SCIENCE

Eddy currents in a rectangular toroid

Philip C. French, MA, CPhys, MlnstP

Indexing terms: Eddy currents, Electromagnetic theory

Abstract: Eddy currents flowing in a metallictoroid of rectangular section modify the mutualresistance and mutual inductance between a pairof windings around the metal. These effects can beused to determine the metal's conductivity. Thispaper presents a two-dimensional analysis of theprocess, using a Green's function method to deter-mine the magnetic field, from which the resistanceand inductance are derived. The results are com-pared both with a one-dimensional theory andwith experimental data.

1 Introduction

Lynch, Drake and Dix [1] devised an experiment todetermine the conductivity of a nonmagnetic metal speci-men. In the shape of a toroid of rectangular section (likea large washer), the metal is enclosed within an air-coredunity-ratio toroidal transformer, wound with a dual-startwinding. When alternating current is applied to theprimary winding, eddy currents flowing in the metalmediate a mutual resistance, together with a reduction inthe mutual inductance. The toroids are used as primarystandards for determining the conductivity of metalsamples distributed by the NPL for calibrating eddy-current conductivity meters.

Lynch et al. used an essentially one-dimensionalanalysis of the eddy-current flow within the core, withsemi-empirical corrections for the finite width of thespecimen. This approximation breaks down when theskin depth becomes comparable to the dimensions of thespecimen. The present work is intended to provide amore exact theory with a wider range of validity.

2 The experiment

The apparatus is described fully in Reference 1. Thespecimen (Fig. 1) is a flat annulus of mean radius r,having a rectangular cross-section of breadth b andthickness t. Around this core a former carries a dual-startwinding (each winding having N turns) of the same meanradius r, set at a distance a from the metal. The usualnominal measurements are r = 150 mm, b = 80 mm,t = 10 mm and a = 8 mm; giving the windings a cross-sectional area of 2496 mm2 of which the core occupies800 mm2, or about 1/3. N varies between 30 and 180.

Paper 5124A (S8), first received 21st May and in revised form 9th Sep-tember 1986

The author is with the Department of Mechanical Engineering, Uni-versity College London, Torrington Place, London WC1E 7JE, UnitedKingdom

Two related quantities are measured in the experi-ment. First, the mutual resistance R due to eddy currents

Z i

x.pFig. 1 Geometry of the specimen and coil former

in the core (the data used here have been corrected foreddy currents in the windings). Secondly, the mutualinductance M' = M + Ma-r; where Mair is the fixed partdue to flux in the airgap, whereas M is due to the flux inthe metal, which is partially screened out by the eddycurrents.

3 Formulation

Within the metallic core, of constant permeability /i andconductivity a, Maxwell's equations lead to a diffusionequation for the magnetic field H

(1)

Introducing sinusoidal time dependence at angular fre-quency co -4 ff/s (so displacement currents may safely beneglected) and taking account of the symmetry leads tothe complex scalar Helmholtz equation:

(2)

where the physical field H is given, in cylindrical polarcoordinates, by

H(p, <£, z, t) = Re [if (x, (3)

The local y-axis is parallel to 4> and the local Cartesianco-ordinate x = p — r. The imaginary constant a2 isrelated to the electromagnetic parameters and to the clas-sical skin depth <5 by a2 = ia>fia = 2i/52. H(x, z) is definedwithin D = [(x, z): | x | < b/2, | z | ^ t/2] with Dirichletboundary conditions on C = dD of the form H(x, z) \e =Hc(x). In a straight solenoid, Hc would be a constant, but

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 4, APRIL 1987 309

in the present apparatus the current turns are closer atthe inside radius. For windings wrapped tightly aroundthe core (a = 0), Hc(x) = Ho r/(r + x), where H0 is thepeak field at the mean radius, which is related to the peakcurrent / 0 by Ho = Nljlnr. However, in the actualapparatus, a and t are of the same order, and so this fieldmay be slightly distorted in the airgap. Because of this,and for comparison with previous work using straightinductors [2-7], it is simplest to use a uniform boundarycondition in the form Hc = fiH0, where f$ is a realnumber /? = 1 + y, with \y\ <£ 1. Choosing a value for /?amounts to finding an effective mean radius r' = r/fi. Theinductor is then notionally unrolled to an equivalentstraight one, with a bending correction /?.

