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Journal of Nondestructive Evaluation, Vol. 14, No. 4, 1995 Interaction of Phase-Shifted Fields of Two Single-Turn Coils Situated Above a Conducting Medium E. N. Derun, 1 A. A. Kolyshkin, 2 and R6mi Vaillancourt 3 Received February 3, I995 An analytical solution is obtained for the problem of the interaction of two phase-shifted electro- magnetic fields generated by two coaxial single-turn coils carrying alternating currents of the same frequency, but of different amplitudes and nonzero phase difference, qt. Two cases are considered: coils situated above a conducting half-space and above a two-layer medium. Numerical results show that ~ is the most important parameter. If the values of ~0 and of the other parameters are chosen properly, then the curve representing the change in impedance can lie in any quadrant of the complex plane. These results can be used for developing more sensitive and more selective eddy current testing methods. KEY WORDS: Eddy current NDE; phase-shifted fields; Hankel transform. 1. INTRODUCTION Eddy current methods are widely used to control the quality of materials. One of the simplest mathemat- ical models describing the interaction of an eddy current probe with a conducting medium consists of a single- tuna coil carrying an alternating current of frequency ~0 above a uniform conducting half-space with conductivity 0-. The analytical solution to this problem is well known. (1,2) The method of solution of this simple prob- lem is generalized (3,4~ to the case of a multilayer medium and to the case of a coil of finite dimensions. Analytical solutions are also known (5,6) for a single-turn coil (or a coil with finite dimensions) located inside a multilayer metal tube (or encircling a multilayer tube). One can assert that, at present, the theory and com- putational methods for encircling coils, internal axial coils and surface-scanning coils are fully developed.( 7~ tDepartment of Automated Control Systems, Riga Technical Univer- sity, Riga, Latvia, LV 1010. 2 Department of Applied Mathematics, Riga Technical University, Riga, Latvia, LV 1010. 3 Department of Mathematics and Statistics, University of Ottawa, Ot- tawa, ON, Canada K1N 6N5. An interesting approach (8~ was suggested in the case a coil is excited simultaneously by several currents with different frequencies. However, considerable mathemat- ical difficulties impose some restrictions on the appli- cation of this method. In order to find new methods and capabilities of eddy current testing it is useful to consider the interac- tion of the phase-shifted fields of two single-turn coils situated above a conducting medium. In this paper, we present an analytical solution to this problem for the cases where the coils are situated either above a uniform conducting half-space or a two-layer medium. Compu- tational results for the induced change in the coil im- pedance are presented. 2. MATHEMATICAL ANALYSIS 193 We consider two coaxial single-turn coils, called Coil I and Coil 2, situated in free space (region Ro) at heights hi and h2, respectively, above a uniform con- ducting half-space (region R1) of conductivity o- and rel- ative magnetic permeability/z, as shown in Fig. 1. In the following, except for the regions R1 and Rz, the sub- 0195-9298/95/1200-0193507.50/0 I995 Plenum Publishing Corporation

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Page 1: Interaction of phase-shifted fields of two single-turn coils situated above a conducting medium

Journal of Nondestructive Evaluation, Vol. 14, No. 4, 1995

Interaction of Phase-Shifted Fields of Two Single-Turn Coils Situated Above a Conducting Medium

E. N. Derun, 1 A. A. Kolyshkin, 2 and R6mi Vai l lancourt 3

Received February 3, I995

An analytical solution is obtained for the problem of the interaction of two phase-shifted electro- magnetic fields generated by two coaxial single-turn coils carrying alternating currents of the same frequency, but of different amplitudes and nonzero phase difference, qt. Two cases are considered: coils situated above a conducting half-space and above a two-layer medium. Numerical results show that ~ is the most important parameter. If the values of ~0 and of the other parameters are chosen properly, then the curve representing the change in impedance can lie in any quadrant of the complex plane. These results can be used for developing more sensitive and more selective eddy current testing methods.

KEY WORDS: Eddy current NDE; phase-shifted fields; Hankel transform.

1. I N T R O D U C T I O N

Eddy current methods are widely used to control the quality of materials. One of the simplest mathemat- ical models describing the interaction of an eddy current probe with a conducting medium consists of a single- tuna coil carrying an alternating current of frequency ~0 above a uniform conducting half-space with conductivity 0-. The analytical solution to this problem is well known. (1,2) The method of solution of this simple prob- lem is generalized (3,4~ to the case of a multilayer medium and to the case of a coil of finite dimensions. Analytical solutions are also known (5,6) for a single-turn coil (or a coil with finite dimensions) located inside a multilayer metal tube (or encircling a multilayer tube).

