THE MILLER OPERATIONAL AMPLIFIER S SETTLING RESPONSE

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1 Optmzatn n rf deves desgn appled n wreless mmunatns, C. Alr, & D. H. Cvarrubas, 9-6 THE MILLER OPERATIONAL AMPLIFIER ETTLING REPONE E. Ruíz-May & F. andval-ibarra nwbush Mex.A. P.I. de C.V. V. Carranza nt., Aguasalentes Ags, Mex ruz@snwbush.m CINVETAV-Guadalajara Unt Av. Centífa 45, 4500 Zappan JAL, Mex sandval@ts-desgn.m ABTRACT Ths paper presents a thrd-rder transfer funtn t mdel the settlng respnse f the Mller peratnal amplfer. The amplfer was smulated (spe) and desgned ardng t desgn rules f a standard.5μm CMO fabratn press. The prpsed mathematal mdel -based n desgn parameters under the desgner s ntrl- s the best settlng apprah reprted up t nw. KEYWORD: Mdelng, rut thery, amplfers. PAC: Bh; 84.0.Bv; 84.0.Le. INTRODUCTION The Mller peratnal amplfer, als alled pamp, s ne f the mst mprtant bas blks fr analg sgnal pressng. Flterng applatns, A/D and D/A nverters, -H ruts and ΣΔ mdulatrs are sme examples f the pamp-based analg applatns sgnal pressng. These applatns generally rrespnd t the tme dman and that s the reasn why tme-based mdelng was develped as a researh feld sme deades ag. Mdelng s mprtant beause t nsttutes a desgn tl n the desgner s atvty. Hene, desgners an ptmze ther desgn by nrpratng addtnal desgn parameters r, alternatvely, by addng wrst ase analyss. One the whle desgn satsfes gven spes, the fabratn f the system s mmnly apprved. The desgn press based n mdelng s a lw-st atvty and wdely remmended due t ts smplty.. PROBLEM DEFINITION Fg. shws the CMO-type Mller pamp, whh s mpsed by a dfferental nput stage and an utput gan stage. Here, physal and eletral haratersts f the nput transstrs are mathed (M n and M n ). By assumng the lnear urrent-vltage haraterst f the MO transstr, t s easy t demnstrate that the utput vltage f the pamp s gven by 0 Jurnal f Appled Researh and Tehnlgy

2 The mller peratnal amplfer s settlng respnse, E. Ruíz-May, et, 0- V DD M p M p4 M p6 C p v,dff C C v ut v n M n M n v - n v bas v bas M n7 Fgure. The Mller pamp. Here C p and C are parast apatanes v ut A0 = (v n vn ) s s () ω ω where the s-alled lw-frequeny pen-lp gan fllws A 0 g g mn mp6 = () g g g g dsn dsp4 dsn7 dsp6 In ths result, [g m ]= and [g ds ]= are bas parameters f the MO transstr []. Fg. shws an ntrns ple ω p that s due t the parast C p and als t g mp. Thus, () s a vald mdel f and nly f the ndtn s <g mp /C p s satsfed. On the ther hand, ω s the ple f the nput stage gven by (g dsp4 g dsn )/C, whle the utput ple gven by g g dsn7 dsp6 ω = () C depends n bth the utput lad (C ) and the whle utput ndutane(g dsn7 g dsp6 ). Frm the pnt f vew f rut analyss, Fg. a s a lumped rut that mdels aurately t (). That fat explans why several authrs have used that rut t analyze the settlng perfrmane f the Mller pamp. Hwever, the rut atually mdels n hgh perentage just t the nn-mpensated Opamp. In prate, even when mpensatn rtera are satsfed, the rut shwn n Fg. a des nt mdel aurately the rrespndng pamp s perfrmane. Thus, n ths paper we demnstrate that an equvalent thrd-rder mdel nt nly mathes the spe smulatn f the lumped equvalent rut, but als mathes the respnse btaned frm spe smulatn f the mpensated Mller pamp at transstr level Vl. 5 N. August