Once H has been found, the mutual resistance R andmutual inductance M can be found from the time aver-ages of the dissipated power P = J • J/2a (where J = curlH) and of the stored energy E = ^/iH • H, respectively,

1

M SfJIdH* dH

H*H dV

dH* dH\(4)

(5)

One-dimensional solution

Thomson [2] and Scott [3] solved the one-dimensionalHelmholtz equation which applies for the infinite straightplate (b -> oo, r -*• oo):

with the uniform boundary condition Hc = f$H0 to arriveat

H(z) = flH0cosh (<xz)

(7)cosh (%<xt)

The corresponding mutual resistance and mutual induc-tance are

pcoMo pcoM0 ( sinh 0 — sin 0R = n X\0) = 0 Vcosh 0 + cos 0

(8)

M =0 0 0 + cos 0

In the above expressions 0 = t/S, and M0 = nbtN2/2nr isthe limiting value of the mutual inductance at low fre-quency (when the flux is not attenuated by the metal).When 0 > 10, the functions x{0) and iJ/(0) are both of theform 1 + e with |g | < 2 x 10~4; replacing them by 1leads to the simplification

R = coM =0

(10)

and hence to explicit equations for the conductivity interms of R or M:

a = nr2R2 4n3fi-2M2

For p, Lynch et al. [1] choose the value /? = ^ where

(11)

This averages the field over the supposed length of theeddy-current path, including both the larger top and

* This formula is given incorrectly in the printed text of Reference 1.

310

bottom faces and the smaller inside and outside ones.With the usual dimensions, ^ = 1.0305, giving anincrease of about 6% in the metal's conductivity asinferred by measuring R or M.

The formulas above have indeed been used for finiteplates [3, 4]. However, as pointed out in References 1and 4, they underestimate by neglecting the flow ofcurrent from one face to the other. Lynch et al. [1]allowed for this by multiplying the expressions for R andM by (b + t)/b. Although appropriate when 0 is large,this factor cannot account for the curved paths takenaround the corners, nor particulary for the interferencebetween upper and lower corners when 0 is small. So inReference 1 the factor is modified to (b + t — mS)/b,where m is a small number which may be determinedempirically, or, by a geometrical argument, is said to be 2when 0 is large (0 > 10). At 0 = 10 this additional correc-tion amounts to about 2% in R and M, or 4% in theconductivity; at smaller values of 0, it is not clear whatthe value of m should be.

5 Two-dimensional solution

A separation-of-variables solution is available [5, 6] inthe form

H(x, z) = PH0\COS^"Z\

|_cosh (fort)

n = 0n cos (Tnz) cosh (V(a2 + T2)x) (13)

where Tn = (2n + l)n/t and the series coefficients Qn aregiven by

Qn =(-l)"4a2

tTn(<x2 + T2) cosh (V(a2 +(14)

This single-series solution has the merits of being anextended version of the one-dimensional solution, and ofeconomy. It could be used directly to derive the induc-tance M, and (via the curl operation to give the currentdensity J), the resistance R. However, it lacks the mani-fest symmetry in x and z that might be expected, andinvolves functions of complex arguments which make itawkward for calculation. Alternatively, a symmetricdouble-series solution may be obtained in several ways;more coefficients must be evaluated, but with moderncomputers the additional labour has become irrelevant.Hammond [8] described the rather restricted method ofRoth, Stoll [5] demonstrated a finite Fourier transformmethod, and Silvester [7] introduced the useful conceptof eddy-current modes. An alternative approach is to usea Green's function expressed as an eigenfunction expan-sion, in a similar manner to Caldwell and Zissermann[9].

The Green's function method used below to solve theHelmholtz equation can easily be generalised to handleother forms of Hc should a fuller analysis including theairgap, or experimental measurements of the field nearthe surface of the metal, become available. The value of/? chosen is that which gives the same mean value of H2

for the undistorted and distorted or constant fields overthe surface, p2 = V^i • T h e motivation for averaging thesquare of the field, rather than the field itself, is that themeasured quantities R and M are proportional to thedissipated power P and stored energy E, respectively,each of which depends on H2.