One can assert that, at present, the theory and com- putational methods for encircling coils, internal axial coils and surface-scanning coils are fully developed.( 7~

tDepartment of Automated Control Systems, Riga Technical Univer- sity, Riga, Latvia, LV 1010. 2 Department of Applied Mathematics, Riga Technical University,

Riga, Latvia, LV 1010. 3 Department of Mathematics and Statistics, University of Ottawa, Ot-

tawa, ON, Canada K1N 6N5.

An interesting approach (8~ was suggested in the case a coil is excited simultaneously by several currents with different frequencies. However, considerable mathemat- ical difficulties impose some restrictions on the appli- cation of this method.

In order to find new methods and capabilities of eddy current testing it is useful to consider the interac- tion of the phase-shifted fields of two single-turn coils situated above a conducting medium. In this paper, we present an analytical solution to this problem for the cases where the coils are situated either above a uniform conducting half-space or a two-layer medium. Compu- tational results for the induced change in the coil im- pedance are presented.

2. M A T H E M A T I C A L ANALYSIS

193

We consider two coaxial single-turn coils, called Coil I and Coil 2, situated in free space (region Ro) at heights hi and h2, respectively, above a uniform con- ducting half-space (region R1) of conductivity o- and rel- ative magnetic permeability/z, as shown in Fig. 1. In the following, except for the regions R1 and Rz, the sub-

0195-9298/95/1200-0193507.50/0 �9 I995 Plenum Publishing Corporation

Page 2: Interaction of phase-shifted fields of two single-turn coils situated above a conducting medium

194 Derun, Kolyshkin, and Vaillancourt

\

P2

0 ',< o~ i

h z N

/ / / x cy, g

Z / )

)

| Coil 1

| Coil 2

R o 0

/ / / / / R / ) y

Fig. 1. Two single-turn coaxial coils in free space, Ro, above a uniform conducting half-space, R~.

scripts 1 and 2 will refer to Coil 1 and Coil 2, respec- tively.

Coil 1, of radius Pl, carries the alternating current

il (r,q~,z,t) eo- = Iel (r, z) e j~ e~ (1)

where

[e (r, z) = 11 6(z -- hi) 6(r - Pl) (2)

o) is the frequency, /1 is the current amplitude, 6 (0 is Dirac's delta function, and e~ is a unit vector in the az- imuthal direction referred to a system of cylindrical po- lar coordinates (r, q~, z), centered at the origin, 0.

Coil 2, of radius P2, carries the current

i 2 (r,q~,z,t) % = I~2 (r,z) e j~'+~ e~ (3)

where

[~2 (r, z) = I26(z - h2) 6(r - P2) (4)

I 2 is the amplitude of the current and ~0 is the phase difference. Note that the frequency, w, is the same in Eqs. (1) and (3)._

Due to the axial symmetry of the problem, the vec- tor potential A(r, q~, z, t) has only one non-zero com- ponent (which is independent of q0,

A (r, q~, z, t) = A (r, z, t) e~ (5)

By the quasistatic approximation, ~ which is a usual as- sumption in eddy current testing, Maxwell 's equations reduce to the following equation:

2s = ~otZO---~ - IXo~I e (6)

where ~o is the magnetic constant and/~ is the ~-com-

ponent of the external current density. Since Eq. (6) is linear, one can use the superposition principle.

We shall first solve Eq. (6) in the absence of Coil 2, with I ~ = I~(r, z) (see Eq. 2). Second, we shall solve Eq. (6) in the absence of only Coil 1, with I e = I~(r, z) (see Eq. 4). Finally, the sum of these two solutions will give the solution to Eq. (6) in the presence of both coils.

In order to solve Eq. (6) we assume that

(r,z, t) = A (r,z) e j~ (7)

It then follows that the functions Ao(r, z) and Al(r, z) satisfy the following equations in regions R o and RI, re- spectively:

~A o = -tXoI ~ 8 ( z -hOS( r -pO z > 0 (8) AA~ + k 2 A~ = 0 z < 0 (9)

where k 2 = -jwcr~0/x. Equations (8) and (9) are coupled by the boundary

conditions

OAo [ 1 0 1 4 1 1 (10) 1o[_.=o = All~=o Oz ~=o ~ Oz ~=o

at the interface between R o and R 1. Moreover, we assume that the functions A o and Ax satisfy the following con- ditions at infinity:

