On the Instability of Current Followers in Potentiostat Circuits J. E. Davis, Departments of Pathology and Medicine, Washington University; Division of Laboratory Medicine, Barnes Hospital, St. Louis, Mo. 63110
E. Clifford Toren, Jr.’ Departments of Medicine and Pathology, University of Wisconsin. Madison. Wis. 53706
An equivalent circuit for the input impedance of the current follower (current-to-voltage converter) is developed which is characterized by an inductive element. Potentiostat stability requires the smallest possible value of feedback resistor consistent with adequate sensitivity and noise. Small signal and transient effects for several circuit forms are compared.
DeFord ( 1 ) presented three fundamental circuits for measuring current flow in an electrochemical cell. One of those was the current follower configuration, current-tovoltage converter, which had the advantages of 1) working electrode at virtual ground, 2) single input operational amplifier (Chopper Stabilized), and 3) voltage drop across measuring resistor not. detracting from cell controller drive voltage. Its obvious disadvantage was the requirement that it be able to supply the full cell current. Booman and Holbrook ( 2 ) presented a computer analysis of the current follower configuration and showed that some measuring resistor values lead to instability. The instability is a disadvantage whose mechanism is not so obvious. An analysis of the causes for this instability is the subject of this discussion. Several means of relieving the instability are noted.
transformed variables be written upper case, the input voltage and current expressions for the current follower can be obtained by inspection of Figure la:
E,
A u t h o r t o whom r e p r i n t requests s h o u l d b e addressed.
(1) D . D . DeFord, presented at the 133rd Natlonal Meeting, American Chemical Society, San Francisco, Calif.. 1958. (2) G. L. Booman and W. B. Holbrook, Anal. Chem., 37, 795 (1965). (3) M . E. Van Valkenburg, “Network Analysis,” 2nd ed., Prentice Hall, Englewood Cliffs, N.J., 1964.
-GE,
where E, is the voltage a t the working electrode and
I,,
=
E, - E , Rm
-
E,
+ GE,
(2)
Rm
The input impedance, 2, is derived from Equations 1 and 2:
The result is a low impedance whenever G is sufficiently large but the impedance will increase with increasing R, and increasing frequency, s. Analogous to that for a voltage follower, the transfer function, Equation 4, shows the output voltage is directly proportional to the input current until the gain, G, approaches l, near the unity gain crossover frequency.
THEORY A typical application of the current follower, CF, to a potentiostat is shown in Figure la. A simple analysis of the circuit assumes that no current flows into the operational amplifier, F, and that the input is a virtual ground. Therefore, the output will be such that e, = iwRm.However, in fact, the input is not virtually at ground potential and deviates increasingly as the frequency increases. Qualitatively, this latter characteristic resembles that of an inductor-i. e., the voltage across an inductor increases with frequency for a constant level of current. So that a quantitative analysis can be made, and the CF will be replaced by an equivalent circuit which contains an inductive element implied in the following. The input impedance of the current follower is calculated by means of Laplace Transforms ( 3 ) and involves the following assumptions: 1) the system is linear, 2) the output impedance of the operational amplifier is zero, 3) the input impedance of the operational amplifier is infinite, and 4) the transfer function of the operational amplifier is adequately described as G = K (1 + m)-l-i.e., a simple rolloff in which K is the dc gain and 117 is the break point frequency, f,. Employing the convention that Laplace
=
E _, -- - G E , I, E, GE, R,
+
- Rrn
=-
1
+ 1/G
(4)
A reviewer suggests that the previous sentence may be misleading because the proportionality is in error by 1% when the frequency is two decades below the unity gain frequency and by 10% at one decade and that these frequencies are not particularly “near” the unity gain frequency. The equivalent circuit, Figure l b , was chosen somewhat arbitrarily. The operational amplifier output is symbolized by the voltage generator, V. The amplifier gain, K , and break point frequency, f,= T - ~ ,are seen to directly affect the equivalent circuit. That circuit represents an impedance which can be calculated using the Laplace transformations:
Zeq =
- R,(1
1
1
-+ R,
1
R,IK
+ STR,/K
+ TS)
1 + ~ s + K
(5)
This was written and simplified by inspection of the series-parallel elements of Figure l b . The equivalance of right-hand-sides of Equations 3 and 5 justifies the choice of the equivalent circuit.