To obtain the Green's function, consider the differencefield F defined by F(x, z) = Hc(x) - H(x, z), which obeys

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 4, APRIL 1987

the differential equation

(15)

with homogenous boundary condition F(x, z) |c = 0. Thenormalised eigenfunctions and eigenvalues (umn, Amn) ofthe homogenous system

V2«mn = AmMwmn (16)

with boundary condition umn \c = 0 are given by

x cos (Tnz) + 4 ^m, n

where Smn is defined by

cos (Bmx) cos2 (Tnz)lJ

c _ —7t

x _

l)(2n

(28)

(29)

The volume element is in the form dV =(r + x) dx dz dcf). Since integrals which are odd in xvanish,

umn = -fc cos (Bm x) cos (Tn z)

lmn = -(B2m + T2)

(17)

(18)

R = (30)

(31)

where Bm = (2m + l)n/b and Tn = (2n + l)n/t (as before).For later convenience, define Rmn = — kmn. The Green'sfunction obeying

(V2 - a2)G(x, z; x', z') = <5(x, x')<5(z, z') (19)

is given by the eigenfunction expansion

u*n(x, z)umn(x', z')

- o f(20)G =

m

i.e.

bt

£ cos (Bmx) cos (Tnz) cos (Bmx') cos (TnzQ

The general solution for H(x, z) is then

H(x, z) = Hc(x) — G(x, z; x', z') xJJD'

[f 771 - «2 W * ' ) l dx' dz' (22)

which in the case where Hc = /?H0 leads to

These equations can be used directly to compute theresistance and inductance, or implicitly to deduce theconductivity. In particular,

in a form suitable for iterative solution.

6 The low-9 limit

When 0 is small (i.e. 0 <t n/y/2 or of ^ n/2fit2), simpleapproximations may be made by taking only the leadingterm in the series (m = n = 0) and neglecting — a4 = 4/<54

by comparison with /?QO t o obtain

"00 —256fi2M0liob2t2f 2t2f2

00 "00fio " " n\b2 + t2)

M ~ p2M0(i - s00) * p2M0\i - -Trr-r-r, f^iVl

(33)

( 3 4 )

or, inverting these to give explicit expressions for the con-ductivity,

a =t2)R

256/32 M0fib2t2f2t2f2

Z ( - Vm+"(Pmn + iQmn) cos (Bmx) cos (Tnz)l (23) ° ~ %^hH2nf V 1 ~v, n _|

M

(35)

(36)

where the real coefficients Pmn and Qmn are given by

p = 4 / ^m" (2m + l)(2n + l)(R2

n + 4/<54)

2/?mw/^2

y m " (2m + l)(2n + l)(i?2n + 4/<54)

(24)

(25)

Note that the double-series solution used by Stoll [5]and Silvester [7] is complementary to this one; there Hitself, rather than F, appears as the eigenfunction expan-sion, and hence the coefficients are different.

The integrands appearing in the expressions for themutual resistance R and mutual inductance M are

^ ^ = 4/?2tf2 X Smn B2m sin2 (Bm x) cos2 (Tn z) (26)

ox ux m.n

d-lTir = 4P2fio I S"» T2n cos2 (Bm x) sin2 (Tn z) (27)

H*H = 02H2[ 1 - ^ Z( - l ) m + l i p m n cos (Bm x)

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 4, APRIL 1987

Note that this limiting form implies an approximatelylinear relationship between R and o for 6 <£ n/y/2.

1 Results and discussion

Computer programs to calculate the field H(x, z), theresistance R(o) and the inductance M(o) have been imple-mented on several computers. The work involved insumming the series is not excessive; indeed the latter twoprograms have been used on the 8-bit BBC micro-computer. Using an accelerated iterative method, theconductivity o(R) can be found with approximately fivetimes the computat ion required to determine R(o).

Fig. 2 is a contour plot of the physical magnetic fieldH (or the eddy-current streamlines) for 9 = 4, at theinstant of peak surface field, normalised to 100 units. Arelatively thick plate (b = 2t) shows the field distributionin the corners. These shapes may be compared with therectangles used by Beland [10] for the time-dependentferromagnetic case.

