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IEEEJOURNALOF SOLID-STATECIRCUITS,OL.SC-9,O.6,DECEMBER1974 353

Macromodeling of

Operational

In tegra ted Circu it

Amplifiers

GRAEME R. BOYLE, BARRY M. COHN; DONALD O. PEDERSON, FELLOW, IEEE, AND

JAMES E . SOLOMON , MEMBER, IEEE

Abstracf—A macr om odel h as been d evelop ed for in tegr at ed

ci rcu i t (IC) op amps which prov ides an exce ll en t p in -for -p in repre-

sen ta tion . Th e m odel elem en ts a re t hose wh ich a re common t o

mos t cir cu it s im u la t or s. Th e m a cr omod el is a fa ct or of m or e t h an

s ix t imes les s complex than the or iginal ci rcu i t, and prov ides s imu-

la ted cir cu it r esp on ses t ha t h ave r un t im es wh ich a re a n or der of

m agn it ude fa st er a nd less cost ly in com pa rison t o m odelin g t he

op amp a t t h e e le ct r on ic devi ce le ve l.

Exp r es sion s for t h e va lu es of t h e e lement s of t h e macromodel a r e

developed s ta r t ing from values of typica l re sponse cha racte r is t ics

of the op amp.Examples a re g iven for th ree represen ta t iveop amps.

I n addit ion , t h e per formance of t h e macromodel in lin ea r a nd non -

lin ea r s ys tems is p r es en t ed . For compar is on , t h e s imu la t ed cir cu it

p er forma nce wh en m odelin g a t t he device level is a lso d em on -

strated.

1. INTRODUCTION

I

NTEGRATED circu it (IC) simulators have proven

t o be a useful tool i;o the IC design engineer . None-

theless, their widespread acceptance in the design

of large-scale in tegra ted circuits and IC subsystems has

been im peded by excessive simu lat ion cost s an d incr eas-

ing convergence problems. Presen t simulators model

sem icon du ct or devices a t t he p-n ju nct ion a nd 2-t er min al

element level. Becaqse c,f the la rge number of these de-

vices in lar ge-sca le IC yyst em s, t he an alysis can surpasst he com pu ter ’s m em or y ca pa bilit y, sim ula tor cir cu it -size

capability, or the inheren t numer ical accuracy of the

computer . Even if ~n adequa te simulator and computer

a re available, the required simulat ion t ime makes the

a na lysis fin an cia lly im pr act ica l. Thi~ pa per descr ibes

one solut ion to this problem: macromodels which have

been developed for IC’S such as operat iona l amplifiers

a nd compara t or s .

The idea and use of macromodels in elect ron ic circuit

design is ver y common at the system level. For example,

in developing an analc)g signal processor , one migh t

u tilize a nu mber of ideal volt age amplifiers, int egr at or s,

Ma nus cr ip t r ece ive d Augu st 2, 1974; r evis ed Augu st 16, 1974.

Th is r esea rch wa s sp on sor ed in p ar t by t he J oin t S er vices E lec-

t ron ics P rogr am u nder Con tr act F 44620-71-C-0087 a nd by t he

Na ti ona l Sci ence Founda t ion unde r Gran t GK-17931 . Th is p ape rwa s p re se nt ed a t t h e I nt er n at ion a l S olid -S ta t e Cir cu it s C on fer -

ence, Phi ladelphia , Pa. , February 1974.G. R. Boyle and D. O. P eder son a re wit h t he Depa rt men t of

E lect r i ca l Enginee r ing and Computer Sciences and the E lect ron ics

Resea rch Labora tory , Univers ity of Ca li forn ia , Berke ley, Ca li f.B. M. Coh n is wit h I nt e)l Cor por at ion , S an ta Cla ra , Ca lif.

J . E . Solom on is wit h N at ion al S em icon du ct or Cor por at ion ,San ta Cla ra , Ca li f.

a nd ot her su bsyst em block s. In effect , a va riet y of zer o-

or der cir cu it m odels a re u sed. To deter mine th e a ct ua l

syst em per forma nce, a pr ot ot ype cir cu it is con st ru ct ed

a nd test ed a t t he device level. Th e size a nd com plexit y

of t oda y’s in expen sive IC’S a re la rge; t her efor e, t he cost

of u sin g pr esen t simu la tor s for design a nd eva lu at ion

can be very large. The cost for large IC’S can only be

ju st ified if ver y la rge ma nu fa ctu re is a nt icipa ted. Th e

cos ts a n d ot her p roblem s ca n be r elieved by t he clevelop-

m en t of m acr omodels for IC’S wh ich p rovide a n a dequ at e

pin -for -pin r epr esen ta tion of t he IC. For digita l IC’S,

logic sim ula tion a n d macr omodels h ave been d evelop ed

for digita l logic blocks [1], [8]. For analog IC’S, th is

p ap er d es cr ibe s a ver y effe ct ive macromodel t h at h a s be en

developed for IC op a mps [2], [3], [9].

Th e a im of m acr om odelin g is t o obt ain a cir cu it m odel

of an IC or a por t ion of an IC which has a significant ly

r eclu cecl complexit y t o p rovicle for sma ller , les s cost ly

sim ula tion t im e, or t o permit t he s im ula tion of la r ger IC’S

or IC systems for the sa me time a nd cost . In the macro-

model for IC op amps shown in Fig. 1, a reduct ion of

approximately 6 in branch and node count has been

a ch ieved wh ile pr ovidin g a ver y close a ppr oxim at ion t o

t he a ct ua l per formin g op amp, i.e., a ccu ra te m odelin g oft he in pu t a nd out pu t ch ar acter ist ics, differ en tia l- a nd

common-mode ga in ve r su s fr equency charact er is t ics , qui-

e scent dc cha ract e ris t ics , offs et cha ract er is t ics , and la rge -

s igna l cha ract er is t ics , s uch as slew r a te, ou tpu t volt age

swin g, a nd sh or t-cir cu it cu rr en t lim it in g. F ur th er , s in ce

much of a simulat ion run is involved with itera t ive

a na lysis t o a n equ ilibr ium cir cu it solu tion , t he r ed uct ion

of 60 to 80 p-n junct ions in an actua l op amp to the 8

junct ions in the macromodel of Fig. 1 indicates bet ter

h ow much fa st er a nd ch ea per ca n be t he sim ula tion u sin g

t he m acr omod els in st ea d of device-level moclels. Th e r e-

su lt s wit h amplifier s, t im er s, a nd filt er s t ha t a re cit ed in

th is paper show that a reduct ion in t ime of 6 to 10 istypical.

In ma ny design or eva lua tion sit ua tion s, it is n ot nec-

essary to model an op amp in all of its performance

ch a ra ct er is tics . F or example, maximum shor t -cir cu it cu r -

r en t limit ing may not be of in terest . If the elements in

t h e macromodel wh ich p rovid e t h is fea t ur e a re e lim ina ted ,

fur ther simplifica t ion of the macromodel is obta ined. As

an example, the simulat ion t ime of the filter in Sect ion

IV is reduced by a factor of 1.4 if the cur ren t and voltage

lim it er s a re om it ted .

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354 IEEEJOURNALOF SOLID-STATEIRCUITS,ECEMBER1974

7 (Vcc)

1

RCIJ Rcz Vc+I11c,

:Rp

+va–6

3.(+)

@

-~ --INPUT STAGE IN TERSTAGE OUTPUT STAGE

Fig.1. Cir cu it d ia gr am of t he op amp macr omod el.

II . MACROMODEL DEVELOPMENT

The circuit model for an IC op amp which is devel-

oped in t his pa per is sh own in F ig. 1. Th e con figu ra tion ,

wit h a su it able ch oice of pa ramet er s a nd elem en ts, a ccu -

ra tely models a broad class of IC op amps. For a given

Op amp, t he m odel pr ovides a n essen tia lly pin -for -pin

cor respon den ce wit h t he op amp, a nd a ccu ra tely r epr e-

sen ts t he cir cu it beh avior for n on lin ea r dc, a c, a nd la rge-

s igna l t r ans ient r es pon s es .

