Proposal of a TEM to TE 01 mode converter for a microwave nondestructive inspection of axial flaws appearing on the inner surface of a pipe with an arbitrary diameter

Abstract

This study discusses the characteristics of a mode converter to generate microwaves in the TE₀₁ mode for the long-range inspection of pipes. Numerical analysis was conducted to evaluate the effect of the pipe diameter and the number of coaxial cables inserted into the pipe on the efficiency of converting microwaves from the coaxial TEM mode to the circular TE₀₁ mode. The results reveal that the most efficient frequency range for TE₀₁ mode propagation strongly depends on the number of cables and pipe diameter. Subsequent experiments were performed using brass pipes with a length of 5 m and diameters of 39.0 and 74.0 mm. The results of these experiments are consistent with the results of the numerical analysis: the pipe with a diameter of 39.0 mm showed a clear signal from an axial slit 1.5 m away from the mode converter and the pipe with a diameter of 74.0 mm showed almost no difference between signals with or without the slit.

Keywords

Nondestructive testing axial cracks TE modes metal pipe time of flight

1. Introduction

Since piping systems play an important role in large-scale plants such as nuclear and oil ones, precise nondestructive inspection is indispensable for their maintenance [1]. Although conventional nondestructive testing methods such as eddy current testing and ultrasonic testing have high precision, applying these methods to long pipes sometimes requires a long time and large expense. Therefore, a nondestructive testing method using microwaves has been proposed to enable quicker inspection of pipes [2–5]. This method detects flaws by evaluating the time of flight (TOF) of microwaves propagating inside a metal pipe.

Previous studies have demonstrated that the method is effective in detecting artificial wall thinning in a pipe with a length of approximately 30 m [6–8]. However, these studies have also revealed that the magnitude of reflection signals due to artificial slits significantly depend on both the orientation of the slit and the mode of microwave propagating inside the pipe [9] and that microwave propagating as TE modes is suitable to detect axial cracks.

To address this issue, a converter from the coaxial TEM mode to the circular TE₀₁ mode with an inner diameter of 19.0 mm was proposed based on numerical analysis and tested in a pipe [9]. Even though reflections were confirmed from the axial slit in this experiment [9,10], since the applicability of this approach for other diameters has not been evaluated yet. Therefore, in the study, numerical analysis was conducted to evaluate effect of the diameters of the mode converter profile on the efficiency of converting the TEM mode to the TE₀₁ mode. After evaluating the detailed characteristics of the mode converter, experiments were conducted to verify the results of numerical analysis.

2. Numerical analysis

2.1. Analysis system

Figure 1(a) shows the numerical analysis model based on the results of the previous study [9]. The model simulates a mode converter that generates circular TE₀₁ microwaves inside of a pipe. Several coaxial cables are inserted into the pipe with even interval in the circumferential direction. The coaxial cables carry microwaves in the coaxial TEM mode with the same amplitude and phase. In the model the pipe diameter, D, and the number of coaxial cables, p, are chosen as parameters and then the electromagnetic fields are evaluated on the evaluating planes.

Table 1
Analytical conditions

Diameter D [mm] Frequency [GHz] Number of coaxial cables p L [mm]

11.0 36–51 2, 4, 6 50

39.0 12–27 2, 4, 6, 8 50

57.5 9–19 2, 4, 6, 8, 16 50

74.0 6–16 2, 4, 6, 8, 16 100

Diameter D [mm]	Frequency [GHz]	Number of coaxial cables p	L [mm]
11.0	36–51	2, 4, 6	50
39.0	12–27	2, 4, 6, 8	50
57.5	9–19	2, 4, 6, 8, 16	50
74.0	6–16	2, 4, 6, 8, 16	100

Figure 1(b) illustrates the profile of the coaxial cable. The coaxial cable consists of three layers: a core wire with a diameter of 0.8 mm, a dielectric with an outer diameter of 2.2 mm, and an outer conductor with an outer diameter of 3.0 mm. The distance between the central axis of the pipe and the core wire, Δr, is set to be 0.3D.

Fig. 1.

Numerical analysis model. Note that, whereas the model shown in this figure has six coaxial cables, the numerical analysis actually evaluated the number of the coaxial cables. Perfectly matched layers (PML) are situated to simulate the infinite area.

