On the Active Compensation of Operational Amplifier Based VCVS

REFERENCES Mos o he p e iously epo ed designs ollow as special cases o ill L. B. Jackson, “An analysis o limi cycles due o mul iplica ion ounding his gene al sec ion which is shown o exhibi minimum sensi i - in ecu si e digi al (sub) il e s,” m P oc. 7 h Ann. Alle on Con . Ci cui and Sys em Theo y, pp. 69-78, Oc . 1969. i y. In ac , al hough some new ci cui s uc u es esul , he main I21 J. L. Long and T. N. T ick, “An absolu e bound on limi cycles due o objec i e o ou wo k is o ie all he ad hoc pape s abou oundo e o s in digi al il e s,” IEEE T ans. Audio Elec &ous ., ol. compensa ed ampli ie s oge he and o p o ide guidelines o AU-21, pp. 27-30, Feb. 1973. [31 T. L. Chang, “A no e on uppe bounds on limi cycles in digi al il e s,” wha should be done o ob ain low-sensi i i y ac i e compensa ed IEEE T ans. Acas ., Speech, Signal P ocessing, ol. ASSP-24, pp. 99- 100, VCVs’s. As a consequence, he pape also gi es a amewo k o Feb. 1976. I41 D. C. Munson, J ., and B. Liu, “Na ow-band ecu si e il e s wi h e o compa ing exis ing (and in some cases new) al e na i e designs spec um shaping,” in P oc. IEEE In . Con . Acous ., Speech, Signal o VCVs’s. P ocessing, Washing on, DC, pp. 367-370, Ap . 1979. PI -, “Low-noise ealiza ions o na ow-band ecu si e digi al il e s,” II. ACTIVE COMPENSATION SCHEMES FOR MINIMUM IEEE T ans. Acow ., Speech, Signal P ocessing, ol. ASSP-28, pp. 41-54, SENSITIVITY Feb. 1980. 161 - “ROM/ACC ealiza ion o digi al il e s o poles nea he uni ci cle,” IEEE T ans. Ci cui s Sys ., ol. CAS-27, pp. 147-151, Feb. 1980. The a ailabili y o low-cos dual ampli ie s is one o he mos [71 A. I. Abu-El-Haija, K. Shenoi, and A. M. Pe e son, “Digi al il e s nc- impo an mo i a ions o he de elopmen o ac i e compensa- u es ha ing low e o s and simple ha dwa e implemen a ion,” IEEE ion schemes. All o he known con igu a ions use wo ope a- T ans. Ci cui s Sys ., ol. CAS-25, pp. 593-599, Aug. 1978. PI E. I. Ju y, Theo y and Applica ion o he Z-T ans o m Me hod. New ional ampli ie s and i appea s easonable o de i e s uc u es Yo k: Wiley, 1964. which only equi e a pai o ac i e elemen s. I should be clea ha any VCVS based on wo ope a ional ampli ie s is a second-o de ac i e-R il e which exhibi s a low-pass cha ac e is ic, i.e., a wo-in eg a o s ci cui wi h a ans- e unc ion gi en by On he Ac i e Compensa ion o Ope a ional Ampli ie Based VCVS J. L. HUERTAS AND A. RODRIGUEZ-VAZQUEZ B and and Schaumann [6] ha e shown ha all second-o de ac i e-R il e s can be de i ed om he gene al s uc u e depic ed in Fig. l(a). The ans e unc ion denomina o associa ed wi h bo h possible ou pu s ua, and us*, is A ~s ac -A uni ied ea men o he ac i e compensa ion o VCVS ealized by ope a ional ampli ie s is p esen ed. I is a summa y o low- D(s)=2+s(c,GB,+y(JGB,)+GB,GB,(c,y,-cc,y,). (2) sensi i i y ci cui s uc u es which can be ei he magni ude o phase Rou ine calcula ions shown ha he condi ion o minimum compensa ed. The pe o mance o such a ci cui s is discussed in de ail. sensi i i y wi h complex poles is [6] Expe imen al da a o some o hem a e also included. c(Jy(J = 0. I. INTB~DUCTION Ful illing ha condi ion is equi alen o opening he local The high- equency oll-o o he ope a ional ampli ie im- eedback loop o one in eg a o . Because he con igu a ion is poses a p ac ical limi a ion o hei use ulness in RC-ac i e il e symme ical we may ake c0 = 0 and conside al e na i ely ei he design. Wi h he ad en o low cos in e nally