Figs. 3 and 4 show the I D , 2D and low-0 limit formsof R(o) and M(o) for a specimen of the usual dimensions,

311

at a frequency of 1 kHz, for conductivities ranging fromnil up to 60 MS/m. The corresponding range of 6 isroughly 0-5. Note that the maximum resistance, and thegreatest discrepancy between the one- and two-

Fig. 2 Magnetic field at 8 = 4 (x horizontal, z vertical)

2.18theta

3.08 3.77 4.35 4.87

12 24 36 48conductivity, MS/m

Fig. 3 Mutual resistance versus conductivity at 1 kHz (low 6)b = 80 mm; t = 10 mm; r = 150 mm; N = 120; /= 1 kHz

2DIDlimit

60

20

16

2.18theta

3.08 3.77 4.35 4.87

x

12

•acZ 8a

12 24 36conductivity, MS/s

48 60

Fig. 4 Mutual inductance versus conductivity at 1 kHz (low 6)b = 80 mm; t = 10 mm; r = 150 mm; N = 120; /= 1 kHz

2DIDlimit

dimensional (ID and 2D) theories, occur close to thepoint 6 = n/Jl. Here the behaviour of the function R(o)changes from being of order R a. a to R <x l/yjo. Asexpected, the linear approximation is useful for 0 <| n/

There is still a 7% difference between the 2D and IDvalues for the resistance at 6 = 5, but the mutual induc-tances are virtually identical for 6 > 4. With a smallerratio b/t, the divergence is greater.

Figs. 5 and 6 show the increasing agreement betweenthe ID and 2D theories at larger values of 6

100

80

GE8*60

16.9theta

23.8 29.2 33.7 37.7

20

12 24 36conductivity, MS/m

48 60

Fig. 5 Mutual resistance versus conductivity at 60 kHz (high 6)b = 80 mm; t = 10 mm; r = 150 mm; N = 30; /= 60 kHz

2DID

250i16.9

theta23.8 29.2 33.7 37.7

24 36conductivity, MS/m

Fig. 6 Mutual inductance versus conductivity at 60 kHz (high 6)b = 80 mm; t = 10 mm; r = 150 mm; N = 30 ; /= 60 kHz

2DID

(approximately 5-40). In general the ID results decreasetowards the 2D results as 0 rises. These two Figures arefor the normal operating frequency of 60 kHz. Althoughthe mutual inductance traces are indistinguishable fromone another, and are in any case too small to be useful,for these larger values of 6 there is still a discernible dif-ference in the mutual resistance, amounting to about 1 %at d = 20.

Tables 1 and 2 contain experimentally determinedvalues of the mutual resistance R, together with(temperature-corrected) conductivities inferred fromthem. In order of decreasing conductivity the sevenmetals are copper, aluminium, duralumin, aluminiumalloys T and 'B', brass and iital', a lithium-aluminiumalloy. Table 1 covers frequencies below 4 kHz, giving low

312 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 4, APRIL 1987

Table 1: Measured mutual resistance and inferred conductivity (low 0)

Metal

CopperCopperCopperCopperCopperCopperCopperAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumDuraluminDuraluminDuraluminDuraluminDuraluminDuraluminAlloy JAlloy JAlloy JAlloy JAlloy JAlloy BAlloy BAlloy BAlloy BBrassBrassBrassLitalLital

T, °C

20.820.820.820.820.820.820.820.820.820.820.820.820.820.820.820.820.820.820.820.820.620.620.620.620.620.620.620.620.620.420.420.420.420.4

N

120120120120120120120120120120120120120120120120120120120120120.120120120120120120120120120120120120120

f, kHz

0.40.60.8

(

I.O1.31.62.0D.8I.O1.31.62.02.53.01.01.31.62.02.53.01.31.62.02.53.01.62.02.53.02.02.53.02.53.0

6, mm

3.262.672.312.071.821.641.472.992.682.352.121.901.701.552.962.602.352.101.881.722.932.652.372.121.942.892.592.322.123.072.752.512.982.72

e

2.943.594.144.635.275.856.533.123.483.974.404.925.506.023.213.664.064.535.075.553.433.804.244.745.193.483.884.344.753.283.664.013.183.48

R. mQ

14.06417.55219.99822.10025.00527.74831.17426.14929.42033.21336.41840.32244.90849.26832.37636.93640.57744.82549.71454.35641.93546.31651.21356.65061.75950.92656.57862.52167.91967.40075.34081.85081.53089.660

Conductivity2D

59.6759.4159.3359.2559.2159.1659.0935.5335.4435.3735.3335.2935.2835.2628.9728.9028.8628.8228.7928.7722.6622.6122.5722.5622.5418.9318.8918.8718.8513.4413.4113.4111.4111.39

1Dm = 2.0

64.4162.7461.7361.1060.6160.3660.1638.0337.4836.9136.5436.2336.0435.9330.9630.4430.0829.7629.5229.3924.1223.7923.4823.2523.1220.1219.8219.5819.4314.3814.1714.0212.2012.06