3 The mller peratnal amplfer s settlng respnse, E. Ruíz-May, et, 0- V,dff V ut a) v dff - g mn, v dff C R R C g mp6 v, dff V,dff C V ut b) v dff - g mn, v dff C R R C g mp6 v, dff V,dff C R z V ut ) v dff - g mn, v dff C R R C g mp6 v, dff Fgure. a) end-rder equvalent lumped rut t mdel (); b) equvalent lumped rut t mdel a mpensated pamp; ) Thrd-rder equvalent lumped rut t mdel the Mller pamp nludng a resstr R z. COMPENATED OPAMP When the pamp s phase margn (φ) satsfes the ndtn 45 <φ<60, t s well knwn that stablty s satsfed fr thse ases where the pamp s used n lsed-lp nfguratns; therwse, t must be rretly mpensated. By addng a apatr (C ) between bth stages, stablty wuld be btaned beause the lw-frequeny ple s mved twards lwer frequenes; that predure s als alled the splt-ple tehnque. Applyng KCL t the rut shwn n Fg. b, we fund that the transfer funtn s gven by v v ut dff RR g mn(g mp6 sc ) = (4) s [ R R g C R (C C ) R (C C )] s R R C [ C C ] mp6 ( C C ) Frm ths result we an see a zer, whh ertanly affets the frequeny respnse f the pamp; that s vsualzed by measurng the margn phase at the 0-db frequeny. In rder t elmnate the zer effet, a resstr (R z ) s seres nneted wth C. Table shws the wdth f the transstr fr eah MO deve (see Fg. ), where the length fr all deves s L=μm. Other useful desgn parameters are als presented n the same table. Taken nt aunt the equvalent rut shwn n Fg., we fund that the transfer funtn s gven by vut vdff gmng mp6 CC RzC = R z R C C C C RzC RC R z = s gmp6 R gmp6 s s RCR CR zc CR CRz R RC C RzC C C RC R C (5) 05 Jurnal f Appled Researh and Tehnlgy

4 The mller peratnal amplfer s settlng respnse, E. Ruíz-May, et, 0- where the zer s ertanly elmnated by prpsng R z =/g m,p6 s prpsed. One the stablty s btaned, sme authrs pstulate that a send-rder mdel, equvalent t (), s a suffent representatn t mdel the mpensated pamp fr small-sgnal lnear analyss. uh an apprah s nt a rret ne beause, as we an see n (5), the system s atually a thrd- rder mdel. In fat, the zer effet des nt appear any mre but R z plays an mprtant rle n the transfer funtn, where we are assumng that the denmnatr s mdeled by D(s)=s k s k sk, beng eah nstant k (=,, ) easly dedued wth the help f (5). Deve W (μm) Parameter Value M n, M n 87.5 C 5.0pF M p, M p4, M p6 0.0 C L.0pF M n V DD 5.0V M n7.0 V bas, V bas.5v Table Parameters f the mpensated Mller pamp 4. ETTLING ANALYI In rder t determne the ndtns fr a sutable settlng analyss, the lumped rut shwn n Fg. - mathematally mdeled by (5)- wll be analyzed. The rt lus analyss ndates that the dmnant ple, ω, remans almst n the same pstn as ndated by (). That s true beause that ple s the funtn f bth the Mller apatane C m and the resstr R : ω = RCm gmp7r RC (6) On the ther hand, Table shws the value f eah sngularty, whle Fg. llustrates spe results. As we an see, there s a frequeny f max n whh the thrd-rder mdel fts sutably t the transstr-based respnse btaned frm spe smulatns. The advantage f ths graphal mparsn s that fttng nludes the pamp entre frequeny range,.e. frm DC t the 0dB gan frequeny. The latter wll be als useful t shw the apabltes f the prpsed mdel. Thus, the tme respnse f the mpensated Mller pamp s btaned wth the help f the fllwng mdel: Fgure. pe results f the ruts shwn n Fg. b and, where eah ne s labeled as Mdel- and Mdel-, respetvely Vl. 5 N. August

5 The mller peratnal amplfer s settlng respnse, E. Ruíz-May, et, 0- ngularty nd rder mdel rd rder mdel ω ω z 0 ω 7.7 MHz.59 khz 4.8 MHz 557 MHz.5 khz GHz. GHz Table. Ple/zer values dedued frm the equvalent mdels v(t) = = p t A e = A 4 pt B e t A B 4 5, 0 t T, t > T mn mn (7) where the nstants are alulated frm ntal ndtns. In rder t analyze the settlng respnse, n the fllwng the pamp s used n a fllwer nfguratn. Durng the slewng perd, the nput stage s urrent lmted, whh means that the lp wrks as an pen rut; ths perd s mdeled by (5). Next, by applyng the prpertes f nvlutn t s pssble t dedue, frm (5), the fllwng dfferental equatn: d v(t) d v (t) k kv(t) Akv = dt dt = n (t) (8) Then, by substtutng the nput vltage n (8), v n (t)=v(t)-v (t), the dfferental equatn f the lsed lp nfguratn s btaned. Next, n a fllwer nfguratn the pamp s nput vltage s mpsed by tw mpnents: the utput respnse, v (t), and the step sgnal, v(t). Hene, (8) an be rewrtten as fllws: d v(t) d v (t) k k( A)v(t) Akv(t) = dt dt = (9) As a nsequene, t slve the thrd- rder dfferental equatn, the mplementary slutn fllws = pt v = C e (0) where C are nstants that are btaned als frm ntal ndtns, whle p are the ples f the system. In ths way, ples an be fund by slvng the fllwng haraterst equatn: p k p k p k (A ) 0 () = By knwng the ples values, the general slutn s alulated ne the step sgnal s fully desrbed: 07 Jurnal f Appled Researh and Tehnlgy

