#### Abstract

Analysis of electromagnetic wave coupling to thin-wire structures plays a very important role in electromagnetic compatibility (EMC). In this paper, a hybrid method, which is integrated parabolic equation (PE) and two-potential integral equation (TPIE), is presented to analyze the coupling problems in terrain environments. To model the realistic scenarios, PE based on the split-step Fourier transform (SSFT) technique is applied to solve the three-dimensional field distribution to obtain the external excitations for the wires. According to the boundary conditions, the high-precision TPIE solved via the moment method (MoM) is developed to simulate the induced currents on the wires. The hybrid method takes the terrain influences into account and provides a more reasonable result compared to the traditional approaches. Numerical examples are given to demonstrate correctness of the proposed method. Simulation experiments of field-to-transmission lines with different frequencies, radiation source heights, conductor radii, and lengths, in a realistic scenario constructed by a digital map, are carried out to investigate the coupling properties.

#### 1. Introduction

Conducting wire structures are common in power facilities and communication systems, such as the aerial power lines, communications cables, and wire antennas. The study of high-amplitude EM field coupling to wire structures plays a very important role in EMC and electromagnetic interference (EMI) analysis [1–3]. Usually, a conducting wire with a radius far less than its length and the working wavelength can be viewed as thin-wire structure. High-precision numerical methods [4, 5] and efficient approximate circuit approaches [3, 6, 7] have been developed to analyze the coupling problems of thin-wires, where the ground is regarded as an infinite plane for simplifying the model to be easily solved. However, the facilities and devices are actually located in a complicated, large scale, and realistic environment. To get a more accurate assessment, the environmental factors like irregular terrains, buildings, and atmosphere usually need to be considered for the reflection and shadowing effects of electromagnetic waves [8, 9]. The traditional numerical methods are obviously insufficient for these scenarios because of the heavy burden from a huge number of unknowns. Besides, the equivalent circuit approaches are difficult to extend to consider the environmental influences due to the complexity.

Parabolic equation (PE) is derived from the Helmholtz equation by means of separating the forward and backward terms, which was first proposed for the solution of electromagnetic waves propagation along the earth’s surface in 1946 [10]. In recent years, PE has been developed to various applications of wave propagation prediction because of its superiority in computational efficiency [11–16]. Unfortunately, PE is not suitable for fine-structured objects because of the large step sizes and nonconformal mesh dissecting. Accordingly, some hybrid algorithms, such as the PE-RO method [17] and PE-FDTD method [18], were proposed to deal with the multiscale propagation problems. In comparison, the MoM based IE methods are applicable for objects with arbitrary shape via choosing appropriate basis functions [19] and have been widely used in the analysis of EM scattering and antenna radiation problems [20–23]. According to the thin-wire approximation, the induced currents on an arbitrarily shaped thin-wire can be assumed to only flow along its axis direction. Using the boundary conditions constructed by the tangential field components, TPIE is expressed in terms of auxiliary potentials, and the induced currents can be obtained via the pulse-basis function expansion and point matching technique.

In this paper, a hybrid PE/TPIE method is presented to model electromagnetic wave coupling to thin-wire structures in terrain environments. In Section 2, PE and TPIE are briefly reviewed, and the hybrid method is also described in this section where the impressed fields provided by PE in a realistic environment are used to excite the thin-wires. The wires are equivalently placed in free space by neglecting the quadratic coupling with the ground. TPIE expressed by Green’s function is developed to solve the wire currents through the boundary condition. A nonuniform mesh technique combined with spatial interpolation is applied to connect the mesh points of the two algorithms. In Section 3, we firstly discuss the typical scenarios where the wires are placed above flat ground, compare the hybrid method results with those obtained from full-wave numerical method, and illustrate the accuracy of PE in calculating of irregular terrain wave propagation through simulations. Simulation experiments of field to conductors in a complex environment are carried out, and the coupling properties under different parameters settings are investigated. Some conclusions are presented in Section 4.

#### 2. Theory and Formulations

##### 2.1. Parabolic Equation (PE) Method

In Cartesian coordinates, assume that the paraxial direction of PE is fixed at axis and the time-dependence is ; the reduced function obtained from the electromagnetic field component via eliminating the fast varying phase term in the direction is given by Then, the three-dimensional (3D) PE with Feit-Fleck approximation derived from the Helmholtz wave equation can be expressed aswhere is the refractive index of the medium and is the wavenumber in free space.

We define the two-dimensional (2D) fast Fourier transform (FFT) and its inverse transform (IFFT) asBy introducing the well-known SSFT technique [11], the solution of (2) can be written aswhere the operators and represent the FFT and IFFT, respectively, and the symbol denotes the conjugation.

Using (4), the field distribution in the whole computational region can be solved via a step-by-step iterative approach when the initial field is provided. However, the upper and lower boundaries should be specially treated. A shift-map method [16] is employed to handle the terrain boundary, and a Hamming absorption window is applied to truncate the upper boundary of computational domain.

