I I v -I L ----_ -_ _ _ 2 2 I D E A L VOLTAOE AW.FlER Fig. 1. RC active filter with ideal voltage amplifier.The choice of D(s) may be so made that the pole sensitivity due to change in the gain of the voltage amplifier is minimum. Considering a secondorder transfer function, letHswhere the poles s1 and s: are at -r k j x . The H may be assumed equal to unity without loss of generality so thatFor this second-order transfer function, let D(s) be of first order having a real negative root. Therefore,Then.In terms of the RC admittances,.
(9)For realizing yl(s) and y2(s), K has been assumed equal to unity.minimized.The sensitivity of T(s) with respect to K may now be calculated [ 2 ] and =-
K x b 2Thus, the real and imaginary parts of the sensitivity may be separated as sq. = ~ 2 K p 2 -2pr + r2 + x' ( K + P For stability of bandwidth, the real part of the root sensitivity should be made minimum by a suitable choice of p . Differentiating Sqr with respect to p and equating to zero, we obtain the conditionSince p is real and positive,as for a narrow-band filter, r << x . Similarly, for the stability of the center frequency, the imaginary part of the root sensitivity can be minimized This leads to the same condition, i.e.,Substituting this value of p into (13), the root sensitivity for a narrow-band filter with K = 1 is obtained asTo achieve the stabilization of center frequency and bandwidth, the foregoing process of minimizing the root sensitivity can be adopted for the synthesis of an active RC filter employing a voltage amplifier. Hakim [4] has shown a different technique to obtain a prescribed pole sensitivity by realizing a conjugate complex phantom zero using a control parameter K which is not the gain of the amplifier but is some function of the gain of the amplifier. The freedom to prescribe the pole sensitivity is also overshadowed [3] by the inconvenient structures resulting.The Horowitz decomposition [SI minimizes sensitivity but this minimization occurs only for certain structures, basically Linvill configurations. Therefore, the Horowitz technique is not general [3] for minimizing the pole sensitivity of different active network configurations. The technique presented here is general and the pole sensitivity can be calculated and minimized with respect to any parameter of the realized network. This technique can also be used to obtain the prescribed pole sensitivity for other structures.The method is applicable to low-pass and high-pass filters as well.
REFERENCES[I] S. S. Hakim, "RC active filters using an amplifier as the active element," Proc. Inst.[2] P. J. McVey, "Sensitivity in some simple RC active network," Proc. Inst. Elec. Eng.,
[31 R. W. Newcomb, A c h e Integrured Circuit Synthesis.Englewood Cliffs,
N. J.:[4] S. S. Hakim, "Synthesis of RC active filters with prescribed pole sensitivity,"
Proc.[SI 1. M. Horowitz "Optimization of negative-impedance conversion methods of active
