Proposal of evaluation method of opposite side depth on steel plate using magnetic field of square wave on direct current bias

Abstract

As for the alternating magnetic flux leakage testing of non-destructive inspection, the alternating magnetic field with high frequency is used. Therefore, this inspection method is not used for inspection of the opposite side defect in the steel plate. In this paper, the inspection method of the opposite side defect in the steel plate is proposed by this inspection method using the magnetizing current of the square wave on direct-current bias. This inspection principle is investigated by 3-D nonlinear FEM analysis taking account of magnetizing properties of minor loop and eddy current in the steel plate.

Keywords

Alternating magnetic flux leakage testing AC square wave on DC bias magnetic field 3-D nonlinear FEM minor loop

1. Introduction

In recent years, the ultrasonography inspection method is used to detect the opposite side defect in the steel wall in the petrochemical plant. However, since this inspection method is the contact inspection method, the long inspection time is needed. On the other hand, the high speed and non-contacting inspection technique using alternating (AC) and static (DC) magnetic field is proposed [1]. However, the inspection equipment of this technique becomes large-sized since two kinds of excitation power supply with a direct current and an alternating current are needed. In this paper, an electromagnetic inspection method using a simple excitation power supply with a rectangular magnetic field on direct-current bias is proposed for detecting the opposite side defect on the steel plate. The impression magnetic field using this square wave is generable with a simple power unit. Moreover, this proposed inspection method is better detection accuracy than the static magnetic flux leakage testing method [2] generally used for the inspection of the opposite side defect. The proposed inspection method is investigated by the flux density using the 3-D nonlinear FEM analysis taking account of magnetizing properties of minor loop and eddy current in the steel plate.

2. Proposed electromagnetic sensor model

2.1. Electromagnetic inspection model

Figure 1 shows the 1/2 domain of the proposed model for inspecting the opposite side defect in the steel plate. This inspection sensor is composed of the exciting coil yoke (lamination of silicon steel plates) for the exciting current of AC square wave on DC bias and a search coil. The AC exciting current of the square wave is 50 Hz and 0.5A_p-p. And the DC bias current is 2.5A as shown in Fig. 2. As for the search coil in this sensor, the x-direction of the flux density (B _x) of the leakage flux on the surface of the steel plate is detected. The inspection steel plate is used as the spheroidal graphite cast iron (FCD 450). The dimension of the steel plate is 150 × 150 × 3 mm. The conductivity of the cast iron is 1.6 × 10⁶ S/m. The width (D_w), the length (D_l), and the depth (D_d) of an opposite side depth are 0.5, 100 and 1 mm, respectively. The distance (lift-off) between the yoke and the surface surface of steel plate is equal to 0.1 mm. In this inspection method, the proposed sensor is moved near the defect in the x-direction by 1 mm steps.

Fig. 1.

Inspection model of the 1/2 domain.

Fig. 2.

Exciting current of AC square wave with DC bias (AC:50 Hz, 0.5A_p-p, DC bias: 2.5A).

2.2. Method of analysis

In this proposed inspection method, since the magnetic field using AC square wave on DC bias is impressed to the steel plate, the flux density and eddy current inside the steel plate is calculated using the 3-D nonlinear FEM taking account of minor loops. The magnetic field is analyzed using 3-D edge-based hexahedral nonlinear FEM and the step-by-step method taking account of hysteresis (minor loop) and eddy current in the steel plate [6,7]. To obtain the steady state result, the calculation is carried out during three period (=40 steps). The time interval of the step-by-step method is chosen as 1.25 × 10⁻³ s, when the exciting frequency is equal to 50 Hz.

