## 3 Theory Analysis of Pattern Reconfigurations
\begin{array}{l}M=S_{11}-S_{11}\Gamma_{L2}S_{22}-S_{11}\Gamma_{L3}S_{33} \\ +S_{11}\Gamma_{L2}S_{22}\Gamma_{L3}S_{33}+S_{12}^{2}\Gamma_{L2}-S_{12}^{2} \Gamma_{L2}\Gamma_{L3}S_{33}\\ +S_{13}^{2}\Gamma_{L3}-S_{13}^{2}\Gamma_{L2}\Gamma_{L3}S_{22}\\ \\ N=E_{1}(\theta,\varphi)(1-\Gamma_{L2}S_{22}-\Gamma_{L3}S_{33}+\Gamma_{L3}S_{33} \Gamma_{L2}S_{22})\\ +E_{2}(\theta,\varphi)S_{12}\Gamma_{L2}(1-\Gamma_{L3}S_{33})\\ +E_{3}(\theta,\varphi)S_{13}\Gamma_{L3}(1-\Gamma_{L2}S_{22})\\ \\ P=1+S_{11}\Gamma_{L1}[\Gamma_{L3}S_{33}+\Gamma_{L2}S_{22}-\Gamma_{L2}S_{22} \Gamma_{L3}S_{33}]+\Gamma_{L2}S_{22}\Gamma_{L3}S_{33}\\ -\Gamma_{L2}S_{12}^{2}\Gamma_{L1}-\Gamma_{L3}S_{13}^{2}\Gamma_{L1}-S_{12}^{2} \Gamma_{L2}\Gamma_{L3}S_{13}\Gamma_{L1}\\ -\Gamma_{L2}S_{22}-\Gamma_{L3}S_{33}+S_{13}^{2}\Gamma_{L3}\Gamma_{L1}\Gamma_{L 2}S_{22}+S_{12}^{2}\Gamma_{L2}\Gamma_{L1}\Gamma_{L3}S_{33}\end{array} \right.. \tag{7}\]  
The \(\Gamma_{i}\) (\(i=1,2,3\)) is the reflection coefficient of each port, which is given by  
\[\Gamma_{i}=\frac{Z_{L,i}-Z_{0,i}}{Z_{L,i}+Z_{0,i}} \tag{8}\]  
where \(Z_{L,i}\) (\(i=1,2,3\)) is the load impedance of each port, and \(Z_{0,i}\) (\(i=1,2,3\)) is the characteristic impedance of each port. In this paper, the impedance of Port 1 is matched, i.e., \(50\)\(\Omega\). Therefore, \(\Gamma_{1}=0\). \(Z_{L,2}\) and \(Z_{L,3}\) are \(Z_{2}\) and \(Z_{3}\), as shown in Figure 3a, respectively. The characteristic impedance of Port 2 or Port 3 is equal to \(87.5\)\(\Omega\). The simulation results show that the input impedance of Port 2 or Port 3 is always inductive and the real part of the input impedance is so small that it can been neglected. The return loss and the gain at \(\phi=90^{\circ}\), \(\theta=-20^{\circ}\) of the proposed pattern reconfigurable antenna are shown in the contour maps (Figure 4a,b, respectively). In Figure 4, the horizontal axis and the vertical axis are the reactance value of load \(Z_{2}\) and load \(Z_{3}\). In order to alleviate the contradiction between the return loss and the gain, the two points in Figure 4 are selected to reconfigure the radiation pattern of the proposed antenna. In these two points, the return loss of the proposed antenna is approximately \(10\) dB, and the gain reaches approximately \(5\) dB. The value of load \(Z_{2}\) and \(Z_{3}\) can be reconfigured among Point 1 and Point 2 shown in Figure 4 by changing the operating modes. Each RF MEMS switch has two states, i.e., an up state (0) and a down state (1). The input impedance of the Part 2/Part 3 is \((0.0808+j1.1140)\)\(\times\)\(87.5\)\(\Omega\)\(\approx\)\((7+j98)\)\(\Omega\) in the \"0\" state, and \((0.05+j0.3503)\)\(\times\)\(87.5\)\(\Omega\)\(\approx\)\((4+j31)\)\(\Omega\) in the \"1\" state according to the simulated results, the value of the \(87.5\)\(\Omega\) is the characteristic impedance of the port. Thus, the load conditions of the driven patch have four kinds of combination, as shown in Table 4. Fortunately, if the real part of the  
Figure 4: The return loss and gain distribution versus \(Z_{2}\) and \(Z_{3}\). (**a**) Return loss. (**b**) Gain at \(\phi=90^{\circ}\), \(\theta=-20^{\circ}\) (The horizontal axis and the vertical axis are the reactance value of load \(Z_{2}\) and load \(Z_{3}\)). input impedance (in RF MEMS switch \"0\" or \"1\" states) is neglected, the reactance value of the two points in Figure 4 will be reached by changing the RF MEMS switch operating modes among (1,0) and (0,1), i.e., Point 1 corresponding to RF MEMS switch state (1,0) and Point 2 corresponding to (0,1). Therefore, the main lobe direction of the proposed pattern reconfigurable antenna can be steered by changing the RF MEMS switch operating modes. The simulated gain results of the proposed RF MEMS pattern reconfigurable antenna are shown in Figure 5.