## 3 Theory Analysis of Pattern Reconfigurations  
An antenna array can be fully described by its scattering parameters and the active element pattern of each radiating element [16]. The proposed pattern reconfigurable antenna in this paper can be viewed as the degenerated antenna array shown in Figure 3a. (The whole structure of the reconfigurable antenna can be divided into three parts shown in Figure 3a, i.e., the left part, middle part, and right part. These three parts constitute an antenna array with three elements. However, the three elements in this array are not identical; both the left and right parts did not equip a separate microstrip feed line. Therefore, the reconfigurable antenna is equivalent to a degenerate antenna array.) Thus, the active element pattern, the scattering parameters, and the signal flow diagram method can be employed to analyze the proposed antenna appropriately. The active element pattern of an element is defined as its radiation pattern when all other elements are terminated in matched loads [16]. In this paper, a method combining an active element pattern and a signal flow diagram [17], is employed to analyze the pattern reconfigurations of the antenna-based RF MEMS switches. The proposed antenna can be divided into three parts. As shown in Figure 3a, the driven patch and the feed line comprise Part 1, and Part 2 (namely load \(Z_{2}\)) consists of an assistant patch, a RF MEMS switch, parasitic patch 1, and parasitic patch 2, and the composition of Part 3 (namely load \(Z_{3}\)) is the same as Part 2. The parasitic patch in Part 2 and Part 3 are used to slightly tune the frequency of the overall antenna structure and extend the operating bandwidth. Without any loss of analysis precision, the parasitic patches are involved in loads \(Z_{2}\) and \(Z_{3}\), as shown in Figure 3. Moreover, the port associated with each part is marked in Figure 3a as well. The proposed antenna can be modeled as an equivalent scattering parameter matrix \(S_{D}\), which consists of a regular scattering parameter matrix \(S_{r}\) and the electric field intensity matrix \(E\) of active element pattern. The equivalent scattering parameter matrix \(S_{D}\) is defined as  
\[S_{D}=\left( \begin{array}{cc}S_{r}&E^{T}\\ E&0\end{array} \right) \tag{2}\]  
and the electric field intensity matrix \(E\) is  
\[E=\left( \begin{array}{cc}\mathbf{E}_{1}(\theta,\varphi)&\mathbf{E}_{2}( \theta,\varphi)&\mathbf{E}_{3}(\theta,\varphi)\end{array} \right) \tag{3}\]  
where \(E_{i}(\theta,\varphi)\) is the electric field vector of each active element pattern, and \(E^{T}\) is the transpose matrix of \(E\). As shown in Figure 3, the driven patch (part 1) is terminated by load \(Z_{2}\) and load \(Z_{3}\). The regular scattering matrix \(S_{r}\) includes three ports, and \(S_{32}\) and \(S_{23}\) are all approximately equal to zero because the couplings between load \(Z_{2}\) and load \(Z_{3}\) can be neglected. According to the symmetry of the designed antenna overall structure, \(S_{13}=S_{12}\) and \(S_{31}=S_{21}\), respectively. Thus, \(S_{r}\) is defined as  
\[S_{r}=\left( \begin{array}{ccc}S_{11}&S_{12}&S_{12}\\ S_{21}&S_{22}&0\\ S_{21}&0&S_{33}\end{array} \right). \tag{4}\]  
The signal flow diagram of the proposed antenna is shown in Figure 3b, all of those parameters are extracted from a full wave simulation. The \(\Gamma_{i}\) is the reflection coefficient of Port No. \(i\). Using the active element pattern and Mason rules, the return loss \(RL\) and gain \(G(\theta,\varphi)\) of the proposed antenna are calculated as follows:  
\[RL=20\log\left(\frac{M}{P}\right) \tag{5}\]  
\[G(\theta,\varphi)=20\log\frac{N}{P\sqrt{\sum\limits_{n=1}^{3}\left|I_{n}e^{i \varphi_{n}}\right|^{2}}} \tag{6}\]  
Figure 3: Divided parts of the antenna and its signal flow diagram. (**a**) Divided parts of the antenna. (**b**) Equivalent signal flow diagram of the proposed pattern reconfigurable antenna. where \(I_{n}e^{i\varphi_{n}t}\)is the signal current applied to Ports \(n,M,N,\) and \(P\), which are defined as follows:  
\[\left\{