Moment method analysis of a broadband circular array antenna

0 Introduction
Among the numerous analysis methods for linear antennas and linear antenna arrays, the moment method has been widely used due to its high calculation accuracy, fast calculation speed, and strong applicability. Usually, when the moment method is used to analyze the mutual coupling between array elements, the array element is approximated as a thin wire model. At this time, the linear antenna current flows approximately along the axis of the vibrator. This analysis method can obtain relatively accurate calculation results when the array radius r<0.01λ. However, when the size of the antenna vibrator is relatively large, this method will obtain incorrect results.
In order to realize the mutual coupling analysis between thicker vibrators, a method that can accurately model and analyze thicker wire antennas is proposed on the basis of the moment method. This method uses the sine difference basis function and the point matching method, and uses the global wire antenna integral kernel for the calculation of the Pocklington equation to achieve accurate analysis of the antenna. In order to verify the correctness of the analysis method, the array antenna designed in this paper is simulated and tested respectively, and the expected results are achieved.

1 Moment method based on sine difference basis function and point matching method
In the calculation of the moment method of wire antenna, the selection of basis function has a great influence on the realization of fast antenna analysis. This paper adopts the following form of basis function:

In the formula, zn is the midpoint of the nth segment, △n is the length of the segment, An, Bn, Cn are three unknown quantities, one of which is used as the variable to be calculated, and the expressions of the other two variables can be obtained by using the current and charge continuity equations. Since this basis function can simulate the actual current of the antenna well, using this basis function can obtain faster convergence characteristics.
Figure 1 shows the moment method model of a thick wire antenna.

Substituting the current expressed in equation (1) into the Pocklington equation, the following formula is obtained using the point matching method
:

Where m=1, 2, 3, ..., N represents the number of segments; (ρm, zm) is the coordinate of the midpoint of the mth segment; is the unit vector of the current direction of the mth segment; w represents the angular frequency; μ is the spatial magnetic permeability; In(z') is the current expression on the mth segment; K(z-z') is usually called the integral kernel of the wire antenna, which is expressed as follows:

Substituting expression (1) into (2) and considering that the array element adopts a thicker wire antenna, it can be seen that the following three integral formulas will be involved in the process of formula expansion and arrangement:

When the field point is far away from the source point, the numerical method can be used to calculate formulas (4), (6), (7), and (9) to obtain accurate results, but when the field point is very close to the source point, the problem of singular point region integration must be dealt with.
For the problem of singular point processing, the literature provides a method for processing the problem of singular point region integration.
In the area near the singular point, the following processing is performed on formulas (4) and (7):

The integral of the second term in formula (10) changes relatively smoothly and can be calculated using the usual numerical integration method. The following transformation is performed on the first term.
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is the second kind of complete elliptic integral. The
first two terms in formula (15) can be calculated with the help of elliptic integrals. The calculation of the last term, by substituting into, we can see that its change is gentle, and its accurate result can be calculated using the common numerical integration method.

2 Antenna array design and test
We now design a broadband circular array antenna, which is a 6-element array (see Figure 2). According to the design requirements, the array element uses a thicker wire antenna, the array element radius is: a=0.035 m, the total length of the array element is: l=0.45 m, the array radius is: R=0.3 m, and the antenna operating frequency is 300 MHz~500 MHz.

Since the antenna works in a wide frequency band, for the convenience of verification and without loss of generality, this paper compares and analyzes the array antenna working at two frequencies, 300 MHz and 500 MHz.
2.1 Analysis of omnidirectional radiation characteristics when 6 array elements are fed with the same amplitude and phase
We designed and simulated the array antenna using the moment method based on the sine difference basis function and the point matching method, and CST software, and compared them with experimental tests. Its H-plane radiation pattern is shown in Figure 3:

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As can be seen from Figure 3(a), when the frequency f=300 MHz, the directional pattern obtained by the moment method numerical calculation is almost the same as the CST simulation directional pattern. The directional pattern obtained by the test fluctuates on the entire circumference, but its non-circularity is only about 0.2 dB. In Figure 3(b), when the operating frequency is 500 MHz, the three results are quite consistent. However, due to the increase in the electrical size of the array radius, the non-circularity of the directional pattern obtained by the moment method numerical calculation, CST simulation, and experimental test has deteriorated. Due to the influence of the manufacturing process of the antenna vibrator and the inconsistency of the channels of each array element, as well as the influence of the test environment, the measured directional pattern is slightly worse than the numerical calculation and software simulation, but its non-circularity is within an acceptable range in engineering.
2.2 Analysis of beamforming directional characteristics when 6 array elements are fed with the same amplitude and different phases The moment method
based on the sine difference basis function and the point matching method combined with the genetic algorithm performs beamforming on the antenna array at two frequencies of 300 MHz and 500 MHz, and the phase distribution of each port of the antenna array is obtained as shown in Table 1.

The antenna H-plane pattern is shown in Figure 4. As shown in Figure 4(a), when the frequency f=300 MHz, the numerical calculation results of the moment method are consistent with the CST simulation results. Compared with the experimental test results, the sidelobe direction is shifted and the tail lobe level is changed. In Figure 4(b), when the frequency f=500 MHz, compared with 4(a), the difference between the experimental test results and the moment method calculation results and the CST simulation results is larger, but from the key parameters affecting the antenna performance, such as main lobe direction, main beam width, main-side lobe level ratio, etc., the results of the three cases are not much different, and have little impact on the performance of the antenna.

3 Conclusions
In this paper, a moment method based on sine difference basis function and point matching method is proposed to solve the problems existing in the analysis of the mutual coupling characteristics of array antennas, and the global linear antenna integral kernel is used for the calculation of the Pocklington equation. This method is used to analyze and design the broadband circular array antenna, and the results are compared with the CST simulation results and experimental test results, achieving good consistency, thereby verifying the effectiveness of the analysis method.