drop. The interest wm in determination of micromolar concentrations of ions, in the presence oj other reducible ions in large excess. The instrument was switched to the cor trolled-potential mode, and a 10,000-ohm precision resistor was used for current measurement. Amplifier #6 wa8sset a t a gain of 100. One microamperc of current thuh gave a pignal a t the (Jutput of amplifier 16 of I volt. The timing circuit and drop-removal hammer were used as described above. Cadmium in 10-6Jf concentration was determined in the presence of millimolar Cu+2 and Pb+2. The reproducibility was rt39; for the micromolar concentrations and =kl% for concentrations of up. A complete analysis of the data and the methods used will be communicated a t a later date. The instrument is thus capable of being used for a variety of electrochemical techniques. It is not an inexpensive system a:; the choice of components was made with performance as the primary criterion. It is the feeling of the authors that slavish
ACKNOWLEDGMENT
We thank J. H. Christie for use of his data for the ramp chronopotentiometry. LITERATURE CITED
(1) Alden, J. P., Chamber, J. S., Adams, R.
S., J . Electroanul. Chem. 5,152 (1963). (2) Booman, G. L., ANAL. CHEM.29, 213 (1957). (3) Ibid., 31, 10 (1959). (4) Christie, J. H., North A~nericaiiAviation Science Center, Canoga Park, Calif., unpublished work, 1963. ( 5 ) &Ford, D. D., 133rd Meeting, ACS, dan Francisco, 1958.
4001
I
~
1
1
1
S E C O N D S
(6) General Electric Co., Semiconductor 1 Products Department, Schenectady
Figure 6. Chronopotentiograms for NaNOz in 18M HzS04 Top curve is a t constant current, bottom is using trapezoidal current
duplication of this instrument would be unwise, in part because new and better components are constantly being developed. For example, specification sheets for the new Philbrick SP656 operational amplifier indicate that it should give much better performance and lower noise figure than the UP-1-2.
E.Y., Specificatim Sheet No. 60 11 (February 1963).
(7) Hoffman, H., Jaenicke, W., Z. Electrochem. 6 6 , 7 (1962). (8) Kelley, M. T., Fisher, D. J., Jones, H. C., ANAL. CHEM.32, 1262 (1960). (9) Osteryoung, R. A., Lauer, G., Anson, F. C., Zbid., 34, 1833 (1962). (10) Osteryoung, R. A,, Lauer, G., Anson, F. C., J . Electrochem. Soc. 110. 926
(1963j.
-4. Philbrick Research Inc., Boston, Mass., Specification Sheet
(11) G.
No. P2/SL/SM/1162.
RECEIVEDfor review June 14, 1963. Accepted August 9, 1963. Division of Analytical Chemistry, 144th Meeting, ACS, Los Angeles, Calif., April 1963.
EIect ro a nci Iytica I Cont roI led-PotentiaI lnstrumenta tion GLENN L. BOOMAN cind WAYNE B. HOLBROOK Phillips Petroleum Co., Atomic Energy Division, /dah0 Falls, ldaho
b Application of elementary concepts of feedback theory, electrochemical cell models, and specific measurement techniques is shown to place the design of electroanalytical controlled-potential instrumentation or a nonempirical yet extremely pract cal basis. The Bode attenuation diagram approach is applied to controller and cell transfer function optimization, with the resulting circuitry givei for fast potentiostatic studies, diverse types of three electrode cell polarography, and controlled-potential coulometry.
I
for controlled-potential coulometry, polarography, and various other electrochemical techniques has been characterized by a highly empirical approach. A rather large number of instrument designs have been published with practically no indication as to accuracy of potential control, degree of stability, or response speed for following or correcting reference input, cell, or load resistor voltage changes. Thrh works of Will (ZS), Gerischer and Sttubach (I?‘), and Bewick et al. (4) hove been largely ignnred during the recent surge of NSTRUMENTATION
interest in the application of readily available operational amplifiers to many types of measurements with threeelectrode cells. The fact that the design of these control systems and the interpretation of the experimental results depends on knowing the transfer function of the electrochemical cell is unescapable. However, these transfer function data for the cells to be used with particular controllers have not previously been published. In some cases, limitations on cell resistance have been noted, again on an empirical basis, with the criterion being whether it works and not how well. The necessary criteria for general control-system performance and stability have been discussed in numerous books (9, 18,2,9). Experimental measurement methods suitable for electrochemical systems have not previously been presented. Adequately specifying mercury-pool cell parameters for high speed controlledpotential coulometry has been a major difficulty in adaptation of this technique. These remarks also apply t o the many three-electrode polarographic techniques. Automatic feedhark control systems
are the “sine qua non” of recent controlled-potential methodology. To be useful, the control system must have sufficiently fast response to the desired reference value, small error, and stability. The chemist must have an understanding of control-systems operation because the most important design parameter of the system, the electrochemical cell, has tremendous leverage on the ability of the apparatus to meet the above criteria. Fortunately, the application of a few elementary concepts of feedback control theory gives an understanding of the kind of potential control that can be attained, and the equipment requirements, including its limitations, for a particular measurement problem. A need has arisen for a general approach to the feedback-control problem associated with electroanalytical controlled - potential instrumentation. Methods for solving the design problem have previously appeared in only fragmentary form. Part I considers the problem of feedback control from the electroanalytical chemist’s viewpoint, emphasizing the graphical transfer function approach and giving methods for VOL. 35, NO. 12, NOVEMBER 1963
1793
measuring cell, amplifier, and complete system transfer functions. To show the various phases of the solution for a particular measurement problem, the general steps in solving a typical polarographic potentiostat problem are outlined in Appendix I, and the steps for a coulometer potentiostat problem are outlined in Appendix 11, including experimental transfer function measurements on a stirred mercurypool coulometer cell. In electroanalytical studies, an arbitrary division can be made between the use of feedback control systems with polarographic microelectrodes primarily used as a concentration measurement probe and those with large electrodes used primarily for exhaustive electrolysis. The instrument for the polarographic application is commonly called a potentiostat and that for the latter application is called a controlled-potential coulometer. Both of these applications use potentiostats, the coulometer applicat,ion differing in always requiring a current integrator and a high current-output amplifier to drive the larger electrode systems. Cell design considerations for both the microelectrode in an unstirred electrolyte and for the larger electrodes in a stirred electrolyte are discussed in Part 11. Use of electrical cell models and hydrodynamic and geometric factors are covered. A precision stirrer assembly is shown. Considerations and circuitry basic to both types of potentiostats are discussed in Part 111. Particular examples of polarographic potentiostats are also given in Part 111. Examples of controlled-potential coulometers are discussed in Part IV along with special considerations for the integrator circuitry. The approach presented is of general applicability to electroanalytical controlled-potential systems and is presented as a technique to be applied to optimize the electrochemical measurement for the conditions of electrolyte resistance, cell geometry, and signal response which the individual problem requires. A word of encouragement is perhaps in order for those workers showing an interest in potentiostatic methods for measurements of fast reactions. There have been a number of recent papers and review articles relegating potentiostatic methods to the basement on the basis of instrumental complexity as compared to coulostatic, harmonic a x ., voltostatic, constant current, and in general most any other approach. These criticisms can be answered by noting that the total number of tubes and transistors, if you include the ever-present oscilloscope and accessory signal-generating equipment, is hardly different for any of these methods. Further, a common complication of nearly all methods is the
1794
ANALYTICAL CHEMISTRY
measurement of a faradaic current that is much smaller than the transients introduced during double-layer charging. These charging-current transients are identical in the coulostatic approach and the potentiostatic approach, with the latter having the advantage of not requiring current-voltage approximations in the theoretical treatment. The potentiostatic method gives kinetic constants directly, allowing the dependance on voltage to be studied after the rate constants are obtained, instead of having to develop approximate equations to apply to varying voltage experimental data to obtain the information of primary interest. More emphasis should be given to the complementary information capabilities of each experimental approach rather than trying to eliminate one in favor of another. I t is true that use of potentiostats requires an acquaintance with some fundamentals of feedback-control theory. Clarification of these basic principles has been attempted in this paper. PART I.
APPLICATION OF AUTOMATIC CONTROL THEORY
The field of automatic control includes the science of organizing the necessary component parts into a system for stability and purposeful action. The primary concern is the interrelations among the various components of a system and the synthetic behavior of the complete device. Accurate performance can be obtained by a feedback system through continually working toward correction of the measured error. However, a dangerous condition of unstable operation can occur when high-gain amplifiers are used in systems having significant response-time delays. Large sustained oscillations or erratic control may take place, giving results of no value. Increasing the amplification of the error signal will not increase the control accuracy unless adequate steps are taken t o ensure stable operation. The system must have an adequate margin of stability and must recover rapidly and smoothly from shocks of irregular inputs or from severe disturbances. Stability and accuracy are, however, mutually incompatible requirements. Recently developed methods of analysis circumvent the previous struggling with high-order differential equations as a method of determining system behavior. A chemist can tackle this servo analysis problem without being a highly trained mathematician or a skilled servomechanism experimenter. These newer methods are well proved and they are an essential part of modern engineering practice, as exemplified by the complex feedback-control systems of all types of aerospace vehicles as well as
the simple feedback control systems in electronically-regulated power supplies, feedback-stabilized amplifiers, and many other diverse applications. The electroanalytical controlled-potential application is distinct from most instrumental approaches in that an essential design component of the feedback system, the electrochemical cell, is a variable over which the chemist wishes to exercise a considerable freedom of parameter values. This requires the ability to look a t the control problem in broad outline and to understand the meaning and the remarkable power of the fundamental concepts of control theory. Although the solution of differential equations is no longer necessary, the correct description of the system dynamic performance by differential equations is basic to all methods of design and analysis. The differential equation does not need to be handled directly as graphical manipulation of the algebraic transform sufices. The same mathematical tool-i.e., use of the Laplace transform method-is as invaluable in understanding the experimental procedure as it is in developing an understanding of the electrode processes themselves. The Laplace transform method reduces a problem in differential equations to one of algebraic operation. In the case of servo analysis, finding the time function response is seldom necessary because the behavior is fully determined by the transform function. Thus, the system requirements can be translated to a set of requirements on the transform equations, or more specifically, on the transfer function when the input characteristics are specified. The transfer function is a complete description of the dynamic and static properties of a system and may be represented as a mathematical expression of the frequency response, or the time response of the system to a specified input signal. The study and design of a system by means of the transfer function are the fundamental techniques in servomechanism engineering. The transfer function is a very convenient parameter both experimentally and theoretically, being defined as the output-toinput signal ratio. In designing a system with a given cell configuration, the synthesis problem involves obtaining the desired changes in the system transfer function by suitable modification of the physical elements of the system. The design criteria are a stability margin giving little hunting or overshoot, adequate response time, and sufficiently small error. To meet stability requirements, the amplifier gain must decrease with increasing frequency. consequently, the error introduced by this reduction in gain becomes greater a t higher fre-
