Finite Amplifier Gain and Bandwidth Effects in Switched-Capacitor Filters

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358 IEEE JOURNAL OF SOLID STATE CIRCUITS VOL. SC 15 NO. 3 JUNE 1980

Finite Amplifier Gain and Bandwidth Effects inSwitched-Capacitor Filters

GABOR C. TEMES, FELLOW IEEE

AbstractThis work discusses the effects of finite operational-amplifier gain and bandwidth on the response of the most widely usedswitched-capacitor filter section. Formulas are derived for the mini-mum acceptable values of the dc amplifier gain and the unitygain fre-quency under specified conditions.

I. INTRODUCTION

IG. 1(a) illustrates the circuit diagram of a switched-capacitor filter (SCF) section which is used as a generalbuilding block in almost all SCl?s [1]- [7]. Fig. 1(b) showsthe timing diagram, including the location of the samplingpoints tn.l, tn, etc. The response of the operational ampli-

fier can be described by the Laplace-transform relation

A ~O(S) _ ~ _ -AouoA(s) =

v(s) S+(70 s+uoHence, in the time domain,

1 duo tU t =-+uo t - ~ ~t .

(1)

(2)

In (1) and (2), A. is the magnitude of the dc gain, U. is theunit y-gain radian frequency, and -0. is the pole radian fre-quency. It follows from (2) that, unless v(t) contains impulses,v.(f) is a continuous function of time,

If U. is infinite, the relation between the z-transforms ofthe output sequence Uo,. ~ U. (tn) and the input sequencesvi,n~ ZJj(tn), i= 1,2, ., lVis

V()(z) - * j CiV~ z .a i=l

(3)

The corresponding formula will now be derived for the casewhen co. < CQ. The calculations are straightforward butlengthy, and only the most important steps will be described,Two transients have to be considered. The first occurs afterthe moving contacts switch to the right (R transient), thesecond one aft er they switch to the left (L transient ). Forsimplicity, we consider first the case N = 1; the results willthen be generalized to N >1.

The analysis will assume that the amplifier is linear, and willneglect such other important limiting factors as the slew rateand settling time of the amplifier.

II. R TRANSIENT

This is the transient taking place for fn_l < t< tn-1/2. romFig. l(a)

Manuscript received March 9, 1979; revised November 7, 1979.The author is with the Department of Electrical Sciences and Engi-

neering, University of California, Los Angeles, CA 90024.

5VI

2

La:;. .

QVa

v A s +

+ o

(a)

kT/2 +T/z +T/z +T/z ++ T/2 +1 I I

k l 1 / 2 tn 1 / 2 ~I n 3 2

tn., t tn+l

(b)

Fig. 1. Switched-capacitor filter section. (a) Circuit diagram. (b) Tim-ing diagram.

Uo t =u. t + u(t). (4

Also, for N= 1, charge conservation requiresl

Ca[ (t) - ,n-l] = Cl[u(f) - ul, n_l]. (5

From (2), (4), and (5), we obtain

C (f) + /1+ -_ _ ljo t = n-lkwo dt \ kAowhere

(6

(7

The finite-de-gain effect becomes negligible if kAo >>1, i.e.A. >>1 + C1/Ca. Hence, the required dc gain is very large ifCa


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TEMES: SWITCHED-CAPACITORFILTERS

gives2

UO t)=On_l[l -e-kw~ t-tn.l l +vo,n-le-kuo t-tn-l < 8

Next, 5) and 2) give

Us t) = On_l [1 - 1 - lc)e-kUO t-fn-l)]

+Uo, n_.l l - l c e-kwo t-tn.l

9)

The values of the voltages at the end of the R transient,vo,n_112 and va, n-1/2~ can be obtained from 8) and 9),respectively, by setting t - tn_l equal to T/2.

