ACKNOWLEDGMENT
This work was supported in part by the National Science Foundation. J. W. Bixler is also indebted to the Ethyl Corp., E. I. du Pont de Nemours & Co., and the Procter and Gamble Co. for financial aid during the course of the studies.
LITERATURE CITED
(1) Bard, A. J., private communications, 1963. (2) Ph.D. thesis. IJniversitv ~, Bixler. J. W.. of Minnesota (1963). (3) Bruckenstein, s., Bixler, J. W., ANAL. CHEM.37, 786 (1965). (4) Bruckenstein, S.. Naeai. T.. Ibid.. 33, 1201 (i96ij. ’ (5) Lord, S. S., Jr., O’Neill, R. C., Rogers, L. B., Ibid., 24, 209 (1952). I
I
(6) Meyer, F. R., Ronge, G., 2. Angew. Chem. 52, 637 (1939). (7)Xisbet, A. R..Bard. A. J.. J . Electroanal. Chem. 6. 332 i1963). (8) Shain, I., ‘“Treatise ’on Analytical Chemistry,” I. M. Kolthoff and P. J. Elving, eds., Part I, Chap. 50, Interscience, New York, 1963. .
I
RECEIVED for review September 3, 1964. Accepted March 12, 1965.
Optimum Stabilization Networks for Potentiostats with . Application to a Polarograph Using I ransistor Operational Amp1ifiers . r
GLENN L. BOOMAN and WAYNE B. HOLBROOK Phillips Petroleum Co., Atomic Energy Division, ldaho Falls, ldaho
b The selection of impedance values in the cell-current-measuring circuit to obtain maximum bandwidth with particular control amplifier characteristics, is discussed and application is made to a polarograph with transistor operational amplifiers. Methods are described for determining the circuit values for the best possible response time with a wide stability margin and tabulated data are given for the transistor polarograph circuit. These optimum values ensure measurements and control to the desired accuracy and simplify the electrochemical cell design for use in many experimental techniques.
I
vacuum tube equipq e n t , transistor amplifiers greatly increase the circuit reliability, useful operating time, and signal-to-noise ratio of many measurement circuits. The replacement of vacuum-tube operational amplifiers in polarographic potentiostat circuits by their transistor counterparts has been accomplished with no problems. Many circuits are used to obtain potential control of a 3-electrode cell ( 2 , 3, 5, 8 ) . The selection of the transistor circuit, described in this paper, was weighted towards use as a versatile research instrument rather than for a specific routine use (8). A follower amplifier with high-input impedance was selected to minimize the I R drop when used with high-resistance reference electrodes and to prevent the polarization of low-tesistance, micro reference electrodes. The necessary bandwidth shaping is then easily accomplished operationally in the highgain stage. Mixing of offset, sweep, and N CONTRAST T O
various function generator signals is readily obtained a t the high-gain amplifier input, with the external signals referred to ground level. The unity-inverter, compensating circuit for measuring the cell current was chosen because only one booster amplifier is required for fast pulse or controlled - potential coulometric applications. For measurements in which a 0.1- to 10-megohm load resistor is desired, or in cases where boosters are not required the circuit can be simply changed to the “current-follower” mode (5). The methods of obtaining stability in closed-loop systems containing an electrochemical cell have been discussed previously (2). However, the selection of optimum impedance values in the cell-current-measuring circuit, as influenced by the available control amplifier characteristics, has not been discussed explicitly in the literature. d basic understanding of how the cell geometry and electrical parameters affect the available range and accuracy of electrochemical measurements is needed by the worker making these measurements. These interactions are discussed from the servomechanism viewpoint, using a very simple, but extremely useful and practical approach, well substantiated experimentally. The chemist must have sufficient knowledge of what parameters under his control are important to the control system performance to be able to evaluate and communicate the special problems that occur in terms the electronic specialist can understand. Optimum is defined here as being the fastest possible risetime obtainable across a certain double layer capacity with a given total cell resistance, and
given control amplifier characteristics. This optimum performance is to be obtained by the addition of passive networks and through adjustment of the reference electrode position in the electrochemical cell. Because the presence of faradaic currents reduces the response time of the control system, this “optimumJ’ performance will occur when the faradaic contributions approach very small values. This condition exists out of necessity in many trace analysis applications. I n kinetic studies, working a t low concentration levels decreases the signal-to-noise ratio. Hence the “optimum” conditions described in this paper should be considered as a limiting case-Le., the best attainable response time as the faradaic current is reduced toward zero. The effect of faradaic current on the response time is included in this discussion. A detailed discussion of the design and construction of the polarograph using transistor operational amplifiers, lowvoltage power supplies, and a controlled-potential plug-in unit is given below. The calculation of the stabilizing network components for the potentiostat and the determination of the optimum value for the reference to measuring electrode resistance is discussed explicitly. Knowing this optimum value can simplify the cell design for many measurements. Placement of the Luggin capillary probe close to the measuring electrode may not be required or even be desirable. If the proper uncompensated resistance can be obtained with the reference probe located away from the measuring electrode, the location of the probe becomes less critical, more robust electrodes can be used, and special VOL. 37, NO. 7, JUNE 1965
795
designs or shielding become unnecessary. Optimum values for the stahilising networks and the reference to measuring electrode resistances for voltage compensating and current follower modes are tabulated along with instructions for their use. Electrochemists who desire to design similar instruments for research or routine use will find the discussion of the calculations and procedures adequate for application to amplifiers having characteristics that do not coincide with those used for the determination of the bahulated data. Merely by knowing the conductivity of the electrolyte and the radius of a spherical micro electrode, the best possible response time can he immediately obtained for control amplifier systems similar to those used in the calculation of the tabulated values. Further, the position of the reference electrode for this optimum response if the time is also obtained-i.e., optimum uncompensated resistance is more than the total cell resistance, then the positioning is noncritical and a position well removed from the measuring electrode is indicated. However, if t,he total cell resistance is more than the optimum value of uncompensated cell resistance, the reference electrode positioning becomes critical. A very important result of these considerations is that zero uncompensated resistance, even if attainable, is never desirable in fast response systems. At high frequencies, the double-layer capacity effectively shorts out the control system, making it ineffective. These considerations, along with the faradaic current considerations are discussed.
