An investigation of horizontal axis coils for eddy current inspection

R. Clark and L.J. Bond

The possible use of horizontal axis eddy current coils for the inspection of fast breeder reactor primary vessels is being investigated. An approach which considers both theoretical modelling work and experimental work has been adopted. Owing to the lack of analytical theory for the horizontal coil case, the initial work has concentrated on determining the impedance change of a horizontal axis air-cored coil when it is brought close to a conducting half-space. An analytical solution (due to Burke) has been considered, along with a newly developed approximate model, for a range of different coils above various conducting media. The approximate model assumes a uniform H field at the material surface and has been extended to consider layered half-spaces such as stainless steel over liquid sodium. The trends exhibited by the impedance change values obtained from both theories compared well with the experimental data.

Keywords." eddy current, horizontal axis coils, impedance change, conducting half-space, stainless steel/liquid sodium

The inspection of nuclear power plants is one of the most demanding fields in which nondestructive evaluation (NDE) is required. The proposed next generation of nuclear reactors in Europe are liquid-metal cooled fast breeder reactors (LMFBRs). Liquid sodium will most likely be the liquid metal coolant used. With these new plants come new inspection problems. The initial phase of an investigation which considers the use of eddy current techniques for inspecting the primary austenitic stainless steel vessel of the LMFBR is described.

Riaziat and Auld m have suggested that horizontal axis coils may be better for flaw detection and less sensitive to lift-off variation than the more often used vertical axis coils. This factor has led to the consideration of the use of horizontal axis coils rather than vertical axis coils for the inspection problem being investigated.

An analytical theory due to Burke [2] for the case of an air-cored horizontal axis coil above a homogeneous conducting half-space has been used to produce impedance change results. The impedance change occurs as the coil is brought close to the half-space. The results have been compared with experimental data obtained using a simple configuration and with values obtained using a newly developed approximate theory for the half-space system.

The approximate theory is a simple mathematical model for predicting the response of an air-cored horizontal axis coil to the presence of eddy currents in a stratified conducting half-space. The aim is to start to understar_d the effect of the austenitic stainless steel/liquid sodium

system on induced eddy currents. A primary (inducing) magnetic field which is uniform in the surface of the material half-space has been considered. The only variation in the field is a change in the magnitude with increasing depth into the material (Figure 1).

Analytical theory Burke [El has put forward two analytical expressions which can be used to calculate the impedance change of a horizontal axis air-cored coil as it is brought close to a homogeneous conducting half-space. One expression is a perturbation expansion expression which is valid only

=Y t O

× t 2

j C o i l

= H Surface

~ H

Uniform field region

p

x

Fig. 1 Geometry assumed for the approximate model for eddy currents in a stratified half-space

0308-9126/89/060331-08/$03.00 © 1989 Butterworth & Co (Publishers) Ltd

NDT International Volume 22 Number 6 December 1989 331

for small values of the skin depth 6 and the other is an exact expression which is valid for all values of 6.

The perturbation expansion expression has the form

AZ = _2 i~#0( _N_ )2

f0 ~' ds x ~ sin2(st)M2(sal, sa2)R(sd ) (1)

where AZ is the impedance change, N is the number of coil turns, t is the coil half-length, a~ is the coil inner radius, a 2 is the coil outer radius, d is the distance from the coil axis to the material surface, s is the variable of integration,

f j ~a2 M(sax, sa2)= d(sa)sall(sa)

al

R(sd) = Ko(2sd ) + (i - 1 )SbPrK~(2sd) +. . .

The exact theory expression has the form

AZ=2kO/~o( _N_ )2 \ t [ a 2 - a l )

f o ds x ~ M2(sal, sa2) sin2(st)

f f ' exp( - 2ed) (2) prC~

dp ~(pr~ + ~1) X

where = (p2

~1 = ( 0~2

+ $2) 1/2

+ i~a/2) 1/2

p is the variable of integration and the other terms are as defined for (1).

Both of these expressions were coded ready for use on a computer. To confirm the accuracy of these programs, results were produced using Burke's coil and material data. The program results are compared with Burke's theoretical results in Figure 2 for both expressions. Both of the programs written produced results which verified Burke's plots.

The differences between the plots for each expression were accounted for as follows. The method for determining M(sa~, sa2) was different in our programs to that used in Burke's programs. When calculating the exact solution values, different numerical integration routines were used in the two programs, although the routines all achieved a comparable accuracy. Despite these differences, the results compared well for each expression.

