A low-sensitivity active RC low-pass filter

IEEE TRANSACTIONS ON AUDIO AND ELECTROACOUSTICS, AUGUST

312

1973

For (25) to hold, the 1 , l entry of C must benegative Of course, in programming either algorithm, one definite and C must have positive determinant, i.e.,would want to take full advantage of the fact that a number of the component’ tests involvereal poly175 + 702, + 702,’ > 0, nomials, and some simplification may then be possible. 122

I=1

References [ l ] C. Farmer and J. B. Bednar, “Stability of spatial digital filters,” Math. Biosci., vol. 14, p . 113-119, 1972. H. Justice, “Stability [2] J. L. Shanks, S. Treitel, and andsynthesis of two-dimensional recursive filters,” ZEEE Trans. Audio Electroacoust., vol. AU-20, pp. 115128. June 1972. [31 T. S. Huang, “Stability of two-dimensional recursive filters,” IEEE Trans. Audio Electroacoust., vol. AU-20, pp. 158-163, June 1972. [41 H. G. Ansell, “Oncertain two-variable generalizations of circuit theory, with applications to networks of transmission lines andlumpedreactances,” IEEE Trans. Circuit Theory, vol. CT-11, pp. 214-223, June 1964, t51 S. H. Lehnigk, Stability Theorems for Linear Motions, with an IntroductiontoLyapunov’sDirectMethod. Englewood Cliffs, N.J.: Prentice-Hall, 1966. E.A. Guillemin, Synthesis of Passive Networks. New York: Wiley, J957. A. Cohn, “Uber die Anzahlder Wurzeln einer algebraischen Gleichung in einer Kreise,” Math. J., vol. 14, pp. 110-148,192.2. M. Fujiwara, “Uber die Algebraischen Gleichungen, deren Wurzeln in einem Kreise oder in einer Halbebene liegen, Math. J., vol. 24, pp. 160-169, 1926. [ 9 ] E. I. Jury Theory and Application o f thez-Transform Method. h e w York: Wi!ey, 1964. [ l o ] S. Barnett. Matrzces In Control Theory. London, England: Van Nostrand Reinhold, 1971. [ l l ] K. J. Astrom, IntroductiontoStochasticControlTheory. New Y2rk: Academic, 1970. reciprocalzeros of a real [12] E, I. Jury, A noteonthe circle,” IEEE polynomialwith respect to theunit Trans. CircuitTheory, vol. CT-11, pp. 292-294, June 1964. [13] B. D. 0. A2derson and E. I. Jury, “A simplified SchurCohntest, ZEEE Trans. Automat.Contr., vol. AC-18, pp. 157-163, Apr. 1973. [14] D. D. Siljak, “A4gebraic criteria for positive realness on the unit circle, in Proc. 2nd Int. Symp. Network Theory, Yugoslavia, July 1972. [15] F. R. Gantmacher, The Theory o f Matrices. New York: Chelsea, 1959. H. A. NourEldin, “Eir,neues Stabilitatskriteriumfur abgetaste Regelsysteme, Regelungstechnik, vol. 7,pp. 301-307,1971. E. I. Jury and .G. Gupta,“Alternatecoefficientsconstraints for stabllity test,” Proc. IEEE (Lett.), vol. 55, pp. 1769-1770, Oct. 1967.

5.

The first inequality is obviously satisfied. To check the second, set x = cos 6 . The inequality which has to be satisfied becomes

(175 + 1 4 0 ~ -) (125 ~ +1 0 0 ~> ) ~0,

-1< X

) 0,

- 1S X

1

and clearly this inequality holds.

VI.

