Project supported by the National Natural Science Foundation of China (Grant No. 51577004).
Abstract
This paper presents a closed expression of the layered-plate factor used to calculate the coil eddy-current impedance over the multi-layer plate conductor. By using this expression, the general series of eddy-current impedance can be written directly without solving the undetermined constant equations. The series expression is easy to use for theoretical analysis and programming. Experimental results show that calculated values and measured values are in agreement. As an application, when the bottom layer of the layered plate is a non-ferromagnetic thin layer conductor and the product of the thickness and conductivity of the layer remains unchanged, using the layered-plate factor expression proposed in this paper, it can be theoretically predicted that the eddy-current impedance curves corresponding to different thin layer thickness values will coincide.
In industrial eddy-current non-destructive testing, the parameters and surface status of the layered conductor material can be determined by measuring the eddy-current impedance of the coil. The eddy-current analytical methods for an air-core current-carrying cylindrical coil vertically placed above layered conductor have been investigated earlier. In 1968, Dodd and Deeds[1] derived the integral-type analytical solution of magnetic vector potential and the eddy-current impedance of an air-core cylindrical coil carrying a sinusoidal current placed above a two-layer conductor. This paper has been widely cited since its publication. Based on the Dodd–Deeds model, Luquire et al.[2] obtained the integral expression of the impedance for the coil above the multi-layer conductive material by using a mathematical induction method. Cheng et al.[3] used the 2 × 2 matrix recursion method to calculate the eddy-current impedance of the coil above layered conductors, furthermore, this method also could be used to calculate the eddy-current field of arbitrary multi-layer conductors. Theodoulidis et al.[4] replaced the infinite region of Dodd–Deeds model by a finite-radius coaxial cylindrical region and took the magnetic vector potential to be zero on the cylindrical surface, and therefore, the series-form analytic solution of the sinusoidal eddy-current field has been obtained. The method in Ref. [4] not only greatly improved the efficiency in calculating eddy-current impedance, but also could be used to obtain approximate analytical solutions of some eddy-current problems.[5–7] Furthermore, Fan et al.[8] obtained the generalized reflection coefficients of the multi-layer plate conductor, and then obtained the series expression of coil eddy-current impedance by using the recursive method.
The derivation method of Lipskaya–Vanyan formula[9,10] is used in this paper. The method is that the solution of the second order homogeneous ordinary differential equation is divided by its derivative. Starting from the undetermined constant equations, using the natural logarithm identity and the definition of hyperbolic tangent function repeatedly, then a general series expression for eddy-current impedance for air-core cylindrical coil placed above an arbitrary multi-layer conductor is obtained for the first time. Compared with previous results, this series expression needs no iteration or solving linear equations, which is easy to use for theoretical analysis and programming.
2. Eddy-current model for the air-core cylindrical coil and layered plate conductor
As shown in Fig. 1, the central axis of the air-core cylindrical coil is perpendicular to the conductor plate and the coil is uniformly tightly wound. r1, r2, h, W, and l are the coil inner radius, outer radius, height, turns, and lift-off (the distance between the coil bottom plane and the multi-layer plate conductor upper plane) respectively. A cylindrical coordinate system Oρϕz is established with three coordinate unit vectors, i.e., eρ, eϕ, and ez. The origin point O is located at the interface between the multi-layer plate conductor upper plane and the vacuum region; and the z axis coincides with the central axis of the coil. The phasor current in the coil is I, with frequency f and angular frequency ω = 2πf, and the reference direction of I has a right-hand screw relationship with ez. N is the number of layers of multi-layer plate conductor. The N layers are numbered along the −ez direction respectively as: Layer 1 (z1 < z < 0), layer 2 (z2 < z < z1), ⋯, layer N (zN < z < zN − 1), where z1 = −h1, z2 = −h1 − h2, ⋯, zN = −h1 − h2 − ⋯ − hN. The conductivity, relative permeability, and thickness of the n-th layer are σn, μrn, and hn, n = 1,2, …, N.
Fig. 1. Model of sinusoidal eddy-current problem composed of N-layer plate conductors and a current-carrying coil.
