CIRCUIT THEORY A N D APPLICATIONS. VOL. 10, 377-403 (1982)

LETTERS TO THE EDITOR

ACTIVE SWITCHED-CAPACITOR NETWORK REALIZATION FOR nth-ORDER VOLTAGE TRANSFER FUNCTIONS:

A SIGNAL-FLOW GRAPH APPROACH

CENGIZ KARAGOZ AND CEVDET ACAR

Faculty of Electrical Engineering, Technical University of Istanbul, GumiiSsuyu, Istanbul. Turkey

1. INTRODUCTION

The realization of the active switched-capacitor (SC) filters has been studied extensively, but not much work has been done for the direct realization of high-order SC filters.'-'

In this letter, using signal-flow graphs, a general synthesis method is given for the realization of nth-order voltage transfer functions in the z -domain by active SC networks using only biphased switches, capacitors and ideal operational amplifiers.

The switches are assumed to be controlled by a two-phase, non-overlapping clock of frequency f c = 1/27 = 1/T, and have a 50 per cent duty cycle with equal on- and off-time periods. Furthermore, those are closed on the even 2k7 times and on the odd (2k + 1 ) ~ times are denoted by e and o respectively.

2. THE SIGNAL-FLOW GRAPH REPRESENTATION O F THE BASIC CIRCUITS

The two basic circuits and the corresponding signal-flow graphs which will be used in the synthesis of a class of active SC filters are shown in Figures 1 and 2.

Writing the nodal charge equations'*2 in the z-domain for the node n of Figure l (a ) at the even and odd time instants, we obtain the following equation from which we can derive the signal-flow graph shown in Figure l (b) :

VZ"t = [ l - z - ( 2 - 1 ) ] vgut +- CI v; c3

where VZut and V ; denote, respectively, the z-transforms of the discrete time voltages sampled at the even times at the output and input terminals 0 and 1.

The signal-flow graph of the capacitive summing amplifier of Figure 2(a) can be obtained in a similar way:

provided that

c,, = c, (3)

where C,, and C, are, respectively, the sum of the capacitor values connected to the nodes n and p , the input terminals of the operational amplifier. The signal-flow graph corresponding to equation (2) and hence to the circuit of Figure 2(a) is shown in Figure 2(b).

The signal-flow graphs shown in the figures represent the basic circuits at the even time instants. If we assume that the discrete input voltages of the basic circuits in Figures l (a ) and 2(a) are such that

V: = z - ' / ~ v ~ , i = 1 , 2 , 3 (4)

0098-9886/82/040377-27$01.00 @ 1982 by John Wiley & Sons, Ltd.

Received 16 October 1981 Revised 23 March 1982

378 LETTERS TO THE EDITOR

C c3

1 - z - i A-1)

(a) (b)

Figure 1. The basic active SC circuit and the corresponding signal-flow graph

then simultaneously

( 5 )

for the discrete output signals. Therefore the signal-flow graphs and hence the even transfer functions determined over these signal-flow graphs provide all the information we need.

- 1 / 2 v:,, = z v:",

3. REALIZATION PROCEDURE

It is assumed that the nth-order voltage transfer function has already been given in the following form:

v:", B,$ + B,-,z ~ I + . . . + B ~ Z + B~ H(z)=----= VFn Z " + A , , - ~ Z " - ' + . . . + A l z + A o

f

V O " I f

+2-cw v,

(a)

'G + c3 / C f - c, /c,

Figure 2. The capacitive summing amplifier derived by discrete signals and the corresponding signal-flow graph

LETTERS TO THE EDITOR 379

One of the signal-flow graphs of equation (6) is obtained by using the approach in Reference 8, and shown in Figure 3(a). Note that this graph is composed of the elementary signal-flow graphs in Figures l (b) and 2(b). Hence the active SC network realizing (6) can be obtained by interconnecting the basic active SC circuits of Figures l (a ) and 2(a). This network is shown in Figure 3(b) when all the coefficients of H ( r ) are positive. Design equations which give the values of the capacitor ratios are determined by equating the branch transmittances of Figures 1 (b) and 2(b) to the corresponding branch transmittances of Figure 3(a):

(7a)

(7b)

