SCIENCE

Surface impedance of saturating iron intravelling fields

E.M. Deeley, B.Sc, Ph.D., C.Eng., M.I.E.E., and Prof. B.J. Chalmers, Ph.D.,D.Sc, C.Eng., F.I.E.E.

Indexing terms: Magnetic fields, Induction motors

Abstract: The step-function approximation to the B-H curve for saturating iron is used here to examine thebehaviour of the surface impedance in the presence of travelling surface fields. The method is an extension ofthat employed by other authors for pulsating fields and uses the concept of a wavefront which sweeps into themedium twice every cycle. The magnitude and phase of the surface impedance is computed for both sinusoidalelectric and magnetic surface fields, from which it is found that, over the permissible range of wavelengths, thechange in both surface impedance magnitude and penetration is small. The surface impedance angle increasesfrom its value for pulsating fields, although, for most practical situations, this increase will not be more than afew degrees.

Principal symbols

Bs = saturation flux density of ironh — instantaneous depth of wavefront using pulsating

theoryS — maximum depth of flux penetration using pulsa-

ting theory</>i> a i = phase angles of effective impedancec = conductivity of ironAs = surface vector potentialAo = maximum value of As

Es, Hs = tangential electric and magnetic fields at surfacek = wavenumber, equal to 2TC/Iv = velocity of travelling field relative to solid ironco = angular frequency of travelling field at surface of

solid ironC = dimensionless constant, equal to kA0/Bs

De, Dh = surface impedances, relative to C = 0

Suffixes13

= fundamental component= third harmonic component

1 Introduction

The problem of a travelling field moving relative to solidsteel occurs in both rotary and linear induction machineshaving unlaminated steel in the secondary member. Theformer are of practical interest in view of their goodmechanical properties, suitability for high speeds, andspecial high-impedance characteristics which may beadvantageously utilised in voltage-controlled motors, suchas for fan-load drives. Linear motors frequently have solid-steel reaction members or solid backing iron behind a thinconducting sheet. In other related problems, it is desired toknow the losses produced in solid iron or steel com-ponents, with a view to their reduction as well as to esti-mation of the temperature rise.

The mechanism of loss, torque or force production insuch devices rests upon the production of eddy currents in

Paper 3876A (S8), first received 25th October 1984 and in revised form 22nd Feb-ruary 1985

Dr. Deeley is with the Department of Electronic and Electrical Engineering, Uni-versity of London, King's College, London WC2R 2LS, and Prof. Chalmers is withthe Department of Electrical Engineering and Electronics, University of Manches-ter, Sackville Street, Manchester M60 1QD, United Kingdom

the saturating iron medium. The nonlinear nature of theB-H characteristics of iron has led to the use of variousforms of approximation of this characteristic, to facilitateanalysis.

The mode of flux penetration for sinusoidal surfacemagnetic field H has been analysed [1, 2] in a one-dimensional situation using the so-called limiting nonlin-ear' approximation to the B-H characteristic of iron, in theform of a step function with saturation density Bs. In thisapproach the penetration of flux into the medium is char-acterised by a wavefront, penetrating from the surfacetwice per cycle, behind which the electric field is uniformthroughout the medium and in front of which it is zero.This method was further developed, and correlated with arange of measurements, by Agarwal [3].

Lim and Hammond [4] developed a more refined solu-tion of this problem, using the Frohlich equation to rep-resent the B-H characteristic, and derived a universal losschart. The one-dimensional analysis was subsequentlyapplied to induction-machine problems by many workers(e.g. References 5-7), effectively assuming a purely tangen-tial field within the secondary member so that, in conse-quence, field wavelength did not enter into the analysis. Itwas effectively considered that adjacent secondary ele-ments experience the same amplitude of pulsating field butwith a time-phase displacement. That is, it was assumedthat the pulsating-field theory could be applied to eachindividual secondary element. Two papers (References 5and 8) have commenced by setting out their initial state-ment of the induction-machine problem in terms of a trav-elling field. However, when proceeding to solve using thelimiting nonlinear theory they have, as pointed out recent-ly [9], neglected radial field components in the secondarymember. Their analyses and results then became identicalto those of the pulsating-field analysis. A true two-dimensional solution of the travelling-field problem insaturating iron has not hitherto been produced.

