peak areas plotted against the sample size gave straight lines for both AsC13 and SnC14. Three solutions of AsC13 and SnC14 were prepared by mixing measured quantities of the pure liquids together in a n inert atmosphere. The results of the integration of the peak areas are shown in Table I. The errors, which never exceed 6y0,are believed to result primarily from the premature vaporization of the samples as they are being injected. By comparing the chromato-
grams of the mixtures with those of pure compounds, it was found that about 0.1 mg. of SnC14 emerged with the AsC13 peak and the amount of contamination was almost independent of sample size. Thus, greater accuracy is achieved by using larger samples, up to 15 mg. S o attempt was made to compensate for the contamination error. The accuracy and precision of the analysis indicate that gas chromatography holds considerable promise for the analysis of these and similar compounds.
LITERATURE CITED
(1) Dal Nogare, Stephen, Safranski, L. W., ANAL.CHEM.30,894 (1958). (2) .Tadmor, J., "Chromatographic Reviews, Michael Lederer, ed., p. 223, Elsevier, London, 1963.
JAMES E. DENNISON~ HARRY FREUND Department of Chemistry Oregon State University Corvallis, Ore.
Present address, Western Electric Go., P. 0. Box 900, Princeton, N . J.
Simplified Use of Transfer Functions in Analysis of Operational Amplifier Electroanalytical Instrumentation SIR: Recently, Booman and Holbrook ( I , 2 ) have demonstrated the importance of a detailed analysis of the gain-frequency characteristics of operational amplifier circuits for controlled-potential electroanalytical instrumentation. It was shown how to obtain optimum performance for any cell-amplifier circuit after analyzing the circuit parameters by suitable combination of cell and amplifier transfer functions. The quantitative application of these concepts is relatively complex and instruments such as xave analyzers are essential for measuring cell transfer functions properly. The use of these same concepts for the qualitative characterization of the circuits is a valuable aid in the preliminary selection of circuit configuration and in the specification of necessary amplifier characteristics. The approach involves formulating expressions for the potentials a t various points in the circuit in terms of simplified transfer functions, without explicitly defining the frequency dependence of the components. These are then used to relate the applied potential to the cell potential and to evaluate the performance of the circuit in compensating for IR drop in the cell and in the current measuring device, etc. The rapid development of solid state operational amplifiers has made it feasible to use many of the possible circuit configurations described previously (3). In many cases, chopper stabilization no longer is required, and both inputs of the operational amplifier may be active. Such circuits can be evaluated conveniently using this simplified approach. Although obvious expressions are obtained for most of the single amplifier circuits, the interrelations between the components are not so straightforward with multiamplifier circuits. The use of the transfer function approach frequently reveals aspects of the circuit performance which are not 1768
ANALYTICAL CHEMISTRY
drop in the current measuring device (load resistor) is properly compensated, and the control amplifier can be stabilized if required. From the properties of an operational amplifier, eo =
I Figure 1 .
J
Single amplifier potentiostat
GI
control amplifier inverting input to amplifler e2 non-inverting input to omplifler eo amplifier output C counter electrode R reference electrode W working electrode RL load resistor or current measuring device V signal generator el
FORMULATION OF TRANSFER FUNCTION EXPRESSIONS
Using a simple example, the circuit of Figure 1 can be analyzed. As discussed previously (S), this is a particularly useful circuit because the signal generator can be grounded, the IR
(1)
where eo is the amplifier output potential, el and e2 are the potentials a t the inverting and noninverting inputs, respectively, and G1 is the transfer function of the operational amplifier used as the controller. For these purposes, GI can be considered as the gain of the amplifier. Next, the input potentials el and e2 must be defined in terms of the other circuit parameters.
E
obvious from inspection of the circuit diagram. Analysis of all of the circuits published previously (3) indicated that five of them were not suitable for the intended purpose. This paper shows how such errors can be detected easily. This general approach to the analysis of multiamplifier circuits permits a qualitative evaluation of their characteristics in applications where the high frequency ax. performance is of importance. The expressions obtained include terms for frequency-dependent transfer functions of the electrolysis cell and of each amplifier. From a qualitative knowledge of how each of these components responds at any particular frequency, the expressions can be used to estimate the circuit performance when rapid response is required.
- (el - e2)G1
=
el=E+V
(2)
ez = 0
(3)
(eo
- iRL - V ) A
(4) Here E is the cell potential (between the reference and working electrodes) , V is the potential of the signal generator, i is the cell current, RL is the load resistor, and A is the cell transfer function. The cell transfer function is the ratio of the reference electrode potential to the counter electrode potential, both with respect to the potential of the working electrode (2)-i.e., the ratio of the cell potential E to the total potential across the cell. Equations 1 4 can be combined to obtain E(&+
1) =
-v
(1 + -kl) - -i;
(5)
Whenever the control amplifier gain Gl is high, Equation 5 will reduce to E=-V (6) and this circuit configuration will function as a controlled potential instrument. At higher frequencies the gain of the amplifier decreases, and the terms containing 1/G1 cannot be neglected.
