Design of switched-capacitor filters using non-ideal op amps

Design

of Switched-Capacitor

Non-ideal

Filters

Using

0~ Amps

by M. A.TAN

Department of Electricul 55455, U.S.A.

Engineering,

University

of Minnesota,

Minneapolis,

MN

C.ACARt

Department of Electrical and Computer Engineering, CA 95616. U.S.A.

University of California, Davis,

and M. s. GHAUSI College

of Engineering,

University

of California,

Davis,

CA 95616,

U.S.A.

ABSTRACT : A general synthesis procedure is given j?)r the realization of switched-capacitor ,jilters in the z-domain. The method uses biphased switches, capacitors and non-ideal op amps with$nite gain-bandwidth product. The design includes at the outset the effticts of the op amp gain-bandwidth product. Two examples are given which have been tjer$ed by a computer program.

I. Introduction In the last decade, a great variety of synthesis techniques for switched-capacitor filters have been proposed (l-4). In most of these papers, the operational amplifier has been assumed to be ideal. It has been shown, however, that the effects of finite gain and finite bandwidth of op amps in SC filters might give rise to distorted frequency response and/or to even an unstable circuit (5-9). Moreover, the SC filters designed under the assumption of ideal op amp cannot be used for frequencies over about 20 kHz (4,8). Sanchez-Sinencio et al. (3,4) have studied special cases of second-order SC filters which are based on the simulation of RLC networks by using the op amp pole. They use the op amp pole to simulate grounded and floating inductors. Their design thus starts with a standard passive RLC circuit synthesis. In that design either prewarping or the condition of wOr << 1 must be met. If this condition is not met, the active SC filter may not be stable for high Q realizations due to the type of transformation used in the design. In the above case, it is also not possible to realize directly any given biquadratic transfer function in the z-domain. A general synthesis technique which takes the finite gain-bandwidth product (GB) of op amp into account in the design, to the best of our knowledge, has not been given yet. t Visiting faculty from Technical

University

of Istanbul,

Turkey.

M. A. Tun et al.

B On (closed) Off (open)

m:odd

9”

-.--A-

9” On (closed Off

I mevend-

(open 1

d”

- +e

FIG. 1. Odd and even clock phases. This paper presents a general technique for designing SC filters using finite gainbandwidth product. The technique presented here utilizes the signal-flow graph approach. At first a z-transformed relationship between the input and output voltages of the op amp is obtained under a particular input waveform assumption. Then a SC integrator using an op amp with finite GB is discussed and its signalflow graph in the z-domain is given. A capacitive summer and its signal-flow graph are also presented. Using these basic circuits, first a second-order and then nthorder SC filter are realized using a signal-flow graph design strategy (1). In this study, the switches are assumed to be ideal and controlled by a two phase non-overlapping clock of frequency ,fc = 1/(2T) as shown in Fig. 1. The switches are assumed to have a 50% duty cycle with equal on and off time periods, T. Note that 4’ is used to denote the even clock phase which keeps the @ switches closed during the following intervals : & = [(n-

l)T,nT]

for II an even integer.

Similarly, @’ denotes the odd clock phase, which keeps the 4” switches closed during the following intervals : C/I: = [(m-

l)T, mr]

for m an odd integer.

& and 4: are the nth phase of 4” and mth phase of 4”, respectively. 4” = (id?>

k odd integer

4” = U&,

k eveninteger

Note that

and @‘U@ = R. The intervals of 4” and 4” succeed one another alternately. & ends at an even multiple of T, i.e. at the time instant nT and 4: ends at an odd multiple of T, i.e. at mT. Any voltage signal in the circuit, v(t) is partitioned into two components denoted by the even part v”(t) and the odd part a”(t) so that v(t) = v”(t)+z’“(t),

56

t E 4”cJ@,

Pi‘(r) = 0,

tE @,

c”(t) = 0,

tE@.

