Operational transconductance amplifier-based nonlinear function syntheses

1576 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 24, NO. 6, DECEMBER 1989 Ope a ional T ansconduc ance Ampli ie -Based Nonlinea Func ion Syn heses EDGAR SANCHEZ-SINENCIO, SENIOR MEMBER, IEEE, JAIME RAM~REZ-ANGULO, MEMBER, IEEE, BERNABE LINARES-BARRANCO, AND ANGEL RODR~GUEZ-VAZQUEZ, MEMBER, IEEE Abs uc -We show ha he ope a ional ansconduc ance ampli ie (OTA), as he ac i e elemen in basic building blocks, can be e icien ly used o p og ammable nonlinea con inuous- ime unc ion syn hesis. Two e icien nonlinea unc ion syn hesis app oaches a e p esen ed. The i s app oach is a a ional app oxima ion and he second is a piecewise-linea app oach. Tes ci cui s ha e been ab ica ed using a 3-pm pwell CMOS p ocess. The lexibili y o he designed and es ed ci cui s was con i med. We will p esen a numbe o nonlinea OTA ci cui s and will discuss wo nonlinea analog unc ion syn hesis ech- niques based On hese OTA basic One syn hesis app oach app oxima ion unc- ions and he o he uses a piecewise-linea app oxima ion. Ac ual ci cui implemen a ions will be p esen ed as well as he expe imen al esul s om se e al 3-pm p-well CMOS , 1. INTRODUCTION es p o o ypes. ATELY, se e al au ho s [1]-[5] ha e been success ully L using he ope a ional ansconduc ance ampli ie (OTA) as he main ac i e elemen in con inuous- ime ac i e il e s. The OTA is a p og ammable de ice’ and has only a single high-impedance node, in con as o con en- ional op amps. This makes he OTA an excellen de ice candida e o high- equency and ol age (o cu en ) p o- g ammable analog basic building blocks. The applicabili y o OTA’s as componen s in he design o linea ne wo ks has been ex ensi ely discussed else- whe e [l], [6] and will no be epea ed he e. The objec i e o his pape is o examine he applicabil- i y o OTA’s as he basic elemen s in he design o nonlin- ea ne wo ks. The e is no much epo ed in he li e a u e on he use o OTA’s o designing nonlinea componen s [7], [8]. Excellen con ibu ions [9]-[ll], [16] a e epo ed o nonlinea ci cui s dealing wi h pa icula impo an nonlinea p oblems. In hs pape , a he han y o ackle a speci ic p oblem, we ocus ou a en ion on a gene al app oach dealing wi h nonlinea basic building blocks using OTA’s as he main ac i e elemen s. No hing special was done o op imize he ci cui pe o mance bu a he o explo e he po en ial and applicabili y o he OTA-based nonlinea sys em app oach. Manusc ip ecei ed Feb ua y 4, 1989; e ised Augus 8, 1989. E. Sinchez-Sinencio and J. Rami ez-Angulo a e wi h he Depa men o Elec ical Enginee ing, Texas A&M Uni e si y, College S a ion, TX 77843-3128. 11. BASIC BUILDING BLOCKS In his sec ion we in oduce he OTA-based undamen- al nonlinea building blocks in ol ed o he syn hesis p ocedu es. A. Mul iplie Block A wo-inpu ou -quad an mul iplie has an ou pu cu en gi en by I, = KMVIV* whe e he mul iplie cons an K, has uni s o ampe es pe squa e ol . I V, and V, can ake any posi i e o nega i e sign, he mul iplie is called a ou -quad an mul iplie . This mul iplie is ep esen ed in Fig. l(a). The co espond- ing OTA-based implemen a ions a e shown in Fig. l(b) and (c). The iangula block labeled a ep esen s a signal a enua o (wi h an a enua ion ac o a); i s unc ion is o equalize he maximum ol age swing o V, and V,. - V,,,, is he usual bias con ol o he OTA. An ac