CIRCUIT THEORY AND APPLICATIONS, VOL. 11, 397425 (1983)

LETTERS TO THE EDITOR

SECOND-ORDER ACTIVE R FILTERS WITH ZERO POLE Q SENSITIVITIES DUE TO GAIN-BANDWIDTH PRODUCT'S?

CEVDET ACAR AND M. ALI TAN

Chair of Circuits and Systems, Faculty of Electrical Engineering, Technical University of Zstanbul, Gumii$suyu, Istanbul, Turkey

INTRODUCTION

The realization of the second-order active R filters has been studied extensively by many authors,'-1° but not much work has been done on the realization of the filters with low sensitivity performance."

The purpose of this work is to give a synthesis method for the realization of the second-order active R filters with zero pole Q sensitivities. In active R filters, the sensitivities due to active parameters (gain-bandwith products) are more important than the sensitivities due to passive parameters (resistors) since the tolerances of the active parameters are larger than the tolerances of the passives. Hence, in this work, the variations in the gain-bandwidth products of the operational amplifiers (opamps) are taken into consideration. The method presented in this paper is based upon the signal-flow graph model corresponding to the state variable equations of the transfer function to be realized and the elementary active R circuits, resistive summer and/or summing amplifiers.

REALIZATION PROCEDURE

Let the transfer function which is to be realized by active R circuits be given in the following form:

B2s2+B1s +Bo T(S)=s2+(op/Qp)~ +o, 2

where the B p are, respectively, the coefficients of the numerator polynomial, Q, is quality factor for the pole pair and w, is the pole frequency. Let the state-variable equations realizing T ( s ) be given, in general, as

where xl(t) and x2(t) are the state variables, e(t) and y(t) are the input and output signals, respectively. The ui,s, b,s, c,s and d are constants depending on the circuit parameters. The state-variable equations

t This paper has been presented in part at the 6th European Conference on Circuit Theory and Design, Stuttgart, W. Germany, in September 1983.

0098-9886/83/040397-29$01.00 @ 1983 by John Wiley & Sons, Ltd.

Received 8 Fetriary 1983 Revised 8 June 1983

398 LETTERS TO THE EDITOR

in (2) realize T ( s ) provided that

(34 alla22-al2a2' = u p

-(ail +a221 = o p / Q p (3b)

(34

b l c i + b 2 ~ 2 - ( ~ 1 1 + ~ 2 2 ) d = B 1 ( 3 4

d=B2 (3e) In order to evaluate the a+, bs and cis satisfying (3a)-(3d), we have to choose four appropriate

parameters as an independent set, as we have four equations and eight unknowns. There exist several sets of independent parameters, and in our study all , a12, c1, c2 were chosen to simplify the formulation of the dependent parameters as given by the following equations"

2

(a 1 1 ~ 2 2 - ai2a2i)d - aiibzcz - a22bici + a1262ci + a2ib 1 ~ 2 = Bo

a 2 2 = - ( a i i + w p / Q p ) ( 4 4

where

A signal-flow graph model corresponding to the state-variable equations of (2) is shown in Figure 1. This graph realizes T ( s ) when equations (3) or (4) are satisfied. Note that the signal-flow graph is composed of the elementary signal-flow graphs as shown in Figures 2(a) and 3(a). These elementary signal-flow graphs correspond, respectively, to the elementary active R circuit and resistive summer of

Figure 1. The signal-flow graph of the state-variable equations realizing T ( s ) of (1)

LE7TERS TO THE EDITOR

G2 + Q v l t b t ----ut

vo v2 v3

G3

399

vo

1- s

G1 +GB - P

G

c;3

Gn -GB - b"'-+

(a) =!F (b)

Figure 2. The basic active R circuit and its signal-flow graph. Here G , = Gl + G2 + G, and G. = G3 + Gb

Figures 2(b) and 3(b)' provided that the open-loop gain of the operational amplifier is characterized by

A = GB/s ( 5 )

where GB is gain-bandwidth product and s is the complex frequency. The active R filter realization of T ( s ) can easily be achieved by interconnecting the elementary circuits in Figures 2(b) and 3(b). Further- more the resistors must be assigned such values as to obtain the proper branch transmittances.

In Table I the values of the branch transmittances of Figure 1 realizing various type of second-order filter characteristics are shown. The values of the independent parameters in the table (i.e. branch transmittances a l l , a ~ ~ , c ~ and c2) are chosen arbitrarily and the values of the others (i.e. branch transmittances a22, C Z ~ ~ , b l , 62 and d ) are determined from equations (4). The active R filters obtained are shown in Figures 4-8. The resistor values are determined by equating the branch transmittances of Figure 1 to the corresponding branch transmittances of Figure 2(b) and/or Figure 3(b), and they are tabulated in Table 11.

