An Improved RCS Calculation Method for Power Lines Combining Characteristic Mode with SMWA

Abstract

The radar cross-section (RCS) of power lines has an important significance for detection of the power lines. The method of moments (MoM) can calculate the RCS of power lines. However, the efficiency of the MoM is limited by the time-consuming computing process, as well as the expensive storage overhead. In order to enhance the efficiency and reduce the storage of the RCS calculation of power lines, we propose an RCS calculation method that combines the characteristic mode (CM) with a Sherman–Morrison–Woodbury formula-based algorithm (SMWA), which is referred to as a CM-SMWA. CMs are used as the basis functions for reducing the dimension of the MoM impedance matrix, and the SMWA is applied to directly solve the CMs-reduced matrix equation, which can reduce the computational time and storage. The numerical results demonstrate that the proposed method can obtain the RCS of power lines, with different incident angles and different polarizations, at a higher efficiency. At 35 GHz, compared with the conventional MoM, for a typical LGJ50-8 power line with a length of 0.276 m, the computation time is reduced by 62.4% and the memory occupation is reduced by 96.4%.

1. Introduction

Power lines pose the most serious threat to the safety of helicopters [1,2], and the detection of the power lines has always been a focus of the research on helicopter collision avoidance. Millimeter wave radar is an effective means of all-weather power line detection [3,4,5,6,7]. The characteristics of stranded power lines can generate Bragg scattering in the millimeter wave band [8]. In the process of power line research based on millimeter wave radar, according to the characteristics of power line scattering, a detection algorithm based on the Bragg feature and polarization feature was developed [9,10]. Further, with the rise of machine learning and deep learning algorithms, learning algorithms were developed. The developed algorithms include support vector machines (SVM) [11] and convolutional neural networks (CNN) [12]. The Bragg scattering pattern-based machine learning (ML) detection method provides an impressive detection performance.

However, in the case of complex terrain, a large number of Bragg scattering samples of power lines are needed to improve the generalization of the Bragg scattering-based ML detection method. Acquiring a large number of samples through an actual experiment is not practical due to expensive costs. Thus, it is necessary to explore efficient algorithms to obtain a large number of radar cross-section (RCS) samples of power lines with different types and specifications.

High-frequency approximation methods, including geometric optics (GO) and physical optics (PO), have been used for efficiently calculating the RCS of the power lines [13,14]. However, the calculation accuracy of GO and PO is unsatisfactory. Although the above two methods can obtain the angle position of the Bragg scattering, they cannot reflect the amplitude information of the Bragg scattering of the power lines accurately.

Compared with the high-frequency (i.e., GO, PO, etc.) approximation methods, the method of moments (MoM) [15,16,17] based on integral equations can provide more accurate RCS results by converting integral equations into matrix equations. However, the computational and storage complexity of the MoM are O(N³) and O(N²), respectively, where N represents the number of basis functions. Therefore, using the MoM makes it difficult to obtain the RCS of power lines with an electrically large size. In order to improve the efficiency of the MoM, a variety of fast iterative algorithms has been developed, such as the multilevel fast multipole algorithm (MLFMA) [18,19] and fast Fourier transform (FFT)-based methods, including the adaptive integral method (AIM) [20,21], precorrected-FFT (P-FFT) [22], and integral equation-FFT (IE-FFT) [23], and others. For solving the RCS samples of power lines, the RCS for multiple incident angles and multiple polarizations is usually required. However, the above methods belong to a type of iterative solvers and need to re-solve the matrix equation for plane waves with different incident angles and polarizations. It will be inefficient, especially for the electric field integral equation (EFIE), due to its poor matrix behavior.

Compared with the iterative method, the direct solution method can avoid unexpected iterative convergence problems and is suitable for electromagnetic scattering problems with multiple incident angles and multiple polarizations. The characteristic mode (CM) method [24,25,26,27] is an efficient and direct solution algorithm which significantly reduces the number of basis functions by constructing macro basis functions, and so the dimension of the impedance matrix is significantly reduced. Therefore, the matrix inversion time and storage memory are correspondingly significantly reduced. However, as the electrical size of the target increases, the reduced matrix method required by the CM method becomes larger and larger, which makes it very time-consuming and takes up a lot of memory when solving for electrically large-sized power lines.

To mitigate this problem, an efficient direct solver based on the characteristic mode (CM) [24,25,26,27] and the Sherman–Morrison–Woodbury formula-based algorithm (SMWA) [28,29,30,31] is proposed in this paper. The CM is used as the basis function for reducing the dimension of the matrix significantly, and the fast direct solver, SMWA, is applied to solve the CM-reduced matrix equation efficiently. The numerical results demonstrate that the proposed method can obtain the RCS of power lines of different incident angles and different polarizations with less memory and computational time. The innovations of this work are as follows:

(1)

An efficient solution algorithm combining the CM with the SMWA is proposed.

(2)

The proposed CM-SMWA is applied to the RCS simulation of power lines.

The rest of this paper is organized as follows: Section 2 introduces the calculation of the RCS of the power lines based on the EFIE and the MoM. Section 3 introduces using the CM method to reduce the dimensionality of the impedance matrix. Section 4 elaborates the proposed RCS calculation method in detail. Section 5 shows the performance of the proposed RCS calculation method. Section 6 draws the conclusions.