DISCUSSION The above circuit will be discussed first, particularly as it affects a typical potentiostat. Final discussion will conANALYTICAL CHEMISTRY, VOL. 46, NO. 6, MAY 1974 * 647
fen R, Figure 2a. Figure l a .
Simple current follower type potentiostat
Equivalent circuit for analysis of stability
3
,ogGAlN
t
Equivalent circuit for current follower and electrochemical cell
Figure l b .
sider application of damping capacitors ( 4 ) and operational amplifier configurations with a low input impedance. Referring to the equivalent circuit Figure l b , it can be seen that the potentiostat is actually controlling the voltage across an RLC circuit. It is not surprising, therefore, that the configuration might tend toward LC oscillation. T h e degree of instability can be cakulated only with detailed knowledge of the specific cell and potentiostat operational amplifier characteristics. Some years ago, Booman and Wolbrook (2) provided a detailed numerical analysis of the current follower-potentiostat configuration for a part,icular type of operational amplifier and stabilization network. Those results provide a quantitative example for the complete circuit to be compared against the following of the current follower alone. Even though the in. ! i s i d m i current follower and potentiostat were stable, the ic1n:bination might not be stable; in which case, it is not psssible io identify one or the other as the cause. By unili.iri1 a i d i n g t h e nature of the interaction, modern servo tiitor,v c:m provide means by which the interaction can be coritrolled or, indeed, may suggest that other circuits be considerpd where control of the interaction is more easily accomplished. For the purpose of stability analysis, consider a signal voltage injected between ground and the CF equivalent circuit of Figure l b . The resulting circuit with the neglect cf R, (but not R,/K) is shown in Figure 2a. Our rationale is t o obtain a circuit which is easy to analyze by Bode plots even though it looks quite different from the original circuit. Nevertheless, if the circuit is unstable, the original circuit will be also. In the usual Bode plot formalism, the magnitude of the ratio of feedback to input impedance is compared to the magnitude of the amplifier gain. The ICLC components of the input dominate the shape of the ratio curve. The undamped natural frequency, on, and the damping, c, are:
As a n example, let R , = lo4, f o = 1 rad/sec, K = 106, and C = 1 pF. The resonant frequency is lo4 rad/sec and the damping is approximately R,/200. Critical damping would result if R, were 200 ohms, which is quite large for a n uncompenhated resistance. If R , were 20 ohms, then the damping LI, ulci he only 0.1, and the response a t reso( 4 ) Richard Bezrnan and P S McKinney, Anal Chem., 41, 1560 (1969).
