The ωσ method: A new contactless comparison method for measuring electricalconductivity of nonferromagnetic conductorsM. B. Miletić, P. M. Nikolić, D. Vasiljević-Radović, and A. I. Bojičić Citation: Review of Scientific Instruments 68, 3523 (1997); doi: 10.1063/1.1148317 View online: http://dx.doi.org/10.1063/1.1148317 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/68/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Lorentz force sigmometry: A contactless method for electrical conductivity measurements J. Appl. Phys. 111, 094914 (2012); 10.1063/1.4716005 Measurement of electrical conductivity in nonferromagnetic tubes and rods at low frequencies Am. J. Phys. 77, 949 (2009); 10.1119/1.3184154 Development of eddy current microscopy for high resolution electrical conductivity imaging using atomic forcemicroscopy Rev. Sci. Instrum. 79, 073705 (2008); 10.1063/1.2955470 An oscillating coil system for contactless electrical conductivity measurements of aerodynamically levitated melts Rev. Sci. Instrum. 77, 123904 (2006); 10.1063/1.2403939 Contactless measurement of electrical conductivity of Si wafers independent of wafer thickness Appl. Phys. Lett. 87, 162102 (2005); 10.1063/1.2105992

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.171.233.72 On: Mon, 24 Nov 2014 23:14:38

The vs method: A new contactless comparison method for measuringelectrical conductivity of nonferromagnetic conductors

M. B. Miletic, P. M. Nikolic, D. Vasiljevic-Radovic, and A. I. Bojicica)

Joint Laboratory for Advanced Materials of SASA, Beograd, Knez Mihailova 35, Yugoslavia

~Received 28 May 1996; accepted for publication 19 May 1997!

A novel, very simple, eddy current method for measuring the electrical conductivity ofnonferromagnetic conductors is described. This method is relative, i.e., we measure the ratio ofconductivities of two samples of the same shape and size~this shape and size may be arbitrary!. Forone of them, conductivity is known, this is the reference sample, whereas the conductivity of theother is unknown and it is to be determined. Consequently, in this method it is necessary to have areference sample of known conductivity. On the other hand, in this method no other calibration isnecessary; the ratio of two conductivities we obtain directly as the ratio of two frequencies, withoutany calibration. The principle of this method is as follows: It turns out, based on Maxwell’sequations, that if we have a coil through which a sinusoidal current of an angular frequencyv flowsand in whose magnetic field a given conductive, nonferromagnetic sample is placed, the effectiveinductance~i.e., the ratio of the reactive part of the impedance to the frequencyv! depends onv andon the conductivity of the samples only through their productvs. Therefore, if it is found that witha sample of conductivitys1 at frequencyv1 the effective inductance is equal to that of a samplewith conductivity s2 and frequencyv2 ~the samples being of the same shape and size, bothnonferromagnetic, and both placed in the same position relative to the coil!, this means that equalityv1s1 5 v2s2 is valid. Hence, ifs1 is known ands2 is unknown,s2 can be calculated. ©1997American Institute of Physics.@S0034-6748~97!04208-1#

I. INTRODUCTION

Contactless methods for measuring electrical conductiv-ity based on the generation of eddy~Foucault! currents havebeen known for a long time.1 Since the response of the ap-paratus to the presence of a conductive body~a sample! in avarying magnetic field depends, in general, in a complex wayon conductivity and on other factors~geometry, frequency,etc.!, an empirical calibration or complex calculations~i.e.,solutions of Maxwell’s equations! are necessary.

Here, we describe a novel, very simple, eddy currentmethod for conductivity measurements of nonferromagneticconductors. We call this method the ‘‘vs method’’ for areason that will become obvious in the following text.

Thevs method is a relative method, i.e., we measure theratio of the unknown conductivity of a given sample to theknown conductivity of a reference sample. Both samplesmust be of the same shape and size, but this shape and sizemay be arbitrary; both samples must be nonferromagnetic. Inthevs method, neither calibration nor solution of Maxwell’sequations~or any other calculations! are necessary. The ratioof two conductivities we obtain as a ratio of two frequencies,without any calibration or calculation.