Design

qf Switched-Capacitor

Filters Using Non-ideal

Op Amps

Moreover all the voltages in the SC network are assumed to be sampled instant for k an integer. Hence, the sampled value of v(t) is

at t = kT

v(kT) v’(kT)

= v”(kT)+v”(kT),

ke N

= 0,

k an odd integer k an even integer.

v”(kT) = 0,

Hence, two discrete time sequences, even and odd are defined and their z-transforms are taken separately. In order to obtain a transfer function in the z-domain for the op amp, the input signal of the op amp used in synthesis is assumed to be zero on each phase of 4” and constant on each phase of 4”. In other words, v;(t)--v:(t)

= 0

tE&,

k even integer

(la)

tE&,

(lb)

and v;(t)-u:(t)

= vz(kT)-vz(kT)

k odd integer

where vi(t) and u:(t) are, respectively even components, and vi(t) and v:(t) are odd components of v,,(t) and v,,(t) which are non-inverting and inverting input voltages, respectively. Let the Laplace transformed transfer function of the op amp with finite GB be given as

(24 where VO is the output voltage of the op amp, VP and V, are the transformed voltages of non-inverting and inverting inputs of the op amp, respectively. GB is the gain-bandwidth product, p is the dominant pole of op am and s is the complex frequency variable. Equation (2a) is expressed in the time domain as

vdt> = vdtde

PC’ ‘d+GB

s

‘e p(’ “[II,,(Z)-z~&)]dr,

t 3 t,sR.

(2b)

f,,

This integral expression obviously shows that v,,(t) is continuous for all t > t, so long as [v,(r)-v,(r)], the input voltage difference of the op amp, is piecewise continuous. Recall the o-input condition in (1) that vi(t)-2$(t)

= 0,

From (I a) and (2b), the even component integer can be written as v;,(t) = vi[(k-

t)ge

tE&q.

of output

-I’(‘~ck ‘)r),

voltage,

tE4k0

vi(t) for k an even

@a)

hf. A. Tan et al.

and hence for t = kT, k an even integer, we have v”,(kT) = v’;l[(k- 1)Tje Taking

the z-transform

PT.

(3b)

of (3b) yields Vi(z) = z ‘12e mpTVi(z).

Recall the o-input

condition

in (1 b) again that

From

= v;(kT)-v;(kT),

fEd$

(34

(2b) and (1 b), the odd component

of v”(t) is obtained

v;:(t)-z$‘(t) for k an odd integer. for t,, = (k- 1)Tas

(3c)

v;(t) = v~[(k-l)~e~P(‘~(k~‘)T)

+ F[v;(kT)-vz(kT)] tE &

x (1 -e~mP(r (k I)‘)),

and the value of vi(t) for t = kT, k an odd integer, v”,(kT) = v’,[(kTaking

the z-transform V;(Z)

Combining

= z

can be written

l)TJe~mPT + c:[v;(kT)-v,(kT)]

(1 -e

(4a)

as “‘).

(4b)

of (4b) yields ':2e pT V\(z) + y[V;(z)-V,o1(1

(4~) and (3~) in order to eliminate

-e

the P’i component

“‘j. yields

The response of the op amp in the z-domain to the input signal (which is zero on 4: and constant on each &) is thus given by (5a). In other words, if we operate all the op amps which are to be used in SC filter systhesis under o-input condition, their responses can be taken into account by (5a). In most cases, p the pole of the op amp is much less than the sampling frequency ,f; = 1/2T. Thus it can be assumed that 2pT << 1. Under this assumption, (5a) becomes I’;(z) = GRT&&(z)-

V;(z)].

(5b)

It can also be shown that the output voltage waveform of the op am becomes piecewise constant in &, and piecewise linear in 4” as illustrated in Fig. 2 under the assumption of 2pT << 1.

II. Basic Circuits and Their Signal-Flow Graphs Two basic circuits, an SC integrator with non-ideal op amp and a capacitive summer are introduced in obtaining second-order filters.

58

Design of Switched-capacitor

Filters Using Non-ideal

Op Amps

FIG. 2. (a) Typical op amp input waveform satisfying the o-input condition of op amp, (b) the output waveform of op amp corresponding to (a), (c) the odd and (d) even components of the output voltage of the op amp for 2pT CC1.

SC intcgrutor The basic SC integrator shown in Fig. 3a. Writing this circuit, we obtain (C, +c

which is used in the proposed synthesis technique is nodal charge equations at & for the nodes n and p in

)V,“(t)-C,U;[(k-

1)7-j = 0, TE&,

(C+ C’)tl”(t)-

CzF[(k-

Since v,,(t) and v,(t) are constant and v*(t), we have

k: odd

(6)

1)7J = 0.

on 4: independent

of the input

voltages

G(t) = G(kT),

Pa) tE&,

k: odd

z&‘(t) = v;(kT). The @ switch in the input Vol. 323. No. I, pp. 55-~72.1987 Prinkd m Great Bntam

v,(t)

of the op amp in the circuit

(7b) of Fig. 3a ensures

the 59

M. A. Tan et al.

C” =c, +c-

c,=c,+c+ (a)

(b)

FIG. 3. (a) SC integrator using non-ideal op amp and (b) its signal-flow graph.

input voltage difference of the op amp to have zero value on 6. Thus the o-input condition is satisfied by the passive subnetwork of the circuit of Fig. 3a. Substituting (7) into (6), yields (C, + C)v;(kT)-

C,u”, [(k-

l)fl

= 0,

@a) k: odd

(C,+C+)u;(kT)-C,v;[(k-l)T]

Taking

the z-transform

= 0.