Ai, OAi - - ---~ 0 as r ---> oo i = 0 , 1 (11) Or

A o - - ~ 0 a s z - - 4 ~ A 1 - - > 0 a s z ~ - oo (12)

The solution to Eqs. (8)-(12) can be found (1,2~ by means of the Hankel integral transform. In region R0, the in- duced vector potential, A~ a, which represents the reac- tion of the conducting half-space to the single-turn Coil 1 with current il, has the form

A ~ ( r , z ) = (13)

mIlplf a -q 2 A/x+q J1 (Apl) J1 (Ar) e a(z+h) dA

where q = ~/A z - k 2, J l (~ is Bessel 's function of the first kind of order 1 and the subscript 01 to A~)d(r, z) on the left-hand side of Eq. (13) means that Eq. (13) cor- responds to the case where only one coil, with current il given by Eq. (1), is present.

Similarly, if in Eq. (8) we replace h 1, Pl and/1 by h2, /92 and I2eJ% respectively, we obtain the induced vec- tor potential, -oz~"d, which represents the reaction of the conducting half-space to the single-turn Coil 2 with cur- rent i2, of the form

Page 3: Interaction of phase-shifted fields of two single-turn coils situated above a conducting medium

Interaction of Two Single-Turn Coils 195

<b

h 2

P2

}<

Z

P~ I | Coil 2

| Coil 1

h ~ Ro

>y

Fig. 2. Two single-ram coaxial coils in free space, R0, above a con- ducting two-layer medium with conductivities 0-, (region R~) and o- 2 (region R2). The depth of region R~ is d.

A~.a (r.z) = (14) O2

/Zo 12 P2 efO fo ~ a /x -q 2 h/z+q J~ (hp~) Jl(hr)e a(~+h) dh

in region R o. Finally, the induced vector potential, A~ "~ (r, z), in

the presence of both coils, is given by the sum of Eqs. (13) and (14):

Aig d (r,z) = Ag]a(r,z) + A~=2d(r,z)

#oI~p~ fo = Atz-q 2 ~l*+q J' ('~P') J,(Ar)e-a~+hO dA

+ /*~ /2 P2 e~ Jo= A/z-q J (Ap2) Jz(Ar)e A(~2~ dA 2 Acz+q

(15)

Let us compute the induced change in impedance, Z ~nd, in Coil 1:

j r 1" z~oa = # ]~ A~a %. dl (I 6)

where C is the contour of the coil. It follows from Eqs. (15) and (16) that

Zmd = joOl~oTrPl s A IX -_____5_q h/z+q ~ (hPl) e 2,~h, dh

I2 e:* j l ~ h /z -q (17) +Jm/z~ ~rP: ~ a tx+q

J1 (Ap~) ,J~ (Ap2) e-a(h,%)dA

Using the following dimensionless variables

hp~ h 1 s - ~ = P~ ~-~1~ot* a = - -

[~ Pl

H h2 P2 I 12 hi P Pl I1

we obtain from Eq. (17) that

Zind = (.olJ~o~TPlZ 0 ( 1 8 )

where

SI.L-- ~S2AF)

Sl~-~/s~+j + j t l p e;~ foo = SlX+k/s2+j J1 ( ts) J, (tips) e ~(t+m*ds

Equations (18) and (19) give the induced change in im- pedance of Coil 1 if both coils carrying currents (1) and (3), respectively, are situated above the conducting half- space.

The case of multilayer media can be treated in a similar way. For example, if Coil 1 and Coil 2 are sit- uated above a nonmagnetic two-layer medium with/z~ = /x 2 = 1 and parameters as shown in Fig. 2, then the induced vector potential, A~ nd, in region R o has the form

A~ nd (r,z) = C, (A)AJ 1 (Ar) e -a~ da

f: ~- e J~b C 2 ( A ) M 1 (/~y) e -kz dh

where

(20)

c,. (a)

IZoI~p,J~ (Apj) e-ah,[(q,+q2)(A-q,)+(q,-q2)(A +ql ) e 2q/]

2A [(A+q~)(ql+q2) - (h-qt)(q2-qO e-2qp]

for i = 1, 2 and q, = ~/.)t 2 + jcoo-,./x 0. Using Eqs. (16) and (20) and the following dimensionless variables

hi h d Of = - - /~ : P l ~/O)OrX ]'~0 S : ~ /91 '}/ : - -

O1 tO1

6 0"2 H h2 I /2 P2

0"1 hi [1 P Pl

we obtain the change in impedance of Coil 1 in the form

Zind ~- O)TTI,-L0p I Z 0 (21)

where

Zo = J t foo ~ D(s) j2 (fls) e -2~ds

+jl3IpeJr fo ~ D(s) J~ ( is) J~ (tips) e -~L+m" ds

(22)

and

Page 4: Interaction of phase-shifted fields of two single-turn coils situated above a conducting medium