Table 2: Measured mutual resistance and inferred conductivity (high 0)

Metal

CopperCopperCopperCopperCopperCopperCopperCopperCopperCopperCopperCopperCopperCopperCopperAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumAluminiumDuraluminDuraluminDuraluminDuraluminDuraluminDuralumin

T. °C

20.020.020.020.020.020.020.020.020.020.020.020.020.020.020.020.020.020.020.020.020.020.020.019.819.819.819.819.819.819.820.020.020.020.020.020.0

N

120120120606060606030303030303030

120120120606060606030303030303030

120120120606060

f, kHz

4.07.5

15.04.07.5

15.030.060.0

4.07.5

15.030.060.090.0

120.04.07.5

15.04.07.5

15.030.060.0

4.07.5

15.030.060.090.0

120.04.07.5

15.04.07.5

15.0

6, mm

1.040.760.531.030.760.530.380.271.040.760.530.380.270.220.191.340.980.691.340.980.690.490.351.340.980.690.490.350.280.241.481.090.771.481.080.77

e

9.2512.6617.92

9.2612.6717.9225.3635.87

9.2412.6517.9325.3835.9143.9250.61

6.969.52

13.476.969.53

13.4719.0526.95

6.959.52

13.4819.0826.9733.0538.19

6.418.77

12.406.428.78

12.41

R, mQ

44.73161.76287.86711.17415.43221.95731.18344.224

2.8003.8655.4877.788

11.04513.55115.67257.20879.398

113.19014.29219.84328.29840.24757.146

3.5794.9647.070

10.05014.27217.50520.22162.96087.581

125.03815.72121.87131.236

Conductivity2D

59.0559.0159.0659.1559.0859.1259.1559.1858.8758.8659.1759.2759.3059.1658.9135.2135.1935.2035.2735.2135.2035.2135.2235.1135.1335.2135.2635.2635.2835.3328.7328.6828.6728.8028.7528.71

1Dm = 2.0

59.8659.6159.5059.9659.6759.5559.4759.4359.6759.4559.6059.5959.5559.4759.3635.8135.6435.5335.8635.6735.5235.4535.3935.7135.5935.5435.5035.4435.4435.4729.2629.0928.9629.3329.1629.00

a. MS/m1Dm = 2.6

61.4260.3859.7359.3459.0858.9958.9536.4236.0735.7035.4635.2735.1935.1629.6629.3329.0928.8928.7528.7023.1222.9122.7022.5722.5019.3019.1118.9518.8613.7613.6213.5311.6911.60

a, MS/m1Dm = 2.6

59.0158.9959.0759.1159.0659.1259.1759.2258.8358.8459.1759.2959.3359.3059.2135.1535.1635.1935.2035.1935.1935.2135.2335.0535.1135.2135.2735.2735.3035.3528.6628.6628.6628.7328.7228.70

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 4, APRIL 1987 313

Table 2: — continued

Metal

DuraluminDuraluminDuraluminDuraluminDuraluminDuraluminDuraluminDuraluminDuraluminAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy JAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BAlloy BBrassBrassBrassBrassBrassBrassBrassBrassBrassBrassBrassBrassBrassBrassBrassLitalLitalLitalLitalLitalLitalLitalLitalLitalLitalLitalLitalLitalLitalLital

T. °C

20.020.019.919.919.919.919.919.919.919.819.819.819.819.819.819.819.819.819.819.819.819.819.819.820.020.020.020.020.020.020.020.C20.020.020.020.020.020.020.019.819.819.819.819.819.819.819.819.819.819.819.819.819.819.820.120.120.120.120.120.120.120.120.120.120.120.120.120.120.1