quencies. At a frequency approximately 1 / 1 ~ of the open-loop gainbandwidth product, errors of about 9% are obtained. For Jignals of similar t y p T e . g . , a step-input signal-the functional accuracy cttn be improved by special bandwidth shaping. With pulse or step waveforms, the risetime or time constant of the system is a useful parameter. The time constant for a critically damped system is equal to the period corresponding to the unity-gain open-hop crossover frequency divided by 2 ~ .The response time of any system can thus be quickly estimated or, conversely, the bandwidth required of a system for a specified response time can be easily obtained. The absolute value of the error voltage is depender t on the signal amplitude. Hence, with an error specification independent of signal size, by using the minimum signal amplitude required for a given measurement problem, the requirements on the amplifier are less exacting and more easily attained. Bode Diagrams. A logarithmic graph of the transfer function us. frequency completely describes most control systems of interest to the electroanalytical chemist and in the form of a Bode diagram it is a convenient basis for estimating system performance and for designing changes in the system for improving performance characteristics. A Bode attenuation diagram easily can be drawn for the net open-loop response once the cell transfer function has been obtained. The amplifier transfer function usually is obtainable from the manufacturer's specifications or it c m be measured as described below. I n cases where the amplifier load impedance is less than the output impedance of the amplifier, the amplifjer transfer function will differ from the published values and should be experimentally measured. Because the gain is plotted on a logarithmic scale, the total gain of series elements is the sum of the component gains at each frequency, with the gains being added and attenuation components subtracted considering the unity gain axis as zero for the summing operation. As a further consequence of the log gain-log frequency Bode plot, the phase shift of the network is obtained directly from the slope of the plot for the systems of interest. A slope corresponding to a change of gain of 10 for a decade change in frequency is characteristic of 90" phase shift. A change of 100 in gain for a decade change in frequency is characteristic of 180' phase shift. Thus, the stability of the system and the resronse time can be determined by inspection of the slope and frequency values in the region of net open-loop gain equal to unity, The Nyquist stability criterion phrased
Figure 1. General schematic for three electrode cell-controller system
in terms of the Bode diagram states (8): The closed-loop system is stable if at the frequency where the log magnitude of the open-loop transfer function is equal to zero, its phase angle differs from -180" by less than 180'. The frequency region and the amount of phase correction required are immediately apparent from the Bode diagram, indicating directly the kind of circuit change required to meet the given specifications or the best performance available from given components. A phase margin of 35" or more is the usual design goal in the crossoverfrequency region. For applications where a wide variety of cell conditions are encountered, and fast response is not required, it is desirable to have a potentiostat which does not require readjustment of the gainfrequency curve for each measurement. One approach is to make the control system have a phase shift of considerably less than 90' from very low frequencies to beyond its unity gain point. This can be accomplished with a multi element RC feedback network. For example, an RC transmission line has a 45' phase characteristic. This is also true of distributed, finite-length RC circuits (19). Of course, as the phase shift of the cell approaches 90°, it is added to any phase shift of the controller resulting in an underdamped system response. The gain-bandwidth product also would be greatly reduced. A second approach is to incorporate some type of automatic gain control circuitry for adjusting the ampliier gain and phase characteristics in some near optimum manner as the external feedback (cell) characteristic changes. Transfer Function Data. The transfer function data for an amplifier -cell system can be obtained in a number of different ways. The particular cell, amplifier, and circuit configuration determine which method is most suitable. Direct calculation from equivalent circuit resistance and capacitance values is acceptable from d.c. to several hundred cycles per second. Measurement of the capacitance of a large mercury pool electrode and resistance measurements of the electrodes are discussed below. The cell transfer function may then be calculated based on a chosen electrical equivalent cell model (see discussion on cell models, Part 11). At frequencies
between 20 C.P.S. and to somewhat greater than 1 Mc. per second, use of commercial narrow-band wave analyzers makes possible the direct measurement of transfer functions without resorting to equivalent electrical models. This has been done here with DME and large mercury-pool cells. Of course, comparison of measured transfer function data with the expected data based on an equivalent electrical model will give important information on the actual potential control a t the measuring electrode electrical double-layer region. Although the open-loop transfer function is the parameter used in diagnosis and synthesis, the closed-loop gain gives a direct measure of how well a circuit is operating. Also the open-loop gain can be obtained from the closed-loop gain and phase shift measurements in the frequency region where the gain is between about 0.1 and 10 with use of a Nichols Chart (9). After dividing out the open-loop gain of the amplifier alone, obtained in a separate measurement, the cell transfer function data would result. By using the recently developed current-probe wave analyzer technique (BQ), both the open-loop and closed-loop transfer functions can be easily obtained for a number of circuits of interest to the electroanalytical chemist. This method is especially convenient and rapid for circuits incorporating a follower amplifier because of the resulting low impedance point for inserting the test signal. The current probe is also a very convenient method of introducing a low impedance signal into any feedback circuit for measuring the gain of system components. These approaches to transfer function measurement are perfectly general for any three electrode cell, controller system. Figure 1 shows the separation of the over-all system into two components, cell and controller. The controller may be composed of any number of amplifiers and passive elements, interconnected in subordinate positive and negative feedback loops, with only three terminals required to define the system, be it complex or simple. This, of course, is a direct consequence of the cell being defined experimentally as a three terminal device. Transfer Function of Electrochemical Cell. The transfer function of a three electrode cell can be measured by several methods. Measurements are best made with the controller amplifier connected, because the transfer function is to some extent potential dependent. In the frequency region where the transfer function is to be measured, the open-loop gain of the controller must be low to directly give transfer function values relatively independent of cell configuration and impedances. Converting the potentiostat operational amplifier to a 10-second integrator and VOL 35 NO. 12, NOVEMBER 1963
0
1795
Figure 2. Measurement arrangement for cell transfer function a t controlled potentia I
using a summing circuit to the counter electrode makes this measurement rather simple. The measuring arrangement is shown in Figure 2.
For the actual cell measurements reported in this paper, a HewlettPackard Model 302-A wave analyzer was used in the beat frequency oscillator (BFO) mode to provide the oscillator signal and to measure the input and output voltages a t the subsidiary and reference electrodes, respectively. The 3-db. bandwidth of less than 3.5 c.p.s. of the wave analyzer makes gain measurements possible in the presence of the large amounts of noise encountered in typical systems. In BFO operation, the oscillator and tuned voltmeter sections of the wave analyzer automatically track together over the frequency range of the analyzer. Transfer Function of Amplifier SysBecause differences in amplifier load and input impedance can change the transfer function, measurements must be made either with the electrochemical cell circuit or with reasonable substitute impedances connected. As shown under cell impedance measurements, the mercury-pool cell terminal impedances were reqistive in the frequency region of interest, allowing a resistor of equivalent size to substitute for the cell loading effect on the amplifier output. The same argument applies to the reference electrode impedance, where use of a resistor equivalent in size to the reference electrode
resistance allows the attenuation and phase shift characteristic of the amplifier input circuit to be measured, either separately or as a part of the amplifier transfer function. Figure 3 shows the circuit arrangement for measuring amplifier open-circuit gain with the electrochemical cell of interest coniiected into the feedback loop. As in the cell transfer function measurements, the \+rive analyzer \YW used to furnish the oscillator signal and to measure the input and output voltages. I t will be noted that in using this circuit, the gain obtained is that from the amplifier input to output terminals, the direct effect of the cell source impedance (reference electrode resistance) is not measured. A measuring circuit using equivalent cell resistances, shown in Figure 4, can include this effect by varying the input resistor shown. A feedback resistor network is ubed to prevent amplifier d.c. saturation and to keep lorn frequency
tems.
Table 1. Open-Loop Response of Philbrick USA-4J Operational Amplifier with Coulometric Cell as Output Load (510 ohms)
Frequency,
kc.
0.05 0 10 0.25 0.50 1. 0 5.0 10.0 50.0 .io . 0
1796
Gain magnitude 11,000 6,400 3,100 1,600 817 178 97 44.5 15.8
output voltage Phase during shift, test, degrees volts 60 1.OO 59 0.15 73 0.82 0.82 82 84 0.80 88 0.74 94 0.090 103 0 051) 0.022 110
ANALYTICAL CHEMISTRY
-
--
Figure 3. Measurement arrangement for amplifier transfer function with cell in feedback loop
noise to a reasonable level. The design characteristics of most operational amplifiers limit the voltage swing a t the higher frequencies. This requires the measurements to be made with small signals, Checking the output waveform on an oscilloscope will show if the signal size is such to cause waveform distortion and the input signal can be reduced to minimize this distortion. The Hewlett-Packard Model 302-il wave analyzer was used for gain measurements in the automatic frequency control (AFC) mode. To obtain phase shift measurements, a Hewlett-Packard Model 200-CD wide-range oscillator was used as the signal source, with the wave analyzer being used for measuring voltage gain and supplying a reconstructed input signal a t the tuned frequency to an Ad-Yu Electronics Model 202 phase-shift meter. The oscillator voltage was used as the reference phase signal. The indicated phase angle is corrected by subtracting the measured phase shift of the wave analyzer. Any possible amhigiiit,y R S to
E i n- - -TO -
1
I300
--
MFD
Figure 4. Measurement arrangement for amplifier transfer function with simulated cell impedances
which quadrant the phase angle mas in is resolved by the direction of change of phase angle with frequency. Higher frequencies give higher phase shift for most of these systems. For help in stabilizing network design for coulometers not using a power output stage, the data in Table I, obtained by the circuit shown in Figure 4, are applicable to systems using Philbrick USA-4J operational amplifiers with a series load of approximately 500 ohms. To show the effect of amplifier load, data were also obtained with no load on the amplifier output except for feedback network. The gain and phase shift increased to 6500, 72'; 11,300, 85'; 5620, 90'; 645, 114'; and 250, 119" a t 250, 500, 1000, 5000, and 10,000 c.p.s., respectively. Closed-Loop Transfer Functions. Closed-loop voltage gain and phase shift data are easily obtained for the complete electrochemical cell, operational amplifier system by placing a lowimpedance (50 ohm) signal generator in the lead between the amplifier output and the cell counter electrode and measuring the ratio of the oscillator voltage to that voltage developed between the amplifier output and the amplifier common terminals (Figure 5). Floating, required of either the measuring equipment or the controller,
Figure 5. Measurement of cell-controller system closed-loop transfer function