The circuit of Fig. 1 is subdivided into three stages.

The in put sta ge consists of idea l t r ansist or s ~1 a nd Q,z

a nd t he a ssocia ted sour ces an d pa ssive elements, Th is

st age pr odu ces t he n ecessa ry lin ea r a nd n on lin ea r dif-

ferent ia l-mode (DM) and common-mode (CM) inpu t

ch ar act er ist ics. F or con ven ien ce, t he st age is design edfor unity voltage gain. The stage can be designed to

pr ovide desir ed volt age a nd cu rr en t offset s. As br ou gh t

ou t in t he n ext sect ion , t he ca pa cit or CB is u sed t o in tr o-

duce a second-order effect for the slew ra te [4], a nd thb

capa cit or Cl in t rod uces a s econ d-or der effe ct t o t he p ha se

response.

The DM and CM voltage gains of the op amp are

pr ovided by the linea r intersta ge a nd out put sta ge ele-

m en ts con sist in g of Gm , G,., R%, Gb, and Roz. Th e fu nc-

t ion of ea ch elem en t is pr esen ted in t he n ext sect ion . Th e

dom inan t t im e consta nt of the op a mp is pr odu ced with

t he in ter na l feedba ck ca pa cit or C2. A feedba ck con nec-

t ion in the m acr om odel is u sed for Ca in or der t o pr ovide

t he n ecess ar y a c ou tpu t r es ist an ce ch an ge wit h fr equ en cy.

In addit ion , the two nodes of C2 can be made ava ilable

to the out side wor ld in order tha t the circuit designer

can in t roduce the same compensa t ion modifica t ion as

might be added to the actual op amp. Not ice the com-

plete isola tion th at exists bet ween the input a nd the in-

ter ior sta ges. This leads t o a sim plifica tion of the fre-

qu en cy an d t h e s lew i-a t e p er forman ce s.

The ou tput st age provides the proper dc and ac ou t-

pu t resist ance of the op amp. The elements Dl, Dz, Rc,

[

Q,2

Q9

39 k R5

Q,, Q,O

3k R4

* 1Q13SA

6

Q,T100

~ R8

:j;:~,c. I INPUT I ,N:;TX

NETWORK STAGE

+&

1°

OUTPUT

STAGE

F ig. 2. Cir cu it dia gr am of t he ICL8741 op amp.,.,,,..,

and Go produce the desired maximum short -circuit cur -

ren t . The element s D3, Vc and 114j VE are voltage-clamp

cir cu it s t o pr odu ce t he desir ed maximum volt age excu r-

sion.

Th e circuit m odel of F ig. 1 ha s been developed using

two ba sic macr omode lin g t ech niqu es : s implifica t ion an d

bu ild -u p. I n t he s implifica t ion t ech niqu e, r ep re sen ta t ive

por tion s of op amp cir cu it ry a re su ccessively simplified

by u sin g sim ple idea l elem en ts t o r epla ce n um er ou s r ea l

elements. Thus, the fina l model using this approach

bea rs a st r ong resemblan ce t o t he r eal circuit . In Fig. 1,

t h e in pu t s ta ge de sign is a n example of t he s implifica tion

t ech niqu e. I n t he bu ild-u p t ech niqu e, a cir cu it con figu -

r at ion com posed of idea l elem ent s is pr oposed to meet

ce rt a in ext ern a l cir cu it s pecifica t ion s wit hou t n ece ss a rily

resembling a por t ion of an actua l op amp circuit con-

figura tion. The build-u p technique is em ployed in t he

d evelopment of t he ou t pu t s ta ge.

To illu st ra te t hese a spect s fu rt her , con sider t he sch e-

mat ic diagram shown in Fig. 2 of the 741-type op amp

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BO1-LEtC1l.:ACROMODELINGOFOPERATIONALAMpL~~lERs 355

TABLE I

DI:SICiN I?CjUziTIONSFOR THE OP ANIPhL4cRoMoD,m,

V, = ~ = 25.85 mV for 300 K

1s, = I.sD, = 1~~4 = 8.10-” A

R, = 100 k!l

C.=+-c,R

1.2=1.–+

~, = IcI / IBI

(32 = IC2/ IB2

R. = 200/IRB

‘s2=d’ +%)1—= v*/Icl

9?n1

R ., = l/ 2m fo ~BC ,

c, =

l ?p =

Ga =

$ tan Ad

( Vcc + V. E )’/(F’, – VCC(.21CJ – V. E IE E)

I /Rc ,

G1

c. =R ,, (C iMRR)

R ,, = Ro-.c

R oz = Rout — Rol

avDRC1Gb = –—

R2ROZ

Ix = (21 JGbR , – Isc

I

R0,18C

SD1= ISD2 = Ix exp — — v,

(In tcr sil ICL8741 ). Th is t ype is t he m ost common , gen -

era l purpose IC op amp. In developing a macromodel

using the simplifica t ion technique, the circu it ry em-

ployed for biasing can be replaced with ideal passive

elem en ts (pu re cu rr en t a nd volt age sou rces).

Sim ila rly, t he a ct ive loa cl a nd ba la nce-t o-u nba la nce

convert er in t he input stage can be replaced with idea l

elemen ts. Fin ally, it is n ot n ecessa ry t o u se composit e

t ransistor s in the input stage. Thu s, as sh own in Fig. 1,

a , s im ple d iffer en t ia l s ta ge ca n be p rop os ed t o m oclel a c-

cu ra tely t he n on lin ea r in pu t ch ar act er is tic of t he op amp .

Th e op amp m acr om odel is developed k eepin g in m in d

exist in g IC sim ula tor s. Th erefor e, t he m odel con ta in s

on ly elem en ts wh ich a re common t o m ost IC sim ula tor s

(i.e., r es is tor s, ca pa cit or s, in du ct or s, d ep ende nt cu r ren t

s ou rces, in depen den t s ou rces , d iod es , a nd bip ola r t ra n-

sistor s). In addit ion , effor t is made to minimize the

n umber of p -n ju nct ion s. Th ese n on lin ea r elem en ts m ak e

n ecessa ry it er at ive an alysis t o obt ain t he equ ilibr iu m

sta te of the circu it . A reduct ion of the number of non-

lin ea r elem en ts lea ds t o sm aller sim ula tion t im e.

F or t he in pu t st age, ou r in vest iga tion s sh owed t ha t a t

lea st fou r idea l ju nct ion s wer e n ecessa ry t o pr ovide t he

n ee de d b ala n ced , n on lin ea r beh a vior in t h e macr omodel.

It wa s det er min ed t hat t he sim plest a rr an gemen t is t ha t

of Fig. 1 where the four idea l ju nct ions were obtained

wit h two idea l t ra ns is tor s, ea ch m ocleled wit h t he lowes t

order Ebers–Moll (E–M ) t ransistor model which in-

cludes two ideal p-n junct ions and two dependent cur -

r en t sou r ce s.

For the out pu t stage, a simplified model of an actua l

op a mp does n ot pr ovide t he best solu tion . A st rippecl-

down class-AB stage with ideal t ransistor s leads to a

br an ch cou nt , of over 13 in com pa rison wit h 11 br an ch es

in t he ou tpu t st age of F ig. 1. In a cldit ion , t he cla ss-AB

stage must be augmented with voltage limiters in the

dr ive cir cu it ry t o lim it t he volt age excu rsion a t t he t ra n-

sistor bases to t he supply potent ia ls. It was foun d tha t

t he idea lized bu ilt -u p p roced ur e p rovid es a n ou tp ut s ta ge

wh ich is con sid er a bly s impler .

III. PARAMETERS AND ELEMENT VALUES

OF THE MACROMODEL

I n t h is s ect ion , exp res sion s a re d evelop ed t o r ela te t he

performance of the op amp and the macromodel to the

parameter s and elements of the macromodel. A sum-

mary of all design equa t ions is present ed in Table I.

