Modeling Multi Frequency Eddy Current Sensor

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Modeling multifrequency eddy current sensor interactions during verticalBridgman growth of semiconductors

Kumar P. Dharmasena a) and Haydn N. G. Wadley b) Intelligent Processing of Materials Laboratory, School of Engineering and Applied Science,University of Virginia, Charlottesville, Virginia 22903

Received 13 March 1996; accepted for publication 22 February 1999

Electromagnetic nite element modeling methods have been used to analyze the responses of twoabsolute and differential eddy current sensor designs for measuring liquidsolid interfacelocation and curvature during the vertical Bridgman growth of a wide variety of semiconductingmaterials. The multifrequency impedance changes due to perturbations of the interfaces locationand shape are shown to increase as the liquid/solid electrical conductivity ratio increases. Of thematerials studied, GaAs is found best suited for eddy current sensing. However, the calculationsindicate that even for CdTe with the lowest conductivity ratio studied, the impedance changes arestill sufcient to detect the interfaces position and curvature. The optimum frequency for eddycurrent sensing is found to increase as the material systems conductivity decreases. The analysisreveals that for a given material system, high frequency measurements are more heavily weightedby the interfacial location while lower frequency data more equally sample the interface curvatureand location. This observation suggests a physical basis for potentially measuring both parametersduring vertical Bridgman growth. 1999 American Institute of Physics.

S0034-6748 99 02306-0

I. INTRODUCTION

The Bridgman method has become a widely used tech-nique for the growth of bulk single crystals from the melt. 1 5

Important semiconducting materials such as CdTe, GaAs,and Ge are all grown by variants of this technique. 3,4,69 Inthe vertical variant of the Bridgman method, an axisymmet-ric quartz, pyrolytic boron nitride ( p-BN), or graphite cru-cible containing the charge material is positioned in the hotzone of a vertically oriented, multizone furnace with a care-fully designed and controlled axial temperature gradient. Acrystal is produced by rst melting the charge in the hot zoneof the furnace and then either vertically translating the fur-nace with its associated temperature prole relative to thestationary crucible or vice versa. In either case, a solid crys-tal is nucleated at the bottom of the crucible and a liquidsolid interface propagates along the crucibles length ideallyresulting in a single crystal sample.

The yield and quality of single crystal material grown inthis way is a sensitive function of the thermal elds withinthe charge during the growth process. These can affect com-pound semiconductor liquid stoichiometry, 10,11 uid ow

patterns in the melted part of the charge,4,1214

the velocityi.e., time-dependent position and curvature i.e., the shapeof the liquidsolid interface, 1519 the kinetics of secondphase precipitation e.g., Te particles in Cd depleted CdTeduring cooling, 20 and the levels of residual stress induceddefects after cooling. 21,22 Analytical models that attempt topredict the evolution of these quantities during growth runshave emerged. 5,14 However, detailed experimental validation

of these models has been handicapped by the lack of experi-mental techniques for noninvasively monitoring many of these quantities as solidication progresses through thecrucible. 6,23

For instance, in order to validate predictive thermalmodels, 14 the temperature elds within both the melt and thegrowing crystal should be continuously measured throughoutgrowth. However, thermocouple arrays cannot be locatedwithin the crucible without seriously affecting the growth

conditions especially for systems like CdTe or GaAs whichhave a high vapor pressure at the melting temperature .Readings from thermocouple arrays on the outer surface of the ampoule are unreliable because radiative heat transfer inthe furnace environment can preferentially heat the thermo-couple. In addition, if the thermal conductivity of the mate-rial is low e.g., as with GaAs or CdTe and/or the ampoulediameter is large, the temperature within the charge may sig-nicantly differ from that at the outer surface of the ampoulewhere the thermocouples must be located. Several groupshave attempted to image the emitted infrared IR radiationusing infrared cameras. 24 However, unless the crucible andcharge are both transparent at the growth temperature, the IRmethod at best provides only an indication of surface tem-perature. This approach must also contend with many otherdifculties associated with limited access to the furnacework area, the stray radiation from nearby heating ele-ments, and uncertainties in the effective emissivity of thesample surface.

An alternative approach is to attempt the noninvasiveobservation of the time-varying interface position and there-fore solidication velocity and liquidsolid interface curva-ture throughout the growth process. These data would pro-

a Electronic mail: [email protected] Electronic mail: [email protected]

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 70, NUMBER 7 JULY 1999

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vide valuable information for validating/extending predictiveprocess models and for improving the growth process. Itmight also create the possibility for applying new feedback control approaches to crystal growth. 25

The rst step in developing this interface sensing ap-proach is to identify noninvasively measurable physicalproperties of the semiconductor that vary signicantly be-tween the melt and the crystal. Ideally, the changes in theseproperties accompanying solidication would be large com-pared to those associated with other effects e.g., changes of melt stoichiometry, liquid temperature, and uid ow . Onepossibility is the use of optical techniques for visual identi-cation of the liquidsolid interface. This is based upon thelarge difference between the solid and liquid optical reec-tion coefcients of many semiconducting materials. 26 Unfor-tunately, their implementation in a vertical Bridgman furnaceis hindered by the high background light intensity within thefurnace, the frequently opaque nature of the crucible/ ampoule e.g., p -BN or carbon coated quartz , and the oftenpoor optical transmission of the charge material at its meltingpoint. However, several other possibilities exist for semicon-ducting materials because of sometimes large differences inelectrical conductivity monitorable with eddy current sens-ing techniques ,2737 elastic constants via laserultrasonics ,38 density via x-ray radiography ,39 and specicheat perhaps measured with a photoacoustic method .

In this, and several related articles, 4042 the use of eddycurrent sensors for monitoring the vertical Bridgman growthof semiconducting materials is explored. The eddy currenttechnique exploits the sometimes very large electrical con-ductivity differences between the solid ( s) and liquid ( l)phases of many semiconducting systems. 43,44 Since both theabsolute conductivity and the conductivity difference are

likely to affect the performance of this sensing approach, thestudy explores the application of eddy current methods to avariety of semiconductors, Table I. Silicon, though not com-mercially produced by a Bridgman method, is included in thematerials analyzed to span a broader conductivity range, andto establish the sensor performancetest material conductiv-ity relationship.