can be accomplished with a low capacity isolation transformer such as the Elcor 350-watt Model L-131. For the closedloop transfer function measurement the Ilewlett-Packard Model 302--4 wave analyzer was used as a tuned voltmeter and tracking oscillator in the BFO mode. Figure 5 snowb the circuit connections. The measurement of the closed-loop transfer function of the amplifier alone may be made lvith the cell removed. Referring to Figure 5. the oscillator would be connected directly between the amplifier input and ouput, without attenuation. I t is of interest to note that this measurement circuit is the same configuration AS the groundedoutput follower discussed below. K i t h some circuits, capacitive loading of the output amplifier, even a3 low as the 40 pf. of the Elcor isolation transformer, may cause internal-loop oscillations. Measurements on a circuit with a Krohn-Hite DCA-10 power amplifier output stag: n-ere made without difficulty ITith the amplifier floating. Oscillations were encountered in measuring another ciacuit configuration with a DCA-10 outpiit stage where the measuring equipment was isolated. Resistile (16) or inductive isolation of the capacitive load would be required where oscillation occnrred. The lowest value of gain that can be measured by this technique iq limited by the ratio of the amp1 fier output impedance to the cell impedance a t the frequency of measurement, as these impedances appear in series across the oscillator output in this measurement circuit with the \ oltage being measured across the amplifier output impedance. The clo5ed-loop gain has to be 5 to 10 times greater than this impedance ratio. This is not a limitation for most controlled-potential sysi,ems in the gain region down to values of less than 0.1 which COT ers the region of interest. Coulometer Cell Mercury-Pool Capacitance. Because of the large value of this capacitance, both in absolute value and in relation to the other cell electrodes, many of the techniques, including the bridge and coulostatic methods, are not applicable. The current-interrupter technique was used, with the measuring-reference electrode voltage us. time slope being measured within 10 microseconds after the cell circuit was opened by operation of a mercury relay conne1:ted to a small d.c. power supply. Voltage--time traces uere pliotographed from a Tektronix Type 532 oscilloscope having :t Type D vertical preamplifier. Cell cirrents of from 10 tn 40 ma. were used, with uranium(V1) or iron(II1) present in the 0.5N sulfuric acid electrolyte. PTith the counter electrode in an 8-mm. diameter tube, interpretable results could not be ob-
tained because of the extremely nonuniform current distribution in the cell. Uniform current distribution was obtained by placing a loop of platinum wire in the cell equidistant from the mercury pool surface and by not using a fritted-glass counter-electrode separator. Capacitance values from 100 to 400 pf. were measured. The usual formula, C = io/(dV/dt)t-o, was used. The reference-measuring electrode d.c. potential was measured with a Hewlett-Packard 425-A microvolt ammeter having the input resistor removed (input impedance greater than 100 meg.). These current interrupter measurements in 0.5M sulfuric acid electrolyte gave pool capacitance values of 320, 160, and95pf. a t -0.2, --0.7, and -1.0 volts us. the saturated calomel electrode with uranium(V1) added to give cell currents of 10 to 50 ma. These capacitance values were obtained with the stirrer rotating a t 1800 r.p.m. and the mercury level even with the top of the conical section of the stirrer. Measurements of the pool capacity with the stirrer off indicate that the pool area is 60% larger with the stirrer running. This is expected from the outward force that the rotating stirrer exerts on the mercury, which causes a slight gap a t the top of the conical section and some dishing of the mercury against the cell ~11s. Cell Resistance. The counter and reference electrode resistanceq were determined by placing a General Radio Type 1432-P decade resistance box in series with the electrode to be measured and the mercury pool electrode, adjusting the capacitively-coupled a x . voltage to 10 mv. across the total circuit while adjusting the variable resistance box to give 5 mv. across the cell or serieq resistance. !Then this condition obtains. the electrode resistance equals the box resistance as the mercury-pool electrode resistance can be neglected with usual cell designs. Csing the Hewlett-Packard Type 302-A wave analyzer in the beat frequency oscillator mode, General Radio Type 874 coaxial attenuators between the analyzer output and cell, and measuring the two voltages on the wave analyzer, the resistance values obtained for the coulometer cell counter and reference electrodes were 360 and 3720 ohms, respectively. These values were independent of frequency over the range of the wave analyzer, 50 c.p.s. to 50 kc. per second. Approach to Design of More Complex Potentiostats. As tliscuwd above, analytical expresiono combined with the Rode diagram allow syiteiii stability and response to be determined in a relatively simple manner. In practice, the Bode diagram method ha5 proved suitable for potentiostat systems of greater than 1 megacycle bandwidth. For more complex systems, or where the simplifying assumptions are not valid, or
if more detailed information is desired other methods must be used. In any case, the differential equations describing the system components are the starting point. These can be written by inspection of most parts of the circuit or can be obtained from the experimental transfer function measurements where necessary. At high frequencies, the effect of distributed impedances and delay times makes experimental measurement or confirmation of transfer functions almost imperative. As an example, the capaGitance of a very small wire is 0.1 pf. per cm. length even when far removed from neighboring objects; that of a sphere is 1.1pf. per cm. radius, giving an irreducible 50 pf. for a 20-inch diameter component. With the determination of a set of system-describing equations, the closed- or open-loop transient or steady state response can be evaluated. For the control systems of interest for electrochemical potentiostats, many of the digital calculation methods are unsuitable as they involve matrix calculations of equations with a very large range of time constants. In many cases these matrix calculations lead to intolerable rounding errors. A careful consideration of circuit values is required to make use of matrix methods such as are involved in preparing the system equations for the Runge-Kutta iterative method, or for methods of matrix solution for the steady state equations. The approach of least dificultj is the calculation of the open-loop, steady state transfer function. The subtraction of two nearly equal numbers is seldom required; when required it is usually in known places which can be easily monitored. From this transfer function, the closed-loop, steady state and transient responses are obtainable by straightforward calculation or they can be evaluated by comparison with known or desired response curves. The disadvantage of this method is that a nongeneral computer program must be used involving considerable complex algebra. Using compiler language-e.g., Fortrari-and complex algebra subroutines, this amounts to little more than copying the iystem equations. For some cases, especially thoseinvolving low frequency controllers, analog computer studies are advantageous, allowing rapid optimization. PART 11.
CELL DESIGN CONSIDERATION
.\ti cffrutii c wll rail kw designed oiily nith itii under*tandiiig of the effects of rlectrode placement, electrode types, anti ytirring on the ease and accuracy of potential control and electrolysis speed in cells u4ng stirring. For accurate potential control, a symmetrical electrode arrangement is required to prevent potential gradients on the measuring
VOL. 35, NO. 12, NOVEMBER 1963
1797
Figure 6. Front and side view of coulometer cell and stirrer apparatus
electrode due to varying path length
I R drops in the electrolyte. The reference electrode tip must be placed close enough to the measuring electrode to make the solution IR drop between it and the measuring electrode negligible. For exhaustive electrolysis or electrode concentratiou methods, cell design optimizing requires high masstransfer rates, usually effected by highspeed mechanical stirring and large electrode area to solution volume ratios. Noise sources including that generated hy the stirrer, affect the potential control accuracy and the required controller bandwidth. A logical approach to cell design is possible by considering the equivalent electric circuit aualogs of the cell components with their connecting impedances. A precision stirred mercury pool coulometer cell design is presented along with performance data. Cell Analogs. Analogs are useful in setting up physical systems and interpreting their boundary conditions because known systems can be compared with unknown. Converting the electrode-electrolyte systems to an electrical equivalent is very useful as electric circuit analysis methods have been developed to a high degree. The equations which mathematically describe the system performance can be written when the basic operation and physical laws governing the process are understood. The effect of the faradaic impedance on the cell transfer function is more difficult to visualize than are the usually predominant effects of the electrode double-layer capacitances and interelectrode resistances. The transmission line analogy to the faradaic electrode processes is well established but not generally used, apparently because of a reluctance to drop the frequency d e pendent single RC analog. The choice of the single RC analog makes the physical interpretation of response to a step 1798
ANALYTICAL CHEMISTRY
function extremely difficult. The transmission l i e equations are identical to the ditrusion equations; they make it easy to visualize the response to a step function; they allow approximations to be made for stability and transient analysis using linear network theory. At any constant frequency, the transmission line(s) can be replaced by an equivalent circuit of a series resistor and capacitor. These are simply related to the transmission line R/C ratio per unit length (2). Barker (2) has discussed equivalent transmission line circuits for the faradaic impedance when there is a slow electrode reaction, adsorption of either or both the oxidized and reduced species, and approximate analogs for the case of a preceding chemical r e action. Any particular reaction mechanism scheme can be accurately modeled by simple extension of this approach. Bewick et al. (4) have described in detail equivalent cell models showing how the placement of the electrodes in the cell affects the as. potential control of the measuring electrode. Component designs are intimately related in a potentiostat system as in any feedback control system. Slight changes in relative electrode positions can make a large difference in a m p E e r requirements for a given control error. With cells using mechanical stirring or forced oouvection, the amplifier bandwidth and gain required for the chosen control accuracy is determined by the magnitude and type of offset signals generated by the interface motion. A high tolerance stirring apparatus can effect savings in the required electronics as well as improving the possible control accuracy. For calculation of the equivalent resistance between a spherical electrode and the reference electrode prohe tip, the equations given in Appendix I can be used. Stirrer Design for Controlled-Potential Coulometry. Obtaining quantitative oxidation or reduction by controlled-potential coulometry in times of 15 to 30 minutes requires the use of small solution volume to electrode area ratios and high stirring rates. To obtain completion of the electrolysis to within 1%in this length of time is not difficult. Magnetic stirrers driven at about 600 r.p.m. work well with mercury-pool measuring electrodes. To obtain completion of the electrolysis to better than 0.1% in the 15- to 30-minute time, stirring rates in the vicinity of 1800 r.p.m. are required with the mercurypool electrodes. Some of the first designs to accomplish this high rate of mass transfer, resulted in splashing of the electrolyte throughout the cell and severe oscillations of the mercury pool. Although highly precise, quautitative coulometry was obtained, adjusting the
rigure
1.
rrecision conicai piosric
stirrer
mechanically driven stirrer for proper operation was critical and the large variations in mercury electrode area and position made precise potential control impossible. Workers using high-speed magnetic stirrers found that a bearing support system was required. Studies a t ORNL (2.5’) resulted in the selection of a small, &t, disk-shaped stirrer, mounted perpendicular t o the shaft. The stirrer dkk was corrugated slightly by squeezing the hot glass with a pliers. Splashing was avoided by having the disk diameter much smaller than the diameter of the pool. No attempt was made to obtain smooth stirrer operation; the glass shaft was positioned by a loose Teflon bushing and connected to the 1800-r.p.m. motor shaft with flexible tubing. Considerable oscillation of the mercury surface occurs. To prevent oscillations .of electrode potential, a symmetrical stirring arrangement is required; magnetic bars or mechanically driven paddles cannot he used. The mercury pool must he kept as quiet as possible to allow a constant spacing of the reference electrode to the mercury pool. A symmetrical electrode arrangement is required to minimiae potential gradients across the measuring electrode. In view of the ~uccessof the O W L stirrer design, studies were made of symmetrical disk and cone shaped stirrers mounted on a high precision ball hearing assembly as shown in Figure 6. The flat smooth disk stirrers were unsatisfactory as large amplitude oscillations of the electrolyte-mercury interface occurred at rotation speeds n e c w
more complex electrode shapes. A cast plastic block may be less expensive than machined Teflon. A smaller bearing would enable the use of a straight angled hole through the cell cover and bearing holder. Placing the mercury pool lead through a hollow shaft motor to the rotating connection is not recommended. Approximately 35 mv. peak-to-peak of 60-cycle voltage and harmonics were measured across a shaft used in this manner. This voltage would cause an undesirable modulation of the measuring electrode potential.