Approximate model Using Maxwell's electromagnetic field equations as the building blocks, equations can be obtained in terms of E, the electric field intensity, and H, the magnetic field intensity, which describe the electromagnetic field in an infinite, homogeneous and isotropic medium TM. The equation in terms of H has been used as the basis of the new model:

OH V 2 H =/20" - - (3)

~t

In order to keep the model simple, the form of the H field

1.00

0.99

0.98

x

0.97

0.96

0.95

\ , ~, j Exact

\ - - i

~ I00 Hz

_

, ; / ...~.F/" ,,,,2" .., i. ~ Per tu rba t i on expans ion

.~,ff~/,'" Coil above an aluminium al loy 500 kHz . ,/. I & / ha l f -space I (~"~.,~'~" "~ - - - - - Burke

- 1....~,.7 / . . . . . UCL ,~f.~1 A R Change in res is tance " X I n d u c t i v e reactance o f coil

near material X 0 I n d u c t i v e reactance o f coil

in a i r I 1 I

0.005 0.010 0.015 0.020

AR/X 0

Fig 2 Comparison of results produced by Burke and UCL for the analytical theory of Burke

was chosen to be

. ( 0 0 ) ,4, H(x)

given the coordinate system in Figure 1. This essentially reduced the problem to one dimension.

Since the field is assumed to vary sinusoidaUy with time, the partial differential equation for H can be written as an ordinary differential equation in the form

d2H -kZH (5)

d x 2

where k2=koap since O/&-,ico. The corresponding equation for E is given by

dZE -- k2E (6)

dx 2

The general solutions for these equations are

H i = A i r +k,x + B i e -k ' x (7)

and

Ei = - -~i (Ai e+k~x- Bie -k'x) (8)

where i denotes the layer that is being considered in a stratified half-space and A and B are coefficients that need to be evaluated.

In order to determine the H and E fields in a stratified half-space, the first step is to evaluate the coefficients A and B by applying the boundary conditions to the

332 NDT International December 1989

problem. There are three sets of boundary conditions:

• a t x = 0 , H = H o a n d E = E o (9) (Ho and Eo are the surface field values)

• at t=t., Hr. =H~.+ and E t . - = E t . + (10) (the tangential components of the H and E fields are continuous at the layer interfaces)

• as x--* oo, H and E---,0 (11) (otherwise the field has infinite energy)

When evaluating the coefficients, the algebra involved is complex. In order to help with the accurate determination of the expressions for the coefficients, a computer package called REDUCE was usedtak The package, based on standard LISP, can be used to perform algebraic operations accurately. It deals with variables and constants in the same way as other computer languages, except that when REDUCE evaluates a variable, the variable can stand for itself.

The value of H on the central axis of a solenoid is given by

NI He = ~ - sin 0 (12)

where N is the number of turns in the solenoid, I is the current through each turn, 21 is the length of the coil and

• - l / c ° i l half-length'~ O=tan t c o i l ~ )

The field outside a finite-length solenoid decreases with increasing distance from the coil. Although this is an obvious physical phenomenon, the mathematical descrip- tion of the field outside a finite-length solenoid is complex and it is most often evaluated numerically.

In this model a new approach has been put forward based on the theory of Brick and Snyder TM. This analytical approach again makes simplifying assumptions about the electromagnetic field generated by the coil. Assuming magnetostatic conditions, the expression for H in the z direction outside the coil and at large radial distances from the coil is

(nIa2~( 1 r2"~ -3,2 H= \ 212,it +77 ) (13)

where n is the number of turns per metre, a is the coil radius and r is the radial distance from the z axis to the external point.

A point was considered at a large radial distance from the coil. At this point it was assumed that the field (H) value had decreased to 0.1% of the value of H on the central axis of the solenoid. Using (13), the value of r at this point was determined. This value of r was then referred to as roo, the point at which the field could be considered to have decreased to zero.

For simplicity it was assumed that the field decayed from the coil axis to r®. The decay mode was taken to be of the form 1/r, as indicated by the Biot-Savart law. This decay was verified to some extent by the work of Simkin and Trowbridge [61 who have used a similar approach in the electromagnetic software package TOSCA. The expression used to determine the external field at a point is

H = ( 1 - l-~Hc (14) \ r r~ /

The reason for this formulation was the need to determine the value of H at. the surface of the conducting half-space beneath the coil such that the first boundary condition could be imposed.