Note that, to realize the highpass, notch and allpass filters of Figures 4, 7 and 8, the sub-signal-flow graph consisting of the branches incoming to the output nodes 3 in Figure 1 is realized by a resistive summer. Hence we can obtain realizable active R filters provided that H G 1/2. If we use a summing amplifier of wide-band type instead of a resistive summer, no restriction will be imposed on H.

Gl'GCI

(a)

Figure 3. Resistive summer and its signal-flow graph. Here Go = G1 + G2 + G3 + G4

P

0

0

Tabl

e I.

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es of

the

bra

nch

trans

mitt

ance

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re 1

real

izin

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ures

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eter

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-wP

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62

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-- 20,

QP

H

LETI'ERS TO THE EDITOR 401

"i

1;' - Figure 4. Highpass filter

G1 2

"0

Figure 5. Bandpass filter

Figure 6. Lowpass filter

402 LElTERS TO THE EDITOR

5 2

vi

vo

Figure 7. Notch filter

SENSITIVITY CONSIDERATIONS

In the following w , and Q, sensitivities are taken into account to understand and improve the sensitivity behaviour of the proposed filters.

The pole frequency, wp can be written as 1/2

w p = (a11a22-a12a21)

The sensitivity of wp due to the active parametet is defined as

Gb2 G 2 2 ‘d

I‘ Figure 8. Allpass filter

Tab

le 1

1. T

he v

alue

s of

the

cond

ucta

nce

in th

e ac

tive R

filt

ers of

Figs

. 4’8

\-

1-2H

0

GP1

0 0

4Qi +

4HQ

p- 1

4Qi

GB

2-w

P

4HpQ

,’ G

B1 -~

4Q

i - 1

50

G

B ->

’-

2 Q

,

P

0

w

404 LETTERS TO THE EDITOR

where GBk is the gain-bandwidth product of the kth opamp. S h B , can be written as follows:

where S 2 and S&, are, respectively, the sensitivities of up and aii due to aii and GBk :

The Sz;s are evaluated from (6) as follows:

Note by inspection of Figure 2 that the transmittances of the branches incoming to nodes 1 and 2 in Figure 1 are, respectively, proportional to GB1 and GB2, the gain-bandwidth products of the operational amplifiers used in the design. Hence the Sa&ks are

s&, =S2& = 1 (1la)

S2& = S 2 & = o (1lb)

S2dZ = S26, = 0 (1 14 s2iz = s2fj2 = 1 (114

Substitution of (10) and (11) into (8) yields

This means that, in the proposed realization, pole frequency sensitivities are independent of up and Qp, and they have constant values of 1/2. Furthermore, selection of the independent parameters a11 and a12

does not affect the pole frequency sensitivities. The quality factor, Q, for the pole pair is

Qp = -(a 1 la22 - a 12a21)~ '~/(~ 11 + a221

The sensitivity of Q, due to GBk is defined as

This sensitivity can be written as follows

where S z ; is the sensitivity of Q, due to ai,:

LETTERS TO THE EDITOR 405

The S:;s are evaluated from (13), (4a) and (4b) as follows:

Substitution of (11) and (17) into (15) yields

Equations (18) and (19) mean that sensitivities of the Q factor due to the gain-bandwidth products of the operational amplifiers used in the synthesis are functions of a l l as well as wp and Q,. If all is chosen so that lalll >>wp/(2Q,) then the sensitivity performance.of the proposed active R filters due to active parameter variations becomes bad. It is very interesting to note that S",s, and S",s, are zero provided that all = -wp/2Q,. Hence, the values of the ai,s which make the pole Q sensitivities zero are

In other words, to make the pole Q sensitivities zero, all and a22 should be equal to each other with a common value of -wp/(2Q,). Furthermore azl and a12 should be chosen so that their product is a constant of value -w;(4Qa - 1)/(4Q;). The other parameters, the bis and cis and d realizing T ( s ) can be evaluated from (4).

The values of conductances in Table I1 provide zero pole Q sensitivities for the active R filters shown in Figures 4-8 since they satisfy the condition (20a). So these filters have the best performance from the Q sensitivity point of view. Note that the range of conductance values in Table I1 is large and, in the worst case, Gnl/Gbl = 20Q; for GB = 100,. This can make the design unattractive, depending on Qp, from a manufacturing standpoint. But alternative choices of the independent parameters, a12, c1 and c2 can reduce this ratio, for example, to Gnl/Gbl = 20Qp or to its minimum value.