2. Calculating the RCS of Power Lines by MoM

The MoM can solve electromagnetic scattering problems by converting the integral equation corresponding to the problem into a matrix equation [15]. The advantages of MoM lie in two aspects: high computational accuracy and good adaptability to three-dimensional targets of any shape. The following introduces the RCS calculation process of power lines based on the MoM.

Figure 1 shows the petal-like envelope on the cross-section of the power lines. When calculating the electromagnetic scattering of the power lines, we can use the petal-like envelope on the surface of the power lines as the boundary and divide the space into $V_1$ and $V_2$, which denote the outside and interior of the conductor of the power lines, respectively. A scattered field $E^s$ in the $V_1$ will be produced due to the incident wave that illuminates the perfect electric conductor (PEC), and the total field is the superposition of the incident field $E^s$ and the scattered field $E^i$. The electric field $E_2$ inside the power lines, as an ideal conductor, is zero. The ${\epsilon }_0$ and ${\mu }_0$ represent the permittivity and permeability of the free space. $\sigma$ is the conductivity and $\sigma =\infty$ for the PEC. $\widehat{\mathbf{n}}$ is the normal vector facing the outside of the perfect conductor’s surface.

The EFIE can be constructed as follows [17]:

$$\widehat{n}\times \widehat{n}\times (E^i+E^s)=0$$

(1)

When calculating RCS of power lines, the incident wave is a plane wave, and the electric field $E^i$ can be represented as follows:

$$E^i(r)=E_0{e}^{-jk·r}$$

(2)

where the $E_0$ is the amplitude of the electric field of the incident wave, $k=\widehat{k}k$, $\widehat{k}$ represents the incident direction, and k denotes the wave number.

According to the relationship between the scattered field and the induced current [17], the electric field $E^s$ in the free space generated by the induced current is:

$$E^s(r)=-j\omega A(r)-\nabla \varphi (r)$$

(3)

where j is an imaginary unit, $\omega$ is the angular frequency of the incident wave, $\nabla$ represents gradient, and $A(r)$ and $\varphi (r)$ represent the vector potential function and scalar potential function, respectively. The detailed definitions of $A(r)$ and $\varphi (r)$ are:

$$A(r)={\mu }_0{\displaystyle \underset{s}{\int }G(r,r^{\prime })J(r^{\prime })}d{s}^{\prime }$$

(4)

$$\varphi (r)=-\frac{1}{j\omega {\epsilon }_0}{\displaystyle \underset{s}{\int }G(r,r^{\prime }){{\nabla }^{\prime }}_s·J(r^{\prime })d{s}^{\prime }}$$

(5)

In Equation (4), $r$ and $r^{\prime }$ are the position vectors of the field point and the source point, respectively. The Green’s function $G(r,r^{\prime })$ is given by the following equation:

$$G(r,r^{\prime })=\frac{e^{-jk|r-r^{\prime }|}}{4\pi |r-r^{\prime }|}$$

(6)

Putting Formulas (4) and (5) into Formula (3), Formula (3) can be rewritten as:

$$E^s(r)=-j\omega {\mu }_0{\displaystyle \underset{s}{\int }G(r,r^{\prime })J(r^{\prime })d{s}^{\prime }}+\nabla \frac{1}{j\omega {\epsilon }_0}{\displaystyle \underset{s}{\int }G(r,r^{\prime }){{\nabla }^{\prime }}_s·J(r^{\prime })d{s}^{\prime }}$$

(7)

Substituting Equation (7) into Equation (1), the EFIE of the PEC can be obtained as follows:

$$\widehat{n}\times \widehat{n}\times (-j\omega {\mu }_0{\displaystyle \underset{s}{\int }G(r,r^{\prime })J(r^{\prime })}d{s}^{\prime }-\nabla \frac{1}{j\omega {\epsilon }_0}{\displaystyle \underset{s}{\int }G(r,r^{\prime }){\nabla }^{\prime }·J(r^{\prime })d{s}^{\prime }})=\widehat{n}\times \widehat{n}\times E^i$$

(8)

where $J(r^{\prime })$ represents the surface current.

The MoM can solve (8) to obtain the surface current $J(r^{\prime })$. Specifically, the scatterer is discretized into interconnected sub-domains, and the basis functions are constructed to describe the distribution of the electromagnetic field in the sub-domains. The RWG has the properties of simplicity and proximity to the real electric current, which has become the first choice for the MoM in modern electromagnetism [32]. By using the RWG as the basis function to linearly expand the induced current, the induced current on the PEC surface can be expressed by the following equation:

$$J(r)={\displaystyle \sum _{n=1}^N{I}_n{f}_n(r)}$$

(9)

where $f_n(r)$ represents the n-th RWG basis function [31].