In a previous paper [10], one of the present authorsdescribed an extension of the one-dimensional techniqueto the two-dimensional problem, using a formulation interms of magnetic vector potential, to describe the mecha-nism of flux penetration in two dimensions. For the one-dimensional situation it can be shown that, by taking theline integral of the magnetic field from the surface down tothe wavefront and back again,

«->'-£ (1)

IEE PROCEEDINGS, Vol. 132, Pt. A, No. 4, JULY 1985 171

where As is the surface vector potential and h is the instan-taneous depth of the wavefront. This provides a relationbetween Hs and £s for dAJdt), and hence an expression forthe surface impedance Z = EJHU where Ex and Hl arethe fundamental components of Es and Hs. This procedureis further developed in the present paper to arrive at acomplete solution of the travelling-field problem, includingflux penetration and surface impedance. The results enableall aspects of performance, such as loss and force, to beevaluated in terms of surface quantities.

In the presence of a nonlinear medium, it is not possiblefor both H and E fields to be sinusoidal. For devices whichoperate under either current-forced conditions, or substan-tially sinusoidal rotating MMF in AC machines, attentionhas often been concentrated on the boundary condition ofsinusoidal H at the secondary surface [1-9]. However,since terminal voltages are commonly sinusoidal, there isalso interest in the case with sinusoidal E or total flux. Inmany practical cases, actual conditions lie between theextremes of sinusoidal H and sinusoidal E, with associatedharmonic voltage components across the stator leakageimpedance. Some attention has accordingly been given tothe pulsating-field problem with sinusoidal E [11, 12].After correction of an error in Reference 12 (where thesurface impedance for sinusoidal E appears to be a factorof two too large), the results differ only slightly from thosewith sinusoidal H. Table 1, derived from References 3 and

Table 1: Comparison of pulsating-field analyses with sinus-oidal H and sinusoidal E

flux density throughout the medium has constant magni-tude Bs so that neighbouring equipotentials are equallyspaced along their length. Two other properties of theequipotentials are also important in the remainder of thisSection. The first of these, which has been demonstratedpreviously [10], is that an equipotential subtends equalangles on either side of the wavefront at the point where itcrosses it. This condition is necessary to satisfy flux contin-uity. The second property, demonstrated in Appendix 6.1,is that lines drawn normal to a family of equipotentials arestraight lines.

2.1 Sinusoidal electric surface fieldSuppose that the vector potential As on the surface of ahard saturating medium is given by

As = Ao sin (cot — kx) (2)

so that the surface electric field (into the plane of thediagram) is

dAEs = — - 1 = — Ao co cos (cot — kx) (3)

Quantity

Tangential flux inmedium, per unit length6

Surface impedance

Phase angles

= Function >

BSOJ6(HfUjB laY12

E2o/U)BS

atuBs 62

BS6(H.BJacuy2

(HJoujBs)y2

EJairBs\I<J6{wBJaH.y12

ujBJoEy<P,a,

< Multiplier

H sinusoidal

0.9481.3420.5550.50.9481.3421.4141.0531.8981.3421.801

26.6°

E sinusoidal

11.3560.5430.54311.3561.35611.8381.3561.838

23°

This is a sinusoidal electric field travelling in the x-direction.

The instantaneous surface distribution of vector poten-tial As, as given by eqn. 2, is shown over a distance equalto one half wavelength at time t = 0 at the top of Fig. 1.This distribution determines the points at which a familyof equipotentials, such as those illustrated by short brokenlines in Fig. 1, reach the surface, and, by using the proper-

X=-7T/2kA=1

11, compares the relevant items of interest in the analysisof an induction machine by the two methods. The slightlylower secondary phase angle under the sinusoidal-£ condi-tion is noted.