A N A L Y T I C A L CHEMISTQY. VOL. 46,
branch
IogXEO
c
648
z1.0
, L C F tr rr
NO. 6 . MAY 1974
Figure 2b. Bode plot for stability analysis
nance would be ten times larger than for critical damping. In general, a resonance with damping less than 0.3 is unusable. At the resonant frequency, l o 4 rad/sec, the impedance of the double layer capacitor is 100 ohms and the impedance of the inductance is also 100 ohms. The CF series resistance is only 0.01 ohm, a value so small that external series resistance must be supplied. From Figure 2b, it can be inferred that decreases in damping will cause the ratio curve to intersect the gain curve a t a n excessive rate of closure, or to otherwise “ring”. Even if the damping were satisfactory, a n increase in the bulk solution resistance, Rs, can cause intersection at a n excessive rate of closure. Finally, the natural frequency, an, can shift to intersect a t an excessive rate of closure. An excessive rate of closure may not result in frank instability; however, a large rate of closure, greater than 20 db/decade, is generally associated with ringing or slowly damped oscillations which may be objectionable. Manipulation of other circuit parameters affects the damping in the following way. Increasing the unity gain frequency of CF ( i . e , increasing the product f,K) increases the stability by increasing the damping. The resonant frequency also increases. Increasing the double layer capacitance increases the damping but decreases the resonance frequency. Increasing the measuring resistor, R,, decreases the stability and lowers the resonance frequency. The foregoing effects go as the square root of the primary change. In contrast, the damping is directly proportional to the uncompensated resistance, R,. Any attempts to decrease R, will lessen the stability. This circuit configuration requires a definite uncompensated resistance to function without oscillation or excessive ringing. Figure 2b also shows the relative need for a high gain and wide bandwidth amplifier, OA2. Not a great deal can be done to tailor its transfer function in a helpful way. But this is meant, Equations 6 and 7 show the bandwidth, f,K, to be the relevant parameter and that is difficult to increase. Instead of the simple transfer function of Figure 2b, an even more complicated roll-off must still be associated with an inductive input impedance whenever the gain is decreasing with frequency. What artifacts can the CF configuration introduce? For a fixed amplitude sine wave, the current will be abnormally large as the resonant frequency is approached. This assumes the damping is less than one--i.e., a small uncompensated resistance. For a step input voltage, the cell current will overshoot and ring. In both cases, the actual voltage across the double layer will be substantially different from the input voltage. The recovery of the current follower from large signal transients will, in general, take longer than predicted from the small-signal analyses. Two conditions occur. First, the
1I
Figure 3.
1
Equivalent circuit for current follower with damping
capacitor, Cf output voltage may limit, in which case insufficient current is returned to the summing junction, and the CF input voltage increases. Thus, the double layer-capacitance takes longer to charge because of the current limitation. There can be brief bursts of oscillation as the CF recovers because the transfer function can be grossly altered near overload conditions. A second condition arises when the output voltage of the CF is “slew limited”--i.e., it is changing as fast as it can and further increases in input voltage will not cause it to change any faster. Until the output voltage increases sufficiently to provide an appropriate feedback current, the input voltage will be large. The first condition is handled by a “bounding” circuit while the second is handled by “clamping” the input. Both approaches may be necessary to provide adequate transient recovery and they are well described in the operational amplifier literature. Bezman ( 4 ) presented an analysis of the current follower wherein the measuring resistor is shunted with a capacitor for control of the damping. T h a t the circuit can be critically damped for a true current source input has little bearing on the overall damping of the potentiostat-current follower combination. An equivalent circuit for the input impedance is shown in Figure 3. A current flowing into the impedance will generate a voltage which can strongly interact with the potentiostat. The stability of the overall circuit can be analyzed in a manner analogous to that for Figure l b . However, the analysis is not as simple and is greatly affected by the relative values of Cdl to Cf. For very small values of Cf, performance is similar to that already described. As Cf is increased, the ratio curve will break and take the branch labeled CF trim in Figure 2b. That branch can cause high frequency oscillation if it intercepts the amplifier transfer function at an excessive rate of closure. For small Cf, the intercept will be influenced by the value of R, and R,. The CF trim might be used to cancel some of the resonance effects at w,; however, in general for small R,, further increases in Cf would not promote stability, contrary to normal experience with other circuits. The stability that Bezman ( 4 ) (Table I of reference 4 ) achieved was probably due to substantial uncompensated resistances coupled with only modest bulk solution resistances. The transient response for this circuit is substantially the same as the conventional CF. Consider a low input resistance for the operational amplifier, F, in Figure la. Such a resistance, R,,, would be represented as a resistor shunting the input to ground and the current flowing through that resistor to ground as &,. Use of Kirchoff‘s law and Equation 1 results in Equation 8.