Table I shows that for most semiconductors, the electri-cal conductivity of the liquid is many times that of the solidat the melting point. The principle underlying the applicationof an eddy current sensor approach to crystal growth is basedon the observation that the eddy current density induced at apoint within a test sample by the electromagnetic eld of an

alternating current ac excited coil is proportional to thesamples electrical conductivity at that point. Since the elec-trical conductivity of liquid semiconductors exceeds that of the solid, higher eddy current densities are expected to existwithin liquid regions of a solidifying charge. Sensors basedupon this principle have been previously proposed for mea-suring solidication conditions and temperature proles dur-ing the Czochralski growth of GaAs and Silicon. 35,36 Theyare widely used in other types of high temperature materialsprocessing, e.g., for determining internal temperatures withinaluminum alloy extrusions 27 and for the measurement of di-mensional changes during hot isostatic processing. 45

The response of an eddy current sensor is a complicated

function of the eddy current distribution induced within thesample by the uctuating electromagnetic eld of an excita-tional coil. This will be affected by the geometry of the ex-citing coil which governs the electromagnetic elds distri-bution within the test material , the coils excitation currentfrequency, the fraction of material solidied in the interro-gated volume, the shape of the boundary between the solidand liquid regions, and the respective conductivities of thesolid and liquid. In the eddy current technique, the distribu-tion of eddy currents induced in the sample is sensed fromtheir effect on the impedance of either the exciting coil or aseparate pickup coil. It will be a sensitive function of thesensors design and test frequency as well as all the materialand growth parameters listed above. Experimental methodsmight be used to perfect the sensor design, optimize the testfrequency, and develop data analysis protocols. However, itis costly and time consuming to equip a crystal grower witha variety of eddy current sensors, and experimentally designa sensor approach. Furthermore, a denitive validation of theresponse is almost impossible because of the lack of inde-pendent observations of the solidication front.

An alternative approach to sensor design has been pur-sued here. The responses of several sensors have been simu-lated using electromagnetic nite element techniques for a

FIG. 1. Schematic diagram of a dual coil eddy current sensing arrangement.A seven-turn driver coil is used to excite an electromagnetic eld. Either a

single coil or a pair of opposingly wound coils are used to pick up theperturbed ux.

TABLE I. The electrical conductivities of selected solid ( s) and liquid( l) semiconductors close to their melting points.

Electricalconductivity

Semiconductor material

CdTe a GaAs b Sib Ge b

s S/m 1200 3.0 104 5.8 104 1.25 105

l S/m 6600 7.9 105 1.2 106 1.4 106

l / s 5.5 26.3 20.7 11.2

aReference 43.bReference 44.

3126 Rev. Sci. Instrum., Vol. 70, No. 7, July 1999 K. P. Dharmasena and H. N. G. Wadley



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variety of liquidsolid interface locations/curvatures, and for

several different materials systems. This allows quantitativerelationships to be obtained between growth parameters suchas the liquidsolid interface location/shape and measurablequantities of an eddy current sensors response e.g., thefrequency-dependent test coil impedance . This approachalso has the advantage of allowing anomaly free protocols tobe designed for deducing the growth parameters from mea-sured experimental data, and provides guidelines for evalu-ating their potential for other material systems.

Here, the simulated responses for two candidate eddycurrent sensor designs are obtained for each of the four ma-terial systems given in Table I. The results are used to inves-tigate the effects of changing either the interface position or

its shape on the sensors complex impedance in the mostexperimentally accessible 200 Hz2 MHz frequency range.It is shown that the impedance change due to a perturbationof the interfaces position or shape is greatest for GaAs thematerial with the highest liquid/solid electrical conductivityratio and is least for CdTe with the lowest conductivityratio . The sensitivity to both location and shape has beenfound to depend strongly upon frequency. The frequency formaximum sensitivity to interface shape change increases asthe test materials conductivity decreases. Thus, the best op-erating frequency range is unique to each material. At hightest frequencies, the sensors response is shown to be domi-nated by the interface location and is almost independent of

interface shape. At lower frequencies, both shape and loca-

tion contribute to the predicted impedance. This provides thephysical basis for the possible discrimination of the locationand shape contributions to eddy current sensor responses andthus their independent monitoring during vertical Bridgmangrowth.

II. EDDY CURRENT SENSOR DESIGN CONCEPTS

The physical basis of all eddy current sensor approachesis electromagnetic induction. A coil carrying an alternatingcurrent is rst used to create a uctuating electromagneticeld. 46 Eddy currents are then induced in any conducting

FIG. 2. a Finite element model geometry; b nite element mesh in interface region.

TABLE II. Electromagnetic skin depths a in several solid ( s) and liquid( l) semiconductors.

FrequencykHz

Skin depthmm CdTe GaAs Si Ge

10 s 205.5 29 20.9 14.210 l 65 5.7 4.6 4.3

500 s 29 4.1 3.0 2.0500 l 9.2 0.8 0.65 0.6

2000 s 14 2.1 1.48 1.02000 l 4.6 0.4 0.32 0.3

aFor cylindrical samples, Eq. 1 can be used only if the skin depth issignicantly smaller than the radius of the cylinder. Signicant errors incalculated eddy current densities arise if the skin depth is larger than, or isof the same order of magnitude as the sample radius Ref. 48 .

3127Rev. Sci. Instrum., Vol. 70, No. 7, July 1999 Eddy current sensor



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medium placed within this eld. These currents create a sec-ondary electromagnetic eld which perturbs the primary eldand changes the inductance of the coil. If the impedances of

other circuit elements are small compared to the coil induc-tance, the impedance of a test circuit containing the coil willbe directly related to the coils inductance and those proper-ties of the test sample that control the eddy current distribu-tion within it e.g., its electrical conductivity or magneticpermeability . By analyzing the coils impedance, it is pos-sible to infer the electrical conductivity/magnetic permeabil-ity and even its spatial distribution within the region of thetest material sampled by the electromagnetic eld. 2736 Thiseld depends upon the coils geometry its number of turns,diameter, axial length, etc. and the extent of penetration of the primary eld into the sample. The latter is governed bythe test materials electrical conductivity , the magnetic

permeability , and by the angular frequency 2 f ,where f is the frequency in hertz of the excitation current.