Figure 8. Electrical rotating shaft
connector
for
sary for rapid electrolysis. The 45' tapered cone, Figure 7, was very satisfactory. Angles between 30" and 60' were also satisfactory. With a 1-inch diameter cone in a 1 binch i d . cell, rotation speeds near I800 r.p.m. were most suitable. The initial cell current was found to be a 1.5 power function of stirrer rotation speed ii the range from 600 to 3000 r.p.m., slowing that the effective electrode area was a function of stirrer velocity (also see capacity Corrugations measurements in Part.):I on the surface were undesirable, causing breakup of the mercury. At 1800 r.p.m., there is a small motion of the mercury surface but insufficient to prevent placement of the reference electrode probe close to the surface. If a symmetrical counter electrode is used, positioning of the reference electrode becomes quite noncritical for cell currents below about 100 ma. Movement of the electrolyte is very smooth; no splashing occurs. Because no wash-down occurs, splashing or spray due to a deaeration purge cannot be tolerated. Satisfactory deaeration was obtained by running the water-saturated gas stream over the surface of the electrolyte during a 15-minute prereduction step a t +0.086 iolt us. the saturated calomel electrcde in a 0.5N sulfuric acid electrolyte. Another laboratory (20)has used a stream of gas a t sufficient pressure to depress the surface of the electrolyte about 6 mm. t o give effective deaeratio 1 without splashing or spraying electrolyte. Prereduction of oxygen directly also has been suggested as another approach (SO). Having a ball bearin'; close to the cell toll, as necessitated for precision stirring, iiitroduces a comp1i:ation for the counter and reference e ectrodes, and for the sweep gas lead .in connections. Various approaches befides the one described are possible and may prove to be more desirable. An electrode block holder split vertically ISbeing used for
Coulometer Cell and Precision Stirrer Construction. The cell was constructed of medium-wall borosilicate glass tubing having a 1.500inch precision bore (Fischer and Porter Co.). The bottom of the cell was made flat, using care not t o curve the edges, to ensure reproducible height of mercury upon constant volume addition. The over-all length of the cell was 4.0 inches. The stirrer apparatus is shown in Figure 6. Readily available standard mechanical instrument components (PIC Design Corp., East Rockaway, N. Y.) are used with the exception of the stand and plastic components. The shaft is connected to the Bodine KYC-26, 1800 r.p.m., 0.56-inch ounce torque, synchronous motor shaft with a precision flexible coupling having a molded neoprene body with metal ends. To obtain a solid mounting, the support stand was made of 0.5-inch stainless steel. The motor and bearing mounting plates are electrically isolated from the stand by a polystyrene sheet. Surfacing of the stand and the polystyrene may be necessary for good alignment. The mounting plate screws are isolated from the stand with plastic bushings and washers. Connection to the mercury pool is made through a 0.063-inch diameter platinum wire, 0.4375-inch long force-fit into the bottom of the Kel-F stirrer, a mercury connection to the stainless steel shaft, and a rotating stainless steel cup through mercury contact to a stationary stainless steel disk. Details of this rotating cup device are shown in Figure 8. Three threaded platinum inserts were placed into the edge of the stationary disk to effect reliable contact with the rotating mercury after high resistance developed in this connector. Three platinum inserts were also placed in the rotating cup. The resistance was reduced to 0.04 ohm. The clear plastic split-ring cover was glued to the rotating cup with epoxy cement. Heating the Kel-F stirrer by immersion in hot water causes sufficient expansion to allow easy insertion of the steel shaft. Stirrer details are shown in Figure 7. The cone shaped section is 1.00-inch diameter a t the top of the cone; the 45' sloped section extends to intersect the section starting 0.25 inch beIow the top of the cone sloped at 13' from horizontal. The bottom nas sloped a t 13' to prevent solution from being trapped.
.1 Side View Cut Line
L
1 Side View Cut Line
Figure
9.
Electrode holder block de-
tails
The length of shaft below the tapered section possibly could be shortened but this wasnot studied. The 3-inch 0.d. Teflon s d i t block for the electrodes was designed to bring the necessary tubes out of the block a t 90' to the shaft, thus allowing the lower bearing to be placed very close to the cell and eliminating stirrer wobble and other nonconcentric motion. Machining requirements are shown in Figure 9. The glass connecting tubing was precision ground to an outside diameter of 0.25 inch, allowing easy clamping in the 0.25-inch diameter holes in the Teflon block. The reference electrode tip was taken from a Beckman No. 39270 fiber-tip calomel electrode, and positioned close to the mercury pool and the counter electrode fritted glass separator. Flow rate in the counter electrode tube was reduced to about 0.5 ml. per 24 hours by introducing a 0.5-inch length of a fine slurry of submicron silica (Cab-0-Sil, Cabot Corp., Boston, Mass.) in 0.5M sulfuric acid directly above and into the frit. Use of ultrafine frits may be suitable without the silica plug. Connection of the 0.25-inch glass tubes in the cell block to the counter and reference electrode compartments was made with Tygon tubing. Performance for Precision Cell. Using the coulometer cell described, electrode potential disturbances were very small. With a IO-second time constant controller or with the controller temporarily disconnected, the maximum noise measured with the HewlettPackard 302-A wave analyzer in its 20cycle to 50-kc. per second range was 0.25 mv. r.m.s. at 20 to 30 C.P.S. deVOL 35, NO. 12, NOVEMBER 1963
* 1799
Table II. Potential Difference across the Surface of a Mercury Pool, Showing Dependence on Total Cell Current
Total cell current,, ma. 1 2 4
Potential difference, mv.
Effective resistance t o pool, ohm
6 6
6 12 28
.5 . 6
Table 111. Potential Differences between a Point in Solution Close to the Counter-Electrode Separator and the Maximum Current Density Point Directly under the Separator, Showing Dependence on Total Cell Current
Total cell current, ma.
Potential difference, mv.
resistance, ohms
1 3
2
2 0
Effective
6
creasing to about 0.07 mv. a t 50 c.p.s with peaks a t 60, 90, 120, 150, and 180 C.P.S. of 0.12, 0.05, 0.06, 0.05, and 0.03 mv. r.m.s., respectively. The noise was less than 20 pv. r.m.s. above 200 C.P.S. These measurements were made with the 1800 r.p.m. stirrer operating in a 0.5M sulfuric acid electrolyte a t -0.3 volt os. the saturated calomel electrode. A cell-controller bandwidth of about 0.1 C.P.S. would seem entirely adequate for a cell having this low noise level. Measurements of noise across the load resistor indicate that it is a serious consideration, a t least for those controller circuits required to perform a direct compensation of load resistor voltages. Using the simple coulometer circuit without bandwidth restriction to effect cell potential control, and the same solution conditions as above, the noise voltage across a 1000-ohm load resistor measured 26 mv. r e m s .with a HewlettPackard 400-H ax. voltmeter with 10C.P.S. to Pmc. bandpass. Adding a 10-pf. capacitor across the load resistor decreased the broadband noise to 16 mv. r.m.s., which is still a considerablc modulation of the desired d.c. potential. A cell-controller bandwidth of about 100 C.P.S. is required to reduce this disturbance to about 5 mv. r.m.s. This bandwidth, defined for the control systems under discussion, is the frequency where the controller gain divided by the cell attenuation is unity. A further experiment to measure 1800
e
ANALYTICAL
CHEMISTRY
noise across a 1000-ohm load resistor involved the reduction of oxygen in 0.5M sulfuric acid at -0.7 volt us. the saturated calomel electrode. With the mercury level even with the top of the cone-shaped stirrer, the totai cell current was 8 ma. with 45 mv. r.m.s. broadband noise measured on the Hewlett-Packard 400-H voltmeter. Lowering the mercury level 2 mm. gave 4.5 ma. and 110 mv. r.m.s., and lowering the stirrer 5 mm. (which is about the end of the 45' tapered section) gave 2 ma. and 20 mv. r.m.s. Thus, the mercury level should be even with the top of the stirrer. Potential Distribution. I n a considerable amount of recent work with three electrode cells, both in polarographic and coulometric studies, the effect of nonuniform current distribution on the meacuring electrode has not been considered. In several instances (Zl), the reference electrode has been positioned a t the furthest point from the counter electrode, with subsequent loss of compensation ability for any significant amount of voltage drop in the cell. Analytical work is possible with these cells, because the large resistive drop a t the counter electrode separator and connecting tubing as well as the voltage drop a t the counter electrode are compensated. The resistive drop in the cell is far from negligible. This is shown for the precision coulometer cell of Figure 6 in Figure 10 and by the data in Tables I1 and 111. These data and experiments in which the pool potential wa4 varied betlveen -0.2 and -0.9 volt z's. the saturated calomel electrode indicate that the observed potential differences are predominantly caused by resistive drop in the electrolyte which is essentially independent of the nature of the current-voltage function a t the mercury pool electrode. The direction of the potential drop is such to give the highest current density a t a point directly under the counter electrode separator. With nonsymmetrical cells, this point is where the potential shouId be controIled. Control of another point will cause this highest current density point to have a potential such that an interfering electrode process can occur. In the example shown, the desired potential is as large as 0.125 volt from ideal. With many electrolytes having less conductivity than 0.531 sulfuric acid, the need for a symmetrical arrangement and proper reference electrode placement is even morc critical. Ideally, a symmetrical counter-measuring electrode would be used and the reference electrode tip would be placed on the mercury surface a t the point of highest current density if any potential gradient remains. The placement and type of reference electrode tip have been studied extensively (12). Resistive sheet models or large scale electrochemical cells are very
Figure 10. Potential distribution a t surface of ring-shaped mercury pool electrode with unsymmetrical counter electrode Total cell current 40 ma. Pool diameter 1.5-inch o.d., 1 .O-inch i.d. Small circle shows position of the counter electrode fritted gloss separator, situuted 4 mm. obove the pool surfoce.
useful to define electrode current distribution and cell potential gradients. A suitable symmetric counter electrode was made for the cell used in this coulometric work by cutting a toroid of 5-mm. 0.d. glass tubing in half and fusing a fritted glass, ring-shaped disk separator to the bottom. Tubing from the top serves as the support and connection to the electrode reservoir. Use of this electrode reduced the mercury pool potential differences to less than 2 mv. at a total cell current of 50 ma. Coulometer Cell Potential Distribution Measurement. A small Beckman saturated calomel electrode with a fiber tip, Beckman No. 39270, was connected to a small polyethylene tube, tapered to about 1-mm. diameter and filled with 0.5M sulfuric acid electrolyte. By using a 1.5-inch i.d. cell filled nearly t o the top, the probe could be positioned by hand a t the desired points. The counter electrode separator was inch from the mercury pool surface. The potential difference between this movable probe and the fixed probe normally in the cell was measured with a Hewlett-Packard Model 425-8 d.c. Microvolt Ammeter (input resistor removed); line power for the voltmeter being isolated from the potentiostat circuit by an Elcor Model L-131, 350-watt power Isoformer (40 pf. isolation from ground, 0.1-pf. interwinding capacitance). The potentiostat was a single Philbrick USA-UT operational amplifier. PART
111.
POTENTlOSTATSi COMPONENTS AND SYSTEMS
The requirements placed on a potentiostat include impedance matching of the input circuit to the usually high impedance reference electrode. sufficient gain and bandwidth to meet performance specifications with the
expected range of cell and external circuit parameters, and impedance matching of the output stage to the measuring electrode-counter electrode system with sufficient current and 7.oltage capability to meet the external needs. These requirements are indiv dually discussed b ~ l o w . Usc of the plus one gain or potentiometric circuit configuration is shown to be a useful solution for the input voltage folloa er requirement. The relation of gain and bandwidth to cell parameters and response requirements is given. Booster circuits or power amplifiers and output impedance matching is discussed with specific reference to commercially available amplifier-. I n all cabeb, the design of a potentiorequiring stat bystem is a com~~romise careful weighting of the following factors: convenience of use, initial and operating coqts, versatility or universality, speed, sensitivity, reliability, simplicity, ease of s e n icing, and ease of fabrication. Kot only must each of the units of the system work well for good overall operation, but they must be designed together. Optimizing requires a give and take between cell design and electronic complexity. In the potentiostat system discussed in this section, convenience of use a t moderate cost has been obtained a t the expense of some bandwidth restriction. Three-terminal ampliiiers are used exclusively, permitting direct use of this less expensive class of operational amplifiers and eliminsking the need for separste carrier-type c1.c. amplifiers. Grounded-Output Follower Circuit. A predominant charxteristic of electroanalytical instrummtation presently in use is the operational summing circuit, effecting the conibination of a reference-nieawring electrode potential uith the various voltage waveforms necessary for the particular application. Recause reference electrode design usually entails a small, high resistance probe (Luggin-Haber capillary) and because the current carried in the reference-measuring e1ei:trode path must be small in compariscn to the total cell current, impedance matching is required before this signal can be applied to the relatively low impedance summing circuit. Where precise control of d.c. level is required, chopper-stabilized circuitry is preferred. The early development of a chopperstabilized follower circuit by DeFord (11, 14) was an important step in increasing the utility of operational amplifier instrumentation. However, several disadvantage. of this early approach are apparent i f which the common-mode error is the most serious. With the chopper amplifier floated between the d e. amplifier differential inputs, a symmetrical input circuit, necessary for high common mode re-
noise is 55 pv. r e m s .at 60 c.p.s. with the input shorted, rising to 500 pv. r.m.s. with 1 meg. from input to ground. The amount of charge required to bring the stabilizing capacitor to the input voltage value (CV product) must be supplied by the voltage source (reference electrode). For a usual maximum input value of 3 volts this corresponds to 150 pC, well within the capability of the reference electrode.