(Xb)

of (8) yields

(C, +c~)V::(z)-z~“2c,Vq(z) (C,+C+)V;(z)-z

= 0,

(9a)

“‘C,VC,(Z) = 0.

(9b)

From (9) and (5b), we obtain 1 V;(z) = [I-(z-,),V’,(z)-GBT:

n

V;(z)+GBTm;

P

V;(z)

(IO4

where c, A c,+c-

(lob)

c, A c2+c+.

(1Oc)

and

The signal-flow graph corresponding to (10) is illustrated in Fig. 3b. In other words, it corresponds to the circuit of Fig. 3a for the voltages sampled at the ends of 4;s. Note that, in the signal-flow graph of Fig. 3b, the branch transmittances incoming to the node Vi are proportional to GB the gain-bandwidth product of the op amp and T, the half clock period. Note also that the negative transmittance corresponds to the SC subnetwork connected to the inverting input of the op amp. It is Journaldthc 60

Franklin

Pergamon

lnit~tutc

Journals

Ltd.

ofSwitched-Capacitor

Des@

Filters Using Non-ideal

(a) FIG. 4. (a)

Op Amps

(b)

Capacitive summer and (b) its signal-flow graph.

proportional to the capacitance connected between node 1 and node n, and inversely proportional to the total capacitance which is connected to the inverting input of the op amp. Similarly, the positive transmittance is proportional to the capacitance connected between node 2 and node p, and is inversely proportional to the total capacitance connected to the non-inverting input of the op amp. The voltages U, and u2 are voltages of the voltage sources that their values are known at kT for k at even integer. They could therefore be the output of the op amp in this SC integrator.

Cupacitizre summer For the capacitive u;(kT)

summer

c,

shown in Fig. 4a, we have

= dP~C(kT) T

C2

+ Fl:‘_(kT) T

for k an even integer

(1 la)

where

(1lb) Taking

the z-transform

of (1 la) yields

C, G(z) = c C(z) + I

c2V”,(z). c.

I

This equation corresponds to the signal-flow graph of Fig. 4b for the voltages sampled at the end of &. The basic circuits of Figs. 3a and 4a can be used to obtain a second-order SC filter employing non-ideal op amp.

IIl. Synthesis of a Second-order

Block

In this section, a synthesis procedure is given to obtain a general biquadratic SC filter employing finite GB op amp by using the basic circuits of the preceding section and the signal-flow graph approach. Let a second-order transfer function H(z) be given in z-domain as Vol. 323. No I, pp. 55-72, 19X7 Pnntcd I” Great Bntatn

61

M. A. Tan et al.

FIG. 5. Signal-flow graph realizing H(z) of (12) B,z2+ B,z+ H(z) =

B,

-z2m+A+7 1

0

(12)

where Bis and A,s are coefficients of the numerator and denominator polynomials, respectively. A signal-flow graph realization of H(z) for (12) is given in Fig. 5. Here the coefficients X, and X2 are scaling factors. The signal-flow-graph of Fig. 5 is composed of two different types of basic signal-flow graphs; one is similar to that of Fig. 3b, the other is similar to that of Fig. 4b. If the circuits of Figs. 3a and 4a corresponding to a sub signal-flow graph of Figs. 3b and 4b are connected to each other according to the signal-flow graph of Fig. 5, the active SC circuit realizing H(z) is obtained. This circuit is shown in Fig. 6. The design equations which give the values of capacitors can be easily determined by equating the branch transmittances of Figs. 3 and 4 to the corresponding branch transmittances of Fig. 5. For the filter shown in Fig. 6, the capacitor ratios are Cl2 c,-

[B2(A,

=

[Ao+A,+ 11x2 GB,T-[B,(A,+Ao)-B,-BBo]X,Xz-[A,+A,,+l]X2’

C,? C:

(BI

c:=

l/X*

A,+2 CT = GB2T-A,-2’ ~~~

c34 c 44 -

(13a)

(13b)

