that can ary by mre than 50% due t transstr parameters aratns and temperature changes, the clsed-lp gan s mre accurate, especally f same type f mpedances are used. Nrmally, the rat f mpedances s sgnfcantly mre precse than the abslute alues f cmpnents. Bth, rats f resstrs and rats f capactrs fabrcated n the same ntegrated crcut can hae msmatch errrs as lw as % Abslute alues f the cmpnents may ary by mre than 0%. Thus, desgnng amplfers wth feedback leads t rbust gans n the presence f manufacturng aratns

9 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj (a) (b) Fg... Inertng amplfer: a) schematc f the crcut and b) the lnear macrmdel assumng that the OPAMP nput mpedance s nfnty and that the utput mpedance s zer. Anther mprtant bseratn s that the dfferental ltage ( x n Fg. b) at the OPAMP nput s deally zer wth nfnte gan. A 0 r The OPAMP nput ltage x s ery small f A s large enugh. It fllws that the larger the pen-lp gan f the OPAMP, the smaller the sgnal wll be at ts nput. x x A - 9 -

10 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj The nputs f the OPAMP can be cnsdered as a rtual shrt crcut. We use the wrd rtual because the ltage dfference between the tw nput termnals ( + and - ) s ery small but they are nt physcally cnnected. In ths crcut, the nn-nertng termnal s grunded, and the nertng termnal has the same ltage as the nn-nertng termnal due t the rtual shrt-crcut when the OPAMP s pen-lp gan s ery hgh. Snce x = 0 (rtual shrt-crcut apprxmatn), the nput current becmes = ( - x ) / = /. Snce the nput mpedance f the deal OPAMP s nfnte, flws thrughut leadng t an utput ltage equal t =. Hence, the clsed-lp ltage gan s / = - /, whch agrees wth Eq..9.. (.0) (a) (b) - 0 -

11 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg... esste feedback amplfers: a) nertng cnfguratn and b) nn-nertng cnfguratn. Nn-nertng cnfguratn.the feedback s stll negate. If was cnnected t the pste termnal, the crcut wuld becme unstable and useless fr lnear applcatns, whch wll be elabrated n the fllwng sectns. The clsed-lp ltage gan f the nn-nertng cnfguratn can be easly btaned f the rtual shrt prncple s used. Due t the hgh gan f the OPAMP, the ltage dfference between the nertng and nn-nertng termnals s ery small. Hence, the ltage at the nn-nertng termnal f the OPAMP s equal. The current flwng thrugh and (twards the real grund) s therefre equal t ( - - 0) / = /. Takng ths bseratn nt accunt, the utput ltage can be expressed as. (.) Accrdng t the abe equatn, the ltage gan / s greater r equal than. - -

12 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj An mprtant characterstc f the nn-nertng cnfguratn s that ts nput mpedance s deally nfnty. Hence, seeral stages can be cnnected n cascade wthut ladng ssues. A specal case f the nn-nertng cnfguratn s the buffer cnfguratn depcted n Fg..5. Fg..5 OPAMP n unty-gan buffer cnfguratn. - -

13 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj.. Amplfer wth Multple Inputs and Superpstn. and mplement a ltage dde such that the nput ltage at the nn-nertng ( + ) termnal s. (.) If a rtual shrt crcut at the OPAMP nputs s assumed, usng KCL at the nertng termnal f the crcut (wth - = + ) leads t. (.) Fg..6. Amplfer cnfguratn wth tw nput sgnals appled t the nn-nertng and nertng termnals. - -

14 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj The utput ltage can be determned usng Eqs.. and., whch after algebrac rearrangement ges. (.) 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 ltage s a lnear cmbnatn f all nput sgnals. If seeral nputs are appled t the lnear OPAMP crcut, then the utput can be btaned cnsderng each nddual nput sgnal ne at a tme (by replacng ltage surces wth a shrt crcut t grund and current surces wth an pen crcut). Mathematcally, the utput s lnear cmbnatn can be wrtten as - -

15 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj K K,.., K N N N K jj j (.5a) where K K N are the ltage gans frm each nput t the utput. Then:,,..,,0..0 0,,..0 0,0,.. N N. (.5b) Let us apply ths prncple t the crcut wth tw nputs n Fg..6. The crcut can be analyzed by applyng ne nput sgnal at a tme: If s cnsdered, s set t zer as n the equalent crcut f Fg..7a

16 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj (a) (b) Fg..7 Equalent crcuts fr the cmputatn f the utput ltage usng superpstn: a) fr and b) fr

17 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj - 7 -

18 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Generalzatn f basc cnfguratns. N j f j j. (.6) j f j j x =0 - + (a) (b) Fg..8. a) Analg nertng adder and b) ts equalent crcut fr analyzng the utput ltage due t j. N An analg nn-nertng adder s depcted n Fg..9a. Smlarly t the preus case, the utput ltage can be fund usng superpstn. Fg..9b shws the equalent crcut fr the j th -nput sgnal wth all ther nputs set t zer

19 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj j j j ( j j j N N X X ) j. (.7a) (a) (b) Fg..9. a) Nn-nertng adder wth multple nputs and b) ts equalent crcut fr the j th nput. Snce many cmpnents f the nn-nertng 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 - 9 -