e.
in
Figure 1 1. follower
Grounded output voltage
jection, is not obtainable. Even if symmetry could be obtained, the differential input design would be limited by unmatched nonlinearities of the two input triodes. With some operational amplifier designs, such as the Philbrick USA-3, drastic modifications t o the printed circuit board are necessary to effect floating of the chopper amplifier section, and the unit is not designed for good differential operation (16). In commercial data system use, this impedance matching requirement has been met by a special unity gain configuration using 3-terminal floating amplifiers. The grounded-output or potentiometric circuit configuration shown in Figure 11 overcomes the common mode requirements of the differential amplifier circuit by use of the operational amplifier as a null device. The performance is shown by a gain within 0.1% of unity up to 5 kc. with either 1- or 10-volt r.m.s. input signal increasing to plus 0.3% over unity gain a t 10 kc. The rise time of the USA-3 follower, connected as shown in Figure 11, was 2.6 microseconds with source impedances up to 10,000 ohms. Capacitance, C, was 50 pf. This rise time is the time for the follower to reach 90'% response to a step voltage change a t the input. With a 100,000-ohm source impedance, the rise time is 10 nlicroseconds. With this high source impedance, the capacitance was removed, the response being overdamped with no capacitance added. The need of B separate, floating power supply is not a serious disadvantage. In fact, the problem of bias or noise signals appearing in power supply wiring common to two or more operational amplifiers is eliminated. A high degree of regulation is not required of this added d.c. supply; the units used have only shunt VR tube regulation. Total cost of the isolated power supply was $88.50. The input resistance of the USA-3 follower circuit was measured as 22,000 meg. per volt without d.c. offset correction. Output d.c. offset is 40 wv. with the input grounded and 80 pv. with 1 meg. from input to ground. Output
Grounded-Output Voltage Follower Construction. This voltage follower mas constructed using one Philbrick CSAL3h13amplifier, two Elcor (Elcor, Inc., 1225 W. Broad St., Falls Church, Va.) AG-300-20 300-volt, 20-ma. d.c. supplies, and one Elcor AX6-4KT 6.3-volt, 4-ampere a x . bu[lplS. The equivalent zener diode-regulated supplies mould also be suitable. Total capacity t o ground of these isolated suppliei is about 80 pf. Ko offset correction circuit was used. The wiring diagram is shown in Figure 11. The numbers correspond to the terminals of the Blue Ribbon connector on the USA-3113. One side of the 6.3-vok filament winding was connected to pins 3, 4, and 5 in the normal manner. The offset connection, pin 9, was connected to the common lead, pin 4. The input cable shield is connected to pin 4 to provide a guard potential. The value of the capacitor, C, is 50 pf. with source impedances below 10,000 ohms decreasing to zero with a source impedance of 100,000 ohms. The output load resistor was usually the 10,000-ohm resistor of the following adder circuit. The 1000ohm resistor between pin 6 and ground removes the capacitive load to the power supplies from the amplifier output stage, improving the step response during the first microsecond of potential change (16). The 10,000-ohm resistor and capacitance, C, serve to reduce the amplifier bandwidth, effectively giving the follower a critically damped response. lf7hen used in a typical closedloop amplifier system. a i in summing circuits, these stabilizing uniti 1%ill usually not be required and may be detrimental as the attenuation characteristics of other sections of the feedback loop will be controlling the response characteristic. No modification of the operational amplifier is required; all wiring is external to the unit and essentially amounts to only connecting the poaer supply, input, and output leads. Bandwidth Noise and Gain Considerations. The required bandwidth for a potentiostat depends on the response time desired for the external input signal, the measuring electrode potential error that can be tolerated, the frequency-amplitude characteristic of the potential disturbances generated by changing geometry and concentration gradients in the cell, and the frequency-amplitude characteristics of the voltages across the load resistor, cell resistance, and other series resistance, caused by variations in cell current. VOL. 35, NO. 12, NOVEMBER 1963
1801
The voltage variations that occur external to the measuring-reference electrode section, caused by cell current changes, are reduced in importance relative to the other sources by the gain of the control system when the circuit configuration uses only the measuringreference electrode signal as the error signal source. That part of the cell current fluctuation signal appearing across the load resistor has appeared directly in the controller input circuit for most of the operational type controllers previously used (21) as well as the three-amplifier circuit in this paper. The simple, single-amplifier potentiostat used in the coulometer circuits discussed here does not have this limitation. For cells using solid electrodes, noise, nhich can result from area changes during anodic electrode dissolution, gas bubble formation and break-away, electropolishing effects, erosion of rough spots, and other posqible geometry changes affects the bandwidth requirements of the potentiostat. The system ri.e time must be considerably smaller than the time region of interest for faradaic current measurement. For example, with a l-microsecond systein time constant, a 100-mr. signal produces a masimum current of 100 ma. with a 1-gf. electrode capacitance and requires 11.5 microseconds t o decay to 1 Fa. The current (or voltage) in a first order system changes a factor of 10 for every 2.3 time constants. Amplifier and/or resistive cell noise becomes the limiting factor in detecting small faradaic currents a t fast response times. Proper impedance matching for minimum input circuit noise has been discussed with reference to a squarewave polarograph circuit (IO)and in the general application t o voltage amplifiers (16). Thermal or Johnson noise can be calculated from the formula: E, = 1.29 X
(RF)l’a
where E,, is the r.m.s. noise voltage generated across a source of R ohms in a bandwidth of F C.P.S. Peak-to-peak noise is approximately 5 times the r.m.s. value, For intermediate speed potentiostats, this is not a limitation on potential control, but may be for current measurement. Using a 100-meg. reference electrode with a 10-kc. bandwidth, the calculated noise is 0.13-mv. r.m.s. With a 10-kc. reactance of a 1-pf. double-layer capacity of 16.7 ohms, the noise current is 8-ma. r.m.s. The d.c. gain required of a potentiostat circuit to give reasonable cell voltage control is very large, especially where high cell or series resistance is present. Several circuits proposed for cell potential control using low gain solid state or vacuum-tube amplifiers do not have the necessary excess gain to compensate the cell attenuation with 1802
ANALYTICAL CHEMISTRY
s a c i e n t gain remaining to enforce the desired control potential. For example, if an output voltage of 100 volts is required t o furnish current to the cell and associated series resistance and control to 1 mv. is specified, an amplifier gain of At100,000 would be necessary. tenuation of the error signal in the amplifier input circuitry would cause a proportionate increase in the required gain. Control-potential data obtained on instruments d t h greater than 10 to 20 mv. peak-to-peak modulation of the d.c. potential would be expected t o be significantly different from that obtained on instruments with good
I CELL
~
I
{
I I I
Figure 12. Impedance matching circuit for matching 1 - to 2-watt output operational amplifiers to 50-ohm or less cells
potential control. Interference effects from electroactive materials in nearby potential regions, especially those showing irreversible behavior, are greatly influenced by voltage uncertainties of this magnitude. Instruments having potential control to within 10 mv. are certainly a requisite for adequate method development work for problems of even slight degree of complexity. Impedance Matching, Power Requirements, Power Amplifiers. Most workers that use operational amplifiers to drive polarographic type cell loads do not match the amplifier to the load. Booster circuits are used in cases where a suitable passive impedance matching network gives the required transient cell current to the usual low impedance electrochemical cell. The circuit shown in Figure 12 furnishes larger currents to the cell, essentially by the step-down transformer technique. The use of wide bandwidth transistors to drive low voltage cell loads is another obvious solution. Circuitry of this type is being developed, giving an inherent power matching to the low impedance load. Considering only the power required to produce a signal across a 1-pf. doublelayer capacity, it is calculated that 6-mw. equivalent is required per volt of
1-kc. signal, and &watt equivalent is required per volt of 1-mc. signal. The power dissipated in the series cell resistance undoubtedly would be greater than this, amounting to nearly Zkw. per volt at 1 mc. for a 50-ohm series resistance. For pulse signals, peak current requirements are very large if microsecond-region responses are desired. Again using the l-pf., 50-ohm cell, 5-kw. per volt applied signal is required to supply the peak current for a 0.1-microsecond time constant. This requirement, plus the capacitances throughout the driver amplifier and associated leads, results in high power circuits being prerequisites for high speed studies. A commercial 10-watt vacuum tube power amplifier having a 1-Mc. bandwidth and operational gains of -1 and $10 is the Krohn-Hite Corp. (580 Massachusetts Ave., Cambridge 39, Mass.) Model DCA-10. Pulse currents of 1300 ma., d.c. currents of k140 ma., and output voltages to *140 volts are available. A 50-watt model of reduced bandwidth is also available from the same company. Solid state d.c. amplifiers with power outputs to 2 kw. are available ( I S ) having a low frequency response up to the several kilocycle range. Commercial power supplies can also be used as the output stage. This approach has the advantage of high current output, and has been effected (8) with a 60-volt, 15-ampere, 0.1millisecond rise time transistorized power supply (Electronic Measurements Co., Regatran Model TO 60-15). Tests in our laboratory show that the Power Designs Model 4005, 1-40 volts, 0-500 ma. transistor power supply is an effective power booster, with a rise time of about 10 microseconds in a unity gain configuration. A water-cooled, 1-kw., series-parallel emitter-follower circuit was built in the Authors’ laboratory to supply 10 amperes at 0 to 100 volts while driven by a small 1-watt output operational amplifier. Efficient forced air transistor coolers now available would remove the inconvenience of the water connections. These high power output units enable controlled-potential electrolysis of gram amounts of electroactive species and make possible removal of large amounts of an interfering species before analysis for a constituent which is often present as a minor component. Magnetic amplifiers and silicon-controlled rectifier circuits should be considered for high reliability instruments, in view of the bandwidth requirements of less than 1 C.P.S. for accurate controlled-potential coulometry with low noise stirring and circuit configurations having the load resistor voltage out of the input circuit. Increased reliability of polarographic potentiostat circuits
would be expected through the use of magnetic amplifiers. KO special difficulty would be expec Led in adapting commercial magnetic amplifier circuits to d.c. and low frequency polarographic applications. ,111 tubes and transistors could be eliminated w,th perhaps the exception of one impedance matching device a t the input from the reference electrode. Potentiostat Amplifier System. A large number of circuli configurations can be used for potentiostatic measurements, Potentiostat circuits using amplifiers interconnected in many different ways have aFpeared in the literature (11). The three-amplifier system given below is ii practical compromise for measurenients down to about 50 microseconds giving considerable convenience and versatility and using three-terminal amplifier8 exclusively. Several simple potentiostat circuits can be effected with three-terminal amplifiers when floating power supplies are used. Figure 13A Ehoms a follower amplifier used alone to effect potentiostatic control, This is useful where a minimum circuit is desired. It is interesting in that it shows a high gain operational amplifier can be considered as a high quality, unity gain, cathode follower merely by chaniing the reader's viewpoint. Figure 13B 5howi a simple modification of this ciIcuit to provide
A.