,1X,

GB2T-(B,-BB?A,)X,-l/X2’

C33

C14 C 44

-B2A

GB2T-(B,-B2A,)X,-l/X2’

C 23

62

--olX,J’2

GB,T-[Bz(A,+Ao)-B,-B,]X,X2-[A,+Ao+l]X2’

C32 C,L

+Ao)-B,

=

(13c)

(134 Q3e)

B2 I-BZ-l/X,’ I/X,

1-B2-l/X,’

(lk)

Des&m qfswitched-Capacitor

Filters Using Non-ideal

Vp Amps

Cl4

FIG. 6. Second-order

SC filter using non-ideal

op amp realizing H(z) of (I 2).

Note that if C;, CT, Cl and C,, are chosen freely other capacitances can be calculated by means of (13). The SC circuit corresponding to the signal-flow graph in Fig. 5 and the realization of H(z) in (12) is depicted in Fig. 6, provided that the coefficients Ais, Bis and X,, XZ satisfy the following conditions : (14a)

x, > 0, B>+l/X,

(14b)

B, > 0,

(14c)

x2 > 0,

(14d)

B,--A, B,(A,

d 1,

+AO)-B,

3 0,

(14e)

- B0 3 0.

(140

The conditions (14aa14c) are due to capacitive summer. If these constraints are to be avoided an active summer including a wide-band amplifier should be used. The synthesis procedure can be used even though one or both of the inequalities (14e and 14f) are reversed. In this case, to obtain the SC filter, the terminal of the switched capacitor corresponding to the branch whose sign is changed must be connected to the other input of the op amp. For example, if (B, - B2A ,) is negative, then C, 3 should be connected to the inverting input of the second op amp by 4” switch, and (13c-13e) would, respectively become

63

M. A. Tan et al.

Cl3 -= C,

(B, -&A 11x1 GB2T-(B,

-B,A,)-A,

-2’

C23 = .~~~_~~~~) 11x2 c,+ GB2T- l/X, CM p= CI

(15b) A,+2

GB,T-(B,

-B,A,)X,

-A,

(15c)

-2’

Note that a 4’ switch is used at the output of the filter in Fig. 6 because this network realizes the H(z) of (12) for the assumption that the output and input voltages are sampled at the ends of 4;s. When B2 = 0, there is no switchless capacitor path on 6 between the input and output of the filter. In this case, it is not necessary to employ any sample-and-hold circuit at the input and also the capacitive summer could be removed by choosing the parameter Xi to be unity. As an example consider the design of a filter which has the following s-domain bandpass characteristic H(s) = This and kHz ing,

2027.9s ?+641.28s+

(16)

1.0528 x 10’.

filter has the following parameters : J; = 1633 Hz, quality factor Q = 16, a peak gain of 10 dB at fO. For the design, let the sampling frequency be 8 (i.e. z = 2T = 125 ms). Applying the bilinear transform to (16) after prewarpyields the following z-domain transfer function (10) : H(z) =

O.l219(z1) (z+ 1) ~~~~ z2 -0.5455z+O.9229

(17)

The active-SC filter realization of H(z) in (17) is shown in Fig. 7. Note that this filter contains op amps with finite gain-bandwidth products of GB, = GB2 = 10’ rad/s, and the capacitor values are calculated by choosing X, = 10, X2 = 1, C: = Ch4 = 1OpF and C, = CT = 5pF: Cl2 = 2.6290pF

CZ3 = 2.18lOpF

C,3 = 1.4503pF

C34 = 1.2852pF

c,‘J = 1.5666pF

C33 = 1.5166pF

(18)

C32 = 2.1565pF.

As a second example,

consider

the following

z-domain

0.1953(2-

transfer

function

:

1)

H(z) = z’-- 0.5255~ + 0.9229 . This filter also has the same center frequency, f0 = 1633 Hz, the quality factor, Q = 16, and a peak gain of 10 dB at f,,. The active-SC filter realizing this function 64

Journal

of the Franklin lnbt~tutc Pergamon Journals Ltd.

Design of Switched-Capacitor Filters Using Non-ideal Op Amps

C2 5pF

B $

Q Y” I

T

=_

c,,= 2.16pF

=

FIG.

O.l219(z-l)(z+l) 7. The active-SC filter realizing H(z) = Z2_. 5455z+o,9229 which has So = 1633 Hz, Q = 16 and a peak gain of 10 dB at f,.

