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FourElectrode Null Techniques for Impedance Measurement with High ResolutionHerman P. Schwan and Clifford D. Ferris

Citation: Review of Scientific Instruments 39, 481 (1968); doi: 10.1063/1.1683413 View online: http://dx.doi.org/10.1063/1.1683413 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/39/4?ver=pdfcov Published by the AIP Publishing

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THE REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 39, NUMBER 4 APRIL 1968

Four-Electrode Null Techniques for Impedance Measurement with High Resolution* HERMAN P. SCHWAN

Electromedical Division, The Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104

AND

CLIFFORD D. FERRIS Department of Electrical Engineering, University oj Maryland, College Park, Maryland 20742

(Received 20 March 1967; and in final form, 27 December 1967)

At low frequencies, the normally difficult measurement of the dielectric properties of conducting materials is severely compromised by electrode polarization. This problem at the electrode-sample interface arises from the modulation of the normal dc boundary potential by the passage of alternating current. A solution, permitting a conductance resolution of 1: 1()li for frequencies between 10 cps and 1 kc, is to use a second noncurrent-carrying pair of electrodes to measure the voltage across the sample and to employ a null technique for obtaining the required precision. Several four-electrode null techniques are proposed, each having certain relative merits. In all cases the resolution in capacitance is shown to be governed by a combination of the resolution in conductance, the sample properties, and the frequency of the measurement.

INTRODUCTION

THE electrical properties of materials are usually measured with a two-electrode apparatus, the elec-

trode pair serving both to carry current and to measure voltage. This simple technique is satisfactory as long as the voltage between electrodes is a satisfactory approxi-mation to the voltage across the sample. However, for materials of relatively high conductivity, such as physio-logic fluids, cell suspensions and tissues, electrode polariza-tion causes the approximation to weaken, and at suffici-ently low frequencies, to fail.l,2 Corrections can be applied to the measurements, and to the reactive measurements particularly, but when electrode polarization becomes severe, the corrections are not only large but uncertain.

Four-electrode measurement techniques have been used to eliminate the problem of electrode polarization by pro-viding a second pair of electrodes, non-current-carrying, with which to measure the voltage across the sample. But usually there is still the problem of insufficient resolution to permit determination of dielectric constants at low frequencies. Null techniques, however, can be combined with the four-electrode principle to increase its resolution while preserving the relative freedom from electrode polarization.3 ,4

This paper first outlines the principles of electrode polarization and resolution as they enter into the deter-mination of electrical properties with two- and four-terminal techniques. Next it proposes several four-electrode null techniques of high resolution which are not, or which

* This work wa~ supported by NIH grant #HE-01253. 1 H. P. Schwan in Physical Techniques in Biological Research, W. L.

Nastuk, Ed. (Academic Press Inc., New York, 1963), Vol. 6, p. 323-406.

2 H. P. Schwan, Biophys. 3, 181 (1966). 3 H. P. Schwan and C. D. Ferris, Proc. 16th Ann. ConI. EMB

(Baltimore) 5, 84 (1963). 4 P. L. Hill and H. P. Schwan, Proc. 19th Ann. Conf. EMB (San

Francisco) 8, 10 (1966). 481

are to a lesser extent, affected by electrode polarization considerations.

ELECTRODE POLARIZATION

Usually dc boundary potentials, exist between electrode and sample. As a current density, i, is applied through the interface, the boundary potential adjusts to a some-what different value which may be greater or smaller than the original potential, Vo, depending upon the direction of the current. If the applied current is sinusoidal, it will evoke a periodic modulation of the "zero current" bound-ary potential Vo which is not in phase with the evoking current. The magnitude of this modulation potential, V, is found to be a monotonic function of i, with V = 0 for i= O. Thus we can develop this functional relationship in a series expression,

