An Improved Combination Method of MoM and UTD for Calculating the Radiation Characteristics of Antenna Arrays Mounted around Electrically Large Platform

Abstract

An improved method, namely, combined UTD with MoM, is studied to calculate the disturbed pattern of array antennas mounted around an electrically large platform. Each array unit is fed in turns in an MoM processor and then substituted into the UTD process to pursue the calculation of the disturbed pattern. When solving the current coefficient in MoM, the linear superposition principle is used to realize the separate feeding of each array unit. The proposed method can avoid unreliable UTD results when an equivalent phase center of the antenna is sheltered, and it can improve the RMS error from more than 15 dB to about 1 dB in the T-shaped example given. No large amount of computing resources are needed for the proposed method; the accuracy, efficiency and necessity are illustrated by examples given.

1. Introduction

In computational electromagnetics, the Uniform Geometrical Theory of Diffraction (UTD) method is a highly efficient method dealing with electrically large problems. UTD method finds its application in problems of electromagnetic scattering and radiating appearing in complex environments [1,2,3,4]. In the popular UTD method, analytical patches can be used to construct large-scale urban environments to analyze electromagnetic problems and parametric surfaces such as Non-Uniform Rational B-Spline (NURBS), and can be used to model arbitrary curved environments to calculate problems such as electromagnetic scattering.

Array antennas are widely used in wireless communication and modern radar system due to their high power, high gain and fast beam scanning performances. When the array antenna is mounted on electrically large platforms, its pattern will be disturbed. When using the UTD method to analyze the disturbed radiation of the array antenna, hybrid methods are usually required [5,6,7,8,9] since UTD cannot deal with calculation of current distribution on complex antennas. These hybrid methods possess good precision in calculation; however, iterative procedures are always needed to modify matrix equations, which will lead to great increase in computational resource demand and rapid decrease in computational efficiency as the electrical size of antenna and platform increases.

In order to improve the computational efficiency, some modified hybrid methods are presented with practical precision maintained [10,11]. In the UTD method, electromagnetic fields are supposed to be carried by various kinds of rays traveling from the source point to the observe point. Thus, in these methods, the electromagnetic fields can be approximated as radiated by an equivalent point source which is usually located at the phase center of the actual antenna. Obviously, compared to mall antennas with a regular radiating element such as dipole or monopole, the practical array antenna is always complex and its phase center is hard to find; therefore, it is not proper to describe the antenna with one equivalent point. Moreover, because the position of the antenna on the platform is changeable, there is a chance that the phase center is occluded so that some of the observation points may have no rays to reach, which results in serious distorted or false calculated distribution of fields.

In this paper, High Order Moment Method (HOMoM) [12,13,14,15,16] is used to simulate the antenna, and the quasihybrid scheme in [11] is used to calculate the disturbed pattern of antenna arrays mounted on electrically large platforms. In order to avoid the influence of the phase center as well as keeping coupling information between array units, the linear superposition principle is used to realize the separate feeding of the array unit in solving the current coefficient by the moment method, and then the final electromagnetic field is obtained by the sum of importing each feeding unit in turns as source points into the UTD process. The impedance matrix in MoM is not modified while the array units are separately fed, but the coupling information between array elements is preserved.

The accuracy and efficiency of the proposed method are illustrated by example of the disturbed pattern of a shipborne array antenna, and the necessity is illustrated by a typical T-shaped structure, in which the RMS error is improved from more than 15 dB to about 1 dB. The presented method can be applied to deal with the radiation problem of array antennas mounted around electrically large platforms with good efficiency.

2. Improved Quasihybrid Scheme

The quasihybrid scheme used in this paper can be described briefly in Figure 1 as follows:

In this scheme, the antenna is modeled and calculated by MoM, and the resulting radiation field is then imported to the UTD method for further calculation.

In the MoM process, radiation of the antenna is treated by converting the electromagnetic equation into a matrix equation as follows:

$$\left[Z\right]\left[I\right]=\left[V\right]$$

(1)

where $[V]$ is the excitation column vector and $[Z]$ is the impedance matrix of antenna units. When the current column vector $[I]$ is solved, the radiation of the antenna will be obtained.