Zi”
L
V -
i
=
Rf
1
+ G + R,/R,,
(9)
These results show that the input .impedance, Zin, can be effectively limited where G is small-ie., a t high frequencies. However, the noise may increase if Rill is smaller than the cell impedance, which result can be concluded from Equation 11 ( 5 ) . e, = (1 1
+ $)e,, +~ nm/
~
i
,
(11)
The input noise voltage and current of the operational amplifier are e, and in, respectively, and e, is the total output noise. At high frequencies, the cell impedance will depend largely on the uncompensated resistance, R,, and the potentiostat. The value of the uncompensated resistance necessary to damp the equivalent LC circuit adequately will be reduced, even to zero, by the lowering of the amplifier input resistance. Indeed, in the earlier example, a 50-ohm resistor, R,,, would give critical damping and obviate the need for any uncompensated resistance for the stabilization of the current follower. Examination of Equation 10 shows a loss in signal bandwidth nearly in proportion to the reduction in Z,. The loss is moot, since the cut-off frequency is still above the undamped natural frequency of the equivalent LC for the cell-current follower combination. Nevertheless, the bandwidth could be restored by increasing the gain, K . The limit to the increase would occur when submerged poles approached the open-loop portion of the Bode plot. It is unlikely that the bandwidth of the conventional circuit could be exceeded because the operational amplifier manufacturer limits the bandwidth in order to submerge secondary poles so that the amplifier will be “unconditionally stable”-i.e., approximate a simple transfer function G = K (1 T S ) - ~ . An increase in the gain would lower the input impedance as can be seen from Equation 9. Also, the double layer capacitance would be charged faster-i.e., better transient response, as a result of the increased gain. Note, however, the gain would have to be changed each time the value of the feedback resistor was changed. With appropriate changes in K and shunting R, with R,,, Figure l b remains a viable equivalent circuit. An alternative method for achieving the same changes in the equivalent circuit would be reduction of the value of R, with the CF followed by another amplifier to make up for the loss in sensitivity. This choice may increase noise more than the low input impedance CF. While the lower input impedance can improve transient response, it is still necessary to provide clamping and bounding if the current excursions are expected to be larger than the CF can handle. Notice that the slew-limiting problem is minimized by restricting the output voltage excursions-Le., smaller feedback resistance, less sensitivity. D. K. Roe (6) has presented a current follower with a low input impedance. The reduction was accomplished by unique circuitry which also allows simple control of the gain.
+
CONCLUSIONS An equivalent circuit for the input impedance of the The input impedance derived from Equation 8 and the resulting transfer function are shown in Equations 9 and 10.
( 5 ) Lewis Smith and D. H . Sheingold, Analog Dialog. 3, No. 1 (1969), Analog Devices, Inc., Cambridge, Mass. (6) D . K Roe, Chem. Instrum.. 4 , 15 (1972).
ANALYTICAL C H E M I S T R Y , VOL. 46, NO. 6, MAY 1974
649
current follower provides insight to the mechanism of interaction between the CF and the potentiostat. The essential characteristic of that impedance for all forms of the current follower is an inductive element which may complicate the stabilization of the potentiostat. The smallest possible value of feedback resistor consistent with adequate sensitivity and noise, should be utilized.
ACKNOWLEDGMENT The helpful criticisms of J. E. Harrar and C. L. Pomer-
nacki were gratefully received. Received for review June 27, 1973. Accepted December 12, 1973. Work was initiated at the time one of us, J. E. Davis, was a Fellow of the Petroleum Research Fund, administered by the American Chemical Society, grant number 2907-A3. This .work was continued in part by grants from the National Science Foundation, GP9608 and GP26505, and the National Institutes of Health, GM10978.