Th e det erm in at ion of t he elem en t valu es of t he ma cr o-

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356

model proceeds from the input , t r ansfer , and output

ch ar act er ist ics of t he op amp.

The Input S tage: Icl and CB

The value of the necessa ry collector cur ren t of the

first stage is established by the slew r ate of the op amp.

If the op amp is connected as a voltage follower , the

pos it ive going s lew ra t e L%+is

(1)

where an n-p-n stage has been assumed [4]. From a re-

a r r angemen t of t h is exp res sion

For a qu iescent situ at ion , equ al collector cu rr en ts a re

u sed in t he in pu t st age ICZ = IcI.

Th e n ega tive goin g slew r at e SE - is sm aller beca use of

the cha rge-storage effect s in the input stage which is

modeled by C~ [4].

or

(3)

21C,c. = ~B_ – c,. (4)

If SE’ < Sri-, the macromodel should be modified to

u se p-n -p t ra nsist or s in t he in pu t st age. In t he equ at ion s

above #R,+and 81?- s hou ld t h en b e in t er ch anged.

In a ddit ion to th e tr ansien t slew r at e effect s, th e ele-

men t CE a lso in tr odu ces a desir able modifica tion t o t he

ac response of the CM gain of the macromodel.

The Trans is t or Parameters

Th e va lu es of 131a nd #Z for t he two idea l t ra nsistor s

a re obt ain ed fr om th e specifica tion s for th e a ver age in-

p ut bia s cu rr en t IB a nd t he desir ed level of in pu t cu rr en t

offset IBOJ.

IB1=IB+LB$, I.,=IF~. (5)

pl=~, & = ~. (6)

Th e volt a ge offs et Vo, for t he m acr omodel is p rod uced

by specifyin g differ en t sa tu ra tion cu rr en ts Is for t he t wot ra nsist or s. Assu me a given va lu e for Isl of QI

(7)

where VT = hT / q = 0.02585 V a t T = 300 K. A sim ila r

expr ession h olds for Icz = Icl

VBE2 .Ic, = 1s, exp ~

T(8)

IEEEJOURNALOF SOLID-STATEIRCUITS,ECEMBER19

vos = VBE1 – vBE2

= V.ln*. (9S2

This leads to

I S2 “-%-=1+%1 “T he In pu t S tage: R ., an d R ,l

Va lu es for t he r esist or s R,l = R ,z a re der ived fr om

t he r equ ir ed va lu e of t he O dB fr equ en cy j~ ~E of t he fu lly

compensated op a mp. The O dB frequency is approxi

ma tely th e pr odu ct of t he DM volta ge goin g a ~~ a nd t h

–3 dB cor ner fr equ en cy fs t E of t he ga in fu nct ion

fOdB = aVDj3dB. (11

Th e cor ner fr equ en cy ca n be est im at ed u sin g a Miller

effect a pp roxima tion in t h e in t er ior s ta ge.

The DM voltage gain a t very low frequencies is

(12

av~ = (G.R J (G5ROJ . (13

G. is chosen to be equal to l/R,l in order to obta in

con ven ien t slew r at e expr ession a s in (1). Th e la st t hr e

exp re ss ion s le ad t o

or

R., = ‘—-.2~f0‘BC2

(14

(1

Alt er n at ely, a r ela t ion sh ip b etween fod~and Sn+ can b

wr it ten u sin g (1).

‘0 ‘B = 2TR f;IcJ “(1

The value of R.l is usu ally sma ll, of t he or der of 2/g~

Rcl and RC2 should be small in order that sa tura t ion o

t he in pu t sta ge (a nd con comit t ent Ia tch up of th e op am

model) is a voided wit h m axim um in pu t. Th e r esist an ce

R,l a nd R.2 in t he in pu t st age a re int roduced t o pr ovid

a degree of freedom with respect to slew rate and O dfr equ en cy, a nd t o sim ula te bet ter cer ta in op amps wh ic

u se em it ter r esist or s for slew r at e en ha ncem en t, e.g., t h

LM118. R.l is found from the I)M voltage gain of th

fir st st age, wh ich for con ven ien ce is t aken t o be u nit y.

v5_ DIR., + @,R ., ———

= A= 1. (1

Vi*~ + (A + ORI + : + (B2 + DR.2

.

Th e offs et volt age is

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BOYLEe~Uz.:MACROMODELINGOFOPERATIONALMPLIFIERS

T he Inpu t S tage: IBB a nd R n

The value of the dc current source in the input stage

for equ a l collect or cu r rent s is

(19)

The resis t or RE is a dded t o pr ovide a fin ite GM in pu t

r esist an ce. Beca use t he cu rr en t sou rce IDE is oft en r ea ]-

ized wit h a n n -p-n t ra nsist or , t he r esist an ce R B is taken

a s it s ou t pu t r es is ta n ce

(20)

where VA is the ear ly voltage of the device. VA for a

sm all n-p-n t ra nsist or is typica lly 200 V.

T he In pu t S tage: Cl

To in tr odu ce excess ph ase effect s in t he DM amplifier

r esponse, another capacitor Cl is added in the input

stage. The second pole of the DM gain funct ion is lo-

ca ted a t

p, = – l/2Rc,C,. (21)

Notice tha t there is no in teract ion amongst the three

ca pa citor s beca use of t he u se of u nila ter al devices a nd

stages.

The excess phase a t ~ = f. ~Bdu e t o t he n on dom in an tpole pz is

%rfOB 2C,

’44 = ‘a n-’ IP21 = t an -’ (2r f0 .,)(2RCIC,) = t an -’ ~.

(;2)

Th e ph ase m ar gin of t he DM open -loop r espon se is t hen

& = 90° – A~. (23)

The n ecessa ry va lu e of Cl to pr odu ce t he excess ph ase is

C’, = $ t an A+. (24)

DC Pow er Dv ain

To model the actual dc power dissipat ion of an op

amp, a r esist or RP is in tr odu ced in to t he m acr omodel.

For the circu it of Fig. 1, in a quiescent sta te, the power

d is sipa tion is

P, = Vcc21c1 + VEJE, +(Vcc + v..)’

R,(25)

‘Th e n ecess ar y va lu e of RP to p rod uce t his d iss ipa tion is

R= =(v,, + VEE)

P. – Vcc21c1 – VE.IEE’(26)

In a typical op amp, most of the cur ren t drain from

t he volt age su pplies is du e t o diode-r esist or cu rr en t de-

fining pa ths. Therefore, a s the supply voltages are

357

changed, the current drain var ies almost linear ly and

RP will con tin ue t o model a ccu ra tely t he power clis sipa -

tion.

The Interstate: G., R ,, and G,,,,

As in dica ted ea rlier , t he coefficien t G. of t he volt age

depen den t cu rr en t sou rce Ga va is ch osen equ al to l / R~l

for con ven ien ce. Sim ila rly, t he va lu e of Rz or G~ can be

a rbit ra rily ch os en . On ly t he p rodu ct is det ermin ed by t heDM ga in . F or “a ct ive r egion ” con sid er at ion s, t he ch oice

of R2 is n ot impor ta nt . H owever , it m ust be kept in min d

th at t he volt age r espon se a t n ode b is lin ea r wit h R2. If

too large a value of v~ is developed dur ing a t ransien t

excu rsion t hr ou gh t he a ct ive r egion of t he op amp, a con -

s ider able disch ar ge or r ecover y t im e ca n be en cou nt er ed

a fter t he a ct ive r egion excu rsion . To pr even t t hese dis-

charge delays in rela t ion to actual op amp behavior , a

sma ll value of R2 should be used. Empir ically, a value

of 100 k ~ is fou nd t o be a ppr opr ia te.

