v DD Z D M 3 c vn - s v P Z L0 s s s s c v SS c

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1 IEEE TRANSATIONS ON IRUITS AND SYSTEMS, VOL. 38, NO., OTOER 99 A General Relationhip etween Amplier Parameter and it Appliation to PSRR Improvement Eduard Sakinger, Member, IEEE, Joef Goette, Member, IEEE, and Walter Guggenbuhl Senior Member, IEEE Abtrat A general relationhip between the gain of multiterminal amplier i derived in thi paper. It reveal a ontraint for the imultaneou improvement of the ommonmode and the powerupply rejetion ratio of the imple operational amplier. Thi ontraint an be relaxed by either adding a upplementary input terminal to the amplier iruit or uing a fully dierential deign. The method an be ued to improve the powerupply rejetion ratio in operational amplier. Several implementation of a twotage operational amplier illutrate thi tehnique. A I. Introdution N important parameter of operational amplier i the dierential gain, A d, whih i the ratio between the ingleended output voltage and the dierene of the two input voltage. Additional peiation deribe the ignal tranfer from the upply terminal to the output, A dd and A, and the ommonmode gain from the input terminal to the output, A m. For an ideal amplier thee paraiti gain hould be zero, and it i the aim of a good amplier deign to ahieve thi goal a nearly a poible. We how in thi paper, however, that thee gain are not independent ofeah other and that their um i loe to unity in pratial onguration. Expreed in term of the (frequenydependent) ommonmode rejetion ratio, MRR() =A d ()=A m (), the (frequenydependent) powerupply rejetion ratio, PSRR + () = A d ()=A dd () and PSRR () = A d ()=A, and the dierential gain, A d (), we nd more exatly that MRR() + PSRR + () + PSRR () = Z L or (a) A d () Z L + Z out Z L A m ()+A dd ()+A ()= (b) Z L +Z out where Z out denote the output impedane of the onidered operational amplier and Z L i the referene load for whih the dierential gain and the variou rejetion ratio are meaured. Equation (b) tate that in abene of Manuript reeived February 5, 99; revied February, 99. Thi work wa upported in part by the Fondation Suie pour la Reherhe en Mirotehnique, No /. E. Sakinger wa with the Eletroni Laboratory, Swi Federal Intitute of Tehnology Zurih, he i now with AT&T ell Laboratorie, Holmdel, NJ J. Goette and W. Guggenbuhl are with the Eletroni Laboratory, Swi Federal Intitute of Tehnology Zurih, ETH Zentrum, H892 Zurih, Switzerland. a load, Z L!, the ommonmode gain and the gain from the power upplie to the output um to unity. Simple operational amplier model often aume that only the dierential ignal i amplied, but aording to () thi i phyially impoible. A related reult i found in lter theory, f. e.g. Fialkow and Gert [] and Hilberman [2], where it i hown that for a paive threeterminal network, having no internal onnetion to ground, the voltage tranfer funtion from one input to the output i the one' omplement of the tranfer funtion from the other input to the output. More generally, [2] prove, baed on Kirhho' and Ohm' law, that a multiterminal network with no onnetion to the ommon terminal exhibit omplementary tranfer funtion. We ue a more general method namely a gaugeinvariane argument to etablih our relationhip () and it generalization. With thi relationhip, whih eem not to be well known in amplier theory, we derive impliation for the deign of dierential amplier featuring good rejetion of the paraiti ignal. In pratie () implie that at leat one of the paraiti gain i in the order of magnitude of unity. In the laial twotage operational amplier heme thi uually onern the gain from one powerupply terminal. Ripple or other paraiti ignal at thi upply terminal are, at mid frequenie, fully tranferred to the output through the integrator apaitor. Thi weakne of the laial operational amplier heme i well known. A remedy againt thi paraiti ignaltranfer path ha been propoed in [3] by introduing a aode truture and an additional terminal. We how that thi olution i jut a peial realization of a more general idea whih i baed on relationhip () and that alternative and even better reult are obtainable. Thi paper i trutured a follow: In Setion II we outline our argument that lead to () and extend the reult for more advaned amplier onguration inluding operational amplier with an additional input terminal, dierential amplier with dierential output, dierential amplier with balaned output, and dierential dierene amplier (DDA) [4] [5]. In Setion III we illutrate the ue of relationhip () by pei iruit example: adding an input terminal with unity gain to the output of the amplier iruit yield mall value of the remaining paraiti gain. With a balaned output amplier, even if only one output i ued, even better performane in rejeting paraiti ignal i ahieved. In the Appendix we preent the derivation of the general relationhip for multipleinput