Substituting Equation (9) into Equation (8), and using the Galerkin method, the matrix equation after EFIE discretization can be obtained as follows:

$$ZI=V$$

(10)

where each element in Z and I are easily derived from Equation (8) [17]:

$$Z_{mn}=jk\eta {\displaystyle \underset{T_m^{\pm }}{\int }{\displaystyle \underset{T_n^{\pm }}{\int }G(r,r^{\prime })}}\left(f_m(r)·f_n(r^{\prime })-\frac{1}{k^2}\nabla ·f_m(r){\nabla }^{\prime }·f_n(r^{\prime })\right)d{s}^{\prime }ds$$

(11)

$$V_m={\displaystyle \underset{T_m^{\pm }}{\int }{f}_m\left(r\right)·E^i(r)ds}$$

(12)

where $\eta =\sqrt{{\mu }_0/{\epsilon }_0}$ represents the wave impedance of the free space. The coefficient $I_n$ is obtained from the known $Z_{mn}$ and $V_m$ and, finally, $J(r)$ is obtained. Then, the $E^s(r)$ can be obtained by Equation (7), and so the RCS can be obtained according to the following equation:

$$\sigma =\frac{4\pi r^2{\left|E^s\right|}^2}{{\left|E^i\right|}^2}, r\to +\infty$$

(13)

3. Characteristic Mode Method

3.1. Characteristic Mode

The CM method [24,25,26,27] is a new, powerful tool for solving and analyzing the electromagnetic radiation and scattering problems of arbitrary electromagnetic structures. The CM is an inherent mode applicable to any electromagnetic structure, independent of external excitation, only related to the shape and material of the structure itself, and it can reveal the inherent electromagnetic properties of the structure itself. The CM method can be combined with the MoM [15,16,17] and reduce the size of the matrix and reduce the amount of inversion calculations.

The impedance matrix $Z$ in Equation (10) is a symmetric matrix, which can be decomposed into two Hermitian matrices, $R$ and $X$, as follows:

$$\mathbf{Z}=\mathbf{R}+jX$$

(14)

where $R$ and $X$ are the symmetric matrices, and the definitions of $R$ and $X$ are as follows:

$$\left\{\begin{array}{c}\mathbf{R}=\frac{Z+Z^H}{2}\\ \mathbf{X}=\frac{Z-Z^H}{2}\end{array}\right.$$

(15)

where H represents the conjugate transpose.

The generalized eigenvalue equation between $R$ and $X$ can be constructed as follows [24,25]:

$$X·J_n={\lambda }_{n}R·J_n$$

(16)

where ${\lambda }_n$ is the eigenvalue of the mode and $J_n$ is the eigenvector of the mode, representing the mode current corresponding to ${\lambda }_n$.

Since the modal significance of the CM is high when ${\lambda }_n$ is small, we take the CM current $J_n$ with a small eigenvalue to reduce the characteristic dimension, and thus reduce the calculation cost. By obtaining the CM of the PEC, Equation (10) can be approximated as follows:

$$Z^{CM}\alpha =V^{CM}$$

(17)

where $Z^{CM}=J^{T}ZJ$ and $V^{CM}={(J^{CM})}^{T}V$.

In the case of many basis functions of the RWG, we divide the target into groups to reduce the cost of calculating the CM [26,27]. The binary-tree-based grouping method is adopted along the z of the power lines. The number of groups is B, which is an integer power. Figure 2 is a schematic diagram of two-level binary tree groupings.

After dividing the power lines into B groups, the $Z$ can be divided into the B × B sub-matrices as follows:

$$Z=\left[\begin{array}{cccc}{Z}_{11}& Z_{12}& \cdots & Z_{1B}\\ Z_{21}& Z_{22}& \cdots & Z_{2B}\\ ⋮& ⋮& \ddots & ⋮\\ Z_{B1}& Z_{B2}& \cdots & Z_{BB}\end{array}\right]$$

(18)

According to Equations (14)–(16), the CM $J_i$ is obtained for the grouped near-field matrix $Z_{ii}$, i = 1,…B, which can greatly reduce the calculation time of the CM, as follows:

$$Z_{ii}^{CM}=J_i^T{Z}_{ii}{J}_i$$

(19)

where $J^T$ is the transposed matrix of J, and the CMs for all groups are calculated to obtain $J_1\sim J_B$, which are then multiplied by the matrix $Z$ to obtain:

$$Z^{CM}=J^{T}ZJ=\left[\begin{array}{cccc}{J}_1^T& & & \\ & J_2^T& & \\ & & \ddots & \\ & & & J_B^T\end{array}\right]\left[\begin{array}{cccc}{Z}_{11}& Z_{12}& \cdots & Z_{1B}\\ Z_{21}& Z_{22}& \cdots & Z_{2B}\\ ⋮& ⋮& \ddots & ⋮\\ Z_{B1}& Z_{B2}& \cdots & Z_{BB}\end{array}\right]\left[\begin{array}{cccc}{J}_1& & & \\ & J_2& & \\ & & \ddots & \\ & & & J_B\end{array}\right]$$

(20)

If we assume that $Z_{ij}^{CM}={(J_i^{CM})}^T{Z}_{ij}{J}_j^{CM}$ and $V_i^{CM}={(J_i^{CM})}^T{V}_i$, it is easy to obtain:

$$\left[\begin{array}{cccc}{Z}_{11}^{CM}& Z_{12}^{CM}& \cdots & Z_{1B}^{CM}\\ Z_{21}^{CM}& Z_{22}^{CM}& \cdots & Z_{2B}^{CM}\\ ⋮& ⋮& \ddots & ⋮\\ Z_{B1}^{CM}& Z_{B2}^{CM}& \cdots & Z_{BB}^{CM}\end{array}\right]\left[\begin{array}{c}{\mathsf{\alpha }}_1\\ {\mathsf{\alpha }}_2\\ ⋮\\ {\mathsf{\alpha }}_B\end{array}\right]=\left[\begin{array}{c}{V}_1^{CM}\\ V_2^{CM}\\ ⋮\\ V_B^{CM}\end{array}\right]$$

(21)

The matrix $Z$ with the size of N × N can be divided into $Z^{CM}$ with the size of $N_{CM}\times N_{CM}$, where N represents the number of the RWG basis functions and $N_{CM}$ represents the number of significant CMs. The dimension of $Z^{CM}$ is much smaller than that of $Z$. The calculation cost of the inversion of the matrix $Z^{CM}$ is greatly reduced. After $\mathsf{\alpha }$ in Equation (21) is calculated, the current coefficient of the RWGs, $J\mathsf{\alpha }$, can be calculated. Finally, the RCS is calculated according to Equation (13).