6 The mller peratnal amplfer s settlng respnse, E. Ruíz-May, et, 0- t Vstep, 0 t Tmn Tmn v (t) = () Vstep, t Tmn where V step s the step magntude, and T mn =/f 0dB s the tme n whh the step sgnal reahes ts fnal value. Then, sne the system s sngulartes are generally f the fllwng lass: p p, = α ± jβ = σ () t s smple t verfy that the mplementary slutn fllws [ A s( βt) A sn( βt) ] σt αt v (t) = Ae e (4) ne the nput sgnal presents a lnear haraterst, see (), the partular slutn must be a frstrder slutn, r equvalently v (s) = A t A (5) p 4 5 In rder t btan the partular slutn (5) must be substtuted n (9) Vstep A k v (t) = t p T A (6) mn k A Thus, the general slutn n the rse regn, 0 t T mn, s αt αt v(t) = Ae e [ A s( βt) A sn( βt) ] V T step mn A k t A k A (7) where A (=,, ) are alulated frm ntal ndtns as well: A k AVstep A v(0) (8-a) k (A ) T = mn AVstep σa αa βa = v'(0) (A )T (8-b) mn σ ( α β )A αβa = v' ' (0) (8-) A Furthermre, fr t T mn the nput sgnal s V mn, and the partular slutn wuld be a nstant. By assumng v (t)=d, t results that v (t) = A V A step (9) and the general slutn fr t T mn fllws Vl. 5 N. August

7 The mller peratnal amplfer s settlng respnse, E. Ruíz-May, et, 0- AVstep σ(t T ) = mn αt v (t) Be e [ B s β(t Tmn ) B sn β(t Tmn )] A (0) where B (=,, ) are funtn f ntal ndtns that are assated t the tme between bth tme perds,.e. t=t mn. 5. REULT Bas data that defne the sngulartes are σ= rad/s, α= rad/s and β= rad/s, whle small-sgnal parameters f the pamp are gven n Table. In ths desgn, the urrent bas f the dfferental stage s I BIA, = 84μA, whereas the utput stage drves a urrent I BIA, = 4μA. ne the 0dB frequeny s apprxmately 0MHz, the rse tme f the nput vltage (step sgnal) s f the rder f 00ns. Fg. 4a shws a mparsn between the prpsed thrd mathematal mdel and the transstr level spe smulatn. It s lear that the mathematal mdel fts rretly the spe respnse. Even mre, by hangng the rse tme f the nput sgnal t 50ns, t s easy t verfy that the mathetal mdel s respnse rrespnds t that btaned frm smulatn,.e. bth urves present the same settlng tme value. Hwever, a small dfferene durng the slewng perd appears n bth respnses, frtunately even when the thrd-rder mathematal mdel presents an versht, t des nt affet the whle settlng respnse (see Fg. 4b). Parameter Value Unts g mn g mp6 g ds g ds4 g ds7 g ds6 C db C db4 C db7 C db6 C gs F/area F/area 0 F/area F Table. Bas parameters f the Mller pamp fr settlng analyss 09 Jurnal f Appled Researh and Tehnlgy

8 The mller peratnal amplfer s settlng respnse, E. Ruíz-May, et, 0- (a) (b) Fgure 4. Cmparsn between the thrd-rder mathematal mdel and the spe respnse. When the rse tme f the nput sgnal s f the rder f 00ns r hgher, bth urves math eah ther (a); hwever, fr lw values f the rse tme (fr example 50ns), a tny dfferene appears between urves(b) 6. CONCLUION In ths paper, bas netwrk analyss was used t prpse a thrd-rder transfer funtn as a sutable desgn apprah t mdel the tme respnse f the mpensated Mller peratnal amplfer. The amplfer was desgned ardng t desgn rules f a standard.5μm CMO fabratn press, and smulated wth the help f the transstr mdel (LEVEL=) gven by the manufaturer. mulatn results based n bth transstr level desgn and ts lumped equvalent rut rrespnd t the respnse f the thrd-rder mdel whh makes f ths prpsal a sutable desgn tl nt nly fr the IC desgner, but als fr the begnner. The results btaned frm ths prpsal rrespnd t the best settlng mdel reprted up t nw [], []. Vl. 5 N. August 007 0

9 The mller peratnal amplfer s settlng respnse, E. Ruíz-May, et, 0- ACKNOWLEDGEMENT. One authr (E. Ruíz-May) s grateful fr the fnanal assstane gven by CONACyT, Mex. 7. REFERENCE [] Randal L. Geger, Phllp E. Allen and Nel R. trader, VLI: Desgn Tehnques fr Analg and Dgtal Cruts, MGraw-Hll, 999 UA (Chapter ) [] C. T. Chuang, IEEE J. f ld-tate Cruts, C-7 (98) 74 [] Jung-Chen Lee and J.H. Nevn, IEEE J. ld-tate Cruts, C- (986) 478 Jurnal f Appled Researh and Tehnlgy