metrically by Grunwald and Berkowitz (3) and more recently by Spivey and Shedlovsky using an electrolytic conductance method (9). The two methods give good agreement over a wide range of ethanol-water mixtures. The pKa of acetic acid in 45% ethanol-water and 76% ethanol-water mixtures were determined using the differential potentiometric method. These results are shown in Table IV. The results at 45% EtOH agree with the results of both Grunwald and Berkowitz and Spivey and Shedlovsky while the results in 76% EtOH agree with the values of Spivey and Shedlovsky. The comparison is shown in Figure 2. The
agreement of the values obtained by the differential potentiometric technique and the conductiometric technique indicates that the method is applicable in ethanol-water mixtures. The acid dissociation constant of diphenylmethylacetic acid, dimethylphenylacetic acid and triphenylacetic acid were then determined in 76% ethanol-water. The diphenylmethylacetic acid and the dimethylphenylacetic acid were determined in 45% ethanol-water as well. The triphenylacetic acid was not soluble to 10~2 less than 76% ethanol-water and thus could not be measured by the differential potentiometric procedure. The results are given in Table IV.
(9) . O. Spivey and T. Shedlovsky, J. Phys. Chem. 71, 2171 (1967).
Received for review April 5, 1969.
34567891011A/in
Accepted May 29,1969.
A Study of Operational Amplifier Potentiostats
Employing Positive Feedback for
¡R
Compensation
Theoretical Analysis of Stability and Bandpass Characteristics
I.
Eric R. Brown1 and Donald E. Smith2 Department of Chemistry, Northwestern University, Evanston, III. Glenn L. Booman Idaho Nuclear Corporation, P. O. Box 1845, Idaho Falls, Idaho
The stability and bandpass characteristics have been
investigated theoretically for an operational amplifier potentiostat which utilizes positive feedback for compensation of ¡R drop in three-electrode cell configurations. Major sources of instability are identified and a reasonably efficient stabilization procedure is elucidated which greatly suppresses previously noted stability margin degradation associated with the use of the positive feedback loop. The calculations indicate that a substantial improvement over previously noted performance with 100% ¡R compensation can be realized through the use of high performance operational amplifiers and appropriate stabilization techniques.
To effect of ohmic potential loss (iR drop) compensation in electrochemical relaxation techniques, a number of workers have considered the addition of a positive feedback loop to conventional potentiostats for three-electrode cell configurations (7-77). Most of these studies have employed potentiostats constructed from operational amplifiers (7-7). Ideally, the positive feedback loop permits addition of a signal equal to the iR drop to the potentiostat input voltage. Although noteworthy success was realized in most cases, it was noted by some workers that degradation of the potentiostat stability margin accompanied the use of the positive feedback loop (3, 5, 6, 10, 11). Maintenance of a safe stability margin was achieved, at the cost of notable reduction in bandwidth, by operating with something less than the ideal of 100% iR compensation (5, 7) and/or by utilizing damping capacitors in the feedback of the control amplifier (6). The damping Present address, Research Laboratories, Eastman Kodak Co., Rochester, N. Y. 14650. 2 To whom correspondence should be addressed. 1
60201
83401
capacitor approach enabled realization of 100% iR compensation with low and moderate frequencies (6). Although the crudeness of this stabilization method was acknowledged, together with the possibility that a more sophisticated approach might avoid the attendant severe loss of bandpass (6), no attempt at improvement has been reported to our knowledge. To realize faster potentiostat response while maintaining a safe stability margin with 100% iR compensation, it appeared essential to obtain more detailed insight into the factors controlling stability and bandpass through an appropriate theoretical analysis of the potentiostat-cell response characteristics. Results of such a study are given below for a potentiostat in the current-follower mode (2) which employs positive-feedback iR compensation. The stability-bandpass calculations are modeled after the treatment of Booman and Holbrook (7, 2), which explicitly accounted for positive feedback sufficient to negate effects of iR drop in the load resistor
