2254 IEEE TRANSACTIONS ON MAGNETICS, VOL. 31, NO. 3 , MAY 1995

Analytical Solutions in Eddy Current Testing of Layered Metals with Continuous

Conductivity Profiles T. P. Theodoulidis, Student Member, IEEE, T . D. Tsiboukis, Member, IEEE,

and E. E. Kriezis, Senior Member, IEEE

Abstract-In this paper exact analytical expressions for the impedance of a cylindrical air-core coil above a layered metal structure whose conductivity varies continuously with depth are presented. Although the model is general, attention is focused on three conductivity profiles: the linear, the quadratic and the exponential. The derived expressions for the impedance change for each profile could provide a useful tool for the solution of the inverse problem: that of determining the conductivity from variable frequency measurements of the impedance. Further- more, the obtained final formulas contain elegant mathematical functions and show a substantially higher computational effi- ciency with respect to existing methods.

I. INTRODUCTION HE eddy current nondestructive method is a powerful T tool in today's industrial practice for measuring the

constant conductivity of metals. In most cases this in- volves the measurement of the impedance of an air-core circular coil of rectangular cross section, driven by a con- stant amplitude alternating current, above a conductive infinite slab with a plane surface. For this configuration of wide applicability, analytical solutions have been pre- sented in [l] and [2] providing both the various electro- magnetic field quantities and the impedance of the coil.

However, continuous conductivity profiles appear also in many applications. Some of these include the thermal processing of aerospace engineering materials in order to obtain added strength or the solidification in a casting pro- cess. Especially in the second application the detection of spatial variations in solidification structures depends upon the detection of the conductivity changes between the solid and liquid state. In such cases the application of the eddy current method seems to be again the only method for the detection of the conductivity profiles.

Only a few things are known about the impedances of eddy current probe coils produced by such geometries where the conductivity is expected to vary with distance from the sample surface. Apart from the possible use of the Finite Element Method this problem can be effectively treated by two independent methods that were both pre- sented in [3]. In the first method, which was also pre-

Manuscript received April 12, 1994; revised November 21, 1994. The authors are with the Department of Electrical & Computer Engi-

neering, Aristotle University of Thessaloniki, Thessaloniki, 54006 Greece. IEEE Log Number 9409090.

'

sented in [4], the medium is subdivided into a number of parallel homogeneous layers for the approximation of a continuously varying conductivity profile. By taking a suf- ficiently large number of such layers of vanishing thick- ness and by implementing the algorithm given in [5] any desired degree of precision may be obtained in the com- putation of the electromagnetic fields and therefore the impedance of the coil. Despite its generality this method being numerical in nature, is not very elegant, since it can not describe straightway the effects of the various param- eters of the model to the solution.

An alternative choice is the second method presented in [3] which consists of taking special forms of conduc- tivity variations and trying to express the solutions of the differential equations in terms of known mathematical functions. This approach was also favored by Wait in his classical work for the propagation of electromagnetic waves in stratified media [6] where he uses some analyt- ical results in order to provide solutions for the reflection coefficient of an inhomogeneous medium. In [3] the prob- lem of determining the impedance of a cylindrical air-core coil over a nonmagnetic metal half-space with an arbitrary near-surface conductivity profile, is formulated in terms of the solution of a one-dimensional ordinary differential equation. Following this formulation, an exact solution for the case of a near-surface conductivity profile that var- ies as a hyperbolic tangent with depth is reported.

In the present paper the same analytical formulation is followed and expanded in order to treat cases that involve a layered metal half-space. The geometry considered is that of a double layer medium whose top layer conductiv- ity varies continuously with depth and bottom layer con- ductivity remains constant. On the boundary of the two layers the conductivity function of depth is allowed to be discontinuous. This extra degree of freedom in the for- mulation presented in [3], which requires a continuous function that describes conductivity of the whole half- space, is the main contribution of this paper. The imped- ance difference calculated is between the impedance pro- duced by the layered half space and by a half-space of the base material.

By using this expansion we provide exact analytical so- lutions for the cases when the surface layer conductivity is an exponential, a linear or a quadratic function of depth.

0018-9464/95$04.00 0 1995 IEEE

THEODOULIDIS et al.: ANALYTICAL SOLUTIONS IN EDDY CURRENT TESTING 2255

Results for the eddy currents induced in each case are also presented.

The final expressions for the impedance change for each profile and the results given can assist in the solution of the inverse problem, Le., that of determining conductiv- ity from variable frequency measurements of the imped- ance. The least squares algorithms can be applied once the unknown conductivity profile is approximated by one of the profiles studied and the solution of the forward problem is already in hand.