Figure 14. networks
Three-amplifier potentiostat with booster, dummy cell, and stabilizing
voltage offset and bcan to a low current load. The two operational amplifiers have a common power supply, isolated from ground. With these two circuits the load resistor has ground reference, making cell current measurement or integration possible rTith the usual grounded equipment. Two possibilities exist for using a power amplifier with the basic circuit of Figure 13A. The most direct method is to place the booster amplifier in series with the voltage amplifier. This approach has been used with the coulometer circuits shown in Part IV, but is not always feasible, as floating a physically large power supply gives considerable output capacitive loading which limits the possible high frequency response. A second method places the input of the power amplifier between the preamplifier output and ground. The stability analysis of this latter configuration is interesting, as differences of transfer functions are involved. A stabilizing network is required between the two amplifiers. By using a power amplifier gain of 1 1, the loop gain is increased by 10 times, allowing use of an inexpensive 10-volt output transistor amplifier as the preamplifier with a 100-volt capability across the cell and load resistor.
+
8.
Figure 13. A. E.
Grounded output potentiostat
Simple one-amplifier potentiostat Simple potentiastat with voltage scat
I
The circuit of Figure 13A has been successfully used in a number of coulometers as shown in Part IV. The circuit of Figure 13B is being constructed with transistor amplifiers for use as a d.c. polarograph. A plus one amplifier will be used to match the load resistor impedance to the recorder system and suitable offset, sweep potential, and switching circuits are being added. A High-Performance Potentiostat for Polarographic Application. The potentiostat circuit shown in Figure 14 is well suited for a variety of controlled-potential polarographic applications, especially those involving pulsed, square-wave, sine-wave, or fast-sweep methods. For high sensitivity d.c., DME polarography, the power amplifier is not required and the load resistor and compensator input resistors can be removed. Features of this rather simple design include the use of only one power amplifier, summing inputs having ground reference-alloBhg the desired control voltage functions to be easily inserted-and provision for compensation of the effective referencemeasuring electrode resistance by the positive feedback loop. Only three chopper-stabilized operational amplifiers are required. A 90% rise time of 8 microseconds is obtained across a 1-pf. plus 1-ohm dummy measuring electrode, with a 20,000-ohm reference electrode resistance, a 50-ohm series cell resistance, and a 10-ohm load resistor, Circuit Details. The circuit configuration of this controller, using three Philbrick USA-3M.13 operational amplifiers and a Krohn-Hite DCA-10 power amplifier, is shown in Figure 14. The 10, 10K, 50K, and lOOK ohm resistors are General Radio Type 500-B, 500-5, 500-T, and 500-U, respectively, being *o.05y0 wire wound and especially designed for high-frequency use. The variable capacitors are small ceramic trimmers. Amplifier 1 is connected as a grounded-output follower as described above (see Figure 11). D.c. offset, sweep, pulse, or other waveforms are summed with the output of VOL 35, NO. 12, NOVEMBER 1963
1803
ance with the unregulated supply was entirely adequate. Unless the USA-3M3 amplifiers are mounted in a position well separated from other units and where room air can freely circulate, forced air cooling should be used. .
.
Figure 15. Pulse response of potentiostat circuit of Figure 14 Upper:
Response to 1.5-volt pulse across dummy measuring electrode Lower; Response ocra~s1 0-ohm load resistor, showing f 4 0 0 - m a . peak charging current Horizontal scale i5 10 fiseconds per division in both figures
the follower. The booster, B, following USA-3M3 amplifier 2 is the Krohn-Hite power amplifier used in the +10 gain position. The C, R, and M on the dummy cell terminals following the booster refer to the counter, reference, and measuring electrodes. The load resistor, RL, is 10 ohms. Bmplifier 3, also a Philbrick USA-3M3, is connected as a unity inverter, effecting load resistance compensation. To compensate also the measuring-to-reference electrode probe tip effective resistance, shown as 1.0 ohm in Figure 14, either the inverter gain could be made greater than unity, or the ratio of the resistors between the inverter and follower amplifier outputs and amplifier 2 input could be changed. Measurements of the performance of this system were made with d.c. power supplied to amplifiers 2 and S from an unregulated, RC filtered ==! 300 volt supply. Perform-
I I Kc
However, increasing the series cell resistance increases the cell attenuation and reduces the system bandwidth. With a 500-ohm series cell resistance, the above system response is about four times slower, but can be adjusted to give a more nearly critical damped response than with the 50- or 10-ohm series resistance. Considerable simplification in potentiostats designed for pulse responses in the 0.02- to 50-microseconds (1% pulse droop) region and sine wave response in the 25-cycle to 30-mc. region (-3db. points) can be obtained by use of the current transformer probe technique. The cell load resistor and its compensating amplifier circuit then can be eliminated. Commercially available probes, such as the Tektronix P 6016 or the Hewlett-Packard AC-SlF, are essentially a transformer with a singleturn primary. The series impedance introduced in the external circuit is less than 0.05 ohm and 0.05 ph. per primary turn, It is also apparent that a summing network will reduce the available controller gain. As shown in the example, the external signals should be summed using resistor values large compared t o those in the control loop to conserve bandwidth. By eliminating the load resistor amplifier from this circuit, the loop gain will be essentially doubled. A commercial function generator is available (Exact Electronics, Inc., 445 South Second =Ive., Hillsboro, Ore., Type 250 function generator) giving precise square, triangular, sine, and ramp mveforms in the 0.001 to
Experimental pulse measurements on this system are demonstrated by an oscilloscope response (Figure 15) which shows a *400-ma. peak charging current for the l-pf. cell capacitance and the 8-microsecond rise time across the 1-pf. plus 1-ohm dummy measuring electrode. The 4.2-microsecond time constant exemplified by the pulse response corresponds to a unity gain crossover point of 38 kc., agreeing well with the Bode diagram (Figure 16). To further improve the response time for this circuit configuration requires the use of wider bandwidth amplifiers. Such amplifiers are now in the developmental stage a t several analog computer companies. It should be noted that in the circuit analysis given in Appendix I, the uncompensated resistance was very small due to the close placement of the reference electrode probe tip to the measuring electrode surface. When this distance cannot be precisely controlled, the resulting time constant, or the accuracy of the positive feedback loop in compensating this resistive voltage drop, may be the limiting factor on the system response time. With the particular amplifiers used, no further improvement in system response time would result from using a smaller value of series cell resisbncee.g., 10 ohms in place of the 50 ohms.
ON
OFF
E --i1 7
e----
I
I
lOKc
IOOKC
I Mc
LOG FREQUENCY
Figure 16. Bode diagram for open-loop response of polarograph potentiostat circuit of Figure 14 A. 6. C.
1804
System response without sfobilking network USA-3M3 response with stabilizing network System response with stabilizing network
ANALYTICAL CHEMISTRY
Figure 17.
Controlled-potential coulometer
Figure I t3.
10,000 C.P.S. region which make the construction of such signal input circuits unnecessary. PART IN.
COULOMETER APPLICATIONS OF POTENTIOSTATS
Coulometer Design. .The simple two-amplifier coulometers described below fulfilled the need for an instrument suitable for high precision, controlled-potential coulometry (5,6 , 2 3 ) . The grounded output configuration is used in the potentiastat sections. Through the use of high quality operational amplifiers, cool operation obtained by small fans and perforated panels, and solid state rectifiers, long life of the electronic components is to he expected. Maintenance, when required, is facilitated by immediate access to any component or wiring connection. By use of cutouts in the mounting panels, no components or terminals are hidden. Controls are a t a minimum, without sacrificing performance in any way. It is apparent that a two-amplifier, controlled-potential coulometer also could be built with the integrator section floating rather than the potentiostat. This is usually not desirable because for the integrator output voltage measurement an instrument such as a digital voltmeter, recorder, or any other voltage measuring device is used which often requires ground reference on one input to avoid loading and noise errors. However, floating the integrator may be preferable if a high-current, fast-response potentiostat is required as the capacitance to ground of the isolated power supply appears across the potentiostat output. This capacitance would be less with the low power requirements of the integrator than it would be for the large power supplies and amplifiers of the high speed potentiostat.
Side views of simple coulometer with doors removed
on the integrator allows precise integration to be accomplished without the special precautions to avoid groundloop common paths necessary when a single power supply is shared. The circuit components that were used gave ample isolation of the potentiostat. Small d.c. leakage to ground does not affect the circuit performance as this as well as the capacity appears across the amplifier output. When using the Krohn-Hite DCA-10 power amplifier as a booster, additional eapacity isolation was required to maintain stability of the power ampIifier internal feedback loop. The 250-pf. capacity of the internal DCA-10 power supply to ground was reduced to 40 pf. by placing an Elcor L-131, 35O-watt, power Isoformer in the 115-volt a s . line lead. Use of the Krohn-Hite amplifier as a booster stage allows up to 140 nm. of d.c. current to be furnished t o the ceIl in either direction and current peaks of over *300 ma. Three coulometer designs are presented using the same basic circuit to show the ease of obtaining a design to fit a given operational amplifier and output requirement. Circuit descriptions are given in outline only, because wiring details will vary with the particular components and with which options such as fixed or variable control potential are chosen. These circuits ean be used with many different types of operational amplifiers. By using only three-terminal amplifiers in the designs, a wide choice of inexpensive operational amplifier types results and no internal modifications of the amplifier unit are required. Controlled-Potential Coulometer. I. The circuit schematic for this instrument is shown in Figure 17. Figure 18
the instrument with the doors open. Every wiring connection, tube, and other part is accessible by opening the doors. The power supply, P in the diagram, for the single fixed measuringreference control potential is an Elcor Isoply Model AZ-10-150, 10-volt, 150ma. zener-regulated d.c. supply. The only controls are the cell OFF-ON switch and the integrator reset button. The operational amplifiers are Philbrick Type USA-4JT having aluminum circuit boards, and Teflon standoff construction. The lO-pf., 200-volt polystyrene integrating capacitor is a Film Capacitors, h e . , No. AZ-2-10M. For low leakage, a Microswitch was used for integrator reset. Four Trans Electronics, Inc. (Canoga Park, Cali.) Model RS-305, 300-volt, 50-ma. regulated d.c. power supplies furnished power for the operational amplifiers, two for the integrator and two for the potentiostat. Capacitance-to-ground of two of the Trans Electronics Model RS-305 power supplies is 250 pf. with the case floating and 4000 pf. with the case grounded. The load resistor, R,, is. a General Radio 2~0.05% precision wire wound 1000-ohm Type 500-H. Four Internationa1 Resistance Co. Type VW2J +l%, 1-meg., wire wound resistors are wired in series for the integrator input resistance. The 100-ohm current balancing resistor is a single turn, Ohmite Type AB. The other resistors are carbon composition. Ground loop problems were avoided by separately bringing all ground wires to a wiring lug on the amplifier chassis next to the chopper socket. The cabinet was oonnected to ground a t only one point. A later modification of this instrument was an added switch to change the control potential as required for proVOL. 35, NO. 12, NOVEMBER 1 9 6 3
* 1805
IO MFD
cedures using a prewave electrolysis.