The depth at which the magnetic eld intensity or in-

duced eddy current falls to a value 1/ e ( 0.368) from that atthe sample surface, is dened as the skin depth, , givenby47,48

2

. 1

Values of this skin depth at three readily accessible testfrequencies 10 kHz, 500 kHz, and 2 MHz are given inTable II for the four materials listed in Table I. For eachmaterial, the magnetic permeability has been taken to be thatof free space ( 4 10 7 H/m). Large skin depthsgreater penetration of the eld into the sample are obtained

FIG. 3. Calculated absolute sensor impedance curves for the liquid and solid states of a CdTe, b GaAs, c Si, and d Ge.




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when the conductivity and/or the frequency are low. Foreach of the four materials, it is possible to probe to differentdepths below the sample surface by varying the sensors ex-citation frequency. If the electrical properties were to vary

with radial position in a cylinder e.g., if the liquidsolidinterface were curved , the sensors frequency-dependent re-sponse will be perturbed from that of a sample with no radialgradient in properties i.e., one with a at interface and in-sight might be gained about the interface shape. Table IIsuggests that the best range of frequencies to reveal sucheffects are test material dependent.

Several experimental approaches have been developedfor eddy current sensing. In the simplest, 33 a single coilsenses small sample induced perturbations to its own eld.When this approach is applied to the crystal growth environ-ment, anomalous changes in coil impedance can accompanytemperature changes of the coil. These arise from the change

in the coils ac resistance due to the windings temperature-dependent resistivity. They can be reduced by using coil ma-terials with a low thermal coefcient of conductivity, butthey cannot be eliminated. This drawback can to some extent

be overcome with a dual coil system. In this approach, onecoil is excited with an alternating current to create the pri-mary electromagnetic eld, and a second coil is then used todetect sample induced perturbations to the eld.

Figure 1 shows a schematic diagram of a two-coil sys-tem embodiment. The transfer impedance of such a sensor, Z , is given by

Z V s / I p , 2

where I p is the phasor excitation current in the primary ordriving coil and V s is the phasor voltage induced across theterminals of the secondary coil. The transfer impedance of

FIG. 4. Imaginary component of impedance vs frequency for the liquid and solid states of a CdTe, b GaAs, c Si, and d Ge.




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such a system is relatively insensitive to the resistance of thecoils if, a the induced voltage, V s , is measured with a highimpedance instrument and b the current, I p , is continu-ously monitored e.g., by observing the voltage across a pre-cision resistor placed in series with the primary coil . Thisenables operation of the sensor at high temperatures withoutthe need for either correcting for coil temperature or poten-tially invasive coil cooling. 46 With two-coil systems likethese, the primary coils axial length can also be made much

longer than the secondary coil, so producing a relatively uni-form eld in the sensing region.For a two-coil system, the ratio of the induced voltage in

the secondary coil to the current in the primary coil can beconveniently obtained from its multifrequency gain-phase re-sponse measured with a network analyzer. 46 To simplify in-terpretation of this type of data, the resulting complex im-pedance components are usually normalized with respect tothe empty coils impedance measured in the samples ab-sence. The results can be presented in the form of impedanceplane diagrams which are plots of the real and imaginarycomponents of the impedance the abscissa and ordinate, re-spectively as a function of frequency. The resulting imped-

ance curves are also functions of the sample and sensor ge-ometries and the electrical/magnetic properties of the testmaterial.

Figure 1 shows two encircling eddy current sensor de-signs selected for detailed study. Both have the same seven-turn primary coil for excitation. One uses a single-turnpickup coil located at the primary coils midpoint, while theother uses two opposingly wound pickup coils located nearthe ends of the primary coil. The single pickup coil sensorarrangement is called an absolute sensor, while the twopickup coil design sensor is referred to as a differentialsensor.

To envision the way such sensors might be used to

monitor solidication during vertical Bridgman growth, con-sider a control volume element in the vicinity of either sen-sor as indicated by the dashed area in Fig. 1 . The length of this control volume is determined by the effective axialrange of the electromagnetic ux created by the primary coil.Its penetration into the sample is controlled by the skin depthin the material under investigation. The induced eddy cur-rents will be inuenced by the volume fractions of the solidand liquid within this control volume. This changes if either

the sample containing a liquidsolid interface remains sta-tionary and the sensor is translated, or if the sensor/sampleare both stationary and the growth furnace is translatedcausing the solidication front to propagate along the

sample . For this latter scenario, the region initially sampledwould be liquid whereas at the end of growth after the in-terface has moved through the control volume , the sensedregion would be fully solid. It is shown that the sensorsresponse will always be bounded by its response to these twostates, and all measured responses during growth must liebetween these two extremes.

A single secondary coil sensor design will be most re-

sponsive to the eddy current density closest to the secondarycoils location. It is likely to be relatively unaffected by eddycurrents excited far from the coil. The sensitivity of a sensorto the presence of an interface could potentially be improvedby using a pair of opposing wound secondary coils locatedabove and below an interface. The response of such a differ-ential sensor will be dependent on the difference in samplecreated eld perturbation at the two coil locations; commoncontributions to the two coils induction will be canceled outin this conguration. The sensitivity of a differential sensormeasurement is likely to vary with the spacing between thetwo pickup coils providing an additional degree of freedomfor sensor design.

FIG. 5. Magnetic vector potential contours for liquid CdTe at frequencies of a 10 kHz, b 500 kHz, and c 2 MHz.




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III. FINITE ELEMENT SIMULATION MODEL

The differential equation governing eddy current genera-tion in conducting materials is derived from Maxwells equa-tions and can be expressed in terms of an unknown magneticvector potential A . For a sinusoidal current of angular fre-quency , the governing equation is

2A j A J s , 3

where, J s is the source current density. For inhomogeneousproblems, the boundary conditions require that A and its nor-mal derivative be continuous across all material interfaces.

If the spacing of the primary coil turns is small com-pared to their diameter, it can be assumed that the coil iscircular symmetric i.e., the helical effect of the coil can beignored . The primary coil can then be modeled as a series of circular loops of a known radius spaced a distance apartequal to the pitch of the primary coil winding. Since thesample encircled by the eddy current sensor is contained in acylindrical crucible, the entire geometry allows axisymmetriccalculations rather than full three-dimensional simulations.Considerations of the cylindrical geometry Fig. 1 show that

only one half of an axisymmetric plane must be analyzed,Fig. 2 a . In addition, no electromagnetic ux can cross theaxis of symmetry, and hence a zero vector potential bound-ary condition can be specied on the axis.