2 2 IEEE TRANSATIONS ON IRUITS AND SYSTEMS, VOL. 38, NO., OTOER 99 multipleoutput amplier from whih we obtain the reult of Setion II. II. Gain Relationhip for Variou Amplifier Type The eletrial notationonvention adopted in thi paper loely follow [6]. Signal are generally deignated a a ymbol with a ubript. The ymbol a well a the ubript are eah either upper or lower ae aording to the following onvention illutrated in the ae of a voltage ignal: v X denote the total intantaneou value of the voltage, V X denote the operatingpoint value of the voltage, wherea v x tand for it mall ignal value v X V X. Finally, the (bilateral) Laplae tranformation of v x i denoted by V x : v x (t) V x (). For a proofoutline of () we onider a onventional twoinput oneoutput amplier with input v P and v N, output v O, and two power upplie v DD and. Note that thee amplier have no eparate ground terminal. We aume that the amplier iruit i biaed into an operating point where the amplier i deribed by an equivalent lumped, linear, timeinvariant iruit. To be able to deribe the eet of time varying power upplie, the equivalent linear iruit i aumed to have four (mall) ignal input v p, v n, v dd, and v, and one output v o. A i ommon pratie in amplier theory, weintrodue the dierentialmode voltage v d = v p v n and the ommonmode voltage v m = (v p + v n )=2 to deribe the ignal input. In term of thee ignal and the upply voltage, the linear timeinvariant iruit i deribed in the Laplae domain by V o = A d ()V d + A m ()V m + A dd ()V dd + A ()V (2) where A i (), i 2 fd; m; dd; g, denote the dierentialmode, the ommonmode, and the powerupply gain, repetively. Our argument that () mut hold i baed on a gauge tranformation: eaue the origin of the potential ale againt whih the voltage are meaured annot be xed abolutely, we expet that all phyially meaningful equation remain invariant if thi origin i globally hifted (thi i alled gauge invariane or gauge ymmetry) [7, p.22]. Applied to any equation haraterizing the amplier, thi mean that the addition of the ame arbitrary timevarying voltage ignal to all terminal voltage hould yield the ame equation. For (2) the addition of a gauge ignal v gauge (t) V gauge () lead to V o + V gauge = A d ()V d + A m ()(V m + V gauge )+A dd ()(V dd + V gauge )+ A ()(V + V gauge ) and ubtrating from thi equation the amplier' original deription (2) lead to an equation ontaining the gaugetranformation ignal V gauge (), V gauge = (A m () +A dd ()+A ())V gauge. Thi in turn lead, beaue v gauge (t) and hene V gauge () are arbitrary mallignal quantitie, to the following relationhip between the paraiti amplier gain for open iruit load (Z L!) or ideal amplier in the ene of Z out : A m ()+A dd ()+A ()=: (3) In amplier pratie, the uual deription i not in term of the gain A i (), i 2fd; m;dd; g, aintrodued above, but intead, the dierentialmode gain, A d (), together with ommonmode and powerupply rejetion ratio, MRR(), PSRR + (), PSRR (), are ued. In term of thee rejetion ratio we an expre (3) alternatively by MRR() + PSRR + () + PSRR () = A d () : (4) Thi i the eential part of () and hold for openiruit load (Z L!) or ideal amplier in the ene of Z out. More generally, it approximately hold for amplier/load ombination for whih jz out =Z L j in the relevant frequeny range. The more general relationhip preented in the Appendix an be peialized for onrete amplier a follow: Operational Amplier with a Referene Input. For operational amplier whih have an additional input terminal v REF,we obtain A m ()+A dd ()+A ()+A ref () = (5) where A ref () i the tranfer funtion from the additional terminal to the output. We note from (3) that for a imple operational amplier at leat one of the paraiti gain mut be nonzero, wherea for an operational amplier with a referene input we obtain an additional degree of freedom for the hoie of a nonzerogain path. Therefore, if the additional terminal an be hoen o that the gain to the output i unity, the remaining paraiti gain add to zero whih i the neeary ondition that thee gain beome zero individually. Example 3 and 4 in Setion III illutrate that thi favorable ituation atually exit in real iruit. If the referene terminal v REF i onneted to a noiefree potential againt whih the ignal are dened, then A ref () ha no inuene on the output ignal, i.e. we havev ref and in turn A ref ()V ref. Note that deigning uh a referene terminal into the amplier iruit mut not deteriorate the ignal gain A d (). Equation (5) lead to a generalization of (3) for nite load or nonzero output impedane of the ampli er. Thi reet the pratial ituation where the gain A d (), A m (), A dd (), and A () are meaured with a nite load impedane Z L to ground. Thi ituation i now analyzed by onneting one ide of the load Z L to the amplier' output terminal and onidering the other ide, whih for uual gain meaurement i onneted to an external ground, a an additional input terminal v REF of the amplier iruit. With Z out denoting the amplier' output impedane, the gain from thi newly introdued terminal to the output i then een to be A ref () =Z out =(Z L + Z out ). Inerting thi into (5) lead to A m ()+A dd ()+A ()= Z L Z L +Z out ; (6) whih generalize (3) for loaded amplier. Equation (6) eaily tranform into (a) when it i rewritten in term of the dierential gain and the rejetion ratio.