20 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj j j j j j j N X g j g g g... g j j g j... g N g X (.7b) where g = /. The numeratr (g j ) s dentfed as the admttance f the element cnnected between the nput sgnal and +j. The denmnatr represents the parallel cmbnatn f all elements attached t +. Equatn.7b can be expressed wth a shrter equatn: j j N g g j g X (.8) Once +j s btaned, the utput ltage generated by j can be fund snce +j s the ltage at the nn-nertng termnal; hence, t fnd the utput ltage fr ths nput s straghtfrward as j = (+ f / ) +j. Takng nt accunt all the nput sgnals and applyng the superpstn prncple, t can be shwn that the erall utput ltage s a lnear cmbnatn f all nputs: - 0 -

21 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj - - N j j j N X f N j X N j j f N J j f g g g g g g. (.9) In the nn-nertng adder cnfguratn, each nput sgnal has a cntrbutn t the utput ltage that depends n all resstrs unlke the case where nertng tplgy ccurs. The nput mpedance fr each nput depends n the array f resstrs. Fr nstance, the nput mpedance seen by the j th -nput sgnal s X N j j j j. (.0) Smlar expressns can be btaned fr all nput resstances at the ther surces.

22 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj - - Desgn Example. Let us cnstruct a crcut that mplements the fllwng equatn: 5 ) ( 0 ) ( 0 t t t Assume that nly supply ltages f +/-5 V are aalable f needed. The crcut n Fg..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 usng a cmbnatn f nertng and nn-nertng OPAMP crcuts. When usng the crcut n Fg..0, the desgn prcedure cnssts f fndng the prper resstance alues. Thus, accrdng t Eq..9, the fllwng equatns must be sled: g g g g g g g g g g g g Snce the desgn space cnssts f three equatns and fe unknwns, tw cndtns can be added t sle the set f equatns. Fr nstance, desgn cnsderatns n nse and

23 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj pwer cnsumptn may requre specfc alues fr sme resstrs. Wthut such restrctns, yu hae mre freedm t chse the resstance alues. Fg..0. Nn-nertng adder example crcut t sum up tw nput sgnals and a DC ltage. Sle t yurself! Fnd the expressn fr V: Fg. 0 s a Very gd test sample! - -

24 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj.. Amplfers wth Very Hgh Gan/Attenuatn Factrs. The amplfer n Fg.. wth a feedback resstr netwrk can be used t ncrease the effecte feedback resstance. Obseratns:. Snce the nertng termnal s at the grund ptental due t the rtual shrt at the OPAMP nput, the resstr s cnnected between nde x and (rtual) grund. x (.). Als, as a result f the rtual grund at the OPAMP s nput, the current flwng thrugh s equal t x / = - = - /.. Substtutng the abe relatns nt the expressn f the current prdes an equatn that relates the utput ltage and nput ltage t each ther n terms f the resstrs: - -

25 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj x. (.) After rearrangng Eq.., the ltage gan becmes ) (. (.) x - + Fg... esste ltage amplfer fr hgh-gan applcatns

26 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Furthermre, ltage y s an attenuated ersn f. Mre specfcally, these ltages are related by the fllwng expressns: ) ( y x, (.) ] ) [( ) ) (( y. (.5) The clsed-lp ltage gan can be btaned based n the abe relatnshps between the ltages: x x y y. (.6) Ths ltage gan can be ery hgh because t s determned by the multplcatn f three terms, makng t equalent t a -stage amplfer. Ntce that the nput mpedance f ths nertng crcut s equal t.

27 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg... esste amplfer wth a mre cmplex T-cnfguratn fr large gan factrs. Very large attenuatn factrs. The crcut n Fg.. uses a T-netwrk at the nput fr sgnal attenuatn. The ltage x s an attenuated ersn f the ncmng sgnal, and the ltage gan s adjusted t the prper leel by the typcal nertng cnfguratn wth dependence n the resstances at the nertng termnal. Assumng agan a rtual grund at the nertng termnal f the OPAMP, the nput attenuatng factr s determned by and the parallel cmbnatn f and. The ltage gan frm x t depends n the rat f resstrs and. Therefre, the utput ltage s gen by: - 7 -

28 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj ( ) x x. (.7) The frst factr n Eq..7 s the result f the ltage dder at the nput f the structure, whle the secnd factr s the result f the nn-nertng amplfcatn f x. Yu can als use the duble T-netwrk frm Fg.. fr large nput attenuatn factrs. It s left t yu t fnd the ltage gan f the crcut n Fg.. wth a duble T-netwrk at the nput. Fg... esste amplfer usng a T-cnfguratn fr large attenuatn factrs.