2 Theory

a*——— _ _ Y x 'V>^- -°A

o.2— ——"* k"!-^-- • -o.2Q _^^sr -=- —*• wavefront «

Fig. 1 Equipotentials and wavefront for sinusoidal surface electric field

ties of the equipotentials just outlined, the complete poten-tial distribution in the medium can be determined. Thisdistribution has the general form illustrated in Fig. 1. Inthis Figure the arrows indicate the direction of the fluxdensity, and the long broken line shows the position of thewavefront. In an analogous manner to pulsating fields, thewavefront reaches its maximum depth (and also effectivelyvanishes) directly beneath a point on the surface where the

The variation of E and H at the surface of a medium vector potential reaches an extremum value, another wave-having a step function B-H characteristic, to which the front commencing from the surface of the medium at theresults of this paper are confined, will be found by con- same point. This is illustrated in Fig. 2 for one completesidering the variation of magnetic vector potential A, both cycle of surface vector potential involving two wavefronts,on the surface and within the medium. The general form of the whole distribution moving to the right with velocity v,the vector equipotentials will first be described for the two equal to co/k. Wfcen a complete wavefront has moved pastconditions to be considered hereafter, namely sinusoidal a given point on the surface, the medium below that pointelectric surface field and sinusoidal magnetic surface field. is left magnetised in a direction parallel to the surface, asThe wavefront method will then be used to derive a illustrated by the regions M in Fig. 2. The next wavefrontgeneral expression from which the surface impedance foreither of these conditions can be found. direction of travelling wave

If x is the direction of the travelling wave and y rep- . . . . . . . . xresents depth into the medium, the problem is two dimen- i:r^- '^ '- ' '^ ' ' /V/^J_'j l \^sional in the x-y plane. The flux distribution in the ::: i"-- 'V^^rtf":V"riI^f------- 'rI^^""-------~-medium can therefore be represented by a single com- —-*.?---= -—•jt.-rponent of vector potential A in the Z-direction. The Fig. 2 Equipotentials and wavefronts for one complete wavelengthassumption of a step B-H characteristic implies that the sinusoidal E

172 IEE PROCEEDINGS, Vol. 132, Pt. A, No. 4, JULY 1985

sweeps through this region, eventually reversing the direc-tion of magnetisation.

The potential distribution in the regions M is easilydetermined, varying linearly with depth y below thesurface. The potential in the regions of curved equipo-tentials on the other side of the wavefront can be found byconstructing straight lines normal to the equipotentials asshown in Fig. 1. The position of the wavefront can then bedetermined by equating the value of potential in the tworegions at the wavefront, which is equivalent to ensuringthat the equipotentials subtend equal angles on each sideof the wavefront as required for flux continuity. Thisenables the wavefront depth, and the spatial variation inpotential between the surface and wavefront, to be calcu-lated. This in turn enables the temporal variation in poten-tial to be found, and a line integral of magnetic field willthen be taken around a region within which the total eddycurrent can therefore be calculated. The path of this integ-ral within the medium will be taken normal to the equipo-tentials so that the only nonzero magnetic field along thepath is at the surface. This procedure will be followed inSection 2.3 and in the Appendix.

2.2 Sinusoidal surface magnetic fieldWhen Hs is sinusoidal, the surface electric field is not, andit is not therefore possible to specify in advance the vectorpotential variation along the surface. Certain features ofthe potential distribution can, however, be seen from amodified form of eqn. 1:

Hs = oh'v *£d

(4)

where v is the velocity of the travelling field and h' is anaveraged depth, modified according to the shape of theintegration path. The surface magnetic field is again zerofor extremum values of As (i.e. when dAJdx is zero) andvaries sinusoidally with distance between these limits.

The wavefront reaches its maximum depth (shown atthe left-hand end of Fig. 3) for an extremum value of As inthe same way as for a sinusoidal surface electric field. To

x =-ir/2kA=1 x=0 A=0

= tr/2kA=-1

0.8 — - ' , -0 .6 - - " " ' -0.A- - '0.2 - - - ~~

0 — - - T L * ~

-0.6

-oFig. 3 Equipotentials and wavefront for sinusoidal surface magnetic field

the left of this point a new waveform will therefore befound to be penetrating from the surface in a mannersimilar to that shown in Fig. 2. This can also be seen at theright-hand end of Fig. 3, where the wavefront reaches thesurface at the point where A equals — 1.