3 SAKINGER ET AL: A GENERAL RELATIONSHIP 3 Operational Amplier with Dierential Output. For dierentialoutput operational amplier there are paraiti gain from the ommonmode input and the upply input to the dierentialmode output a well a to the ommonmode output voltage. Denoting by A o i (), i 2 fm; dd; g and o 2 fdm; mg, the gain from the input i to the output o, the following two relationhip between the paraiti gain are valid: m()+ dd ()+ ()=; A m m()+a m dd ()+A m ()=: (7a) (7b) From (7) it i een that the gain to the dierentialmode output, i (), an individually beome zero, but that at leat one of the gain to the ommonmode output, A m i (), mut be nonzero. Operational Amplier with alaned Output. alanedoutput amplier are amplier with dierential output that have an additional input, v AL, to ontrol the ommonmode output voltage v OM. With a notation orreponding to that ued in (7), the paraiti gain relationhip for thee amplier are m()+ dd ()+ ()+ bal() =; A m m()+a m dd ()+A m ()+A m bal() = (8a) (8b) where A o bal (), o 2fdm; mg denote the gain from the balane input to the repetive output. Uually A m bal () i et to unity (the ommonmode output voltage v OM trak the balane ignal v AL ), allowing the remaining paraiti gain in (8b) to beome zero individually. A i hown by Example 5 in Setion III, the gain A m m(), A m dd (), and A m () atually beome very mall for a pratial ampli er with A m bal (). Even if the ignal of only one output terminal i ued againt the referene v AL, an amplier with a very good iolation of the upply terminal ignal i obtained. Thi property hold in the frequeny range where A m bal (), whih extend for the onidered example from d to an upper frequeny limit. Dierential Dierene Amplier. The dierential dierene amplier (DDA) deribed in [4] and [5] i an amplier with four ignal input deignated v PP, v PN, v NP, and v NN, one ignal output v O, and twopower upplie v DD and. Ideally, it amplie the dierene between the port voltage (v PP v PN ) and (v NP v NN ), i.e. the dierential ignal v D = (v PP v PN ) (v NP v NN ), and uppree the three ommonmode ignal v P = (v PP +v PN )=2, v N = (v NP +v NN )=2, and v N = [(v PP v PN )+(v NP v NN )]=2. For thi amplier the relationhip between the (mallignal) paraiti gain i A p ()+A n ()+A dd ()+A ()= (9) where A i (), i 2fp; n; dd; g, denote the gain from the repetive input to the output. Note that the gain from the ommonmode voltage v d, A d (), doe not enter into the relationhip. A for the imple operational amplier (f. (3)), at leat one of the paraiti gain in (9) mut be nonzero. III. Diuion and Example In the following we illutrate our reult with pei iruit example, and diu priniple whih aim to improve the rejetion of paraiti ignal. eaue the relationhip i impler in term of gain than in term of rejetion ratio, mot of the diuion i in term of gain and we take are that the dierential gain of ompared amplier are equal. Exept for Example, whih analytially demontrate the bai relationhip (6), our example fulll jz out =Z L j in the relevant frequeny range. Thu the term that aount for nonzero output impedane are uneential, and the implied relationhip given in Setion II applie. A mentioned above it i aording to (3) not poible to deign a imple operational amplier that ha all paraiti gain, i.e., the ommonmode gain and powerupply gain, equal to zero. Therefore, the frequently ued aumption for ideal operational amplier model that only the dierential ignal i amplied i phyially impoible. We further onlude that if there i no powerupply dependene of the output, the ommonmode gain mut be unity, or, tated otherwie, the ommonmode rejetion ratio and the dierential gain mut be idential. Traking eet of thi kind are frequently oberved in operational amplier. Note that (4) i a relationhip among omplexvalued funtion; data heet of atual amplier, however, peify only the magnitude of the dierential gain and the rejetion ratio. y applying the triangle inequality to (4), four relation among the orreponding frequenyrepone magnitude are obtained whih imply variou traking eet. 2 Example 2 below illutrate traking for a pratial amplier. The relationhip (3) developed in Setion II and it extenion to amplier with an additional input terminal that allow for the redution of the paraiti gain i now illutrated by ome pratial example. While the rt example illutrate (3) by analytial expreion for the variou gain of a imple buered dierential amplier, the next example onern the Spie imulation of the gain of a twotage general purpoe operational amplier propoed in the literature [6]. Example 3 to 5 Of oure, orreponding tatement are true if two other paraiti gain are zero and, in turn, the rejetion ratio of the remaining gain i idential to the dierential gain. 2 The relation are of the form =w =x +=y +=z where w i one of the four parameter ja d ()j, jmrr()j, jpsrr + ()j, and jpsrr ()j, and x, y, z repreent, for a eleted w, the remaining parameter. From thee inequalitie we onlude that, at any given frequeny, it i impoible for exatly one of the four parameter to be muh wore (i.e. muh lower) than the other, beaue eah invere parameter i upper bounded by the um of the remaining invere parameter. Furthermore, if two of the four parameter are muh wore (i.e. muh lower) than the other, thi implie that the magnitude of the latter parameter mut be very imilar. Thi i explained by the fat that aording to the above inequalitie thee parameter are approximate upper bound for eah other. Thi traking eet an be oberved in many amplier iruit in the midfrequeny range and i alo found in the data heet of many pratial amplier.