3.2. An Extended Method for Computing CMs

The groups of the basis functions in Section 3.1 do not overlap and cannot guarantee the current continuity at the boundary, which further affects the accuracy of the RCS calculation. In order to reduce the current singularity caused by region division, each group needs to be extended [33].

As shown in Figure 3, the dotted line part at the edge of the near-field impedance matrix is the extension area, and the extension length is $\delta \lambda$. Generally, the extension of $\delta \lambda$ is made as small as possible to ensure that the increase in the amount of required calculation is within an acceptable range.

In Figure 3, the groups have been extended. First, the CM is calculated with the extended group, then the CM in the extended part is removed and the eigenvector $J_i$ of the eigenvalue ${\lambda }_i$ is reserved in the CM of the unextended block [33] to improve the accuracy of the CM method.

4. CM-SMWA for Calculating RCS

The performance of the MoM is limited by the low computational efficiency and high storage memory requirement. Similarly, the performance of the CM in reducing the matrix is also limited by the low computational efficiency. This is because the simulation of power lines requires a large number of RWG basis functions. The purpose of this paper is to find a more efficient method for fast and accurate RCS calculation.

To this end, we propose a CM-SMWA based on an RCS calculation method, which combines the CM with the SMWA [28,29,30,31] to reduce the memory and computation time of the traditional CM method [26].

4.1. Single-Level CM-SMWA

The CM-SMWA of a single-level binary tree partition divides the RWG basis functions into two groups, namely, group 1 and group 2, and the corresponding matrix in Formula (10) becomes:

$$\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{21}& Z_{22}\end{array}\right]\left[\begin{array}{c}{I}_1\\ I_2\end{array}\right]=\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right]$$

(22)

where $Z_{11}$ and $Z_{22}$ represents the self-impedance matrices of the RWG basis functions in group 1 and group 2, respectively. $Z_{12}$ and $Z_{21}$ are the mutual-impedance matrices between the RWG basis functions of group 1 and group 2, which are low-rank matrices that can be compressed by the adaptive cross-approximation (ACA) algorithm [34,35]. $V_1$ and $V_2$ are the voltage vectors corresponding to group 1 and group 2. $I_1$ and $I_2$ are the current coefficients corresponding to the two groups.

According to Equation (19), the SMWA is introduced into the CM and divided into two small matrices to obtain:

$$\left[\begin{array}{cc}{J}_1^T& 0\\ 0& J_2^T\end{array}\right]\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{21}& Z_{22}\end{array}\right]\left[\begin{array}{cc}{J}_1& 0\\ 0& J_2\end{array}\right]\left[\begin{array}{c}{\mathsf{\alpha }}_1\\ {\mathsf{\alpha }}_2\end{array}\right]=\left[\begin{array}{cc}{J}_1^T& 0\\ 0& J_2^T\end{array}\right]\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right]$$

(23)

where $J_i$ represents the retained CM. $Z_{12}$ and $Z_{21}$ can be compressed by the ACA algorithm [34,35] as:

$$Z_{12}\approx A_{12}{B}_{12}, Z_{21}\approx A_{21}{B}_{21}$$

(24)

The effective rank of $Z_{12}$ and $Z_{21}$ is much smaller than the size of N/2. Substituting Equation (24) into Equation (23), the following equation can be obtained:

$$\left[\begin{array}{cc}{J}_1^T{Z}_{11}{J}_1& J_1^T{Z}_{12}{J}_2\\ J_2^T{Z}_{21}{J}_1& J_2^T{Z}_{22}{J}_2\end{array}\right]\left[\begin{array}{c}{\mathsf{\alpha }}_1\\ {\mathsf{\alpha }}_2\end{array}\right]=\left[\begin{array}{cc}{Z}_{11}^{CM}& J_1^T{A}_{12}{B}_{12}{J}_2\\ J_2^T{A}_{21}{B}_{21}{J}_1& Z_{22}^{CM}\end{array}\right]\left[\begin{array}{c}{\mathsf{\alpha }}_1\\ {\mathsf{\alpha }}_2\end{array}\right]=\left[\begin{array}{c}{J}_1^T{V}_1\\ J_2^T{V}_2\end{array}\right]$$

(25)