3. Magnetic vector potential in upper half space generated by the single-turn circular phasor current
3.1. General solutions and undetermined constant equations
In the cross-section of the air-core cylindrical coil shown in Fig. 1, a single-turn circular current I is taken arbitrarily on the plane S′ : z = z′. Magnetic vector potential A is suitable for solving the axisymmetric electromagnetic field problem, which can be written as A(ρ, ϕ, z) = Aϕ(ρ, z)eϕ.[11] In order to facilitate the calculation, the whole field region is truncated by a long straight coaxial cylindrical surface whose radius ρ = b (b ≫ r2, b ≫ |zN|), and let Aϕ(b, z) = 0 on the cylindrical surface.
Based on the assumptions given above, neglecting the displacement current, and then following the analytical methods in Refs. [1] and [4], the general solutions for circumferential components of the magnetic vector potential in each region can be written as follows:
where , , i , Reαnm > 0, μ0 = 4π × 10−7 H/m; z0 = 0; λm is the m-th positive zero point of J1 (λ) which denotes the Bessel function of the first kind of order one. C01m, C02m, D02m, Cnm, Dnm, and D03m are undetermined complex constants which satisfy the following linear equations:
where
There are 2(N + 2) equations and 2(N + 2) undetermined constants in Eqs. (5)–(12). In theory, whatever the number N of layers is, all undetermined constants can be solved from Eqs. (5)–(12).
3.2. Derivation of the layered-plate factor expression
Let
Solving Eqs. (5) and (6), we obtain
As can be seen, if LNm has been derived, C01m and C02m can be obtained and then C1m and D1m can be solved from Eqs. (7) and (8). All of the undetermined constants will be obtained by the continuous equivalent transformation with a similar process to the above.
Now, by using the derivation method of Lipskaya–Vanyan formula,[9,10]LNm can be derived. The ratio of Eq. (8) to Eq. (7) is
Using the natural logarithm identity x = exp(ln x), equation (17) can be converted into
Further, using the definition of hyperbolic tangent function thx and hyperbolic cotangent function cthx, equation (18) can be written as
Because the inverse function of y = cthx is x = arcthy, equation (19) can be changed into
Using the identity transform
equation (20) can be written as
To solve C1m/D1m, we take the ratio of Eq. (10) to Eq. (9) and obtain
Imitating the derivation process of Eq. (20), equation (22) can be written as
Similarly,
can be obtained successively, and expressed as
Finally, substituting ln(CNm/DNm)/2 into ln(CN − 1,m/DN − 1,m)/2, ln(CN − 1,m/DN − 1,m)/2 into ln(CN − 2,m/DN − 2,m)/2, …, until ln(C1m/D1m)/2 into Eq. (21), we obtain
In this paper, LNm is named layered-plate factor, and it is a complex function determined by thickness, relative permeability, and conductivity of each layer.
3.3. Magnetic vector potential above the multi-layer plate conductor
The region above the multi-layer plate conductor is the measurement region, so the most important step is to solve the magnetic vector potential in this region.
Substituting Eqs. (14)–(16) into Eqs. (1) and (2), the series expression for circumferential components of the magnetic vector potential, under the excitation of single-turn circular sinusoidal current, can be obtained as follows:
Based on Eq. (26), the series expression for circumferential components of the magnetic vector potential in the upper half space, under the excitation of eddy-current in the multi-layer plate conductor, can be written as
4. Series expression of eddy-current impedance for air-core cylindrical coil
The impedance[11] for the air-core cylindrical coil corresponding to eddy-current is
where nden = W/[h(r2 − r1)] is the density of turns of the air-core cylindrical coil, V is the occupied region of the air-core cylindrical coil. ΔZ is called eddy-current impedance for short, which is the difference between Z (impedance for air-core cylindrical coil placed above a multi-layer plate conductor) and Z0 (impedance for the air-core cylindrical coil in the infinite vacuum). Substituting Eq. (27) into Eq. (28), the series expression of eddy-current impedance can be obtained as follows:
where
In this paper, Cm is named coil factor, determined only by the coil parameters W, r1, r2, and h.