Taking a dipole array antenna shown in Figure 2 as an example, Equation (1) can be written as:

$$\left[\begin{array}{ccccc}{Z}_{11}& \cdots & Z_{1i}& \cdots & Z_{1n}\\ ⋮& \cdots & ⋮& \cdots & ⋮\\ Z_{i1}& \cdots & Z_{ii}& \cdots & Z_{in}\\ ⋮& \cdots & ⋮& \cdots & ⋮\\ Z_{n1}& \cdots & Z_{ni}& \cdots & Z_{nn}\end{array}\right]\left[\begin{array}{c}{I}_1\\ ⋮\\ I_i\\ ⋮\\ I_n\end{array}\right]=\left[\begin{array}{c}{V}_1\\ ⋮\\ V_i\\ ⋮\\ V_n\end{array}\right]$$

(2)

where $V_i$ is the excitation of unit i, $Z_{ij}$ is the mutual impedance between unit i and unit j and $I_i$ is the current coefficient of unit i to be calculated.

In the UTD process, electromagnetic fields are supposed to be carried by various kinds of rays traveling from the source point to the observe point. These rays are approximated as transmitted from the equivalent point of the antenna array, which is usually located at the phase center or the geometrical center, as shown in Figure 3.

Obviously, compared to mall antennas with regular radiating element such as dipole or monopole, the practical array antenna is always complex, and its phase center or even geometrical center is hard to find; therefore, it is not proper to describe the antenna with one equivalent point.

Moreover, because the position of the antenna on the platform is changeable, there is a chance that the phase center is blocked so that some of the observation points may have no rays to reach, which results in a seriously distorted or falsely calculated distribution of fields. As can be seen in Figure 4, the path between the equivalent point and the observation point is blocked by the scatterer; thus, no field is transmitted. However, it can be seen clearly that the observation point is still in the area of the antenna aperture. A distorted result will be obtained under this condition.

To avoid the invalid calculation, the linear superposition principle is used to realize the separate feeding of each array unit in the MoM procedure.

Since the matrix equation in Equation (2) is a linear equation, if the excitation vector is split into the superposition of column vectors as in Equation (3), separate excitations for each unit can be achieved.

$$\left[\begin{array}{ccccc}{Z}_{11}& \cdots & Z_{1i}& \cdots & Z_{1n}\\ ⋮& \cdots & ⋮& \cdots & ⋮\\ Z_{i1}& \cdots & Z_{ii}& \cdots & Z_{in}\\ ⋮& \cdots & ⋮& \cdots & ⋮\\ Z_{n1}& \cdots & Z_{ni}& \cdots & Z_{nn}\end{array}\right]\left(\left[I_1\right]+\cdots +\left[I_i\right]+\cdots \left[I_n\right]\right)=\left[\begin{array}{c}{V}_1\\ ⋮\\ 0\\ ⋮\\ 0\end{array}\right]+\cdots +\left[\begin{array}{c}0\\ ⋮\\ V_i\\ ⋮\\ 0\end{array}\right]+\cdots +\left[\begin{array}{c}0\\ ⋮\\ 0\\ ⋮\\ V_n\end{array}\right]$$

(3)

where $\left[I_i\right]={\left[I_1^i\cdots I_i^i\cdots I_n^i\right]}^T$ is the current column vector of the array when only unit i is excited by $V_i$, and they obey that:

$$\left[\begin{array}{ccccc}{Z}_{11}& \cdots & Z_{1i}& \cdots & Z_{1n}\\ ⋮& \cdots & ⋮& \cdots & ⋮\\ Z_{i1}& \cdots & Z_{ii}& \cdots & Z_{in}\\ ⋮& \cdots & ⋮& \cdots & ⋮\\ Z_{n1}& \cdots & Z_{ni}& \cdots & Z_{nn}\end{array}\right]\left[\begin{array}{l}{I}_1^i\\ ⋮\\ I_i^i\\ ⋮\\ I_n^i\end{array}\right]=\left[\begin{array}{c}0\\ ⋮\\ V_i\\ ⋮\\ 0\end{array}\right]$$

(4)

$I_i$ is the current coefficient of unit $j$ when only the i-th antenna unit is excited by $V_i^{}$, and this unit is brought into the UTD calculation process as the transmitting point. Together with the field of the whole array calculated based on Equation (4), $E{r}_{MoM}^i,E{d}_{MoM}^i$, the ray field $E_{UTD}^i$ at the observation point is then calculated by UTD process, as illustrated in Figure 5.