i e a enua o can be implemen ed in CMOS echnology [15]. The signal le el in he mul iplie is es ic ed by a ew hund ed milli ol s o V;, and V z. Al hough no indica ed in Fig. 1, assume he powe supplies o he OTA’s a e V,, and -Vss. The wo op ions o Fig. l(b) and (c) allow us o change he sign o K,. Thus o he ci cui o Fig. l(b) we ob ain B. Lina es-Ba anco is wi h he Depa men o Elec ical Enginee ing, amen o de Elec bnica y Elec omagne ismo, Uni e sidad de Se illa, Texas A&M Uni e si y, College S a ion, TX 77843 and wi h he Depa - (24 41012 Se illa. Spain. Elec omagne ismo, Uni e sidad de Se illa, 41012 Se illa, Spain. A. Rod iguez-Vbquez is wi h he Depa amen o de Elec bnica y ’The ou pu cu en I,, o an OTA due o a di e en ial inpu ,~ is I,, = g,,,c>,,. and g,,, is a ol age (cu en ) con ollable pa ame e [l], [6], and IEEE Log Numbe 8931198. JO2 = - s,,V1= - K( VIz + VSST) Vl (2b) whe e K is a p ocess- and geome y-dependen cons an , [71. 0018-9200/89/1200-1576$01.00 01989 IEEE SANCHEZ-SINENCIO e al. : OTA-BASED NONLINEAR FUNCTION SYNTHESES Vl -VBIAS 1577 2 The ou pu cu en Io becomes 1, = lo, + zO2 = [ K( - BIAS + &si-) - K( u 2 - 'BIAS + 'SST)] '1 (4) (5) Z, = - aKV1V2 = KMVlV2, K, = - uK. A simila analysis o he ci cui o Fig. l(c) yields Zo = aKViV2 = KMViV2, KM = uK. (6) The e o e, we can make he sign o KM posi i e o nega- i e. B. Di ide Block A wo-inpu di ide has an ou pu whxh is he a io o he wo inpu s, mul iplied by a cons an K, wi h dimen- sions in ol s: Vl V, = K,- 2 (7) A symbol o he di ide is shown in Fig. 2(a), whe e n and *We ha e assumed equal K's and h eshold ol ages y's o he OTAs. (a) (b) Fig. 2. Di ide : (a) symbol and (b) OTA implemen a ion d s and o nume a o and denomina o , espec i ely. The co esponding ci cui implemen a ion using he mul iplie symbol is shown in Fig. 2(b). Analysis yields 11 = g,V1 (84 I2 = K,VoV2. (W (94 By Ki chho s cu en law (KCL) we ob ain Il + Z2 = gmV, + K,VoV2 = 0. Thus, he esul ing ou pu signal is p opo ional o he a io o he inpu signals Obse e ha KR can also be ei he nega i e o posi i e and V2 = 0 is no allowed o a oid ou pu sa u a ion (V,). A close look a he ci cui o Fig. 2(b) is equi ed o s udy s abili y. Assuming he dominan dynamic elemen is a pa asi ic capaci ance C' a he ou pu , (9a) is modi ied as z1 + I, = SCPV, (loa) which yields Thus, V, has a pole loca ed a K, s, = 7 V2. LP Hence, in o de o he ci cui o be s able, i s pole mus be in he le -hand plane (LHP), which dic a es ha 3 K,V2 < 0. (104 A summa y o he s abili y condi ions a e pic o ially indica ed by he hype bolas o Fig. 3. C. Squa ing and High-Powe s (Exponen ia ion) Blocks A one-inpu squa e has an ou pu p opo ional o he squa e o he inpu : 30bse e ha we a e applying linea ea men o a nonlinea ci cui . This linea analysis is alid i o a ce ain ins an x, V,( ,) and b( ) a e conside ed cons an ; hen V, will e en ually each i s solu ion in&- penden o he ini ial condi ions. 1578 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 24, NO. 6, DECEMBER 1989 VO 4, Kn< 0 (a) (b) Fig. 3. S abili y egions o di ide . (a) S able o K,,, > 0 and V2 < 0. (b) S able o K, < 0 and V2 > 0. Fig. 4. Exponen ia ion ( aising o a powe ) ope a ion: (a) squa e , (b) cubic, and (c) p h. (a) (b) Fig. 5. Squa e oo e : (a) implemen a ion and (b) OTA implemen a ion. The implemen a ion o he squa e is ob ained by simply using a mul iplie wi h