(1) G. L. Booman and W. B. Holbrook, Anal. Chem., 35, 1793 (1963). (2) G. L. Booman and W. B. Holbrook, ibid., 37, 795 (1965). (3) J. W. Hayes and C. N. Reilley, ibid., 37, 1322 (1965). (4) D. Pouli, J. R. Huff, and J. C. Pearson, ibid., 38, 382 (1966). (5) G. Lauer and R. A. Osteryoung, ibid., 38, 1106 (1966). (6) E. R. Brown, T. G. McCord, D. E. Smith, and D. D. DeFord, ibid., 38, 1119 (1966). (7) R. R. Schroeder, Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1967. (8) J. W. Hayes and . H. Bauer, J. Electroanal. Chem., 3, 336 (1962). (9) . E. Peover and J. S. Powell, J. Polarog. Soc., 12, 64 (1966). (10) H. Gerischer and K. E. Staubach, Z. Elektrochemie, 61, 789 (1957). (11) P. Valenta and J. Vogel, Chem. Listy, 54, 1279 (1960). VOL. 40, NO. 10, AUGUST 1968
·
14
c
Amplifier 1—Control amplifier: Analog Devices Model 210 in this work Amplifier 2—Voltage follower amplifier: Philbrick Model Q25AH Amplifier 3—Current measuring amplifier (current follower): Analog Devices Model 210 Various symbol notation given in notation definitions. For this work Rx R Rz Ry =
which is employed in potentiostats configured in the voltagefollower mode (/, 2). The present treatment employs a simple modification of the Booman-Holbrook approach which allows consideration of 100% compensation with a current follower potentiostat. Effects of bandpass and input impedance characteristics of all potentiostat amplifiers are examined. A relatively simple, but reasonably efficient, stabilization approach is made apparent which gives due consideration to a variety of destabilizing factors—e.g., double-layer capacity, reference electrode resistance, amplifier input impedances, etc. The calculations provide guidelines regarding realizable bandpass as a function of the cell transfer function, including moderate and high cell resistances characteristic of many organic solvents. Since the present work was submitted, a paper by Koopman (72) appeared which analyzed in a similar, but less detailed, manner the response of a positive feedback potentiostat (nonoperational amplifier type). ANALYSIS OF POTENTIOSTAT-CELL SYSTEM STABILITY
In this section a theoretical stability analysis is presented for an operational amplifier potentiostat employing positive feedback iR compensation. The stability behavior is characterized in terms of the well-known Bode plot of log gain vs. log frequency (7, 2, 7, 13, 14). This type of application of automatic control theory has been discussed at great length in this journal (7, 2, 15, 16) and elsewhere (7, 14, 17-19). Accordingly, a detailed description of the basic theoretical (12) R. Koopman, Ber. Bunsenges. Physik Chem., 72, 43 (1968). (13) H. W. Bode, “Network Analysis and Feedback Amplifier Design,” Van Nostrand Co., New York, 1945. (14) H. Chestnut and R. W. Mayer, “Servomechanisms and Regulating System Design,” Vol. 1, 2nd ed., Wiley, New York, 1959. (15) I. Shain, J. E. Marrar, and G. L. Booman, Anal. Chem., 37, 1768 (1965). (16) D. T. Pence and G. L. Booman, ibid., 38, 1112 (1966).
1412
·
ANALYTICAL CHEMISTRY
=
=
fundamentals appears unnecessary. The calculation approach simply will be outlined by stating the potentiostat configuration, cell analog, system equations and associated transfer functions and indicating how they are utilized to calculate open-loop gain and system accuracy. Major emphasis will be placed on results of the calculations. One item of background information worth recalling is that a region of primary interest on the Bode plot of potentiostat-cell system open-loop gain vs. frequency is the frequency where the gain becomes unity: the so-called unity gain cross-over frequency of the system. The stability of the system is related to the slope of the gain curve at the unity gain cross-over frequency, because this slope is related to the phase shift of the system. When the phase shift becomes equal to or greater than 180° at unity gain, a net positive feedback situation exists and instability is the result. A slope of —40 db per decade of frequency (or larger) in the Bode plot is indicative of this situation. Any roll-off of gain less than —40 db per decade is indicative of a nominally stable system. A —20 db roll-off means a 90° phase shift with a 90° phase margin (the difference between 180° and the actual phase shift). In general, a phase margin of 50° to 60° is necessary for stable, yet critically damped, transient response frequency also gives a (18). The unity-gain cross-over qualitative indication of the system bandpass. In general, the higher the unity-gain cross-over frequency the wider the bandpass, other factors held constant. The system bandpass is considered quantitatively in the next section. Basic Elements of Open-Loop Gain Calculation. PotentioThe potentiostat circuit whose performance is of stat. concern here is shown in Figure 1. It is almost formally identical to a circuit already described so that the previous (17) E. R. Brown, Ph.D. Thesis, Northwestern University, Evans-
ton,
111.,
1967.
