ELECTRIC POTENTIAL IN EDDY CURRENT TESTING
H. Hoshikawa, K. Koyama, and M. Maeda
Nihon University, Izumicho Narashino Chiba 275-8575, Japan
ABSTRACT. This paper indicates that eddy current is induced not only by induction electric field but
also by electric potential gradient. The potential gradient plays a major role in inducing eddy current
around discontinuities in the test material. The potential is generated by conversion of induction electric
field where the eddy current is smaller than that induced by induction electric field.
INTRODUCTION
It is well known in eddy current testing that the alternating current in the exciting coil
induces eddy current in the test material by electromagnetic induction. However, little has been
studied about why the eddy current is diverted away from discontinuities. The authors have
studied how eddy current is induced by electromagnetic induction and clarified that electric
potential gradient plays a major role in inducing eddy current especially around discontinuities.
When the exciting coil and test material are arranged axi-symmetrically in
electromagnetic induction, no electric potential is generated in the material and the eddy current
is induced only by induction electric field. As a result, attention has been concentrated to the
eddy current induced by induction electric field and the role of electric potential has hardly ever
been taken into consideration in eddy current testing while a lot of reports have been published
on the analysis of eddy current testing [1-2].
When the exciting coil and test material are not arranged axi-symmetrically, some of the
induction electric field by the exciting current is converted to electric potential gradient. The
gradient causes eddy current to flow in different directions from the induction field in the area
around a discontinuity. It is the gradient that diverts eddy current away around discontinuities [3].
Thus electric potential plays an important role in inducing eddy current in the test material.
EDDY CURRENT IN ELECTROMAGNETIC INDUCTION
Electric current density JQ in an exciting coil of volume V generates magnetic vector
potential A
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/$20.00
375
(1)
where r is distance and ju magnetic permeability. A is a vector parallel to J0 near the
exciting coil. Magnetic flux density B is correlated to A.
B = VxA.
(2)
Variation of magnetic flux density induces electric field E .
Combining Equations (2) and (3) leads to Equation (4).
VxE = Vx(-dA/dt).
(4)
Equation (5) is derived from Equation (4).
where (/) is scalar potential. Eddy current J is induced by E in conducting material.
j = a. E = a • (-dA/dt - V0)
( a: conductivity)
(6)
Equation (6) indicates that eddy current is induced by electric field which comprises of
induction electric field (-dA/dt)
and potential gradient V0. In the meantime, eddy current
has to satisfy the continuity law.
V•/ =0
(7)
Induction field induces the eddy current parallel to the exciting coil while potential gradient
induces the eddy current in any direction. Thus it is the potential gradient that induces eddy
current at places around discontinuities and far from a tangential exciting coil.
Based on Ohm's law, eddy current causes electric potential drop.
V0
('.' p = I/a: resistivity)
(8)
In other words, eddy current is induced so that potential drop by eddy current is equal to the
difference between induction field and potential gradient. If the test coil and test material are
arranged axi-symmetrically, eddy current is induced only by the induction electric field and
there exists no electric potential. If the test material includes discontinuities and the system is not
axi-symmetric, the eddy current becomes smaller and potential gradient is generated as the
remainder of subtraction of potential drop from induction electric field.
V<f> = -dA/dt-p-J
(9)
Equation (9) indicates that the conversion of induction field to potential gradient occurs where
the eddy current is smaller than that induced by induction field.
In electrostatic field, electric potential is defined as the integral of electric field with respect
to an arbitrary path. On the other hand, electric potential in AC electromagnetic field cannot be
376
defined as the same way because AC electric field comprises induction field and potential
gradient as given by Equation (5) and is not conservative. Thus the authors have defined AC
potential (/) as the integral of the potential gradient given by Equation (9) with respect to an
arbitrary path in the material.
p-J}-M
(10)
Because of the current continuity law, eddy current has to circulate continuously in the
material even against the induction electric field. It is potential gradient that causes eddy current
to circulate against the induction field or in directions of which the induction field has no
component. Equation (10) indicates that the remainder of subtraction of potential drop from
induction field is converted to electric potential, the gradient of which causes eddy current to
circulate at places far from tangential exciting coils and around discontinuities in order to satisfy
the current continuity law.