(1) The first term of this expression will obviously dominate for low values of i and establish the linear response char-acteristics of the electrode boundary observed for low i. In accord with Ohm's law, the "polarization impedance" of the electrode appears as Zp= al. In practice, one always observes the sum of two polarization impedances, corre-sponding to the two electrodes. The limits of linearity in Eq. (1) are typically reached at current density levels of about 1 mAl cm2 for 1 kc, but there is a frequency dependence.26

Since readjustments of electrode boundary potentials to changes in i do not occur instantaneously, the evoked alternating potential lags in phase relative to the current. Hence, the polarization impedance is a complex quantity. Its reactive component is observed approximately to compare in magnitude with its resistive component, so

I; H. P. Schwan and J. G. Maczuk, Proc. 18th Ann. Conf. EMB (Philadelphia) 7, 24 (1965).


482 H. P. SCHWAN AND C. D. FERRIS

FIG. 1. Principle of four-electrode tech-nique. The potential V registered ?y .the electrodes E' is not affected by polanzatlOn. Hence the impedance of the sample confined by th~ dashed equipotential surfaces equals Vii.

that the phase angle of the polarization impedance is usually not very different from 45 and varies but little with frequency.6 Polarization impedances are usually con-veniently characterized by a series combination of a polar-ization resistance, Rp, and a polarization capacitance, Cpo

The effects of electrode polarization upon the deter-mination of sample properties can be discussed conven-iently with the aid of the approximations1,7

R= R8+ Rp+Rp(wRC)2; C= c.+ (1/w2R2Cp). (2) Rand C are the measured total resistance and capacitance, R. and c., the true sample values, and Rp and Cp the polarization resistance and capacitance. The equations apply provided that the polarization impedance is smaller than the sample impedance.

To minimize electrode polarization, the electrode sepa-ration may be increased to increase the sample resistance, particularly as the frequency is lowered. This procedure, however, leads to difficulties caused by complicated con-tributions of stray field components to the sample impe-danceS and by the increased requirement for sample volume. Other techniques1,9 permit the extension of the frequency range of correct sample impedance information by about one decade. The four-electrode techniques to be discussed below permit, in principle, elimination of errors resulting from electrode polarization at any frequency.

RESOLUTION Conductance resolution, Sa, and capacitance resolution,

Sc, may be defined

Sa=G/lG, Sc=C/lC, (3) where G and C are total measured parallel conductance and capacitance representing electrode and sample imped-ance, and lG and lC are their smallest detectable changes. It has been previously stated that the resistive and ca-pacitive resolution may be interrelated byl,lO

wlC=lG. (4) In order to achieve acceptable resolution in C, a high resolution in G often must be provided. Consider for

8 H. Fricke, Phil. Mag. 14, 310 (1932). 7 H. P. Schwan, Z. Naturforsch. 6b, 121 (1951). 8 H. P. Schwan and J. G. Maczuk, Rev. Sci. Instr. 31, 59 (1960). 9 H. P. Schwan, Ann. N. Y. Acad. Sci. 148, 191 (1968). 10 H. P. Schwan, G. Schwarz, J. G. Maczuk, and H. Pauly, J.

Phys. Chern. 66,2626 (1962).

example a sample of biological interest, blood of normal cellular concentration. Its resistivity is very approxi-mately 100 Q cm and its dielectric constant about 1500. Both quantities are nearly frequency independent up to 100 kc. Thus, in order to measure the dielectric constant at 1 kc with an accuracy of 10%, a conductance resolution of 10H must be available. Such high resolutions are im-possible to achieve with direct reading techniques. Thus, null techniques must be used for the study of the dielectric properties of good conductors at low frequencies.