##### 2.2. Two-Potential Integral Equation (TPIE)

For a thin-wire conductor placed in free space with known impressed excitation, the boundary condition is imposed on the wire, givingwhere and denote the incident field and scattering field, respectively, and is the tangential vector along the wire.

By introducing magnetic vector potential and scalar potential expressed by line current and charge density , we obtainwhere and are permeability and permittivity in vacuum, respectively. denotes the distance between the source point and field point.

To realize the integral in (6), we divide the thin-wire into* N* segments and rewrite the integral in a sum of these segments. Each segment consists of a starting point , a midpoint , and an end point (see Figure 1).

The unknown current on the wire is expressed as a linear combination of pulse-basis functions in MoM solution:where the pulse-basis function is defined asExpressed in terms of only unknown current coefficients, (6) can be rewritten in a matrix form via point matching technique [24]:where and denote the impedance and excitation voltage, respectively. The* LU* factorization can be used to solve above matrix.

##### 2.3. Combination of Two Algorithms

The currents on the wires can be solved via (9) when incident field is provided. In the traditional processing, the ground is generally treated as flat under plane wave assumption. This simplification is effective in saving costs and makes calculations easier, but it ignores the influences of environments. The accuracy would be decreased for some realities, such as the cases involving irregular terrains, because electromagnetic waves usually produce nonnegligible reflection, diffraction, and shadow effects when encountering obstacles. In this paper, a region-level algorithm, SSFT-PE method, is introduced to model the field distribution in complex scenarios, and the high-precision TPIE algorithm is used to solve the induced currents on the thin-wire structures. A schematic diagram of the hybrid method is shown in Figure 2.

For solving the currents, it is necessary to determine the tangential field components at the middle point of each segment. We can use spatial interpolation to connect the grids points of PE and TPIE. A satisfactory result will not be achieved for a thin-wire placed arbitrarily in the PE region if interpolated directly in the coarse grids. For the sake of improving the accuracy of solution, we would expect to use a global fine mesh, but it greatly increases the amount of unknowns. In this paper, we present a nonuniform mesh technique to reduce the interpolation error, so as to provide an accurate excitation for the wires.

In PE region except for the critical region containing thin-wire structures, coarse grids adapted to the variation of environment are used, while local fine grids are applied to handle the critical region. The incident field on an arbitrarily placed thin-wire can accurately be obtained via interpolation if the local fine grids are meticulous enough. Through this processing, the computational efficiency can be effectively improved while maintaining sufficient precision compared to the global fine grid system. Assuming the wires are placed horizontally above ground, we can use bilinear interpolation:where denotes the field values at the unknown point ; , , , and are PE grids points around point , all of which have known values.

#### 3. Results and Discussion

This section begins with numerical experiments to validate the hybrid method compared with full-wave method, followed by simulations to illustrate the accuracy of PE model in calculating irregular terrain wave propagation. The field-line coupling problems in complex scenarios in the presence of irregular terrains are discussed. All calculations are performed on a workstation with a configuration of six processors and 16 GB memory. The configured processor is Intel(R) E5-2620 v3 with a dominant frequency of 2.4 GHz.

##### 3.1. Algorithm Verification

Several examples are presented to validate the hybrid method. In the first example, a finite straight conductor is placed horizontally above an infinite PEC flat ground and illuminated by plane wave (see Figure 3). The conductor has a length of 9.62 m and a radius of 0.001 m and is placed at a height of 1 m from the ground. The incident plane wave (PW) with unit amplitude is polarized along the direction, with incident angle of , 5°. The simulation frequency is 300 MHz and the mesh sizes for PE model are set to m. The currents induced on the conductor are shown in Figure 4. As shown, the results of the hybrid method are consistent with those of IE method. In addition, the simulation results are also presented when the wire is excited by both plane wave and voltage excitation, which is often discussed in the problem of wire antenna radiation. The voltage with unit amplitude is set to the middle of the wire. The two curves coincide with each other, as shown in Figure 4.

Figure 6 shows the simulation results of a curved conductor at different location heights (see Figure 5). The incident plane wave with predominant polarization parallel to the* y* axis has a frequency of 300 MHz. The angle of incidence is set to , 5°. As shown, the results of the hybrid method are consistent with those of IE method.

A dual-conductor case is also discussed (see Figure 7). Two straight wires with a spacing of 1.42 m are placed parallel to the ground and illuminated by a plane wave with an incident angle of 5°. The frequency is set to 300 MHz. Both of the wires have the same length (9 m) and radius (0.001 m), and their heights from the ground are 6.8 m and 7.4 m, respectively. The currents induced on the dual-conductor are presented in Figure 8. In this test, the dual-conductor was divided into 1024 segments. The mutual coupling between the two conductors can effectively be characterized via mutual-impedance in TPIE solution. The results are consistent with those of IE method (see Figure 8).