I f a secon d volt age-con tr olled cu rr en t sou rce is in tr o-

d uced acr os s Rz, the CM voltage gain response can be

in tr odu ced. Th e CM volt age ga in in t he in pu t st age fr om

vi. to v, is approximately unity since Rn is la rge. Th e

CM voltage gain from the input to VOis then approxi-

mately

The differen t ia l voltage gain fr om the input to v~ is

v~~~. G.R , = EL R,. (28)

v,n~~ c1

The CM reject ion rat io (CMRR) is the rat io of the two

ga in s [5]

1

CMRR = a: = Rc,G; ”(29)

Therefore,

———““ = (CM;R)R.,

(30)

The dominant behavior of the CM frequency response

will be approximately the same as the DM frequency

r espon se except t ha t t he pr esen ce of t he ca pa cit or Cm i n

t he in pu t st age in tr odu ces a t ra nsm ission zer o in t he CM

gain funct ion at – l / RBCB.

T he Ou tpu t S tage: R ~l, R U,, an d Gb

The output stage provides the desir ed dc a nd a c out -

pu t r esist an ces a nd t he ou tpu t cu rr en t a nd volt age lim i-

t at ion s. F rom F ig. 1, it is seen t ha t t he ou tpu t r esist an ce

a t ver y low fr equ en cies for t he qu iescen t st at e is

R 0“ t = R,, + R ,,. (31)

At h igh fr equ en cie s, RO, is sh or ted ou t by t he (cu rr en t)

Miller -effect ou tp ut ca pa cit an ce a cr os s it d ue t o Cz. Th e

effect ive sh un t ca pa cit an ce is C,fi e CZ( 1 + R2 Gb). For

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358

th e situ at ion wher e a la rge loa d r esist an ce is pr esen ted

to the macr omodel, the corner frequency of the ou tpu t

imped an ce is

(32)

For frequencies well above th is value, the ou tpu t re-

s is ta n ce is RO1.Therefore,

R,, = R , . . . . (33)

Wit h th is va lue est ablish ed, Roz fr om (31) a nd G~ fr om

(13) a re

R,, = Ro., –R ,, (34)

and

aV~RC1

“ = Z,E,”(35)

T he Ou tpu t S tage: Cu rren t L im itin g

In th e ou tpu t st age of F ig. 1, t he desir ed out pu t-cu r-

r en t lim it in g is pr ovided by t he elem en ts GcV6, Rc~ Dl,

D2, and Rol. Th e R c, GCV6 combin at ion is a n equ iva -

len t t o a volt age-con tr olled volt age sou rce (wh ich is n ot

a va ila ble in sim ula tor s su ch a s pr ogr am SP ICE). T hu s,

V,..t = V6 a ls o app ea r s a cr os s Rc. If bot h of t he volt age-

clamp diodes D3 and D4 are off, the maximum

to the ou tpu t is the ra t io of the potent ia l a cross

and RO1

V. = V, I n $:

Ix Maximum cu rr en t t hr ou gh D, or D,.

18~, Sat u ra t ion cu r re nt of d iod es D,, D,.

current

Dl, D2,

(36)

(37)

Since Rol is known, 1~~1 can be established once Ix is

determined. In Fig. 3, a reduced por t ion of the ou tpu t

sta ge is shown which applies for a posit ive ou tpu t ex-

cursion, and where a very small load resist ance is as-

sumed. A Th 6ven in equ iva len t for G~v~ a n d R02 is u sed.

Th e Th 6ven in op-en -cir cu it a va ila ble volt age is a r~vi..

An idea l volt age-con tr olled volt age sou rce V. is a lso u sed

in place of GCV6 and Rc. Assume fir st t ha t t he volt age

a vDv,i. is n ot la rge. Th e ou tpu t cu rr en t flowin g t hr ou ghthe re sis t or Rol t hen pr odu ces on ly a sm all volt age dr op.

The polar ity of the volt age v. is such as to forwar d bias

diode D,. If the voltage drop across Ro, is sma ll, t he

cu r r en t t h rough D1 is very small and can be neglected.

As a v~vi, in cr ea ses, so does IL a nd the voltage dr op

a cr oss Rol. As t he la tter a ppr oa ch es t he ‘(ON” volta ge

of Dl, i.e., t he volt age for a ppr ecia ble cu rr en t t hr ou gh

the diode, the increasing cur ren t from the source IDI

flows t hr ou gh t he d iod e. I L is t hen a ppr oximat ely lim it ed

beca use of t he expon en tia l in cr ea se of 1~1 wit h r es pe ct

IEEEJOURNALOF SOLID-STATEIRCUITS,ECEMBER197

v~ ROI

-m-

DI ~IXlL~ +

R02

+ +v~

avdvinV.

. . .

F ig. 3. S implified cir cu it d ia gr am of t h e ou t pu t s ta ge

to I~Rol. Th e a ppr oxim at e lim it in g con dit ion is fou nd

from

I. = I.., exp 9T

Th e lim it in g va lu e of

con dit ion a t th e input .

r en t fr om Gbvb is I~~x

Im ax = Ix

(38

lx is det erm in ed by a n over dr ive

Th e s hor t-cir cu it a va ila ble cu r-

+ I scs = 21c,R ,G ,,. (39)

A t ypica l va lue of &.X is 100

F rom t he equ at ion a bove,

I –I –IxSD1— ,SD2—

A.

@d-R*) ’4F or la rge r equ ir ed va lues of Rol, the value of ~fl~l can

be ext remely small which may lead t o numerica l diffi-

cu lty. In many applica t ions when the ou tpu t resist ance

is n ot cr it ica l, a smaller value of Rol can be us ed neglect -

in g t he exa ct r ea liza tion of Ro–~c, e.g., if lsD1 = ~~1,

R,, C-S ~v~n ~. (41Sc 81

R oz is t hen in cr ea sed t o Rout – ROI, a nd G~ is decr ea sedto main ta in the same value of the G~ Roz product.

I n or der t o a pp roxima te well a volt age-con tr olled volt -

age sou rce , Rc must be very sma ll. If the volt age drop

across Rc is to be only 1 percen t of VD1 or VD!2 ,

R.=&100IX

In ~ IK .SDl

(42

Th e n ecessa r y va lu e for t he volt age-con tr olled cu rr en

sou rce GCV6 is

(43

Th e Ou tp ut S t age: V oltage L im it in g

Th e ou tput volta ge excur sion is lim ited by th e volta ge

sou rce -d iode clamp combina t ion s Vcj D3 and VE, D 1

shown in Fig. 1. With a large, posit ive ou tpu t volt age

su ch a s t o for wa rd bia s D3

v+ut = Vcc – Vc + VD3

l..+=Vcc–Vc+V.ln~- (44

As indica ted a bove, the diode curr en t is limited to the

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BOYLE43U1.:MACROMODELINGOFOPERATIONALMPLIFIERS 35’9

TABLE II

CIRCUITDATA .4NDGUMMEL-POON TRANSISTORPARAMETERSFORTHE ICL8741OP AMP

CLrCUit Data-.

Guom.1-Poon Tr an sis tor ?a r.m et ms

‘E1.mmt Nodes value.mMODEL_13NPi_ NGP_ 0FII=209. _ 8QM=2. 5__

,?~RLI= 670 .._

02 17 1.0< + RC=300. CC7=1.417P TF. I.ISN TX= 405. N

R2 Ez 16 50. { + CJE. O.55P CJC=O. S6P 1s. 1.26 E-15 V2=178.6

—R3— 02 18 l.o K____+. CZ= 1653._ IK=l.611M _NE.2. o PE=c .60_

Fib 02 08 3.0 K + MS=3. PC. O.45 ric=3.

04 05 39. K .MO13EL

—zz——

SNP2 NGP 13FM=400. BRM=6. I Ril= 185.

12 26 77. + _.RC= 15. CC S.3.455P_. iF=O.76N__TR. 243. N_

Q7 12 25 22. + CJE=2.80P CJC=l.55P IS= 0.395:-15 IVA.267. O

32 2s 100. + C2= 1543. IK=lO. OOH

—~: —

NE=2. O

02 21 50. K

PE=o . 60

.—.+ —-—— .ME=3. Pc=o. k5RIO Zk 27 40. K

XC=3. _.!400EL 8PN1 PGP QFM= 75. ERM=3.8

Rli

Ru= 500.