An additional resistor in series across the low voltage power supply gave the second fixed source potential. Placing the control-potential supply in the reference electrode lead has also been done. The 60-C.P.S. a.c. modulation introduced in the control-potential is 5 mv. for each 10,000 ohms of reference electrode resistance, using one of the shielded Elcor transformer supplies with 0.1 pf. interwinding capacitance. With the measured resistance value of 3720 ohms for the fiber-tip junction, calomel electrode, a negligible 1.35 mv. results, Without using a power booster, the USA-4J amplifier can supply over 25 ma. of reduction current to a 500-ohm cell in the circuit described. This is ample current for many methods. Limiting the cell current with series resistance in the cathode lead and a control point chosen close to the beginning of the current-voltage plateau allows samples containing up to 100 microequivalents to be quantitatively reduced in less than 20 minutes. Performance of Simple Coulometer : Coulometric Determination of Uranium(V1). The data obtained for the controlled-potential coulometric determination of uranium with this simple two-amplifier coulometer indicate that precision is as good as that obtained with more complex instruments. The relative standard deviation, based on 14 analyses a t an 8.4microequivalent level, was =k0.05%, and for six analyses a t a 43-microequivalent level was rtO.O18%. Analysis conditions were a prereduction with a watersaturated helium sweep over the solution a t 4-0.085 volt us. the saturated calomel electrode for 20 minutes in a 0.5M sulfuric acid electrolyte followed by reduction of uranium(V1) to (IV) a t -0.45 volt for 20 minutes. Blank and background corrections were made. Solution purging was not used. Controlled-Potential Coulometer. 11. A coulometer very similar t o that described above, but using two Philbrick UPA-2 and one Krohn-Hite DCA-10 power amplifiers has been used in developing controlled-potential coulometric methods. All amplifier interconnections can be made with banana plugs with these models. Figure 19 shows the wiring connections and phase correction networks used in the potentiostat section. The integrator was wired as in (I) above. Because of the particular networks used to match the cell-amplifier characteristics, the switching could not be done as in Figure 17 but required the lead connected to the reference electrode during operation to be connected to the UPA-2 amplifier output when the switch is in the off position. An additional voltage divider could be used if switching across both amplifiers mas desired. 1806
ANALYTICAL CHEMISTRY
c%
4-30 Pf
M-45 pf
800 pf M Figure 19. Coulometer potentiostat using UPA-2 and DCA- 10 amplifiers The Krohn-Hite DCX-10 d.c. to 1-mc., 10-watt power amplifier allowed d.c. currents of up to *140 ma. and pulse currents of 1300 ma. to be supplied to the cell. This unit has a selfcontained power supply. To obtain slightly improved system response, the amplifier was used in the +10 gain position. A.c. line power was furnished through an Elcor Model L-131 low capacity isolation transformer to prevent Z output loading. The variable capacitors shown in the wiring diagram are small ceramic trimmers. Controlled-Potential Coulometer. 111. A two-amplifier coulometer circuit similar to Coulometer I above was constructed using Philbrick USA-3 printed circuit cards mounted directly in cutouts in a 12l/4 X 19 inch relay rack panel. All other components including switches, potentiometers, and connectors were also mounted on this panel. Elcor type AG300-20 VR-tube regulated power supplies were used for both the potentiostat and integrator giving 50 pf. total shunt capacity from the potentiostat power supply to ground. With a 1000-ohm load, the potentiostat can furnish 10 ma., using a 40,000-ohm current balancing resistor in the amplifier output stage. Terminals are furnished for series power amplifier if higher currents are required. Two offset-potential circuits, using an Elcor AS1240 zener-regulated power supply, are provided allowing direct switching between the pre-electrolysis and measuring steps. This instrument could be considered as a minimum cost, high precision coulometer, with the expense for components being less than $500 and operational tests giving the same high precision as that obtained with the other designs. Integrator Considerations for High Precision. Even though the operational amplifier resistance-capacitance integrator has been shown to be a simple, inexpensive method capable of
*
better than 0.01% precision in both the analog computing field and in application to electrochemical cell current integration, many workers have suggested other approaches. In evaluating these various methods, it is of interest to note that the voltage-tofrequency converter plus digital counter technique (1, $9) uses an RC operational amplifier as the voltage to frequency conversion element, thereby limiting its precision to less than the direct analog method capabilities, a t the same time adding complexity. Some workers have not used the RC operational amplifier method because they felt it is a high impedance method. This misconception is understandable as the amount of current going into the integrator is small; a 10-fif. capacitor charged t o 100 volts contains only 1 mC of charge. However, to obtain this small current, current dividers having small values of cell series resistance are required as evidenced from published circuits using this type of integrator. Further, in the low current application ( 7 ) , if an input resistance is not used, the d.c. input resistance is essentially zero, being the dynamic input summing point impedance of the operational amplifier. Two recent developments in integrators are of considerable interest to the electroanalytical chemist. For applications where a precision of about 0.1 t o 1% is satisfactory, the electrolytic mercury column integrator is a highly reliable, simple device. The integrated current value is obtained by the change in position of the mercury meniscus caused by the electrochemical transfer of mercury across a small length of electrolyte in a small, precision-bore tube. This low-noise, no-tube, notransistor device (Curtis Instruments, Inc., ,751 Lexington ilve., Mount Kisco, S. Y.) is available in various sizes applicable for much of the analytical coulometric range.
The other integrat'or is a military design of high precision for very low drift application in guidance systems ($7). I t uses analog-to-digital voltage conversion followed b:y a digital integrator (adder), ther2by eliminating drift and obtaining tk,e full accuracy capabilities of the digital voltmeter, crystal controlled timar combination. Precision of 0.00170 should be possible, approaching the limitation of voltage measurement. In slow systems, an adding machine could be used as the digital integrator. A filter could be uied to smooth the input signal. To obtain highly :murate, preci.qc integration by the RC operational amplifier technique, drift and temperature effects must be considered. A high quality polystyrex capacitor will drift about 0.0170 in 3 minutes when charged t o 100 volts because of the resistive limitation 0:' the dielectric. The temperature coeficient of polystyrene capacitors is about -0.1% per O C., requiring teniperature control to 1 5 " C. for 0.1% integration precision. The offset potential a t the amplifier input also affects .the drift rate by supplying a current to the integrating capacitor equal to tke offsct voltage divided by the intepator input resistance. For exampk, a 2-pv. offset across a 1-meg. inpuj, resistance produces 2 ppa., a drift of 0.017, per hour for a 10-pf. capacitor. Increasing the amplifier input reaizkance decreases the leakage current. The following provides a calculation of the minimum resistance. The minimum load reaistance equals the product of the operational amplifier equivalent input bias and the analysis time divided by the product of total sample coulombs and the required pr3cision. As an example, for 20-pv. amplifier offset, 30-minute integration Lime, 6 coulombs, and a precision of O.Ol%, the minimum load resistance is 60 oh:ns. Fortunately the requirements for high precision of a large series load resistor and a high integrator input resistor arc compatible. Of course, little improvement in drift characteristic is gained by using an integrator input resistence much larger than the amplifier input resistance. For this reason, a 4-meg. resistance is used in the circuit shown in Figure 5 with the 1-meg. operational amplifier input resistance. I n evaluating operational amplifiers for use as RC integrators, the input resistance, offset, and d.c. gain are of primary importance and do not meet coc.lometric requirements in many cases, especially in many of the small sdid state amplifiers. Several very good solid-state and vacuum tube units are available. Polystyrene is presently the best dielectric material for integrating apacitors. The man ifacturers of precision polystyrene capacitors have ob-
tained dielectric absorptions less than 0.050/0, insulation resistances greater than 5 X 1OI2 ohms-microfarads, retraces less than 0.02% on temperature cycling, and long term stabilities better than 0.2% in a year. With the load resistor and integrator input resistance considerations given above, and because offset leakage currents are independent of capacitor size, a large capacitor size is indicated. Because of the expense and volume of large polystyrene capacitors, values of 1 to 10 pf. are a reasonable compromise, still allowing 0.0170 precision t o be obtained. Frequency dependent errors of operational amplifier circuitry can be considered in terms of the basic amplifier and feedback network parameters (25). For an integrator, the virtual gain will be -t/(RC), where t is the integrating or pulse input interval. To integrate a 10-microsecond pulse with about 1% accuracy would require an amplifier with a bandwidth of approximately 3 mc. and the integrating time constant chosen to give a virtual gain of unity or less. For this particular example, the formula gives an integrator RC time constant of 10 microseconds minimum. APPENDIX I. POTENTIOSTAT CONTROL SYSTEM ANALYSIS
To obtain a properly damped, fast response time with potentiostat circuits, the design of stabilizing networks as influenced by the electrochemical cell of interest must be considered carefully. The electrochemical cell and controller transfer functions are required. The cell transfer function can be obtained directly by measurement or can be calculated if some of the experimental parameters are known. For calculations pertaining to spherical mercury drop electrodes, the equation giving the voltage drop between two concentric spheres is useful (3) :
where Vir is the resistjive voltage drop caused by a current density, io,flowing through a solution of conductivity, k, with radius, r, of the inner sphere, and u, the separation between spheres. For the total cell resistance, the separation distance can be assumed to be much larger than the microelectrode radius, giving an expression for the cell resistance, R:
with reasonable values for the radius O f a DhfE and conductivit,y O f an acid electrolyte, a cell resistance of about 20 ohms is calculated. Equation 1 is useful for calculating the effective, uncompensated reference electrode tir, to measuring electrode surface resistance for a given electrode-reference probe
separation. This obtained expression 1s :
R=
U
47rTk(T
+ u)
(3)
For a separation distance of 0.1 times the drop radius, this simplifies to: 0.007
R=-&-
(4)
Again assuming an acid electrolyte, k = 0.1 (ohm-cm.)-l, and an average DME size ( T = 0.05 cm.), the calculated effective resistance value is about 1.4 ohms. (For maximum conductivity sulfuric acid, k = 0.83 (ohm-cm.)-l, giving a significant improvement.) With a double layer capacity, C, of 20 pf. per sq. cm., the capacitance of a 0.5cm. radius electrode is about 0.6 pf. It should be emphasized that a perfect controller cannot produce a rise time better than this time constant mithout special shaping networks or positive feedback of a signal proportional to the cell current. The best time constant for the system just calculated would be 1.4 X 0.6 = 0.84 microsecond. As a reasonable approximation to these calculations, the dummy cell values shown in Figure 14 were chosen. K i t h these values and for the control circuit shown, expressions can be written for the voltage transfer functions from the pon-er amplifier output to the inputs of the unity gain noninverting follower amplifier and the unity gain inverting amplifier. These complex number transfer functions can be denoted as A and B, respectively, and one can solve for the difference signal ( A - B ) appearing a t the control amplifier input:
B =
IOCS
1
+ ( R + 60) CS
(6)
The symbol S has the usual significance as the differential operator d/dt, and is equivalent to j 2 ~ ffor steady-state signals; with j denoting the vector direction and f the frequency. The transfer function for the follower amplifier can be simplified by representing the USA-3 transfer function by K / S which it is in the high frequency region of interest. The follower transfer function, TF, then follows as:
By similar algebraic manipulation, the transfer function for the unity gain inverter is:
Note the factor of two reduction in bandwidth due to the feedback network of the inverter. The transferfunction of the Bohn-Hite DCA-10 power amplifier can be expressed B s :
VOL. 35, NO. 12, NOVEMBER 1963
1807
0.3
I 60 72
,
aoa IOOCps
I KC
LOG
10 Kc FREQUENCY
6. C.
D.
I
I
I
I
150 200 300 500 LOG FREQUENCY cps.
-
I
1000
\\ '
2000
Figure 2 1. Bode diagram for the closed-loop response coulometer potentiostat of Figure 19
of
i
I Mc
100 KC
Figure 20. Bode diagram for open-loop response coulometer potentiostat of Figure 19. A.
I
100
Cell attenuation UPA-2 response System response without stabilizing networks System response with two stabilizing networks
of
Ln
t
J
0
>
The individual transfer functions may now be combined to obtain the net open-loop transfer function. This gives: 1'F (net open-loop) = TIME Figure 22. Transient response showing closed-loop sponse of coulometer potentiostatof Figure 19
re-
Each horizontal division i s 0.5 msecond; each vertical division is 20 mv.