the right and left compartments of a DYMEC, 8048-0029, combining case as shown in Figure 1. The groundedoutput, voltage follower is maintained as a separate instrument outside of the combining case, which permits short connections to the high-impedance reference electrode. Controlled-Potential Plug-In Unit. T h e schematic circuit for the polarograph is shown in Figure 2 . All the circuit components with the exception of the electrochemical cell and the amplifiers, are contained in the controlled-potential plug-in unit, Figure 3. All the operating controls are
located on the front panel of the plugin unit. The power supplies, P.S.-1 and P.S.-2, are inexpensive Isoply, Model AS12-40, units made by Elcor, Inc. (Falls Church, Va). The precision resistors used in the plug-in unit are General Radio Co., Type 500. The polarograph circuit component values are as follows: A-1. voltage follower amplifier; A-2, A-3, A-4, A-5, 3-terminal amplifiers; C,, hypass impedance; C,, C,, stabilizing network; C,, 0.25 pfd.; C,, 10 pfd.; D,, D?, IN1522A, Zener diodes; R, currentlimiting impedance; R2,R,, Re, R,,,
DESIGN AND CONSTRUCTION OF POLAROGRAPH
Transistor Operational Amplifiers. T h e transistor operational amplifiers selected for the polarograph are Model DY-246OA amplifiers and Model DY-2461A- M4, + 1 Gain, plug-in units made by D Y M E C , a division of the Hewlett-Packard Co. (Palo Alto, Calif.). The power supplies for each amplifier are self cont,n.ined ...-.
The Model DY-2460A amplifier with the Model IIY-2461A- M4, +1 gain. plug-in unit gives a + I gain, noninverted, grounded-output, vokage follower. The polarogralih design requires four 3-terminal oiierational amplifiers. The required anrlilifier arraugement is obtained by a simlde modification of the Alodel IIY-2461A plug-in unit. For convenience the input and output terminals were made available on the front of the plug-in unit of t,he follower amplifier which is used in the reference electrode cirruit. The det,ails of thcse modifications are given in Appendis I. The four-&terminal operational amplifiers are conveniently housed in 796
ANALYTICAL CHEMISTRY
P.S.4
& 2V
P.9-2
Figure 2.
Polarograph circuit
X SCAN
tional resistance can be added externally, in the measuring electrode lead. The double-layer capacitance of the measuring electrode surface is represented by C,. The load resistance is renresent,ed hv R. The impedance values, R,, Rz,and C,, are experimentally measured quantities. The method of measnring these values has been discussed in another paper (2). Calculation of Stabilizing Network Components for Voltage Compensating Mode. I n this laboratory a Fortran-IV program for the IUM7040 computer was written and used to make the following calculations. T h e calculations are made for the circuit shown in Figure 2 and the cell impedance representation shown in Figure 4. T h e calculations are based on the Bode diagram method of establishing stabilizing networks discussed in a n earlier paper (2). For a signal entering the cell system, A and B are the cell transfer functions at R and M , respectively. The error signal is determined by the transfer function A-B. The transfer function, G,, is the nominal open-loop gain characteristic of the control amplifier, A,. The complex formulas for A , B , A-B, and G3are: ~~
Figure 3. Polarogroph plug-in unit
50,oOO ohms; R3, Ra, Rs, R,, 10,000 ohms; R , stabilizing networks; R,,, load resistor; R,2. 500,000 ohms; R l s , Rz,, 160 ohms; R,,> 1000 ohms, 10-turn pot.; R,,, R,,, 100,000 ohms; Rls, 200,000 ohms; Rls, 50,000 ohms, trimpot.; R L o ,10 - megohms; R,, 100 megohms; Rn, 10,000 ohms, 10-turn pot.; S,,St,Sa SPST switch; S,, Sa, DPST switch, Sa,SPDT switch. Amplifier A 4 is used as a n active filter, primarily for d.c. polarographic measurements. The plug-in unit is constructed to occupy the center compartment of the DYMEC, 8048-0029, combining case as shown in Figure I . All the interconnecting wiring, to the amplifiers, is through a pair of Amphenol Blue Ribbon, Ifipin, connectors. The plugin unit can easily he removed from the combining case and all the components are readily accessible for maintenance. The stahilizing network components, C,, Ca, and R,, are on a plug-in unit for rapid and easy component replacement. Signal interactions and noise pick-up are prevented by separation of critical wires, shielding, single-point grounding, and by the individual power supplies in each amplifier unit. The drift rate of the integrator is less than Zmv. per hour. The input impedance of the voltage follower is given as 10'0 ohms and 50 pf. by the manufacturer. Connections to the electrochemical cell are through General Radio Co., GR-874-PLT, coaxial connectors and General Radio Co., GR-874R22L, coaxial cables. All other outside connectors, such as recorder, signal generator, are Amphenol, Series 91, audio connectors. DESIGN OF STABILIZING NETWORKS