11. THEORETICAL ANALYSIS The geometry considered is shown in Fig. 1. A circular

coil of rectangular cross section is located above a layered nonmagnetic and conductive half-space. The conductivity of the upper layer is a continuous function of the depth z while the conductivity of the bottom layer is constant.

The magnetic vector potential A in the air is the sum of 'the primary potential due to the harmonically varying source current of the coil I exp ( jut) in an unbounded space and the secondary potential originated from the eddy currents induced within the conductive material

(1)

Since we assume an axial symmetry both of these po- tentials have only an azimuthal component and lack axial and radial ones. The expressions for the primary field A,, for a N-turn coil and the cylindrical coordinates centred at the axis of the coil, were given in [ l ]

A = A, + A,,

1 " 1

- exp (-az) [exp (ah,) - exp (ah,)] da

1 OD1

- [a(z - h2)1 - exp [-a(z - h , ) ] } da

1 " 1

Af.'(r, z) = 5 (pi01 j, 2 1 0 - 2 , rl)JI(ar)

2 so a3

(2)

Aiv2@, z ) = - (pio) - I(r2, rl)Jl(ar){2 - exp

(3)

Afq3(r, z ) = - (pio) j, 2 1 0 - 2 , rl)JI(ar) exp (az) 2

- [exp ( -ah l ) - exp (-ah,)] da (4) where

and

The procedure to derive the above expressions was to solve for the magnetic vector potential for a single delta- function coil and then approximate the coil by the super- position of a number of delta-function coils. This super-

1.2 2 r l

I h ' z

r 1.3

Fig. 1 . Eddy current testing of a layered half-space whose top layer con- ductivity is a continuous function of depth.

position led to an integration over the coil cross section when the current distribution in the delta-function coils was let to approach a continuous current distribution.

On the other hand, due to the axial symmetry the sec- ondary field A,, can be found from the solution of the dif- fusion equation which for an isotropic, linear and in- homogeneous conducting medium reduces to the form:

Following the method of the separation of variables, by setting A(r, z ) = R(r)Z(z) we take for the r dependence the ordinary differential equation

d2R(r) 1 dR(r) dr2 r dr

+ - - + ( a2 - - r12) R(r) = 0 (8)

which has the solution:

R(r) = CJ,(ar) + DYl(ar) (9) where JI and Yl are the Bessel functions of the first order and of the first and second kind respectively. Due to the divergence of Yl at the origin, D = 0 at all regions.

To find out the z dependence we have to solve the or- dinary differential equation

-- - (a2 + jwpa (Z))Z(Z). d 22(z)

dz

The solution of this equation, for a continuous conduc- tivity function a ( z ) is the main problem in the exposed approach. In the following we shall present exact solu- tions of (10) when a ( z ) varies exponentially, linearly and quadratically. For the moment we shall assume that the solution of (10) can be written as the linear combination of two linearly independent solutions Fl( f ( z ) ) and

Z(Z> = CFI(f(2)) + BF2(f(z)) (1 1)

where the function f ( z ) originates from the possible trans- formations of the independent variable, that take place during the process of the solution of (10). To proceed with the analysis we note at this point that if U = 0

Z(z) = C exp ( -az) + B exp (az) (12)

~ _ _ _ - -- I r-.

2256 IEEE TRANSACTIONS ON MAGNETICS, VOL. 31, NO. 3. MAY 1995

and if U is constant

Z(z) = C exp ( - a l z ) + B exp (a l z ) { = - 2 exp (%)a and

k3

m

A3 = so B? exp (a lz )J l (ar) da (16)

where we have set BI = C3 = 0 due to the requirement that the electromagnetic fields vanish for z -+ f 03. Ap- plication of the appropriate boundary conditions on the boundary interfaces z = 0 and z = - c leads to a 4 X 4 linear system containing C I , C2, B2, and B3 as unknowns. The solution of this system is given in the Appendix.