Closed form theoretical solutions to electromagnetic in-duction problems are in general limited to simple geometriesand are based on simplifying assumptions for the geometryfor example, the sample is assumed to be innitely long,

cylindrical etc. , and its electrical properties are assumed uni-form throughout the interrogated volume. 49 Electromagneticnite element modeling provides a convenient tool to evalu-ate eddy current sensor responses for the sensor and samplegeometries encountered here provided the electromagnetic

properties i.e., the electrical conductivity and the magneticpermeability of the sample material are known along withthe sensors excitation frequency. 5055

The problem modeled consisted of a cylindrical 76 mmdiam sample containing one of ve interface locations andve interface curvatures. Two of the interfaces had convexshapes dened by a convexity parameter z / D

0.167, 0.333 where z is the interface curvature height

on the axis and D the sample diameter , one interface wasat ( 0.0), and the remaining two were concave( 0.167, 0.333). The nonplanar interfaces were hemi-spherical surfaces of differing radii of curvature. In order tominimize the effects of mesh size on the solution, all veinterface shapes were incorporated into one nite elementmodel and the same nite element mesh see Fig. 2 b wasused for all the calculations. The different models corre-sponding to each interface shape/position were built fromthis mesh by changing the assigned material properties i.e.,the electrical conductivity of the elements in the mesh tocreate regions of solid, liquid, or air.

In order to account for the skin effect at high frequen-cies, the nite element mesh was rened in the interfaceregion with an increased number of elements concentratedtowards the edge of the charge. As a result, the elements withthe smallest depth 0.038 mm were placed along the outersurface of the sample. These element sizes were smaller thanthe skin depth at the highest frequency analyzed 2 MHz forthe most conductive sample condition liquid Ge . Themodel had a total of 913 grid points and 1007 triangular andquadrilateral elements. Additional mesh renement wasconstrained by the limitations of the commercial electromag-netic analysis package 56 used for the creation of the axisym-metric nite element model. However, this step was not con-sidered to be important since calculations were performedwith and without the sample using the same element mesh toobtain normalized impedance values. The output of themodel allowed calculation of the inductive reactance of thecoil. The model did not incorporate the capacitive reactanceor the ac resistance of the coils, nor the impedance contribu-tions of other test circuit elements.

The nite element code solved Eq. 3 for the magneticvector potential A ( r , z), where r is the radial and z the axialposition subject to a prescribed source current applied loaddistribution and boundary conditions. The applied load forthis problem was the driving current in the multiple turnprimary coil. This was specied as a point current at each of

the seven grid points corresponding to the location of each of the seven turns on the primary coil. Since each calculationwas normalized with respect to the empty coil condition, theactual value of current in the primary coil was not importantand for convenience was taken to be unity.

The magnetic vector potential obtained from the niteelement calculations can be directly used to obtain the sen-sors transfer impedance see Ref. 57 for details . For anabsolute sensor, it can be shown that

Z 4 2 f N sr s

I pIm A ave j Re A ave , 4

FIG. 6. Variation of the imaginary component of impedance with the liquid/

solid conductivity ratio for the absolute sensor.




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where r s is the secondary coil radius, N s is the number of turns in the absolute secondary coil, f is the test frequencyand A

aveis the average vector potential over the cross section

of the secondary coil wire.For an axially separated differential sensor

Z 4 2 f r s

I p N s 1 Im A ave j Re A ave 1

N s 2 Im A ave j Re A ave 2 , 5

where N s 1 and N s 2 are the number of turns at the two loca-tions of the differential secondary. All of the calculated coilimpedances were normalized with respect to the coil imped-ance at the calculation frequency. This was obtained by re-placing the relevant electromagnetic properties and of

the solid and liquid region elements of the charge bythose of the air elements, and repeating the nite elementanalysis.

IV. ABSOLUTE SENSOR RESPONSE

A. Homogeneous liquid or solid states

The simplest problems to analyze are the initial and nalstates of a growth run when the sensor observes either onlythe melt prior to solidication or only the solid aftercompletion of growth . In this case, the test material wasassumed to have a uniform conductivity as dened in TableI. Figure 3 shows the absolute sensors calculated normalizedimpedance response for the liquid circles and solidsquares states of CdTe, GaAs, Si, and Ge at 13 frequencies

FIG. 7. Normalized impedance curves for three positions of a at interface for the absolute sensor a CdTe, b GaAs, c Si, and d Ge.




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between 200 Hz and 2 MHz. The impedance data for boththe liquid and solid states of all four materials fall on thesame characteristic comma-shaped curve. The shape isalmost identical to that expected for an innite conductingcylinder contained in a long solenoid. 47,49 The size of thiscurve can be characterized by its high frequency intercept I with the normalized imaginary impedance component axis.This is a function of the sample and pickup coil diameters,and is independent of the test material conductivity. 27 For an

innitely long cylinder contained in a long solenoid, fringeeld effects are insignicant and I 1 ( d s / d c)2 1 ll

factor, where d s is the sample diameter and d c the diameterof the secondary coil. Both the liquid and solid forms of allfour test materials intercept the imaginary axis at the samepoint because the intercept is independent of the test materi-als conductivity in the high frequency limit. In this limit allconductors totally exclude the penetration of ux into thesample. This well understood phenomenon is the basis foreddy current dimensional sensing 27 and could be exploited invertical Bridgman growth e.g., to detect debonding of thesolid from its ampoule during cooling .

The only difference between the sensors response to

either a solid or liquid test material is a shifting of frequencypoints along the impedance curve. A decrease in conductiv-ity, associated for example with solidication, causes the im-pedance at a xed frequency to move counter clockwisearound the curve because the sample becomes less inductive.This also explains why the length of the impedance curvecalculated up to 2 MHz decreases as the test material con-ductivity decrease. In the limit, as the conductivity of the testmaterial approaches zero, the normalized impedance at eventhe highest frequencies would be located at (0 j), i.e., atthe upper left corner of the impedance plane, which is thesame as the no sample or coil in air situation.