The analytical conditions are shown in Table 1. Four diameters were selected: 11.0, 39.0, 57.5, and 74.0 mm to evaluate the generated TE mode. The frequency bandwidth was 10 or 15 GHz with a step of 0.5 GHz, which was determined to be above the cutoff frequencies of the TE₀₁ mode for the respective D. The cutoff frequencies of the major modes are shown in Table 2. The distance between the evaluating planes and the incident plane, L, was determined to be greater than the guided wavelength at the respective minimum frequency. Maximum number of cables, p, is determined based on geometrical restriction.

Numerical analysis was conducted in the frequency domain by the 3D finite element method using the commercial software COMSOL Multiphysics v.5.0 and v.5.2a with its RF module. The governing equation was as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle \text{rot}({\mu}_{\text{r}}^{-1}\text{rot}∼\boldsymbol{E})-k_{0}^{2}\left({\varepsilon}_{\text{r}}-\frac{\text{j}{\sigma}}{{\omega}{\varepsilon}_{0}}\right)\boldsymbol{E}=0\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle k_{0}={\omega}\sqrt{{\varepsilon}_{0}{\mu}_{0}}\end{eqnarray}$ (2) where E denotes the electric field, ϵ₀ the permittivity of vacuum, 𝜇₀ the permeability of vacuum, ϵ_r the relative permittivity, 𝜇_r the relative permeability, σ the electrical conductivity, and ω the angular frequency, respectively. In the analysis, these values are set to be 𝜇_r = 1, ϵ_r = 1.687, and σ = 0 for the dielectric and 𝜇_r = 1, ϵ_r = 1.000 and σ = 0 for the air. The number of elements and the computational time were approximately 1.7–3.4 million and 5–26 hours (Windows 10, Intel Xeon E5-1620, 128 GB RAM), respectively, which depended on mainly the diameter of the pipe and the frequency bandwidth.

Table 2

Cutoff frequency (GHz)

Mode	D = 11.0 mm	D = 39.0 mm	D = 57.5 mm	D = 74.0 mm
TE₁₁	15.97	4.51	3.06	2.37
TE₂₁	26.50	7.47	5.07	3.94
TE₀₁	33.24	9.38	6.36	4.94
TE₃₁	36.45	10.28	6.97	5.42
TE₄₁	46.13	13.01	8.83	6.86
TE₅₁	55.66	15.70	10.65	8.27
TE₀₂	60.86	17.17	11.64	9.05
TE₆₁	65.07	18.35	12.45	9.67
TE₈₁	83.69	23.61	16.01	12.44
TE_16,1	156.70	44.20	29.98	23.29

2.2. Results and discussion

Figure 2 shows the energy of the microwaves converted into the TE₀₁ mode normalized by the input energy. The figure shows that the ratio of microwaves in the TE₀₁ mode strongly depends on both the diameter of the pipe, D, and the number of coaxial cables, p. In general, the ratio is high until the cutoff frequency of the TE₄₁ mode (p = 2, 4) or TE₀₂ mode (p = 6, 8, 16). This result is likely due to the geometrical symmetricity of the mode: the analysis models of p = 2 and p = 4 are two-fold and four-fold rotationally symmetrical, respectively, allowing the formation of a four-fold rotationally symmetrical mode such as the TE₄₁ mode. The TE₀₂ mode is axisymmetrical and therefore can be formed in any rotationally symmetrical model. The cutoff frequency of the TE₀₂ mode generally lies between those of the TE₅₁ and TE₆₁ modes and is higher than that of the TE₄₁ mode in a circular waveguide. Accordingly, the maximum frequency for efficient transmission of the TE₀₁ mode, f _max, lies as follows: $\begin{eqnarray}\displaystyle f_{\max }=\min (f_{n1},f_{02}) & & \displaystyle\end{eqnarray}$ (3) where f _n1 denotes the cutoff frequency of TE_n1 mode, f ₀₂ the cutoff frequency of TE₀₂ mode, n the minimum multiple of p above 2. The difference of between f _max and the cutoff frequency of TE₀₁ mode, Δf, is described as follows: $\begin{eqnarray}\displaystyle {\rm\Delta}f=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{c({\rho}_{n1}\prime -{\rho}_{01}\prime )}{{\pi}D} & (p\lt 6)\\[10.0pt] \displaystyle \frac{c({\rho}_{02}\prime -{\rho}_{01}\prime )}{{\pi}D} & (p\geq 6)\end{array}\right. & & \displaystyle\end{eqnarray}$ (4) where c denotes the speed of light in vacuum, D the inner diameter, and 𝜌_ab ′ the b-th root of the derivative of a a-order Bessel function of the first kind, respectively. The maximum frequency bandwidth is obtained when p ≥ 6. Empirically, a frequency bandwidth of about 5 GHz is required to inspect pipes with a high precision [8,9], which means the present method is available to the pipe diameter less than 60 mm, as shown in Fig. 3.