If $V_1^{CM}=J_1^T{V}_1$, $V_2^{CM}=J_2^T{V}_2$, $A_{12}^{CM}=J_1^T{A}_{12}$, $B_{12}^{CM}=B_{12}{J}_2$, $A_{21}^{CM}=J_2^T{A}_{21}$, $B_{21}^{CM}=B_{21}{J}_1$, the following equation can be obtained:

$$\left[\begin{array}{cc}{Z}_{11}^{CM}& A_{12}^{CM}{B}_{12}^{CM}\\ A_{21}^{CM}{B}_{21}^{CM}& Z_{22}^{CM}\end{array}\right]\left[\begin{array}{c}{\mathsf{\alpha }}_1\\ {\mathsf{\alpha }}_2\end{array}\right]=\left[\begin{array}{c}{V}_1^{CM}\\ V_2^{CM}\end{array}\right]$$

(26)

Due to the compression of the CM, $A_{ij}^{CM}$, $B_{ij}^{CM}$, and $Z_{ii}^{CM}$ are much smaller than $A_{ij}$, $B_{ij}$, and $Z_{ii}$. By adding the inverse of the near-field impedance matrix to the left and right of Equation (26), Equation (27) can be obtained:

$${\left[\begin{array}{cc}{Z}_{11}^{CM}& 0\\ 0& Z_{11}^{CM}\end{array}\right]}^{-1}\left[\begin{array}{cc}{Z}_{11}^{CM}& A_{12}^{CM}{B}_{12}^{CM}\\ A_{21}^{CM}{B}_{21}^{CM}& Z_{22}^{CM}\end{array}\right]\left[\begin{array}{c}{\mathsf{\alpha }}_1\\ {\mathsf{\alpha }}_2\end{array}\right]={\left[\begin{array}{cc}{Z}_{11}^{CM}& 0\\ 0& Z_{11}^{CM}\end{array}\right]}^{-1}\left[\begin{array}{c}{V}_1^{CM}\\ V_2^{CM}\end{array}\right]$$

(27)

After calculation, the following equation is obtained:

$$\left[\begin{array}{cc}1& {\hat{A}}_{12}^{CM}{B}_{12}^{CM}\\ {\hat{A}}_{21}^{CM}{B}_{21}^{CM}& 1\end{array}\right]\left[\begin{array}{c}{\mathsf{\alpha }}_1\\ {\mathsf{\alpha }}_2\end{array}\right]=\left[\begin{array}{c}{\hat{V}}_1^{CM}\\ {\hat{V}}_2^{CM}\end{array}\right]$$

(28)

where:

$${\hat{A}}_{12}^{CM}=Z_{11}^{CM-1}{A}_{12}^{CM},{\hat{V}}_1^{CM}=Z_{11}^{CM-1}{V}_1^{CM}$$

(29)

$${\hat{A}}_{21}^{CM}=Z_{22}^{CM-1}{A}_{21}^{CM},{\hat{V}}_2^{CM}=Z_{22}^{CM-1}{V}_2^{CM}$$

(30)

In Equation (28), 1 represents the identity matrix. Using the SMW Formula [36], the following equation can finally be obtained:

$${\left[\begin{array}{cc}1& {\hat{A}}_{12}^{CM}{B}_{12}^{CM}\\ {\hat{A}}_{21}^{CM}{B}_{21}^{CM}& 1\end{array}\right]}^{-1}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}0& {\hat{A}}_{12}^{CM}\\ {\hat{A}}_{21}^{CM}& 0\end{array}\right]{\left[\begin{array}{cc}1& B_{21}^{CM}{\hat{A}}_{12}^{CM}\\ B_{12}^{CM}{\hat{A}}_{21}^{CM}& 1\end{array}\right]}^{-1}\left[\begin{array}{cc}{B}_{21}^{CM}& 0\\ 0& B_{12}^{CM}\end{array}\right]$$

(31)

Putting Equation (31) into Equation (28), the following equation can be obtained:

$$\left[\begin{array}{c}{\mathsf{\alpha }}_1\\ {\mathsf{\alpha }}_2\end{array}\right]=\left[\begin{array}{c}{\hat{V}}_1^{CM}\\ {\hat{V}}_2^{CM}\end{array}\right]-\left[\begin{array}{cc}0& {\hat{A}}_{12}^{CM}\\ {\hat{A}}_{21}^{CM}& 0\end{array}\right]{\left[\begin{array}{cc}1& {\widehat{B}}_{21}^{CM}{A}_{12}^{CM}\\ {\widehat{B}}_{12}^{CM}{A}_{21}^{CM}& 1\end{array}\right]}^{-1}\left[\begin{array}{c}{B}_{21}^{CM}{\hat{V}}_1^{CM}\\ B_{12}^{CM}{\hat{V}}_2^{CM}\end{array}\right]$$

(32)

where $I_1\approx J_1{\mathsf{\alpha }}_1$ and $I_2\approx J_2{\mathsf{\alpha }}_2$ are the approximate current coefficients of the RWG basis functions.

4.2. Multi-Level CM-SMWA

Although the single-layer binary tree can reduce the computer storage space to a certain extent, the matrix is still large, and the effect of reducing the amount of calculation is limited. In order to reduce the amount of calculation as much as possible, it is necessary to further group the binary tree. For a multi-level algorithm (assuming L levels), the steps of the CM-SMWA algorithm are as follows:

Step 1: Generate the bottommost block diagonal matrix and find the CM:

$${\lambda }_{i,m},J_i^L=eig\left(imag(Z_i^L),real(Z_i^L)\right)$$

(33)

where $Z_i^L$ represents the bottommost near-field impedance matrix (a total of $2^L$), m represents the dimension of the bottommost matrix $Z_i^L$, eig ( ) represents the generalized eigenvalue solution, and real () and imag () represent the real and imaginary parts of the input, respectively. ${\lambda }_{i,m}$ are sorted from small to large, the eigenvectors corresponding to the $K_{\min }$ smallest eigenvalues in the eigenvalues ${\lambda }_{i,m}$ are retained, and the CM ${\widehat{J}}_i^{L^{}}$ is obtained. Finally, the $K_{\min }$ mode currents (eigenvectors) $J_i$ are retained, and the other eigenvectors are removed.