The basic equation of eddy current analysis using the A − 𝜙 method is given by: $\begin{eqnarray}\displaystyle \text{} & \displaystyle \text{rot}({\nu}∼\text{rot}∼\boldsymbol{A})=\boldsymbol{J}_{0}-{\sigma}\left(\frac{\partial \boldsymbol{A}}{\partial t}+\text{grad}∼{\phi}\right) & \displaystyle\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \displaystyle \text{div}\left\{-{\sigma}\left(\frac{\partial \boldsymbol{A}}{\partial t}+\text{grad}∼{\phi}\right)\right\}=0 & \displaystyle\end{eqnarray}$ (2) where A is the magnetic vector potential, 𝜙 is the scalar potential, v is the reluctivity, J _o is the current density and σ is the conductivity. The Newton-Raphson (N-R) method is used for the nonlinear iteration. The changes $\partial\boldsymbol{A}$ and $\partial\phi$ of A and 𝜙 are obtained by solving the following equation: $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}cc@{}}\displaystyle \frac{\partial \boldsymbol{G}_{\boldsymbol{i}}}{\partial \boldsymbol{A}_{\boldsymbol{k}}} & \displaystyle \frac{\partial \boldsymbol{G}_{\boldsymbol{i}}}{\partial {\phi}_{\boldsymbol{j}}}\\ \displaystyle \frac{\partial {\eta}_{\boldsymbol{m}}}{\partial \boldsymbol{A}_{\boldsymbol{k}}} & \displaystyle \frac{\partial {\eta}_{\boldsymbol{m}}}{\partial {\phi}_{\boldsymbol{j}}}\end{array}\right]\left\{\begin{array}{@{}c@{}}{\delta}\boldsymbol{A}_{\boldsymbol{k}}\\ {\delta}{\phi}_{\boldsymbol{j}}\end{array}\right\}=\left\{\begin{array}{@{}c@{}}-\boldsymbol{G}_{\boldsymbol{i}}\\ -{\eta}_{\boldsymbol{m}}\end{array}\right\} & & \displaystyle\end{eqnarray}$ (3) where G _i and 𝜂_m are the residues of (2) and (3), respectively. $\partial\boldsymbol{G}_{\boldsymbol{i}}{∕}\partial\boldsymbol{A}_{\boldsymbol{k}}$ includes the derivative term of reluctivity $\partial\nu/\partial B^{2}$ . The minor loop is modeled using these upper hysteresis curves of the steel plate (FCD450) as shown in Fig. 3. It is assumed that the obtained B and H are at the point b (H _min, B _min) on the upper curves as shown in Fig. 4. If the calculated flux density B _c at N-R iteration is larger than B _min, then B _c should be located at the point d (H _d, B _c) on the lower minor loop. The H _d on the lower minor loop is given by the following equation if the upper loop is symmetrical with respect to the middle point e: $\begin{eqnarray}\displaystyle H_{d}=H_{g}+2(H_{e}-H_{g}). & & \displaystyle\end{eqnarray}$ (4) The v _d and $\partial \nu_{d}/\partial B_{c}^{2}$ , which are necessary in N-R iteration, are given by $\begin{eqnarray}\displaystyle \text{} & \displaystyle v_{d}=\frac{H_{d}}{B_{c}} & \displaystyle\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \displaystyle \frac{\partial v_{d}}{\partial B_{c}^{2}}=\frac{1}{2B_{c}^{2}}\left(\frac{\partial H_{g}}{\partial B_{g}}-v_{d}\right) & \displaystyle\end{eqnarray}$ (6) where $\partial {\nu}_{d}/\partial \boldsymbol{B}_{c}^{2}$ at the point d is obtain using the property that $\partial\boldsymbol{H}/\partial\boldsymbol{B}$ at the point d is equal to that at the point g. The conditions of calculation and experiment are listed in Table 1.

Fig. 3.

Hysteresis curves in steel plate (FCD450).

Fig. 4.

Modeling of minor loop.

Table 1

Conditions of calculation and experiment

Exciting coil	50 Hz, 2–3A, 70 turns
Search coil	40 turns
Lift-off	0.1 mm

Steel plate (FCD450)	Width in the x-direction = 150 mm,
	Length in the y-direction = 150 mm,
	Thickness in the z-direction = 3 mm,
	Conductivity σ = 1.6 × 10⁶ S/m

	Width in the x-direction (D _w) = 0.5 mm,
Opposite side defect	Length in the y-direction (D _l) = 100 mm,
	Depth in the z-direction (D _d) = 1 mm,

Nodes and elements	66417, 59280

Convergence criterion	N-R method: 1.0 × 10⁻³ T,
	ICCG method: 1.0 × 10⁻³

Time interval	Δt =1.25 × 10⁻³ s

3. Calculation and measurement results of opposite side defect inspection

Figure 5 shows the distribution of flux density in the steel plate when the exciting current is 2A as shown in Fig. 2. Figure 5(a) shows the display domain. Figure 5(b) shows the distribution of flux density in the steel plate without the defect. This figure denotes that the flux density is distributed uniformly in the steel plate. The maximum value of the flux density is 0.79 T in the surface domain of the steel plate. Figure 5(c) shows the distribution of flux density in steel plate with an opposite side defect. This figure illustrates that the flux density is distributed so that the opposite side defect position is bypassed. Therefore, the distribution of the flux density in the surface domain inside steel plate near the defect is increased. Figure 6 shows the distribution of the flux density in the steel plate when the exciting current is 3A as shown in Fig. 2. Figure 6(a) and (b) show the distribution of flux density in the steel plate with and without the defect. Figure 6(a) denotes that the flux density in the steel plate is distributed uniformity. Figure 6(b) illustrates that since the flux density is distributed so that the opposite side defect position is bypassed, the flux density in the surface domain inside the steel plate near the defect is increased. Moreover, the flux density in the surface domain near the defect is approached to the magnetic saturation. Then, as for the distribution of the flux density, these values in Fig. 6(a) and (b) are increased from them in Fig. 5(b) and (c). Figure 7 shows the distribution of the flux density near the search coil on the steel plate. Figure 7(a) shows the display domain. Figure 7(b), (c) and (d) show the distribution of flux density near the search coil on the steel plate with and without the defect when the exciting currents are 2A and 3A, respectively. These figures denote the distribution of the magnetic flux between the magnetic poles of the yoke on the surface of the steel plate. Figure 7(b) and (c) illustrate that the magnetic flux which permeates the surface domain of the steel plate without the defect is decreased, if the exciting current is increased. The maximum values of the flux density near the search coil on the steel plate without the defect are 9.1 × 10⁻⁴T and 1.6 × 10⁻³T when the exciting currents are 2A and 3A as shown in Fig. 7(b) and (c). Moreover, Fig. 7(c) and (d) illustrate that if the exciting current is same value of 3A, the magnetic flux which permeates the steel plate is decreased when there is the defect. The maximum values of the flux density near the search coil on the steel plate with and without the defect are 1.6 × 10⁻³T and 3.2 × 10⁻³T when the exciting current is 3A, respectively. This is, because the magnetic flux bypassed the defect and is concentrated to the surface domain near the defect inside the steel plate as shown in Fig. 6(b). Figure 8 shows the inspection results of the square wave magnetic field B _x in a search coil on steel plate with an opposite side defect. This figure denotes the difference value ΔB _x of flux density in search coil between 2A and 3A of the square wave magnetic field as shown in Fig. 2. The proposed sensor is moved in the x-direction from −5 mm to 5 mm by 1 mm steps. This figure illustrates that the detection signal ΔB _x is increased near the defect position. Moreover, the figure shows that the calculated results are in agreement with the measured results. In addition, the initial magnetization curve is used in conventional electromagnetic field analysis. However, it is understood that the analysis considering the minor loop is more useful than the analysis considering only the initial magnetization curve from Fig. 8.