The magnitude of the magnetic field at this point iszero, and about this point it varies sinusoidally with x.There is a discontinuous change in W at this point, so thatit follows from eqn. 4 that there is also a discontinuouschange in the derivative dAJdx to satisfy the magnetic fieldvariation. The equipotentials shown in Fig. 3 are consis-tent with these conclusions, and can be contrasted withthose for the sinusoidal surface electric field.

This argument shows that a discontinuous change insurface electric field can be expected twice per cycle. Thisis, of course, well known to occur in the one-dimensionalpulsating situation, but can now also be seen to be afeature when a travelling field is present. The maximum

normal flux density at the surface is seen to occur at thesediscontinuities, in contrast to the situation with sinusoidalsurface electric field, where the maximum normal flux wasfound to be in the centre of the half-wavelength pattern.

2.3 Vector potential within the mediumSuppose that the vector potential As on the surface of ahard-saturating medium is given by

As = Ao g(cot — kx) (5)

where g is a periodic function. The surface electric field isthen given by

E=-dt

= —Ao cog'(cot — kx) (6)

The function g' is therefore a cosine for sinusoidal E at thesurface, but has no simple form for sinusoidal surface H.

Using these expressions and the properties described forequipotentials in a medium having a step B-H curve, thetime derivative of the vector potential A is found inAppendix 6.2 at a depth y below the surface. This gives theelectric field, and hence the eddy-current density at depthy, and, by forming the line integral of magnetic fieldaround a path which includes the surface and extendsdown to the wavefront, an expression for the surface mag-netic field Hs can be obtained. This is described in Appen-dix 6.3 and gives

(9max + 9)9'

+ (i-cy2)vri/2

X I — 2 (1 - CV2)(1 + 71 - cy2)J (7)

where Eo = — coA0 and C is a dimensionless constantequal to kA0/Bs.

3 Field solutions and surface impedance

3.1 Sinusoidal surface electric fieldsFor the case of sinusoidal surface E, the functiong'(cot — kx) is simply cos (cot — kx), and the solution for Hs

is straightforward. For C = 0, eqn. 7 reduces to

H =2coB,

(1 -I- sin cot) cos cot (8)

which is just the expression obtained for the one-dimensional pulsating field problem. It should be notedthat eqns. 7 and 8 are valid for only one half of the cycle,corresponding to the period during which the flux patterndepicted in Fig. 1 passes the point x = 0. This correspondsto the interval of time for which — n/2 < n/2, and Hs is theresulting surface magnetic field during the half cycle ofsurface electric field. During the next half cycle the magni-tude of Hs is the same, but the sign is reversed. For asurface electric field Ex cos cot the corresponding magneticfield can be written as

Hs = Ht cos (cot - (f>i) + H3 cos (3cot — cf>3) + •• •

For pulsating fields (C = 0), the magnetic field componentHl has been shown to be [11]

cfE2

° 5 4 3

and the phase angle 4>l to be 23° under these conditions[11, 12]. The surface impedance magnitude can thereforebe written as

1EE PROCEEDINGS, Vol. 132, Pt. A, No. 4, JULY 1985 173

: = o>= 1-838<rEl

as in Table 1. The maximum permissible value of C isunity, since the maximum normal flux density at thesurface is then equal to Bs, the satiuration flux density ofthe medium. The solution of eqn. 7 enables H^ and Ze tobe calculated for any value of C2 between zero and unity,and these values, relative to those for C = 0, are given inTable 2 together with the phase 4>l. The parameters usedare

LI "7

k l n e

In Table 3 the amplitude and phase of the third-harmonic

Table 2: Fundamental component of surface magnetic fieldand surface variation

~C* ^ 0? ^e

00.10.20.30.40.50.60.70.80.91.0

1.0001.0001.0001.0001.0001.0021.0041.0091.0141.0271.058

23.023.824.625.526.527.628.930.334.235.537.8

1.0001.0001.0001.0000.9990.9980.9960.9910.9850.9740.945

Table 3: Third-harmoniccomponentof surface magnetic field

C2 ke3 0° "

00.10.20.30.40.50.60.70.80.91.0

0.2360.2380.2430.2520.2650.2840.3080.3390.3850.4510.590

-90.0-85.7-81.2-76.6-72.0-67.6-63.5-59.9-57.0-55.2-57.1

component of Hs is given for the same range of values ofC2, where

k^=—^-= O)

Eqn. 22 can be written in the form

The function f(cot), which corresponds to the surface mag-netic field waveform, is illustrated for three different valuesof C2 in Fig. 4, for one cycle of the surface electric field,El cos cot.

