# Design and Analysis of Pattern Reconfigurable Antenna Based on RF MEMS Switches
## 3 Theory Analysis  
Calculating the current density distribution on the patch and then determining the antenna's radiation direction using the directivity function is the conventional approach to determining the radiation direction of a microstrip patch antenna. However, it is difficult to obtain the current density at each point on the patch and, even if the current density is obtained, it has to be integrated to obtain the radiation pattern, which makes this method very difficult to use to calculate the radiation direction. The directivity function can be expressed as follows [19]. \[P(\theta,\varphi)=\int\mathbf{J}_{s}(x^{\prime},y^{\prime},z^{\prime})e^{|j \hat{k}_{0}(x^{\prime}\sin\theta\cos\varphi+y^{\prime}\sin\theta\sin\varphi+z^ {\prime}\cos\theta)|}\cdot dv^{\prime}, \tag{5}\]  
where \(P(\theta,\varphi)\) represents the directivity function of the antenna, \((x^{\prime},\ y^{\prime},\ z^{\prime})\) represents the coordinates of the radiation source point, \(k_{0}\) represents the wave number of free space, and \(v^{\prime}\) represents the region where the current density \(\mathbf{J}_{s}\) exists. Since the above method finds it difficult to solve the radiation direction of the antenna, we proposed the electric field vector addition method to solve the radiation direction of the  
Figure 8: S\({}_{21}\) of the MEMS switch loaded with the fan-shaped microstrip line. (**a**) Insertion loss; (**b**) Isolation. Figure 7: Structure diagram of the MEMS switch. antenna. Figure 9 shows the electric field distribution of the antenna and the coordinate system used for the vector superposition method calculation (for the sake of simulation and observation, the MEMS switch was replaced by a metal patch here). As shown in Figure 9, the radiation characteristics of the antenna are mainly generated by the gap between the main patch and the sub-patch and the dielectric layer. Whether the MEMS switch on the left is in the up or down state, or the two MEMS switches are combined into other modes, similar results will be obtained. In order to facilitate the solution of the vector superposition method, a coordinate system was established, as shown in Figure 9.  
The vector superposition method was used to analyze the radiation characteristics of the antenna, where the far-field of the main patch, the sub-patch, and the parasitic patch can be analyzed by the equivalent magnetic current model [19]. In this model, the vector potential functions \(\mathbf{F}_{m}\), \(\mathbf{F}_{a}\), and \(\mathbf{F}_{b}\), generated by the equivalent magnetic current, represent the potential situation on the main patch, the sub-patch, and the parasitic patch, respectively. The vector potential functions \(\mathbf{F}_{m}\), \(\mathbf{F}_{a}\) and \(\mathbf{F}_{b}\) can be expressed by Equation (6) [8]. \[\left\{ \begin{array}{l}\mathbf{F}_{m}\propto-\frac{\varepsilon}{2}\frac{1}{| \mathbf{R}|}e^{-|\mathbf{k}_{0}|\mathbf{R}|}.\frac{\sin(k_{0}h\sin\theta\cdot\cos\varphi)} {k_{0}h\sin\theta\cdot\cos\varphi}.\frac{\sin\left(\frac{1}{2}k_{0}w_{m}\cos \theta\right)}{k_{0}\cos\theta}\\ \mathbf{F}_{a}\propto-\frac{\varepsilon}{2}\frac{1}{|\mathbf{R}-\mathbf{d}_{1}|}e^{-|\mathbf{ k}_{0}|\mathbf{R}-\mathbf{d}_{1}|}.\frac{\sin(k_{0}h\sin\theta\cdot\cos\varphi)}{k_{0}h \sin\theta\cdot\cos\varphi}.\frac{\sin\left(\frac{1}{2}k_{0}w_{m}\cos\theta \right)}{k_{0}\cos\theta}\\ \mathbf{F}_{b}\propto-\frac{\varepsilon}{2}\frac{1}{|\mathbf{R}-\mathbf{d}_{2}|}e^{-i(k_ {0}+\pi)|\mathbf{R}-\mathbf{d}_{2}|}.\frac{\sin(k_{0}h\sin\theta\cdot\cos\varphi)}{k_{ 0}h\sin\theta\cdot\cos\varphi}.\end{array} \right.