Cell Impedances. An approximate representation of the electrochemical cell impedances is shown in Figure 4, which is quite suitable for most cells helow 1 to 10 kc. and for specially designed cells to greater than 1 mc. T h e faradaic component is assumed t o he sufficiently small to have negligible effect on the system response. T h e sum of the resistance of the counter electrode separator and the solution resistance between the counter and reference electrodes is represented by R,. The sum of the resistance of the measuring electrode-e.g., a mercury capillary-and the solution resistance between the reference and measuring electrodes is represented by Rf. This resistance, R,, is referred to as the uncompensated cell resistance which is composed of inside-cell and outside-cell components. For cases where this resistance needs to be increased, addi-
C 9
~i
iR T c'
iR -
Figure 4. Cell impedance representation
(4) where: K = 5 X lo' = amplifier d.c. gain T , = 1/(2* X 0.0018) T, = 1 / ( 2 r X 13) Ta = 1/(2r X 144)
T , , T,, Ta are break frequency characteristics of the nominal open-loop gainbandwidth characteristic of the particular control amplifier chosen. The open-loop gain, C , of the system without the stabilizing network, Cz, C,, and R7 is:
The open-loop gain, C , is computed at IO frequency levels per decade. The levels are multiples of 1.0, 1.2, 1.5, 2.0, 2.5, 3.2, 4.0, 5.0, 6.4, and 8.0 starting at 1 cycle per second. The computation is continued to the next higher decade until the magnitude of C is less than or equal to 0.1. The frequency a t this magnitude is Fa. The frequencies F, and F, are:
F,
=
0.01 X Fs
F, = 0.1 X Fa
(6)
(7) T o ensure sufficient summing point
network resistor accuracy for 0.0570 precision resistors and to test for sufficient gain, the following tests are made. First, the difference signal is tested to ensure that it is greater than 0.1570, which is arbitrarily chosen as the lower limit of signal significance with 0.05Y0 resistors.
A-B - > 0.0015 A
(8)
Next, the difference signal is tested to ensure a gain of at least a factor of two for amplifier A , at the frequency F,.
I n these formulas ( A - B ) / A is computed at the frequency F,. The transfer function, C', is then computed as a log gain-log frequency function from F, to Fa. I t is computed at the same frequency levels per decade as mentioned above and is:
where C8 = magnitude of the open loop gain C , at Fa. The desired control amplifier transfer function, D, may now be computed as:
D
=
C'/(A-B)
(11)
This transfer function, D, is now classified as D,,D1,D3, ,D+, ,or Ds. These functions are shown in Figure 5. The classification is made according to the phase shift angle, $, as follows: VOL. 37. NO. 7, JUNE 1965
797
Ca
=
0
(17)
w
=
2~F3
(18)
K2
=
magnitude of Dz at Fz
The open-loop gain, COL, of the system with the computed stabilization network components is:
COL
=
D'(A-B)
K3 = l/(Cz X lo4)
Figure 6. Polarograph, current-follower mode
(19)
Where the transfer function, D', is computed for classification DZas:
where:
R
P
(21)
Determination of Optimum Value for Uncompensated Cell Resistance, Rz. An approximation of the optimum value for Rz is obtained by setting the cell, R2C1,time constant equal to the control system time constant and solving for R 2 :
D' is computed for the classifications D1, D3, D4,and Dg as: Transfer function, D, classi-
Figure 5. fication
D1: 6 > 45" a t F1 and 6 < 20" at Fa D2: All 6 > 45" D3: All 6 < 20" D4: 6 > 45" a t F1 and F3 but contains 6's < 20" Ds: 6 < 20" a t Fl and 6 > 45" at Fa The phase angle limits are arbitrarily chosen to separate the five cases. The break-frequency values of the classified functions of D are determined as follows:
D,: BF1 = lowest frequency that 6 45"