Having determined H, it needed to be related to the changes in resistance and inductance of the measuring coil due to the presence of the conducting half-space. The eddy currents in the material set up ohmic losses. In order to maintain the field in the material, more power needed to be drawn from the supply. Hence there was an apparent increase in the supply resistance (ie coil resistance):

1 ( ' dH dH* A R = a ~ J , o , dx d~-dv (15)

where AR is the change in resistance, I is the peak current and H* is the complex conjugate of H (this is required since H is complex but AR is real). It also needs to be stated that

Rcoi!= R o + AR (16)

where Rco . is the coil resistance in the presence of the half-space and R o is the coil resistance in air.

A similar argument follows for the inductance change. The flux-carrying capacity of the material was reduced owing to the eddy currents. This resulted in a reduction in the coil inductance:

AL= fvo, H . H * dv (17)

where AL is the change in inductance. Also

Lcoil = L o -- AL (18)

The problem encountered with this approach was that the two H fields in Equations (15) and (17) were different. If the eddy current generation is considered step by step, it becomes clear that the primary magnetic field, as determined earlier, is responsible for the eddy current induction and is the H field in Equation (15). The secondary flux field, which leads to a reduction in the flux linking the coil and thus the reduction in the coil inductance, is the H field in Equation (17). This idea of two parts to the problem linked by the eddy current distribution indicates why it is so important to consider both the resistive and the inductive components of the coil impedance change.

It is clear that the surface value of H for the secondary field must now be determined such that the equations can be evaluated. No solution was found to this problem in electromagnetic theory, so, on the basis of the work performed, a simple expression for the secondary surface H value was put forward (Figure 3a). Since the only variable between the tests on a certain piece of material was the frequency, an expression of the form

Hso = Hpof (19)

was proposed, where Hso is the secondary surface H value, Hpo is the primary surface H value and f is the frequency.

Initially the value of Hpo was considered when evaluating the AL results. When the results were plotted against frequency and compared with the experimental values, the trends exhibited in the two cases were mirror images of one another, ie for the experiment AL became more negative (Figure 3b). At a particular frequency the two

NDT International December 1989 333

-kx H = H e p p0

Primary field

a

AL

Hso Metal surface

-kx Hs= HsO e

Secondary field

fco

Hp..O

x

Frequency

-----'-"~""~1D model

Experiment

b

Fig. 3 (a) Primary and secondary fields. (b) Previous AL characteristic

plots crossed. This frequency was calledfco, the crossover frequency. At this point it was assumed that Hs0 and Hpo had the same magnitude but were in opposite directions. Since there would be a phase difference between the primary and secondary fields, it did not mean that at fco there was no overall field.

The expression proposed for H~o was

C]co Hso = Hpo ~ -

l i f \ l l 2

.. H~o= Hpot~o~o) (20)

where & is the skin depth and f is the frequency. This expression appeared to produce reasonably good results over the frequency range considered, although it must be remembered that it did require a set of experimental data for it to be of use and the value of f~o was material- dependent.

The final step required for determining AR and AL involved the evaluation of the integrals. For the model to be valid, field uniformity was necessary. This was considered to be the case directly underneath the test coil where the majority of the energy associated with the eddy currents was concentrated. Hence the volume over which the integrations were performed was taken to be

d r = 21-2a.dx (21)

where 2l is the coil length and 2a is the coil diameter.

Thus Equations (15) and (17) become

1 f ; drip dH* dx AR = ~i ~. 21. 2a dx " dx (22)

where Hp is the primary field, and

A L : ~ ' 21"2a .to ~ Hs 'H* dx (23)

where H, is the secondary field. Furthermore, if the half-space is stratified,

1 r ~, dRip " dH% AR = i~" 21" 2a Jo dx a~ dx dx

1 r ~ dH2p" dH~'p + a ~ ' 21- 2a .j, dx

, dx dx

+ ~ ' 2 1 " 2 a f t ~dH3p'dH*p dx (24) 3 2 dx d x

A similar expression was obtained for AL.

This theory constitutes the approximate model for the eddy currents in a stratified half-space. The effect of the assumptions made will be discussed when analysing the results.