equal inpu s, as shown in Fig. 4(a). To ob ain an exponen ia ion ( aising o a powe ) ci cui wi h an inpu V, and an ou pu o be p opo ional o VIP, whe e p is an in ege g ea e han 2, we equi e (p + 1)/2 mul iplie s o p odd and p/2 mul iplie s o p e en. Fu he mo e, since he p oposed mul iplie s a e o he ansconduc ance ype, he ou pu s mus be con e ed in o ol ages o use as he inpu s o subsequen mul iplie s. This can be done by connec ing an equi alen esis o a he ou pu . An equi alen esis o using an OTA [5], [6] is implemen ed by connec ing he ou pu o he nega i e OTA inpu and g ounding he posi i e OTA inpu . An example o p = 3 is shown in Fig. 4(b). I should be e iden ha a simila p ocedu e can be ollowed o ob ain an exponen ia ion o any o de p; his is symbolically illus a ed in Fig. 4(c). D. Squa e-Roo e Block A one-inpu squa e oo e has an ou pu wi h he nega- i e o posi i e squa e oo o an inpu ol age mul iplied by a cons an o a p ope pola i y, e.g. Fig. 5(a) shows he implemen a ion o he squa e oo e , whe e he ou pu V, is gi en by , VO o = K,- (134 whch yields A mo e de ailed desc ip ion o he implemen a ion is shown in Fig. 5(b). The ci cui will be s able i , a e a pe u ba- ion, he ou pu e ol es owa ds he desi ed ou pu alue. Assume he inpu is ixed a = U,. To s udy he s abili y and he dynamics o he ci cui , a pa asi ic capaci ance Cp a he ou pu is again conside ed. Using he KCL a he ou pu node esul s in he ollowing nonlinea di e en ial equa ion: c - dV0 = g,U, + K, ,' d which can be ew i en as 1 dVn d -.L eg a ing bo h sides o (14b) and sol ing o Vo( ) (when ( g, /KM 1 us < 0) yields 1- whe e K. = hence Fo (g,/K,)U, > 0 he solu ion yields an unbounded ou pu . I is concluded ha a s able squa e- oo e ci cui is ob ained when (gm/K,)y < 0, i.e., KRV, > 0. Fu he - mo e, he pola i y o Vo can be de e mined acco ding o (15b) o (1%). Fig. 6 shows he condi ions o s able ope a ion o he squa e oo e . E. Piecewise-Linea Func ion Gene a o s Diodes in e connec ed wi h OTA's can simula e ideal diodes, hence allowing he c ea ion o a piecewise-linea app oxima ion o any desi ed nonlinea unc ion. The ideal basic building blocks o a piecewise-linea unc ion ap- p oxima ion a e shown in Fig. 7. High- equency imp o e- -. SANCHEZ-SINENCIO e 01. : OTA-BASED NONLINEAR FUNCTION SYNTHESES 1579 *.." Fig. 6. S abili y egions o squa e oo e men s o his basic block a e discussed in he Appendix. No e ha I, = 0 un il he b eaking poin ( ol age e e - ence V,) is eached. The slopes o he linea segmen s a e p opo ional o he g,'s. The diodes can be implemen ed wi h MOS ansis o s wi h hei ga e and d ain ied o- ge he . I a s ep ype inpu -ou pu cha ac e is ic is needed o implemen discon inui ies in he unc ion app oxima- ion, he linea OTA o Fig. 7 can be subs i u ed by an OTA compa a o which ideally simula es a la ge g, and a sa u a ion (ou pu ) cu en o IBIAS. 111. NONLINEAR FUNCTION SYNTHESES We p esen wo app oaches o nonlinea unc ion syn- heses. The i s app oxima ion uses a polynomial ap- p oach, and he second app oxima ion a piecewise-linea app oach. The i s is a a ional app oxima ion ha has he gene al o m o a polynomial unc ion o o a a io o polynomials, i.e., whe e i is a posi i e in ege numbe . In ac , he exponen i can be a ac ional exponen o he o m p/q, whe e p and q a e nega i e o posi i e in ege s. Assume an elemen K,xP/q needs o be implemen ed. This