FOUR-ELECTRODE TECHNIQUES Four-terminal ac techniques, such as discussed later,

permit, in principle, elimination of polarization problems. Consider for example the four-terminal device shown in Fig. 1. The alternating current, which is conducted through the sample with the aid of the electrodes E,'will evoke ac polarization potentials at these electrodes. However, the electrodes E' will not pass any current through the volt-meter V, provided that its input impedance is very high in comparison with the sample impedance. Hence no ac boundary potentials can develop at the voltage electrodes E'. These electrodes will, therefore, register correctly the potentials of the equipotential surfaces which reach the electrodes. Thus, the impedance of the sample, which is confined by these equipotential surfaces is correctly de-termined by dividing the measured voltage by the current i. The same argument cannot be applied to the dc case since dc boundary potentials exist even in the absence of a current and are most certainly not proportionally related to it. It is, therefore, apparent that both 2- and 4-terminal impedance techniques are affected by dc polarization potentials. It is a frequent misconception that 4-terminal impedance measuring devices which operate at dc avoid errors due to polarization.

The simple circuit shown in Fig. 1 is of no use for the determination of the dielectric properties of conducting samples. Its resolution is at best about equal to the accuracy of current and voltage measurement, i.e., hardly better than 1%. A resistive resolution of 100 is, however, by far too low for the determination of the dielectric properties of conducting samples at low frequencies as discussed above. Null techniques appear, therefore, as the

I

][

FIG. 2. Principle of four-electrode null technique. The two amplifiers, I and II, are assumed to have equal gain and input im-pedances much higher than that of the sample between the electrodes to amplifier I.


IMPEDANCE MEASUREMENT 483

only means to further improve the resolution. Several null techniques which are based upon the four-electrode principle and provide high resolution will be discussed in the following paragraphs.

A first proposal which comes to mind is shown in Fig. 2 and has been suggested by us before. l The total current of the generator is passed first through the sample cell and then through a variable parallel RC network. The components of this network are adjusted until the po-tentials applied to the two voltage amplifiers I and II are identical. This is noted by a null of the difference potential V between the two amplifiers. Under conditions of balance the potential across the RC network is identical to the one between the two equipotential surfaces which reach the electrodes connected to the amplifier I. Hence, since the same current passes through both systems, the imped-ance of the RC network is identical with the one of the sample bounded by the aforementioned equipotential surfaces. The argument assumes that the currents through the two amplifiers can be neglected, i.e., are very small in comparison to the current passing through the sample and RC network. This calls for amplifier input impedances which are very high in comparison to the sample imped-ance. Since we are here primarily interested in the in-vestigation of conducting samples, this condition can be easily satisfied. Grounding is best provided at the point indicated. This permits the use of a single ended generator and amplifier I I and use of network components which are grounded. However, amplifier I must be of the differential type, its position with regard to ground being determined by the sample network arrangement. Very high common mode rejection of amplifier I is an important consideration since the potential received by amplifier I is not to be modified by its elevation above ground. The common mode rejection figure directly reflects the desired resolution. If, for example, a resolution of 100 000 is desired, then, obviously, errors in the potentials received and registered by the amplifiers I and II should not be greater than 1 part in 105 This means that the common mode rejection of amplifier I should be characterized by

FIG. 3. Four-electrode sub-stitution null technique.

02

Cv To Ampl. ]I

0 "1 ZCD [ ~c 1- [ Zec FIG. 4. Four-electrode bridge + technique. Readings of R~ and "-0 Cv Rv C. are taken with V connected ZAB :fA first to B and then C. The ...L

sample impedance is calculated from these readings. Rs

a factor of 105 or more. A further consideration which applies to the amplifiers is concerned with the errors which result in shunting the sample and network imped-ances by the amplifier input impedances. The resultant errors are usually small since we demanded above, for different reasons, that the input impedances be high in comparison with the impedances of the sample and the network. Moreover, if the two amplifiers have identical input impedances, both in magnitude and phase, no error will result, since errors which arise from the two amplifiers equally apply to the sample and RC network and, there-fore, cancel out. For highest precision, an operational modification of the principle is suggested which is shown in Fig. 3. First the adjustable R.C. network is balanced against the unknown sample; then the sample cell is replaced by a circuit which duplicates the R.C. combina-tion. This means that corresponding working and probe electrodes must be connected together. This new RC com-bination is balanced against the original balance settings of R. and Cv. The original dials Rv and Cv are not changed during this procedure. The properties of the RC network are then equal to those of the unknown. Errors resulting from insufficient common mode rejection, unequal am-plifier gains, shunt effects of the input amplifiers, and other electronic asymmetries largely cancel out and are of no consequence with this substitution technique.