E x p e r i m e n t a l w o r k

The experimental work performed was designed to provide data with which both the Burke theory and the results from the approximate model could be compared. A simple set-up was used (Figure 4).

Initially two single-layer coils were wound on separate Perspex rod formers using coated copper wire. The details of the two coils are as follows:

Coil 1

140 turns rod diameter 10 mm coil length 39 mm wire diameter 0.25 mm

Coil 2

200 turns rod diameter 10 mm coil length 29 mm wire diameter 0.122 mm

The coils were connected to a Wayne Kerr 6425 precision component analyser using Kelvin clips attached to the instrument leads. The Kelvin clips automatically made the correct connections to a lead eliminator circuit within the instrument which, as far as possible, eliminated the effect of the extra capacitance due to long connecting

Wayne Kerr 6425 impedance analyser I X-Y

chart recorder

Get plots of R versus L (resistance versus inductance for coil)

Horizontal coil wound on a ooo~ Perspex former

Block of material

Fig. 4 Experimental set-up

334 NDT International December 1989

leads. The instrument was microprocessor-based and provided a direct read-out ofthe impedance of a component which was connected between the two output terminals of the instrument. An X - Y chart recorder was linked to the impedance analyser such that a hard copy of the results could be obtained. The plots obtained were of coil resistance versus coil inductance.

The metal specimens used were a block of 316 stainless steel and a sheet of copper. Two layered specimens were also considered; one consisted of 1.62 mm of copper on a stainless steel block and the other was 6.35 mm of stainless steel on an aluminium alloy block.

The test procedure involved bringing each coil close to each specimen, one at a time, at several frequencies in the range 100 Hz to 100 kHz. Given the ultimate inspection requirements (ie a reasonably large skin depth), a frequency range of 100 Hz to 100 kHz was considered to be the most useful for initial investigation. In each test a lift-off of 0.143 mm was maintained. The values for the change in resistance and the change in inductance of the coil were recorded.

R e s u l t s a n d d i s c u s s i o n

Several test cases have been considered using both computer models and by performing some simple experiments. The results presented have been chosen to demonstrate the use of two different coils, the study of two different materials, the effect of variations in lift-off and the study of two different layered specimens. Most of the results are presented as plots of AR versus frequency and AL versus frequency rather than as impedance plane diagrams. This was done in order to make the results for the two components of the impedance change clearer and easier to understand.

Figures 5a and 5b illustrate the resistance and inductance characteristics when the 140-turn coil is brought close to a 316 stainless steel block over a range of frequencies. The trends demonstrated are as expected: AR becomes more positive with increased frequency and AL becomes more negative with increased frequency.

The approximate model and experiment agree quite well over the range of frequency considered when considering AR, although the two plots do start to diverge at high frequency (100 kHz). At high frequencies the skin effect phenomenon in the coil wire becomes significant tv]. This results in an increase in the coil resistance and a decrease in the coil inductance, both of which are independent of the eddy currents induced in the metal block. This was not considered in the approximate model and would thus account for some of the model underestimation of the AR value. The results obtained from the Burke exact solution were an overestimation of the experimental AR values at high frequencies. The trend exhibited though was similar to that of the experimental data. The reason for the difference was considered to be due to errors with the exact solution computation and errors introduced when performing the experiments. It should be noted that the experimental results obtained for frequencies below 1.5 kHz were subject to large possible errors (AR -t- 105 %, AL___41%).

The AL results (Figure 5b) show good agreement across the entire frequency range considered. The results produced

0.25

0.20

0.15

0.10

<3

0.05

- - O - B u r k e Exact - - o - E x p e r i m e n t - - x - Approximate Model

. . . . . x----x- . . . . "~

-0.05 100 1000 I 0 000 100 000

a Frequency (Hz) (log scale)

-0.2 X

X E1

-0.4

D -0.6 x - I -

.~ x F'I

<3 -0.8

Burke Exact -1.0 D Experiment

x Approximate Model ]

-1.2

- 1 . 4 I I I I I 100 1000 10000 100000

b Frequency (Hz) (log scale)

Fig. 5 (a) AR and (b ) zSL versus frequency for 31 6 stainless steel ( 140-turn coil )

by the approximate model theory compare well, thus indicating the usefulness of the simple expression developed.