is ob ained as shown in Fig. 8(a). The exponen ia ion blocks a e o he ype o Fig 4(c). I a nega i e - p/q is needed, an addi- ional di ide is used as shown in Fig. 8(b). Nex we discuss a piecewise-linea app oxima ion syn- hesis app oach. Ths app oach consis s o adding ( ans- conduc ance) gain segmen s ha ha e null con ibu ion un il a e e ence ( h eshold) ol age is eached. A simple bu illus a i e example is shown in Fig. 9 whe e a con ex cu e is app oxima ed. The b eakpoin s occu a yl, yz, and V,?. The slopes a e gi en by so = 0, I, = 0, Fig. 7. Piecewise-linea (PL) unc ion gene a o building blocks 'i~l~~ ol Exponen la lon Block Exponen la lon Block (a) (b) Fig. 8. F ac ion powe exponen ia ion: (a) KJ"4 and (b) V'-P/Y. 3 > V 2 > l (a) (b) Fig. 9. Con ex cu e piecewise-linea app oxima ion: (a) , e sus ] cha ac e is ics, and (b) ci cui implemen a ion. No e he con inuing inc ease in magni ude o he slope as he inpu ui inc eases, hus o ming a con ex cu e. The app oxima ion will imp o e as he numbe o segmen lines inc eases. Obse e ha by combining he basic building blocks o Fig. 7, a bi a y unc ions wi h a iable posi i e and nega- i e slopes and b eakpoin s can be app oxima ed. Fu he - mo e, he slopes and b eakpoin s a e ol age p o- g ammable, which gi es an addi ional lexibili y in he unc ion app oxima ion design p oblem. No e ha i a esis i e load simula ed wi h an OTA is used, he slopes become a ios o ansconduc ances which p o ides a e y o V,] > 1580 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 24, NO. 6, DECEMBER 1989 Fig. 10. Ci cui diag am o he OTA [3] used Fig. 12 Nonlinea i y mul iplie e o : ixed VI =1 V and a iable ian- gula wa e o V . Fig. 11. La ge-signal cha ac e is ics o mul iplie . V, = k {0.75, 0.50, 0.25.0.0) V. good empe a u e compensa ion [18] and accu acy im- p o emen . In s ingen applica ions whe e minimum em- pe a u e dependence is equi ed, he use o a esis i e- load OTA is needed. One example o an a bi a y unc ion app oxima ion con aining nega i e and posi i e slopes is discussed in he nex sec ion. De ails on he p ac ical conside a ions o he OTA-based piecewise-linea ci cui s a e gi en in he Appendix. Fig. 13. Nonlinea i y mul iplie e o : ixed = 1 V and a iable VI. IV. EXPERIMENTAL RESULTS Se e al es ci cui s con aining OTA's and ansis o s connec ed as diodes we e ab ica ed using a 3-pm p-well CMOS p ocess h ough (and hanks o) MOSIS. The lin- ea ized OTA used o syn hesize he di e en nonlinea analog unc ions is epo ed elsewhe e [3]. I s schema ic is shown in Fig. 10. The OTA has an a ea o 220x700 pm2 and consumes 10 mW o 5-V supply ol ages. In all he examples (unless o he wise indica ed) he ou pu cu en was measu ed ac oss a lOO-kQ load esis o . A. T ansconduc ance Mul iplie The s uc u e used is as shown in Fig. 1. The measu ed alue o IKMI is 3.3 pA/V2. The ou pu cu en was measu ed ac oss a 100-kQ load esis o . The la ge-signal cha ac e is ics o he mul iplie a e shown in Fig. 11. V, was held cons an (a 0.0, 50.25, 0.50, and kO.75 V), while he inpu V, a ied be ween & 1 V. The nonlinea i y e o is shown in Fig. 12. Fo V,, a iangula 2-V peak- o-peak signal was applied, while keeping V, equal o 1 V. SANCHEZ-SINENCIO e d. : OTA-BASED NONLINEAR FUNCTION SYNTHESES 1581 4250 4750 250 1250 2250 3250 750 1750 2750 3750 quency Spec um o he mul iplie ou pu ol age: V, =1 V and Vz = 2sin? X 103 . Fig. 