## 2 Proposed Array Antenna  
Figure 1 shows our design approach and geometry of the proposed array antenna with two radiators consisting of a patch and a monopole wire, where the two radiators share a common square ground plate with a width of \(l_{g}\). The first array element, the patch radiator, is designed to form a rectangular ring shape with inner and outer widths of \(w\) and \(l\), which is printed on a thick FR-4 cuboid substrate (\(\varepsilon_{r}\) = 4.5, tan\(\delta\) = 0.018) with a thickness of \(t\). Another array element, the monopole wire antenna of a length of \(h\), is located in the center of the ground plate, and the two ports of the array elements fed by SMA connectors have an extremely narrow distance \(d\) from each other. This proposed array geometry can control the mutual coupling between two elements by adjusting the antenna parameters \(t\) and \(w\), while maintaining good matching characteristics for both elements. Notice that in terms of the antenna geometry, this proposed array is different from the previous research [32; 33; 34; 35; 36; 37; 38; 39] due to its distinct feeding mechanism and radiating modes. To observe the electromagnetic coupling between the two elements according to the tuning parameters of \(t\) and \(w\), we design an equivalent circuit model using a two-port network for the proposed array, as illustrated in Figure 2. In this circuit model, two ports of 50-\(\Omega\) impedance are connected directly to both elements, which are fed via pins represented by inductances of \(L_{\eta 1}\) and \(L_{\eta 2}\). For the elements, the monopole is expressed by a series circuit of \(R_{m}\), \(L_{m}\), and \(C_{m}\), and the patch is composed of the parallel lumped elements of \(R_{p}\), \(L_{p}\), and \(C_{p}\). The monopole and patch circuits are coupled through the series coupling capacitance \(C_{c}\) and the inductive coupling coefficient \(k\). The mutual couplings of \(C_{C}\) and \(k\) can be adjusted by the array geometry parameters of \(h\) and \(w\). Detailed values of the lumped elements are \(L_{\eta 1}\) = 13.8 nH, \(L_{\eta 2}\) = 0.36 nH, \(L_{m}\) = 6.6 nH, \(L_{p}\) = 0.38 nH, \(R_{m}\) = 5.1 \(\Omega\), \(R_{p}\) = 11 \(\Omega\), \(C_{m}\) = 30.8 pF, \(C_{p}\) = 16.9 pF, \(C_{c}\) = 2.1 pF, and \(k\) = 0.28. Figure 3 presents a comparison of mutual coupling effects of the proposed arrays according to the electric and magnetic coupling strengths. The solid line indicates the simulation result of the mutual coupling for the proposed array, and the dashed line is calculated from the designed circuit model using both electric and magnetic couplings. The mutual couplings after removing either the inductive coupling coefficient \(k\) or the capacitive coupling \(C_{c}\) are specified by dotted and dash-dotted lines, respectively. For a more precise observation of the electromagnetic coupling effects, we calculate Equation (1) of the mutual couplings at the target frequency (1.6 GHz) in accordance with the variations of \(C_{c}\) and \(k\) from the circuit model, and the result is shown in Figure 4:  
\[S_{12}(C_{c},k)\ =\ \frac{(-1.5+\beta.8)\cdot 10^{11}C_{c}+(-1.2-\beta.4)k+C_{c}k^{2}(2.0-\beta.9)\cdot 10^{11}}{((-3.7+\beta.2)\cdot 10^{12}C_{c}+ (-7.6+\beta)\cdot 10^{11}C_{c}k+(2.5+\beta.9)\cdot 10^{12}C_{c}k^{2}+(-0.1-\beta.4)k^{2}+8.6-\beta.2)}. \tag{1}\]  
Figure 1: Design approach and geometry of the proposed array antenna: (**a**) design approach of the proposed array antenna; (**b**) side view of the proposed array antenna. Figure 2: An equivalent circuit model with a two-port network for the proposed array. In addition, we examine the relationships among the coupling elements and the antenna geometry. Figure 5a,b illustrate the coupling capacitance \(C_{c}\) and coefficient \(k\), respectively, according to the adjustment of the width \(w\) for the inner patch and the thickness \(t\) for the thick substrate. These two parameters can easily control the coupling capacitance from 0.4 pF to 2 pF and the coupling coefficient from 0.35 to 0.85.