3.2 Sinusoidal surface magnetic fieldFor the sinusoidal surface H, the differential equation hasbeen solved for pulsating fields (C = 0) by Agarwal [3],but there is evidently no simple solution of eqn. 7 forC ^ 0 . The form of g has therefore been found numericallyusing a finite-element method. The forms of the surface Ewaveform (i.e. g') for three values of C2 are shown in Fig. 5,normalised to a maximum amplitude of unity for C2 = 0.The corresponding waveforms for surface vector potential

are shown in Fig. 6. The amplitude and angle of the funda-mental and third harmonic components of the surface elec-

1-0r

-jr/2

3ir/2

TT/2

C2 = 0.9

-1.0

Fig. 4 Surface magnetic-field waveform

cut=3ir/2

Fig. 5 Surface electric-field waveform

1.0

Fig. 6 Surface vector potential for sinusoidal Hs

trie field given by

Es = Ei cos (cot + a3) + • • •

are given in Tables 4 and 5 by the normalised parameterskhl and kh3, which are the field amplitudes divided by

The maximum value of C2 occurs when the maximumnormal flux density given by eqn. 13 in the Appendixequals the saturation flux density Bs. Because of the non-linear form of the surface electric field, and hence surface

174 IEE PROCEEDINGS, Vol. 132, Pt. A, No. 4, JULY 1985

Table 4: Fundamental component of surface electric fieldC2 *„, a, Dh

00.10.20.30.40.50.6

1111111

.000

.002

.004

.006

.007

.008

.008

26.6027.1327.7128.3629.1029.9430.86

1.0001.0031.0071.0111.0131.0141.014

Table 5: Third-harmonic component of surface electric field

00.10.20.30.40.6

0.2340.2380.2430.2510.2680.277

9.248.868.698.89

11.3513.81

vector potential, this occurs at a value of C2 of less thanunity. The numerical solution of eqn. 7 indicates amaximum value of C2 of approximately 0.62, and themaximum value of C2 quoted in Tables 4 and 5 is there-fore 0.6. From Table 1 the magnitude of the surface imped-ance for pulsating fields is given by

=o) — 1.801coBs

The magnitude for various values of C2, relative to that forC = 0, is given in Table 4 by the parameter Dh.

4 Conclusions

The magnitude and angle of the effective surface imped-ance of a medium having a step B-H characteristic havebeen found under travelling-wave conditions, for bothsinusoidal electric and sinusoidal magnetic surface fields.Tables 2 and 4 show that, for both situations, there is onlya small variation in the magnitude of the surface imped-ance as the wavelength of the travelling field is reducedfrom infinity (corresponding to a pulsating field) to itsminimum value, as measured by the dimensionless con-stant C2. For sinusoidal electric field at the surface (Table2) there is a decrease of about 5% from that for pulsatingfields, while for sinusoidal Hs (Table 4) there is an increaseof about 1.5%. Extreme conditions, corresponding tonormal flux density becoming equal to the saturationvalue Bs, are most unlikely to be approached in practice,however, so that in practical situations the change in mag-nitude of the surface impedance from that for pulsatingfields can be regarded as negligible. There is some increasein the third harmonic content of the surface field as C2

increases.

The change in the surface impedance angle is moremarked than the magnitude change. For a sinusoidal elec-tric field there is an increase from 23° to 37.8° for themaximum value of C2 (unity), while for a sinusoidal Hs

there is a lesser increase from 26.6° to 30.9° when C2 isequal to 0.6. This is close to the maximum value of C2 inthese conditions, as the normal flux at some point on thesurface then approaches Bs.

The depth of penetration <5 defined in Table 1 for pulsa-ting fields is directly related to the maximum value ofsurface vector potential for both situations. For the sinus-oidal electric field Es, the surface vector potential is com-pletely defined by Es itself, while, for the sinusoidalmagnetic field Hs, it can be seen from Fig. 6 that, althoughthere is a small change in the shape of the vector potential

waveform as C2 is increased, the change in the maximumamplitude is negligible. It follows that in both situationsthe change in <5 over the full range of variation of C2 is alsonegligible.