compensa ed dual we, o us2 as he ou pu o he compensa ed VCVS. Hence o h ope a ional ampli ie s, he design o ac i e-compensa ed ol age- con olled ol age sou ces (VCVS) is cu en ly he mos pop& we shall call ype-1 s uc u es he ci cui con igu a ions wi h u,,, as he ou pu o he compensa ed ampli ie . Simila y, we shall way o o e come such a limi a ion. In ha sense, many ad hoc call ype-II s uc u es he con igu a ions wi h u,,z as he ou pu . ci cui con igu a ions ha e been epo ed o ealizing com- Equa ions (4) and (5) show, espec i ely, he ans e unc ion pensa ed ampli ie s bu a clea s a emen o he connec ions o bo h ypes o s uc u es among hese s uc u es is missing and i emains di icul o selec anyone o hem o a speci ic applica ion. In pa icula , mos ype-l: ,GB, + GBP,w, (4) discussion ail o poin ou wo impo an conside a ions. The i s one is a de ailed exposi ion o he di e en c i e ia which can be used o achie ing an imp o emen o he ampli ie ype-II: T,,(s)+ SC,% + GB,G&(w, + CRY,) equency esponse. The second one being a discussion o he I D(s) sensi i i y p ope ies o he p oposed compensa ion schemes. (5) This pape p esen s a uni ied app oach o he gene a ion o ac i e-compensa ed VCVs’s. S a ing om a second-o de ac i e- whe e o ensu e s abili y we mus impose he ollowing condi- ions: R low-pass sec ion, a p ocedu e is o mula ed o inding ci cui implemen a ions o ealize an ampli ie ei he wi h ampli ude y. > 0 and c,y,<O. (6) (maximally la magni ude) o wi h phase ( --) 0) compensa ion. I is wo h no ing [6] ha all pa ame e s a e di e ences o esis o a ios, i.e., e.g. (yi,c,)=xi-yi, wi h OG(xi,y,)Gl, Manusc ip ecei ed Ap il 10, 1981; e ised Feb ua y 3, 1982. The au ho s a e wi h he Depa men o Elec ici y and Elec onics, Uni e - howe e , in o de o a oid la ge sensi i i y due o di e ence si y o Se ille, Se ille, Spain, e ec s, we shall choose ei he xi o y, always equal o ze o. ‘BW’ can be conside ed as a meaning ul measu e since o phase compensa- Assuming ha , he ci cui in Fig. l(b) shows he gene al s uc u e ion only ci cui s uc u es wi h Q 2 I a e o in e es . o any low-sensi i i y compensa ed VCVS and p o ide de ini ion 009%4094/82/0700-0497$00.75 01982 IEEE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-29, NO. i’, JULY 1982 497 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 06,2020 a 13:35:17 UTC om IEEE Xplo e. Res ic ions apply. 498 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-29, NO. 1, JULY 1982 "i i I ““1 (4 (b) Fig. I. (a) Schema ic ep esen a ion o a gene al second-o de ac i e-R il e (b) Gene al ci cui s uc u e o a low-sensi i i y compensa ed VCVS. o he pa ame e s in e m o speci ic esis o a ios as ollows: LYi gi 3 c,=--- I i gj ,i4 Jj' i=1,2 jzz1 y;,L- Gi+3 3 2 Gj i Gj+Go 3 i=1,2 j=l j=4 Yo’ 6 GO 2 G,+Go j=4 (7-a) (7-b) (7-c) whe e ei he g,(G,) o g,+3(Gi+3) is always equal o ze o depending on he sign o ci( y,), i = 1,2. No e ha since he summa ions a e pe o med by a esis i e node, he ollowing addi ional in e connec ion equi emen s can be de i ed: ampli ie one : O~yo-~{(1--sgnyl)yl+(1--sgny2)y2)~1 ampli ie wo : O< -i{(l-sgnc,)c,+(l-sgnc,)c,} <I (8-a) (8-b) (8-c) (8-d) which mus be conside ed when making a pa icula design. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 06,2020 a 13:35:17 UTC om IEEE Xplo e. Res ic ions apply. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL.CAS-29,N0. 