The time constant of the R transient is

_ca+cl

R Cau o lo)

Thus, t_R >> 1/~0 if Ca


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360 IEEE JOURNAL OF SOLID-STATECIRCUITS, VOL. SC-15, NO. 3, JUNE 1980

Therefore, the replacements Cl ~ CP and VI(t) ~ UP(t) shouldbe performed in all formulas. The resulting transfer functionsare thus, from (14),

vi z 1 (1 - S)z

~ CjVi Z= Ca z - 1)(Z - e)

i=l

Va(z) 1 (1-a +e)z-e. -

~ C , Vi ZCa (z - 1)(Z - e)

jzl

23

It should be noted that one of the inputs vi(t) can, in fact,be chosen as the output UOt). Consider, e.g., the circuit ofFig. 2. Its transfer function, from (23), is

v~ z= cl

(1 - ti)z

v, (z) Ca(z - l)(Z - e)+c~(l - S)z 24

If Cl = Cz = Ca, the transfer function of the circuit with anideal amplifier is that of a negative delay: V./ V i = -z 1. Dueto the finite gain and bandwidth, the actual transfer function

is, from (24),

Vo(z) -(1 - (5)Z .VI (z) z-(e+6)z+e

25

From (25), the actual loss and phase of the section can be pre-dicted.

The formulas given in Sections II and III can also be used toderive the transfer function for sections in which some of thecapacitors Ci [Fig, 1(a)] are switched in opposite phase. Suchsections include the simulated inductors of Hosticka et al .[7 ] and Temes et al . [8] . As an example, Fig. 3 illustratesthe simulated inductor described in [7]. The charge enteringduring the nth clock interval is

dnz-) - q(nl q = cl( ,n - /2)

+ C2(l, n - O n-l/2 . 26

Now vO,n-1/ and vn-1/z = vo n-1/2 - va n-112 can be foundfrom (8) and (9) setting t tn l T / 2, Then, taking thez-transform of the resulting relation and substituting VO(z)and Va z from (14), we obtain

Q(z)=2c1{[z +(l-@][z-l][z-e]VI (z)

+(1 - @ [ l -8 +e z+el

++(1 6) Z}[(Z - 1)2(Z - C)]-l (27)

where the usual condition Cl = C2 = 2Ca was assumed [7] andwhere @~ (1 + k) exp (-lctio T/2).

The circuit of Fig. 3 simulates an inductor only at signalfrequencies a satisfying U T < < 1. Hence, in (27) the substi-tution z-l = e-jwTs 1 - j~l may be made. Carrying out this

substitution, and assuming also that 8, e, c , and U T are allmuch smaller than unity, the first-order approximation

z =1-8 2C1VI (z) l-e j uT 28results. Hence, only the value of the equivalent inductorchanges by a factor (1 - e)/(1 - 6). The Q is not affected, and

no other parasitic elements appear.

.

CzFig. 2. Delay circuit.

J+q+~~

c.q

++ -

VI c1 v A s .+o+.

Fig. 3. Simulated inductor circuit.

VI. CONCLUSIONS

The derivations given in the paper show that the effects offinite dc gain A. and finite unity-gain radian frequency tio willbe negligible in the general SCF stage of Fig. 1(a) if the follow-ing conditions hold:

k AO > >1

6 Q [1 - lc(l - e-wOT2)] e-kwOT2


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IEEE JOURNAL OF SOLID-STATECIRCUITS, VOL. SC-15, NO. 3, JUNE 1980 361

[6 ] R. W. Brodersen, P. R. Gray, and D. A. Hodges, MOS switchedcapacitor filters; Proc. IEEE, vol. 67 pp. 61-75 , Jan . 1979.

[7] B. J. Hosticka and G. S. Moschytz, Switched-capacitor simula-t ion of grounded inductors and gyratorsj Electron. Lett., vol. 14,pp. 788-790, NOV. 23, 1978.

[8] G. C. Temes and M, Jahanbegloo, Switched-capacitor circuitswhich are bilinearly equivalent to a floating inductor or FDNR,Electron. Lett., vol. 15, pp. 87-88, Feb. 1, 1979.

Gabor C. Temes (SM66-F73) received the

Dipl. Ing. degree from the Technical Universityof Budapest, Budapest, Hungary, in 1952, theDipl. Phys. degree from Eotvos University,Budapest, Hungary, in 1954, and the Ph.D.degree in electrical engineering from the Univer-si ty of Ottawa, Ont., Canada, in 1961.