Once we have determined the vector potential, we can calculate any electromagnetic field quantity. In the eddy current testing the two most important quantities are the eddy currents themselves and the probe coil impedance. The eddy currents are derived from

where I , and K, are the modified Bessel functions of the first and second kind respectively and of order Y =

2Ja2 + pk,/k3. 2 ) If the conductivity varies linearly, i.e., if a ( z ) = kl

+ k22, then by defining the variable

a' + pkl + pk2z ( ~ k , ) ~ ' ~ l =

the solution of (10) becomes

Z(z) = CAi({) + B Bi({) (22)

3 ) Finally if the conductivity varies quadratically, i.e., where Ai and Bi are the Airy functions.

if ~ ( z ) = kl + k,z + k3z2,

then by introducing the variable transformations

2k3({ - Z ) = k2

J = -jwa(z)A(r, z ) (17) and

while the coil impedance can be calculated from 2 = V/Z where

(Pk2I2 4Pk3

k4 = a2 + pkl - ~

the differential equation (10) takes the form j h * in rA132(r , z ) dr dz. (18) j2xwN

I/= ( A 2 - hl)(r2 - ri) h i ri

The final expression for the impedance is = (k4 + pk312)Z({) (23)

d r 2 In order to solve the above differential equation we fur-

ther apply the transformations

+ [2 exp ( - a @ , - h , ) ) - 2

+ (exp ( -ah2) - exp ( - a h l ) ) 2 ] C l ( a ) } da (19)

This is the basic result, which gives the impedance of a coil located above a two-layer metal whose upper layer conductivity varies continuously with depth. We are now concerned with the solution of (10) in order to identify the functions involved in the expression of C l ( a ) . This differential equation is solvable in terms of known math- ematical functions for many forms of ~ ( z ) . By simple transformations of the independent variable it reduces often to the Bessel, the Mathieu or the Kummer equation [ 7 ] , [8]. Next we study three simple conductivity varia- tions from the possible alternatives.

1 ) For an exponential conductivity variation in the up- per layer, i.e., if ~ ( z ) = kl + k2 exp (k3z), if we make the substitutions

8

Then the solution of (23) is given by

where Z ( t ) satisfies the differential equation:

which has the solution:

where M and U are the Confluent Hypergeometric or Kummer functions.

THEODOULIDIS et al.: ANALYTICAL SOLUTIONS IN EDDY CURRENT TESTING 2257

111. NUMERICAL TREATMENT AND RESULTS An efficient computation of the higher transcendental

mathematical functions, encountered in the preceding analysis, is undoubtedly very useful. In the present paper these functions have been computed as follows: The Airy and the Bessel functions, have been computed by using a portable package for the computation of Bessel functions of complex argument and real nonnegative order [9]. The computation of the Confluent Hypergeometric functions has been based on the approximations given in [ 101. The computation of the modified Bessel functions which arise in the expressions for the exponential profile has been done by using the following relations with the Confluent Hypergeometric functions, since for kl # 0 they are both of complex argument and order and the use of [9] is not recommended :

o lSlml ‘d

Linear

Quadratic

1.5 ‘ I I 0 1 2

z Imml Fig. 2. The conductivity variations studied, as a function of depth.

= r ( l + v) exp (z)

(26)

where I’ denotes the gamma function. Except the gamma function of complex argument, an-

other function that has been computed efficiently was the Struve function H, of orders 0 and 1 , which arises in the computation of the Bessel function integral:

Finally the computation of the infinite integrals has been performed by mapping the infinite range of integration to a finite interval and proceeding with application of adap- tive quadratures.

Proceeding now with the results we focus our attention in the calculation of the impedance difference produced by a half space of the base material and the impedance produced by a layered half space. Both impedances are taken from (19). In the first case Cl(a) contains functions that are solutions of (10) for constant conductivity equal to that of the base material, while in the second case it contains functions that are solutions of (10) for each variable conductivity profile that is examined. The final expression for the impedance difference is

Before we proceed with the main results that exhibit the performance of each profile it is very interesting to com-

pare the analytical results obtained for the exponential profile with those obtained with the numerical method presented in [3]. One of the conductivity variations that was studied in [3] had the exponential form

a(z) = u2 + (al - u2) exp ( -~/0.410-~) (30) which is shown in Fig. 2 together with the linear and quadratic variations that will be studied in the following:

(32)

For all three cases the base has been taken to have the conductivity u2 = 3.766 lo7 S/m, while the conductivity of the surface layer, which is c = 1 mm thick, has been chosen to be half that value, uI = 1.883 lo7 S/m, at z = 0. The coil has the following parameters: N = 580, rl = 1 .3 mm, r2 = 3.3 mm, hl = 0.5 mm and h2 = 7.8 mm.

A comment must be made about the exponential profile studied. It is easily seen from Fig. 2 that the variation examined in [3] does not include a bottom layer of con- stant conductivity, but the form of the variation is such that the conductivity of the upper layer tends to a2 in a continuous way. Thus the model presented here has to be simplified, either by moving the interface z = - c to in- finity or by omitting the second layer in the above anal- ysis.