196 Derun, Kolyshkin, and Vaillancourt

-0.2 -0.1

~ = - ~ ,

= -re/2

= - r e / 4

\Im Z o = 3~/4

~ .'~.',-s 13 . . . . 014' ' i i ' ' / I R e Z 0

~ = 0

Fig. 3. Z 0, as a function of/3 and q/, in the case of a conducting half-space, with c~ = 0.1, /x = 1, H = 0.9, p = t, I = l.

/b Im Zo -0.05 | 0.05 0.1 0.15 0.2 0.25 0.3

. . . . . . . . . . . . . ,e o

~ f . / = -r~/4 ~/=0

Fig. 4. Zo, as a function of/3 and 0, in the case of a conducting half-space, with c~ = 0.1, ~ = 1, H = 0.9, p = 1, I = 0.5.

D ( s ) =

- ( s - s 2 ~ ) ( ~ / s - U + + j 6 - ~/s-~+j)e -2~" s~2+:

3. NUMERICAL RESULTS AND DISCUSSION

The change in impedance, Z0, is computed for the cases of a conducting half-space (Eq. 19) and a two- layer medium (Eq. 22), for different values of the par- ameters. The ten dots on each curve in Figs. 3-10 cluster in the direction of increasing/3 as indicated by an arrow.

For the case of a conducting half-space, Zo is given by Eq. (19) as a function of/3 and of the phase shift, q/, as shown in Figs. 3-6, for c~ -= 0.1, ~ = 1 and H = 0.9. The values of the other parameters are p = 1, I

Page 5: Interaction of phase-shifted fields of two single-turn coils situated above a conducting medium

Interaction of Two Single-Turn Coils 197

/~Im Z o

-0.6 -0.4 -0.2 ~ ~ ~ 0.8

" ~ ' L , . . . . . . . . . . . " 1 % 2 ~ 6 ~ ' ~ ' ' >ReZo

~i/__~rC/2 - 0 . 6 / ~ / 1 ; ~'~

W_ r c / 4 ~ _ l . 0 I - 0 . 8 ~ ~ g/4

Fig. 5. Zo, as a function o f t and ~p, in the case of a conducting half-space, with ~ = 0.1,/x = 1, H = 0.9, p = 1, I = 2.

0.1 -o.1 S

I , , , ,

C"

-0.3

gt = -rt/2 /

/ tg = -rt/4

I m Z o

- • 0 . • 0.3

-0.5-0"4 / / / ~g = rt/2

-0.6 / ~ = n/4

~ = 0

~ ' - ~ R e Z 0

Fig. 6. Zo, as a function offi and ~p, in the case of a conducting half-space, with a = 0.1, / ~ = 1, H = 0.9, p = 2, I = I.

= 1 in Fig. 3 ;p = 1, I = 0.5 in Fig. 4 ; p = 1, I = 2 in Fig. 5; and p = 2, I = 1 in Fig. 6. Note the dis- tinctive peculiarity of the curves representing Z 0 if there is a phase shift (~ r 0) in comparison with no phase shift.

Figures 3-6 show that the most important param- eter is the phase shift, ~p. This fact can be explained as follows. On the one hand, the energy accumulation in Coil 1, which depends on the phase difference, ~9, be- tween currents i 2 and il, affects the real part of Z o. On

the other hand, the medium demagnetizing action on the coils, which also depends on 0, affects the imaginary part of Z 0.

It is seen in Figs. 3~5 that the curve representing the change in impedance can be located in any quadrant of the complex plane (in contrast with the case tp = 0) provided the parameters of the problem (and especially the phase difference, O) are suitably chosen. In particular (see, for example, Fig. 5), for any given value of/3 one can find a value of ~ such that either ReZ o = 0 or ImZ 0 = 0.

Page 6: Interaction of phase-shifted fields of two single-turn coils situated above a conducting medium

198 Derun, Kolyshkin , and Vai l lancourt

Im Z o = 37c/4

= - 3 ~ 4 ~ = ~/2

~ = -/i;/2 ~ / ~ ~=- g/4

= - n / 4 i ~ -~ 0 -

Fig. 7. Z0, as a function of/3 and ~0, in the case of a two-layer medium, with c~ = 0.1, H = 0.9, y = 0.1, ~ = 0.8, p = 1, I = 1.