4 Gain [d] 4 IEEE TRANSATIONS ON IRUITS AND SYSTEMS, VOL. 38, NO., OTOER 99 v DD v P M M 2 I Z M 3 vn I 2 Z 2 Z L vo Fig.. Shemati of a imple MOS dierential tage whih i buered by a oure follower. how three approahe for the redution of the paraiti gain with the help of an additional terminal. Thereby, the Example 3 and 4 ue onventional ingleended ampli er and obtain improved rejetion in the mid frequeny range. In ontrat Example 5 make ue of a dierential output truture and give rejetion improvement even down to d. To ompare the variou amplier iruit on the bai of gain (rather than on the bai of rejetion ratio) the amplier iruit in Example 2 to 5 have been deigned to have an equal dierential gain of 85 d. Finally, Example 6 ompare thee amplier in a negative feedbak iruit and how that the amplier of Example 5 outperform the other preented olution. Example (Simple Amplier) In order to illutrate (6) by analyti expreion we onider the imple dierential tage with two load impedane buered by a oure follower a hown in Fig.. We aume that the MOS tranitor are imply modeled by linear voltageontrolled urrent oure with real tranondutane g m for the dierential tage tranitor and g m3 for the follower tranitor, repetively. Then the relevant frequeny repone are eaily derived to be A d () =g m g m3 2 Z L Z out Z L + Z out ; A m () = g m g m3 +2g m Z S Z L Z out Z L + Z out ; A dd () =g m3 A () = Z L Z out Z L + Z out ; g m g m3 + ZL Z out +2g m Z S Z S2 Z L + Z out () where the output impedane i Z out = Z S2 =( + g m3 Z S2 ). Subtituting the frequeny repone given in () for the orreponding variable in (6) how that the derived relationhip i atually fullled. 2 Example 2 (Two Stage Op Amp) Wherea the above example, made for analytial demontration purpoe only, ould be onidered an overimpliation, the next example i loer to pratie. It i the rt example in a erie whih v DD v N vp vo Fig. 2. Shemati of a two tage MOS operational amplier with nhannel input pair (from [6, Fig. 8.32]) A () A dd () Magnitude Repone A m () Frequeny f [Hz] L Z L =(Z L + Z out ) Fig. 3. Magnitude repone of the ommonmode gain A m(), the poitive powerupply gain A dd (), and the negative powerupply gain A () for the two tage op amp of Fig. 2. Alo hown i the term jz L =(Z L + Z out)j (dahed line) whih indiate that it value i loe to unity below khz jutifying the ue of the implied equation (3) or (4). illutrate ome improvement trategie that are baed on our relationhip. Intead of ymbolially omputing the relevant frequeny repone we havenow to reort to numerial omputation. We onider the two tage MOS operational amplier, taken from [6, p. 396], whih i hown in Fig. 2 and onit of a dierential tage aaded with an integrator. A Spie omputation of all parameter in relationhip (6) for the omponent value of the deign example in [6, pp. 393.] numerially onrm our relationhip. The frequeny dependene of the parameter involved in (6) are plotted in Fig. 3. The mot intereting feature in thee plot i that A dd () i approximately equal to unity over a broad frequeny range whih orrepond to a loe traking of the dierential gain and the poitivepowerupply rejetion ratio. The weak poitive upply rejetion illutrated by thee urve i well known in operational amplier pratie (f. e.g. [8]) and i attributed to the virtual ground at the integrator input whih trak the hange of the poitive upply voltage v DD. Thee variation are fully tranferred to the

5 Gain [d] Gain [d] SAKINGER ET AL: A GENERAL RELATIONSHIP 5 v DD v DD vo v N vp v REF vo Fig. 4. Shemati of a aode MOS operational amplier uing an auxiliary terminal (from [3, Fig. 2]) A ref () A () Magnitude Repone A m () A dd () Frequeny f [Hz] A dd;2 () * Fig. 5. Magnitude repone of the ommonmode gain A m(), the gain from the referene terminal A ref (), the gain from the poitive powerupply A dd (), and the gain from the negative powerupply A () for the aode op amp of Fig. 4. For omparion, the poitive powerupply gain that reult for the original operational amplier of Fig. 2 i alo hown by the urve labeled A dd;2 () (dahed line). output through the integrator apaitor and the integrator load at medium and high frequenie. The obervation of A dd () and low value of the A m () and A () i onitent with our relationhip (3). 