It is also shown from the similar analysis that both pole-Q and pole frequency sensitivities with respect to the passive parameters are less than 1/2 in magnitude when the Q sensitivities are zero.

CONCLUSION

The active R filters are realized by using a signal-flow graph model of the state-variable equations corresponding to a general biquadratic transfer function, T ( s ) of (1) and the elementary circuits of Figures 2 and 3. The independent branch transmittances which are a l l , a12, c1 and c2 give flexibility to the design. In the proposed filters, the pole frequency sensitivities, S&, are independent of w,, Q, and also all , a12, c1 and c2. They have constant values of 1/2. This means that the choice of independent parameters does not affect the pole frequency sensitivities. Moreover, the pole-Q sensitivities, .SSB,s are dependent upon w,, 0, and all . They will considerably increase if lalll >>wp/(2Q,). Furthermore, the pole-Q sensitivities will be zero provided that a l l = a22 = -wp/(2Q,). The values of conductances in Table I1 provided zero pole-Q sensitivities for the active R filters shown in Figures 4-8 since they are evaluated such that the condition all = az2 = -op/2Q, is satisfied. Finally, the spread in the values of the elements depends on choice of the independent parameters a12, c and c2.

406 LETTERS TO THE EDITOR

REFERENCES

1. C. Acar, ‘Active R filter realization for the second-order voltage transfer function: signal-flow graph approach’, Proc. Fourth

2. C. Acar and T. Ozker, ‘Active-R network realization for n th-order voltage transfer functions: a signal-flow graph approach’,

3. C. Acar, ‘Realization of nth-order lowpass voltage transfer function by active R circuit: signal-flow graph approach’, Electronics

4. H. K. Kim and J. B. Ra, ‘An active biquadratic building block without external capacitors’, ZEEE Trans. Circuits and Systems,

5. A. K. Mitra and V. K. Aatre, ‘Low-sensitivity high-frequency active R filters’, IEEE Trans. Circuits and Systems, CAS-23,

6. R. Schaumann, ‘On the design of active filters using only resistors and voltage amplifiers’, AEU, 30, 245-252 (1976). 7. M. A. Soderstrand, ‘Design of active R filters using only resistors and operational amplifiers’, Znt. J. Electronics, 40,417-432

8. S. Srinivasan, ‘Synthesis of transfer functions using the operational amplifier pole’, Znt. J. Electronics, 40, 5-13 (1976). 9. R. Schaumann, ‘Low sensitivity high frequency tunable active filters without external capacitors’, IEEE Trans. Circuits and

10. J. R. Brand and R. Schaumann, ‘Active R filters: review of theory and practice’, IEE J. Electron Circuits. Syst., 2,89-lOl(l978). 11. M. A. Tan and C. Acar, ‘Sensitivity minimization in signal-flow graphs’, Int. J. Cir. Theor. Appl. 10, 19-25 (1982).

International Symposium on Network Theory, 221-226, Ljubljana (Yugoslavia), Sept. 1979.

AEU, 32,463-464 (1978).

Letters, 14,729-730 (1978).

CAS-24,689-694 (1977).

670-676 (1976).

(1976).

Systems, CAS-22, 39-44 (1975).

A NEW ACTIVE COMPENSATED DIFFERENTIAL INTEGRATOR WITHOUT MATCHED OPERATIONAL AMPLIFIERS

A. M. SOLIMAN Electronics and Communications Engineering Department, Faculty of Engineering, Cairo University, Giza, Egypt

AND

M. ISMAIL*

Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, Canada, R 3 T 2NZ

INTRODUCTION

The single operational amplifier (opamp) differential integrator’ finds wide use in many circuit applications. It is well known that the finite and complex gain nature of the opamp degrades significantly the performance of the differential integrator.’ Recently, an active compensated differential integrator has been reported in the literature.’ In this paper a novel active phase compensated differential integrator is introduced. The proposed network has the advantage that the phase compensation condition is independent of the gain bandwidth of the opamps employed in the circuit. Experimental results are included.

NEW CIRCUIT

The proposed phase compensated differential integrator is shown in Figure 1. Straighforward analysis of the circuit yields the following expression for the output voltage Vo in terms of the inverting input voltage V1 and the non-inverting input voltage VZ.

-+- 1+-

* Dr. M. Ismail is now with the Dept. of Electrical Engineering, University of Nebraska, Lincoln, N E 68588-0511, U.S.A.

Received 9 December 1981 Revised 1 June 1983