(18) W. C. Carter, Instruments and Control Systems, 40, 107 (Jan. 1967). (19) A. Bewick, A. Bewick, M. Fleischman, and M. Liler, Electrochim. Acta, 1, 83 (1959),
explanation of its operation (6) is applicable. The only fundamental innovation is capacitor Cj which represents a new stabilization approach as shown below. Cell Analog. Figure 1 also indicates the chemical cell impedance analog employed in this work in the form of components Ri, R ', R¿, and C2. The impedances represented by these components are identified in the notation definitions. One should note the explicit consideration of the reference electrode resistance. Although this resistance is often quite large due to the use of fine frits, asbestos plugs, and the like, it has been considered explicitly in potentiostat response calculations on only a few occasions (7,19). The ohmic resistance to be compensated by the positive feedback loop is represented in Figure 1 as a single resistor, R2. One should not take this as meaning that consideration is being given only to the solution ohmic resistance between the working electrode and the tip of the reference electrode probe. In actual practice a second source of ohmic resistance to be compensated often exists in the form of an internal resistance within the working electrode—e.g., the resistance of a dropping mercury electrode capillary—which is physically located on the electrode side of the double-layer capacity. Such a resistance between C2 and the current amplifier (amplifier 3) input is not shown explicitly in Figure 1 which was designed primarily to represent a reasonably precise cell impedance analog without necessarily representing the physical placement of the impedance components. Failure to represent explicitly the internal electrode resistance does not influence the accuracy of the impedance analog because, as is well known, the impedance associated with a capacitor interposed between two resistors (RCR circuit) is electrically equivalent to a capacitor in series with a single resistance of magnitude equal to the sum of the resistances in the RCR circuit. Thus, resistance R2 is taken as the sum of ohmic resistance components to be compensated. The cell analog of Figure 1 represents the interfacial impedance as composed of the double-layer capacity only, thus neglecting contributions of the faradaic impedance. The influence of the faradaic impedance on potentiostat response characteristics has been discussed at some length (2). With a conventional three-electrode potentiostat one effect of the faradaic impedance is to increase the iR drop (increase i) and thus degrade the response (2). This effect should be of no consequence with the potentiostat under consideration provided 100% iR compensation is employed. A second effect of the faradaic impedance is to reduce the phase shift associated with the interfacial impedance transfer function relative to the 90° value characteristic of a pure capacitive impedance. This phase shift reduction normally should have a stabilizing influence on the potentiostat-cell system (perhaps not when the faradaic current has a large absorption component where phase shift reduction may be small). We have observed empirically such an effect with both dummy cell simulation and actual electrolytic cell observations (20). For example, in ac polarographic measurements the observed potentiostat stability was almost invariably greater at dc potentials in the vicinity of the faradaic wave than at potentials where only double-layer charging current exists. Thus, because they neglect the faradaic impedance, the calculations to follow may be taken as considering the worst case as far as stability is concerned, with the possible exception of some cases involving negative impedance electrode effects (27, 22).
Transfer Functions and Potentiostat-Cell System Open-loop Gain. To obtain the open-loop transfer function of the three-amplifier potentiostat-cell system shown in Figure 1, it is necessary to know the transfer functions of the various (20) D. E. Smith, E. R. Brown, T. G. McCord, H. L. Hung, and J. R. Delmastro, unpublished work, Northwestern University, Evanston, 111., 1965-67. (21) R. Tamamushi, J. Electroanal. Chem., 11, 65 (1966). (22) R. Tamamushi and K. Matsuda, ibid., 12, 436 (1966).
amplifiers and the chemical cell. It has been shown that the product of the combined transfer functions is a good approximation to the open-loop gain of the potentiostat system (16). From examining the Bode plot of this function, critical parameters in the system can be identified and modified to optimize stability and response. The relevant transfer functions and open-loop gain expressions are readily formulated from the set of eight independent system equations governing the response of the network in Figure 1. These equations, which are listed in the appendix, allow calculation of the eight unknown network voltages (see Figure 1) from which the transfer functions and open-loop gain are obtained. Because the system equations are linear they are most efficiently solved by determinants. Rather than write the resulting cumbersome transfer function solutions in conventional algebraic form, they normally will be expressed in a more compact form using the solution determinants. Although of eighth-order, these determinants are easily expanded as the majority of elements are zero.
The transfer function F//e0 relating the signal at the voltage follower output to that at the control amplifier output (the negative feedback signal) may be written
F/
_ ~
Dej
_
(1)
De0
where DEf and Deo are eighth-order determinants defined by Equations A21 and A22 of the appendix. The transfer function F0/e„ seen at the current measuring amplifier output is
E°
Deo
B*
=
—
(2)
=
dZ
ea
and the transfer function Eije0 seen at the center tap of the iR compensation potentiometer Rc (the positive feedback signal) is
^
=
ß
=
e0
^
(3)
D,0
DEo and De¡ are determinants given by Equations A23 and A24 of the appendix. Finally, the transfer function relating the control amplifier output and the inputs to its resistive summing network, e0/(eiu + F, + Ef), may be written
_e_o_ (e¡„
Q
=
+ Ei + Ef)
=
_gl_ [(1
-
G1)RCp
+
(4)
3]
Equation 4 is obtained simply by combining Equations A1 and A5 of the appendix. To express the open-loop gain Gv of the potentiostat-cell system of Figure 1, we consider the loops as open at the inputs to the control amplifier summing network resistors. For such an open-loop configuration, the open-loop gain is defined
as