2 The following Example 3 and 4 illutrate way to improve the poitive upply rejetion through the introdution of a upplementary ontrol terminal a indiated in the above diuion. Example 3 (aode Op Amp) In the iruit illutrated in Fig. 4 the return point of the integrator apaitor i iolated from the poitive upply terminal by a ommon gate tage a propoed in [3, Fig. 2]. In the midfrequeny range the \A dd () path" of the iruit in Fig. 2 i now replaed by a path with an A ref () tranfer funtion from the gate of the ommongate tage to the output. Thi i the neeary ondition for low value of the other paraiti gain A dd (), A (), and A m (). The reult of L v N v REF vp Fig. 6. Shemati of a urrent injetion MOS operational amplier uing an auxiliary terminal A dd;2 () R A ref () A () Magnitude Repone A dd () Frequeny f [Hz] A m () Fig. 7. Magnitude repone of the ommonmode gain A m(), the gain from the auxiliary terminal A (), the gain from the poitive powerupply A dd (), and the gain from the negative powerupply A () for the urrent injetion op amp of Fig. 6. For omparion, the poitive powerupply gain that reult for the original operational amplier of Fig. 2 i alo hown by the urve labeled A dd;2 () (dahed line). a imulation in Fig. 5 how that thi goal i ahieved by thi iruit modiation. We reall that onneting the referene terminal to a noiefree voltage oure doe not add any diturbing ignal omponent to the output ignal although the gain A ref () i near to unity. We note, however, that thi aode truture (f. Fig. 4) require an additional noiefree d upply and ha the diadvantage of limiting the input ommonmode voltage range. 2 Example 4 (urrent Injetion Op Amp) An alternative method for the reation of a \gain one path" from a upplementary terminal to the output i hown in Fig. 6. onneting a apaitor from the input of the urrent mirror to ground in the laial two tage operational amplier of Fig. 2 i an eaily appliable olution to the problem whih doe not inuene the ommonmode voltage range and doe not require a upplementary d upply. In thi onguration, injet a orretion urrent via the urrent mirror into whih ompenate for the variation at the integrator input with v DD. y hoing an idealized model of the path from L

6 Gain [d] Gain [d] 6 IEEE TRANSATIONS ON IRUITS AND SYSTEMS, VOL. 38, NO., OTOER 99 v DD Dierential Amplier M M 2 v IN v IP ommonmode Feedbak Fig. 8. Shemati of a balanedoutput MOS operational amplier. 2 4 Magnitude Repone A m bal () v OP v ON v AL v in R H HH Fig.. R 2 H v out v in R H HH R 2 Noninverting amplier onguration. A dd;2 () Magnitude Repone H A dd;4() R + R 2 I A dd;3 () v out 6 8 A m m () A m dd () 2 A dd;5 () 2 A m () Frequeny f [Hz] Fig. 9. Magnitude repone of the ommonmode gain A m m(), the gain from the balane terminal A m (), the gain from the poitive powerupply A m dd bal (), and the gain from the negative powerupply A m () for the balanedoutput op amp of Fig. 8. the ground (referene) terminal to the output lead to A ref () = (=p d )=[( + =p d )( + =g m )] where p d denote the amplier' dominant pole magnitude and g m the urrentmirror tranondutane. Small value of the other paraiti gain in the frequeny interval limited by p d and g m = are now poible aording to (5). Spie imulation of the repetive iruit hown in Fig. 7 illutrate that in the midfrequeny range where the gain from the referene terminal approahe unity, the poitive power upply rejetion i ubtantially improved (i.e. the gain A dd () i lowered). The frequeny range for A dd uppreion an be enlarged by enlarging the urrentmirror tranondutane g m a i een from the above expreion for A ref (). 2 Our next example propoe a fully dierential amplier with a balanedoutput, imilar to [9], to improve the rejetion of paraiti ignal. Example 5 (alaned Output Op Amp) The iruit deign tart from the original dierential amplier of Fig. 2 and ymmetrially applie thi deign to a balanedoutput amplier. The ommonmode output voltage i regulated with a dierential dierene amplier (DDA) [4] [5] to an external voltage v AL by ening the two dierential output line. The obtained iruit i hown in Fig. 8. The reulting ommonmode gain hown in Fig. 9 indiate that A m bal () i unity over a broad frequeny range whih, in on Frequeny f [Hz] Fig.. Magnitude repone of the poitive powerupply gain in the noninverting amplier of Fig.. The ubript indiate the number of the example where the amplier i introdued. trat to the ingleended ounterpart of the aode and urrentinjetion operational amplier, extend down to d. It fore the remaining paraiti gain A m m(), A m dd (), and A m () tovery mall value in thi broad frequeny range. The dierential