Figures 6a and 6b illustrate the results for the 200-turn coil when it is brought close to a 316 stainless steel block. The results show reasonable agreement over most of the frequency range although at high frequency the agreement is less definite. This was especially true when considering the Burke exact data. The differences were attributed to computational errors and experimental errors as discussed earlier. The results presented in Figures 7a and 7b are for a copper ~aalf-space, ie a material with a higher

NDT International December 1989 335

0.5 / 0.4 ~ B u r k e Exact

--{3-- Experiment A - - x - -Approx imate Model

0.3

0.2

0.1

o~

I

X ~

-0.1 I I I I I 100 1000 10000 100000

a Frequency (Hz) (log scale)

b

ol

-1

-2

-3

X

\

---O--Burke Exact I i ~ E x p e r i m e n t Model ~\ - - x - -Approx ima te

X'-~X__~

-4 I I I I I 100 1000 10000 I00000

Frequency(Hz) (log scale)

Fig. 6 (a) AR and (b) z~L versus frequency for 316 stainless steel ( 200-turn coil )

conductivity. As with the previous results, the trends exhibited by both of the models compare well with the experimental data.

Using the 140-turn coil above the 316 stainless steel block, an investigation of the effect of lift-off variation was performed (Figure 8). These results indicate that although the horizontal coil is sensitive to lift-oK the impedance changes associated with varying degrees of lift-off using these simple air-cored horizontal coils are small compared with those obtained with a conventional ferrite-cored vertical axis coil in the same situation. It must be stated that the signals (AR and AL) obtained from these

a

0.05

0.04

0.03

0.02

0.01

ol 100

(

J ---O-- Burke Exact / - -O-- Experiment / - - x - - Approximate M o d e l /

/ ///

-- X JX~

~x~X~

I 0 0 0 I 0 0 0 0 1 0 0 0 0 0

Frequency(Hz) (log scale)

-0.21

-0.4

Burke Exact Experiment

- - x - - A p p r o x i m a t e Model

-0.6

- 0 . 8 "1-

-1 .0

-1 .2

- 1 . 4

x

\ x

~ X - - - X . . . . . . . . X - - -

-1 .G 100 1000 10000

b Frequency (Hz) (log scale)

100000

Fig 7 (a) ARand (b) AL versus frequency for copper (140-turn coil )

horizontal axis coils are, in general, smaller (m~ and pH) than those obtained from conventional vertical axis coils (£2 and mH). The addition of a ferrite core would considerably increase the horizontal coil signals. Despite the small signals, Figure 8 indicates that small impedance changes can be measured using a simple experimental configuration and that the values obtained compare favourably with those obtained from theoretical work based on an approximate model of the system and based on a detailed analytical description of the system.

In order to demonstrate the use of the approximate model for the consideration of a stratified half-space, two layered

336 NDT International December 1989

1.000 0.020

x

0.995

0.990

0.985

0.980 I

0.975 0

- - 0 - Burke Exact ~' Experiment

- -X-- Approximate Model \~

I I I

0.002 0.004 0.006 0.008 0.010

ARIX o

Fig. 8 Impedance plane diagram; 316 stainless steel; l ift-off variation ( 140-turn coil )

specimens have been investigated. A 1.62 mm thick layer of copper on a 316 stainless steel block was the first specimen studied. As with the previous results, the approximate model does not reproduce the experimental values exactly, but the trends exhibited provide a useful first approximation to the experimental values of AR and AL (Figures 9a and 9b). Given that the horizontal coils produce small signals anyway, the differences between the model and experimental values are, in many cases, very small. Similar comments can be made about the second layered specimen, which consisted of a 6.35 mm thick layer of 316 stainless steel on an aluminium alloy block. Figures 10a and 10b illustrate the good agreement between the model and experimental results. In both cases the 140-turn coil was used.

The results presented illustrate some of the work performed during the initial phase of an evaluation of the use of horizontal axis coils for eddy current inspection. The relatively good agreement between the experimental data and the data predicted using the exact theory of Burke and the newly developed approximate model looks promising for the future.

The assumptions included in the approximate model obviously had an effect on the accuracy of the results produced, but the work has demonstrated that, given the ease of programming and the speed of calculation relative to the Burke exact theory, the approximate model Offers a useful first approximation for the values of AR and AL. In saying this, it is indicating that the assumptions made when developing the approximate theory hold up quite well when using the model for the situations considered. The determination of the bounds of the approximate model will need to be considered in future work. When using both models, it must be remembered that the input data are of great importance, since any inaccuracies in those data will carry through to the results produced.