29 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj.5. C Crcuts: Integratrs and Dfferentatrs. Basc ntegratrs. If the feedback resstr f the standard nertng amplfer s replaced by a capactr, then we btan the lssless ntegratr shwn n Fg... The analyss f ths crcut can be perfrmed n the frequency dman wth an mpedance f the capactr equal t /(jc ). The transfer functn f the nertng cnfguratn s, as n the preus cases, determned by the rat f the mpedance n feedback and the mpedance at the nput, leadng t the fllwng result: H( s ) sc (.8) where s = j. Ths crcut has a ple at the rgn (when = 0). The magntude respnse has extremely hgh alues at lw frequences, and t decreases wth a frequency at the rate f - 0 db/decade. The phase s +90 (-70) degrees at all frequences. Ntce that the magntude f the ltage gan s unty at = /( C ). Usually ths cnfguratn s nt used as a standalne crcut, but t s a key buldng blck fr hgh-rder flters and analg-t-dgtal cnerters

30 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg... Lssless nertng ntegratr. In general, the cmbnatn f resstrs and capactrs leads t the generatn f ples and zers. Fr example, the crcut mplementatn f a frst-rder flter s dsplayed n Fg..5a. Ths crcut s als knwn as a lssy ntegratr because the resstr dscharges the capactr (.e., ntrduces lsses) whle njects charge nt C. The gan 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 (/jc ) and. At frequences at whch the mpedance f the capactr can be gnred, the lw-frequency gan (- / ) depends nly n the rat f the tw resstrs. At ery hgh frequences, the mpedance f C dmnates the feedback and the crcut behaes as the lssless ntegratr shwn n Fg

31 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj. wth a hgh-frequency gan defned by -/s C. The erall ltage gan f the crcut n Fg..5a at any frequency s gen by: H( s ) sc. (.9) 0lg0 / > (a) (b) Fg..5. Frst-rder lw-pass flter: a) crcut schematc and b) sketch f ts magntude respnse. 0 db P C u lg - -

32 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Equatn.9 cnfrms that the lw-frequency gan s determned by the rat f the resstrs. The ple s lcated at = /( C ), and fr frequences beynd ths frequency, the ltage gan decreases wth a rllff f 0 db/decade. The man dfferences between ths crcut and the passe lw-pass flter (ltage dder wth a resstr and a capactr) are twfld: a) n the acte realzatn (wth OPAMP) the lw-frequency gan can be greater than unty by adjustng the rat f the resstrs, whle n the passe flter the gan s always less than ne r equal t unty; b) the OPAMP allws us t cnnect the crcut t the next stage wthut affectng the transfer functn, whch s thankfully a result f the lw utput mpedance f the OPAMP. A typcal magntude respnse btaned wth ths crcut s pltted n Fg..5b. If the unty-gan frequency s anther mprtant parameter f the crcut n Fg..5a. It can be btaned by takng the magntude (r squared magntude) f Eq..9, and equatng t t. The resultng equatn can be sled fr the unty-gan frequency, leadng t the fllwng result that yu shuld erfy fr yurself: - -

33 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj u. (.0) C Fr >>, ths frequency s apprxmately gen by u = / C. Ntce that fr < the slutn s magnary, meanng that the unty-gan frequency des nt exst. In fact, yu cannt fnd any frequency where the gan s unty fr an attenuatr (wth DC gan less than ). Fr <, the lw-frequency gan s less than unty, and cnsequently there s n unty-gan frequency. - -

34 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj A general frst-rder transfer functn can be mplemented by usng the tplgy n Fg..6. Snce the elements are cnnected n parallel, t s ery cnenent t fnd the ltage gan as the rat f the equalent nput admttance and the equalent feedback admttance as fllws: g sc sc H s g sc sc. (.) Wth ths transfer functn fr the crcut n Fg..6, yu can desgn the fllwng flters: (a) Lw-pass flters f C s remed. The ple s frequency s gen by /( C ). (b) Amplfer f C and C are remed. Gan = - /. (c) Amplfer f the resstrs are remed. Gan = - C /C (nt ery practcal because ery hgh resstance alues are needed, especally fr lwfrequency applcatns) (d) Hgh-pass f s remed. The ple s frequency s at /( C ), and the hgh frequency s equal t -C /C. - -

35 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg..6. General frst-rder flter. C The hgh-pass transfer functn can als be realzed f a seres cmbnatn f a capactr and resstr s used at the nput, as shwn n the Fg..7. Lw-frequency sgnal cmpnents are blcked by the capactr due t ts hgh mpedance at lw frequences. At hgh frequences, the capactr behaes as a shrt crcut, and the gan s dctated by the rat f the resstrs as n the standard nertng cnfguratn. Usng cnentnal crcut analyss technques, the erall transfer functn can be btaned as C - + H s sc s C sc. (.) A DC zer and a ple lcated at =/( C ) can be bsered frm the abe transfer functn. After the ple s frequency the ltage gan appraches - /

36 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg..7. Frst rder hgh-pass flter usng a seres capactr fr nput. Nn-nertng ntegratr. A nn-nertng ntegratr can be mplemented wth the crcut n Fg..8. It s usually an expense mplementatn because the tplgy requres matched elements (.e., hgh-precsn cmpnents). The transfer functn can be btaned by ntng that the ltage at the nn-nertng termnal s the result f a ltage dder between and C, such that + / n = /(+s C ). The ltage at the nnnertng termnal s then amplfed by a factr f plus the rat f the feedback mpedance /(sc ) and the resstr. The resultng transfer functn yelds: ) s C sc sc H ( s. (.) - 6 -

37 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Ths crcut behaes as a nn-nertng ntegratr f C = C, such that the ltage gan decreases wth a rll ff f -0 db/decade when the frequency ncreases and ts phase shft s -90 degrees at all frequences. Fg..8. Nn-nertng lssless ntegratr