output ignal i almot ompletely independent of the upply and ommonmode voltage in thi fully ymmetri iruit, whih mean that the gain i (), i 2 fm;dd; g, are pratially zero. Furthermore, the paraiti gain with repet to a ingleended output (one of the output line with repet to v AL ) are igniantly improved a ompared to the aode and urrent injetion operational amplier example. Like the urrentinjetion operational amplier, orretion urrent are upplied to the integrator apaitor in order to ompenate for the variation of the integrator input at v DD. However, in thi ae the urrent are upplied by the urrent oure M and M 2 whih are ontrolled by the ommonmode feedbak iruit. 2 Example 6 (loed Loop iruit) We ue the amplier preented in the Example 2, 3, 4, and 5 in a noninverting, ingleended, amplier onguration with feedbak ahown in Fig., and ompare their powerupply rejetion performane. Note that the amplier load of the balanedoutput amplier i reprodued at the nonued output terminal. The Spie imulation of the reulting gain from the poitivepower upply in Fig. learly indiate that the balanedoutput amplier of Example 5 outperform the other amplier deign. Similar reult have been obtained for the gain from the negativepower upply

7 SAKINGER ET AL: A GENERAL RELATIONSHIP 7 but are not preented here beaue in the onidered amplier the negative power upply rejetion i le ritial. 2 IV. Summary and onluion We have hown that a ontraint exit for the gain whih are meaured between the input terminal of an amplier and it output. Applied to a imple operational amplier with a dierential input, two upply terminal and one output, thi mean that a relationhip exit between the o alled paraiti gain (i.e. the ommonmode gain and the gain from the upply terminal to the output): their um approahe unity in pratial iruit. Therefore, upply and ommonmode rejetion ratio annot imultaneouly aume high value. Thi onluion i onrmed by pratial operational amplier deign where uually one of the upply terminal i ritial and exhibit a relatively high paraiti tranfer to the output. Our derivation of the preented general relationhip i baed on the phyial law of gauge invariane of the eletrial potential. Adding to the amplier iruit a upplementary referene terminal whih i onneted to a noiefree voltage and the ue of dierentialoutput truture add a degree of freedom to the above mentioned gain relationhip and an be ued to maximize ommonmode and powerupply rejetion imultaneouly. The pratial ue of thi idea i demontrated with four dierent implementation of a bai twotage MOS operational amplier. Deigning the upplementary terminal uh that it tranfer to the output i unity inreae the upply rejetion of the ritial terminal. While two of the preented deign how thi improvement only in the midfrequeny range, a dierentialoutput amplier with ommonmode feedbak exhibit thi favorable property down to d even when ued a a ingleended output amplier. Adding an external feedbak to thee amplier onerve the improvement of the openloop performane a i demontrated with a noninverting amplier onguration. Wherea we have mainly onidered amplier with two ignal input, we would like to point out that the reult are more general. Moreover, generalization with repet to the number of powerupply line and output terminal are traightforward and we expet that our idea are appliable to other iruit onguration a well. Appendix Appendix: Derivation of the General Relationhip An eay way to etablih the relationhip in Setion II i to ue a gaugeinvariane argument to obtain the relationhip for a more general ituation whih an eaily be peialized for the needed amplier type. Thi i bet done in avetorpae formalim. Toavoid onfuion with the eletrial notationonvention we repreent the ariing vetor by a ymbol delimited by a left vertial bar and a right angular braket (e.g. jai). 