Since the basic statement of thevs method, i.e., that ifEq. ~4! is valid, then Eq.~5! also must be valid, results fromMaxwell’s equations almost without approximations~at leastif the working frequencies are not too high!, it may be con-cluded that thevs method is, in principle, very exact.

II. BASIC STATEMENT

If a given conductive body, sample S, is placed in amagnetic-fieldB of a coil ~it will be denotedLw , the ‘‘work-ing coil’’ !, through which a sinusoidal current of an angularfrequency2 v flows, eddy currents will appear in the sample.The magnetic field of these currents will be superposed onthe magnetic-fieldB and we obtain, in the neighborhood ofthe sample,~of course, in the interior of the sample, too!, andtherefore, in the interior of the coil, some resulting fieldB8different from the original fieldB. Therefore, if sample S isplaced in the vicinity of the coil, its effective inductanceL8~i.e., the ratioX/v, whereX. is the reactive part of the im-pedance, which can be measured at the terminals of the coil!will be different from the inductanceL of the coil alone.WhereasL is independent ofv ~of course, we assume thatcoil capacitance, influence of skin effect, etc., can be ne-glected! L8 depends onv, i.e., L8 5 L8(v).

Assume that sample S is not ferromagnetic so that themagnetic permeabilitym may be taken as unity

m51. ~1!

As it is known, if Eq.~1! is valid, and on the condition thatdisplacement currents in the interior of the sample are neg-ligible in comparison with conduction currents, it followsfrom Maxwell’s equations that the complex amplitude offield B8, denotedB08 , in the interior of S, satisfies the fol-lowing equations:

div B0850; DB08524p j vsB08

c2 ~2!

~j is the imaginary unit;s is electrical conductivity of S;c isvelocity of light!.a!Electronic mail: ebozinju@bbg.etf.bg.ac.yu

3523Rev. Sci. Instrum. 68 (9), September 1997 0034-6748/97/68(9)/3523/5/$10.00 © 1997 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.171.233.72 On: Mon, 24 Nov 2014 23:14:38

We assume that the wavelength 2pc/v is many timeslarger than the linear dimensions of the sample, the coil, andthe distance between them; consequently, magnetic-fieldB8outside the sample and near the coil at any moment is almostidentical to a static magnetic field. Therefore,B08 in theneighborhood of the sample satisfies, almost perfectly, theequations of a static magnetic field

div B0850; curl B0850. ~3!

On the surface of the sample,B08 must satisfy the boundaryconditions; since Eq.~1! is valid, the boundary conditions forB08 are that B08 must be the same on both sides of thesurface.3

Equations~2! and~3! and the boundary conditions on thesurface of S determineB08 . Sincev ands are present~only!in Eq. ~2! and only as the productvs, B08 depends onv ands also through the productvs only. Hence, the flux ofB08through the coil and, thus,L8(v) depend onv ands throughthe productvs only.

Suppose that we have two samples, S1 and S2, of thesame shape and size but of different materials, and thus, ofdifferent conductivities,s1 ands2, respectively. Concerningtheir magnetic permeabilitiesm1 andm2, we assume that Eq.~1! is valid, i.e., that S1 and S2 are not ferromagnetic.@Wesuppose that the very weak deviation from condition~1!,characterizing normally paramagnetic and diamagnetic con-ductors (um 2 1u , 1023), may be neglected. Some obser-vations concerning the hypothetical case of a very strongparamagnetic sample are given in additional remark~2! inSec. VI.#

If we first place S1 in the field of the coil ~in someexactly determined position in regard to the coil! at somefrequencyv1, the effective inductance of the coil will besomeL18(v1). If, then, we exchange the samples~i.e., if weremove S1 and place S2 in the same position!, the effectiveinductance will beL28(v1), which is different fromL18(v1).Now, we change the frequency to some valuev2 at whichwe obtain

L28~v2!5L18~v1!. ~4!

If Eq. ~4! is valid, this means, as stated above, thatB08 is, inboth cases, the same and, hence,

v2s25v1s1 , ~5!

therefore, ifs1 is known~i.e., if S1 is the reference sample!,and if v1 andv2 are measured,s2 can be very simply cal-culated.