I

Conclusion

Our method is really to be compared against that of Huang. By and large, the methods involve the same sort of calculations, save that weavoid the bilinear transformation component ofHuang’s method. Undoubtedly, thisrepresents a substantial computational load for any butthe simplest two-variable polynomials.On theotherhand, as soon as the twovariable polynomials under test become at all complex, presumably computers will be used to do the checking, and it might well prove the case that programming considerations determine which is the better method. Nevertheless, for those who have some knowledge of the Schur-Cohn and related theory, our method would probably be conceptually preferable. For those whoseknowledgeencompassesHurwitz testing, but not the Schur-Cohn material, Huang’s method might be conceptually preferable.

Correspondence A Low-SensitivityActive RC Low-Pass Filter AHMED M. SOLIMAN

Manuscript received September 20 1972. Theauthor was withthe DePa;tment of Engineering,College of Steubendle,steubenfie, Ohlo 43952. He is now at 2 Dr. SOliman Square, Dokkl-Cano, Egypt.

tional amplifier ( O A ) andtheresults arrived atarecompared with other well-knownlow-pass filters. Experimental results agree with the theoretical ones.

model of operational the amplifier (OA) to determine the upper bound On the frequency that these networks can effectively realize, was developed [ 31.

31 3

CORRESPONDENCE

That is, wo can be tuned by varying R or C or both without affecting Q of the filter. Equations (9) and (10)are the design equations of the filter. Note that K is determined from the specified pole Q. Next, the sensitivity with respect to A2 is derived. Assuming A2 I I A & , it can be seen that for finiteA02

R1 AI\

I

" I

vi

I

where

L

e------

-

-J

Fig. 1. Two O A active low-pass filters.

In this correspondence a new active RC filter that is suitable for realizing a low-pole Q second-order low-pass transfer function is given. The filter has low sensitivities with respect to all passive and active circuit components, and in particular, the Q sensitivities with respect to R and C are zero. The frequency limitations of the circuit arediscussed. II. New Filter

,

Consider the circuit shown in Fig. 1. Exact analysis of the circuit leads to

The above sensitivities are very low for low Q (0.5 < Q < 5) filters. In fact, for most applicationsof low-pass filters, one is interested in Q in the above range. The effect of finite gain of A 1 is discussed next. Substituting from (6) and (7) in (1)and setting A l 2 A o ~one , gets

T(s, =

-

(

1+;I)

s2+

(I+-

A:I);R

K R2C2 .-+

( 201) 1+-

RC 'Z

*

(18)

which realizes a low-pass characteristic having a dc gain, and Q given, respectively, by

UO,

Thus the actualvalues of the dcgain, w 0 and Q of the filterare

In the special case, if

one gets for A01 0 0

>> 1. (21)

JK

=-

RC

I I I. wo and 0 Sensitivities

The 00 and the Q sensitivities with respect to R and C are given by The above sensitivities are very low and approaching zero for A01 >> 1.

IEEE TRANSACTIONS AND ON AUDIO

374

ELECTROACOUSTICS, AUGUST 1973

IV. Frequency Limitations of the Filter

VI. Experimental Results

In this section, the actual values of w o and Q of the filter are derived, takingintoeffectthe rolloff of the OA gain. Assuming that ideal OA's are used for A1 and A2 such as the pA741,

The circuit was built using the pA741 OA and satisfactory results were obtained. For example, when K = 6, R = 1 k a , and C = 0.02 pF (i.e.,' Q = 2.45), fo (measured) = 19.1 kHz. Using (9) and(32), the actual theoreticalvalue f o , = 1 9 . 2 kHz. That is, there is 0.5 percent error between f o (measured) and foa.

K=

VII. Conclusions

GB S + -

A new active RC low-pass filter was given. The sensitivities with respect to all passive and active circuit components are very low. The frequency limitations of the filter are derived using the one-polerolloff model of the OA.

GB

KO

where we have the following. Open-loop dc gain of the OA. Open-loop 3-dB bandwidth of the OA. GB Gain-bandwidth product of the OA.

A.

