PIERS ONLINE, VOL. 7, NO. 8, 2011
766
A New Analytical Method for Calculation of Eddy Current
Distribution and Its Application to a System of Conductor-slab
and Rectangular Coil
T. Itaya1 , K. Ishida2 , A. Tanaka3 , N. Takehira2 , and T. Miki4
1
Suzuka National College of Technology, Japan
2
Tokuyama College of Technology, Japan
3
Ube National College of Technology, Japan
4
Yamaguchi University, Japan
Abstract— This study proposes an analytical method for the eddy current distribution analyses, and provides the eddy current distribution in conductor slab with rectangular coils arranged
perpendicular to the slab. Our analytical method utilizes double Fourier transform to derive a set
of equations for determining the eddy current distribution. The eddy current density is derived
from the analytical solution called stream function. The spatial distribution of eddy current,
which is dependent upon the frequency of coil current, the thickness and velocity of a conductor
slab, is successfully obtained. Our analytical method is usable for calculations of varieties of eddy
current problems, and we demonstrate the eddy current distributions in a conductor slab facing
to a rectangular coil.
1. INTRODUCTION
The eddy current analysis is widely used in solving the problems on magnetic interaction between
an electrically conductive material and an excitation coil that carries an AC current, e.g., the eddy
current problems of magnetic resonance diagnosis in medical field, nondestructive testing (NDT)
and magnetic sensing in industrial measurement field, induction heating in industrial power application field, etc. Numerical methods such as the finite element method (FEM) and/or boundary
element method (BEM) are often employed, and recent increase of computer power have enabled
three-dimensional eddy current analysis [1–3], Recently, eddy current distribution (ECD) imaging
has also been developed by several research groups [4–6]. ECD imaging of a moving conductor
facing to an excitation coil is useful for developing more precise and sophisticated magnetic sensing as well as other varieties of applications. Although the FEM is a potential tool for solving
eddy current problems, it is hardly applied to the eddy current problems with moving conductors because of long computational time. Instead, the authors have proposed a new analytical
method that enables to obtain the exact solution of magnetic field in a moving conductor facing to
a rectangular excitation coil [7, 8]. Using this analytical method, we have developed optional NDT
and magnetic sensing techniques, for instance, the thickness and velocity measurements of moving
conductors. This article presents an analytical method of ECD in a moving conductor slab facing
to a rectangular coil. The ECDs have been obtained using a set of analytical formulae derived by
double Fourier transform of Maxwells equations. A shape function is introduced for analyzing ECD
excited by an arbitrarily-shaped coil [9, 10], and numerical ECD calculations were carried out with
a conventional personal computer. In this article, we demonstrate the ECDs dependent upon the
excitation frequency of the coil as well as the thickness and velocity of the moving conductor slab.
2. THEORETICAL ANALYSIS
Figure 1 shows a schematic drawing of the coil-conductor arrangement studied in this work. The
plane of a rectangular coil is perpendicularly arranged to the conductor slab. In our ECD analyses,
we have simply assumed:
(1) The moving conductor is isotropic and infinitely wide.
(2) The coil is one-turn, and it carries an AC current with a given effective RMS value and angular
frequency ω. The coil wire is assumed to be infinitely thin.
(3) Conductivity σ, permeability µ and conductor velocity v are all constant.
PIERS ONLINE, VOL. 7, NO. 8, 2011
767
z
b
z0
a
I
vy
0
y
x0
v
vx
x
d
(σ , µ
0
. µr )
Figure 1: Arrangement of analytical model.