Step 2: Use the CM to compress the bottommost diagonal matrix:

$${{\rm Z}}_L^{CM}=J_L^T{Z}_L{J}_L=\left[\begin{array}{cccc}{J}_1^T{Z}_1^L{J}_1& & & \\ & J_2^T{Z}_2^L{J}_2& & \\ & & \ddots & \\ & & & J_{2^L}^T{Z}_{2^L}^L{J}_{2^L}\end{array}\right]$$

(34)

If $Z_i^{CM}=J_i^T{Z}_i^L{J}_i$, which represents the self-impedance matrix of the CM in the i-th block, the combination of all inverses can be expressed by the following equation:

$${{\rm Z}}_L^{C{M}^{-1}}=\left[\begin{array}{cccc}{Z}_1^{C{M}^{-1}}& & & \\ & Z_2^{C{M}^{-1}}& & \\ & & \ddots & \\ & & & Z_{2^L}^{C{M}^{-1}}\end{array}\right]$$

(35)

Step 3: The main diagonal matrix blocks of $Z_L^{C{M}^{-1}}{Z}^{CM}$ are taken in the L-1 th level to form the matrix $Z_{L-1}^{CM}$. Each diagonal block of $Z_{L-1}^{CM}$ is similar to the form of Equation (28), and so the inverse matrix $Z_{L-1}^{C{M}^{-1}}$ can be obtained quickly by using the single-level SMWA algorithm.

Step 4: All levels $Z_{L-2}^{CM}\sim Z_0^{CM}$ are found according to step 3, where $Z_{𝓁}^{CM}$ ($𝓁=L-2,\dots ,0$) is the block diagonal matrix of $Z_{𝓁+1}^{C{M}^{-1}}{Z}_{𝓁+2}^{C{M}^{-1}}\cdots Z_{L-1}^{C{M}^{-1}}{Z}_L^{C{M}^{-1}}{Z}^{CM}$ in the $𝓁$-th level. $Z_{𝓁}^{C{M}^{-1}}$ can be quickly obtained by the single-level SMWA algorithm.

Step 5: Using $Z_0^{C{M}^{-1}}{Z}_1^{C{M}^{-1}}\cdots Z_{L-2}^{C{M}^{-1}}{Z}_{L-1}^{C{M}^{-1}}{Z}_L^{C{M}^{-1}}{V}^{CM}$ to obtain $\mathsf{\alpha }={[{\alpha }_1^T,{\alpha }_2^T,\dots ,{\alpha }_{2^L}^T]}^T$. And then the current coefficient $I=J\mathsf{\alpha }$ is obtained.

As shown in Figure 4, a two-level grouping is used, and $Z^{CM}$ is the impedance matrix after the two-level grouping, which is transformed into the multiplication form of $Z_2^{CM}$, $Z_1^{CM}$, and $Z_0^{CM}$ by the SMWA, and the inverse matrix can be efficiently obtained by the SMW Formula [36].

5. Numerical Simulation Results and Analysis

5.1. Simulated Models

We use two examples to illustrate the advantages of the proposed CM-SMWA. One is the simulated simple cylinder, and the other is the simulated power lines. The cylinder is close to the shape characteristics of the power line, which can quickly verify the performance of the proposed method under simple conditions.

(1)

Simulated simple cylinder

Figure 5 shows a cylindrical model used in the experiment, with the diameter $D_c$ = 0.4 m and length $L_c$ = 20 m. In Figure 5, 60°, 90°, and 120°, respectively, represent the incident angles of the electromagnetic wave. The detailed parameters of the simulated cylinder are provided in

Table 1. Parameters of the simulated cylinder.

Model	${\mathit{L}}_{\mathit{c}}$	${\mathit{D}}_{\mathit{c}}$
Cylinder	20 m	0.4 m

.

(2)

Simulated power lines

In order to prevent force or gravity breakage, most power lines with a diameter greater than 10 mm use steel-cored aluminum stranded wire for long-distance transmission. Figure 6 shows a typical power line, LGJ50-8, which has a steel core and six outer aluminum stranded wires. The physical structure of the power lines is generally aluminum stranded or steel-cored aluminum stranded wire.

In Figure 6, $\rho$ represents the distance between two strands, P represents the winding cycle of a single strand, D represents the diameter of the stranded wire, and d represents the diameter of the outermost single aluminum stranded wire. The detailed parameters of the power line are listed in

Table 2. Parameters of the power line LGJ50-8.

Model	Steel Core	Outer Aluminum Strand/d	Diameter D	P	$\mathit{\rho }$
LGJ50-8	1	6/3.2 mm	9.55 mm	138 mm	23 mm

.