14. Fig. 15. Modula ion o wo-inpu sinusoidal signals. Fig. 16. Di ide expe imen al esul s: cons an VI and a ying V2 The ou pu cu en p oduced a iangula ol age signal o 660-mV peak o peak. Sub ac ing hs signal om an ideal iangula wa e, he esul ing peak- o-peak e o signal was 17 mV, which yields a nonlinea i y e o o nea ly 2 pe cen . (The ideal iangula wa e is a scaled e sion o he inpu in such a way ha he ampli ude o he e o signal is minimum.) Repea ing he measu emen bu in e - changing Vl and V2 (VI is a iangula signal o 2-V peak o peak), he esul ob ained is shown in Fig. 13. The peak- o-peak e o signal o 23 mV co esponds o a 3.5- pe cen nonlinea i y e o . The asymme y o he mul i- plie (see inpu s in Fig. 1) yields hs dis o ion di e ence when in e changing he inpu s. Making Vl = + 1 V and V2 a 2-V peak- o-peak sinusoidal wa e o m o 1 kHz, he spec um o he mul iplie ou pu shown in Fig. 14 was measu ed. Obse e ha only he second ha monic, 33 dB below he undamen al, is p esen . Fig. 15 shows he mul iplie being used as a modula o whe e bo h inpu signals a e sinusoidal. B. Vol age Di ide The es ed ci cui has he s uc u e shown in Fig. 2(b) wi h IK,I as be o e in Sec ion IV-A and K, > 0. The expe imen al esul shown in Fig. 16 was ob ained by swi ching VI be ween wo symme ical cons an alues Fig. 17. Squa e expe imen al esul s. (+ 1 V) while a ying V, (V, < 0). This esul ma ches wi h he heo e ical esul s o Fig. 3(a). C. Squa e The squa e is ob ained by simply making VI = V, in he mul iplie discussed in Sec ion IV-A. In his pa icula case, K, is nega i e esul ing in he in e ed pa abola shown in Fig. 17. The inpu ange was 1 V. D. Squa e Roo e The basic a chi ec u e used is he one shown in Fig. 5(b). The inpu signal V, is gi en by V, = A + A COS u and 1582 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 24, NO. 6, DECEMBER 1989 esis o load, empe a u e a ia ions a e minimized. The p og ammabili y and lexibili y o he OTA p o ide he po en ial o design ime- a ying nonlinea ci cui s. The expe imen al esul s e i ied heo e ical p edic ions. Imple- men a ions o o he nonlinea syn hesis app oaches [20] a e easible using he basic blocks he e in oduced. The e a e many impo an a eas o applica ion o nonlinea unc ions [22]. One o hem is in neu al ne wo ks [19], [21] as shown by Mead [17, ch. 61. The p oposed OTA-based building blocks can be inco po a ed in a CAD so wa e [12] o ully exploi hei unc ionali y and e sa ili y. Fig. 18. Squa e oo e expe imen al esul s he ou pu , ob ained in he i s quad an , has he o m APPENDIX PRACTICAL CONSIDERATIONS OF THE OTA-BASED PIECEWISE-LINEAR CIRCUITS In his Appendix some p ac ical conside a ions o he OTA-based Diecewise-linea ci cui s a e discussed aking in o accoun; some nonideali ies o he OTA and o he MOS ansis o used as a diode. A simple modi ica ion o he ci cui s o Fig. 7 ha leads o a d as ic imp o emen in hei high- equency pe o mance is also p esen ed. whe e g,, > 0, K, < 0, A =1 V, =11 kHz, g, = 3.2 pmhos, and K, = 3.3 pA/V2. The expe imen al esul s a e shown in Fig. 18 whe e he lowe ace signal is he inpu and he ou pu is shown in he uppe ace. E. Piecewise-Linea App oxima ion Nonideal Ci cui Elemen s The in ended ans e cha ac e is ic is shown in Fig. 19(a) and consis s o h ee linea segmen s. The indi idual slopes due o each OTA a e indica ed in he lowe pa o Fig. 19(a), and he composed esul ing