N

606030303030303030

120120120606060606030303030303030

120120120606060606030303030303030

120120120606060606030303030303030

120120120606060606030303030303030

/, kHz

30.060.04.07.5

15.030.060.090.0

120.04.07.5

15.04.07.5

15.030.060.04.07.5

15.030.060.090.0

120.04.07.5

15.04.07.5

15.030.060.0

4.07.5

15.030.060.090.0

120.04.07.5

15.04.07.5

15.030.060.0

4.07.5

15.030.060.090.0

120.04.07.5

15.04.07.5

15.030.060.0

4.07.5

15.030.060.090.0

120.0

6, mm

0.540.381.491.080.770.540.380.310.271.681.230.871.681.230.870.610.431.681.230.870.610.430.350.311.831.340.951.831.340.950.670.471.841.340.950.670.470.390.342.171.591.122.171.591.120.790.562.181.591.120.790.560.460.402.361.731.222.361.731.220.860.612.361.731.220.860.610.500.43

e

17.5524.82

6.418.78

12.4217.5624.8630.4435.14

6.008.21

11.606.008.21

11.6016.4023.18

6.008.20

11.6016.4123.2028.4032.77

5.497.51

10.615.497.51

10.6115.0021.20

5.487.50

10.6115.0121.2225.9829.99

4.636.348.974.636.348.97

12.6917.964.636.348.98

12.7017.9622.0025.404.015.487.754.015.497.75

10.9515.484.015.487.75

10.9615.4918.9621.88

/?, mQ

44.46663.190

3.9385.4727.805

11.11015.77219.35722.38371.34599.310

142.06217.82924.82535.50950.60471.970

4.4606.2108.878

12.64117.97522.07725.54377.934

108.173155.025

19.47527.03938.75655.26578.637

4.8796.7649.688

13.80519.64224.12827.91193.029

126.764182.25423.25531.68545.55965.02792.539

5.8207.925

11.38516.24823.13428.41132.865

102.384136.982196.512

25.59334.23149.12270.308

100.2056.4038.561

12.27917.57225.03530.78535.639

Conductivity2D

28.7028.6928.6728.6928.7328.7328.7728.7628.7522.4922.4622.4322.5122.4722.4422.4222.4022.4822.4422.4322.4622.4422.4222.3918.8418.8118.7918.8518.8218.7918.7818.7618.7818.7918.7918.8118.7918.7718.7613.4113.4213.4313.4113.4213.4313.4413.4513.3813.4113.4413.4513.4513.4513.4511.3711.3311.3111.3711.3411.3111.3011.2911.3511.3311.3211.3111.3011.2811.27

1Dm = 2.0

28.9228.8429.2029.1029.0228.9428.9328.9028.8722.9622.8222.6922.9822.8322.7022.6122.5322.9522.8022.6922.6422.5822.5322.4919.2819.1319.0219.2919.1419.0218.9518.8819.2119.1119.0318.9818.9118.8718.8513.8513.6813.6313.8513.6913.6313.5813.5513.8213.6713.6413.5913.5513.5413.5311.8611.5811.4911.8611.5911.4911.4311.3811.8511.5811.5011.4311.4011.3611.34

a. MS/m1Dm = 2.6

28.7028.6928.6028.6728.7228.7328.7828.7828.7722.4322.4422.4222.4522.4422.4322.4222.4022.4222.4222.4222.4622.4522.4222.4018.7918.7818.7818.8118.7918.7818.7718.7618.7318.7618.7818.8118.7918.7818.7713.4313.3813.4213.4313.3913.4213.4313.4513.4013.3813.4313.4513.4513.4613.4611.4711.3011.3011.4711.3111.3011.2911.29

• 11.4511.3011.3011.3011.3011.2911.27

values of 0 (approximately 3-6.5) similar to Fig. 3. Corre-sponding to Fig. 5, Table 2 covers a higher range of 6(approximately 4-50). The conductivities deduced fromthe 2D theory are remarkably consistent across the fre-quency range from 4 kHz upwards, with a spread of nomore than 0.7%; they rise at the lowest frequencies, at

314

which the experiment is known to be less accurate. Inapplying the ID theory, the functions x and \j/ were usedin full. The ID theory, with the recommended valuem = 2 from Reference 1, is not only somewhat high com-pared with the 2D one, but is also less consistent as thefrequency changes. A priori, the result should be indepen-

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 4, APRIL 1987

dent of frequency, except perhaps at very high frequenciesgiving small skin depths in which the surface finish of thespecimen might become significant. Varying the constantm, it is found that the choice m = 2.6 leads to theminimum frequency dependence, and much closer agree-ment with the 2D theory. For 8 < 7 the 2D theory isessential.