I

v;+o

FIG. 8. The active-SC filter realizing H(z) = zZ _o,54552+o.9229

which has f0 = 1633 Hz,

Q= 16andapeakgainoflOdBatf,.

is shown in Fig. 8. The capacitance values are calculated Cg = 30pF and C; = C; = 5pF: Vol. 323, No. 1, PP. 55-72, 1987 Pnnied m Great Brhn

by choosing

X, = X2 = 1,

65

M. A. Tan et al. Example

I 1

IIO -

o-

-IO

-

m

z g -m3

tl El

-30

s 40 :A

-505 200

4

6

8

Ik

2

Frequency

FIG. 9. The simulated

c,j

(Hz

4

6

Ek

1

gain response of the SC-filter shown in Fig. 7.

= 1.1591pF

C12 = 1.4134pF

CZ3 = 5.9351pF

C33 = 1.5165pF.

Note that the total capacitance value (i.e. CC,) is 42.79pF for the first realization and 50.02pF for the second. The spread in the capacitor values (i.e. C,,,/C,,,) is 8 for the first realization and 26 for the second. The spread in the second realization can be reduced to 9 by choosing X, = 10 instead of X1 = 1. Note also that these values are not greater than those which are obtained in the realization of the same center frequency, the same quality factor and the same peak value by the E-type and F-type of the active-SC circuits (10) using ideal op amps. Both of the active-SC filters shown in Figs. 7 and 8 were analyzed using the program SCFIL4 developed by Rudd and Schaumann (11). In our study the switches are assumed to be ideal and the op amps behave as ideal integrators with finite GB whose input and output impedances are respectively infinite and zero. However, in the analysis of the filters using SCFIL4 program we assumed that the switch resistances are 100 Q when they are closed, and that the op amp output impedances are 100 R. The simulated gain responses obtained from the analysis program are depicted in Fig. 9 for the first example, and in Fig. 10 for the second example. A comparison of the ideal and simulated responses shows that they are in close agreement although we use finite resistances for the switches and for the output impedances of the op amps. The difference between ideal and simulated responses which is respectively shown for the first and second examples in Fig. 11 and Fig. 12 appears at the vicinity of the center frequency. As seen from the figures, this difference is too small to cause any problems. The program results verify our technique and show that the proposed circuit in Fig. 6 is suitable for the active-SC filter implementation.

66

Journalofthe Frankhn

Institute Pergamon Journals Ltd.

Design

qf Switched-Capacitor

Filters Using Non-ideal

Example

200

4

6

8

2

lk

2

Frequency

FIG. 10. The simulated

gain response Example

Op Amps

4

6

8k

(Hz)

of the filter shown in Fig. 8.

I

“L

k

Frequency

FIG. 11. The ideal

and

~

simulated

gain

(Hz1

responses

for the SC-filter

shown

in Fig. 7.

Ideal response obtained from H(z). Simulated response obtained from the circuit.

The proposed active-SC filter configuration which contains op amps with finite gain-bandwidth product is capable of realizing high quality factor, Q. If we calculate the coefficients of the denominator polynomial of (12) in terms of the capacitor ratios and the gain-bandwidth products, we obtain Vol. 323, No I. PP 55-72. 1987 Printed in Great Britain

67

M. A. Tan et al.

Frequency

FIG. 12. The ideal ~ ~.-.~‘-

(Hz)

and simulated gain responses for the SC-filter shown Ideal response obtained from N(z). Simulated response obtained from the circuit.

A0 =

1

in Fig.

-(jB2TsJ3 + GB,G&T’~ g 2n

In

(194 2p

A, = -2+GB,Tg

U9b)

2”

where

8.

Cl,_ C2,, and Czp are

c,,

=

c,2+c32+c;

C2n

=

c,,

c,

=

c,3+c23+c:.

+

(204 G

@Ob)

WC)

It has been shown in (10) that when o,,z << I and Q >> 1 the same coefficients be written in terms of resonant frequency, o0 and the quality factor, Q as

can

(214

where z = 2T. Combining (2 1) and (19) yields 68

Journal 01 the Franklin institute Pcrgamon Journals Ltd.

Design of Switched-Capacitor

Filters Using Non-ideal

Op Amps

(22)

It is interesting to note that when wOz << I and Q >> 1, the resonant frequency is independent of sampling period, z = 2T whereas the Q depends upon it. This fact may be used to increase Q by T. Note also that it is possible to adjust wO and Q to the desired values by changing Cs2 or C23 first and then C33. Note further that obtaining very high Q is possible by choosing an appropriate value CX3. Sensitivity analysis from (22) and (23) shows that S&

= S$

= ,/2

s&,

= - l/2

(24a) (24b)

tCB,GB,T’$ 2’

G&T$ 2n

r;B,TgJ

In

_ GB,GB2T2

$?