The Bragg scattering principle of power lines is shown in Figure 7. The echo strength point equation of the Bragg scattering angle can be obtained as follows:

$$\rho \sin {\theta }_n=n\frac{\lambda }{2},n=0,\pm 1,\pm 2,\dots$$

(36)

In Equation (36), ${\theta }_n$ represents the n-th scattering peak angle (n-th Bragg) and $\lambda$ represents the wavelength. The phenomenon of scattering of peaks at a specific incident angle ${\theta }_n$ is called Bragg scattering, and ${\theta }_n=0^{°}$ represents the main lobe perpendicular to the power lines, while other peak angles are called side lobes. Since the electromagnetic wave has phase information, when the received electromagnetic waves have the same phase, the superposition of wave peaks makes the reflection the strongest. Considering the stranded periodic structure of power lines, when the incident wave reaches the surface of the adjacent stranded wire and the wave path difference is an integer multiple of 1/2 a wavelength, the backscattered echoes will be superimposed in phase, that is, the incident angle satisfies the characteristics of Bragg scattering.

5.2. Evaluation Indicators

In this paper, we use the compute time, the memory consumption, and the simulation accuracy to measure the performance of the proposed RCS calculation method. In particular, we adopt the results of the MoM as the reference results and use the root mean square error (RMSE) as the objective function for calculating the simulation accuracy of our proposed method. The RMSE is as follows:

$$RMSE=\sqrt{{\displaystyle \sum _{n=1}^{N_{\theta }}\frac{{\left|\sigma (n)-{\sigma }_s^{MoM}(n)\right|}^2}{N_{\theta }}}}$$

(37)

where ${\sigma }_s^{MoM}$ represents the result of the reference MoM algorithm, the unit is dBsm, and $N_{\theta }$ represents the number of discrete angles.

5.3. Numerical Calculation Comparison

All examples were run on a computer with a processor frequency of 2.6 GHz and 32 GB memory. The error threshold of the ACA was set at 1 × 10⁻³. The angle $\theta$ = 90° was perpendicular to the long axis of the power lines for all simulations.

(1)

Simulated simple cylinder

Figure 8 shows that the CM-SMWA with an extension can obtain the RCS results with a similar accuracy to the MoM method. However, a large extension will lead to an increase in the amount of CM calculation. Thus, we set the extension to 0.1$\lambda$ in this paper.

The CM-SMWA is compared with the MoM and the CM method. The extensions of groups in the CM method and the CM-SMWA are set at 0.1$\lambda$.

Figure 9 shows the comparison of the simulation results of the different algorithms for the cylinder. It can be seen from Figure 9 that the detection results of the CM and CM-SMWA are similar to that of the MoM method. It shows that although the CM is only extended by 0.1, both the CM method and the CM-SMWA method have good accuracy for the cylinder. The difference from the MoM method is concentrated in the position of the minimum value, as shown in the subfigure in Figure 9, which shows that the difference between the CM-SMWA and the conventional MoM is small.

Table 3. Comparison of the calculation time and memory of the different algorithms for the cylinder.

Method	Total Time	RMSE	Z-Matrix Memory	CM Memory
MoM	31 s	--	1104 MB	--
CM method	31 s	0.234	19 MB	4.5 MB
CM-SMWA	28 s	0.230	8 MB	4.5 MB

presents the comparison results of the computing time and memory requirement of the different algorithms. The number of CMs for each group is set at 70, and so the required CM occupies the same memory for both the CM method and CM-SMWA, which was 4.5 MB. As shown in Table 3, compared with the MoM and CM method, the CM-SMWA reduces the memory greatly and has advantages in computing time. Because the target is relatively small, the CM-SMWA saves little time.

(2)

Simulated power lines

In the millimeter waveband, the frequency bands of 35 GHz, 76 GHz, and 94 GHz have the smallest electromagnetic wave propagation loss, and so the power line RCS is mainly concentrated around this band for research.

The LGJ50-8 power line was selected for the numerical simulation. The specific parameters are shown in Table 2. The length of the power line model is selected as two times P, that is, 2 × P = 0.276 m, which is equivalent to two stranded cycles of the power lines in Table 2. The power line model is discretized by the RWG basis function with an average side length of 0.1$\lambda$, and a total of 34,848 RWG basis functions are used. The six level binary tree is used, the power lines are divided into 64 segments, and the number of CMs in each group is fixed at 300. The extension for generating CMs in the CM method and the CM-SMWA is 0.1$\lambda$. The simulation results of the power line RCS with the different algorithms at 35 GHz horizontal–horizontal (H-H) polarization are shown in Figure 10.

From the comparison in Figure 10, it can be seen that the simulation results of the latter two methods are close to the MoM, indicating that the three methods can effectively simulate the power lines. As shown in the subfigure in Figure 10, the positions with relatively large relative errors displayed in the form of dB are also concentrated in the low RCS part and mainly concentrated in the positions below −40 dB, and the absolute error of the RCS is not large.

According to related research [37], the Bragg scattering strengths of different polarization power lines are different. In order to further confirm the characteristics of power lines with different polarizations, V-V polarization simulations are simultaneously carried out in this paper. The extension of each segment for generating CMs is set at 0.1$\lambda$. The V-V polarization simulation is as seen in Figure 10.