7,JULY 1982 499 TABLE I DESIGNEQUATIONSFORTYPE-IANDTYPE-II STRUCTURES; PARAMETER b CORRESPONDSTO GB,/GB, 2 “” phase compensa ion condi ion magni ude compensa ion condi ion III. COMPENSATIONREQUIREMENTS compensa ed design equi es ob aining he highes w,, and Q - Exp essions (4)-(8) allow conside able eedom in choosing he alues compa ible wi h (10). di e en design pa ame e s. In ac , many choices a e possible A magni ude-o ien ed compensa ion scheme has o be di ec ed and some o hem co espond o pa icula cases which ha e been owa ds maximizing he equency ange whe e he magni ude eno ed elsewhe e 111-151. Hence, in o de o ake ad an age o esponse is la . F om (1) his esul s in I -- -- ha eedom we will iden i y me p oblem o wha mu; be op imized in a compensa ed VCVS. Basically, au ho s engaged wi h ha ques ion ha e conside ed wo di e en app oaches, emphasizing ei he magni ude-o ien ed o phase-o ien ed com- WC/W0 Q= (13) J. 1+2g Wo’ pensa ion c i e ia. A design based on he phase-compensa ion mus minimize he Fu he mo e, a condi ion may be de i ed consis ing in maxi- low- equency phase e o [4]. Tha e o , om (l), is app oxi- mixing he 3-dB bandwid h o he ci cui . F om Geige ’s pape [l] ma ely gi en by ha bandwid h is gi en wi h (13) by BW=wo/w. (14) Then a maximum o BW compa ible wi h (13) mus be looked o . F om (9) i is clea ha we mus impose he condi ion Q2i wo in o de o educe he alue o Cp( w) o In he phase-o ien ed case he pu pose o he design echnique is wo old. The emaining phase-e o mus be educed o a minimum while a he same ime ex ending as much as possible he e ec i e bandwid h, BW’, which will be de ined as he equency a which he gain alue is inc eased by 3-dB.’ A e ou ine calcula ions we ge , using (10) BW’= we/m. (12) IV. PRACTICAL CIRCUITS Table I p o ides he design equa ions o ype-1 and ype-II s uc u es. In wha ollows, we shall name hese equa ions in a compac way by using a ma ix-like e minology. Fo ins ance, he equa ion gi ing BW o he ype-II will be e med as (11-7). These design equa ions ha e o be comple ed wi h he in e con- nec ion equi emen s gi en by (7) and (8). Ma ched ope a ional ampli ie s we e no assumed since bo h s uc u e ypes a e sensi- i e o GB misma ch in a di e en way. In ac , we can gene a e di e en compensa ed VCVs’s by conside ing he combina ions o he sign o he pa ame e s in (7) and (8) which a e compa ible wi h he equa ions on Table I as well as wi h he s abili y condi ions gi en by (6). Also, o ype-II s uc u es i is in e es ing o impose he addi ional cons ain : sgn(w,) = w(c,y,) = sgnk, (15) which a oids an unnecessa y limi a ion on he a ainable alue o m, in Table I. F om (11) and (12), i should be clea ha a “good” phase Figs. 2 and 3 show all he esul ing p ac ical ci cui s. In each Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 06,2020 a 13:35:17 UTC om IEEE Xplo e. Res ic ions apply. 500 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-29, NO. 7, JULY 1982 ” 01 00 0 0 ‘6 c /kb M.C.: - =ko-1; G1=O; G2+m ; $ =/ + -1 g5 0 93 G5 “.C.: - =ko-1; G1+= ; G3=O; c = -kob R2 0 I- - - -l l+b +2kob M.C.: $ = ,k, I; G4=0; G4 : %=lk 1; G6=0; G R5 P 0 Fig. 2. Ci cui con igu a ions o ype-1 compensa ed ampli ie s: CI.1; CI.2 [4] CI.3 [5]; CI.4 [4]. Pa ame e b co esponds o GB,/GB,. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 06,2020 a 13:35:17 UTC om IEEE Xplo e. Res ic ions apply. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-29, NO. 7, JULY 1982 501 G5 6 ‘6 4.c.: gl=o; g2- ; ” = q; ” = ko6-+-& dko i ” 02 G3 :- G2 -kob-1 G3 G4 bk M.C.: c, =ko-I; , s 0 -1 2 oajzl n.c.: g4-0; *5+- ; 0 y-E&J 2 “02 =4 P.C.: 0 G4 % cc.: c = -F 0 0 Fig. 3. Ci cui con igu a ions o ype-II compensa ed ampli ie s: CII. 1, CII.2 [I][31 CII.3 and CII.4. Pa ame e b co esponds o GB,/GB,. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 06,2020 a 13:35:17 UTC om IEEE Xplo e. Res ic ions apply. 502 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL.CAS-29, NO. 1, JULY 1982 TABLE II(a) PARAMETERVALUESFORPHASECOMPENSATION; k,,>O; PARAMETER b CORRESPONDSTO GB,/GB* Pi cui .oa a Fe e* signs-and in e conec ion equi emen s. Op imum design pa ame e alues YI Y2 <I 2 m cII.2 kl,Y2)‘0 (C2.Y,F0 Y, 9, = 1 TABLE II(b) PARAMETERVALUESFORPHASECOMPENSATION; k,<O; PARAMETER b CORRESPONDSTO GB,/GB, Ci cui , pa * me e sipns and ince conec ion equi emen s Op imum design pa ame e alues I cI.3 (c1.c2)‘0 (Y,.Y2)‘0 Cl + c2 = 1 Yo-Y,-Y2= I cI.4 Y2>0 k,.c2,Y1)<0 Cl + c2 =-1 Yo-!, =a yoke Yo-Y,-1 yoke -lk,l l+lkol (Y1,Y*)>O k,.c2 n , 4 2 =-1 Y, + Y2 = 1 cII.4 c2>o (C,‘Y,.Y2)<0 Yo-Y,-Y2' 1 Y'Ik lb l-Y2 1 (l+ L) l+ kol l-Yoby, b yokob -1 l+lko -1 -l-c, ----W~kohob+Y~lko~b) l+lkol 1 -!- l+lkol ci cui he ac ual esis o a ios mus be choosen in o de o ul ill A. Phase Compensa ion ei he he phase o he magni ude compensa ion condi ion. Nex , Phase compensa ion condi ions ha e been applied o bo h we will conside he pa icula choices o bo h compensa ion ype-1 and ype-II s uc u es. The esul s a e summa ized in condi ions. Table II(a) o k, > 0 and Table II(b) o k. ( 0. A ew ema ks Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 06,2020 a 13:35:17 UTC om IEEE Xplo e. Res ic ions apply. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL.CAS-29,NO.l,JULY 1982 503 TABLE III(a) PARAMETERVALUESFORMAGNITUDECOMPENSATION; k,>O; PARAMETER b CORRESPONDSTO GB,/GB, TABLE III(b) PARAMETERVALUESFORMAGNITUDECOMPENSATION; k,<O; PARAMETER b CORRESPONDSTO GB,/GB2 Ci cui “o iln”m ‘ie+ipn na ame e alues Y Y 0 Y1 2 Cl ‘2 a e no ewo hy: 1) No e om (I-2) and (11-2) ha 1 c2/ and ] y2 1 ha e o be selec ed as high as possible in o de o a ain a maximum alue o w,. Hence, he in e coMec ion equi emen s in ol ing such a pa ame e ha e been chosen as uni y whene e i does no gi e ise o a con adic ion wi h he design equa ions. 2) The alues o he di e en pa ame e s a e lis ed as unc ions o y. and k,. In mos cases, he designe is ee o choose he pa icula alue o y. bu , om he exp essions o BW’ and $J (see Table I), i should be clea ha y. mus be selec ed as small as possible in o de o op imize he compensa ion. 3) An addi ional cons ain on he minimum alue o y. is de i ed om a mo e de ailed s abili y analysis including he ope a ional ampli ie ’s second pole [7]. I esul s in w2 lW22 2 GB, W2: ” GB,( w21+ w22) + y” GB, ( w2, + w*2)2 ’ ICZYZI (16) whe e w2i is he loca ion o he i h (i = 1,2) ope a ional ampli- ie ’s second pole. B. Magni ude Compensa ion Table III(a) and III(b) depic he esul s o applying he magni ude compensa ion condi ions o bo h ypes o s uc u es. Fo e e y ci cui , he pa ame e alues allowing o a ain a maximum BW a e shown. These pa icula alues ha e been de i ed by a compu e -aided nume ical analysis on he c, and y, alues. V. DISCUSSION Fo con enience, we shall conside sepa a ely wo cases de- pending on he sign o k,. A numbe o conclusions ollow o bo h o hem. A. k,>O F om a phase compensa ion poin o iew he ci cui CI.2 exhibi s pe o mance supe io o ha o ci cui CI. 1. I is due o he ac ha o he same y. he o me allows o ob ain a highe (lowe ) alue o m($), de ined in Table I, han he la e . I mus be ema ked ha Reddy’s ci cui [4] is a special case o ci cui CI.2, o