Present Status of the N-Silicon/Stainless Steel Combination Electrode for Acid Determinations J. P. McKaveney Garrett Research and Development Company, Inc., Division of Occidental Petroleum Corporat/on. La Verne. Calif. 9 1750
M. D. Buck Hach Chemicai Company, P. 0. Box 907. 713 South Duff, Ames. lowa 50010
The N-Silicon/Stainless Steel combination electrode has been examined more critically as both a general acid monitor and selective hydrofluoric acid (HF) analyzer. Metallic ions in their lowest stable oxidation state in an aqueous medium (AI3+, Fe2', Ni2+, Mn*+, Pb2+, and Zn") appear to produce little or no interference in acid measurement. However, metallic ions in reducible oxidation states (Cr3+, Cu2+, Fe3+, and U0z2+) produce readings about 5% high in acid value. The effect is not appreciably concentration-dependent and can be screened out by addition of the reducible ion to the NH4F calibration electrolyte. Applications are given for the measurement of both free and hydrolysis acid (Sb3"rapid hydrolysis) as well as for free acid in the presence of cations ( C d - and Zn") not hydrolyzing at low pH values. The electrode is also applied to the measurement of free HF in phosphoric acid (fertilizer industry) as well as to the type of fluoride complexes (AI3+, Fe31, Si4t) present in strong phosphoric acid produced from phosphate rock.
The combination electrode of a low resistivity N-Silicon anode and stainless steel cathode was introduced in 1970 ( I ) as both a selective monitor for hydrofluoric acid (HF) and a total acid monitor in the presence of a fluoride electrolyte. Since that time, applications have been made for the determination of H F and "03 in titanium sheet and H F in both pickling baths ( Z ) , HCl, &Sod, "03, carbon steel and stainless steel acid treating baths (3), and quite recently to nuclear fuel reprocessing solutions containing principally "03, HF, and fluoboric acids ( 4 ) . The electrode has also been shown to be applicable to the measurement of H F in either HzS04 or H3P04 mixtures (1) J . P. McKaveney and C. J . Byrnes, Anal. Chem.. 42, 1023 (1970). (2) T. S. Yang, Grumman Aerospace Corp., Bethpage, N.Y.. private communication, 1971 ( 3 ) W . A. Straub, W. J. Nolan, and R. G. Theys. Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Cleveland, Ohio. March 9, 1972. paper No. 230. ( 4 ) D. R . Kendall, S. D . Reeder. and S. S. Yamamura, Talanfa, 21, in press, 1974.
650
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, N O . 6, M A Y 1974
( 5 ) .An additional application has been made to the measurement of organic acids including acetic, citric, oxalic, and tartaric (6). The combination electrode appears capable of measuring acids which have K,,, values greater than 1 x 10-6 or slightly lower than acetic acid (K,,, = 1.76 X a t 25.0 "C). The electrode is selective for H F and, when measuring other acids, depends on the use of an excess of NH4F electrolyte for three reasons. First, to produce a weakly ionized molecular species HF by forcing the equilibrium in that direction, e . g . , with hydrochloric acid: XH,+
+ F- + H+ + C1-
+ HF
+ NH,' + C1-
(1)
As such, the electrode responds only to H F or fluoride ions in acid solution and is not influenced by changing ionic strength of salts such as NaCl up to 3.0 molar. Second, as the electrode responds only to H F or free fluoride ion in acid solution, excess fluoride is required to produce current saturation. In the acid ranges covered by the meter, a current plateau is reached above 1.0 gram of NH4F per 100 ml measured. Earlier data ( I ) indicated a current amplification of about four for H F in the presence of excess fluoride. This suggested the anodic etching mechanism for the silicon may involve the HFz- anion (7), viz:
Si
+ 2HF,SiF, + 2H' + 4eSiF, + 2F-(excess) SiFz--+
---t
(2)
(3)
In static solutions, small gas bubbles are seen to form over a period of time on the stainless steel cathode. While very little free hydrogen ion is available with the large excess of fluoride ion, it is suggested that hydrogen is produced through reaction of the ammonium ion a t the cathode:
NH,'
+ le-
-
NH,
+ i/?H,t
(4)
The actual electrochemical procedure appears to be a combination of constant potential electrolysis and am( 5 ) J. P. McKaveney. Ana/. Lett.. 4, 407 (1971) (6) J. P. McKaveney, A n a / . Chim. Acta. 62, 37 (1972). ( 7 ) J. S. Judge.J. Electrochem. SOC..116, 1772 (1971).
