# Design of a Compact Microstrip Decoupled Array
## 2 Materials and Methods
### Decoupling Feed Network  
Considering that the proposed mono-pulse antenna has two units, the feed network needs at least four ports, two for feeding the radiators and two for connecting the sum and difference ports, as shown in Figure 10. The electromagnetic wave at the external port is denoted by e, while \(i\) denotes the one at the internal port, and a and b denote the incident wave and the reflected wave, respectively. The expression of the \(S\) parameter of the antenna feeding system is \(b_{e}=S^{a}a_{e}\). When seen from the external port of the antenna feeding system, if the scattering matrix \(S\) is diagonal, it indicates that the antenna has been decoupled completely. The scattering matrix expression of the array antenna can be formulated as follows  
\[a_{i}=S^{a}b_{i} \tag{5}\]  
The scattering matrix expression of the feeding network is  
\[\left[ \begin{array}{c}b_{e}\\ b_{i}\end{array} \right]= \begin{bmatrix}S_{ee}&S_{ei}\\ S_{ie}&S_{ii}\end{bmatrix} \left[ \begin{array}{c}a_{e}\\ a_{i}\end{array} \right] \tag{6}\]  
where \(a_{i}\), \(b_{i}\), \(a_{e}\), and \(b_{e}\) are column vectors with two elements and \(S_{ee}\), \(S_{ei}\), \(S_{ie}\), and \(S_{ii}\) are 2 \(\times\) 2 matrices. For the ideal feed network, \(S_{ee}=S_{ii}=0\), \(S_{ei}={S_{ie}}^{\mathsf{T}}\), then  
\[b_{e}=S_{ei}S^{a}S_{ie}a_{e}=S^{c}a_{e} \tag{7}\]  
Since the reflection coefficient of a mono-pulse antenna unit is small enough, it can be assumed that there is only coupling but no reflection between elements of the antenna array:  
\[S^{a}= \begin{bmatrix}0&s_{12}\\ s_{21}&0\end{bmatrix} ,\text{ and }s_{12}=s_{21}.\]  
Through simple calculation, the eigenvectors can be obtained as  
\[e_{1}=\frac{1}{\sqrt{2}} \begin{bmatrix}1\\ 1\end{bmatrix} ,\ e_{2}=\frac{1}{\sqrt{2}} \begin{bmatrix}1\\ -1\end{bmatrix} .\]  
The antenna array operates in the mode corresponding to the eigenvector e\({}_{1}\) with the feeds of both ports being of equal amplitude and in-phase and in the mode corresponding to e\({}_{2}\) with the feeds of both port being of equal amplitude and in-phase. Therefore, \(S_{ie}=[e_{1}\quad e_{2}]\) and \(S_{ei}={S_{ie}}^{\mathsf{T}}=S_{ie}^{-1}\). The scattering matrix of the antenna feeding system is diagonal. The two-element microstrip array antennas are decoupled. Figure 10: The feeding network scattering matrix. The scattering matrix of the decoupled feeding network is  
\[S^{c}= \begin{bmatrix}s_{11}&0&s_{13}&s_{13}\\ 0&s_{22}&s_{14}&-s_{14}\\ s_{13}&s_{14}&s_{33}&s_{34}\\ s_{13}&-s_{14}&s_{34}&s_{44}\end{bmatrix} . \tag{8}\]  
The decoupled network of the two-element microstrip array is obtained through matrix calculation, and the decoupled network can be represented by Figure 11.  
\(\theta_{1}\) and \(\theta_{2}\) take on arbitrary values. Figure 11 shows a common decoupling network for a two-element array of antennas. In order to ensure that the decoupling network is better adapted to the miniaturized antenna array, take \(\theta_{1}=\theta_{2}=\frac{\lambda_{g}}{8}\). TL\({}_{4}\) takes \(\frac{\lambda_{g}}{4}+\theta_{2}\), and at this time, the decoupling network in the size of the ring is greatly reduced. However, changes in the size and impedance of the ring cause the imaginary part to arise. On the other hand, the phase at the difference port of the decoupling network cannot achieve reverse superposition. Therefore, an equivalent circuit is used to add an open short branch on the torus, as shown in Figure 12a,b. In this way, the impedance of the port can be matched. The phase-inverse superposition of differential ports is realized by adjusting the equivalent length of TL\({}_{4}\). The equivalent circuit is designed by adding an open branch, and the total size of the equivalent ring is \(\frac{3\lambda_{g}}{2}\), and the corresponding transmission line impedance is \(\sqrt{2}Z_{0}\). At this time, the impedance matching is realized at each port, and the phase inversion superposition is realized at the differential port. Figure 11: The transmission line equivalent model.