Current Follower Stabilization in Potentiostats K. Keiji Kanazawa* and R. K. Galwey ISM Research Laboratory, San Jose, Calif. 95 193
Instabilities in analog potentiostat instrumentation have received considerable attention in recent years. A discussion of many of the sources of instability is given by Harrar (1).As he has discussed, stabilizing the main control amplifier according to Nyquist-Bode criteria, e.g., by phase lead compensation, may not in itself be sufficient to stabilize the potentiostat inasmuch as other sources of instability may be present. A particularly troublesome source of instability occurring in potentiostats using a current follower in the working electrode circuit was identified and studied by Davis and Toren (2). Figure 1 shows a simple potentiostat with the current follower portion indicated in dark lines. Those authors showed that the feedback resistor, R, is reflected in the input impedance as a series resistance-inductance arm. When uncompensated, the inductance can resonate with the cell's double layer capacitance to generate destabilizing phase shifb. We present analytical and experimental results which show that this inductive element can be effectively eliminated by shunting the feedback resistor with an appropriately chosen capacitance.
ANALYTICAL In the circuit of Figure 1, the open loop dc gain of the operational amplifier A1 is denoted by K , and its characteristic response time is denoted by 7 . The input admittance of the current follower can be expressed as K 1 jwRC Yi, = - + j w c + R R 1+jm The simplest equivalent representation of this admittance using frequency-independent, passive elements is shown in Figure 2. It is identical to that presented by Davis and Toren and is included here simply for reference. The inductive component of the equivalent circuit is contained in the complex part of the last term on the right hand side of Equation 1. If the condition
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represented by an input conductance ( K 1)IR shunted by a capacitance C. Bezman and McKinney (3) studied the transfer function eo&,, for the current follower and showed that when condition 2 is satisfied, the follower is overdamped. In particular, their results indicate that for critical damping the appropriate choice for the time constant, (RC)crit,is
T/K. It is not immediately obvious from the equivalent circuit that the two rightmost arms of the network reduce to a resistance of value RIK when RC = 7. It is also not clear how one can operationally determine when condition 2 has been satisfied. Both to demonstrate the inductance cancellation and to provide a criterion for assessing when condition 2 has been satisfied, we made a quantitative study of the input admittance. The phase angle @ between the voltage across the input and the current provided a useful parameter to study since it showed the reactive behavior and was relatively easy to measure. From Equation 1, it is straightforward to show that the phase angle is given by KT - RC(K 1 uW)] tan @ = (3) K 1 W ~ T ( T KRC) Two different behaviors of phase angle with frequency are exhibited, depending upon whether RC is greater or less than T. The condition delimiting these two behaviors is RC = T and is identical with condition 2. When RC < T , the behavior of the phase shift is complicated, being small and positive at low frequencies, increasing to a maximum with increasing frequency, eventually crossing zero and going negative. In terms of the equivalent circuit, one can say that when RC < T , the
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RC=T (2) is met, then the last term reduces to a frequency-independent, real conductance KIR. The input admittance can then be Figure 2. Equivalent circuit representation of the input admittance of the current follower having a feedback resistance R, shunted by a capacltance C
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Simplified block diagram for potentiostat using a current follower for current measurement. The follower portion is indicated by dark lines Flgure 1.
Figure 3. The phase-frequency
relation for the current follower input when no intentional capacltance shunts the feedback resistance Dotted points are experimental,and the solid line represents Equatlon 3
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Figure 5. The phase-frequency
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relation for the current follower input
when RC just exceeds 7 Dotted points are experimental, and the solid line represents Equation 3 0
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Capacitance (picofarads) Figure 4.
The phase shift at 250 Hertz as a function of compensating
shunt capacitance Note the inductive phase shift with small capacitance values, changing to a capacitive shifl for large values
reactive current is dominated by the inductive arm at low frequencies and by the capacitive arm at high frequencies. When RC > 7 , the phase behavior is simpler, being negative for all frequencies. That is, the effective input reactance is capacitive for all frequencies. EXPERIMENTAL For these measurements, the current follower was driven from a function generator (Hewlett-Packard 3310A) through a 100-kQresistance. The generator voltage was then in phase with the current and was used as the reference signal for a gain-phase meter (Hewlett-Packard 3575A). The voltage at the negative input terminal of a operational amplifier was coupled through a non-inverting buffer amplifier to the input of the gain-phase meter. To illustrate the phase behavior and the influence of the compensating capacitor, the operational amplifier, a Date1 405-2, was deliberately degraded to have a measured open loop dc gain of 525 and a characteristic response time s. This was accomplished by shunting its bandwidth of 1.05 X control pin (pin 8) to ground through a parallel combination of a 1.25 MQ resistance and a 20-pf capacitance. A feedback resistance of 159 kQ was used.
RESULTS AND DISCUSSION With no intentional feedback capacitance, the phase shift was found to vary with frequency as shown in Figure 3. The circled points indicate the measured values. The solid line shows the phase relation calculated using Equation 3, assuming a stray feedback capacitance of 15 pf. This excellent agreement gives confidence not only in the measuring technique, but also in the assumption that Equation 1includes the major effects on the input admittance. Over the whole frequency range studied, the phase shift reflects an inductive reactance in the input. To examine the influence of a capacitor C shunting the feedback resistor, the frequency was held fixed at 250 Hertz and the phase shift was studied as a function of compensating capacitance values. In this low frequency regime where w 2 ~ 2
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can be neglected with respect to K , the phase shift will vanish for RC = rK/(K 1).This condition is virtually identical with condition 2 and can be used to determine the proper capacitance value. These results are shown in Figure 4.They clearly indicate a value for the compensating capacitance of 650 pf, which is in excellent agreement with the value of 660 pf calculated from r/R. The phase-frequency behavior of the current was again studied over the same frequency interval as in Figure 3, this time using 675 pf as a compensating capacitance. The resulting phase shift is shown by the circled points in Figure 5. The solid curve indicates the analytical results. Note the twenty-fold increase in the ordinate scaling. We believe the discrepancies between the experiment and analysis arise both from instrumental phase inaccuracies and from the sensitivity of the phase to small capacitive changes. Two potentiostats which had been previously constructed in our laboratory without regard to this compensation exhibited cell-dependent instabilities. When the current followers in these potentiostats were tailored to obey condition 2, the stability of those potentiostats was enhanced. Of course, the stabilization was gained at the sacrifice of bandwidth. The follower bandwidth without the compensating capacitor is given by the characteristic time r/K. When fully compensated, the bandwidth is reduced to a value characterized by 7. This bandwidth limitation will be of great importance in choosing the optimum operational amplifier for current follower application, particularly in high speed potentiostats. In conclusion, we have shown that the inductive reactance of the current follower input can be eliminated by choosing RC = 7. We have further shown that this criterion can be experimentally studied by examining the voltage-current phase shift at frequencies well below 1 / ( 2 m ) .
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LITERATURE CITED J. E. Harrar, “ElectroanalyticalChemistry”, Vol. 8, A. J. Bard, Ed., Marcel Dekker, Inc., N.Y., 1975.(2) J. E. Davis and E. C. Toren. Jr., Anal. Chem., 46, 647 (1974). (3) R. Bezman and P. S. McKinney, Anal. Chem., 41, 1560 (1969). (1)
RECEIVEDfor review November 9,1976. Accepted January 17, 1977.