The imaginary component of impedance i.e., the nor-

malized inductive reactance is plotted as a function of fre-quency for the liquid and solid states of the four test materi-als in Fig. 4. At low frequencies e.g., below 10 kHz forCdTe , each material system gives a null response. Thisarises because the rate of change of the electromagnetic eldwithin the test material is insufcient to induce detectableeddy currents. The sample is effectively transparent to theeld, and the sensors response is similar to that when nosample is present. Figure 5 shows vector potential magnetic

ux contours for liquid CdTe for three test frequencies.Note that at a frequency of 10 kHz, Fig. 5 a , the vectorpotential contours are indistinguishable from those of anempty coil.

Figure 4 shows that beyond this threshold frequency, aclear separation of the liquid and solid impedance curves isseen, and a measurement of the imaginary impedance com-ponent in this region could be used to distinguish betweenthe solid and liquid states. Beyond the threshold frequency,the separation of the curves is seen at rst to increase, reacha maximum, and nally decrease as the frequency is in-creased. For the highest conductivity material Ge , theliquid/solid impedance separation decreases more rapidly as

the test frequency increases because the skin effect moreeffectively expels ux in higher conductivity materials. Thisux expulsion can be clearly seen in the vector potentialmagnetic ux contour plots of Figs. 5 b and 5 c . The

impedance of the sensor in the intermediate range of fre-quencies where detectable eddy currents are excited in thesample but ux expulsion is not complete depends bothonthe test samples diameter and its conductivity. Data col-lected at these frequencies where skin depths are around 0.5times the sample radius is widely used to measure the con-ductivity of test materials of known diameter obtained fromhigh frequency data and to infer sample conditions that af-fect it e.g., temperature .27

FIG. 8. Magnetic vector potential contours for CdTe at a frequency of 500 kHz for a at interface position at a h 12.7 mm, b h 0 mm, and c h12.7 mm.




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The solid and liquid impedance curves represent the sen-sors response to two extremes of a growth process; they arethe upper and lower bounds of all sensor responsesthat could be encountered during growth. All the impedancecurves observed, no matter what interface shape or position,must lie between these bounds. The relative sensitivity of aneddy current solidication sensor to interfacial shape/ position will depend upon the magnitude of separation of the

liquid and solid impedance curves (Im Z ) which is a func-tion of the excitation frequency. Examination of Fig. 4 re-veals that there exists a characteristic frequency where a

maximum impedance separation and thus, sensor sensitiv-ity occurs. Values for Im Z max and the frequencies at whichthey occur are tabulated for each of the test materials inTable III.

Examination of Tables I and III reveals that Im Z maxmonotonically increases with the liquid to solid conductivityratio, Fig. 6. The frequency at which the maximum differ-ence occurs varies inversely with the melt or solid conduc-tivity. Germanium which has the highest liquid and solidconductivities has the lowest frequency where the maximumimaginary impedance change occurs. The highest frequencyoccurs in CdTe which has the lowest liquid and solid con-ductivities. Clearly, the best frequency for operation of an

FIG. 9. Variation of the imaginary component of impedance with frequency for an absolute sensor for ve positions of the at interface a CdTe, b GaAs,c Si, and d Ge.

TABLE III. The maximum difference between the imaginary impedancecomponents (Im Z max ) of an absolute sensor for the homogeneous liquidand solid states and the frequency at which it occurs.

Material CdTe GaAs Si Ge

Im Z max 0.2417 0.3817 0.3625 0.3167Frequency Hz 2.5 105 7.5 103 4.5 103 2.2 103




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eddy current solidication sensor is a material dependent pa-rameter, and varies from one system to another. All of thefour systems analyzed here have a sufciently high conduc-tivity that the frequency of maximum response is wellwithin the range of frequencies experimentally accessiblewith conventional eddy current sensing instrumentation andhave a sufciently large Im Z max value for reliable eddycurrent sensing.

B. Interface position effects

To assess the response of an absolute sensor to the po-sition of an interface, a series of calculations were performedfor ve locations of a at interface. Figure 7 shows calcu-lated normalized impedance curves for three of these posi-tions. The impedances of all four materials are seen to con-verge at high frequency and again approach a commonintercept with the imaginary axis because in the modeled

problem all four materials have the same diameter and there-fore identical ll factors . In the limit, as the test frequencyapproaches innity, the sensors response depends only uponthis ll factor and is independent of the interfaces positionwithin it, even when a liquidsolid interface exists within theinterrogated volume.

The impedance curves are seen to be a strong function of interface position at lower frequencies, Fig. 7. Recall that inthe completely liquid or solid cases Fig. 3 , the sample actedlike an innite cylinder of uniform conductivity encircled bya long solenoid, and changes of conductivity only shifted theimpedances at specic frequencies around a common curve.However, when an interface between dissimilar conductivitymaterials exists within the eld of an eddy current sensor, theobserved response can be viewed as the net effect of simul-taneous interactions with two nite length cylinders of dif-ferent conductivities. Fringing of the eld at the interface

FIG. 10. Normalized impedance curves for three interface shapes for the absolute sensor a CdTe, b GaAs, c Si, and d Ge.




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allows the impedance for a given frequency to move at anonzero angle to the characteristic impedance curve, in effectshrinking its size. The contribution of this effect must de-

pend upon where the interface is located with respect to thesensor. Thus, when the interface is below the center point of the sensor i.e., h 12.7 mm, Fig. 7 , more of the higherconductivity liquid cylinder is sampled by the encirclingeddy current sensor. The solid cylinder is still in the eld of view of the sensor, but contributes less to the sensors re-sponse. Later in a growth process when the interface hasgrown upward beyond the center of the stationary sensor,say to a position h 12.7 mm above the sensors center,more of the lower conductivity solid cylinder is encircled bythe sensor and a lesser contribution is made by the liquidregion.

The consequence of this phenomenon can be clearly

seen from the magnetic vector equipotential contours uxlines . Figure 8 shows the 500 kHz vector potential eld forthe CdTe material system for interface heights of 12.7, 0

and 12.7 mm. Because of the bigger skin depth in the solidphase, the depth of penetration into the solid is always muchgreater than that of the liquid. Since the vector potential iscontinuous across the liquid/solid interface, the eld near theinterface is perturbed from that expected for a homogeneouscylinder of either conductivity. The extent of this perturba-tion depends on the frequency of excitation through the skineffect and the relative position of the interface within thesensing coil. Since the elds are no longer the same as thoseof an innite uniform cylinder, the sensors response departsfrom that of an ideal uniform cylinder Fig. 3 and pro-vides the potential for a method of sensing position.

