# A Novel Magnetic Coupling for Miniaturized Bandpass Filters in Embedded Coaxial SIW
## 2 Materials and Methods
### Resonator 3D EM Simulations  
To perform the study of the performance of a CSIW resonator, let us consider a benchmark configuration with center frequency 1.5 GHz and using a 10-layer LTCC stack-up, whose layout has been already shown in Figure 1. Regarding the dielectric material, the DuPontTM Green TapeTM 9K7 substrate (\(\epsilon_{r}=7.1\pm 0.2\), tan\(\delta=0.0009\) @ 10 GHz) has been chosen due to its very low loss tangent. Each dielectric tape layer has a thickness of 0.254 mm in the green state, while the fired thickness is 0.224 mm on account of the shrinkage effect in the \(z\)-direction after the LTCC firing step. This means that the total thickness of the 10-layer stack-up of the CSIW resonator is 2.24 mm. Since we wanted to boost the miniaturization of the device with respect to standard SIW structures, we set a high value for the slope parameter, which is \(b\!=\!105.5\) mS. As a result, the main parameters of the CSIW resonator become: \(C_{l}=10.75\) pF and \(Z_{0}=61.1\)\(\Omega\). The main dimensions of the compact CSIW resonator considered in this study are shown in Table 1, where \(p_{d}\) is the pitch between adjacent via holes forming the cavity sides. It is worth mentioning that the value of the capacitor plate separation (i.e., \(g_{p}=0.448\) mm) has been achieved by introducing the circular disk plate after two dielectric tape layers below the external top metal layer in the stack-up. Thus, the length of the embedded coaxial line is \(h_{p}=(2.24\) mm\(-2\cdot 0.224\) mm) \(\simeq 1.792\) mm, which corresponds to an electrical length of \(\theta_{0}=8.8^{\circ}\) @ 1.5 GHz. These values emphasize the remarkable degree of miniaturization that can be simply achievable with the CSIW topology. Indeed, the SIW cavity side is just \(9.2\times 9.2\) mm\({}^{2}\), which corresponds to \(0.126\times 0.126\,\lambda_{g}^{2}\) and \(0.047\times 0.047\,\lambda_{0}^{2}\), where \(\lambda_{g}\) is the guided wavelength at 1.5 GHz. The simulated frequency response of the designed device, which has been obtained by using ANSYS HFSS software, is depicted in Figure 3a, and it is compared to the frequency response of a standard TE\({}_{101}\)-based SIW resonator having the same cavity size in Figure 3b. As shown, the use of an embedded CSIW topology enables also the shifting down of the fundamental mode of the SIW cavity resonator (i.e., the TEM mode in a CSIW resonator) with respect to the first spurious mode, which is the TE\({}_{101}\) mode, thus widening the stop-band bandwidth. Indeed, the EM field of the first spurious mode is altered due to the inner via holes and capacitive patch that provoke an increase of its resonant frequency, from 8.79 GHz as fundamental mode of a TE\({}_{101}\)-based resonator to 11.6 GHz as first spurious mode of the CSIW resonator. Thus, in the proposed simulated CSIW structure, the stop-band bandwidth is greater than \(7\times f_{0}\), as can be seen in Figure 3b. Furthermore, Figure 4 shows how the resonant frequency (\(f_{r}\)) and \(Q_{u}\) of the CSIW resonator change as the function of the radius \(r_{p}\) of the capacitive disk, which means modifying the total \(C_{l}\) value. It is evident that the resonant frequency can be controlled over a wide frequency band (i.e., \(>\)1.5 GHz) without affecting the overall resonator dimensions, while keeping a constant \(Q_{u}\) over frequency. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(I_{c}\) & \(h\) & \(h_{p}\) & \(d_{v}\) & \(d_{h}\) & \(r_{p}\) & \(g_{p}\) & \(w_{e}\) & \(p_{d}\) \\ \hline
9.2 & 2.24 & 1.792 & 0.68 & 0.18 & 4.2 & 0.448 & 0.15 & 0.4 \\ \hline \end{tabular}
\end{table} 
Table 1: Layout Dimensions of the Designed CSIW Resonator (Unit: mm). In this context, Table 2 provides the comparison between \(Q_{u}\), area miniaturization factor (MF, which has been defined in [18]) and spurious-free band of the designed embedded CSIW resonator and standard TE\({}_{101}\)-based SIW cavity resonators, centered also at 1.5 GHz and implemented in the very same LTCC stack-up.