02 22 50. K + RC=150. CC S=2.259P TF=27.4N TR=2540. N

.c.— 15 19 30. P__+__ _cJE. o.io P___cJc=i.05P___ Is= 3.i5E-i5 vA.55. il_

&l 10 07 13

10 06 11

—:$—— 14 09 13

Q4 15 09 ii

14 16 17

—.:;— 15 16 18

01 14 16

10 10 Oi

__::___ 09 10 01

010 09 05 08

oil 05 05 02

DNPI + C2= 1764. IK=270. ou NE=2,0 ?E=O.45

ONP1 ME=3. PC= C.45 tlc=3.

oPNl__:M00EL_BPN2_ PGP..._ - BFM=117. _--_--B RM=8.8 RB= ..80. _

8PNi + Rc=i56. TF.26.5N TR=2430. N

6NP1 + CJE=4.05P CJC. Z. 80P Is= 17.6E-15 Vfi=57.94

P.NP1 ——_— .— .——— ——— ——-. .—. ——

13NP1

QPPJ1 + CZ= 478.4 IK=590.7u N:=2. o PE=o.60

BPN1_.+ –— . .._. ME=4 . . . ..—. .PC=O.60 -–—

BNP1 .MOOEL BPN3 PGP

14C=4..

BFH=i3.8 13RM=1.4

ENPi +

RB=1OO.

RC= 80. CC S=2.126P TF. z7.4N T?= 55. N

_Q12 _D4 04 01

0i3A 20 04 0:

Q13B 19 04 01

_fli4 ___Ol 20 26

Q15 20 26 12

Qlb of 15 21

_G:7_ .19 21 23

018 20 27 24

219 20 20 27

_ Q20__.02 24 25

a2i 22 25 12 BPN1 + CJE=i.10P CJC=2.40PQZ2

IS= 0.79E-i5 VA=79.45

15 22 02 BxPi + Cz= 12!9. IK=$o.55u h ’E. z .o P := C .6P

_.-oz3A_., 02 i9 24 BP)/5_” +___ ME=4 . PC= D.60. —___ MC=+. -,

ciz3a 02 i9 15 OF’N6 .MOOEL BPN6 PGP BFII= 19.

Q24 22 22 C2 BNP1

C!w. i.fl RU= 6SC.

[:lNc=ioo, T<r=2G.5N Tfl=2i20. M

cJE. i .9 0P.— cJC=2.40P— Is=0,0063E-15 ‘JA=i67 -l_

+ C2=5?. I+9K IK=80.55U

+

NS=Z. O PE=o.60

Mg.4 . pG. o.60 UC=+.

OPNi + CJE=O.lOP __._ CJC=O.30P__ IS= 2.25E-15

aPN 3—-+ C’2=84.37K IK.5. ooi3N N[=z. o

BPN4 + t+E=3, PC= O.45 !ic=3.

3N?2__ .HOQEL__BPN4_ PGP_ aFM=i4.8 _ BRM=l.5_.

BNP1 + RC=120. CCS=Z.126P

r3NPi + CJE=O. IOP

TF.27.4N

CJC=0,90P Is= 2.25E-15

L3!lPi. _.+.— cz. ?4.37K_ IK=ii:.8U _ !! E=2. O___

3!4?1 + tlc=3. PC. O .45

BNPi

NC=3.

.I{ODFL BPN5 PGP !3Fli= 80.

BPN2

ORM=I.5

__+__– -.. RC=170. _ _.__. _TF=26.5N _

~vA=s 3.55_

PE=O .95

RD=i60. _

TR= 220. N

VA=83 .55

Pi. C.i5 _

!2.0=1100.TR=9550.N—

short current I,yo+.The nece ssa ry

Similarly,

v. = v.. + v.”,- +

The Comp let e Model

bia s volt age is

V, in ~;. (45)SD3

TABLE I II

OP AMPPERF ORMANCEHt iR.kCT~RISTICS

8741 LM741 LM118Device-Level 8741 Data Data

Model Macromodel Sheet Sheet

v InI&~T (46) C, (pF )

I“ D4 SE+ ( v/p s )

A su mm ar y of t he design equ at ion s for t he pa ra met er s

of the macromodel is given in Table I. An example of

th e u se of t hese equ at ion s is given in t he Appen dix. Th e

:st ar tin g poin t is op a mp per forma nce da ta .

Th e p ar ticu la r IC u sed t o illu st ra te t he design p roced -

u re wa s t he object of a n ear lier stu dy [6]. Th e configu -

rat ion of th is IC was established to be that of Fig. 2 and

t h e t r an sis tor s wer e cha ra ct er ized by t h e Gumnlel–P eon

(G-P ) p ar amet er s of Ta ble 11.1

P rogr am SPICEha s been u sed t o es ta blis h t h e per for n l-ance and character ist ics of the op amp [7]. The r esult s

are summarized in Table III, column 1. These values

are used in the Appendix to develop the element and

par am et er valu es of t he m acr om odel. As br ou ght ou t in

t he Appen dix, sever al par amet er s of t he macr om odel

cou ld be chosen a rbit r a rily:

1 A sligh t m odifica tion of t he G–P pa ramet er s of t he ou tpu tt r ans is tor s ha s been made to p roduce a typ ica l leve l of maximumshor t-circuit available current .

SR- (v//.Ls)

I fl (nA)In.. (nA)V., (mV)

Ct iRR (dB)R .., (Q)

R .-.c (Q)

I sc+ (mA)

Z,SC-(mA)

v+ (v)

v- (vj

Pd (mW)

30

0.9

0.72

256

0.7

0.29!?

4.17.10’

1.219.103

16.8

106

566

76.8

25.9

25.9

14.2

–12.7

59.4

300.8990.718

255<1

0.2984.16.10’1.217.10’

16.310656676. S26.226.214.2

–12.759.4

300,670.62

802012.10’

103

20

90

75

252514.0

–13.5—

510071

120622,. ]05

16.10$40

10075

252513

– 13—

T = 300 K(V, = 25.85 mV),

Is, = Ism = 8.10-” A,

R , = 0.1 MQ,

wh er e 1s 1, 1s~3 ar e the satur at ion currents of the fir st

t r an sis tor a nd t h e volt a ge-lim it er cliod es , r es pect ively.

In a ddit ion , t he m ajor com pen sa tion ca pa cit Qr is fixed

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360 IEEE JOURNALOF SOLID-STATEIRCUITS,ECEMBER19

TABLE IV

MACROMODELARAMIJTICRS

8741 LM 741 LM 118

T (K) 300 300 300

Is, (A) ~ .10-16 8.10-16 8.10-16

I~D, (A) 8.10-16 8.10-16 8.10-16

R, (ka) 100 100 100

C, (pF) 30 30 ~

C~ (pF) 7.5 2.41 2.042

& 52.6726 111.67 2.033.103

P2 52.7962 143.57 2.137.103I~E (PA) 27.512 20.26 500R~ (m~) 7.2696 9.872 0.401,$, (A) 8.0925 .10-’6 8.309.10-1$ 8.619.10-’8R., (Q) 4352 5305 1989

R., (o) 2391.9 2712 18846“, (pF) 4.5288 5.460 2.098

~ (kQ) 1,5.363U (pmho) 229.774 188.6 .502.765Gc~ (nmho) 1.1516 6.28 5.028R,, (n) 76.8 32.13 ‘32.13Ro, (Q) 489.2 42.87 42.87G* (mho) 37.0978 247.49 92.792Ix (A) 100.138 —

1~~, (A) 3.8218.10-3’ ~ .10-16 8.10-16

Rc (Q) 0.1986 .10-3 0.02129 .10-3 0.00279 ~10-3Gc (mho) 5034.3 46964 358000Vc (v) 1.6042 1.803 2.803VE (v) 3.1042 2.303 2.803

‘7 ~o device-level model

. macro model

-5{l’I 1 I I I

Q 10 20 30 40

t Ipsec )

F ig. 4. S imula t ed volt a ge followe r s lew r a te p er forman ce u sin g bot h d evice-leve l mod els a nd macr omod els .

by the type of op amp under study or is chosen appro-

pr ia tely. For the case at hand, Cz = 30 pF.