38 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj.6. Instrumentatn Amplfers. In many practcal applcatns, t s desrable t use amplfers wth ery hgh nput mpedance and ery lw utput mpedance. Ths s the case when the sensrs hae large utput mpedance r lmted current delerng capabltes. Fr thse applcatns, the nertng 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 nn-nertng amplfers, as the nes shwn n Fg..9. Here, the ncmng sgnal s assumed t be dfferental and delered by and ; therefre, tw nn-nertng amplfers are used n ths case. In a dfferental system, the nfrmatn s determned by the ltage dfference between the tw nputs ( - ) rather than by the ltage at each nde. The crcut n Fg..9 s cmpsed f tw sngle-ended nn-nertng amplfers. Each amplfer s a partcular case f the crcut shwn n Fg. 7b where = 0, =, = and = 0, leadng t a unty-gan amplfer (buffer) wth ery hgh nput mpedance. Snce the OPAMP utput mpedance s ery small yu can safely cnnect nertng and nn-nertng amplfers at the utput f ths structure f requred. The benefts f such buffers wll be edent n the fllwng chapters

39 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg..9. Fully-dfferental amplfer based n tw buffers

40 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj The tplgy shwn n Fg..0 s mre useful and can prde ltage amplfcatn greater than. Its nput mpedance s als ery hgh and entrely determned by the nput mpedance f the OPAMP. The analyss f ths crcut s straghtfrward f we take adantage f the rtual shrt-crcut prncple. As anntated n Fg..0, the ltages at the tw nertng termnals are equal t and. The current flwng thrugh s = (-)/. Ths current flws thrugh the resstrs, generatng a ltage drp f (-) ( /) acrss each resstr. The utput ltage s then equal t + (-) (/), whle s equal t - (-) (/). The dfferental ltage gan s therefre gen by d d. (.) Fg..0. Practcal fully-dfferental nstrumentatn amplfer nput stage

41 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Ntce n Eq.. that the utput ltage s als dfferental: d = -. Snce the gan n ths equatn s defned as the rat f the dfferental utput ltage and the dfferental nput ( d ), t s knwn as the dfferental ltage gan. Fr cmmn-mde sgnals ( c = = ) appled at the nput f the crcut f Fg..0, the ltage dfference acrss s zer, leadng t = 0. As a result, the ltage drp acrss resstrs s zer, and c = = = c. Hence, the dfferental utput ltage d = - s zer, ndcatng that the cmmn-mde nput sgnals d nt hae any effect n the crcut s dfferental utput sgnal. In ther wrds, ths nstrumentatn amplfer nput stage cmpletely suppresses any cmmn-mde nput sgnal cmpnents when the OPAMPs are deal. In many measurement applcatns, cmmn-mde nput sgnals nterfere wth the dfferental measurement, whch s why ths amplfer tplgy s ery helpful. Fr nstance, the crcut s tlerant t electrmagnetc nterferences that equally affect bth nputs. Ntce that when = = c, the cmmn-mde sgnals are present at each amplfer utput. But, cmmn-mde sgnals that are present at the crcut s nput wll nt hae any effect n the crcut s utput after subtractng the tw dfferental utput sgnals f each amplfer. A ppular sngle-ended nstrumentatn amplfer archtecture s depcted n Fg... There are tw nputs, and the nfrmatn f nterest s n dfferental frm: d = -. A majr adantage f fully-dfferental crcuts s that they hae lw senstty t nse - -

42 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj - - and sgnal nterference that affect bth nputs; e.g., sgnals present at amplfer nputs wth same phase and same magntude. The sngle-ended utput shuld be prprtnal t d, and sgnals that are present at bth nputs wth same ampltude and same phase are cancelled by the dfferental nature f the amplfer. Applyng the superpstn prncple t the crcut shwn n Fg.., t can be shwn that the utput ltage s gen by a lnear cmbnatn f and as fllws: (.5) Usng Eqs.. and.5, the utput ltage can be btaned as (.6) After sme algebra we get the fllwng result: (.7)

43 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj If we ntrduce the cndtns =, = 6 and 5 = 7, ths equatn smplfes t the fllwng utput: 5. (.8) The mprtant prpertes f ths amplfer are:. The nput mpedance s extremely hgh and depends n the selected OPAMP. Therefre, t can be easly cnnected t a number f sensrs regardless f the sensr s utput mpedance. Snce the amplfer s nput mpedance s hgh, t des nt sgnfcantly affect the peratn f the sensr when cnnected;.e., ladng effects are aded.. The utput ltage s senste t dfferental nput sgnals d = - nly.. Cmmn-mde sgnals present at bth nput termnals are rejected by the dfferental nature f the tplgy prded that the resstrs are matched. Hence, accurate resstrs wth lw tlerance specfcatns shuld be selected durng the desgn f ths type f nstrumentatn amplfer. Wth prper desgn, the ablty t reject cmmn-mde nse (electrmagnetc nterference at bth amplfer nputs fr nstance) s a majr adantage f ths archtecture. - -

44 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg... Practcal sngle-ended nstrumentatn amplfer. - -