5. Experiment verification
In order to verify Eq. (29), a three-layer clinging tightly non-ferromagnetic plate has been made which consists of a copper plate, a paperboard, and an aluminum plate successively from top to bottom. The detailed parameters of the three-layer plate and air-core cylindrical coil are listed in Tables 1 and 2, respectively. Let b = 1 m. We pick 32 frequency points between 180 Hz and 6000 Hz and then calculate the sum of the first 300 terms in Eq. (29) of which the MATLAB calculation program is provided in Appendix A. Calculations show that under the above conditions, the calculation values of the eddy-current impedance have at least 4 significant figures. For comparison, we measure Zmeas (impedance for the coil above the three-layer plate) and Z0meas (impedance for the coil in infinity air) respectively with a WK6500B impedance analyzer, and then obtain the eddy-current impedance ΔZmeas = Zmeas − Z0meas corresponding to the 32 frequency points. Figure 2 shows the calculated and measured values of ΔR = ReΔZ as a function of f, respectively. Figure 3 shows the calculated and measured values of ΔX = ImΔZ as a function of f, respectively. Both figures 2 and 3 show that the measured values are in agreement with the calculated values. Figure 2 shows that there is a large deviation between the measured values of the real part and calculated values at the frequency of several hundred Hz. The reason for this phenomenon is that the real part of the impedance represents eddy-current power loss, which is small at a low frequency, and it is obviously affected by the change of surrounding temperature. Hence, one of the measures to reduce the large deviation is to place both the coil and the multi-layer plate conductor into the temperature controlled oil sump.[12]
6.1. Writing method of layered-plate factor expression
According to Eq. (25), the layered-plate factor LNm is composed of N functions shaped like th(t + arthx) and the innermost part of the layered-plate factor is written as
and writing them successively from inner to outer as follows:
Eventually, we obtain
In this way, the layered-plate factor corresponding to N = 1,2,3,4 can be written respectively as
6.2. Properties of layered plate factor
Even though there are multi-value functions arthx and lnx in LNm, both
and
are single-value functions. Hence, LNm is single-valued.
LNm can also be composed of cth(t + arcthx) because
When the coil is in infinite vacuum, μr1 = μr2 = ⋯ = μrN = 1, σ1 = σ2 = ⋯ = σN = 0, so LNm = th(ln 1/2) = 0.
6.3. Application example of layered-plate factor expression
Let the N-th layer (bottom layer) of the N-layer plate be non-ferromagnetic conductor, and . If the N-th layer is a thin-layer conductor and its thickness hN satisfies
then
where λm ≈ (m + 0.25)π (m ≤ 300 in general). Using the following approximate expressions:[13]
we have
where
Hence
As can be seen, LNm is the function of σNhN. It means that if σNhN remains unchanged, LNm will not change. In other words,
where a > 0.
Taking a three-layer plate for example, the parameters of the first layer and the second layer are the same as those listed in Table 1. The third layer is replaced by the following two kinds of thin-layer plates separately:
h3 = 0.01 mm, σ3 = 35 MS/m, h3σ3 = 350 S;
h3 = 0.1 mm, σ3 = 3.5 MS/m, h3σ3 = 350 S.
Let the coil parameters be the same as those listed in Table 2. The results show that when the calculated values of eddy-current impedance are accurate to the fourth place after the decimal point, both the impedance values of these two thin-layer plates are ΔZ = 0.0473 − i0.3609 Ω at f = 6000 Hz.
The parameters of the multi-layer plate conductor can be inversed with the least square method,[14–17] i.e., choosing a set of frequency values f1, f2, ⋯, fn, and then adjusting parameters to minimize
where ΔZmeas (fi) is the measured value and ΔZcalcu (fi) is the calculated value. Numerical calculation shows that when the least square method is used to inverse the parameters of thin-layer metal plate, only the product of the thin-layer metal plate thickness and conductivity can be obtained, i.e., the thickness cannot be obtained directly.[18–20] This phenomenon could be explained by the research results above.
7. Conclusions
In the paper, the closed-form expression and properties of a layered-plate factor are presented. Based on them, it is easy to write a general series expression of eddy-current impedance of the coil. This series expression consists of three independent factors: layered-plate factor, coil factor, and lift-off. The expression has a simple form and is convenient for theoretical analysis. In addition, the method in this paper can also be used to derive the integral solution of eddy-current impedance and analyze the electromagnetic fields of other layered media.