Figs. 3(a) and 3(b) show the real and imaginary parts of the impedance difference, for the exponential profile, as a function of frequency for both the numerical and the analytical method.

The numerical method uses 40 layers of constant con- ductivity to approximate the exponential variation of the conductivity and gives quite a good approximation of the exact analytical solution in the lower frequency range. On the other hand in the higher frequency range the inherent deficiency of the numerical method is observed as the in- duced eddy currents are accumulated close to the surface and the region of interest contains fewer layers. The so-

2‘ a(z) = 01 + (01 - 02) 2.

- I -7

2258

3500

3000

2500

2000

1500

1000

500

0

IEEE TRANSACTIONS ON MAGNETICS, VOL. 31, NO. 3, MAY 1995

-

-

-

-

-

-

, I I I

Re(L.2) [Ohm] 0.05 I

-0.2 I I I

0 10 20 30 40

f [KHzl

(a)

0

-0.2

-0.4

-0.6

-0.6

Analytical

0 10 20 30 40

f [KHzl

(b)

Fig. 3. The impedance difference for the exponential profile as a function of frequency: (a) Real part; (b) Imaginary part.

lution is either to make an adaptive choice of the length of the layers or to simply increase their number, both of which increase the computational time. Comparison of the two methods, concerning CPU time, showed that the an- alytical method, for the best approximation of the Bessel functions of complex argument and order, was three times faster than the numerical method which was using 40 lay- ers. Figs. 4(a) and 4(b) show the real and imaginary parts of the impedance difference produced by the linear and the quadratic profile.

It is easily seen that the quadratic profile has the largest impedance change at all frequencies, while the exponen- tial has the smallest. This confirms the observation done in [3], that the more localized the conductivity change is to the surface, the greater the impedance change. It is also seen that the greatest change occurs for the higher fre- quency considered. This means that at this frequency the eddy currents induced in the layered half space have the greatest difference from those induced in the case of the

Examination of contour plots of the eddy currents shows that they have all the same radial behaviour. The arith- metic mean of the two radii of the coil r = ( r , + r7)/2

' base material.

- Linear

Quadratic

-0.4 0 10 20 30 40

f [kHz1

(a)

Linear

0 10 20 30 40

f [kHz1

(b)

Fig. 4. The impedance difference for the linear and quadratic profiles as a function of frequency: (a) Real part; (b) Imaginary part.

- Constant

Exponential

Linear

Quadratic

-

Fig. 5 . Eddy currents amplitude for the conductivity variations studied, as a function of depth.

has been chosen to be the radial distance for which the behaviour of eddy currents with depth will be examined. Fig. 5 shows the amplitude of the eddy currents induced in the upper layer for the three variable profiles, as a func- tion of depth, compared with the eddy currents induced when the half mace is consisted exclusivelv from the base

THEODOULIDIS et al.: ANALYTICAL SOLUTIONS IN EDDY CURRENT TESTING 2259

material. The amplitude of the driving current was as- sumed 50 PA.

As it was expected, the eddy currents for the constant profile have the greater value at the surface but they go under a stronger attenuation far from it. For the other three cases, the eddy currents for the exponential profile are closer to the eddy currents for the base material since in this case the conductivity variation is smaller. A remark- able difference is observed only close to the surface, since for this combination of conductivities and excitation fre- quency the skin effect is quite strong.

IV. CONCLUSIONS In this paper we have solved for the magnetic vector

potential produced by a cylindrical air-core coil above a layered metal structure whose conductivity varies contin- uously with depth, therefore we are able to calculate any physically observable electromagnetic induction phenom- enon for this situation. We have concentrated our atten- tion in the calculation of such a phenomenon that is of great interest in eddy current testing: the impedance of the coil. Equation (19) is the final expression that provides a rigorous tool for the solution of the forward problem of determining the impedance produced by a conductivity profile which varies with depth. The analytical solution presented for this problem, which so far was treated nu- merically, is exact and reduces the computational time considerably. On the other hand by using (19) we can make further investigations in the contribution of all the variables in the problem and finally assist in the solution of the inverse problem for the determination of the profile from impedance measurements, once this profile is ap- proximated by the variations for which (10) is solved an- alytically.

Three conductivity variations have been studied, but in order to further implement the model presented we have to seek for analytical solutions of ( lo ) , in terms of known mathematical functions, for other conductivity variations a(z). Further work is needed in this direction.

