CH2. The Principles of Eddy Current Testing 2.1. Introduction The eddy current testing method is a form of electromagnetic nondestructive testing based on the principles of electromagnetic induction. The physicals of eddy current testing are shown in figure 2.1. The Maxwell- Ampere law states that an alternating current sets up a time varying magnetic field. When a primary coil excited with an alternating current is placed in close proximity to a conducting surface, the alternating primary magnetic fields from the probe interact with the law. This induced voltage produces eddy currents in the material witch, by Lenz's Law (楞次定律), set up secondary magnetic fields that oppose the ones producing them. The eddy currents, thus, are analogous to a mutually coupled secondary circuit in witch the current flows in a circular direction opposite that of the current in the coil. This mutual inductance causes a change in the impedance of the coil because the load of the secondary circuit, now the eddy currents in the specimen, are referred, or "reflected", back to the primary coil. For instance, the resistive component is a measure of the energy losses within the material since the eddy currents represent a resistive load. Hence, as a coil is brought close to a non-ferromagnetic conductive material (非鐵磁性材料) its resistance, the real component of its impedance, increases. 1
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CH2. The Principles of Eddy Current Testing 2.1. Introduction
The eddy current testing method is a form of electromagnetic nondestructive testing based on the principles of electromagnetic induction. The physicals of eddy current testing are shown in figure 2.1. The Maxwell-Ampere law states that an alternating current sets up a time varying magnetic field. When a primary coil excited with an alternating current is placed in close proximity to a conducting surface, the alternating primary magnetic fields from the probe interact with the law. This induced voltage produces eddy currents in the material witch, by Lenz's Law (楞次定律), set up secondary magnetic fields that oppose the ones producing them. The eddy currents, thus, are analogous to a mutually coupled secondary circuit in witch the current flows in a circular direction opposite that of the current in the coil. This mutual inductance causes a change in the impedance of the coil because the load of the secondary circuit, now the eddy currents in the specimen, are referred, or "reflected", back to the primary coil. For instance, the resistive component is a measure of the energy losses within the material since the eddy currents represent a resistive load. Hence, as a coil is brought close to a non-ferromagnetic conductive material (非鐵磁性材料) its resistance, the real component of its impedance, increases.
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Fig. 2.1. Principles of eddy current testing.
Impedance also consists of a reactive imaginary component, called the
inductive reactance. The work required to establish a magnetic flux in a coil, on the energy stored in the magnetic field of a coil, is directly to its inductance. Thus, when the opposing secondary field from the induced eddy currents decrease the magnetic flux linkages, as shown in figure 2.1, the coil's stored energy decreased which in turn reduces its inductance and 2
inductive reactance. However, when a coil with a certain magnetic field strength, H, is placed over a ferromagnetic material(鐵磁性材料), the lower reluctance in the material allows the flux lines to increase in density within the coil's windings. This, therefore, increases the coil's inductance.
Any material property or geometry variances that affect the eddy current's distribution and the related magnetic fields will always result in a change in the coil's impedance. These changes are monitored by either measuring them directly with an impedance analyzer or, more commonly, by observing the trajectories traced on the scope of a commercial eddy current instrument. (一般以阻抗分析儀或 EddyScope觀察紀錄) Usually, the signals are measured as departures from a reference condition, such as that for the coil placed on a standard specimen (若無理論值可驗證,則需有標準樣本作為比對,一般工場採用此法,可以想見需有多少樣本!). Signals therefore result from changes in one or more of the specimen's properties such as conductivity, permeability, thickness, surface roughness, temperature and, most notably, the presence of defects. A defect, for example, interrupts the eddy currents' path, especially if it is a crack perpendicular to the flow of the eddy currents, changing the secondary impedance, and hence the impedance of the coil. Test setup factors such as liftoff, frequency, and electronic noise also affect the signal. Under most operating conditions, the tests can 3
differentiate between various factors and material properties if they each produce a different signal response. For example, commercial eddy scopes will display relatively distinctive trajectories for different types of deviations. This allows the instrument to be setup in a way that separates an unwanted signal, such as that caused from liftoff, from the signal of interest, such as a flaw. Recorded signals can also be processed later by removing the noisy responses known to be caused by certain factors.
Commercial eddy current instruments are almost exclusively used in a
relative nature where reference standards are used to set the normal operating point. A material similar to the standard is tested by observing the signals that deviate from the reference condition; these indicate flaws or other property changes. A good understanding of these signals and how an instrument detects and displays them is necessary to develop a simple calibration method that would make quantitative impedance measurements possible. Consequently, commercial instruments would be able to perform quantitative NDE (QNDE), such as is now performed with an impedance analyzer in conjunction with an applicable eddy current theory.
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2.2. Theory of Eddy Current Phenomenon
Electromagnetic phenomenon for time varying conditions are governed by Maxwell's equations as follows:
tBE
∂∂
−=×∇ (2.1)
∂
tDJH
∂+=×∇ (2.2)
0=⋅∇ B (2.3)
ρ=⋅∇ D (2.4)
The constitutive relations for an isotropic, linear and homogeneous
medium are given as HB µ= (2.5) ED ε= (2.6) EJ σ= (2.7)
where
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µ = the magnetic permeability in henrys per meter(H/m) ε = the electric permittivity in farads per meter (F/m), and σ = the electric conductivity in seimens per meter (S/m).
Since the divergence of B is zero from equation (2.3), B can be expressed as the curl of another vector, called the magnetic vector potential (MVP) A, given by
.AB ×∇= (2.8) Combining this with equation (2.1) gives
tAE
∂∂
×−∇=×∇ (2.9)
or
0=
∂∂
+×∇tAE (2.10)
Since an irrotational field can be replaced with the gradient of a scalar
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potential function, the solution to equation (2.10) is
VtAE ×−∇=
∂∂
+ (2.11)
where V is the scalar electric potential.