III. APPARATUS

The apparatus that we used is presented in Fig. 1. Itconsists of an ac bridge with a signal generator as a sourceand with a synchronous detector and a galvanometer as theindicator of bridge balance. As the reference signal for syn-chronous detection, the potential drop across the resistorr isused. Thereby, it is attained that, in spite of frequencychanges but without any subsequent adjustment of phase, itis always possible to detect the unbalance of the bridge

caused by an unbalance of inductances separate from thatcaused by an unbalance of resistances~andvice versa!.

IV. PROCEDURE OF MEASUREMENT

The procedure of measurement is as follows: At first, atfrequencyv1 ~arbitrarily chosen! with sample S1, we balancethe bridge by adjustingLv and r v . Now, the inductance ofLv is equal to (Ra /Rb)L18(v1). Then, we remove S1 and putS2 in its place. Now we again balance the bridge, butLvremains fixed at its previous value whilev is changed. Inthis way, we findv2, the frequency at which the bridge isagain balanced. Now, the inductance ofLv is equal to(Ra /Rb)L28(v2). Since the inductance of fixedLv is inde-pendent of frequency, we haveL28(v2) 5 L18(v1).

V. VERIFICATION

We tested thevs method using various metallicsamples. As an example, the results obtained for threesamples~copper, aluminium, and brass! are given in Table I~all values obtained at room temperature!.

The copper sample has been used as the referencesample at various frequenciesv1; the obtained ratios of con-ductivity of copper (sCu) to conductivities of aluminum(sAl) and brass (sbrass) are shown.

The samples were cylinders~diameter 8 mm, length 27mm! made of commercial materials of unknown purity. Theused coil (Lw), whose inner diameter was 11 mm and length3 mm, consisted of 175 turns of varnished copper wire~f0.18 mm!. The samples were fixed in the coil by means of

TABLE I. Verification of the vs method: the ratios of conductivity ofcopper (sCu) to conductivities of aluminum (sAl) and brass (sbrass), ob-tained for various values ofv1.

v1/2p ~kHz! sCu /sAl sCu /sbrass

0.55 1.60 4.111.50 1.58 4.104.35 1.52 4.128.47 1.50 4.21

FIG. 1. Ac bridge and block diagram of the equipment used. S, sample;Lw ,working coil ~i.e., the coil in whose magnetic field the sample is placed!;Lv , variable inductor;r v , variable resistor;r , Ra , Rb , resistors~resis-tances!; SG, signal generator; FM, frequency meter; RSA, reference signalamplifier; BSA, bridge signal amplifier; SD, synchronous detector; and G,galvanometer.

3524 Rev. Sci. Instrum., Vol. 68, No. 9, September 1997 Electrical conductivity This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.171.233.72 On: Mon, 24 Nov 2014 23:14:38

paper inserts~Fig. 2!, coaxially with the coil. The current,I w , in the coil was'40 mA ~effective!. Dissipated power inthe samples@measured asI w

2@R8(v) 2 R#, see additionalremark~1! in Sec. VI# was less than 5 mW in all cases.

Also, we have measured the conductivity of a series ofsamples of cold-pressed silver powder~round plates of 10mm diam and 1 mm thickness! obtained at various pressingpressures.4

VI. ADDITIONAL REMARKS

~1! It is not only the equality of effective inductances,equality~4!, that can be used as an indication that Eq.~5! isvalid. Any other relation or procedure establishing thatB08 isthe same in both cases~i.e., with S1 at frequencyv1 and withS 2 at frequencyv2! can be used as an indication that Eq.~5!is valid. For example, it can be easily shown that if Eq.~5! isvalid @and, of course, Eq.~4!#, then

R28~v2!2R

v25

R18~v1!2R

v1~6!

@R is the active resistance of the coilLw alone;R18(v1) andR28(v2) are its effective active resistances at frequenciesv1

andv2, with samples S1 and S2 in its vicinity, respectively#.Therefore, it is possible to utilize Eq.~6! instead of Eq.~4!~with an appropriate modification of the apparatus!.