He was a member of the faculty of the Tech-nical University of Budapest from 1952 to1956, and was employed by MeasurementEngineering Ltd. , Arnprior, Ont., Canada, from

1957 to 1959. From 1959 to 1964 he was with Northern ElectricR D Laboratories, Ottawa, Ont., Canada. From 1964 to 1966 he wasa research group leader at Stanford University, Stanford, CA; from1966 to 1969 he was a Corpora te Consultant for Ampex Corporation,Redwood City, CA. Since 1969 he has been a Professor at the Univer-sity of California, Los Angeles. Between 1975 and 1979, he was alsoChairman of the Depar tment of Electrical Sciences and Engineer ing.He has published about 90 technical papers. He was coeditor (withS. K. Mitra) and coauthor of Modern Filter Theory and Design NewYork: Wiley, 1973), coauthor of Introduction to Circuit Synthesis and

Design New York: McGraw-Hill, 1977), and a contr ibutor to severalother edited volumes.

Dr. Temes is an Associate Editor of the Journal of the Franklin Zn- titute, a former Editor of the IEEE TRANSACTIONSNCIRCUITTHEORY,a former Vice President of the IEEE Ckcuit s and Systems Society, and aformer Chairman of its Best Paper Awards and Nominations Commit-tees. He was a member of the Fellow Award Committee as well as theCANDE Committee of the IEEE Ckcuits and Systems Society. He wasa cowinner (with H. J. Orchard) of the 1968 Darlington Award of theIEEE Ckcuits and Systems Society.

Phototransistor Optical solator Noise ModelKOY B. COOK, JR., SENIOR MEMBER, IEEE, WEN-CHIN YEH, AND

VICTOR W. RUWE, SENIOR MEMBER, IEEE

AbstractA noise model for the phototransistor opticsl isolator ispresented and used to predict opt ical i so lator noise performance. Re-sults are demonstrated to agree with existing experimental data onphotot ransistor opticsl i solator noise. The model presented includesburst noise, flicker noise, and shot noise.

1. lNTRODLJCTION

HE results of experimental investigations of the noiseperformance of several commercially available photo-transistor optical isolators have been presented in prior publi-cations [1] , [2] . It is the purpose of this paper to present acircuit model which will accurately predict phototransistoroptical isolator noise performance and which will thus be use-ful for computer simulations for low noise circuit design.

An optical isolator is composed of a phototransistor orphotodiode detector optically coupled to a light-emittingdiode (LED) as shown in Fig. 1. The LED not only providesthe bias current to establish the transistor Q-point, but it alsoprovides the input signal to the phototransistor.

The equations which describe the operation of the opticalisolator are also shown in Fig. 1 where

I D = dc LED bias current,I c = dc phototransistor collector current,

I PH = dc photocurrent in the phototransistor due to photonabsorption from the LED,

Manuscript received May 23, 1979; revised February 8, 1980.K. B. Cook, Jr., and W.-C. Yeh are with the Department of Electr ical

Engineering, Auburn University, Auburn, AL 36830.V. W. Ruwe is with the U.S. Army MICOM, Redstone Arsenal, Hunts-

ville, AL 35809.

1PH= kOID

lPH= k AI D

l C = FEk OT D AIC= hfek AIO

Fig. 1. Optical isolator schematic and describing equations.

k. = static current transfer ratio,k = dynamic current transfer ratio.

The static and dynamic current transfer ratios lco and k,respectively, are, in general, dependent on LED current [2].

II. THE MODEL

A noise model for the phototransistor optical isolator is pre-sented in Fig. 2. The right-half is the electrical noise model ofthe phototransistor under ideal LED illumination (one whichexhibits only shot noise in its optical fluctuations) and the left-hrdf is the electrical noise model for the LED. The hybrid-ntransistor equivalent circuit has been used and rm Cm r Cpand gm have the conventional hybrid-n definitions. In thephototransistor, ~ and ~~ are shot noise current generators[3], The contribution of flicker noise in the phototransistoris represented by two flicker noise current generators ~~1 and

0018-9200/80/0600-0361 00.75 @ 1980 IEEE

Finite Amplifier Gain and Bandwidth Effects in Switched-Capacitor Filters

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