As can be seen from Figure 11, which shows the simulation results of the power line V-V polarization RCS with the different algorithms at 35 GHz, the method proposed in this paper can accurately simulate V-V polarization. The comparison results are shown in

Table 4. Comparison of the CPU time and memory requirement of the different algorithms at 35 GHz.

Methods	Total Time	RMSE (H-H)	RMSE (V-V)	Z-Matrix Memory	CM Memory
MoM	17.3 min	--	--	18,530 MB	--
CM method	26.4 min	0.47	0.25	5648 MB	160 MB
CM-SMWA	6.5 min	0.59	0.24	505 MB	160 MB

. As can be seen in Table 4, the calculation time of the CM-SMWA is much less than the other two methods. In terms of memory consumption, the CM-SMWA has great advantages over the other two methods. In terms of accuracy, the CM method is slightly different from the CM-SMWA method, but it is very close. It can be seen that the accuracy of the simulation of the V-V polarization and H-H polarization RCS using our method can be close to that of the MoM method.

Figure 12 compares the simulation results of the V-V polarization and the H-H polarization at 35 GHz. It can be seen from the comparison that the V-V polarization is about 10 dB stronger than the RCS of the Bragg sidelobes (n = ±1) of the H-H polarization. The vertical angle (incident angle $\theta$ = 90° echo) intensity of the power lines is almost the same. Therefore, for this power line, the detection effect of the V-V polarized microwave radar is obviously better at 35 GHz.

For the detection radar of millimeter wave power lines, the selectable band is Ka~W, and 76 GHz can be used as a low-cost or on-chip radar [38], and so it also includes the mainstream research 76 GHz. Due to the shorter wavelength of this band, more basis functions need to be established for the simulation. This makes the calculation and storage of the MoM matrix extremely large, and it is difficult for conventional equipment to simulate. Therefore, the length of the power lines in Example 3 is halved, and the simulation is carried out under the condition of 76 GHz. The length of the power lines is selected as P (with P = 0.138 m), the target is discretized by the RWG basis function with an average side length of 0.1$\lambda$, and a total of 81,822 basis functions are used. The number of levels for the binary tree is L = 7, which means the power line is divided into 128 segments. The number of CMs for each segment is fixed at 300, and the extension of each segment for generating CMs is set at 0.1$\lambda$.

The simulation results of the H-H polarization are shown in Figure 13 and the results of the V-V polarization are shown in Figure 14. It can be seen that the angle with a large error is mainly concentrated at the minimum, and because in actual radar detection the echo of power lines is affected by ground clutter, this “minimum point” error is weaker than the ground clutter, and so these errors are acceptable for the power line simulation.

The comparison of the power line results calculated by the different algorithms at a 76 GHz simulation are shown in

Table 5. Comparison of the CPU time and memory requirement of the different algorithms at 76 GHz.

Methods	Total Time	Z-Matrix Memory	CM Memory
MoM	224 min	99.8 GB	--
CM	171 min	22.6 GB	375 MB
CM-SMWA	41 min	2.1 GB	375 MB

. Our method has obvious advantages in terms of CPU time and storage. Due to excessive memory usage, the total time of the MoM method in Table 5 is the estimated time.

Figure 15 compares the RCS of the H-H polarization and V-V polarization power lines under the condition of 76 GHz. According to the RCS comparison results, the vertical angle (incident angle $\theta$ = 90°) echo of the H-H polarization is more than 10 dB stronger than the V-V polarization. For the Bragg sidelobes with d = 3.2 mm, the V-V polarization is stronger than the H-H polarization, and so it is basically consistent with the conclusion obtained in [37], and it also proves that the power line fast simulation method proposed in this paper can effectively evaluate the power line RCS.

Figure 16 shows the comparison of the simulation results of the H-H polarization and V-V polarization. It can be seen that the V-V polarization of the power line RCS in this paper is still stronger than the H-H polarization at 94 GHz. While the Bragg first sidelobes (n = ±1) have the same levels, the Bragg second and third sidelobe (n = ± 2, ± 3) V-V polarizations are stronger. Therefore, due to higher RCS levels, it is easier to detect V-V polarization power lines at 94 GHz.

According to the power line results calculated by the CM-SMWA at 94 GHz (shown in

Table 6. Comparison of the CPU time and memory requirement of the different algorithms at 94 GHz.

Method	Total Time	Z-Matrix Memory	CM Memory
CM-SMWA	76 min	4.8 GB	524 MB

), the calculation time and memory of the CM-SMWA are still acceptable (116,820 basis functions), indicating that the method in this paper meets the calculation requirements of different frequencies.

6. Conclusions

In this paper, an efficient method for simulating the RCS of power lines, which combines the CM with SMWA, is proposed. The simulation results show that the proposed method has advantages over the conventional MoM and CM methods in both the memory requirement and calculation time of RCS calculations. At 35 GHz, compared with the conventional MoM, for a typical LGJ50-8 power line with a length of 0.276 m, the computation time is reduced by 62.4% and the memory occupation is reduced by 96.4%. The proposed method can quickly complete the RCS simulation with limited computing resources, and it can be applied to simulate numerous RCS samples of power lines. In the future, we will combine the power line echo with the radar echo background to generate radar echo samples and train deep learning algorithms to make the algorithms more effective in detecting power lines. At the same time, we will study how to further improve the computational efficiency of the proposed CM-SMWA algorithm by utilizing the periodicity of the power lines.