After all n units are excited in turn and the calculation is completed, the final radiation field can be achieved by the sum of ray fields calculated by the UTD method of each unit, $E={\displaystyle \sum _{i=1}^n{E}_{UTD}^i}$.

There are two improvements in this calculation: 1. When only antenna element of unit i is excited, the coupling influence of other antenna elements is not ignored, and good calculation accuracy can be achieved while the calculation efficiency is preserved. 2. The radiation point of the antenna does not need to be equivalent to the phase center, and the calculation of the radiation field can still be effectively completed in the UTD procedure when the path between the phase center and the observation point is blocked.

3. Numerical Results

Practical examples are given in this section to illustrate the accuracy, efficiency and necessity of the proposed method.

Case 1. As shown in the Figure 6, an 8 × 8 dipole array antenna with the working frequency of 181.0 MHz is mounted on a ship structure. The ship is placed along the z-axis with a length of 150 m and a width of 16.5 m.

The disturbed pattern on the yoz plane and xoz plane are calculated and compared to those calculated by HOMoM, as shown in Figure 7.

The requirements for computing resources of MoM are taken into account in this example; a parallel strategy is set and used on a high performance platform. It is necessary to point out that to obtain the disturbed pattern, the presented method does not need any high-performance resources beyond a common personal computer with 8 GB RAM. The cost of computing is listed and compared in

Table 1. Comparison of computing cost.

	CPU Used	Memory Cost (GB)	Time Consumed (h)
HOMoM	576	918.1447	2.8687
Presented method	1	2.7246	1.2310

, in which the efficiency of the proposed method can be seen.

It can be seen from the results that the calculations are in good agreement. Additionally, it can be seen that the presented method can be applied to deal with the radiation problem of array antennas mounted around electrically large platforms with good efficiency.

Case 2. Shown in the Figure 8 is a T-shaped structure, which often appears on electrically large platforms such as aircraft tails or ship masts and is always located near antennas. A nine-dipole array antenna with the working frequency of 1.3 GHz is placed in front.

Firstly, the disturbed pattern on the xoz plane is calculated by the phase center-based quasihybrid method and compared to HOMoM, as shown in Figure 9. It can be seen clearly that serious distortion has appeared and been evaluated by RMS error; the numerical difference is as high as 16.9395 dB, which means that the result is no longer reliable. Such distortion is due to the fact that the phase center of the antenna is occluded at most observation points on the xoz plan. In the subsequent UTD calculation, the blocked phase center means the entire antenna is blocked, leading to a lack of field strength.

Such calculation distortion can be prevented by using the presented method in this paper. Because each unit of the antenna is fed and imported into the UTD method as source point in turn, not all the antenna array is blocked; thus, the lack of field strength is eliminated.

Figure 10 shows the disturbed pattern on the xoz plane calculated by the method presented in this paper.

It can be seen that the improved scheme can avoid the invalid calculation effectively and the RMS error is only 0.2758 dB in this example.

4. Conclusions

In this paper, an improved quasihybrid MoM–UTD method is used to calculate the disturbed pattern of an array antenna mounted around an electrically large platform. In the process of solving the current coefficient in MoM, the linear superposition principle is used to realize the separate feed of array units. Each feeding unit is then substituted in turns into the UTD process to pursue the calculation of the disturbed pattern. The presented method does not use one single equivalent source point to replace the antenna; therefore, the influences from the rest of the unfed unit can be considered simultaneously when a fed unit is used. Thus, good calculation accuracy can be maintained. Furthermore, the presented method can avoid the unreliable UTD results due to the shelter of one equivalent point, which can be seen in the example given where the RMS error from is improved from more than 15 dB to about 1 dB.