2.1. Magnetic Flux Density
In our previous papers [9], we have introduced shape function to analyze the magnetic field produced
by an arbitrarily-shaped coil. However, because of high symmetry of rectangular coil, one can simply
express the magnetic flux density for this coil-conductor system in Fig. 1. According to Refs. [7, 8],
the x-, y- and z-components of the magnetic flux density B 1 in the conductor slab are given by the
following equations:
B1x
B1y
B1z
Z∞ Z∞
o
√
sin (bη) hn
ξ 2 ejx0 ξ
µ0 µr I
−( ξ 2 +η 2 −γ )d
2γd
−
(1
+
λ
)
e
+
ν
e
erz
= −
0
0
4π 2
η (ξ 2 + η 2 ) 1 − e2γd
−∞ −∞
n
o
i
√ 2 2
√ 2 2³ √ 2 2
√ 2 2´
+ 1+λ0 −ν0 e−( ξ +η −γ )d e−rz e−z0 ξ +η ea ξ +η −e−a ξ +η e−j(xξ+yη) dξdη, (1)
Z∞ Z∞
o
√
µ0 µr I
ξejx0 ξ sin (bη) hn
−( ξ 2 +η 2 −γ )d
2γd
= −
erz
−
(1
+
λ
)
e
+
ν
e
0
0
4π 2
ξ 2 + η 2 1 − e2γd
−∞ −∞
n
o
i
√ 2 2
√ 2 2³ √ 2 2
√ 2 2´
+ 1+λ0 −ν0 e−( ξ +η −γ )d e−rz e−z0 ξ +η ea ξ +η −e−a ξ +η e−j(xξ+yη) dξdη, (2)
Z∞ Z∞ jx0 ξ
o
√
µ0 µr I
sin (bη) hn
ξe
−( ξ 2 +η 2 −γ )d
2γd
erz
= −j
−
(1
+
λ
)
e
+
ν
e
0
0
4π 2
ηγ
1 − e2γd
−∞ −∞
n
o
i
√ 2 2
√ 2 2³ √ 2 2
√ 2 2´
− 1+λ0 −ν0 e−( ξ +η −γ )d e−rz e−z0 ξ +η ea ξ +η −e−a ξ +η e−j(xξ+yη) dξdη.(3)
Here,
q
ξ 2 + η 2 − jσµ0 µr (vx ξ + vy η) + jωσµ0 µr ,
© 2
¡
¢ª ¡
¢
γ − µ2r ξ 2 + η 2
1 − e−2γd
λ0 = ³
,
´2 ³
´2
p
p
γ + µr ξ 2 + η 2 − γ − µr ξ 2 + η 2 e−2γd
√ 2 2
p
4µr ξ 2 + η 2 γe( ξ +η −γ )d
v0 = ³
.
´2 ³
´2
p
p
γ + µr ξ 2 + η 2 − γ − µr ξ 2 + η 2 e−2γd
γ =
(4)
(5)
(6)
Here, σ is the conductivity of the conductor; d is its thickness; vx and vy are the x- and ycomponents of the conductor velocity, respectively; µr is the relative permeability of the conductor;
µ0 is permeability of vacuum. The quantities ξ and η are the integration variables of the Fourier
transform. The values and depend on the angular frequency of the excitation current, conductor
thickness, velocity, and material properties as well as the coil geometry.
PIERS ONLINE, VOL. 7, NO. 8, 2011
768
2.2. Eddy Current Density
The x- and y-components of ECD in the conductor slab is expressed by the following equations:
µ
¶
∂B1y
1
∂B1z
(7)
J1x =
−
,
µ0 µr
∂y
∂z
µ
¶
1
∂B1x ∂B1z
J1y =
−
.
(8)
µ0 µr
∂z
∂x
2.3. Stream function
We introduce stream function to derive ECDs. The following equation expresses streamline of eddy
current that satisfies Jx = J1x (x, y, z) and Jy = J1y (x, y, z) in the x-y plane is
Re (Jy )
dy
=
.
dx
Re (Jx )
(9)
Re (Jy ) dx − Re (Jx ) dy = 0,
(10)
Therefore,
where, Re(. . .) returns the real part of the arguments and gives the instantaneous value of the eddy
current density. The stream function U (x, y) in the x-y plane is given by
µZ
¶
U (x, y) = Re
Jy dx = k = constant,
(11)
or
µZ
U (x, y) = Re
¶
−Jx dy
= k = constant.
(12)
2.4. Stream Function Including Time t
Since the eddy current in conductor slab is a function of time t, the stream function U (x, y, z, t) is
given by
¶
· µZ
¶
µZ
¶
¸
µZ
√
√ jωt
= 2 Re
Jy dx cos ωt − Im
Jy dx sin ωt = k (13)
U (x, y, z, t) = Re
Jy dx 2e
where, Im(. . .) returns the imaginary part of its arguments and gives the instantaneous value of
the eddy current density. The stream function on the z 0 -plane at time t0 is given by
¡
¢
U x, y, z 0 , t0 = k = constant.
(14)
From Eq. (14), the constant k is obtained by changing (x, y) and connecting point (x, y) with
the equivalent value k. In this way, ECDs on various z-planes are obtained.
