# A Balanced Symmetrical Branch-Line Microstrip Coupler for 5G Applications
## 2 Design of Resonator, Filters and Coupler
According to Equation (3) and for low frequencies, \(Z_{\mathrm{Sh}}\) will be an open circuit. Therefore, \(\mathrm{S_{11}}\) and \(\mathrm{S_{21}}\) at low frequencies will be changed as follows:  
\[ \begin{array}{c}S_{11}=\frac{|\omega_{\mathrm{low}}L_{b}}{Z_{0}+\frac{i\omega _{\mathrm{low}}}{Z_{0}}L_{b}}\\ S_{21}=\frac{Z_{0}}{Z_{0}+\omega_{\mathrm{low}}L_{b}}\end{array}  \tag{6}\]  
where \(\omega_{\mathrm{low}}\) is a low angular frequency. From Equation (6), it is clear that the \(-3\) dB cut-off frequency depends on the values of \(\mathrm{L_{b}}\) so that a cut-off angular frequency (\(\omega_{\mathrm{c}}\)) can be obtained using the following equation:  
\[ \begin{array}{c}\left\{\begin{array}{c}\frac{1}{\sqrt{2}}|S_{21}|_{Max}=| S_{21}|\\ |S_{21}|=\frac{Z_{0}}{\sqrt{2Z_{0}^{2}+2(\omega_{c}L_{b})^{2}}}\\ \frac{1}{\sqrt{2}}|S_{21}|_{Max}=\frac{1}{\sqrt{2}}\Big{|}\frac{Z_{0}}{Z_{0}+ \omega_{c}L_{b}}\Big{|}_{Max}=\frac{Z_{0}}{\left\{\sqrt{2Z_{0}^{2}+2(\omega_{ c}L_{b})^{2}}\right\}_{Min}}=\frac{1}{\sqrt{2}}\end{array}\right.\Rightarrow\\ \frac{Z_{0}^{2}}{2Z_{0}^{2}+2(\omega_{c}L_{b})^{2}}=\frac{1}{2}\Rightarrow \omega_{c}L_{b}\approx 0\end{array} \right.\end{array} \tag{7}\]  
Our target is to obtain a cut-off frequency in GHz; therefore, by choosing a \(\mathrm{L_{b}}\) of less than \(1\) nH, \(\omega_{\mathrm{c}}\mathrm{L_{b}}\) will be near zero. Since the simulation results show that at the cut-off frequency \(|S_{21}|=|S_{11}|\), the second way to obtain the cut-off frequency is as follows:  
\[|S_{21}|=|S_{11}|\Rightarrow\frac{Z_{0}}{\sqrt{2Z_{0}^{2}+2(\omega_{c}L_{b}) ^{2}}}=\frac{\omega_{c}L_{b}}{\sqrt{Z_{0}^{2}+\left(\omega_{c}L_{b}\right)^{2 }}}\Rightarrow\omega_{c}=\frac{Z_{0}}{\sqrt{2}L_{b}} \tag{8}\]  
According to the cut-off angular frequency and based on Equation (8), the value of \(\mathrm{L_{b}}\) can be calculated. Then, using Richards' transformation, the value of the physical length \(\mathrm{l_{b}}\) can be determined. Using the analyzed lowpass resonator, two lowpass filters (LPF1, LPF2) are proposed. These LPFs with their frequency responses are depicted in Figure 2, where all dimensions are in mm. As shown in Figure 2, the cut-off frequencies of LPF1 and LPF2 are close to each other. The LPF1 has the best insertion and return losses of 0.03 dB and 24.9 dB in its passband. To design the LPF2, two shunt stubs were used. LPF2 has the best insertion and return losses of 0.006 dB and 32.7 dB in its passband. All simulation results in this work were obtained using the EM simulator of ADS software. The used substrate was a Rogers RT/duroid 5880 with h = 0.7874 mm, \(\varepsilon_{\mathrm{r}}\) = 2.22 and tan (\(\delta\)) = 0.0009. To obtain a microstrip BLC, we integrated the proposed LPFs. The layout configuration of our coupler is illustrated in Figure 3, where all dimensions are in mm, and we did not change the dimensions of the LPFs in it. Also, another internal shunt stub was added near the isolation port (Port 4), which is able to improve the isolation and shift the operating frequency simultaneously. The overall size of this coupler is 6.4 mm \(\times\) 13 mm = 0.14 \(\lambda_{\text{g}}\)\(\times\) 0.29 \(\lambda_{\text{g}}\), where \(\lambda_{\text{g}}\) is the guided wavelength calculated at the operating frequency. Using the current density distribution, the dimensions of the physical lengths and widths 1\({}_{1}\), 1\({}_{2}\), w\({}_{1}\), w\({}_{2}\), w\({}_{3}\) and w\({}_{4}\) were optimized to improve the coupler performance. The current density distributions of our BLC at the operating frequency for simulating Ports 2 and 3 are shown in Figure 4. As can be seen, the thinner cells with higher impedances have higher current densities than the internal stubs. This is verified through the mathematical analysis of the proposed lowpass resonator. When we simulate Port 2 and Port 3, the upper and lower transmission lines have higher current densities, respectively.