Tables 3 and 4 summarise inferred conductivities, byaveraging the results at different frequencies from Tables1 and 2, respectively. In general the 2D analysis givesgreater consistency across the frequency range and closer

Table 3: Conductivities inferred from resistance measure-ments (low 8)

Metal

CopperAluminiumDuraluminAlloy JAlloy BBrassLital

Conductivity a, MS/m2D

59.30 ±0.1835.36 ± 0.0928.85 ± 0.0722.59 ± 0.0418.89 ±0.0313.42 ±0.0111.40 ± 0.01

Table 4: Conductivitiesments (high 8)

Metal

CopperAluminiumDuraluminAlloy JAlloy BBrassLital

1 D (m = 2.0)

61.59 ±1.4236.74 ± 0.7330.03 ± 0.5423.55 ± 0.3619.74 ±0.2614.19±0.1512.13 ±0.07

1 D (m = 2.6)

59.70 ± 0.8535.61 ± 0.4529.07 ± 0.3422.76 ±0.2319.06 ±0.1713.64 ±0.0911.65 ±0.05

inferred from resistance measure-

Conductivity a, MS/m2D

59.09 ± 0.1335.22 ±0.0528.72 ± 0.0422.44 ± 0.0418.79 ±0.0413.43 ±0.0211.32 ± 0.03

1 D (m = 2.0)

59.58 ±0.1635.57 ±0.1429.02 ±0.1522.70 ±0.1719.03 ±0.1513.64 ±0.1111.54 ±0.17

1 D (m = 2.6)

59.11 ±0.1535.21 ± 0.0728.71 ± 0.0522.42 ± 0.0218.77 ±0.0313.43 ±0.0311.33 ± 0.07

agreement between the low-0 and high-0 regions. In prac-tice the experiment is known to be less accurate at lowfrequencies, where the ID theory could not be expectedto hold, and so the high-0 data should be taken moreseriously. Indeed, the internal consistency of the high-0data is greater than that of the low-0 data; the standarderror of the conductivity being 0.3% of the mean, atworst, for the 2D analysis. At the highest 0 the ID theory(m = 2) approaches the 2D theory within 0.5%. Choosingm = 2.6 gives a close approximation to the 2D theory forlarge 0.

Table 5 contains measured values of the mutual induc-tance for a copper specimen at low frequencies, using120-turn windings. To eliminate the Mair constant termin such measurements, the data recorded are actually thedifferences between M' (equal to differences in M) at aparticular frequency and at the lowest practicable experi-mental frequency, 0.4 kHz. Using a = 59.20 MS/m as theconductivity, the expected differences in M have been cal-culated by ID and 2D methods. The 2D theory is inexcellent agreement (0.14% error at worst), but the IDtheory (m = 2) is adrift by up to 12%. Recalling Fig. 4, it

Table 5: Mutual inductance, measured and calculated (low0j

f, kHz

0.60.81.01.31.62.0

(/W(0.4 kHz) -measured

1.6042.3792.8343.2683.5653.861

2D

1.6022.3792.8363.2723.5703.863

M(f)),fjH1 D (m = 2.0)

1.7242.5222.9793.4083.7023.991

will be appreciated that the failure of the ID theory atthese low frequencies is due to its incorrect prediction ofthe inductance in the DC limit. If the reference measure-ment were taken at a higher frequency, the ID theorywould do better. However, mutual inductance is mostsensitive at low frequencies; as it is also experimentallyunsatisfactory to use the same number of turns at highand low frequencies, this could not easily be carried out.

8 Conclusions

The 2D method interprets the experiment over a widerange of frequencies, conductivities and dimensionswithout empirical correction factors. It predicts the pene-tration of the magnetic field into the specimen, the eddy-current paths, and the mutual resistance and mutualinductance. It is in accordance with one-dimensionaltheory where that is valid, has a simple low-0 limit, and isin excellent agreement with experimental data. At lowfrequencies and/or conductivities it is quick and accurate.At the highest frequencies and conductivities (the thin-skin limit) it is less convenient: a large number of termshave to be taken in the series, while the ID theory pro-vides a good approximation; but not, perhaps, onewholly acceptable for a precision experiment. The consis-tency of the ID method is greatly improved by takingm = 2.6 rather than m = 2, especially at large 0.

The remaining uncertainty, affecting both the presenttheory and that of Reference 1, is the use of a uniformboundary condition and the selection of /?. Any error in /?appears in the final calculation as a term proportional tothe square of the breadth b, and so using a narrowerspecimen could lead to a more precise experiment.Numerical experiments have shown that changing thevalue of fi from 1.015 by ±0.002 causes marked fre-quency dependence. A strong point of the 2D theory, vin-dicating the approximation of uniform surface field andthe value /?2 = y/Pi, is its frequency independence.