Referyces

w

Substitutingfrom (6), ( 7 ) , (9), ( l o ) , ( 2 4 ) , and ( 2 5 ) in ( l ) , andaftersomeapproximations,one gets the following expression for the denominatorof the transfer function:

[l] R P SallenandELKey "A practicalmethod of designin RC a&& filters," IRE !l'kns. 'Circuit Theory, VOL CT-2, pp. 7f-85, I"far. I 1955. c21 IP.K. Mltra, Analysis-andSynthesis of Linear ActiveNetworks. New York: Wiley 1969 131 A, Bud& and D: Petrei?, "Frequency limitations pf active filters usm operationalamlifiers" IEEE Trans. Czrcuzt Theory, vol. CT-$9, pp. 322-328, &ly 19?2.

where P 1 ( S n )=

s; +-Q1

sn

+l)+sn

and

s

=-

+

1

(27)

(. z) Q".] +-

+-

(28)

S

(29)

wo'

According to the Budak-Petrela technique [ 31, it follows that AWO = - -

a0

(30)

Q GB

WO

Poles of Maximally Flat Sharp-Cutoff Low-Pass Filters SUHASH C. DUTTA ROY and RAKESH K. PATNEY Abstract-The poles of a recently discussed [ 21 maximally flat sharp-cutoff low-pass filter are tabulated for various orders of thefilterand various values of the parameters.Loci of these poles arealso sketched for some representative orders. Ithas been recentlydemonstratedthat alow-passfilter characterized by themagnitude-squared function

Thus the actualvalues of w 0 and Q are

IHo'w)/2 = [ ( a 2 - o ; ) 2 m ] /

r

[(a2

Q,=Q

[+ 1

[Q3 + Q -

4)

-

w ; y m + (a;-

1 y m a 2 n1

(1)

>

wo]

GB

(33)

Equation ( 3 0 ) gives the frequency limitationof the filter for specified A W O / W Oand a given Q. For example, for Q = 1, and fora specified maximum allowable change in w 0 and using the same OA, it can be seen that the given filter realizes w 0 , which is twice that realizable using eitherthe negative K Sallen-Key filterorthe positive K Sallen-Key filter having equal R and equal C. a

with n 2m is maximally flat and has a cutoff slope which is much steeper than that of the Butterworth filter. Budak and Aronhime [ l ] investigated the case m = 1 while Dutta Roy [ 2 ] studied the more general case of multiple pairs of zeros, coincident, as in ( l ) , or distinct. For the design of such filters, we need to find out the transfer function H(s). From (l), it is obvious thatH ( s ) will be of the form

H(s) = ( 2 + a;)"/[(a; - 1 y D(s)] (2) where D ( s ) is the product of the left half s plane factors of the polynomial

E ( s ) = D ( s ) D ( - s ) = ( - 1 ) n S z n + [(s2 + o ; ) / ( w ; - 1)12". (3)

V. Stability Analysis

From (26), (27), and (281, and applying Routh's criterion for stability, it is found that for stability

That is, for Q < 1, the circuit is always stable, and for Q the circuit will oscillate if wo

Q3 > GB.

> 1, (35)

As an example, for Q = 2 and using the Fairchild IC, pA741 having GB = 2n X lo6 rad/s,thecircuit will oscillate at frequencies above 1 2 5 kHz.

Unfortunately,it has not been possible to find explicit expressions for the rootsof (3), as is possible for the conventional Butterworth or Chebyshev filters. The roots of ( 3 ) were thereforeobtained by using thestandard IBM subroutine called POLRT, for various values of n, possible allowed values of m(n 2m), and a0 varying from 1.06 to 1.15. Some of the results forlower orders aregiven in Table I.

>

Manuscript received December 5 , 1972. S CDuttaRoyiswiththeDepartment of Electrical Engineering India on leave attheDel Ind'ian' Institute of Technology New Delhi partment of Electrical and Elechonic Enginkering,'University of Leeds, L ~ ~ is. with K ~ Department ~ ~ of Electrical ~ ; Engineering, Aligarh MuUmverslty, Ahgarh, Uttar Pradesh, India.