Estimates of possible errors in control potential because of improper iRL compensation can be obtained from the RI operational amplifier specifications. Even more important, however, is the 1 appearance of the term 1/AG1 in Equation 5 , since the product of the cell transfer function and the amplifier gain must be significantly greater than unity for the circuit to operate properly. I n cells with microelectrodes, the cell transfer function may be close to unity up to 10 kc. or more, but in coulometric Figure 2. TWO amplifier potentiostat with type cells, the signal is attenuated to split load resistor a much greater extent so that control GI control omplifier errors. may appear a t quite low freGz unity gain follower amplifier quencies. Rz RB, Rc R1,
MULTlAMPLlFlER CIRCUITS
This method of analyzing circuit configurations is very effective when considering multiamplifier instruments. For example, it is possible to use this approach to show clearly that one of the circuits previously presented, Figure 15c in Reference (S),did not compensate for I R drop in the load resistor, contrary to Table I1 of Reference (3). A cursory examination of the circuit would lead one to the conclusion that the load resistor was not located in the comparison loop, and thus there would be no interaction. However, using the same approach described above, the input potentials, el and e2, and the cell characteristics for Figure 15c, Reference (S), can be formulated :
- ez)Gl [(iRL+ E)Gz - VI X eo =
el
=
- (el
+ Rz +
-~ Rz V
Ri
e2 = ~ R L E = (eo - ~ R L ) A
(7)
(8)
(9) (10)
where G1 is the transfer function of the control amplifier, G2 is the transfer function of the unity-gain-follower amplifier, R1 and RZ are the bridge resistors in series with the follower output and voltage source, respectively, and the other terms are defined as in the corresponding portions of Figure 1, above. Combining Equations 7-10, the characteristics of the circuit can be described by :
-v
(1 - - RI R2 + RZ) +
of split load resistors would not be inconvenient as suitable ganged switches are available. Another example of the importance of the simplified transfer function analysis of multiamplifier circuits involves the configurations in Figure 16 of Reference (3). In particular, the circuit of Figure 16c is superficially analogous to Figure 15c, using the alternate grounding procedure. From the circuit, one might expect a similar lack of compensation for iRL. The characteristic eauation (assuming G1 >> 1 and G2 = 1) is:
voltage comparison bridge resistors split load resistors (R3 f Rc = RL)
Considering only the low frequency properties of the circuit, G1 >> 1, G2 = 1, and thus,
I n the usual case where R1 = Rz, the actual cell potential differs from the applied potential by the potential developed across the load resistor. Examination of the equations reveals that this occurs because the entire L a t ez, but a t el, potential ~ R appears the potential iRL is attenuated by the bridge R1 and R2. This difficulty could be avoided if the iRL signal a t e2 were attenuated in the same way as at el. This can be accomplished easily in the circuit shown in Figure 2, where the only change from Figure 15c of Reference (3) is that the load resistor RL has been split into R3 and R4. I n this case, the characteristic equation for the circuit is (assuming G1 >> 1, and Gz = 1):
and a useful potentiostat can be obtained by selecting R3/Ra = R1/R2. This, of course, introduces a new class of potentiostats not considered previously (S), but which may be very useful in some circumstances. The use
Thus, except for the trivial case where the cell transfer function is close to unity, control of the cell voltage is not attained. This difficulty appears in all the circuits in Figure 16, and in Figure 206, Reference ( 3 ) . As a result, the circuits as shown are not useful as potentiostats (or galvanostat in Figure 206). However, Equation 14 shows that if the bridge resistors are selected such that RP>> R1,a useful circuit can be obtained, even though the potential of the signal generator is attenuated. illthough the examples discussed here do not make full use of the quantitative treatment of the circuit analysis approach presented by Booman and Holbrook, they do indicate a very valuable qualitative way of checking on the performance of any selected circuit configuration. LITERATURE CITED
(1) Booman, G. L., Holbrook, W. B., ANAL.CHEM.35, 1793 (1963). (2) Booman, G. L., Holbrook, W. B., Zbzd., 37, 795 (1965). (3) Schwarz, W. M., Shain, I., Zbid., 35, 1770 (1963). IRVING SHMN JACKSON E. H A R R A R ~ GLENNL. BOOM AN^ Chemistry Department University of Wisconsin Madison, Wis. University of California Lawrence Radiation Laboratory Livermore, Calif. * Phillips Petroleum Co. Atomic Energy Division Idaho Falls, Idaho Work supported in part by the U. S. Atomic Energy Commission [Contract No. AT(11-1)-10831. Other support from the U. S. Atomic Energy Commission [Contract No. AT-(10-1)-205] through the Idaho Operations Office.
VOL. 37, NO. 13, DECEMBER 1965
1769