Another circuit which avoids the need for a differential amplifier with high common mode rejection is depicted in Fig. 4.11 The null detector is a single ended grounded amplifier V. It operates as the null detector of a bridge. Different parts of the total sample impedance are balanced against other parts of the sample in series with the variable RC network. Denoting with subscripts Band C the imped-ance values of the RC network which apply for the different balance equations when V is connected with the points Band C yields

(B) (C)

RS+ZAB=ZBC+ZCD+ZB,

RS+ZAB+ZBC=ZCD+ZC. 11 K. Schmidt (private communication).

(5) (6)


484 H. P. SCHWAN AND C. D. FERRIS

FIG. 5. Four-electrode null technique circuit. a-current electrodes; b-potential electrodes. Voltages and phases shown for null condition. Use of a shielded transformer is required. The technique is recom-mended for frequencies in excess of 10 cps where satisfactory trans-formers are available. For lower frequencies the technique indicated in Fig. 4 is preferable.

Taking the difference of these equations yields

ZBC=t(ZC-ZB), (7) where ZBC characterizes the sample impedance between the electrodes Band C undistwbed by electrode polariza-tion. The circuit offers all advantages of a bridge circuit. However, it is not a direct reading circuit and the genera-tor must be separated from ground. A limitation is the necessity to achieve a high degree of constancy of the electrode polarization impedances with time. Slight varia-tions of the polarization impedances as the null detector is shifted from position B to C can cause major errors in the calculated sample properties and multiple measure-ments may be needed to reduce such errors.

Additional techniques have been developed by Hanss!2 and Schmitt.!3 Hanss developed a system quite similar to the one proposed by us! but limited to purely resistive measurement. Schmitt's system is somewhat closer to the circuit shown in Fig. 5, designed for determinations of small phase angles at very low frequencies down to 0.01 cps, but of more limited resolution.

ELECTRONICS FOR FOUR-ELECTRODE TECHNIQUE

The basic measuring technique as presented in Fig. 3 entails some shielding and grounding problems indicated before. Floating ground operation is required for amplifier I and the system must be carefully shielded for hum and noise pickup. A satisfactory system can be developed as shown in Fig. 5. A unit-gain difference amplifier is used for the probe amplifier. A simple voltage divider, in com-bination with a differential amplifier I, can be used to form this unit-gain amplifier.

The system presented in Fig. 5 has certain advantages and disadvantages. It is direct reading, as is the system shown in Fig. 3. The system shown in Fig. 5 has a different ground reference point than the one suggested for the system of Fig. 3 and requires a transformer at the input. It is therefore probably useful only above 10 cps. Unless

12 M. Hanss, Ph.D. thesis, Univ. of Paris (1965). 13 O. Schmitt, Ann. N. Y. Acad. Sci. 115, 952(1964).

the two amplifier stages are very carefully matched so that no relative phase shift exists, capacitance measure-ments are meaningless, although resistance determinations may still be made with some loss in resolution. The voltage divider must be frequency insensitive for both voltage and phase over the measurement frequency range. The amplifiers used in the system shown in Fig. 3 must also have identical amplitude and phase response character-istics. Small relative phase shifts may be compensated to some extent by use of an above-indicated substitution technique (Fig. 3).

The systems shown in Figs. 3 and 5 demonstrate a possible resistance resolution of one part in 105 at 1 kc. This value degenerates to about one part in 103 at 1 cps. System noise and detector inadequacies accounted for the loss in resolution. Resolution for C is governed by Eq. 5 and sample properties.