Now, substituting equation (2.5) into equation (2.2) yields
∂∂
+=×∇tDJB µ (2.12)
The displacement current tD ∂∂ is negligible when compared to the current density, J, for the relatively low operating frequencies in eddy current testing (below 10 MHz). Therefore, combining equations (2.7), (2.8) and (2.12) we have
JA µ=×∇×∇ (2.13)
Eµσ= (2.14) 7
Replacing E in equation (2.14) with the relation of (2.11) gives
∂∂
−∇−=×∇×∇tAVA µσ (2.15)
Recognizing the vector identity
( ) ( ) AAA 2∇−⋅∇∇=×∇×∇ (2.16) and choosing the Coulomb gauge , equation (2.15) becomes the following partial differential equation (p.d.e.) for any varying current
0=⋅∇ A
tAJA s ∂
∂+−=∇ µ2 µσ (2.17)
where VJ s ∇−= σ is the source current density in Amperes per square meter, such as an infinite sheet of current over a conductive half-space. Assuming the fields are time harmonic and vary sinusoidally in steady state, sinusoidal eddy currents will exist within the conductor and A can be expressed as
tjeAA ω= m (2.18)
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where ω is the angular frequency. Differentiating with respect to time, we get
AjtA
ω=∂∂ (2.19)
Thus, substituting equation (2.19) into (2.17) results in
sJAjA µωµσ −=∇2 (2.20) where the first term is a function of the density of the induced eddy currents, Je , given by
AjJe ωσ= (2.21) The ability to obtain analytic solutions for the magnetic vector potential from equation (2.20) is usually limited to simple geometries such as a symmetrical coil over an infinite half space medium. This equation also governs the skin effect that will be discussed in the next section.
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2.3. Skin Effect
Equation (2.17) is a diffusion equation that describes the propagation of EM waves in a conductor. Assuming an exciting current sheet is flowing in the x direction over a conducting half-space, the magnetic field intensity, Hy, in the conductor as a function of depth from the surface, z, is found to be
−
−
=
2/12/1
220
ωµσωµσ jzz
y eeHH (2.22) Hence, EM fields rapidly decay exponentially with depth. This effect also applies to eddy currents, where the induced current density, Je, is given by
δδ //0
jzze eeJJ −−= (2.23)
where
µσπδ
f1
= (2.24)
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is skin depth or standard depth of penetration. This is the depth at witch the magnitude of the induced eddy current density decreases by a factor of 1/e from its value at the surface, J0. This current distribution is illustrated in figure 2.2. The eddy currents are confined to a thin layer of the conductor's surface adjacent to the coil. And relationship (2.23) indicates that lower frequencies are required to reach greater depths and detect or measure deeper subsurface flaws.
Fig. 2.2. The skin effect.
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2.4 Forward Solution for the Impedance of a Coil Over a Single-
Layered Half Space The calculation of the impedance of a right-cylindrical air-core coil placed
above a 2-layer flat plate of metals is a canonical (標準的) problem in
quantitative eddy-current inspection. Dodd and Deeds proposed a widely-
used solution for a coil above a two-conductor nonmagnetic metals. Each
layer is characterized by a uniform . We review this paper in the following. σ
III
VI
I
IIz=0
z=-c
z=l
µ 0σ 1
µ0σ2
Coil
Figure 2.3. Coil above a two-conductor plane.
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The coil above a two-conductor plane is shown in Fig. 2.3. We have divided
the problem into four regions. The differential equation in air (regions I and II)
is
0122
2
2
2=−++
rA
zA
rA
rrA
∂
∂∂∂
∂
∂. (圓柱座標) (2.1)
(Note: 由於對稱,所以與角度θ無關;空氣之σ=0,所以無 ) Aj iωµσ
The differential equation in a conductor (regions III and IV) is
012
2
22
2=−+−+ Aj
zA
rA
rA
rrA
iωµσ∂
∂∂∂
∂
∂ (2.2)
Setting A(r,z)=R(r)Z(z) (分離變數法)and dividing by R(r)Z(z) gives
0111122
2
2
2=−−++ ij
rzZ
ZrR
rRrR
Rωµσ
∂
∂∂∂
∂
∂ (2.3)
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We write for the z dependence
ijconstz
ZZ
ωµσα∂
∂+== 2
2
21 , (2.4)
or
ii jzjz BeAeZ ωµσαωµσα −+ +=22 + . (2.5)
We define
ii jωµσαα +≡ 2 . (2.6)
Equation (2.3) then becomes
0111 222
2=+−+
∂∂
∂
∂
rrR
rRrR
Rα (2.7)
The is a first-order Bessel equation and has the solutions
( ) ( rDYrCJR αα 11 += ) (2.8)
Note: Bessel functions arise in solving differential equations for systems with
cylindrical symmetry.
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Note:
Plot BesselJ 1, x , x, 0, 50 Plot BesselY 1, x , x, 0, 50 @ @ D 8 < @ @ D 8 <D
10 20 30 40 50
-0.2
0.2
0.4
0.6
10 20 30 40 50
-0.6
-0.4
-0.2
0.2
0.4
Combining the solutions we have
[ ] ( ) ( )[ ]rDYrCJBeAeRZA zzii αααα
11 +⋅+== − (2.9)
We now need to determine the constants A, B, C, D. They are functions of
the separation “constant” and are usually different for each value of . α α The
complete solution would be a sum of all the individual solutions, if α is a
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discrete variable, but, since α is a continuous variable, the complete solutions
is an integral over the entire range of α . Thus, the general solution is