Since this behavior again originates from the skin effect,

FIG. 11. Variation of imaginary component of impedance with frequency for ve interface shapes for the absolute sensor a CdTe, b GaAs, c Si, and dGe.




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the frequency at which it occurs will be conductivity andthus test material dependent. This can be seen more clearlyby plotting the normalized imaginary impedance componentagainst excitation frequency for each interface position, Fig.9. Again, there exist material dependent low and high fre-quency thresholds below/above which the sensors responseis independent of frequency. However, for each materialthere also exists an intermediate range of frequencies wheresensitivity to interfacial position is a maximum. The sensi-tivity i.e., the difference in impedance for h 12.7 mmand optimal frequency are listed in Table IV for each testmaterial. For the highest conductivity materials like germa-nium or silicon , the curves converge at signicantly lowerfrequencies than for lower conductivity materials such asCdTe. This is because the penetration depth of the electro-magnetic eld becomes innitesimal i.e., approaches the in-nite frequency limit at lower frequencies in higher conduc-tivity materials. From Tables I and IV it is also observed thatthe maximum separation due to interfacial position(Im Z max ), increases/decreases as the liquid / solid ratioincreases/decreases. The frequency at which this maximumposition effect occurs varies inversely with either the liquidor solid electrical conductivity.

Figure 9 shows that in the intermediate range of frequen-

cies, the sensors imaginary impedance component is amonotonic function of interface position. If such a sensorwere used to monitor an interface that moved through thesensor, the sampled fractions of liquid and solid wouldkeep changing within the volume interrogated by the sensor.When the interface was well below the sensor i.e., h

12.7 mm , a larger fraction of the high conductivity liq-uid would be sampled in the limit of h tending to minusinnity, a uniform liquid response like that of Fig. 4 wouldbe obtained . As the interface approached the sensor, a pro-gressively increasing fraction of solid would be sensed,and as the interface moved past the sensor, its responsewould approach that for a uniform solid, Fig. 4. Since theliquid conductivity is always much greater than the solids,the net effect would always be an increase in the imaginaryimpedance as each of the liquids studied gradually turnedinto solid during the growth process.

C. Interface shape effects

During Bridgman growth, the liquidsolid interfaceshape can be concave, at, or convex and can change curva-ture as the governing heat and uid ow conditions evolveduring growth. 14 In order to assess the response of the abso-

FIG. 12. Magnetic vector potential contours for CdTe at a frequency of 500 kHz for three interface shapes a 0.333, b 0, and c 0.333.

TABLE IV. Imaginary impedance component values at the frequency of maximum sensitivity to interface position for the absolute sensor.

Relative interfaceposition, h mm CdTe GaAs Si Ge

12.7 0.610 0.567 0.570 0.5986.4 0.640 0.600 0.602 0.6280 0.682 0.654 0.654 0.6746.4 0.734 0.722 0.717 0.728

12.7 0.778 0.782 0.774 0.773Im Z max 0.168 0.215 0.204 0.175Frequency Hz 4 105 1 104 6 103 3.4 103

TABLE V. Imaginary impedance component values where maximum inter-face shape effect occurs for the absolute sensor.

Interface convexity, CdTe GaAs Si Ge

0.333 0.818 0.808 0.818 0.8240.167 0.786 0.769 0.780 0.7900 0.762 0.740 0.752 0.7650.167 0.738 0.714 0.726 0.7400.333 0.711 0.684 0.696 0.712

Im Z max 0.107 0.124 0.122 0.112Frequency Hz 2 105 3.7 103 2.1 103 1.5 103




14/18

lute sensor to this curvature, a series of calculations havebeen conducted where the interface shape has been allowedto have one of ve shapes specied by a convexity param-eter, z / D , where z is the difference in axial intersection

of the interface central axis and periphery of the sample i.e.,the interface height , and D is the diameter of the sample seeFig. 2 . For each calculation, the point where the interfaceintersected the outer boundary of the test material was xedat the axial location of the secondary coil i.e., h 0 .

Figure 10 shows examples of the normalized impedancecurves for a convex ( 0.333), a at ( 0.0), and aconcave ( 0.333) interface. The shape of the interface isseen to have a small but signicant effect upon the structureof the impedance plane curve. The dependence upon interfa-cial curvature disappears at low and high frequencies andexhibits a similar frequency dependence to the interface po-sition effect. This can be more clearly seen when the imagi-

nary impedance component is plotted as a function of fre-quency for the ve interface shapes, Fig. 11.

The dependence of the intermediate frequency imped-ance upon interface curvature again results from the electro-

magnetic ux interaction with each interface, Fig. 12. In thehigh frequency limit, the ux is conned closer to the samplesurface, and the sensors response is insensitive to the inter-nal interface shape. This limit is most nearly approached inthe higher conductivity materials Ge, Si, GaAs for frequen-cies beyond 1 MHz. For lower conductivity materials such asCdTe, it would be necessary to increase the frequency to-ward 10 MHz in order to obtain an impedance that is almostindependent of interface shape.

At lower frequencies, the sensors imaginary impedanceshows a signicant dependence upon interfacial curvature.The frequencies at which the interface shape effect is a maxi-mum and the magnitude of the impedance changes are both

FIG. 13. Normalized impedance curves for three positions of a at interface for the differential sensor a CdTe, b GaAs, c Si, and d Ge.




15/18

given in Table V for each of the test materials.Although the effect of curvature upon the imaginary

impedancefrequency relationship was similar to that seen

for interface location, the maximum interface shape effectsoccurred at a lower frequency than the interface position ef-fect for all four materials. To understand why this occurs,recall that all of the interfaces meet at the same point on thetest samples outer boundary. It is only when the magneticux penetrates sufciently deep into the material that itsamples the interior solidliquid boundary that each inter-face will differently perturb the ux at the secondary coillocation. In CdTe, this is seen to occur at 500 kHz, Fig. 12.In contrast, the interface position still affects the response of the sensor even when the ux is concentrated very close tothe edge of the crystal, i.e., when operating at higher fre-quencies. It is only when the innite frequency limit is ap-

proached that one loses sensitivity to the position.This analysis of an absolute sensors response has shown

it to be sensitive to both the position and shape of the inter-

face. The sensitivity to both phenomena is frequency depen-dent and a maximum sensitivity exists at intermediate fre-quencies. The analysis has shown that both location andposition effects are coupled in an impedance measurement inthe intermediate frequency range. However, careful measure-ments over a range of frequencies may be able to separatelyresolve the two growth parameters because of their differentfrequency dependencies.