ELECTROMAGNETIC INDUCTION IN TRANSFORMERS
The electromagnetic induction in transformers can substantiate the concept that induction
electric field is converted to electric potential. Electric current in the primary winding generates
induction field in the secondary winding. When the secondary winding is open, the winding
carries no current and generates a high voltage that is measurable by electric instruments. This
fact means that there has to be no electric field inside the secondary winding because electric
field causes current to flow by Equation (6). Thus the induction field in Equation (5) has to be
cancelled out with the potential gradient that is the conversion by Equation (9) of the induction
field inside the secondary winding.
When secondary winding is connected to closed circuits, secondary current circulates
through the circuits. It is obviously the potential gradient by the secondary voltage and not the
induction field by the primary current that causes the current to circulate because the potential
gradient and not the induction field can generate current at arbitrary positions along the circuits.
When the secondary winding is short-circuited, the secondary voltage becomes zero and a large
current flows in the secondary winding. In this case, no potential gradient exists inside the
secondary winding because the induction field is cancelled out with the potential drop by the
large current flowing through the secondary winding as given by Equation (9).
Thus the phenomena inside transformers validate the concept that induction electric field is
converted to potential gradient at places in the secondary winding where current is smaller than
that induced by the induction field.
RESULTS BY FINITE ELEMENT ANALYSIS
Finite element analysis was conducted to clarify the relation between eddy current and
electric potential in the conducting material. Figure 1 shows the sizes of a tangential exciting coil
and a conducting plate used for the analysis. A tangential coil was chosen because it should
clarify electromagnetic induction phenomena where the eddy current just under the tangential
coil circulates in the same direction as the coil winding, while circulating in the opposite
direction at places far from the coil in order to satisfy the current continuity law. The analytical
results were derived under the following condition. The conductivity of the conducting material
is 1.3xl07 S/m and the frequency is 9 kHz.
377
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conducting plate
FIGURE 1. Tangential coil and conducting plate used for the analysis.
©0 • electric potential
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FIGURE 2. Eddy current and electric potential induced at the surface of conducting plate by a tangential coil.
Figure 2 shows the eddy current and the electric potential induced at the surface of the
conducting material. Figure 2(a) indicates that the eddy current circulates along under the exiting
coil winding and in the opposite direction at places far from the coil. Figure 2(b) indicates that
electric potential is developed along the coil and generates potential gradient anywhere around
the coil. It can be assumed that the dominant induction field over the potential gradient induces
eddy current just below the tangential coil and the dominant potential gradient over the induction
field induces the backward eddy current at places far from the tangential coil.
Figure 3 shows the eddy current, the eddy current component by induction field, and the
eddy current component by potential gradient at the surface of the conducting plate. Figure 3(b)
indicates that the induction field induces eddy current component only upward everywhere in
the figure, while Figure 3(c) indicates the potential gradient induces eddy current component
downward. Thus the eddy current circulates continuously as shown in Figure 3(a).
Figure 4 shows the eddy current circulating along a cross section inside the conducting
plate parallel to the tangential coil. The figures have the coordinate enlarged along the coil.
Figure 4 (b) indicates that the eddy current component induced by induction field circulates from
left to right in the figure and decays with the depth from the conducting material surface. Figure
4 (c) indicates that the eddy current component by the potential gradient circulates from right to
left and stays almost constant with the depth. As a result, the eddy current circulates from left to
right decaying very rapidly with the depth.
378
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FIGURE 3. Eddy current distributions at the surface of conducting plate by a tangential coil.