~~= 2 + QwOz.

(249

$ In

2n

,

21,

2p

Note that if QoOz d l/2 then Sg,, ,< 1. In order to get a good sensitivity ance, the quality factor may be chosen to satisfy the following condition Q < 1/2w,z. From

perform-

(25)

(25) it is seen that a large Q implies small coOr.

IV. Nth-order

Realization

We next extend the above method to the realization The nth order z-domain voltage transfer function

cl(z)

H(z) = ~ = VT(z>

K

of the nth-order

z”+B,~,z”~‘+...+B,z+B, z”+A+

,z+ ’ + ... +A,z+A,

filter design.

(26)

is assumed to be given. The signal-flow graph for (26) is shown in Fig. 13. Its branch transmittances fiL and a,, i = 0, 1, . . . , n are defined in terms of B,s and A,s as 6, = B,-..&

i=o,1,2

)...)

n-l

(27a)

(27b) p= Vol.323,No I.pP. 55 72.1987 Printcdin (ircat Britain

1,2 ,...,n 69

M. A. Tan et al.

Fro. 13. The signal-flow graph corresponding

to nth-order

H(z) of (26).

(274

This graph, like the second-order graph in Fig. 5, is also composed of the basic signal-flow graphs corresponding to the active-SC integrator of Fig. 3 and the SC summer of Fig. 4. Hence, the active-SC filter realizing nth order H(z) of (26) can be obtained by interconnecting the active-SC integrators and SC summer according to the graph shown in Fig. 13. The configuration of this filter is shown in Fig. 14, assuming that all of the coefficients, 6s and a,s are positive, and that K < l/2. The design equations which give the capacitance ratios are determined as in the second-order case : (28a)

Go/C = &IWf4)T-&l C,,,/C = 6,/[(GB,+

,)T-

(&+

A^oIWW-&I Cm/C= ~iIW%+ JT-~,l Cl/C = MW,+,)T-(~,+1)1

CdC

C&Z

l)]

(28b)

WC)

=

= C,,jC

= K/(1 -2K)

i= 1,2,...,n-1

(2gd)

WeI (28f)

where GB], j = 1,2,. . . , n is the gain-bandwidth product of the jth op amp, T is the half clock period, and B,, & and a, are the parameters which are functions of the coefficients of H(z) and determined by (27).

V. Conclusion The method presented in this study enables one to realize all stable biquadratic and nth order voltage transfer functions with arbitrary zeros in the z-domain. The proposed procedure is straightforward and simple in that it gives the capacitor

70

Des@

qf Switched-Capacitor

Filters Using Non-ideal

Op Amps

FG. 14. The active-SC filter realizing &h-order H(z) of (26). ratios directly from the coefficients of the transfer function. The finite gain-bandwidth product of the op amp is included in the design. Two examples are included. These examples have also been analyzed using an available computer program. Note that the existence of parameters X, and X2 gives flexibility in the design. It is always possible to optimize the dynamic range and/or the capacitor spread of the proposed filter by choosing A’, and XI so that all the op amps saturate for the same input signal level and/or the spread in capacitor ratios can be minimized. It is also possible to optimize the total capacitance of the filter by giving the smallest capacitances appropriate values permitted by the design technology used. In the proposed filter none of the parasitic capacitances change the circuit topology if the 4” switches between input terminals of the op amps are split into two grounded 4’ switches. In this case, the effect of the parasitic capacitances can be treated as capacitance variation since the parasitic capacitances are in shunt with the capacitors of the filter. Finally, other active SC-filter configurations different from that proposed can be derived in a similar way using a different signal-flow graph model and/or SC integrators with non-ideal op amp.

Acknowledgement The authors wish to thank Professor University simulation

Rolf Schaumann and Mr Eric. P. Rudd of the of Minnesota for the use of their SC-network analysis program, SCFIL4 for the of the examples and the Computer Center of the University of Minnesota

Vol. 123, No. I. pp 55 72, 19x7 Prinlcd in Circa1 Bnlain

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M. A. Tan et al. for running the programs. ECS8403755.

The last two authors

were supported

partially

by NSF grant

References (I) C. Karagbz (2) (3)

(4)

(5) (6)

(7) (8)

(9)

(10) (11)

12

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