Fig. 2.

Transmission characteristic of the TE₀₁ mode.

Fig. 3.

D vs Δf (p ≥ 6).

Fig. 4.

Mode converters.

3. Experimental validation

3.1. Experimental system

To validate the numerical analysis results, mode converters made of brass with inner diameter of 39.0 mm and 74.0 mm were fabricated as shown in Fig. 4 whose diameters are smaller or greater than 60 mm as are discussed in the above section These mode converters had eight coaxial cable connectors (Anritsu, K101F-R) and eight semirigid cables (Anritsu, K118) inserted circumferentially with even interval. According to the numerical analysis, when the TEM mode is excited with the same amplitude and phase from the respective connectors, the TE₀₁ mode is emitted from the end of the mode converter.

Fig. 5.

Experimental system.

Figure 5 shows the experimental system. In this study brass pipes of 1 m length with inner diameters of 39.0 mm or 74.0 mm were used. Five pipes and the mode converter were connected with flanges, as illustrated in Fig. 5, to situate the mode converter in the middle of the pipe, yielding a total length of approximately 5 m. Figure 6 shows the axial slits used in this study to evaluate the mode converters. The axial slits of about 1 mm width were machined by a circular saw and were situated L _s away from the mode converter. The outer surfaces of the slits were covered with aluminum foil during the measurements to simulate of crack not penetrated.

In these experiments, a network analyzer (Agilent Technologies, E8363A) emitted microwaves and measured their reflections. A power divider (ET Industries, D-140-8) was used to split the microwaves. The reflected signals were measured in the frequency domain with a step of 2.5 MHz and transformed into the time domain by inverse Fourier transform. Table 3 shows the experimental conditions of the frequency bandwidth and slit location.

Fig. 6.

Axial slits.

Table 3

Experimental conditions

D [mm]	Frequency [GHz]	L _s [cm]
39.0	12.0–17.0	57, 157
74.0	6.0–9.0	57, 157

3.2. Results and discussion

Figure 7 compares the reflection signals in the time domain observed with and without an axial slit. When D = 39.0 mm, a clear signal due to the slit was confirmed, as shown by the arrows. Moreover, the reflection from the slit was delayed as L _s increased. This result suggests that the TE₀₁ mode propagated with sufficient amplitude and bandwidth. However, when D = 74.0 mm, the difference between reflections with and without a slit was not observed. One of the most plausible reason for this is a small bandwidth for which TE₀₁ mode can propagate, as is discussed in Section 2.2.

Fig. 7.

Time domain signals.

4. Conclusion

In this study, the characteristics of the TE₀₁ mode converter are evaluated, which is used to detect axial cracks on the inner surface of a pipe based on the propagation of microwaves inside the pipe. Numerical results suggested the mode converter can be applied in pipes with a diameter of up to 60 mm. Subsequent experimental results using brass pipes with a length of 5 m with axial slits were consistent with the numerical analysis results: clear signal difference was observed between with and without the slit in the case of D = 39.0 mm, while no difference was observed in the case of D = 74.0 mm.

Footnotes

Acknowledgements

This study was supported by a Grant-in-Aid for Challenging Exploratory Research (15K14298) and JSPS Fellows (14J04906).

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Proposal of a TEM to TE 01 mode converter for a microwave nondestructive inspection of axial flaws appearing on the inner surface of a pipe with an arbitrary diameter

Abstract

Keywords

1. Introduction

2. Numerical analysis

2.1. Analysis system

Table 1 Analytical conditions Diameter D [mm] Frequency [GHz] Number of coaxial cables p L [mm] 11.0 36–51 2, 4, 6 50 39.0 12–27 2, 4, 6, 8 50 57.5 9–19 2, 4, 6, 8, 16 50 74.0 6–16 2, 4, 6, 8, 16 100

3.1. Experimental system

Footnotes

Acknowledgements

References

Table 1
Analytical conditions

Diameter D [mm] Frequency [GHz] Number of coaxial cables p L [mm]

11.0 36–51 2, 4, 6 50

39.0 12–27 2, 4, 6, 8 50

57.5 9–19 2, 4, 6, 8, 16 50

74.0 6–16 2, 4, 6, 8, 16 100