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FIGURE 4. Enlarged eddy current distribution at a cross section of a conducting plate along the tangential coil
Figure 5 shows the eddy current and electric potential at the surface of a conducting plate
with a slit flaw. Figure 5 (a) indicates that the flaw diverts and decreases the eddy current toward
it. As a result, plus and minus electric potentials are developed at both sides of the flaw as shown
in Figure 5(b). Thus the resulting potential gradient forces the eddy current to circulate along and
under the flaw in order to comply with the continuity law of current.
379
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FIGURE 5. Eddy current and electric potential distributions at the surface of a conducting plate with a slit flaw
frequency :9kHz
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FIGURE 6. Enlarged surface eddy distributions around a slit flaw
Figure 6 shows the enlarged eddy current distributions around a slit flaw. Figure 6(b)
shows the eddy current component induced by induction field. The component seems to be
constant toward the flaw. Figure 6(c) shows the eddy current component induced by potential
gradient. It is obvious from the figures that the component develops so as to cancel out the eddy
current component perpendicular to the flaw as shown in Figure 6(a)
Figure 7 shows the eddy current circulating along a cross section inside the conducting
plate parallel to the tangential coil. The figures have the coordinate enlarged along the coil.
Figure 7 (b) indicates that the eddy current component induced by induction field circulates from
left to right in the figure and decays with the depth from the conducting material surface. Figure
7 (c) indicates that the eddy current component by the potential gradient circulates from right to
left near the surface and from left to right under the flaw. As a result,
380
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FIGURE 7. Enlarged eddy current at a cross section along the tangential coil in a plate with a slit flaw.
/x
10
[mm]
(a) eddy current
(b) electric potential
FIGURE 8. Eddy current and electric potential distributions at the surface of conducting plate by a circular
pancake coil
little eddy current circulates close to the flaw and large eddy current circulates under the flaw as
shown in Figure 7 (a).
Figure 8 shows the eddy current and electric potential distributions at the surface of the
conductor by a circular pancake coil of 9 mm outer diameter. Since the coil and the plate are
arranged axi-symmetrically, the eddy current is induced axi-symmetrically and no electric
potential is developed in the plate as shown in the figures.
381
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FIGURE 9. Eddy current and electric potential distributions at the surface of a conducting plate with a slit flaw
by a circular pancake coil
Figure 9 shows the eddy current and electric potential distributions at the surface of the
conductor by the a circular pancake coil when the conductor has a slit flaw at its center. The
figure shows that eddy current is diverted from the flaw and two peaks of plus and minus
potential are developed at both sides of the flaw. Thus the gradient of electric potential around
the flaw diverts the eddy current from the flaw. The difference of the electric potential along
flaws can provide a method to detect the depth of the flaws [4-5].
CONCLUSION
The authors have tried to explain qualitatively the phenomena of electromagnetic induction
in eddy current testing. It has been clarified that some of the induction electric field is converted
to electric potential where the eddy current is smaller than that induced by induction field. The
gradient of electric potential plays a major role in inducing eddy current at places far from a
tangential coil and around discontinuities.
REFERENCES
1.
2.
3.
4.
5.
Dodd, C. V. and Deed, W. E., "Analytical Solution to Eddy Current Probe-Coil
Problems," Journal of Applied Physics, 39, 6, 2829-2838 (1968).
Onoe, M., "An Analysis of a Finite Solenoid Coil Near a Conductor," (in Japanese)
Journal of BEE of Japan, 88-10, 961,1894-1902 (1968).
Hoshikawa, H., Koyama, K., Maeda, M. "Role of electric Potential in Eddy Current
Testing Phenomena," Review of Progress in QNDE, 20A, 380-387 (2002).
Koido, J., Hamanaka, N., Hoshikawa, H., "Fundamental Studies on ACFPD Using
Electromagnetic Induction Exciting Method", Proc. Second Japan-US Symposium on
Advances in NDT, 266-270 (1999).
Hoshikawa, H., Koyama, K., Koido. J., Karasawa, H., "A New Method to Detect Surface
Crack Depth in Metal Using Electromagnetic Induction and Electric Potential," Review of
Progress in QNDE, 19B, 2003-2010 (2001).
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