QuantitiesL8(v), which are present in Eq.~4! dependonly on that component of the fluxB8 through the coilLw ,which is in phase with the current inLw , while the quantitiesR8(v) 2 R, which are present in Eq.~6!, depend only onthe component of the flux, which is in quadrature with thecurrent inLw . Of course, it is of interest, in a general case, tohave a criterion for validity of equality~5!, depending onboth components of the flux~e.g., if the productvs is suf-ficiently small, then the magnetic field produced by eddycurrents in the sample is approximately in quadrature withthe current in the coilLw and, thus, this field almost has noinfluence on the effective inductance ofLw , etc.!.

If both sides of equality~4! are multiplied by j andadded to Eq.~6!, one obtains a complex equality dependingon both components of the flux ofB8 throughLw ,

R28~v2!2R1 j v2L28~v2!

v25

R18~v1!2R1 j v1L18~v1!

v1.

~7!

From Eq.~7! one may obtain two new formulations of theconditions for Eq.~5! to be valid, which depend on bothcomponents of the flux; the condition that the moduli of thecomplex quantities present in Eq.~7! as numerators are pro-portional to the frequencies:

$@R28~v2!2R#21@v2L28~v2!#2%1/2

v2

5$@R18~v1!2R#21@v1L18~v1!#2%1/2

v1, ~8!

and the condition that the arguments of these quantities areequal,

v2L28~v2!

R28~v2!2R5

v1L18~v1!

R18~v1!2R. ~9!

Let us denote byI w the current~effective! in the coilLw . It is clear that the productI w

2 @R8(v) 2 R# 5 W(v) isthe Joule heat, which is generated, per unit of time, in thesample due to the eddy currents.~Of course, we assume thatthere are no other causes of generating heat inside of thesample, e.g., due to hysteresis, which is, in fact, already con-tained in the initial assumptions of thevs method!. If wemultiply both sides of Eq.~6! by I w

2 , and if we denote heatsgenerated in S1 and S2 at frequenciesv1 andv2 by W1(v1)andW2(v2), respectively, we obtain

W2~v2!

v25

W1~v1!

v1. ~10!

This means that if Eq.~5! is valid and ifI w , in both cases, isthe same, the heats generated in S1 and S2, at frequenciesv1

and v2 respectively, are proportional to these frequencies.Therefore, Eq.~10! is also a criterion for the validity of Eq.~5!; this is, in fact, the same criterion as Eq.~6!, but ex-pressed by heats generated in S1 and S2. @If we would set theapparatus so thatI w

2 is inversely proportional to the fre-quency, we would have more simplyW2(v2) 5 W1(v1),instead of Eq.~10!.#

Consequently, it is possible to utilize Eq.~10! instead ofEq. ~6!, using an appropriate calorimeter for measuringW1

andW2.~2! Equalities~4! and~6! are mutually independent@i.e.,

if condition ~4! is satisfied, this does not mean by itself thatcondition ~6! is satisfied, and vice versa#, but both of theseconditions must be satisfied if Eq.~5! is valid. The sameapplies to conditions~8! and~9!. Therefore, ifv1 is chosen,it is necessary to adjust only one parameter,v2, so that bothpairs of these conditions are satisfied. But if the experimentshows that it is not so, i.e., that, e.g., condition~4! is satisfiedfor one value ofv2 and condition~6! requires some othervalue ofv2 in order to be satisfied, this means that at leastone of the initial assumptions for thevs method, given inthe basic statement in Sec. II is not fulfilled.

As an example, let us consider the case when sample S2,whose conductivity we measure, is a sufficiently5 strongparamagnetic~but not ferromagnetic! material, while refer-ence sample S1 is a weak paramagnetic or diamagnetic ma-

FIG. 2. The position of samples relative to the coil in experiments whoseresults are shown in Table I.