ELECTRODES AND MEASURING CELL

The four-electrode cell was designed in the same general configuration as a conventional two-electrode cell previ-ously described by us.! The parameters of the sample are determined directly from the cylindrical cell geometry using the relations

K =1/(ARv) E= ICv/A,

where K is the conductivity, E is the dielectric constant of

'-r~;;;J;;;~r' SET SCREW POLYSTYRENE BLOCK 1.2cm 1.9X8.9XI.2.", L PLATINUM CEMENTED TO TUBE

ELECTRODE I-..., 0.64cm.

--1-.

8.9.m

FIG. 6. Drawing of sample cell arrangement.


IMPEDANCE MEASUREMENT 485

the sample and where I is the separation of the potential probes, A is the cross-sectional area of the cell, C. and R. are the direct readings of the variable elements. There are two additional design criteria associated with the four-electrode measurement: (1) Provision is desirable for changing the separation between the potential probe electrodes to accommodate for variation of sample imped-ance; independence of specific sample properties (di-electric constant, conductivity) from potential electrode spacing will increase the confidence in the results obtained; and (2) there must be sufficient separation between the potential probe electrodes and the principal current elec-trodes such that direct coupling by stray fields or conduc-tion currents does not occur.

Various possible configurations exist for the potential probe electrodes. These include rings mounted concentric to the cylindrical body of the measuring cell as described by Shedlovsky,14 straight wire probes mounted perpen-dicular to the cell body, and recessed electrodes.! After preliminary trials, the recessed electrodes were found most suitable for reasons discussed before.1 ,2

Field plots were made to determine a satisfactory loca-tion for the probes. A desirable arrangement for the probe electrodes is one which permits perpendicular in-cidence of the equipotential lines with the surface of the probes. Location of the probes directly in the wall of the measuring cell in practical cases does not satisfy this criterion. A certain amount of field distortion occurs even when the probes are loaded by an infinite impedance (open circuit). One is then inclined to recess the probes some distance from the cell wall. It must be determined how far the probes should be recessed, as too great a distance will introduce a series impedance (because of the electrolyte bridge) between the sample in the cell and the input amplifiers with concomitant attenuation of the input signal.4

Model electrode configurations were constructed from Teledeltos paper upon which silver electrodes were painted. A 400 cps potential source was connected across the elec-trodes and the equipotential lines plotted using a null technique. A ratio of separation from cell wall to electrode diameter of 2: 1 was found to yield a satisfactory result. For such a ratio the current density caused by the small current through the amplifier is nearly uniform across the electrode. The electrode polarization impedance is there-fore lower than would be the case if the current would be nonuniform.

A diameter of 1 cm was chosen for the probe electrodes and they were recessed 2 cm from the cell walL Figure 6 shows a detailed drawing of the final design for the cell and Fig. 7 is a photograph of the completed celL All of

14 T. Shedlovsky, J. Am. Chern. Soc. 52, 1806 (1930).

FIG. 7. Photo of sample cell arrangement.

the electrodes are fabricated from hard platinum sheet 1.5 mm in thickness. The electrodes are sandblasted and platinized as described below. The diameter of both of the principal electrodes measures approximately 2.5 cm while the separation between the principal electrodes is 10 cm. Holes and plugs are provided along one side of the cell to permit location of the probe electrodes at different distances. A fill plug is provided in the main body of the cell (as shown in the photograph) to permit the removal of air bubbles and facilitate filling the cell. The caps with the electrodes are removable and screwed to the cell body. Leakage is prevented by applying to the threads an appropriate amount of grease or Vaseline.

Before the electrodes are placed into use, they should be properly prepared so that electrode polarization phe-nomena caused by small currents through the electrodes do not affect the correct determination of sample proper-ties. The preparation of electrodes consists of depositing colloidal platinum (platinum black) upon the working surfaces. This is accomplished by the use or conventional electroplating techniques as described bv Kohlrausch15 and Schwan.!

15 F. Kohlrausch, Ann. Physik Chern. 60, 315 (1897).


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