CI =

c2 =

B2 =

B3 =

where

and

APPENDIX A

K{a[(LS - MR) + aI(MP - Le)] - (NS - OR)

- al(OP - NQ)}/DEN (All

-2aK(a1 Q - S)/DEN (A2)

ZZK(U,P - R)/DEN (A3)

2aK exp (a,c)(PS - QR)/DEN 644)

DEN = u[(LS - MR) + al(MP - LQ)]

+ (NS - OR) + uI(OP - NQ) (A5)

S = [F2(f(z))lS = -c (A14)

where the dash symbol denotes differentiation with re- spect to z.

REFERENCES

[I] C. V. Dodd and W. E. Deeds, “Analytical solutions to eddy-current probe-coil problems,’’ J. of Appl. Phys., vol. 39, pp. 2829-2838, 1968.

[2] J. A. Tegopoulos and E. E. Kriezis, Eddy Currents in Linear Con- ducting Media.

[3] E. Uzal, J. C. Moulder, S. Mitra and J . H. Rose, “Impedance of coils over layered metals with continuously variable conductivity and permeability: Theory and experiment,” J. of Appl. Phys., vol. 74, no. 3 ,pp. 2076-2089, 1993.

[4] E. Uzal and J. H. Rose, “The impedance of eddy current probes above layered metals whose conductivity and permeability vary con- tinuously,” IEEE Trans. Magn., vol. 29, pp. 1869-1873, 1993.

[5] C. C. Cheng, C. V. Dodd and W. E. Deeds, “General analysis of probe coils near stratified conductors,” Int. J . Nondestrucrive Tesr- ing, Vol. 3 , pp. 109-130, 1971.

[6] J. R. Wait, Electromagnetic Waves in Stratified Media. New York: Pergamon Press, 1962, pp. 64-85.

[7] G. M. Murphy, Ordinary Differential Equations and Their Solutions. New York: Von Nostrand Reinhold, 1960, pp. 311-379.

[8] M. Abramowitz and I. A. Stegun, Handbook of Marhemarical Func- tions (with formulas, graphs and mathematical tables). New York: Dover, 1970.

[9] D. E. Amos, “A subroutine package for bessel functions of a com- plex argument and nonnegative order,” Sandia Laboratories, Tech. Rep. Sand 85-1018, May 1986.

New York: Academic, 1969, pp. 70-100.

New York: Elsevier, 1985, pp. 60-111.

[lo] Y. L. Luke, The Special Functions and Their Approximations.

Theodoros P. Theodoulidis (SM’94) was bom in Ptolemaida, Greece, on February 1. 1969. He received the diploma in electrical engineering from the Aristotle University of Thessaloniki in 1992.

Currently he is a research assistant at the department of Electrical and Computer Engineering of the Aristotle University of Thessaloniki and he is working toward a Ph.D. degree. His research interests are focused on computational electromagnetics with applications on nondestructive testing and signal processing of eddy current testing data. He is a member of the American Society for Nondestructive Testing and the Technical Chamber of Greece.

Theodoros D. Tsihoukis (S’79, M’81, M’91) was bom in Larissa, Greece, on Feburary 25, 1948. He received the diploma of EE and ME from the National Technical University of Athens, Athens, Greece, in 1971 and the Dr.Eng. degree from the Aristotle University of Thessaloniki, Thessalo- niki, Greece, in 1981.

Since 1981 he has been working at the department of Electrical and Com- puter Engineering of the Aristotle University of Thessaloniki, where he is now a professor.

-r I--

2260 IEEE TRANSACTIONS ON MAGNETICS, VOL. 31, NO. 3, MAY 1995

Prof. Tsiboukis is the author of several books and papers. His research activities include electromagnetic field analysis by energy methods, com- putational electromagnetics (FEM, BEM, Edge Elements, MOM, FDTD, ABC), and adaptive techniques with the FEM. He is a member of various societies, associations, chambers and institutions.

Epameinondas E. Kriezis (M’7 I , SM’82) received the degree in mathe- matics from the University of Thessaloniki, Thessaloniki, Greece. He then received a diploma in electrical engineering and the doctor’s degree from the National Technical University of Athens, Greece.

He worked in the Public Power Corporation until 1974. Since then he has been a professor in the Department of Electrical and Computer En- gineering at the University of Thessaloniki, where he teaches electromag- netic field theory. His research encompasses problems of the field related to eddy currents, scattering related to remote sensing and optics. He has served as chairman of the Department of Electrical and Computer Engi- neering for five consecutive years.

Dr. Kriezis is the author of the books Eddy Currents in Linear Con- ducting Media (Elsevier, 1985) and Efectromagnerics and Optics (World Scientific, 1992). He has been the recipient of the Embirikion Award for Science and Technology and he is a member of the Technical Chamber of Greece and Eta Kappa Nu.