3525Rev. Sci. Instrum., Vol. 68, No. 9, September 1997 Electrical conductivity This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.171.233.72 On: Mon, 24 Nov 2014 23:14:38

terial, i.e., we suppose that condition~1! is satisfied bym1,but not bym2, i.e., thatm2 . 1. We shall also suppose thatthe productvs is not very large~and we also assume thatm2

is not much larger than 1!, so that the depth of penetra-tion of the electromagnetic field in the sample, (c2/vsm)1/2,is significantly larger than the linear dimensions of thesample, which also means that the magnetic field in thesample generated by eddy currents is significantly weakerthan the original fieldB. As is known,3 in this regime, theheating of the sample, due to the eddy currents, is approxi-mately proportional tov2; therefore, the quantityR8(v)2 R is also approximately proportional tov2 and, thus, theratio @R8(v) 2 R#/v increases, too, ifv increases.

Let us suppose that at firstm2 5 1 so that the obtainedvalue of v2 is one on which both conditions, Eqs.~4! and~6!, are satisfied, and thus, the correct value ofs2 is ob-tained: s2 5 v1s1 /v2. Let us say that afterwards,m2 isincreased but everything else remains unchanged. It is clearthat the increase ofm2 will cause an increase of the induc-tance of the coilLw ~similarly, as if we introduce a paramag-netic core inLw!, i.e., L28(v2) will increase; the increase ofeddy currents in S2 due to the increase ofm2 will only par-tially compensate this increase of the inductance. In order toachieve the previous value ofL28(v2), we have to increaseeddy currents by increasingv2. Therefore, ifv2 is deter-mined by criterion~4!, we will have a new~but wrong! valuev2, which is larger than the initial value~the right value!,and this means that we will now, fors2, obtain a wrongvalue, which is smaller than the right value.

On the other hand, the increase of eddy currents in S2

due to the increase ofm2 must cause an increase of heatingof S2, and this means thatR28(v2) 2 R will increase, andthus, the ratio@R28(v2) 2 R#/v2 also increases; in order todecrease this ratio to the previous value, we must decreasev2. Therefore, ifv2 is determined according to criterion~6!,we will have a new~wrong! valuev2, which is smaller thanthe previous~right! value, so that new~wrong! value fors2

will be larger than the right value.Therefore, ifm1 5 1 andm2 Þ 1 and the measurement

is performed in the regime as described, one will obtain dif-ferent values ofs2 depending on which criterion,~4! or ~6!,is used: one of these values will be larger and the second onewill be smaller than the right value. If as a result of measure-ment, we take their arithmetical mean value, we will have adeviation from the right value, which is smaller than half ofthe difference between these two values obtained.

It is clear that ifm1 5 m2 Þ 1, the measurement will beequally right as whenm1 5 m2 5 1 and both criteria,~4!and~6!, must give the same result@of course, criteria~8! and~9!, too#. Only a difference betweenm1 and m2 produceserror; it is obvious that the error will be smaller if S1 and S2

are either both paramagnetic, or both diamagnetic, with simi-lar susceptibility.

~3! As has been stated, the choice ofv1 is arbitrary, andthe result must be independent of that choice~of course, weassume that the ratios2 /s1 itself is completely independentof v1!. The repetition of the measurement for different val-ues ofv1 is a check of the results. Also, the result must beindependent of the position of S1 and S2 with respect to the

coil ~but, of course, the position must be the same for bothsamples!. However, this is so, in general, only if bothsamples are homogeneous and isotropic. On the contrary, if,for example, reference sample S1 is homogeneous and iso-tropic, but S2 is inhomogeneous, the result will, in general,depend on the choice ofv1. In such a case, the result willalso depend on the position of the samples in regard to thecoil. Therefore, it is possible to utilize thevs method as aprocedure for a contactless detection of inhomogeneity oranisotropy of a sample.