Th e r em ain in g va lu es of t he pa ramet er s of t he m acr o-

model a re presented in Table IV, column 1.

IV. COMPARISON WITH DEVICE-I,EVEL iklODELS

Basic Macro model Perf orm tance

Th e valu es for t he m acr omodel of Table IV, colum n 1

wer e u sed t o define an ext er nal model in pr ogr am SPICE .

The same set of computer runs was made as lead to the

op a mp per for ma nce r esult s of Table III, colum n 1. The

r esu lts for t he macr omodel ar e pr esen ted in colu mn 2 of

t his t able. It is seen t hat th e compar ison is excellen t for

bot h sma ll-s ign a l a nd la r ge-s ign a l exper imen t s.

To pr ovide a fu rt her compa ris on , t he la rge-sign al, sle

r at e per for man ce for a voltage follower is sh own Fig.

for bot h t he device-level m odel a nd t he m acr or nodel. I

is seen that the responses are very similar.

The presence of C@ p roduces a step in the init ial response of the voltage follower . From simple theory [4]

t he jump sh ou ld be a ppr oxim at ely

For t h is example,

AVin = 10V and AVOU,= ~ (10) = 2.5V.

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361OYLEetal.: MACROMODELINGOFoPERATIONALmplifierS

20k

.:*W———ig. 5. A m on ost able t im e dela y cir cu it .

From Fig. 4, the observed jump for the macromodel is

3.6 and 3.4 V for t h e d evice-le vel model.

A m ea su re of t he com plexit y of t he t wo op amp models

can be obtained by compar ing the node and branch

cou nt s of ea ch cir cu it . F or t he device-level model, wh er e

each G–P t ransistor model has 2 internal nodes and 7

branches, the tota ls are 81 nodes including the da tum

node, a nd 193 br an ch es . F or t he m acr omodel, wh er e ea ch

E-M transistor model has no in ternal nodes and 4

branches, the tota ls are 16 nodes and 28 branches. The

ra t ios for the two models are 5.1 for the nodes and 6.9

for the branches. The number of p-n junct ions in the

device-level m odel is 52 a nd 8 in t he m acr om odel, a r at io

of 6.5.

The tota l computer cent ra l processing unit (CPU)

t ime on a CDC 6400 to simulate the voltage follower

slew r at e per formance is 39.2 s for t he device-level mod el

a nd 4.0 s for t he m acr omodel, a r at io of 9.8. An a lt er na te

com pa rison is obt ain ed if on ly t he sim ula tion t im es for

t he in it ia l st at e a nd t he t ra nsien t a na lyses a re u sed. Th e

impr ovemen t r at io is t hen 12.0.

F or t he dc a nd a c sim ula tion s, t he device-level m odel

t o ma cr omodel CPU tim es, for t he a na lyses on ly, ha ve

t he r at ios of 3.9 a nd 6.0, r espect ively. Note th at t he im-

pr ovemen t r at io is less for th e dc a na lyses.

A R egenerat iv e T imer

Th e mon ost able t ime dela y cir cuit of Fig. 5 pr ovides

a good t est of t he a bility of t he ma cr omodel to per for m

a s d esir ed wh en deman din g n on lin ea r per formance is r e-

quired. The voltage a nd t iming levels of the circuit ofFig, 5 ha ve been ch osen t o provide a maximum stress on

the op amps with respect to voltage limits, cr it ica l volt-

age switching levels and speed of response, slew-ratelim ita t ions, etc. The output waveforms of the circu it as

pr edict ed by pr ogra m SP ICE u sing both t he device-level

model and the macromodel are shown in Fig. 6. It is

seen t hat t he r esponses com par e closely. The leadin g and

tra iling edges of the ou tpu t pulse differ in t iming by less

than 1 time step of the computer outpu t , i.e., bet t er than

3 percen t of the overa ll pulsewidth. The tota l CPU times

for the simulat ions using the two models a re in the ra t io

of 8.9. If the common outpu t t ime is deleted, the im-

pr ovem en t r at io is 9.6.

Vou! (

15–

[

lo- ),

v)

5- (h

o- , ,,

-5- P

k

-lo– <1$

. device-level model

o mocro model‘h.

-15+ I I [o 20 40 60

t ( , u sec )

F ig. 6. Sim ula ted ou tpu t pu lse r espon se of t im e dela y cir cu it

us ing both dev ice -level model s and macromodels.

100k 100k

399k

10k 10k 10k

v:

— — . T

F ig, 7, “Ring of Thre e” bandpas s filt er .

TABLE V

l?.ESPONSE DATA FOR THE , ’[RING OF THREE” BANDPASS FILTI;R

Design Actual Gain Gain GainCenter Cen ter Ma gn it ude Ma gn it ude Ma gn it udeFrequency Frequency at a t

f., (kHz) f.a (kHz) ~R o.~f.d l.lf,a

1 0.998 11.01 4.806 4.311

(O.998) (11.00) (4.806) (4.311)

2 1.996 12.24 4.898 4.368

(1.996) (12.23) (4.896) (4.367)

10 9.’934 112.1 5,547 4.398

(9.934) (107.1) (5.542) (4,401)

Numbers inparenthesesefero resultsithdevice-levelodel.

An Active R C Filter

To fur th er check t he secon d-or der a c r espon se of th e

macromodel, the simple “Ring of Three” op amp filter

of Fig. 7 was designed for a center frequency of 1 kHz

and a ~ of 10. The frequency response from program

SPICE for the two models is summarized in Table V.

Again, itis seen that the comparison isvery close.

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362 IEEE JOURNALOF SOLID-ST.4TECIRCUITS,DECEMBER19

At higher fr equencies, the phase response of the (1 expr essions of Ta ble I. Th e fin al r esu lt s a rc pr esen te

lllH z) 741-t ype op amps com es in to effect . A r espon se in column 1, Ta ble IV.

compar ison for the two models as shown in rows 2 and From the slew rate performance of the 8741 a s give

3 indica tes that the macromodel is providing the proper in Table III

phase response.

The tota l CPU simulat ion t imes to determ ine the dcs,+ = 0.90 V/ps and SIZ- = 0.72 V/ps.

state a nd t he fr equ en cy r espon se u sin g t he two m odels For the given compensa t ion capacitor of Cz = 30 p

have the rat io of 5.8. If the common output t ime is these values lead to

om it ted, t he r at io becomes 6.8.In t his a pplica tion, th e n on linea r per for ma nce of th e ICI = ;SE+C, = 13.50 PA

op amp is not of major in terest . In order to check on

the improvement of computer run t ime for a reduced C, = ~? – C, = 7..5OPF.

m acr om odel, t he volt age a nd cu rr en t lim it in g cir cu it ry..

Th e a ver age ba se cu rr en t is 256 nA a nd th e desir ed baof Fig. 1 was omit ted. The simulated response of the

filter did n ot ch an ge, of cou rse; h owever , t he t ota l CPUcu r rent offs et is 0 .7 nA.

run t ime was reduced by a factor of 1.4. I,, = 256.3 nA, IB , = 255 .7 nA

l’.MACROMODICL PARAMETERS FOR OTHER IC OP AMPS /3,= 52.6727, /32= 52.7962.

Th e det ailed, p recis e p er forman ce ch ar a ct er ist ics for

an individual op amp as obta ined from the use of the

device-level m odels a re u su ally n@ a va ila ble. Th e pr e-

cision used in the numerical example of th is paper is

employed in or der t o obt ain a n a ccu ra te est im at e of t he

per formance of a macromoclel in rela t ion to a known

reference. Exper imental r esu lt s with actual op a mps

cou ld inclucle s ign ifican t measuremen t inaccu racie s,

Typically, one has a da ta sheet or averagecl exper i-

menta l data for a type of op amp which is to be in-

cluded in a system. As an example of the choice of

ma cromodel pa ra meters in th is situat ion , two fur ther

examples a re given . In Ta ble III, colu mn s 3 a nd 4, mea -

su red t ypica l op amp da ta a re given for bot h t he LM741

and the LMl 18. The macromodel parameters corre-

sponding to these data are given in columns 2 and 3 of

Ta ble IV.