45 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj.6.. Multple Feedback nd-rder Lw-pass Flter. Flters are used n electrncs fr the selectn f nfrmatn that s lcated n a specfc frequency band. A ppular structure s the s-called multple feedback flter tplgy dsplayed n Fg... Tw feedback paths can be bsered n ths crcut: The frst ne due t the admttance Y, and the secnd ne due t Y 5. Bth elements prde negate feedback, makng the crcut stable. The ltage gan can be fund by slng the ndal equatns wth KCL at nde x and the OPAMP s nertng termnal. These equatns can be summarzed n matrx frm: Y Y Y 5 Y 0 x Y Y Y Y Y Y Y5. (.9) The slutn f the system f equatns allws us t fnd the utput ltage f the crcut. Ntce that we are nt wrtng any equatn fr the utput nde f the OPAMP. When fndng the slutn f Eq..9, we can set - = 0 snce there s a rtual grund at the nput f the OPAMP. Frm nspectn, we can bsere that: - 5 -

46 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj. Fr the frst rw, we cnsder the ndal equatn (KCL) 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 Fg.., whch are Y, Y, Y, and Y.. The element, f the matrx asscated wth the frst rw secnd clumn s determned as the negate 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 negate f the admttance(s) cnnected between x and, whch s -Y n ths case.. Fr the rght hand sde term, we cnsder the nput ltage and the admttance cnnected between and x, whch s Y. 5. Fr the secnd rw we cnsdered the ndal equatn (KCL) at nde -. The matrx term n secnd rw and frst clumn s the negate 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 negate f the admttances cnnected between - and. 8. The secnd rw f the rght hand sde clumn s zer because we d nt hae any elements cnnected between - and

47 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg... Multple feedback secnd-rder flter. x Y Y Y Y Y Y Y Y Y Y. (.0) Als, ntce that depends n Y 5, Y and x as fllws: x x Y Y 5 5. (.) 5 ) ( Y Y Y Y Y Y Y Y Y s H. (.)

48 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg.. shws a specfc aspect f the crcut called the secnd-rder lwpass transfer functn. By prperly selectng the admttances, the crcut shwn n Fg.. behaes as a secnd-rder lwpass flter. Frm Eq.. t can be dered that ts transfer functn s H( s ) s C 5 C sc 5 G G (.) G G G GG where the admttances Y, Y and Y were replaced by the cnductances f the resstrs, and the substtutns Y = sc and Y 5 = sc 5 were made. Fg... Multple feedback lwpass flter

49 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj The basc prpertes f the crcut n Fg.. becme edent frm just nterpretng 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 magntude respnse wll reman relately flat dependng n the frequency f the dmnant ple. Fr frequences abe the frst (dmnant) ple, the magntude 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 Fg... Fg... Magntude respnse fr a secnd-rder transfer functn wth tw real ples. Fr better rejectn f hgh-frequency cmpnents the ples are ften lcated clse t each ther as sualzed by the dashed lne n Fg... In ths case, the hgh-frequency - 9 -

50 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj rllff f the magntude respnse s -0 db/decade. Fr the desgn f a secnd-rder lwpass flter t s mre cnenent t express Eq.. as fllws: C C C s s C C C C G G C G G G s s C C G G ) s H(, (.a) r 5 5 C C C s s C C ) s H(, (.b) The ples f the abe transfer functn can be determned by fndng the rts f the denmnatr, whch are:

51 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj G G G C C G G C G G G C G G G C C G G C G G G, P (.5) Dependng n the alues f the cmpnents, the ples can be real r cmplex cnjugates. The cndtns fr these cases are: 5 5 G G G C C G G f cnjugated Cmplex G G G C C G G f eal, P (.6) The phase respnse f Eq.. s als mprtant fr the full characterzatn f the lwpass flter. The nertng flter cnfguratn has a phase shft f -80 degrees at ery

52 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj lw frequences. Ths can be erfed by ealuatng the transfer functn n Eq.. at s = 0. Each ple ntrduces a phase shft f -5 degrees arund ts ple frequency as dscussed n Chapter II. If the ples are far away frm each ther, the phase respnse lks lke the ne depcted by the sld lne n Fg..5. If the system has the tw ples clse t P, then the system s phase respnse has a rllff f -90 degrees/decade arund P due t phase cntrbutn f the tw ples, as exemplfed by the dashed plt n the fgure. Fg..5. Phase respnse f an nertng secnd-rder transfer functn: bde apprxmatn

53 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj.6.. Lwpass Flter Desgn Example: DC Gan = 0 db, p = p = 00 Krad/sec. The key desgn equatn s the desred flter transfer functn n a smlar frm as Eq..b: H ( s) 0 P 0 P. (.7) s P s P s Ps P Nte that the numeratr 0 P s needed t btan the desred DC ltage gan f 0 db. The terms f ths equatn are equated ne by ne wth the terms n Eq..b used t btan the fllwng desgn cnstrants: / = 0, ω P = (0 5 ) = /( C C 5 ), ω P = (0 5 ) = (/ +/ +/ ) / C. Let us desgn the flter based n pwer cnsumptn cnsderatns. T ad the use f ery small resstrs, whch mples ery large currents and hgh pwer cnsumptn, let us fx the smaller resstr t = 0 k. Furthermre, we can use = fr smplcty. Hence, - 5 -

54 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj = = 0 = 00 k, C C 5 = / (ω P ) = / ( ) = 0-0 F, C = (/ +/ +/ ) / ( ω P ) = (. 0 - ) / ( 0 5 ) = F. It fllws that C 5 = C C 5 / C = (0-0 F ) / ( F) = F. Ths desgn s dsplayed n Fg..6a. The crcut was smulated n PSPICE t ealuate the magntude and phase respnses n Fg..6b and Fg..6c. Yu can bsere that the lw-frequency gan 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 = 5,95 Hz)