ans e cha ac e is- ics a e shown in he uppe pa o Fig. 19(a). The ac ual OTA ci cui implemen a ion is shown in Fig. 19(b), whe e an op ional diode and ol age sou ce ha e been added a he OTA (2 and 3) o imp o e he high- equency pe o - mance o he ci cui (see he Appendix o mo e de ails). No e ha he slopes o he ans e cha ac e is ics can be easily modi ied by changing he OTA ol age-dependen ansconduc ances. The expe imen al esul s a e shown in Fig. 19(c). To show u he lexibili y, a chip es ci cui was ep og ammed o implemen a i e-segmen cha ac e - is ic o con e a iangula o a sinusoidal wa e o m. The expe imen al cha ac e is ic is shown in Fig. 30(a), and he inpu and ou pu wa e o ms a e shown in Fig. 20(b). The measu ed THD, a e op imally adjus ing b eakpoin s and slopes, was 1.5 pe cen . The chip pho omic og aph o hese expe imen al esul s is shown in Fig. 21. V. CONCLUSIONS The sui abili y o OTA's as he main ac i e elemen o ob ain basic building blocks o he design o nonlinea ne wo ks was es ablished. Me hods o implemen p ac ical nonlinea ci cui s in a sys ema ic design app oach we e de eloped. Two p ac ical syn hesis app oaches we e in o- duced. Obse e ha o bo h app oaches, e e y ime he ou pu cu en is con e ed in o a ol age by an OTA Simple equi alen ci cui s o an MOS OTA and o a diode-connec ed MOS ansis o a e shown in Fig. 22. The OTA is cha ac e ized by i s ou pu impedance (ou pu capaci ance CO and ou pu esis ance R,) and inpu capac- i ance (C,). The diode-connec ed MOS ansis o , as shown in Fig. 22(b), is modeled by a esis ance4 R, in se ies wi h a ol age sou ce V, ( h eshold yol age o he MOS ansis- o ) and an ideal diode, whe e I, is he peak cu en in he MOS diode. Fig. 22(c) shows he equi alen ci cui o he OTA when used as a wo- e minal esis i e elemen o simula e a g ounded load esis ance [6]. The simpli ied analysis ha ollows assumes ha he ansconduc ance g, o he OTA is equency independen and neglec s he pa asi ic capaci ance o he MOS an~is o .~ Also, e ec s due o OTA and diode esis ance nonlinea i ies a e no conside ed. Low-F equency Conside a ions The low- equency equi alen ci cui o he OTA p eci- sion ec i ie o Fig. 23(a) is shown in Fig. 23(b). gwZ2 and RO2 a e he ansconduc ance and he ou pu esis ance o an OTA used as a load. Pa asi ic capaci ances ha e been neglec ed o hls low- equency analysis. S anda d ci cui 41n ac , alung in o accoun he mobili y deg ada ion [14], R, can be mo e accu a ely e alua ed, i.e., R, 'In mos p ac ical cases he ime cons an R,C, associa ed wi h he MOS ansis o is negligible compa ed wi h o he ime cons an s in he ci cui . (28/p0COx)( L/ W). SANCHEZ-SINENCIO e d. : OTA-BASED NONLINEAR FUNCTION SYNTHESES 1583 1.2 A 4 I "c2 (4 esul s. Fig. 19. Piecewise-linea app oxima ion unc ion: (a) ans e cha ac e is ic, (b) ci cui implemen a ion, and (c) expe imen al analysis shows ha o U, > 0: Equa ion (17) can be simpli ied by assuming l/gm2<< RO2, Rol. Then The ac o RJ( R, + RO1) in he i s e m o he igh side o (18) ep esen s an undesi ed a enua ion. The sec- ond e m in (18) ep esen s an o se added o he ou pu signal whch, added o he o se o he OTA, limi s he minimum ampli ude o he signal ha can be ec i ied. Assuming R, << R,, and R,, = Ro2, an es ima ed alue o he second e m is V,/A 2 whe e A,= gm2RO1 is he ol age gain o he OTA 2. Typical alues V, = 1 V and A, = 200 esul in app oxima ely 5 mV o hs second e m. This is also a ypical alue o he o se ol age o an MOS OTA. To educe (18) o he ideal case, i.e., conside he ollowing p ac ical design conside a ions o low- equency applica ions. 1) Use OTAs wi h high- ol age gain in o de o mini- mize he o se e m in (18) and design he OTA yielding he lowes possible o se ol age. Then limi he minimum inpu signal ampli ude acco ding o (18). 