An Improvement of ClosedForm Formula for Mutual Impedance Computation
 Author: Son TrinhVan, Hwang Keum Cheol, Park JoonYoung, Kim SeonJoo, Shin JaeHo
 Organization: Son TrinhVan; Hwang Keum Cheol; Park JoonYoung; Kim SeonJoo; Shin JaeHo
 Publish: Journal of electromagnetic engineering and science Volume 13, Issue4, p240~244, 31 Dec 2013

ABSTRACT
In this paper, we present an improvement of a closedform formula for mutual impedance computation. Depending on the centertocenter spacing between two rectangular microstrip patch antennas, the mutual impedance formula is separated into two parts. The formula based on synthetic asymptote and variable separation is utilized for spacings of more than 0.5
λ _{0}. When the spacing is less than 0.5λ _{0}, an approximate formula is proposed to improve the computation for closely spaced elements. Simulation results are compared to computational results of mutual impedances and mutual coupling coefficients as functions of normalized centertocenter spacing in both E and Hplane coupling configurations. A good agreement between simulation and computation is achieved.

KEYWORD
Array Antenna , ClosedForm Formula , Microstrip Patch , Mutual Coupling , Mutual Impedance.

Ⅰ. INTRODUCTION
The design of a finite array requires an accurate determination of the mutual impedance between two elements and the mutual impedance matrix of whole array. An accurate approach using the moment method has been proposed for mutual impedance computation [1]. However, the moment method requires that each element be segmented into many basis functions; therefore, this method becomes tedious and time consuming as the number of elements in array increases. Several methods have been proposed that deal with the mutual impedance computation based on simplified models such as the transmission line model [2], and the magnetic current approximation [3]. These methods are much faster but may be inaccurate. Recently, a closedform mutual impedance formula has been proposed, which is based on synthetic asymptote and variable separation [46]. In this formula, only 12 unknown coefficients are determined by matching with the simulated data or measured data. Therefore, this method is very fast and accurate due to its use of a synthetic asymptote form of the separated variables of the centertocenter spacing and the azimuth angle between two elements. However, when the centertocenter spacing is less than 0.5
λ _{0} (λ _{0} is the freespace wavelength), the computational result for very closely spaced elements is incorrect if this formula is used.In this paper, we propose a method to improve the closed form formula for the mutual impedance computation between two very closely spaced elements. The mutual impedance formula is separated into two parts depending on the centertocenter spacing between two elements. When the spacing is more than 0.5
λ _{0}, the mutual impedance is computed by utilizing the synthetic asymptote formula [4]. An approximate formula is proposed when the spacing is less than 0.5λ _{0}. The simulated and computational results agree well in terms of mutual impedances and mutual coupling coefficients between two closely spaced microstrip patches in Eplane and Hplane.Ⅱ. FORMULATIONS
Fig. 1 shows the geometry of two coupled microstrip patch antennas. These antennas are designed to operate at 5 GHz on the dielectric substrate with
ε_{r} =2.55 and thicknessh =1.57 mm. The dimension of patch is determined asW×L =22.6 mm×17.52 mm. The feed point is located at the center ofW with the distancea =5mm. In this work, we only focus on the mutual coupling between two patches in E and Hplane coupling configurations. The mutual impedance formula is separated into two parts corresponding to the spacingr less than 0.5λ _{0} and more than 0.5λ _{0}.1. The Mutual Impedance Formula Based on the Synthetic Asymptote and Variable Separation
When the centertocenter spacing
r between two patches is more than 0.5λ _{0}, the closedform mutual impedance formula based on the synthetic asymptote and variable separation is utilized. In this method, the mutual impedance can be written as a function of spacingr and the azimuth angleφ , as shown in Fig. 2(a). The use of a synthetic asymptote form of separated variables of spacingr and angleφ , gives the following as the mutual impedance between the two elements [4]where
η _{0} is the intrinsis impedance of free space,k _{0} is the free space wave number, and the unknown complex coefficientsC_{n,m} (n = 1/2, 0, 1, 2 andm =0, 2, 4) must bedetermined. These 12 coefficients can be found by matching with the simulated results of mutual impedance between center patch “0” and 12 coupled patches in a skeleton array, as shown in Fig. 2(b). The 12 models consisting of the center patch “0” and each of 12 coupled patches with the respective spacing set in skeleton array are simulated to obtain the mutual impedances at resonant frequency. From 12 values of the simulated mutual impedances, the Eq. (1) can be used to establish a set of 12 independent equations to be solved for the 12 coefficientsC_{n,m} by matrix inversion.2. The Approximate Mutual Impedance Formula for Closely Spaced Elements