3 The peial vetor (; ;:::;) T 3 Thi i Dira' braket notation, f. [, p.8], [, p.7]. i denoted ji. Matrie are written a upper ae, bold fae ymbol; the identity matrix being I. The following general proof i illutrated with the dierentialoutput operational amplier a an example after eah intermediate tep. The onidered amplier iruit have a number of inputterminal the voltage of whih are olleted in the terminalvoltage input vetor jv IT i. To be able to deribe the eet of nonideal (time varying) power upplie, thi vetor ontain, beide the ignal voltage alo the power upply voltage. orrepondingly, the output voltage are olleted in the terminalvoltage output vetor jv OT i. For the amplier of Setion II thi vetor ontain a ingle voltage in the ae of ingleended amplier or a pair of voltage in the ae of dierentialoutput amplier. Generally, the tranfer funtion from the input, jv IT i,to the output, jv OT i, of an amplier iruit i peied by a et of nonlinear dierential equation of the input and the output variable. eaue we are intereted in the ampli er' approximate linear behavior whih reult for mall inputvoltage variation when the amplier iruit i biaed into an operating point, we introdue the orreponding mallignal quantitie (operating point quantitie ubtrated) tranformed to the frequeny domain: jv it ()i and jv ot ()i. The amplier ould now be haraterized by a gain matrix ummarizing the gain from the inputterminal to the outputterminal voltage. However, it i ommon pratie to peify the gain between the input and output mode voltage rather than the terminal voltage; thu we have M OjV ot i = A()M I jv it i : () The matrie M I and M O tranform the terminal voltage to mode voltage and A() i the matrix ontaining the mode gain. Example: The notation i laried with the example of the dierentialoutput operational amplier. Thi ampli er ha two ignal input, v IP and v IN, and two powerupply input, v DD and,thu the input vetor ha four omponent. There are two output, v OP and v ON. Equation () then peialize to Vop = =2 =2 V on dm () m() dd () Adm () A m dm () Am() A m dd () Am () =2=2 A V ip V in V dd A : V The matrie M I and M O tranform the terminal input and output ignal into dierentialmode and ommonmode ignal, while the powerupply voltage are ued diretly. The gain matrix ontain eight value, only the one relating the input dierential mode with the output dierential mode, dm (), i deired. 2

8 8 IEEE TRANSATIONS ON IRUITS AND SYSTEMS, VOL. 38, NO., OTOER 99 If we hift the origin of the potential ale of () by adding a gauge ignal V gauge () v gauge (t), we obtain M O(jV ot i + V gauge ji) =A()MI(jV it i+v gauge ji). We ubtrat from the new equation the original equation and make ue of the fat that V gauge () v gauge (t) i an arbitrary mallignal 4 voltage. The reulting general relationhip in vetor notation i A()M Iji = M Oji : (2) Example: For the dierentialoutput operational ampli er relationhip (2) beome dm () m() dd () Adm () A m dm () Am() A m dd () Am () A = where the two vetor are obtained by umming the olumn vetor of the orreponding mode matrie M I and M O. Rewriting the equation in alar notation yield (7). 2 The above relationhip i orret for any output load onneted to v DD and. In pratie, however, the gain are meaured for an external referene load onneted between the output of the amplier and a noiefree ground voltage V gg. In thi ae the whole ytem, amplier and load, an be viewed a a iruit having one more input terminal than the amplier alone, namely the ground terminal. Thu the relationhip between the gain meaured with an external load an be obtained by extending the formula for the V gg input and alulating the orreponding tranfer funtion, jaggi, from thi input to all output. Equation (), extended for the ground terminal, read M OjV ot i = A()M I jv it i + M OjAggiV gg. If YL denote the load admittane matrix and Zout the outputimpedane matrix of the amplier, we obtain the additional gain a jaggi =[I (I+ZoutYL) ]ji. Performing the gauge tranformation and ubequent derivation a arried out above for thi extended equation lead to the nal and mot general relationhip preented in thi paper: A()M Iji = M O(I + ZoutYL) ji: (3) If we have(i+ ZoutYL) I in the relevant frequeny range, the gain for load and the gain for no load do not dier by muh and we an ue the impler relationhip (2). For ingleended amplier the matrie Zout and YL degenerate to alar and the orretion term in (3) i =( + Z out Y L )=Z L =(Z L +Z out ) a in (6). Example: For our example with the dierential output operational amplier we aume that the two output are independent and that eah output i loaded with a eparate load Z L and Z L2. Thu the outputimpedane matrix and the loadadmittane matrix are Zout = ZO ; YL = Z O2 =ZL : =Z L2 4 There i atually no ontraint with repet to the amplitude of the gauge ignal, ine it doe not diturb the operating point. Speializing (3) for the dierentialoutput operational amplier and the above load ituation yield the following vetor equation dm () m() dd () Adm () A m dm () Am() A m dd () Am () =2 =2 Z L Z L + Z O Rewritten in alar notation thi i m()+ dd ()+ () = A m m()+a m dd ()+A m () = 2 Z L2 A = Z L2 + Z O2 Z L Z L +Z O A Z L2 ; Z L2 +Z O2 Z L Z L2 + : Z L +Z O Z L2 +Z O2 Note that in order for the dierentialoutput gain term to beome zero (rt equation) the ombination of load and output impedane ha to fulll ertain ymmetry ondition, alo ompare Example 6. 