2) Selec he W/L a io o he diode-connec ed MOS ansis o such ha R, << R, is sa is ied. Ths will make ROl/( RO1 + R,) = 1. Since he ou pu impedance R, o an OTA is ypically e y la ge, his condi ion in gene al does no equi e an excessi ely la ge alue o he W/L a io o he MOS ansis o . 1584 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 24, NO. 6, DECEMBER 1989 he signal, and o he pa icula case gm, = gm9. (b) high- equency ci cui; pe o mance) and/o design a “be - e OTA” wi h Smalle CO,. Fig. 20. (a) Fi e-segmen ans e cha ac e is ic, (b) T iangula and sinusoidal wa e o ms a 1 MHz. High-F equency Analysis Conside a ions The mos impo an p ac ical ac o limi ing he high- equency ope a ion o he ci cui o Fig. 23 (see also Fig. 7) is he delay ime , equi ed o discha ge he ou pu capaci ance CO, om he nega i e peak ol age -V. (which is cha ged du ing nega i e hal -cycles) o he alue V, equi ed o he diode o s a conduc ing. Assuming, o simplici y, ha he cu en du ing he nega i e hal - cycle is en i ely supplied o CO, (wi h he cu en in R, conside ed negligible), he ollowing ela ionship o he inpu signal V, = V, sino in ol ing i s equency and he del:y ime , can be de i ed om CJdV /d ) = g,V, sin o as6: c., whe e IMAX = g, is he peak ou pu cu en . Fig. 24 illus a es he ypical wa e o m ha is obse ed in he p ecision ec i ie ci cui o Fig. 23 o high- equency ope a ion when , becomes compa able o he pe iod o Modi ica ion o he Basic P ecision Rec i ie Ci cui o Imp o e High-F equency Ope a ion Fig. 25 shows high- equency imp o ed e sions o he OTA- ec i ie ci cui s including a second diode8 D, and a sou ce V, = ( K, + V,J whe e V,, and V,* a e he h eshold ol ages o D, and D,, espec i ely. In hs ci cui he OTAl ou pu ol age does no go in o sa u a ion, bu is limi ed o a minimum (maximum) alue (- V,,) du ing nega i e (posi i e) hal -cycles. A he beginning o he posi i e (nega i e) hal -cycles, he diode D, will be eady o conduc ion so ha no delay due o cha ging o dis- cha ging o he ou pu capaci ance c,, akes place, hence The addi ion o he diode D, and he bias V, allows a d as ic imp o emen in he high- equency pe o mance o he ci cui wi hou excessi ely inc easing he powe con- sump ion. Addi ional powe dissipa ed s ill needs o be supplied by he ba e y V, which abso bs he ou pu cu en o he OTA h ough D, du ing he nega i e hal - cycles, bu his does no inc ease quiescen powe dissipa- ion. The emaining high- equency limi a ion o he im- p o ed ci cui s is ela ed o he low-pass beha io o he ec i ie . The ci cui o Fig. 26 shows a high- equency equi alen ci cui o he OTA ec i ie (assumed linea o , = 0. ’Whe e lb‘< I = Al,C o IV- I < Vss, o he wise l c I IVssl and Ss is * n wo ks simila o he ca ch diode o he ou-amu e sion 1131 o he 6An addi ional delay, no conside ed in he simpli ied analysis p e- sen ed he e bu ha can be obse ed in Fig. 24, is he ime aken o he he nega i e powe supply o age. .* .~ -2 ou pu signal o each he inpu signal once he diode is conduc ing. same ci cui .