55 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj 00 kω 6.6 pf 0 kω 0.6 nf 00 kω + - (a) 0 0 Vltage Gan (db) E+ E+ E+ E+ E+5 E+6 Frequency (Hz) (b)

56 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj 80 Phase (Degrees) E+ E+ E+ E+ E+5 E+6 Frequency (Hz) (c) Fg..6. Secnd-rder lwpass flter example: a) schematc wth cmpnent alues, b) smulated magntude respnse, and c) smulated phase respnse. The bde apprxmatns are the dashed cures. elatnshp between frequency dman and tme dman. In many cases we are mre nterested n seeng the respnse f the crcut n tme dman; e.g., 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 usng the ntegr-dfferental 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

57 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj equatn by usng the prpertes f the Laplace transfrm. Amng the many ther prpertes f the Laplace transfrm, ne f the fundamental nes s the fllwng: N a s 0 N x a 0 t d x. (.8) dt Ths prperty f the Laplace transfrm s used fr the cnersn f ratnal lnear functns n the s-dman t dfferental equatns n the tme dman. T llustrate ts use, let us cnsder the fllwng s-dman (frequency dman) lwpass transfer functn: n s a0 s s bs b0. (.9) The abe transfer functn can be rewrtten as s b s b s a s. (.50) 0 0 n

58 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj If the Laplace transfrm prperty frm Eq..8 s appled t bth sdes f ths equatn, the tme-dman equalent s btaned leadng t the fllwng secnd-rder dfferental equatn: d dt d dt t b t b t a t. (.5) 0 0 n The next step s t sle ths equatn whle takng the type f nput sgnal 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 gen pass band; hence, lwpass r hgh-pass flterng mght nt be the mst effcent apprach fr sgnal detectn. A band-pass 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 gd example f ths s shwn n Eq.. where the multple feedback transfer functn generates a lw-frequency zer f ne f the tw elements Y r Y s a capactr and the ther ne s a cnductance. A sutable ptn fr

59 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj such a band-pass flter realzatn s shwn n Fg..7. The analyss f the crcut s smlar t the ne used fr the preus lwpass flter, and the transfer functn f ths band-pass flter s C C G G G C C C C G s s s C G G G G C C sg C C s sg C ) s H(. (.5) The abe transfer functn has the desred zer at DC. If the ples are at the same frequency, the magntude and phase respnses can be apprxmated by pece-wse lnear functns as depcted n Fg..8. Fg..7. Multple feedback band-pass flter.

60 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj (a) (b) Fg..8. nd -rder bandpass flter transfer functn: a) magntude respnse and b) phase respnse..7. Crcuts wth Partal Pste Feedback..7.. esste Amplfers wth Partal Pste Feedback. Partal pste feedback can als be used fr the mplementatn f hgh-perfrmance crcuts n applcatns wth demandng specfcatns. Fr nstance, negate resstrs hae t be used fr the desgn f ltage-cntrlled scllatrs t cancel the effects f resstances asscated wth nductrs and capactrs (due t resste lsses). In crcuts wth partal pste

61 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj feedback, bth termnals (nertng and nn-nertng) are part f feedback lps, whch s exemplfed by the crcut n Fg..9 where the ltage at the pste termnal s an example f the utput ltage based n the fllwng ltage dder: x. (.5) The utput ltage s nfluenced by the cntrbutn f (as n an nertng amplfer wth a ltage gan = - / ) and x (as n a nn-nertng amplfer wth a gan f (+ / )). Thus, the utput ltage can be expressed as: x. (.5) earrangng the abe equatn t relate the nput ltage t the utput ltage yelds:. (.55)

62 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj The pste feedback f the crcut n Fg..9 s reflected n the negate term f the denmnatr n the abe equatn. The ltage gan can be ery hgh f ( + ) / ( ( + )) s slghtly less than unty. Ntce that the gan can ptentally be nfnte, whch n a practcal crcut wuld cause the utput t be stuck at the pste r negate supply ltage leel. The stuatn wth a denmnatr n Eq..55 hang a alue clse t zer s undesrable because a small aratn n any f the cmpnents has a ery hgh mpact n the erall ltage gan. Such aratns culd be due t cmpnent manufacturng tlerances, temperature changes, r cmpnent agng effects. Thus, f pste feedback s used, t s gd practce t ensure that negate feedback s dmnant and that cmpnent aratns d nt drastcally affect the crcut s perfrmance. Fg..9. esste amplfer wth negate and pste feedback

63 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj.7.. ealzatn f Negate Impedances. The crcut n Fg..0 uses partal pste feedback snce the resstr lnks the utput ltage and the nn-nertng termnal. T understand the peratn f the crcut, let us fnd the ltage at the nn-nertng termnal. Applyng the superpstn prncple, x s cmpsed f cntrbutns frm and. The frst cmpnent can be btaned by cnsderng and grundng n the analyss, whch can be dne because the utput f the OPAMP s a lw-mpedance nde and s defned by the ltages appled at the OPAMP nputs. The secnd cmpnent s btaned by cnsderng and grundng. The cmbnatn f these tw cmpnents s: x. (.56) The abe expressn fr x can be substtuted nt the nn-nertng gan relatnshp between x and : x. (.57) Wth sme algebra, yu can dere the erall transfer functn as - 6 -