-0.05

-0.1

= - 3n/4 -0.3

= - n / 2 -0.5

Im Z 0

0.05 0.1 0.15 0.2 0.25 0.3

/ t

= - n / 4 ~ = 0

Fig. 8. Z o, as a function of fi and ~0, in the case of a two-layer medium, with a = 0.1, H = 0.9, 3' = 0.1, 6 = 0.8, p = 1, I = 0.5.

The parameter I is also important since the accu- mulated energy and the medium demagnetizing action also depend on its value.

A second set o f computations for Zo is done by means of Eq. (22) for the case o f a two-layer medium and the results are shown in Figs. 7-10, for a = 0.1, H = 0.9, y = 0.1 and ~ = 0.8. The values o f the other parameters are p = 1, I =- 1 in Fig. 7; p = 1, I = 0.5

in Fig. 8 ; p = 1, I = 2 i n F i g . 9; a n d p = 2, I = 1 in Fig. 10. As in the case of a conducting half-space, the most important parameter is ~. By choosing appropriate values of ~0 and of the other parameters, one can get the curve representing the change in impedance in any quad- rant o f the complex plane. These peculiarities o f Zo can be used for developing eddy current testing methods with higher testing sensitivity and finer selectivity.

Page 7: Interaction of phase-shifted fields of two single-turn coils situated above a conducting medium

Interaction of Two Single-Turn Coils 199

~ = r c 0.4

-0.6 - 0 " 4 , 7 0.2 ,,~0.~

= - 3~/4 - 0 .

-0.6 = -re/2

J gt = -~/4

Im Z o

~ : = t g = 3~/4

0.8

~/2

-0.8 ; ~u = rW4

-1.0

~ = 0

F i g . 9. Zo, as a f u n c t i o n o f f i a n d ~, in the c a s e o f a t w o - l a y e r m e d i u m , w i t h a = 0 . I , H = 0 .9 , 3' = 0 .1 , 6 = 0 .8 , p = t , I = 2.

/•Im Z o

_o, ' 03 o4

~ =-3~z/4 ' ) / I " 1 " 1 2 " " ~ ~ ~ ' x ~ - g " ~ ~

�9 -- / : o ; / 2 / = -~/4 -0.6 , , / ~ = rW4

W=0

Fig . 10. Zo, as a f u n c t i o n o f f i a n d ~, in the c a s e o f a t w o - l a y e r m e d i u m , w i t h a = 0 .1 , H = 0 .9 , 3 / = 0 .1 , 6 = 0 .8 , p = 2, I = ! .

4. CONCLUSION

The interaction of the phase-shifted fields of two single-turn coils situated above a conducting medium is studied in the present paper. Two cases are considered: two coils above a conducting half-space and two coils above a two-layer medium. Numerical results show that the phase difference, ~, is the most important parameter. If the values of ~ and of the other parameters of the problem are chosen properly, then the curve representing

the change in impedance can be situated in any quadrant of the complex plane. The results of this paper can be used for developing eddy current testing methods with higher sensitivity and finer selectivity.

ACKNOWLEDGMENTS

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada

Page 8: Interaction of phase-shifted fields of two single-turn coils situated above a conducting medium

200 Derun, Kolyshkin, and Vaillancourt

under grant A 7691 and the Centre de recherches math6-

mat iques o f the Universi t+ de Montreal .

REFERENCES

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2. D. H. S. Cheng, The reflected impedance of a circular coil in the proximity of a semi-infinite medium, IEEE Trans. Instr. Meas. 14(3):107-116 (1965).

3. C. V. Dodd, W. E. Deeds, and J. W. Luquire, Integral solutions to some eddy current problems, int. J. Nondestr. Testing 1(1):29-107 (1969).

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6. C. V. Dodd and W. E. Deeds, Analytical solution to eddy cur- rent probe coil problems, J. Appl. Phys. 39(6):2829-2838 (1968).

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8. V. G. Pustinnikov and S. D. Anisimov, Multiparameter electro- magnetic testing of steel products, Zavodskaya Laboratoria 30(10): 1236-1239 (1964) (Russian).

9. W. R. Smythe, Static and Dynamic Electricity (3rd Ed.) Revised printing (Hemisphere Publishing Corporation, New York, 1989).