It is possible to in t roduce programming into a simu-

la tor t o det ermin e a ut om at ica lly t he m acr omodel pa r am -

eters. This has been done at one locat ion for program

SPICE.In th is situa tion , all tha t is necessary to define a

specific op amp model is to list its ch ar act er ist ics on a n

op amp “model card’) in much the same way tha t one

~urren t ly defines a t ra nsistor model by specifying it s

ch ar act er ist ics on a t ra nsist or model ca rd. Wh en a n op

amp ch ar act er ist ic is n ot in pu ted, a defa ult va lu e is u sed.

A high level of pr ecision is u sed in t his example in or d

to obta in an accur ate compar ison of macr omoclel pe

formance in rela t ion to that of the op amp modeled

the device leve l.

The necessa ry emit ter curren t source for the inp

stage is ~~fl = 27.512 PA. The value of the CM emit t

r es is tor is

R, = 200/ I BB = 7.2696 Ma

where a value of VA = 200 V ha s been used.

For ~1, the assumed value of satura t ion current

8”10-’6A which is a typicalvalue for a small n-p-n I

transistor.o proclucethe desiredinput offsetvoltag,

0.299 mV

IS2=8’0-1’(10) =80’2510-16For a fully compensa ted op amp with a rolloff of –

dB/octa ve, the O dB frequency can be calcu la ted fro

the product of the gain and the value of the frequenc

at which it is mea sur ed providing that the frequency

well a bove t he cor ner fr equ en cy of th e ga in ch ar acter

ist ic. From the da ta of column 1, Table III,

f, dB = (1.219” 103)(103 ) Hz = 1.219 .106 MHz.

Th e va lu e of t he collect or r esist or s of t he fir st st age is

APPENDIX R., = ——1-— = 4352 Q.2~f0dBc2

THE 8741 MACROMODEL The value of the reciprocalof g.,forthe firststage is

In t his Appen dix, a n um er ica l example is u sed t o illu s-

tr ate the development of the pa rameters of the op a mp1

— = 1915 Q.

m acr oln odel. F or t he examp le, t he r es pon se ch ar act er is -9m1

tics of the 8741 op amp are used as determined by sev- The required value of the emit ter resistor is

er al sim ula tor r un s u sin g device-level modelin g. Th e cir -R ., = 0.9814(4352 – 1915) = 2392 Q.

cu it ’ of F ig. 2 t oget her wit h t he t ra nsist or pa ramet er s of

Ta ble I I h as been a na lyzed t o obt ain t he ch ar act er ist ics The final element for the input stage is (71, w hich pro

whicb a re summar ized in column 1, Table III . duces the nondominant pole of the gain funct ion . Fo

Th e developmen t pr ocedur e follows t he sequ en ce of A+ = 16,80°

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BOYLEet d.:MACROMODELINGOF OPERATIONALMPLIFIERS

C, = ~ tan 16,80° = 4.529 pF,

Th e va lu e of t he r esist or RP to s imu la t e power d is sip a -

t ion for *15 V supplies a nd a dissipa t ion of 59.4 mW is

R. = 15,363 kQ,

F or t he in ter st age, Rz is t aken to be 100 k~ and

~a = #- = 229.7 74 /Jld10 .

For a CMRR of 106 ~~ (199.5” 10S)

GG.

CM = CIV!RR= 1.1516 nmh o.

In the ou tpu t st age, the desired dc and ac ou tpu t re-

sist an ces a re 566 ~ a nd 76.8 0, r espect ively. Th er efor e,

R o, = 76.8 Q, R OZ= 489.2 Q.

The value of Go to pr ovide the correct DM voltage gain

of 417. 10s is

aV~R.l

‘b = KR;; = 37 “097”

The maximum cur ren t th rough the diode DI or

Ix = 21ClGbR, – I.c = 100.14A

where the desired value of I~ c is 25.9 mA. Wit h

va lues, the sa tu ra t ion cur rent s of DI a nd D2 are

I SD = Ix exp – ‘~~’z = 3.822.10$’2 A.T

The va lu es for t h e app roxima te volt a ge -con t rolled

a ge sou rce a re

Dz is

these

volt-

l’ort h e volt a ge -clamp cir cu it s, t h e s at u ra t ion cu r re nt s

of diodes D3 and D4 are chosen to be 8.10-16 A. The

volt a ge s ou r ce s s hou ld b e

Vc = 15 – 14.2 + 0.02585 in ~$~~% = 1.604 V

and

V. = 15 – 12.7 + 0.804 = 3.104 V.

AGIciNOWT~E~~~~~~

The au thors are pleased to acknowledge the a id a nd

discussion on th is topic with C. Bat t j es, R. Bohlman ,

S. Taylor , R. Dut ton , and I. Getr eu .

Th e st affs a t t he Com pu ter Cen ter s a t t he Univer sity

of Ca lifor nia j Ber keley, a nd a t Tek tr on ix, I nc., Bea ver -

t on , Or eg., h ave been ext rem ely gen er ou s a nd h elpfu l in

363

the suppor t given for the numerous computer runs neces-

sa ry for t his p roject .

REFERENCES

[11 J . R. Gr een ba um , ‘[Di~ it a l-IC model s for comnute r -a ided

[21

[31

[41

[51

[61

[71

[81

[91

design,” Electronics, vol~ 46, 25, pp . 121-125 ,Dec.’6 , 1973.B. M. Coh n, D. O. P eder son , and J . E , Solomon , “Ma cr o-model ing of oper a t ion a l amplifie rs ,” in ISSCC Dig, Tech.

F’apem, Feb. 1974,pp. 42-43.

D. 0. Pederson and J . E , Solomon , “The need and use ofmacromodels in 1 (3 s ub sys tem des ign ,” in F%oc. 1974 IEEE

Sqmp. Circuits and Systems, p, 488.

J . E . Solom on , “Th e m on olit hic op a mp: a t ut or ia l st udy,”IEEE J. Solid-state Circuits, t h is i ssue, pp . .314-332 .P. R. Gray and R, G. Meyer , “Recen t advances in mono-

l ith ic opera t ional amplifier design,” IEEE Trans. Circuits anti

S~st., vol . CAS-21, pp. 317-327, May 1974,B. A. Woolcy, S.-Y. J . Won g, a nd D. O. P eder son , “A com -pu te r-a id ed eva lu a t ion of t h e 741 ampl ifie r,” IEEE J. SolicZ-

State Circuits, vol. SC-6 , pp. 357-366 , Dec. 1971 .I ,YW. Nagel a n d D. O. P ed cr son , “S imula t ion p rogr am wit hi nt egr a t ed ci rcu it empha sis (SPICE) , “ 131cct ron . Res. La b.,

Univ. of California, Berkeley, Memo ERL-M382, and in

Proc. 16th Midwest &mp. Circuit Theory, 1973.

D. N. P ocock a nd M. G. Kr ebs, ‘{Termin al m odelin g a ndphotocornpensat ion of complex microcircui ts ,” IEEE Trans.

Nucl . Sci ., vol . NS-19, p p. 86-93, Dec. 1972.D. H. Tleleaven and F . N . T rofimenkoff, “Model ing o~er a -

t iona l ampli fi er s for computer -a ided ci rcu i t ana lysi s? IEEE

Trans. Circuit Theor~ (Cor r cs p.), vol . CT -18, pp. 205 -207,J a n . 1971.

G ra eme R. Boyle was born in Echuca, Vic-

t or ia , Au st ra lia , on Oct ober 26, 1949. H er e ce iv ed t h e B .E , a nd M .Eng.Sc. d egr e es ine le ct r ica l e ngin ee ri ng fr om the Un ive rs it yof Melbou rn e, Melbou rn e, Au st ra lia , in

1972 and 1974, re spect ive ly .H e is cu r ren t ly a t t h e Univer sit y of Ca li-

forn ia , Berke ley, work ing toward the Ph .D .

degr ee in t he field of m odelin g a nd com -pu te r s imu la t ion of in t egr a t ed cir cu it s.

