# Design of a Compact Microstrip Decoupled Array
## 2 Materials and Methods
### Miniaturization Antenna  
The relationship between microstrip antenna size and frequency is inversely proportional (here, the equation does not take into account the edge shortening effect), where \(W\) and \(L\) are the antenna length and width dimensions. The designed microstrip antenna operates in TM\({}_{10}\) mode, and the antenna current path is shown in Figure 3. The equivalent current length of the slot is codirectional in the \(L\) direction and inverse in the \(W\) direction. The slot changes the direction of the current and lengthens the effective current path, thereby reducing the resonant frequency and miniaturizing the antenna. \[W=\frac{c}{2f}\big{(}\frac{\varepsilon_{r}+1}{2}\big{)}^{-\frac{1}{2}} \tag{1}\]  
\[L=\frac{c}{2f\sqrt{\varepsilon_{\epsilon}}} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Para** & **Ws** & **d** & **W** & **v** & **v1** & **R** & **k1** & **k2** & **u** \\ \hline Value & 78 & 32 & 23.4 & 17.1 & 16.5 & 3.8 & 6.1 & 8.4 & 0.3 \\ \hline Para & P & L1 & L2 & W1 & W1s & R & R1 & Wm & H \\ \hline Value & 11.2 & 23.15 & 11 & 2 & 1.5 & 4.3 & 11.1 & 0.3 & 1.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Antenna parameters (unit: mm). Figure 2: The designed mono-pulse antenna. In order to ensure that the shape of the antenna radiation pattern is not distorted, it is necessary to ensure that the slot is symmetrical along the \(L\) direction when the antenna surface is slotted, and the antenna current is still reversed in the \(W\) direction, with only the \(L\) direction component. The effect of slotting on the antenna path is shown in Figure 4. Comparing the three current distribution diagrams in Figure 4, the location of the maximum current is similar, but the path of the maximum current value in Figure 4b,c is longer, which indicates that it has the most significant effect of reducing the resonant frequency and the longest effective current path. The value of the current near the U-shaped slots is larger, so it plays the main role in extending the current path. When a U-shaped slot is added to the antenna, the resonant frequency is less than 8% of the bandwidth near the expected resonance point, and the \(S\) parameter is shown in Figure 5. Comparing Figure 4b,c, it can be seen that the function of the rectangular slot is to fine-tune the frequency to the resonant frequency point. The resonant frequency changes caused by the length of the slot and the size change of the microstrip antenna are shown in Figure 6. This is consistent with the above analysis indicating that the longer the antenna slot, the lower the resonant frequency. Figure 4: Effect of antenna slot on surface current path, and the arrows show the current direction. (**a**) Microstrip antenna’s current distribution, (**b**) symmetric slots’ current distribution, (**c**) two pairs of symmetric slots’ current distribution. Figure 3: The current path of the microstrip antenna in TM\({}_{10}\) mode. The feed point location in the antenna design process has some influence on the impedance matching. The input impedance of the microstrip antenna is affected by many parameters. For any feed point, the radiation resistance of the microstrip antenna is  
\[R=\frac{45\pi^{2}\Big{\{}\int_{-\
u/2-d}^{\
u/2-d}\Big{[}\frac{1}{2\pi}\int_{- \infty}^{\infty}g(p)\frac{e^{-\mu\
u}}{e^{\mu\mu}}dp\Big{]}\cos\frac{\pi}{v}(x +d)dx\Big{\}}^{2}}{\Big{(}\frac{v}{A_{0}}\Big{)}^{2}\Big{[}1-0.374\Big{(} \frac{v}{A_{0}}\Big{)}^{2}+0.13\Big{(}\frac{v}{A_{0}}\Big{)}^{4}\Big{]}} \tag{3}\]  
\[g(p)=\frac{\sin(pW_{m}/2)}{pW_{2}/2}-\frac{1}{2}\frac{\sin^{2}(pW_{m}/4)}{(pW_ {m}/4)^{4}} \tag{4}\]  
where the length of the slot is \(v\), the distance from the feed point to the center of the slit is \(d\), and \(p\) is a variable of the Fourier transform.