Figure 11 shows the predicted variation in AR with

0.015

~0.010

<3

0.005

-'-£2-- Experiment - -x - -Approx imate Model

i/ / X / /

/ /

/

/ / /

/

0 I ~ 100 1000 1 0 0 0 0

a Frequency {Hz) (log scale)

100 0 0 0

-0.21

-0.4

-0.6

..t-

"~ - 0 . 8

-..4 <:3

-1.0

-1.2

-1.4 100 1000

b Frequency (Hz)

~ E x p e r i m e n t x Approximate Model

100000 10000

log scale)

x x

Fig. 9 (a) AR and (b) AL versus frequency for copper on 316 stainless steel ( 1 40-turn coil )

increasing 316 stainless steel thickness at 200°C in a 316 stainless steel/liquid sodium system as would be present in an LMFBR. The results were obtained from the approximate model by considering the use of the 140-turn coil at a frequency of 10 kHz. The plot predicts that the contrast between stainless steel and liquid sodium will be sufficient to produce an easily measurable AR value when the thickness of the stainless steel is such that at least some of the eddy currents are induced in the liquid sodium. This can be considered to be the first small step towards indicating the feasibility of this eddy current technique for inspecting the LMFBR primary vessel.

NDT International December 1989 337

0.12 r .~ 35

0.I0

0.08

0.06

~C <3

0.04

0.02

-0.02 L__ 100

a

~ E x p e r i m e n t - - x - - Approx imate Model

/ 1 1 X

I

I

1000 10000

F requency (Hz ) (log scale)

100000

-0 .2

-0 .4

"I-

"~ -0 .6 ~J

b

-0 .8

- I .0

- I .2 .... 100

- _ - - X - - A p p r o x i m a t e - - ~ Exper iment

I I t I I 1 0 0 0 I 0 0 0 0 1 0 0 0 0 0

Frequency (Hz) (log scale)

Fig. 10 ( a ) A R a n d ( b ) A L v e r s u s f r e q u e n c y f o r 3 1 6 s t a i n l e s s s t e e l on aluminium alloy (140- turn coi l )

Conclusions

This paper has described the initial phase of work that has considered the use of horizontal axis coils for the eddy current inspection of austenitic stainless steel pressure vessels. The initial results look promising, the experimental values comparing well with those obtained from two completely different theoretical models for the case of a coil above a metal half-space. Although the new approximate model does not have the accuracy of the Burke exact theory, it does have the advantages of being

Paper received 1

30

25

E

<3

20

15 j 10 -" I I I

5 10 15 20

Thickness (mm)

Fig. 11 AR versus thickness of 316 stainless steel on liquid sodium; 200 C, 10 kHz (140- turn coi l )

easily programmable and quick to use, as well as the capability to consider half-space stratification.

With these two models to complement one another and through a programme of experimental work, it is hoped to provide some answers to the engineering problem of fast reactor pressure vessel inspection using eddy currents.

Acknowledgements

This work has been supported by the Science and Engineering Research Council and the Northern Research Laboratories of the United Kingdom Atomic Energy Authority at Risley, Cheshire, UK.

The authors would like to thank Mr P. French for some helpful discussions.

References

l Riaziat, M. and Auld, B.A. 'Angular spectrum analysis applied to undercladding flaws and dipole probes' Rev Prog QNDE 3A (1984) pp 511-521

2 Burke, S.K. 'Impedance of a horizontal coil above a conducting half space' J Phys D: Appl Phys 19 (1986) pp 1159-1173

3 Libby, H.L. Introduction to Electromagnetic NDT Methods Wiley Interscience, New York (1971)

4 Rand Corporation REDUCE User's Manual version 3.1, ed A.C. Hearn, The Rand Corporation, Santa Monica, CA (t984)

5 Brick, D.B. and Snyder, A.W. 'External dc field of a long solenoid' Am J Phys 33 (1965) pp 905-909

6 Simkin, J, and Trowbridge, C.W. "Three dimensional non-linear electromagnetic field computations using scalar potentials' IEE Proc Pt B 27 (1980) pp 368-374

7 Terman, F.E. Radio Engineering McGraw-Hill, New York, 3rd edition ( 1951 )

Authors

The authors are in the Non-Destructive Evaluation Centre, Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE.

2 June 1989

338 N DT International December 1989