64 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj. (.58) Once agan, the pste feedback s reflected n the negate term f the denmnatr. An mprtant specal case ccurs when = and =, such that the preus equatn smplfes t. (.59) The abe transfer functn allws nn-nertng amplfcatn. The mst nterestng prperty f the crcut n Fg..0 s asscated wth the nput mpedance. Frm Eqs..56 and.59, the nput mpedance s btaned fr the case where = and = as shwn n Eq

65 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj x. (.60) Therefre, the current flwng thrugh s: x. (.6).0. Amplfer wth partal pste feedback. It can be ntced frm Eq..6 that the current flwng thrugh depends n but s ndependent f. Hence, ths crcut can be cnsdered as a ltage-cntrlled current

66 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj surce: The current s cntrlled by the nput ltage and the resstrs =, and ths current s frced t flw thrugh. On the ther hand, yu can dere the expressn fr the mpedance at the nput prt yurself and cmpare t wth ths result:. (.6) The nput mpedance s pste fr <, and negate fr >. Thus, f desred, the crcut n Fg..0 can be desgned wth a negate nput mpedance. A useful crcut that s ften emplyed n the desgn f flters s the negate mpedance cnerter shwn n Fg.., whch s a arant f the crcut depcted n Fg..0. The nput ltage s appled t the nn-nertng termnal f the negate mpedance cnerter, and the utput ltage s = (+ / ). The nput current s = ( - )/, leadng t the fllwng expressn f the nput mpedance:. (.6)

67 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Ntce that the equalent mpedance at the nput s negate. A negate mpedance means that, cntrary t the case f a pste mpedance, the crcut delers current when pste ltage sgnals are appled. The reasn fr ths behar s that the OPAMP crcut wth and amplfes the nput sgnal (wthut nersn) and the utput ltage s greater than r equal t. Hence pste generates >, and snce the element s cnnected between the utput and nput termnals, t generates a current that flws frm t. Fg... Negate mpedance cnerter..7.. Sallen-Key Flter. Pste feedback has been used fr the desgn f flters fr a lng tme. The flter n Fg.. cnssts f fe admttances and an amplfer wth fnte gan K. Snce the amplfer s nn-nertng, the feedback prduced by Y s pste. By

68 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj fllwng the analyss prcedure dscussed earler n ths chapter fr the multple feedback flters, the transfer functn can be btaned by wrtng the admttance matrx as fllws: y y y y5 y y y y 0 K y 0 0 y 0 0 x n y (.6) Y Y x Y y K Y 5 Y Fg... Secnd-rder Sallen-Key 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 gan gen by = K y. The slutn f ths system leads t the fllwng transfer functn fr the flter:

69 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj H( s ) y y K. (.65) y y y y y y y y K 5 Lwpass, bandpass and hghpass flters can be desgned based n the abe transfer functn by selectng the prper elements and cmpnent alues. The specal cases are: ) Selectng Y and Y as cnductances, and Y and Y as capacte admttances, whch leads t a lwpass transfer functn, Y 5 can be remed n ths case, resultng n the flter dsplayed n Fg.. wth the fllwng transfer functn. H ( s) C C (.66) s s C K C C K C C Smlarly, t can be shwn that the cndtns belw lead t band-pass and hgh-pass transfer functns. ) Y and Y shuld be selected as cnductances and Y and Y 5 as capactrs t realze a band-pass flter wth the transfer functn n Eq..65. ) If Y and Y are selected as cnductances, and Y and Y are capactrs, then a hghpass transfer functn s btaned

70 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj 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 = jω nt the transfer functns and plt H(jω) s. ω t bsere the magntude respnses. C K C Fg... Secnd rder Sallen-Key Flter wth pste feedback..8. Practcal Lmtatns f Operatnal Amplfers. Frst at all, we must recgnze that practcal OPAMPs are nt een clse t the deal mdel wth nfnte nput mpedance, nfnte gan, nfnte bandwdth, and unlmted utput current capablty and utput ltage range. The actual parameters and lmtatns depend n the OPAMP tplgy (arrangement and parameters f transstrs, resstrs and

71 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj capactrs as well as technlgy used and pwer cnsumptn). There are many dfferent OPAMPs ffered by endrs such as Texas Instruments, Farchld, Natnal Semcnductr, etc. Althugh the specfc rgns f OPAMP desgn 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 DC Gan, Fnte Input Impedance and Nn-zer Output Impedance. A smewhat mre realstc OPAMP macrmdel s depcted n Fg... Example ranges fr sme parameter alues f cmmercally aalable OPAMPs are: = MΩ - GΩ, = -00Ω, and A = V/V (60-0 db). The gan usually decreases at a rate f -0 db/decade abe the cutff frequency n the 0 Hz khz range. These lmtatns ntrduce errrs n the transfer functn. Nrmally, t s cumbersme t ealuate system degradatns wth analytcal equatns, especally fr cmplex crcuts. Here, we wll btan sme results fr a sngle nertng amplfer stage, but mst f the cnclusns frm the analyss f ths crcut are als ald fr cmplex crcuts