5 References

1 MACLEAN, W.: 'Mathematics of penetration of tangential magneticfield into iron, with square magnetisation curve', J. Appl. Phys., 1954,25, pp. 1267-1270

2 McCONNELL, H.M.: 'Eddy current phenomena in ferromagneticmaterials', Trans. Am. Inst. Elec. Eng., Pt. 1,1954,73, pp. 226-235

3 AGARWAL, P.D.: 'Eddy current losses in solid and laminated iron',ibid., Pt. I, 1959, 78, pp. 169-179

4 LIM, K.K., and HAMMOND, P.: 'Universal loss chart for the calcu-lation of eddy-current losses in thick steel plates', Proc. IEE, 1970,117, (4), pp. 857-864

5 McCONNELL, H.M., and SVERDRUP, E.F.: 'The induction motorwith solid iron rotor', Trans. Am. Inst. Elec. Eng., Pt. Ill, 1955, 74, pp.343-349

6 WOOD, A.J., and CONCORDIA, C : 'An analysis of solid rotormachines. Pt. 4: An approximate non-linear analysis', ibid., Part III,1960, 79, pp. 26-31

7 CHALMERS, B.J.: 'Asynchronous performance of characteristics ofturbo-generators', Proc. IEE., 1962,109A, pp. 301-308

8 VADHER, V.V.: 'Theory and design of linear induction motors withsteel reaction plates', IEE Proc. B, Elect. Power Appl, 1982, 129, (5),pp. 271-278

9 CHALMERS, B.J., and SALEH, A.M.: 'Analyses of solid-rotor induc-tion machines', ibid., 1984,131, (1), pp. 15-16

10 DEELEY, E.M.: 'Flux penetration in two dimensions into saturatingiron and the use of surface equations', Proc. IEE, 1979, 126, (2), pp.204-208

11 RAGHUNATHJI, B., and KRISHNAMURTHY, M.R.: 'Eddycurrent effects in solid unslotted iron rotors', ibid., 1969, 116, pp.612-613

12 LOWTHER, D.A., and WYATT, E.A.: 'The computation of eddycurrent losses in solid iron under various surface conditions'. Pro-ceedings of the first compumag conference, Oxford, April 1976, pp.269-276 (published by Rutherford Laboratory)

6 Appendix

6.1 Normals to equipotentials when B is constantIn two dimensions the normal to an equipotential isdefined by the cross product A x B, where A has only onecomponent A in the z-direction. The equation of a linewhich is everywhere normal to the equipotentials cantherefore be written as

— = A x curl Adk

(9)

where x is the co-ordinate vector in the plane normal to zand k is a parameter of variation. It will be shown that thisis the equation of a straight line.

The components of A x curl A in the xl5 x2 plane areA dA/dxl and A dA/dx2, so

(10)dk dXi

Differentiating,

d2x(

dk2 '=z—(

j dxj \

y(dA

j \dxj

' * -v it

dA

dXi

\dXj

)dk

dA A

dXj

d2Adxj dx.

dAdXj

IEE PROCEEDINGS, Vol. 132, Pt. A, No. 4, JULY 1985 175

However, the summations in eqn. 11 are simply equal toB2, which for a step B-H curve is a constant B2. Thus

dA(12)

Comparison between eqns. 10 and 12 shows that the firstand second derivatives of xt are proportional, so that thevectors dx/dX and d2x/dX2 lie in the same direction. Itfollows that the curvature K given by

K =

dx dhc_

dX X dX2

vanishes everywhere. The line defined by eqn. 9 is thereforestraight.