Ba r ry M. Cohn (S’68) was bor n in S ea t tle,Wa sh ., on Apr il 8, 1949. H e r eceived t heB.S .E .E . d egr ee, gr adu at in g Ma gn a CumLa ud e a nd wit h College H on or s, fr om t he

Un iv er s it y of Wash ingt on , Sea t t le , in 1971.H e r eceived t he M.S . d egr ee in elect rica len gin eer in g fr om t h e Ca lifor n ia I ns tit u teof Tech n ology, P as ad en a, in 1972.He has done power engineer ing for

Sea tt le Cit y Ligh t d ur in g t he summer s of

1968–1970. “F rom 1972-1974 , whi le a t t h eUn ive rs it y of Califor n ia , B erk ele y, h e d id r es ea r ch and pub lis hedon t h e t op ic of macromodeli rr g in t eg ra t ed cir cu it s for comput er -a ided design . Du rin g t he summer of 1973, h e wa s em ployed a s

a n Engin ee r w it h Na tion a l Semiconduct or Corpor a t ion , wher e h ed id r es ea r ch on t h e t op ic of ma cr omod elin g op er a tion a l amp li-fier s. F rom 1973-1974 h e h as been a Resea rch Assist an t a t t heE lect ron ic Resea rch La b a nd a Tea ch in g Assist an t a t t he U ni-versity of California, Berkeley, from which he is currently on

leave and presently employed by Intel Corporation, Santa Clara,

Calif., w here h e d irects CAD op erations.

Mr. Cohn is a member of Tau Beta Pi and Phi Eta Sigma. He

is the recipient of a National Science Foundation Trainceship, a

General Telephone Electronics Grad uate Fellow ship, and nu mer-

ous undergr aduat e s chola rsh ip s.

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364 IEEE JOURNALOF SOLID-STATEIRCUITS,OL.SC-9,O.6,DECEMBER197

Don ald O. Pederson (S’49-A’51-M’56-F’64)

was born in H allock, Minn., on Septem ber

30, 1925. He received the B.S. d egree from

North Dakota Agricultural College (now

North Dakota State University), Fargo, in

1948 and the M .S. and Ph.D. degrees from

Stanford University, Stanford , Calif., in

1949 a nd 1951, r espect iv ely .

From 1951 to 1953 he was a Research

Associate in the Electronics Research Lab-

oratory, Stanford University. From 1953 to

1955 h e wor ked a t Bell Teleph on e La bor at or ies, In c., Mu rr zyHill, NJ ., a nd wa s a lso a Lect ur er a t N ewa rk College of E ngi-neer ing, Newark , N.J . In 1955he joined the E lect r ica l Engineer ingDepar tmen t , Un iver s it y of Cali forn ia , Berke ley, where is is now a

P rofe ss or an d engaged in r es ea r ch in in t egr a t ed cir cu it s an d com -

A High-Performance

put er -a id ed cir cu it an alys is an d d es ign . F r om 1960 t o 1964h e wasD ir ect or of t h e E le ct r on ics Res ea r ch Labor a t or y.

Dr . P eder son is a mem ber of Sigm a Xi a nd E ta Ka ppa Nu . H e

and t hr ee coa uthor s wer e a war ded a Best Pa per Awa rd for a

p ape r p res en te d a t t h e 1963 I nt er na t ion al S olid -S ta t e Cir cu it sCon fer en ce. H e wa s a Gu ggen heim F ellow in 1964 a nd wa s t he

r ecipien t of t he IE EE E du ca tion Meda l in 1969. I n 1974 h e wa

e le ct ed t o t he Nat ion al Aca demy of E ngin eer in g.

James E. Solomon (S’57–M’61) , for a .photograph and b iog raphy

p le as e s ee p . 332 of t h is is su e.

Monolith ic Mult ip lier

Using Act ive Feedback

BARRIE GILBERT , SENIORMEMBER,IEEE

Absfrac f—Since it s concept ion in 1967, the l in eari zed t r an scon -

d uct a nce mult iplier (LTM) h as r a pidly ga in ed a cce pt a nce a s t he

preferred approach to the realization of monolithic analogmultipliers,

a nd it s s im plicit y h as commend ed it for u se in low -cos t m od ula r

d es ign s. Accu r a cie s of t h es e u n it s h ave b een lim it ed t o abou t 0.5 t o

t o 2 p er cent , a n d d rift a n d nois e p er formance h ave gene ra lly b een

worse than tha t possibl e u s ing the dominan t a l te rnat i ve t echn ique

of p uls e-wid th -h eigh t m od ula t ion , Th is p ape r s hows t ha t wh en

ca r efu l a t t en t ion is given t o a ll t h e sou r ce s of e r ror it is p os sible t o

a t ta in a five-fold im pr ovement in a ccu ra cy a nd cor r es pon din gr ed uct ion s in t h e d rift a n d nois e leve ls . Odd -or d er n on lin ea r it ie s

can b e r edu ced t o n egligib le magn it u de s b y t h e u se of a ct ive fe ed -

back , by sub st it u tin g t h e u su a l r es is tive -br id ge fe ed back p at h b y

an amplifie r i d en t ica l t o t h at u sed a s t h e in pu t s tage s.

I. INTRODUCTION

T

HE LINEARIZED t ra nscon du ct an ce mu lt iplier

(LTM) technique’ is now widely used in one or

another of it s several forms, the commonest of

wh ich ar e sh own in Fig. 1. Usin g idea l t ra nsist or s t hese

cir cu it s a re fu ndamen ta lly exa ct a nd in sen sit ive t o iso-

t herma l va ria tion s of t emper at ur e. Compa red t o a lt er na -

t ive mult iplica t ion techniques, the LTM core is ex-t remely sim ple a nd h as in tr in sica lly wide ba ndwidt h.

It is t he ba sis of m an y ot her ‘(fu nct ion al” cir cu it s, su ch

Manuscr ip t r e ce ived May 25, 1974; revised August 11, 1974.

Th is p ap er w as p resen ted at th e In tern ation al Solid -State Circu its

Co nfer en ce, Ph ilad elp hia., P a., Febr uar y, 1974.

The au t hor is w it h Ana log Devi ce s S em icondu ct or , Wilm in g-

ton , Mass. 01887 .

1 B. Gilber t ,, “A new wide-band amp li fi er t echn ique,” IEEE J.

So.M-Stute Circuits, vol . SC-3 , pp. 353 -365 , Dec. 1968.

&1 12 1, 14

(a) J,J4 =+J3

I,

l+%, 1, 14

(b) J 1~=J3J4

“ +X)w-c) (d)

F ig. 1. Ba sic LTM con figu ra t ion s. (a ) “Norma l” form. (b) ‘Tn -

ve rt ed ” form. (c) S pecific ver sion of t he n orma l form h avin glow bet a sens it ivit y. (d ) Common in ve rt ed form , which is con -

venien t but has h igh beta-sensit ivi ty.

as p re cis e two-qua dr a nt d ivid er s, rm s conver t er s, vect or

sum modu les, et c.

This pa per descr ibes a fou r-qu adr an t mult iplier de-

s ign ed t o m ak e fu ll u se of t he pot en tia l a ccu ra cy a ffor ded

by t his t ech niqu e u sin g cu rr en tly a va ila ble m on olit hic

processes. The basic er ror sources are reviewed, an d

method s p re sen t ed t o m in im ize t h eir effect s. A s ign ifica n t

improvement has resulted from the use of an act ive

feedba ck s ch eme wh ich la rgely elim in a tes t h e non lin ea r -

it ies in tr odu ced by t he in pu t amplifier s. Apa rt fr om t his,

t he r es ult in g d es ign is s im ila r t o mos t t ra ns con du ct an ce

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