ON THE STABILITY OF RESISTIVELYVARIABLE CAPACITORS USING GENERAL IMPEDANCE CONVERTERS

In a recent publication Cadosena et al. [1] investigated the use of general impedance converters (GICs) for the simulation of grounded and floating resistively variable capacitors. This investigation resulted in three optimum circuits; shown in Fig. 1. In an attempt to check these variable capacitors, they were used to realize the voltage-divider circuit shown in Fig. 2. Unfortunately, during the course of this experiment, self-sustained oscillations, which overrode the input signals, were observed across the simulated grounded capacitors. These oscillations were found to be of various different frequencies and very sensitive to the presence of the oscilloscope leads connecting to the simulated grounded capacitor. In some cases, oscillations were observed only at the terminals of the simulated grounded capacitor and when the oscilloscope leads were moved to other points in the circuit, oscillations were removed. This raises the question whether the realizations of Fig. 1 are always stable or not? Although this question was discussed by Cadosena et al [1], it is felt here that further investigation is needed and this is the major intention of this paper.


INTRODUCTION
In a recent publication Cadosena et al. [1] investigated the use of general impedance converters (GICs) for the simulation of grounded and floating resis- tively variable capacitors.This investigation resulted in three optimum circuits; shown in Fig. 1.In an attempt to check these variable capacitors, they were used to realize the voltage-divider circuit shown in Fig. 2. Unfortunately, during the course of this experiment, self-sustained oscillations, which overrode the input signals, were observed across the simulated grounded capacitors.These oscillations were found to be of various different frequencies and very sensitive to the presence of the oscilloscope leads connecting to the simulated grounded capacitor.In some cases, oscillations were observed only at the terminals of the simulated grounded capacitor and when the oscilloscope leads were moved to other points in the circuit, oscillations were removed.This raises the question whether the realizations of Fig. 1 are always stable or not?Although this question was discussed by Cadosena et al [1], it is felt here that further investigation is needed and this is the major intention of this paper.

ANALYSIS
Consider the circuit of Fig. l(a).Assuming that the input resistance of the operational amplifier is infinite and its output resistance is zero, then the transfer function of the circuit of Fig. l(a) with the feedback loop disconnected at point M, The three optimum circuits proposed in [1] for realizing resistively variable capacitors.R and C represent the input resistance and the input capacitance of the oscilloscope and cable.Cr" fixed capacitor Ceq" resistively variable capacitor of Fig. 1 is given by 1 + + G(s) 1 z -+--1 + aG(s) (1) where

Vo
s is the gain of the internally compensated operational amplifier where B is the gain-bandwidth product of the operational amplifier, where R and C represent the input resistance and the input capacitance of the oscilloscope and cable, which may be connected at the input terminals of the simulated capacitance in order to measure the performance of the voltage divider circuit of Fig. 2.
By closing the loop and using equations (2)-(4), the characteristic equation of the circuit, T(s) 1 0, can be expressed as s+ C,R Therefore, by equating the real and imaginary parts of (5) to zero, i.e., using the Barkhausen criterion, the frequency of oscillation of the circuit of Fig. l(a) can be expressed as and the condition of oscillation can be approximated by C 1/(1 + Ra/R5) + 1/(1 + R2/R3) Equation ( 7)was obtained on the assumption that B >> 1.It appears from equations ( 6) and (7) that, for practical values of active and passive parameters, it is possible for the circuit of Fig. l(a) to oscillate.This will be investigated in the next section.
Following a similar procedure, the characteristic equation of the circuit of and the condition of oscillation can be approximated by

Req
It appears from equations (9) and (10) that, for practical values of active and passive parameters, it is possible for the circuit of Fig. l(b) to oscillate.This will be investigated in the next section.
and the condition of oscillation can bc approximated by C 2 (2C, + Ceq ) It appears from equations ( 12) and (13) that, for practical values of the active and passive parameters, it is possible for the circuit of Fig. l(c) to oscillate.This will be investigated in the next section.

SIMULATION AND EXPERIMENTAL RESULTS
To check the validity of the conjectures made in the preceding section, the circuits of Fig. 1 were simulated using Pspice Student Version 5.0.A sample of the results obtained is shown in Fig. 3.In all cases, the uA741 model available in the EVAL.LIB was used.A de supply voltage of + 15V was used for the operational DOES THE C RCU T OSC LLqTE Date/Time run-02/01/94  Temperature 27.0 Operational Amplifiers: CA741CE. =50pF, =10nF,C4=1nF, amplifiers. To initiate the oscillation, a noise voltage simulated by a piecewise- linear voltage was used.The Spice format of this noise voltage is Vnoise PWL(0 50V 10N 50V 10.001N 0V).
As a second check for the validity of the conjectures made in the preceding section, the circuits of Fig. 1 were experimentally tested using the operational amplifiers uA741 and LF411.The results obtained, shown in Fig. 4, are in good agreement with the Spice simulations.Thus, both the computer simulation and the experimental results confirm the conjectures made in the preceding section.This means that the proposed realizations for the resistively variable capacitors may oscillate if, for any reason, the equivalent of the input impedance of an oscillo- scope and a cable is connected across the simulated capacitor.

CONCLUSION
In this paper, the stability of the three circuits proposed in [1] for realizing resistively variable capacitors was studied.It has been shown that the three circuits may oscillate for practical values of active and passive parameters and components.It is therefore important for a potential user of these circuits to properly select the value of the gain-bandwidth product of the internally compensated operational amplifiers as well as the values of the resistors and capacitors in order to avoid any potential oscillation.

FIGURE 2
FIGURE 2 Voltage divider circuit used for checking the proposed resistively variable capacitors of
FIGURE 3 (b) Simulated oscillation obtained from Fig. l(b) with: R 1 M, C 120 pF, R R