6.2 Electric field in the mediumThe magnitude of the normal flux density is found fromthe space derivative of eqn. 5, giving

BL = —kA0g'((ot — kx) (13)

Since the flux density in the medium is Bs, the angle, 9,shown in Fig. 7 between the surface and the equipotentials

_L

Fig. 7

- " \• - —cAx

Shift in wavefront position in time At

can be found from flux continuity across the surface, giving

kA0 ,sin u = g'{(Dt - kx) (14)

The potential at time t at a fixed point S (Fig. 7) at dis-tance x along the surface is Ao g(cot — kx), so that at pointQ along a line normal to the equipotentials, and a distanceb from s, the potential at time t is

AQ{t) = Ao g(cot - kx) - bBs

= Ao g(cot — kx) — yBs sec 9

(15)

(16)

The potential at time t at point P, corresponding to apoint on the surface a small distance Ax to the left of S, is

dA

wnere

dAQ

dx

sin 9 d9

(17)

(18)

The flux pattern moves to the right at a velocity co/k sothat, at time t + At, where At is equal to k Ax/co, the whole

pattern will have moved a distance Ax to the right, point Pmoving to P'. The distance between. P and Q can bewritten as c Ax, where c is a numerical factor depending onthe depth of Q below the surface, so that at time t thepotential at P' can be found by simple interpolationbetween P and Q, giving

AM) =c- 1

Using eqns. 15 and 17, this can be rewritten as

'c - 1\ dAr= A0g(cot-kx)-bBs- dx

(19)

x Ax (20)

At time t + At the potential at P' is given by an expressionsimilar to eqn. 15, namely

Ap{t + At) = Ao g(co(t + At) - kx) - b'Bs

where

(21)

d9

= b — y x

db 30 . . .x—(-Ax)

ddx

sin 9

cos2 9

d9(22)

Subtracting eqn. 20 from eqn. 21 gives

A p{t + At) - A pit) = A0co Atg'(cot - kx) - Bs{b' - b)

'c - 1\ dAQ— ^ Axc J dx

Using eqn. 22 and substituting co At/k for Ax, there results

AA sin 9 d9— = Ao cog (cot — kx) + ycoBs —̂: x

cos2 9 dx

c - 1\ dA

dx

and now using eqn. 18 in the third term on the right-handside, and proceeding to the limit,

dA\ i f „ , , ycoBs sin 9= - \ A0cog(cot — kx) H — x — x

dt

where 9 is given by eqn. 14 and

dO kCg"

cos2 9

dx cos 9

(23)

(24)

176

The dimensionless constant C is equal to kA0/Bs. Thisgives the total derivative dA/dt at a point at a depth ybelow the surface, and lying on a line normal to the equi-potentials drawn from the fixed point S on the surface.From earlier remarks it is evident that it is this derivativethat is required to calculate the eddy current lying withinthe integration path which includes the surface point S.

6.3 The integration pathFrom eqn. 16 the potential at Q is

A = Ao g(cot — kx) — yBs sec 9

In the region below the wavefront the potential is givensimply by

A = yBs- AQgmax

so that, for Q to lie on the wavefront, these two potentialsmust be equal, or

IEE PROCEEDINGS, Vol. 132, Pt. A, No. 4, JULY 1985

^ 0 (9max + 9)Bs (1 + sec 0)

(25)

where y0 is the depth at the wavefront.The integration path around a region of eddy current

can now be taken as shown in Fig. 8. This path is of length

by integration and equated to the line integral of the mag-netic field, the only contribution to which is the surfacefield Hs; so

and, substituting from eqn. 5,

•--n coBs sin 6 dd

f ( 2 6 )

Performing the integration, substituting for y0 througheqn. 25, and eliminating terms involving 6 using eqn. 14and 24, there results

oA\ co f (g 9)9'

KX 1 -

d-cV2)-1/2"J{Qmax + 9)9

(1 - + - C2g'2)\(27)

wavefront

Fig. 8 Integration contour

b0 and is wedge shaped on account of the curvature of theequipotentials. If the path length on the surface at S isequal to Ax, the width of the wedge in the x-direction at apoint Q is c Ax, so that the element of area da about P is

da. = c A dy

The total eddy current within the wedge can now be found

Since the impedance can be found at any point on thesurface we can, for convenience, make x equal to zero,which, from eqn. 6, means that the surface electric fieldmust be taken to be — Ao (og'(cot). Equating this toEo g'(cot), the expression for Hs becomes

(9max + 9)9'

r C2

L 2 ( 1 -L{Qmax + 9)9"

C2g'2){\ +Jl- C2g'2(28)

IEE PROCEEDINGS, Vol. 132, Pi. A, No. 4, JULY 1985 111