y. = l/k,. The maximum alue o m is ob ained o he ci cui CII.2 [3]. Ne e heless, he ci cui CI.2 allows o a ain a lowe phase e o and highe BW’ han CII.2 whene e y. o he o me is selec ed, espec i ely, o be A he o he hand, he maximum BW is ob ained o ci cui CII.2 in acco dance wi h Geige ’s pape [l]. Mo eo e , i is impo an o no e ha ci cui s CI. 1 and CII. 1 ha e in ini e inpu impedance, and in addi ion ob aining a BW alue which is no Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 06,2020 a 13:35:17 UTC om IEEE Xplo e. Res ic ions apply. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-29, NO. 1, JULY 1982 42 J 40 . 38 . 36 - 0 34 32 . 30 - 28 . 26 24 _ 22 . 20 - 18 . 16. 14. 12. 10 8- 6- 4- 2 10 20 30 40 50 60 70 100 200 300 400 500 800 1000 (KHz) (4 PHASE T(jw) .” -20 -30 x -40 -50 -60 -70 -80 -90 -100. -110. L -lZO- -130- -140' -150- -160- -170- 00 0 -180 - 10 20 30 40 50 60 70 100 200 300 400 500 800 1000 (b) Fig. 4. Ci cui CII.2 compensa ed ampli ie o k, = 10.9. (a) Magni ude cha ac e is ics. (b) Phase cha ac e is ics. con inuous ace lines co espond o he heo e ical esponses. + co esponds o he measu ed esponse o he uncompensa ed ampli ie . * co esponds o he measu ed esponse o he magni ude compensa ion. D co esponds o he measu ed esponse o he phase compensa ion. (KHz) oo a om he one ob ained by he Geige ’s ci cui (see Table CI.4 allows lowe phase e o s han CII.4. The o me has been 9. p e iously epo ed by Reddy o a = 1, a de ined in Table II(b) B. k,<O [4]. Howe e , o magni ude compensa ion, he maximum BW is ob ained om ci cui CII.4. Fo phase compensa ion, he ci cui s CI.4 and CII.4 yield he Finally, i is in e es ing o no e ha he phase compensa ion maximum alue o m, de ined in Table I. Mo eo e , he ci cui condi ion emains s ill alid o ype-1 s uc u es no ma e wha Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 06,2020 a 13:35:17 UTC om IEEE Xplo e. Res ic ions apply. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL. CAS-29,N0. 7,JULY 1982 505 42~ 40. 38. D 36. 34. 92 30. 28. 26. 24. 22. 20. 18. 16- 14. 12- 10. 8- 6- 4- 2 L 10 20 30 40 50 ho 70 100 200 300 400 500 800 1000 (4 PHASE T(jw) -180 -170 -160 -150 -140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 IO 20 30 40 50 6n 7n 10'1 200 3no 4 )o 500 8on im 03 Fig. 5. Ci cui CII.4 compensa ed ampli ie o k, = - 10.9. (a) Magni ude cha ac e is ics. (b) Phase cha ac e is ics. (KHz) (KHz) he deg ee o misma ch. This is an ad an age when compa ed wi h phase-compensa ed ype-II ci cui s. Qui e in he con a y, o magni ude compensa ion, bo h ypes a e sensi i e o he misma ch, as can be seen on Tables I and III. VI. EXPERIMENTALRE~ULTS As an example o he pe o mance o he di e en ci cui s, we show he magni ude and phase cha ac e is ics o he CII.2 and CII.4 conside ing bo h magni ude- and phase-compensa ion cases. Fig. 4 depic s he magni ude and phase esponse o he ci cui CII.2 wi h k, = 10.9. Al e na i ely, Fig. 5 depic s he co espond- ing cha ac e is ics o he ci cui CII.4 wi h k, = - 10.9. Fo bo h ci cui s we ha e used ~A747 dual ope a ional ampli ie s. ACKNOWLEDGMENT The au ho s wish o hank o he e iewe s o hei sugges ions o imp o e he p esen a ion o his wo k. REFERENCES [I] R. L. Geige , “Ampli ie s wi h maximum bandwid h,” IEEE T ans. Ci cui Sys ., ol. 24, pp. 510-512, Sep . 1977. [2] A. M. Soliman and M. Ismail, “Ac i e compensa ion o ope a ional ampli ie s,” IEEE T ans. Ci cui Sys ., ol. 26, pp. 112-l 17, Feb. 1979. 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