A useful piece of further work would be to measure thefield on the surface of the metal. The theory could thenbe made exact, although it would cease to be strictlycomparable to the one-dimensional theory. The Green'sfunction would be unchanged, but the appropriate Hc(x)would be used in the integral to find F(x, z), which mighthave to be evaluated numerically, requiring more com-puter time. Alternatively, accurate DC measurementscould give an independent calibration, but such measure-ments are awkward to perform and to interpret withtoroidal specimens.

9 Acknowledgments

I am grateful to Dr. Arnold Lynch (University CollegeLondon) for proposing this work, and for several stimu-lating discussions during the course of it; also to Dr.Lynch and to Mr. Tony Drake of the National PhysicalLaboratory (Electrical Sciences Division) for the use ofthe experimental data.

10 References1 LYNCH, A.C., DRAKE, A.E., and DIX, C.H.: 'Measurement of

eddy-current conductivity', IEE Proc. A, 1983, 130, (5), pp. 254-260and ibid., 1987, 134, (4), e. 316

2 THOMSON, J.J.: 'On the heat produced by eddy currents in aniron plate exposed to an alternating magnetic field', The Electrician,1892, 28, pp. 599-600

3 SCOTT, K.L.: 'Variation of the inductance of coils due to the mag-netic shielding effect of eddy currents in the cores', Proc. Insl. RadioEng., 1930,18, pp. 1750-1764

4 REED, M.: 'An experimental investigation of the theory of eddy

IEE PROCEEDINGS, Vol. 134, Pt. A, No. 4, APRIL 1987 315

currents in laminated cores of rectangular section', J. IEE, 1937, 80,pp. 567-578

5 STOLL, R.L.: 'The analysis of eddy currents' (Clarendon Press,Oxford, 1974)

6 STRUTT, M.J.O.: 'Stromverdrangung in rechteckigen Leitern', Ann.Phys., 1927, 83, pp. 979-1000

7 SILVESTER, P.: 'Eddy-current modes in linear solid-iron bars',Proc. IEE, 1965,112, pp. 1589-1594

8 HAMMOND, P.: 'Roth's method for the solution of boundary-value problems in electrical engineering', Proc. IEE, 1967, 114,pp. 1969-1976

9 CALDWELL, J., and ZISSERMANN, A.: 'Magnetostatic field cal-culations involving iron using an eigenfunction expansion', IEEETrans., 1983, MAG-19, pp. 2725-2729

10 BELAND, B.: 'Eddy currents in circular, square and rectangularrods', IEE Proc. A, 1983,130, pp. 112-121

ErrataLYNCH, A.C., DRAKE, A.E., and DIX, C.H.: 'Measure-ment of eddy-current conductivity', IEE Proc. A, 1983,130, (5), pp. 254-260

The apparatus described in the paper we wrote (withC.H. Dix) in 1983 has been modified. Instead of usingtwo different inductors, we now use the same inductorwith or without a specimen; unintended differencesbetween the two inductors are thus avoided, but the tech-nique of bridge measurements becomes more difficult.

The inductor has been modified so that the primarywinding has smaller winding resistance and leakageinductance. We have found, and overcome, errors causedby circulating currents in the inductor windings. We haveexperimental evidence to suggest that the constant k usedin Sections 5 and 11 of our paper should be about5.5/(b + t). A further paper on these changes is in prep-aration.

We take this opportunity of correcting the followingequations which were wrong in Section 5 of the earlierpaper.

R2 = co232M2 (b + tf

a =4nf(b

R2l2/2 NA

1 1 16 1 1+

al2

D, 18 l(D2/b) In l(D + b)/(D - b)] D + b D-b

A.C. LYNCHConsultant on Precise Electrical Measurement

A.E. DRAKEDivision of Electrical Science, National Physical Labor-atory, Teddington, Middx. TW11 0LW

4941A

MOSLEY, K.E.: 'Factors affecting the design and oper-ation of redox batteries', IEE Proc. A, 1986, 133, (6), pp.375-386

On page 383, the horizontal axis of Fig. 17 should read

— t/cf), h/cm

5326A

316 IEE PROCEEDINGS, Vol. 134, Pt. A, No. 4, APRIL 1987