(7c)

clIcfl= 1 ( 7 4

C,,/Cfl = lAll,

C,l/C,3 = C,,/CI3 = 1,

i = 0, 1 , 2 , . . . , n - 1

i = 0, 1 , 2 , . . . , n Cb,/Cfz = lBll,

i = 1 , 2 , . . . , n

Note that if some coefficients of H ( z ) are negative then C,,, C,, capacitors corresponding to the positive transmittances should be connected to the positive input terminals of the summing amplifiers as denoted in Figure 2(a). In this case we need the following additional design equations:

CtnICf= ( C p -Ctp)/Cf (8a)

CtJ Cf = ( Cn - Ct n 1 / Cf (8b)

As an example, the signal-flow graph and the corresponding active SC network realizing the 3rd-order voltage transfer function

N ( z ) 0 * 0 1 6 ( ~ ~ + 3 ~ ~ + 3 ~ + ~ ) D ( z ) z 3 - 1*975z2+ 1.5242 -0.454

H ( 2 ) = - = (9)

are obtained by the above procedure and shown in Figure 4. H ( z ) is the z-domain voltage transfer function of the filter passing from 0 to 100 Hz with 0.5 dB ripple, which falls off monotonically to at least -19 dB at 183 Hz, and it is obtained using the Chebyshev approximation and bilinear transformation9 by taking the sampling rate 1,000Hz. The coefficient sensitivities of (9) due to capacitance ratios are evaluated and tabulated in Table I. These sensitivities due to all capacitance ratios except some of those CI4/C,, i = 1,2, 3 are not greater than one in magnitude, and that the larger coefficient sensitivities due to CI4/C,, can occur depending upon the coefficients. The sensitivity of the actual gain response due to capacitance ratios can be calculated in terms of the coefficient sensitivities in Table I by the following formula"'

where x stands for capacitance ratios.

4. CONCLUSION

The method presented here is straightforward and simple in that it gives the capacitor ratios directly from the coefficients of the voltage transfer functions, H ( z ) . The proposed network is capable of realizing all stable nth-order voltage transfer function, H ( z ) of equation (6) with arbitrary zeros. The coefficients of the numerator and denominator polynomials are independently adjustable by changing the values of the capacitors Cais and Cbjs. At most n + 2 operational amplifiers are required for the realization, and the proposed network contains n identical basic SC-circuits which may be interesting in mass production point of view. It is always possible to optimize the dynamic range of the proposed network by changing the values of the branch transmittance of Figure 3(a) such that the loop and forward path gains remain unchanged. It is easy to see that only the stray capacitances due to the grounded capacitors affect H ( z ) ,

380 LETTERS TO THE EDITOR

CI + C-iF v, "

-l(cb[n-'' ,.

Figure 3. The signal-flow graph realizing H ( z ) of (6) and the corresponding active SC network

I.E7TERS TO THE EDITOR 381

-1 1-

C,, = 0,016

C,,=l

C , = l p-

c,z =o, 0.4 8 I /

5

C,,=0,016

Figure 4. The signal-flow graph realizing the 3rd order voltage transfer function H ( z ) of (9) and the corresponding active SC network

LEITERS TO THE EDITOR

Table I. Coefficient sensitivities due to capacitance ratios

S X S,"' Sp2 sp, s:o SfZ S:', 5;30

ClICi, 1 1 1 1 0.28 0 0.28 CI 1 / CI3 0 0 0 1 0 0 1 G I I c23 0 0 1 1 0 1 1 C3IIC33 0 1 1 1 1 1 1

C24IC23 0 0.33 1 0 -0.50 -1.29 0 C,,/C33 0 0.33 0 0 -0.50 0 0

CI4IC13 0 0.33 1 3 -0.50 -1.29 -3.35

Ca2ICi, 0 0 0 0 0.44 0 -0.56 Ca,ICi, 0 0 0 0 0.43 1 0.43 Gaol ci, 0 0 0 0 -0.13 0 0.87 Ct,lCi, 0 0 0 0 -0.3 1 0 -0.31 Cb,,/ Cf, 0 0 0 1 0 0 0 Ck?,ICi, 0 0 1 0 0 0 0 Cb2ICi2 0 1 0 0 0 0 0 Cbl ci, 1 0 0 0 0 0 0

but assuming that the degree of these strays is either known or can be estimated they can be compensated for by a slight adjustment of the grounded capacitors, Finally, another active SC-network configuration different from that proposed, can be derived in a similar way using a different signal flow graph model andlor basic SC-networks.

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