3. RESULTS AND DISCUSSION
From Eq. (13), one obtained ECDs dependent upon the excitation frequency f , conductor velocity
v and conductor thickness d. The ECDs were calculated by adjusting the integration range of
the integral using the Gauss-Legendre integration method, and the contour plots were obtained by
using MATLAB. In this research, we have calculated the ECDs in metallic aluminum slabs. The
specifications of the conductor slab are given in Table 1, and the dimensions of rectangular coil
are shown in Table 2. The contour plots shown hereafter are the ECDs at z = 0 and t = 0. For
Table 1: Specifications for conductor slab.
Table 2: Specifications for coils.
Aluminum
d = 10 mm
σ = 3 × 107 S/m
µ = µ0 = 4π × 10−7 H/m
Rectangular coil
a = 25 mm
b = 25 mm
z0 = 55 mm
x0 = 0 mm
PIERS ONLINE, VOL. 7, NO. 8, 2011
769
Table 3: Comparison between our technique and FEM.
Proposed method
exact solution
within two seconds
Computational accuracy
Computational time
0.04
-4
20
10
20
-20
15
20
6
y [m]
-2
4
6
-15
-20
-15
0
5
-6
-6
0
-20
15
-4
0
2
0
-5
-10
4
-4
15
4
-15
0.02
0.02
-0.02
-0.02
4
-4
-0.04
5
0
2
0
x [m]
0.05
0
-1
-0.06
-0.1
0.1
-15
-10
-5
10
-0.05
-2
-0.06
-0.1
15
-4
4
-0.04
0
y [m]
0
10
0.04
-1
0.06
10
0.06
FEM
dependent on mesh size
10 times longer
10
-0.05
0
x [m]
(a)
0.05
0.1
(b)
Figure 2: Effect of excitation frequency on ECD of standstill conductor. (a) ECD at f = 100 Hz and (b)
ECD at f = 1 kHz.
0.06
0.06
-4
4
4
-4
0.04
-4
0.04
6
-2
-4
-6
-8
6
-10
0
-4
6
-8
-6
y [m]
6
y [m]
-4
0
-6
0
2
2
0
-2
6
4
-4
0.02
4
0.02
-0.02
-4
-0.02
6
4
2
4
2
0.05
-0.06
-0.1
0.1
-0.05
-4
0
-6
-2
-0.05
-2
0
-0.06
-0.1
-4
0
4
-0.04
4
-0.04
0
x [m]
0.05
0.1
x [m]
(a)
(b)
Figure 3: Effect of sliding motion of conductor slab on ECD in the slab with a thickness of 10 mm. The
excitation frequency is f = 100 Hz. (a) ECD for vx = 10 m/s and (b) ECD for vx = 20 m/s.
0.06
0.06
6
-6
6
-6
-10
8
-8
10
-60
6
40
-20
-60
y [m]
-6
0.05
0.1
-0.06
-0.1
-6
0
4
2
-6
6
-0.05
-4
40
20
-8
-2
0
y [m]
-8
-10
8
(a)
8
6
0
x [m]
-0.04
0
-4
-20
40
-0.05
0
-0.02
-40
-0.06
-0.1
10
60
-0.02
0
2
4
-40
0
20
60
0
-2
-4
0.02
0.02
-0.04
-8
8
-40
-40
0.04
40
40
0.04
0
x [m]
0.05
0.1
(b)
Figure 4: The ECDs of conductor-slab with thicknesses of (a) d = 1 mm and (b) d = 5 mm. The excitation
frequency is f = 1 kHz.
separate computation of the real and imaginary part of the ECDs, in double precision, the CPU
time for one ECD calculation on a 3.2 GHz Core i5 PC is about 1.2 s. The comparison between our
PIERS ONLINE, VOL. 7, NO. 8, 2011
770
technique and the FEM are shown in Table 3. The accurate solution of ECDs can also be obtained
using the FEM. However, it requires a long computational time than our technique.
