# Design of a Modified Compact Coupler with Unwanted Harmonics Suppression for L-Band Applications
## 2 Design Procedure
### Horizontal Resonator Design  
To add harmonics suppression in the designed coupler, several resonators should be added in the main conventional structure. At first, an initial horizontal resonator is presented to obtain initial suppression band. The _LC_ equivalent circuit (LCEC) of the proposed initial horizontal resonator is shown in Figure 3. Additionally, the frequency response of the presented _LC_ circuit is compared with the frequency response of the layout transmission line realization in Figure 4. The applied values of the _LC_ components for the LCEC of the proposed initial horizontal resonator are listed in Table 1.  
Figure 1: BLC operating at 1 GHz: (**a**) conventional structure and (**b**) frequency response. All dimensions are in the millimeter unit. Figure 2: Design procedure of the proposed BLC. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{_LC Horizontal_} \\ \hline \(L_{0}\) & \(L_{1}\) & \(L_{2}\) & \(L_{3}\) & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) \\ \hline
4.1 nH & 0.6 nH & 0.4 nH & 0.3 nH & 0.5 pF & 0.4 pF & 0.2 pF \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(LC\) components values for the LCEC of the proposed initial horizontal. To determine the transmission zeros of the presented LCEC, the transfer function of the proposed initial horizontal resonator should be obtained. The extraction of the transfer function of the presented resonator is explained as follows. At first, the value of the \(Z_{P}\) impedance can be written as shown in (1). \[Z_{P}=L_{1}\,S+\frac{L_{2}\,S+\frac{\frac{1}{\mathrm{c}_{3}}+L_{3}\,S}{\mathrm{ C}_{2}\,S\left(\frac{1}{\mathrm{c}_{2}}\mathrm{s}^{\mathrm{+}}\frac{1}{ \mathrm{c}_{3}}+L_{3}\,S\right)}}{C_{1}\,S\left(\frac{1}{\mathrm{c}_{1}} \mathrm{s}+L_{2}\,S+\frac{\frac{1}{\mathrm{c}_{3}}\mathrm{s}+L_{3}\,S}{ \mathrm{c}_{2}\,S\left(\frac{1}{\mathrm{c}_{2}}\mathrm{s}^{\mathrm{+}}\frac{1} {\mathrm{c}_{3}}+L_{3}\,S\right)}\right)} \tag{1}\]  
Figure 4: Frequency response of the presented \(LC\) circuit, as compared with the frequency response of the layout transmission line realization. Figure 3: LCEC of the proposed initial horizontal resonator. By obtaining the value of \(Z_{P}\), the transfer function of \(H(S)\) can be calculated as written in (2). \[H(S)=\frac{R\,Zp}{\left(L_{0}\,S+\frac{Zp\,(R+L_{0}S)}{R+Zp+L_{0}S}\right)\left(R +Zp+L_{0}\,S\right)} \tag{2}\]  
where in (2), the parameter '\(R\)' represents the 50 \(\Omega\) impedance of the port. The magnitude of the extracted transfer function, \(H(S)\), is illustrated in Figure 5. As seen, the main transmission zero, which is extracted using analyses, equal to 5.5 GHz, is identical to the obtained one by circuit and EM simulations. After obtaining the LCEC of the initial resonator, the microstrip transmission line realization can be achieved. The layout structure and frequency response of the initial horizontal resonator is depicted in Figure 6a,b. Figure 5: Magnitude of the extracted transfer function \(H(S)\). Figure 6: Initial horizontal resonator: (**a**) layout and (**b**) frequency response. All dimensions are in the millimeter unit. As seen in Figure 6b, the frequency response of the initial horizontal resonator is not perfect in the passband and suppression band, so the initial horizontal resonator should be improved. The structure of the improved initial horizontal resonator and its frequency response are shown in Figure 7a,b. To increase the suppression band and to modify the cut-off frequency, a stepped impedance resonator is added into the improved initial horizontal resonator structure, which forms the final horizontal resonator. The layout and frequency response of final horizontal resonator are shown in Figure 8a,b. As can be seen, the frequency response of the initial horizontal resonator is improved and optimized by adding the stepped impedance resonators in two steps, which finally have formed the final horizontal resonator.