2 Intead of the paraiti gain ued above, ommonmode and powerupply rejetion ratio are ometime more pratial. The relationhip (2) and (3) an eaily be tranformed into equation of rejetion ratio by dividing the whole equation by the deired (nonparaiti) gain. In our example of the dierentialoutput operational amplier thi gain i dm (). Furthermore, it i traightforward to peialize the general formula (2) for the amplier type preented in Setion II. Aknowledgement The author would like to thank to Prof. G. S. Mohytz for a helpful diuion and Jane romley for her omment on the manuript. Referene [] Aaron Fialkow and Irving Gert. The tranfer funtion of general two terminalpair r network. Quarterly of Applied Mathemati, X():3{27, April 952. [2] Dan Hilberman. Input and ground a omplement in ative lter. IEEE Tran. iruit Syt., T2(5):54{547, September 973. [3] David. Ribner and Mile A. opeland. Deign tehnique for aoded MOS op amp with improved PSRR and ommonmode input range. J. SolidState iruit, S9(6):99{925, Deember 984. [4] Eduard Sakinger and Walter Guggenbuhl. A veratile building blok: The MOS dierential dierene amplier. J. SolidState iruit, S22(2):287{294, April 987. [5] Eduard Sakinger. Theory and monolithi MOS integration of a dierential dierene amplier. In W. Fihtner, W. Guggenbuhl, H. Melhior, and G. S. Mohytz, editor, Serie in Miroeletroni, Vol.. HartungGorre Verlag, Kontanz, Germany, 989. [6] Phillip E. Allen and Dougla R. Holberg. MOS Analog iruit Deign. Holt, Rinehart and Winton, New York, 987. [7] John David Jakon. laial Eletrodynami. John Wiley & Son, 2nd edition, 962. :

9 SAKINGER ET AL: A GENERAL RELATIONSHIP 9 [8] Paul R. Gray and Robert G. Meyer. MOS operational ampli er deign a tutorial overview. J. SolidState iruit, S 7(6):969{982, Deember 982. [9] Sudhir M. Mallya and Joeph H. Nevin. Deign proedure for a fully dierential foldedaode MOS operational amplier. J. SolidState iruit, S24(6):737{74, Deember 989. [] Paul A. M. Dira. The Priniple of Quantum Mehani. Oxford Univerity Pre, 4 edition, 958. [] Stephen. Haley and Karl Wayne urrent. Repone hange in linearized iruit and ytem: omputational algorithm and appliation. Pro. IEEE, Vol. 73():5{24, January 985. Eduard Sakinger (S'84 { M'9) wa born in ael, Switzerland, on Augut 3, 959. He reeived the M.S. and Ph.D. degree in eletrial engineering from Swi Federal Intitute of Tehnology (ETH), Zurih, Switzerland, in 983 and 989, repetively. In the fall of 983 he joined the Eletroni Laboratory of the Swi Federal Intitute of Tehnology where he invetigated analog appliation to oatinggate devie. Hi dotoral work wa on the theory and MOS integration of a dierential dierene amplier. Sine September of 989 he i working for AT&T ell Laboratorie in Holmdel, NJ, in the eld of artiial neuralnetwork hardware and it appliation to harater reognition. Hi reearh interet inlude analog iruit deign and parallel ditributed proeing. Joef Goette (S'83 { M'9) reeived the diploma in eletrial engineering in 977 from the Swi Federal Intitute of Tehnology, Zurih, Switzerland. From 977 to 982, he wa employed by the Intitute of Applied Phyi of the Swi Federal Intitute of Tehnology where he partiipated in indutrial reearh projet. In 983 he joined the Eletroni Laboratory of the Swi Federal Intitute of Tehnology where he i a Reearh Aoiate. Hi interet inlude variou apet of iruit and ytem, tatitial analyi, algorithm, omputeraided deign, and ommuniation theory. Walter Guggenbuhl (SM'6) reeived the diploma in eletrial engineering in 95 from the Swi Federal Intitute of Tehnology, Zurih, Switzerland. He wa an Aitant in the Department of Eletrial Engineering of the Swi Federal Intitute of Tehnology for about ix year, while puruing hi Ph.D. degree. Thereafter, he joined ontrave AG, Switzerland, where he wa Manager of an R&D Department, involved in the eld of eletroni iruit and ubytem deign. Sine 973 he ha been a full Profeor of Eletroni iruit Deign at the Swi Federal Intitute of Tehnology and i diretor of the Eletroni Laboratory. Hi main interet are lownoie iruit and omputer hardware for ignal proeing.