Simplifying this expression by multiplying and removing insignificant terms gives a close approximation to the actual transfer function which may be written as :
is the equivalent amplifier input resistance. Substitution of this expression into the loop-transfer function equation and simplifying gives: TF =
The second and third terms appearing in the denominator can be removed as they differ from unity at frequencies higher than those of interest. There results:
The first term is now clearly dependent only on the controller, the second only on the cell parameters. These terms are plotted individually and combined on the Bode diagram shown in Figure 16. It is apparent the system without gain-phase shaping is close to instability. An operational feedback network has been added to give the required gainphase relationship without reducing the system output voltage capability as would result from a series insertion of correction networks. The transfer function of the control amplifier after adding the stabilizing network is: TF
=
-(I
+ RlClS)
RzCiS
(14)
Where R1 and C1 are in the series network added across the amplifier and Rz 1808
ANALYTICAL CHEMISTRY
( l + RlCIS) 2000 CIS(1 60 C S )
+
(15)
I t is clear that first order, critically damped response is obtainable by making the time constant equal or larger than the time constant 60C. I t also is apparent that the system unity gain frequency is dependent primarily on R1 and not C1. This fact allow easy experimental determination of optimum values for R1 and C1 after approximate values have been obtained. That is, the square wave response across the 1-ohm plus I-pf. dummy measuring electrode is observed while varying C1 to give optimum square wave response for each chosen value of R,. To ensure stability of the inner feedback loops, the capacitors shown across the amplifiers are chosen with the criterion of minimum value to suppress internal oscillations and a maximum value less than that which will impair the system square-wave response. APPENDIX II. COULOMETER CONTROL SYSTEM ANALYSIS
Application of the Bode diagram method to a controlled-potential coulometer potentiostat circuit is illustrated by Figure 20. In this example, the system
is composed of three series elements; the electrochemical cell, a high gain operational amplifier n ith a constant 90' phase shift characteristic, and a flat d.c. to 1 mc. response power amplifier used in the +10 gain position. Because of the rising characteristic of the cell attenuation curve. a simple RC filter was placed immediately preceding the poner amplifier to produce an attenuation of 20 db. per decade with a breakpoint of 10 kc. per second. The circuit diagram, Figure 19, also h h o n ~the network used to flatten the gain characof the high gain operational er in the frequency region where the net open-loop gain is reduced to unity. The closed-loop gain for this system is shown in Figure 21. The steady-state gain peak occurs at 500 c.p.s. with a value of 1.36. Transient response of this potentiostat-cell system, obtained from oscilloscope photos, is 1 millisecond to obtain a peak response corresponding to 18% orershoot (Figure 22). Settling time to be within 5% of the final value is 17 milliseconds. Reducing the power amplifier gain to 1 and optimizing the shaping networks gives an overshoot of 29% nith response times equal to those above. 9 more damped respon-e could be obtained by using less bandwidth. From the behavior of first order systems, the phase shift in the region where the net open-loop gain is unity must be 90" to obtain critical damping.
LITERATURE CITED
( 1 ) Bard, A. J., Solon E., ANAL.C H E x 34, 1181 (1962). ( 2 ) Barker, G. C., “Transactions of thc
Symposiuni on Ele8:trode Processes,” E. Yeager, ed., pp. 336-44, Wiley, New York! 1961. ( 3 ) Barnartt, S., J . Ekctrochem. Soc. 108, 102 (1961). (2) Bewick, A , , BewicE., A , , Fleischmann, M.,Liler, M., Elect,.ochim. ilcta 1, 53 (1959). (5) Booman, G . Holbrook, W. R., A X A L . CHEN. 31, 10 (19591. ( 6 ) Boonian, G. L., Holbrook, W. B., Rein, J. E., Ibid., 29, 219 (1957). ( 7 ) Booman, (2. L , Morgan, E., Crittenden, A. I,, J . .-Inz. [‘hem. SOC.78, 5533 (1956). (S) Brown, 11. H., U.S. Atomic Energy Conini. Rcpt. IDO-16852, hIay 1963. (9) Chestnut, H I h l a l e r , R. JT ., “Servo-
mechanisms and Iiegulating Systeni Ilesign,” 1-01, 1, 2nll ed., n-iley, Sew York, 1959.
(10) Cockbaine, 11. H . , Atoniic Energy Research Establishment (Gt. Brit.), AERE-EL/R-1528, ?vIay 5 , 1955. ( 1 1 ) DeFord, 11. D. 133rd Meeting,
ACS, San Francisco, Calif., April 15, 1958; and unpublished research. (12) Delahay, P., “New Instrumental hfethods in Electrochemistry,” pp. 391-3, Interscience, New York, 1954. (13) Dynatron Electronirs Corp., 178 Herricks Road, Mineola, K. Y., Bitilctin 126, 1962. (14) Geo. A. Philbrick Rese:trclies, Inc., 127 Clarendon St., Boston 16, Mass., “The Lightning Empiricist,” p. 3, Issue No. 6, October 1958. (15) Ibid., “GAP/R .4pplivation Brief,” I). 1, Ko. D2, April 1, 1960. (16) Ibid., S;. 6140’ AB-261-1, p. 3 , Feb. 1, 1961. (17) Gerischer, H., Staubach, IC., Elektmchern. 61, 789 (1957). (18) ,Grabbe, E. AI., Rarno, S., TVooldndge, D. E., eds., “Handbook of Automation. ComDiitation and Control,” Vol. 1, “Conhi1 Fundnntentals,” Wiley, S e w l-ork, 1958. 1191 Hager. C. I C . Elctlronzcs 32. 44 ‘ (KO.33, Septeniber 4, 1959). (20) Harrar, J. E., University of California, Lawrence Radiation Laboratory, Livermore, Calif., private communication, 1962. (21) Kelley, 11. T., ,Jones, H. C., Fislier, I). J., ASAL. CHEM.31, 485, 956 (1959).
( 2 2 ) Kuo, S. C., “Automatic Control Systems,” Prentice-Hall, Englewood Cliffs, N. J., 1962. ( 2 3 ) Shults, \Y. D., Dunlap, L. B., ASAL. CHEM.35,921 (1963). (24) Spolin, P., Hedett-Packnrd Journcd 14, KO. ,556, pp. 5-8, Jan.-Feb., 1963. (2.5) Tekt,ronis, Inc., Reaverton, Oregon,
“Introduction to Operational Ampli-
fiew,” Service Scope S o . 19,.~ iluril 196:i (26) T‘alley, G. E,, Jr., Wallninn, Ii;! eds., “J’acuuin Tube Amplifiers. Massachusetts Institute of Teclinolopy Radiation IAoratory Series, T:ol. 18, LLIcGraw-Hill, S e w York, 1948. ( 2 T ) JTeber, .T. H., Spnce/Aeronaulics, p. 134. Snvemher 1058. (28) IVill, F. G., Z. Elektrochent. 63, 184 (19.50). (29) Wise, E. S . , L l s a ~CHEN. . 34, l l S l
(1962).
( 3 0 ) Zittel, H. IC., I)unlap, L. 13., Ibid.,
35, 125 (1963).
RECEIVED for review I1eceniber 26, l%2.
Accepted Akugust 29, 1963. Division of Analytical Chemistry, 144th >Ieeting, ACS, I m Angeles, Calif., April 1963. Work supported by U.S.Atomic; Energy Commission under contract no. AT( 10-1)205 through Idaho Operations Ofice.
A Digital Readout Device for Analog integrators E. CLIFFORD TOREN, Jr. and CHARLES P. DRISCOLL Department of Chemktry, Duke University, Durham,
b A circuit, capable of direct digital readout of analog integrators within a relative error and relative standard deviation of .tO.lc%, uses an operational amplifier to reset the integrator a t preset voltages. The unit may b e programmed to read equivalents directly. h A L O G I P U ’ T C G R A I ~ O Rl ~i a ~ e
iecently found extensir iipplication in analytical instrumentation--e.g., as c~~ulonieters.Relatiw standard deviations of = t 0 . 0 5 ~ ofor integration of electrolysis curieiits have been reported ( 1 ) . -1 readout device as accurate as the integrator is necessaiy to achieve the maximum capabilities of the integrator. Rooman ( 1 ) has u-ed manual and recording potentioineters. Kelley, Jones, and Fisher (4, 5 ) have further utilized digital voltmeters. I n thi. work a Ion cost, simple, accurate, and direct readout device is developed to avoid th. inconvenience of manual potentiometers and the high cost of recorders and digital voltmeters. This curcuit is based cm the principles of an instrument origin tted by Higinbotham and Rankowita (S), but is extended to use commercially available operational amplifiers 3
CIRCUIT DESCRIPTION AND OPERATION
Figure 1illustrates the complete circuit for the digital readout unit and block
N. C.
diagrams of aisociated modules necessary for circuit operation. The Integrator module is a conventional circuit used by DeFord ( 2 ) , but modified to switch to various values and types of capacitor, C. The integrators deslcribed by Booman (1) and by Kelley, Jones, and Fisher (4, 5 ) could also be used esentially without modification. The Voltmeter module used by DeFord ( 2 ) has a range of 10 mv. to 10 volts fullscale which ii variable in steps by means of a panel snitch. d potentiometer in parallel with the ha+ IO-mv. movement and external .hunt is wed to perform minor calibration adjustments of *lo% of the iiominal fullscale setting of all ranges. The Discriminator voltage is obtained from the Secondary Standard Voltage module designed by DeFord ( 2 ) . This unit, consisting of a Dekapot (Electro/ Scientific Industries, iLIodel DP-211) in series with O.lyoprecision resi>tors, divides the +300-volt calibrated power supply output (George A. Philbrick Researches, Inc., &Iode1 R-100B) with a linearity of 0.1%. Operational amplifier, OA, without feedback is a voltage crossing detector ( 7 ) used as the sensing element in the digital readout unit. 08 is balanced in the Zero position as shown in Figure 1 by adjusting the Discriminator voltage input such that zero voltage is obtained at pin 6. By applying E , volts, in addition to the bias 1 oltage necessary to balance O A , the output of 0-1 rapidly jumps from -50 to +50 .iolts for a K2-IT7 amplifier or from -100 to
+lo0 volt.. for a C/lOO/B, when the Integrator output attains a value of E , 1 0 003 volt. This jump is sufficient to bring T’, normally held below cutoff, into saturation to activate the relay. Ppoii closure of the relay, C is discharged through R4 to avoid burning the relay contact, and the mechanical register is advanced 1 count. The fraction of the last count is read on the x oltmeter, the fullscale value of rshich is adju-ted to E , volts. A convenient value for coulometric applications i* 9.65 volt> for direct readout in eyuivalcnt-1snitch, SIV> is provided for rmcr+ ing the input terminals of Orl when the polarity of the Integrator output is changed, and for balancing O24. The switch positions +Ill-T and --I?VT refer to the polarity of the signal applied to the Integrator input. voltage divider network of R2 and Ra is used to ensure that VI remains cutoff during balancing or warmup time. CI is placed in parallel with VI to hold the relay clobed for sufficient time to activate the mechanical register. CI need not exceed 1 pf. a. determined by experiment. Thi. provides time of ea. 5 milliseconds to activate the couiiter and to discharge C almo.;t completely-Le., five or more time constants, R4 X C seconds--since C has a maximum value of 10 pf. The abqolute performance of the circuit i- checked wing the high quality, polystyrene dielectric, integrating capacitor, and the digital readout circuit The capacitors are atljuted in the InteVOL. 35, NO. 12, NOVEMBER 1963
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