##### 3.2. Simulations in Terrains Environment

In this section, we present several simulations to test the accuracy of PE in modeling of the radio wave propagation in terrain environments in 2D case. The PE results are compared with those of the full-wave method, i.e., MoM-IE method. The frequency of electromagnetic wave transmitted by a horizontally polarized antenna, with the Gauss pattern, is set to 30 MHz. The incident angle is , 3°, and the beam width is 20°. The antenna is located at an altitude of 200 m. The mesh sizes for the 2D PE model are set to , m, and m. The terrain profiles are divided into 1500 segments in MoM-IE method; i.e., the two methods have the same subdivision scales for the terrain profile.

Figure 9 shows the electric field distribution calculated via PE and MoM-IE method over different 2D terrains. As shown, irregular terrains produce significant reflection and diffraction effects on electromagnetic waves, and the two methods get consistent results. Figure 10 shows the curves of electric field strength changing with the propagation distance at a fixed height of 200 m. Figure 11 presents the curves of electric field strength with altitude at a propagation distance of 3 km. The results of the two methods are consistent with each other (see Figures 10 and 11). It is noted that the computational load of MoM-IE is much larger than that of PE due to the heavy burden from a huge number of unknowns. In this experiment, the CPU computing time for PE is only 6 seconds, while that of MoM-IE is 7 minutes.

##### 3.3. Application Example

There are often complicated scenarios in the presence of irregular terrains in the field of EMC, where we want to gain insight into how high-amplitude electromagnetic waves act on the transmission lines. Unfortunately, it is usually difficult to handle such problems using pure algorithms because of the inherent contradiction between accuracy and efficiency. In such a dilemma, hybrid algorithm becomes an alternative [17, 18]. In this section, we focus on the field-line coupling problems in a realistic terrain environment using the presented PE/TPIE hybrid method. The 3D experimental scenario is constructed by a high-precision digital map (see Figure 12). The transmitting antenna has a Gaussian beam pattern with beam width 4°. The amplitude of the initial field is 5000 V/m. The concerned power transmission lines are marked as red triangles in the map (see Figure 12). The heights of the transmission lines above ground are 30 m, 29 m, and 40 m, respectively. In 3D PE model, we set m and m in the coarse grid system, and m in the fine grid system. In our experiments, all lines are assumed to be along the polarization direction of the electromagnetic wave, which is most conducive to inducing the maximum current.

The operating frequency is set to 60 MHz. The conductors are 20 m in length and 0.007 m in radius. Figure 13 shows the electric field obtained by PE method. From the simulation result we can see that the slopes and undulating mountains produce remarkable refraction, diffraction, and shadow effects. In order to accurately analyze the field-line coupling problem, these effects should be considered. In this paper, we present a hybrid PE/TPIE method, taking the terrain effects into consideration.

Figure 14 shows the currents induced on the lines. In the absence of undulating topographic occlusion in front of* A* and* B*, the contribution of electromagnetic field mainly comes from the direct components and reflection components of the ground. The current on line* B* is smaller than that on* A* because of the diffusion of electromagnetic waves. The line* C* far from the emission source is shielded by the Longquan Mountains, resulting in weak values.

Figures 15–18 show the coupling strength expressed by maximum currents with different frequencies, radiation source heights, conductor radii, and lengths. The currents vary with transmitter frequency when the conductor lengths and spatial locations are fixed (see Figure 15), because of the different sensitivity and coupling ability of electromagnetic waves acting on lines. What is more, the low-frequency waves show stronger coupling effects. In the PE/TPIE method, although PE is not limited to frequency of the emission source, the thin-wire approximation conditions need to be met to ensure that TPIE gets a reliable result for a relatively high frequency. The peak currents are related to the external fields. Hence, different results will be obtained for different radiation source heights (see Figure 16). In addition, all peak currents increase with the radius of the conductor (see Figure 17), but not the length (see Figure 18).

#### 4. Conclusions

Research on high-amplitude electromagnetic field coupling to conducting thin-wire structures is meaningful for EMC. Different from the previous work which almost focuses on the cases of a flat ground, this paper presents a hybrid algorithm that employs PE and TPIE, used to model complicated situations in the presence of irregular terrains. The complicated spatial fields are obtained by PE solution and used to excite the conducting thin-wire structures. Through a nonuniform mesh technique and spatial interpolation, the induced currents are solved via MoM to take the influence of environment into consideration. The presented method is validated by comparing the results with those of full-wave method. A simulation experiment is carried out to analyze the field-line coupling properties in a realistic scene. The presented method has been found to be feasible, and it is open to further improvements to model the complex problems involving terminal devices.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61771407) and the National Science Foundation for Young Scientists of China (Grant No. 61801405).