V. DIFFERENTIAL SENSOR

The essential idea of an axially displaced differentialsensor is to sample the difference in eld perturbation at two

FIG. 14. Variation of the imaginary component of impedance with frequency for a differential sensor for ve positions of a at interface a CdTe, b GaAs,c Si, and d Ge.




16/18

positions along the axis of a sample. By placing two oppos-ingly wound secondary coils at these locations and ensuringthat they are symmetrically located within the primary coil,equal magnitude, but opposite sign voltages are induced inthe coils when a homogeneous sample is present. The intro-duction of an inhomogeneous sample with different conduc-tivities near the two pickup coils will perturb the electromag-netic ux at one coil more than the other, and a nonzeroresultant voltage will be observed. Thus, such a sensor willbe incapable of distinguishing between an entirely liquid orsolid sample because of equal but opposite induced voltagesat the two coil locations , but might exhibit enhanced sensi-tivity to the location and curvature of an interface separatingmaterials of different electrical conductivity.

A. Interface position effects

Figure 13 shows the effect upon the normalized imped-ance curve of moving a at interface through a differential

sensor. For these calculations, the two secondary coils wereplaced close to either end of the primary coil they were 34mm apart . It can be seen that the imaginary component of impedance at rst increased with frequency, reached a maxi-mum, and then decreased for each location. Figure 13 indi-cates that the frequency corresponding to the maximumimaginary component was interface position dependent thefrequency increased as the interface passed upwards throughthe sensor . This can be seen more clearly in Fig. 14 wherethe imaginary impedance component is plotted as a functionof frequency for each h value. At or above the frequency of maximum response, the impedance reached its maximumvalue well after the interface had passed through the centerof the primary coil. The exact location at which this occurredwas determined by the relative magnetic vector potentials ateach secondary coil location. This depends on the test fre-quency and the electrical conductivities of the solid and liq-

FIG. 15. Normalized impedance curves for three interface shapes for the differential sensor a CdTe, b GaAs, c Si, and d Ge.




17/18

uid. As the frequency is further increased beyond this peak,the curves for each interface location continue to remainseparated until very high frequencies are reached, again dueto differences in the fringe eld at the two coil locations.

B. Interface shape effects

Interface shape effects were investigated by changingthe interface shape while maintaining the outer edge of theinterface midway along the primary coil. The normalizedimpedance curves for concave, at and convex interfaces areshown in Fig. 15. The size of the impedance curve was ob-served to increase as the interface curvature changed fromconcave to convex. The sensitivity to interface shape at rstincreased with frequency, went through a maximum at a ma-terial dependent frequency, and then decreased again at highfrequency beyond 2 MHz before the curves eventually con-

verged, Fig. 15 a . This can be seen more clearly in Fig. 16which shows the imaginary impedance components fre-quency dependence. These calculations reveal the existenceof a relatively narrow, material specic, intermediate rangeof frequencies where a strong sensitivity to the interfacescurvature exists. In this region, the imaginary impedancecomponent monotonically increases as the interfaces shapechanges from concave to convex. Above and below this re-gion the sensor has little or no sensitivity to curvature.

If Figs. 14 and 16 are compared, it is again apparent thatthe calculated impedance above 10 5 Hz 107 Hz for CdTe isdominated by the interfaces location while lower frequencydata are sensitive to both the interface curvature and the po-sition. Therefore, data collected over a range of frequenciesmay be sufcient to separately discriminate interface loca-tion and shape. The range of frequencies where the sensor

FIG. 16. Variation of imaginary component of impedance with frequency for ve interface shapes for the differential sensor a CdTe, b GaAs, c Si, andd Ge.




18/18

signicantly responds to interfacial curvature is seen to bereduced with the differential sensor arrangement because of fringe eld effects at the ends of the primary coil. The mag-nitude of the imaginary impedance components change as-sociated either with movement of the interface or a change of its curvature are also signicantly enhanced with the differ-ential sensor design compare the ordinate scales of Figs. 10and 16 . Thus, from a practical point of view, higher quality

information about interfacial curvature and location might beobtained with a differential sensor. However, this methodwould preclude the measurement of conductivity and the fac-tors that affect it e.g., melt composition or temperature .

VI. DISCUSSION

Because inductive contributions to an eddy current sen-sors test circuit eventually become overwhelmed by parasit-ics and other circuit component impedances at high frequen-cies, the eddy current sensing method appears to be mostpromising for high liquid conductivity materials like Ge, Si,and GaAs. Lower conductivity systems such as CdTe would

require careful test circuit design to enable observations atthe high frequencies predicted to be needed for location de-termination. Axially separated differential coils are moresensitive to changes in the position and curvature of theliquid/solid than the absolute sensor design and provide en-hanced discrimination of these two contributions to thesensed response.

ACKNOWLEDGMENTS

This work was performed as a part of the research of theInfrared Materials Producibility Program conducted by aconsortium including Johnson Matthey Electronics, TexasInstruments, IIVI Inc., Loral, the University of Minnesota,and the University of Virginia. The authors are grateful forthe many helpful discussions with their colleagues in theseorganizations. The consortium work was supported by anARPA/CMO contract monitored by Raymond Balcerak.

1 J. C. Brice, Crystal Growth Processes Blackie, London, 1986 .2 F. M. Carlson, A. L. Fripp, and R. K. Crouch, J. Cryst. Growth 68 , 747

1984 .3 S. Sen, S. M. Johnson, J. A. Kiele, W. H. Konkel, and J. E. Stannard,Mater. Res. Soc. Symp. Proc. 161 , 3 1990 .

4 R. A. Brown, AIChE. J. 34 , 881 1988 .5 S. Brandon and J. J. Derby, J. Cryst. Growth 121 , 473 1992 .6 J. N. Carter, A. Lam, and D. M. Schleich, Rev. Sci. Instrum. 63 , 3472

1992 .7

S. Motakef, J. Cryst. Growth 104 , 833 1990 .8 M. Pfeiffer and M. Muhlberg, J. Cryst. Growth 118 , 269 1992 .9 S. McDevitt, D. R. John, J. L. Sepich, K. A. Bowers, J. F. Schetzina, R. S.Rai, and S. Mahajan, Mater. Res. Soc. Symp. Proc. 161 , 15 1990 .