~4! Let us assume that some dependence ofs on v existsand that we want to establish it. If this would be performedby measuring the potential drop as a function ofv, weshould take into account the skin effect; it could happen thatthe dependence of resistance onv due to the skin effect ismuch stronger than the dependence of the resistance due tothe dependences on v. If we would solve the same problemusing the vs method, no corrections for the skin effectwould be necessary. Of course, since thevs method is arelative one, it is necessary to have a reference sample ofknown conductivity. In order to find the dependence ofs2

on v2 in the interval fromv2a to v2b , we should have areference sample whose conductivitys1 is either completelyindependent ofv1, or to know the dependence ofs1 on v1

in a corresponding interval fromv1a to v1b , but the redis-tribution of current density in the samples due to skin effectwould not affect the measurements. It is clear that if equality~5! is valid, then the fields of eddy currents in S1 and S2

would be the same, i.e., the complex amplitude of the currentdensity vector in any point of S2 at frequencyv2 is the sameas the complex amplitude of the current density vector in thecorresponding point of S1 at frequencyv1. On the basis ofthis, it seems that thevs method could be very convenientfor investigating the dependence ofs on v.

~5! The requirement that samples S1 and S2 must havethe same shape and size in some cases can be relaxed. Forexample, if S1 and S2 are flat plates sufficiently broad andthick so that each of them may be considered, relative to thefield of the coil, as an infinite conductive half-space, it isevident that their dimensions need not be equal~if the platesare not sufficiently thick, then their thicknesses must be thesame but other dimensions need not!. Also, if S1 and S2 aresufficiently long round cylinders having the same diameter,their lengths may be unequal, etc.

~6! Let us suppose that the sample is a conductive foil ora thin sheet metal~of course, nonferromagnetic!. Suppose,also, that its thicknessd is much smaller than the other char-acteristic linear dimensions of the coil and sample, and thatdis also much smaller than the depth of penetration of theelectromagnetic field into the material of this sample@i.e.,d ! (c2/vs)1/2]. It is obvious that under such conditionsB8depends ond in the same way as it depends ons. For ex-ample, the presence of a foil of thicknessd and conductivitys in a fixed position in regard to the coil has the same influ-ence onB8 as if this foil were of thickness 2d and of con-ductivity s/2. Thus, if we have two foil samples, S1 and S2,of unequal thicknesses~d1 andd2, respectively!, and of un-equal conductivities~s1 ands2, respectively!, the conditionthat B08 is the same in both cases is, instead of Eq.~5!,

3526 Rev. Sci. Instrum., Vol. 68, No. 9, September 1997 Electrical conductivity This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.171.233.72 On: Mon, 24 Nov 2014 23:14:38

v2s2d25v1s1d1 . ~11!

In the case that both foils are made of the same material, i.e.,if s1 5 s2, we obtain

v2d25v1d1 . ~12!

Hence, ifd1 is known and ifv1 andv2 are measured,d2 canbe calculated. Therefore, thevs method can be utilized as acontactless, calibrationless method for measuring the thick-ness of conductive, nonferromagnetic foils.

We have performed a check of thevs method for thick-ness measurements using Al foils of 0.02 mm thickness. Forexample, if we balance the bridge first with one foil at

v1/2p 5 5 kHz, and then we remove this foil and replace itwith five foils packed together, we obtain the bridge balanceat v2/2p 5 1 kHz, etc.

1For references see, for example, T. Ishida, K. Monden, and I. Nakada,Rev. Sci. Instrum.57, 3081~1986!.

2The frequency range used in this work extends approximately from 0.5 upto 35 kHz.

3For more details about the conditions of validity of Eqs.~1!, ~2!, and~3!,and cited boundary conditions, see, for example, L. D. Landau and E. M.Lifshitz, Elektrodinamika Splosˇnih Sred~Moskva, 1957!, Sec. 45.

4D. M. Todorovicet al., Proceedings of the 6th International Symposiumof the Science and Technology of Sintering~Haikou, China, 1995!.

5We do not know of a good conductor that is so strongly paramagnetic~ordiamagnetic! that one could easily perform an experimental check of theconsiderations given in this additional remark. Of course, this could bechecked out with normally paramagnetic~or diamagnetic! conductors,e.g., with Pt, Al, etc., using more sensitive apparatus than we have.

3527Rev. Sci. Instrum., Vol. 68, No. 9, September 1997 Electrical conductivity This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

129.171.233.72 On: Mon, 24 Nov 2014 23:14:38