As mentioned above, when the spacing
r is less than 0.5λ _{0}, the computation of mutual impedance for closely spaced elements obtained using Eq. (1) is incorrect. Therefore, we propose an approximate formula to improve the computation for two very closely spaced elements. It is worth noting that the mutual impedance between two elements in E or Hplane coupling configuration only depends on the spacingr . Several sampling values of the spacingr are chosen and the simulated results of mutual impedances are obtained through simulation. By fitting some curves to the simulated data, the mutual impedance formulas for E and Hplane can be expressed asfor Eplane coupling configuration, and
for Hplane coupling configuration. By combining Eq. (2) or Eq. (3) with Eq. (1), the mutual impedance between two elements in the E and Hplane, respectively, can be accurately computed even through the centertocenter spacing
r is less than 0.5λ _{0}.Ⅲ. RESULTS AND DISCUSSION
Fig. 3 plots the simulated reflection coefficients versus frequency of two microstrip patch antennas at the centertocenter spacing of 0.5
λ _{0}, . The simulation was conducted by using Ansys HighFrequency Structure Simulator (HFSS) based on the threedimensional finite element method. The two patches operate at the same resonant frequency of 5 GHz.First, the 12 unknown coefficients
C_{n,m} in Eq. (1) must be determined by matching with the simulated data. The 12 coupled patches on the skeleton array are arranged with the fixed sampling points as shown in Fig. 2(b). The 12 coefficientsC_{n,m} are computed and listed in Table 1.Once the 12 coefficients
C_{n,m} of Eq. (1) have been obtained, the mutual impedance between the two patch antennas is calculated by combining Eq. (1) with Eq. (2) for the Eplane coupling configuration or with Eq. (3) for the Hplane coupling configuration. Figs. 4 and 5 show the mutual impedances versus normalized centertocenter spacingr between two patches in the E and Hplane from simulation, and from computation using synthetic asymptote formula, and our proposed formula. Clearly, when the spacingr is more than 0.5λ _{0}, the computation results agree well with the simulation results. When the spacingr is less than 0.5λ _{0}, the results of mutual impedance using the synthetic asymptote formula are very different compared to the simulation results, especially the imaginary part of the mutual impedance. However, the use of our proposed formula to enhance the computation with the closely spaced elements achieves a good agreement between the simulation and computation.Overlapping is avoided by choosing the minimum values of centertocenter spacing
r as 0.3λ _{0} and 0.38λ _{0} for the E and Hplane couping configurations, respectively. Table 2 shows the comparison between the simulation and computation results of the mutual impedance corresponding to the minimum spacingr . Good agreement between our formula and the simulation is observed.The mutual coupling
S_{ab} between two patches expressed in decibels can be defined as [7]where
Z_{aa} is the selfimpedance of the patch,Z_{ab} is the mutual impedance between the two patches, andZ _{0} is the feed line impedance. We typically assumeZ_{aa} =Z _{0} =50. Figs. 6 and 7 show the results of the mutual coupling versus normalized spacingr in the Eplane and Hplane coupled configurations, respectively. Clearly, the computation results using our formula and asymptote formula agree well with the simulation results, except for the greater difference in the Eplane coupling configuration from the asymptote formula when the spacingr is less than the halfwavelength.Ⅳ. CONCLUSION
An improvement in the closedform formula for mutual impedance computation between two very closely spaced microstrip antennas in E and Hplane has been presented. A good agreement was achieved between the simulation and computation. The computational results show that the proposed approach is feasible for application to the design of linear microstrip patch arrays with arbitrary element spacing.

[Fig. 1.] Geometry of two coupled rectangular microstrip patch antennas.

[Fig. 2.] (a) Coordinates of two coupled patches and (b) skeleton array configuration.

[Fig. 3.] Simulated reflection coefficients of two microstrip patch antennas.

[Table 1.] The 12 complex coefficients Cn,m

[Table 2.] Comparison of mutual impedance with the minimum spacing

[Fig. 4.] Mutual impedance versus normalized centerto