10 P. Rudolph, U. Rinas, and K. Jacobs, J. Cryst. Growth 138 , 249 1994 .11 P. Rudolph, M. Neubert, and M. Muhlberg, J. Cryst. Growth 128 , 582

1993 .12 P. Rudolph and M. Muhlberg, Mater. Sci. Eng., B 16 , 8 1993 .13 P. M. Adornato and R. A. Brown, J. Cryst. Growth 80 , 155 1987 .14 S. Kuppurao, S. Brandon, and J. J. Derby, J. Cryst. Growth 155 , 93

1995 .15 T. Ejim, Ph.D. dissertation, University of Virginia, 1983.

16 T. Fu and W. R. Wilcox, J. Cryst. Growth 48 , 416 1980 .17 C. E. Chang and W. R. Wilcox, J. Cryst. Growth 21 , 135 1974 .18 R. J. Naumann and S. L. Lehoczky, J. Cryst. Growth 61 , 707 1983 .19 C. J. Chang and R. A. Brown, J. Cryst. Growth 63 , 343 1983 .20 P. Rudolph, J. Cryst. Growth 128 , 582 1993 .21 C. Parfeniuk, F. Weinberg, I. V. Samarasekera, C. Schvezov, and L. Li, J.

Cryst. Growth 119 , 261 1992 .22 W. R. Wilcox et al. , Acta Astron. 25 , 505 1991 .23 P. G. Barber, R. K. Crouch, A. L. Fripp, W. J. Debnam, R. F. Berry, and

R. Simchick, J. Cryst. Growth 74 , 228 1986 .24 G. Westphal private communication .25 H. N. G. Wadley and W. E. Eckhart, JOM 41 , 10 1989 .26 P. M. Amirtharaj and D. G. Seiler, in Handbook of Optics , edited by M.

Bass McGraw-Hill, New York, 1995 , Chap. 36.27 M. L. Mester, A. H. Kahn, and H. N. G. Wadley, Proceedings of the

Fourth International Aluminum Extrusion Technology Seminar, Chicago,1988, p. 259.

28 A. H. Kahn and M. L. Mester, Rev. Prog. Quant. Nondestr. Eval. 7 , 15991988 .

29 S. J. Norton and A. H. Kahn, Rev. Prog. Quant. Nondestr. Eval. 9 , 20251990 .

30 A. H. Kahn, K. R. Long, S. Ryckebusch, T. Hsieh, and L. R. Testardi,Rev. Prog. Quant. Nondestr. Eval. 5 , 1383 1986 .

31 J. A. Stefani, J. K. Tien, K. S. Choe, and J. P. Wallace, J. Cryst. Growth88 , 30 1988 .

32 K. S. Choe, J. A. Stefani, J. K. Tien, and J. P. Wallace, J. Cryst. Growth97 , 622 1989 .

33 J. A. Stefani, J. K. Tien, K. S. Choe, and J. P. Wallace, J. Cryst. Growth106 , 611 1990 .

34 J. K. Tien, J. P. Wallace, J. Kobayashi, B. C. Hendrix, and B. R. Birming-ham, Proc. Winter Annual Meeting Am. Soc. Mech. Eng. 21 , 1 1990 .

35 J. P. Wallace, J. K. Tien, J. A. Stefani, and K. S. Choe, J. Appl. Phys. 69 ,550 1991 .

36 K. S. Choe, J. A. Stefani, T. B. Dettling, J. K. Tien, and J. P. Wallace, J.Cryst. Growth 108 , 262 1991 .

37 G. Rosen, Ph.D. dissertation, Clarkson University, 1994.38 D. T. Queheillalt, M.S. thesis, University of Virginia, 1993.39 K. Kakimoto et al. , J. Cryst. Growth 94 , 412 1989 .40 K. P. Dharmasena and H. N. G. Wadley, J. Cryst. Growth 172 , 303

1997 .41 H. N. G. Wadley and K. P. Dharmasena, J. Cryst. Growth 172 , 313

1997 .42 K. P. Dharmasena and H. N. G. Wadley, J. Cryst. Growth 172 , 337

1997 .43 H. N. G. Wadley and B. W. Choi, J. Cryst. Growth 172 , 323 1997 .44 V. M. Glazov, S. N. Chizhevskaya, and N. N. Glagoleva, Liquid Semicon-

ductors Plenum, New York, 1969 .45 A. H. Kahn, M. L. Mester, and H. N. G. Wadley, Proceedings of the

Second International Conference on Hot Isostatic Pressing, Gaithersburg,MD, 1990.

46 H. N. G. Wadley, A. H. Kahn, and W. Johnson, Mater. Res. Soc. Symp.Proc. 117 , 109 1988 .

47 H. L. Libby, Introduction of Electromagnetic Nondestructive Test Meth-ods Wiley-Interscience, New York, 1979 .

48 N. Ida, Numerical Modeling for Electromagnetic Non-Destructive Evalu-ation Chapman and Hall, London, 1995 .

49 C. V. Dodd and W. E. Deeds, J. Appl. Phys. 39 , 2829 1968 .50

N. Ida, R. Palanisamy, and W. Lord, Mater. Eval.41

, 1389 1983 .51 N. Ida, K. Betzold, and W. Lord, J. Nondestruct. Eval. 3 , 147 1982 .52 N. Ida and W. Lord, IEEE Computer Graphics and Applications 3 , 21

1983 .53 R. Palanisamy and W. Lord, Mater. Eval. 38 , 39 1980 .54 W. Lord and R. Palanisamy, Eddy-Current Characterization of Materials

and Structures , ASTM STP 722, edited by G. Birnbaum and G. FreeASTM, Philadelphia, 1981 , p. 5.

55 R. Palanisamy and W. Lord, IEEE Trans. Magn. MAG-16 , 1083 1980 .56 MSC/MAGGIE, MacNeal-Schwendler Corporation, Inc. 1990 .57 H. N. G. Wadley and K. P. Dharmasena, J. Cryst. Growth 130 , 553

1993 .


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Modeling Multi Frequency Eddy Current Sensor

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