First, the velocity of conductor slab was set to be vx = vy = 0. Figs. 2(a) and (b) show the
ECDs at f = 1 kHz and 10 kHz, respectively. The direction of eddy currents on either side of the
x = 0 line is opposite each other, which is concordant with our common sense. Upon increasing
frequency f , the magnitude of eddy current increases, and the ECD becomes more concentrated
around the portions where two intense eddy currents arise. Next, we investigated the effect of the
sliding motion of conductor slab on ECDs. Figs. 3(a) and (b) show the ECDs at vx = 10 and
20 m/s, respectively, and vy = 0. The excitation frequency was chosen to be f = 100 Hz. The
contour plots of Fig. 3 indicate that the sliding motion causes asymmetric ECDs. Two types of
eddy currents, one is shown by counter-clockwise vectors and the other by clockwise vectors in
Fig. 3, simultaneously shift towards the −x direction. The ECD shift becomes larger, as the slab
velocity increases. The effect of excitation frequency on the ECD shift was also examined. It
is found that the sliding motion of conductor slab gives rise to no significant change in ECD at
f = 1 kHz, when vx is less than 20 m/s. Figs. 4(a) and (b) show the ECDs of conductor slabs with
thicknesses of d = 1 mm and d = 5 mm. The ECDs are calculated at vx = vy = 0 and at f = 1 kHz.
The magnitude of the surface eddy current of thin conductor is larger than that of thicker slab.
The eddy current is strongly influenced when the conductor slab thickness is smaller than the flux
penetration depth. Since the penetration depth is about 2.68 mm in aluminum at f = 1 kHz, high
eddy current for the aluminum slab with d = 1 mm reflects the penetration effect.
4. CONCLUSION
In this paper, a theoretical solution was derived that exactly displays the eddy current distribution
occurring when a rectangular coil is arranged perpendicular to the conductor slab, which was
until now unsolved. The flow of the eddy current, which arises in the conductor as a function of
excitation frequency, thickness of the conductor slab, and speed of the slab, has been clarified by
this theoretical solution. This analysis method enables the immediate determination of the eddy
current irrespective of whether the conductor slab is moving. Moreover, it is considered useful not
only in sophisticated analyses of eddy currents but also in other types of eddy current analyses.
ACKNOWLEDGMENT
The authors gratefully acknowledge Mr. K. Nawata for his cooperation.
REFERENCES
1. Cacciola, M., S. Calcagno, G. Megali, D. Pellicano, M. Versaci, and F. C. Morabito, “Eddy
current modeling in composite materials,” PIERS Online, Vol. 5, No. 6, 591–595, 2009.
2. Tsuboi, H., N. Seshima, I. Sebestyn, J. Pv, S. Gyimthy, and A. Gasparics, “Transient eddy
current analysis of pulsed eddy current testing by finite element method,” IEEE Trans. on
Magnetics, Vol. 40, No. 2, 1330–1333, 2004.
3. Preis, K., O. Biro, and I. Ticar, “FEM analysis of eddy current losses in nonlinear laminated
iron cores,” IEEE Trans. on Magnetics, Vol. 41, No. 5, 1412–1415, 2005.
4. Theodoulidis, T. P. and E. E. Kriezis, “Impedance evaluation of rectangular coils for eddy
current testing of planar media,” NDT and E International, Vol. 35, 407–414, 2002.
5. Yin, W. and A. Peyton, “Sensitivity formulation including velocity effects for electromagnetic
induction systems,” IEEE Trans. on Magnetics, Vol. 46, No. 5, 1172–1176, 2010.
6. He, Y., F. Luo, and M. Pan, “Defect characterisation based on pulsed eddy current imaging
technique,” Sensors and Actuators, Vol. A, No. 164, 1–7, 2010.
7. Tanaka, A., N. Takehira, and K. Toda, “Analysis of a rectangular coil facing moving sheet
conductor,” IEEJ. Trans. A, Vol. 101, No. 8, 405–412, 1981.
8. Itaya, T., K. Ishida, A. Tanaka, and N. Takehira, “Analysis of a fork-shaped rectangular coil
facing moving sheet conductors,” IET Science, Measurement and Technology, Vol. 3, No. 5,
279–285, 2009.
9. Ishida, K., T. Itaya, A. Tanaka, and N. Takehira, “Magnetic field analysis of an arbitrary
shaped coil using shape functions,” IEEE Trans. on Magnetics, Vol. 45, No. 1, 104–112, 2009.
10. Ishida, K., T. Itaya, A. Tanaka, N. Takehira, and T. Miki, “Arbitrary-shaped single-layer coil
self-inductance using shape functions,” IET Science, Measurement and Technology, Vol. 45,
No. 1, 21–27, 2010.