Fig. 5.

Distribution of flux density in the steel plate when the exciting current is 2 A.

Fig. 6.

Distribution of flux density in the steel plate when the exciting current is 3 A.

Fig. 7.

Distribution of the flux density near the search coil on steel plate with and without the defect.

Fig. 8.

Inspection results of the square wave magnetic field B _x in a search coil on steel plate with an opposite side defect.

4. Comparison with static magnetic flux leakage testing method

Generally, the static magnetic flux leakage testing method (DC-MT) [2–6] is applied for the inspection of the opposite side defect in a steel plate. In this research, the comparison of the detection sensitivity of the defect between the usual DC-MT and the proposed inspection method is investigated by the numerical analysis. In the DC-MT, the same inspection model as shown in Fig. 1 is used, and the direct-current of 3A is impressed. Then, the static leakage flux inside a static magnetic field sensor of the same size as the search coil in Fig. 1 on the steel plate is evaluated. The static leakage flux is calculated by 3-D nonlinear FEM in consideration of an initial magnetization curve in Fig. 3. Figure 9 shows the inspection result of the leakage flux inside static magnetic field sensor on the steel plate with an opposite side defect. Moreover, the inspection results by the proposed inspection method are also shown in this figure. These inspection sensors are moved in the x-direction from −5 mm to 5 mm by 1 mm steps. This figure shows the rate values from the leakage flux in the place without the defect. The figure denotes that the detection sensitivity of the defect by the proposed inspection method is higher than it by the DC-MT method.

Fig. 9.

Comparison results of DC-MT and proposed method (calculated).

5. Conclusions

The results obtained by this research are summarized as follows:

In this proposed inspection method, since the flux density is distributed so that an opposite side defect position is bypassed, the flux density in the surface domain inside the steel plate near the defect is increased. When the lower magnetic field (2A) in AC square wave on DC bias is impressed, the flux density in the surface domain inside the steel plate near the defect is approached in a magnetic saturation. Then, when the higher magnetic field (3A) in AC square wave on DC bias is impressed, since the magnetic field seldom permeates the domain inside the steel plate near the defect, more leakage flux is generated near the plate on the steel plate.

The detection sensitivity of the opposite side defect in steel plate by the proposed inspection method is higher than it by the usual static magnetic flux leakage testing method.

The examinations of detectable defect size and the optimal impression magnetic field, ect. Are future research subjects.

References

Gotoh

and Takahashi

, Evaluation of detecting method with ac and dc excitations of opposite side defect in steel using 3d non-linear FEM taking account of minor loop, IEEE Trans. Magn. 44(6) (2008), 1622–1625.

Wilson

and Tian

G.Y.

, Experiment and simulation study of 3D magnetic field sensing for magnetic flux leakage defect characterisation, NDT&E INT 40(2) (2007), 179–184.

Park

G.S.

and Park

E.S.

, Improvement of the sensor system in magnetic flux leakage-type nondestructive testing (NDT), IEEE Trans. Magn. 38(2) (2002), 1277–1280.

Katoh

Masumoto

Nishio

and Yamaguchi

, Modeling of the yoke-magnetization in MFL-testing by finite element, NDT&E INT 36(7) (2003), 479–486.

Zhang

and Wang

, A fast method for rectangular crack sizes reconstruction in magnetic flux leakage testing, NDT&E INT 42(5) (2009), 369–375.

AI-Naemi

F.I.

Hall

J.P.

and Mo

A.J.

, FEM modeling techniques of magnetic flux leakage-type NDT for ferromagnetic plate inspection, J. Magn. Magn. Mater. 304(2) (2006), 790–793.

Miyata

Ohashi

Muraoka

and Takahashi

, 3-D magnetic field analysis of permanent-magnet type of MRI taking account of minor loop, IEEE Trans. Magn. 42(4) (2006), 1451–1454.