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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.

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 [

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 [

In this paper, a hybrid PE/TPIE method is presented to model electromagnetic wave coupling to thin-wire structures in terrain environments. In Section

In Cartesian coordinates, assume that the paraxial direction of PE is fixed at

We define the two-dimensional (2D) fast Fourier transform (FFT) and its inverse transform (IFFT) as

Using (

For a thin-wire conductor placed in free space with known impressed excitation, the boundary condition is imposed on the wire, giving

By introducing magnetic vector potential

To realize the integral in (

Thin-wire model divided into

The unknown current on the wire is expressed as a linear combination of

The currents on the wires can be solved via (

Hybrid algorithm with nonuniform mesh technique. Coarse and fine grids are applied to different regions.

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:

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.

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

Straight conductor above flat ground. Conductor’s length is 9.62 m, and the height from the ground is 1 m.

Currents on the straight conductor.

Figure

Curved conductor above flat ground. Conductor’s length is 4.19 m.

Currents induced on the curved conductor.

A dual-conductor case is also discussed (see Figure

Two straight wires with a spacing of 1.42

Currents on the straight conductor

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

Figure

Field strength profiles in dB calculated via PE and IE method, respectively. The operating frequency of a horizontally polarized antenna is 30 MHz. The beam width is 20°, and the antenna height is 200 m.

Electric field at a fixed height of 200 m.

Electric field at a receiving distance of 3 km.

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 [

Experiment scene. The concerned power transmission lines are marked as red triangles.

The operating frequency is set to 60 MHz. The conductors are 20 m in length and 0.007 m in radius. Figure

Electric field distribution in dB at frequency of 60 MHz.

Figure

Currents induced on the power lines. The radii are 0.007 m, and the lengths are 20 m.

Figures

The maximum currents at different frequencies. The conductors are 20 m in length and 0.007 m in radius. The radiation source heights are 1 km.

The maximum currents at different transmitting antenna heights. The frequencies are 60 MHz, and conductor radii are 0.007 m.

Maximum Currents with different conductor radii. The frequencies are 60 MHz, and radiation source heights are 1 km.

Maximum currents with different conductor lengths. Conductor radii are 0.007 m.

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.

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

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

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).