72 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj Fg... An peratnal ltage amplfer wth fnte nput resstance, fnte ltage gan, and nn-zer utput resstance. Let us cnsder the crcut shwn n Fg.5a and nclude the effects f bth the OPAMP fnte nput mpedance ( ) and the OPAMP fnte gan. Nte, s mre general than because the nput mpedance s typcally dctated by bth a resste and a capacte part. It s assumed that pen-lp amplfer gan (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. Usng the macrmdel f Fg.. where = 0 (assumng that << F, L ), the equalent crcut can be drawn as shwn n Fg..5b. The transfer functn can be btaned f the ndal equatn at the nertng termnal - s wrtten. Snce s cntrlled by the ltage-dependent ltage surce [ =A ( )], the utput ltage s entrely defned by the ltage acrss the OPAMP nput termnals and the external mpedance elements. The current demanded by F and L s prded by the deal ltage-cntrlled ltage surce, when can take n any necessary alue t sle the equatns. Hweer, a real OPAMP has a specfed - 7 -

73 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj maxmum utput current (that s lsted n the datasheet), and as a cnsequence yu shuld be careful when selectng external resstrs. Befre fnalzng a desgn, t s mprtant t erfy wth calculatns and transent smulatns that the resstr alues are large enugh t ad an excesse current flw that cannt be sustaned at the OPAMP utput termnal. F - + L (a) (b) Fg..5a) Inertng amplfer wth OPAMP nput mpedance and lad mpedance L, and b) ts small-sgnal equalent crcut where = 0. Let us quantfy the effects f and fnte A n the transfer functn f the crcut n Fg..5. The crcut s transfer functn can be dered by slng the fundamental equatn ( = + ) and takng nt accunt that = A ( ) where + = 0 due t the cnnectn t grund: - 7 -

74 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj F F F A s H ) ( ) ( ) (. (.67) The effect f the fnte pen-lp DC gan and fnte nput mpedance n the nertng amplfer can be better apprecated f an errr functn s defned. Frm Eq..67 t fllws that ) ( F F s H (.68) where the apprxmatn s ald when the alue f the errr functn ξ s small (ξ << ), and ξ s defned as F F F F F A A A s ) ( ) ( ) ( ) ( ) (. (.69) If the OPAMP DC gan A s lmted, the assumptn f a rtual grund at the nertng nput s n lnger ald because any utput ltage aratn crrespnds t a fnte aratn f the dfferental nput sgnal gen by /A. The smaller the OPAMP gan, the

75 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj larger the ltage aratns at the OPAMP nput termnals are. Hence, the errr shuld be nersely prprtnal t A, as predcted and cnfrmed by Eq..69. The ltage aratns n the nn-nertng termnal lead t current errrs: Frst, the nput current = ( - - )/ has an errr prprtnal t ( errr = - - / ); secnd, the OPAMP nput mpedance affects the current generated by, leadng t anther current errr cmpnent f errr = - / I. These current errrs are cnerted nt ltage errrs by the feedback mpedance F. Ntce frm Eq..67 that een f the OPAMP nput mpedance s nfnte, a gan errr apprxmately prprtnal t the magntude f the ntended (deal) gan F / and nersely prprtnal t A s present. Fr a gen OPAMP penlp gan A, the larger the magntude f the clsed-lp amplfer gan (.e., the amplfcatn gan yu want t mplement gen by F / ) the larger the errr s. The errr tlerance s applcatn-dependent. Fr nstance, t keep the transfer functn errr belw %, t s requred t satsfy the cndtn: ( s ) A F (.70) An mprtant lmtng factr s the magntude f the desred amplfcatn gan F /. Ntce that Eq..69 can als be rewrtten as

76 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj ) A A F F ( s (.7) The frst factr n the rght hand sde f Eq..7 s the rat f the magntude f the ntended gan er the OPAMP pen-lp gan. The secnd factr shws that t s desrable t mantan the alues f the resstrs well belw the alue f the OPAMP s nput mpedance. Fr nstance, f the OPAMP nput mpedance s = M and the clsed-lp ltage gan s - F / = -0, and we want t mantan the gan errr under %. The estmated pen-lp ltage gan requrement f the OPAMP accrdng t Eq..7, whch s 000 when = M. Examples f the OPAMP gan requrements fr dfferent F / rats are lsted n Table.. Ntce that the gan requrement decreases as F ncreases relate t the OPAMP s nput mpedance. The realzatn f hgh clsedlp gan factrs n cmbnatn wth lw feedback resstance alues usually demands hgh-gan OPAMPs. Table. equrements fr the OPAMP pen-lp gan n the example crcut t mantan a clsed-lp gan errr belw % wth dfferent F / rats

77 Intrductn t Electrnc Crcuts: A Desgn Apprach Jse Sla-Martnez and Marn Onabaj F / A V 00 Ω / 0Ω > kω / KΩ > 00 MΩ / 00 > 00 KΩ 0 MΩ / > 000 MΩ.8.. Effects f Fnte OPAMP Bandwdth. Unfrtunately the OPAMP bandwdth s usually lmted. Fr example, the -db frequency f the μa7 s nly n the -0 Hz range whle the pen-lp DC gan s arund 0 5 V/V. The prduct f the pen-lp DC gan and the bandwdth s defned as the OPAMP s gan-bandwdth prduct (GBW). Fr the OPAMP μa7, the typcal alue f GBW s arund.5 MHz. These parameters and the transfer functn f the μa7 are llustrated n Fg