# A Novel Magnetic Coupling for Miniaturized Bandpass Filters in Embedded Coaxial SIW
## 2 Materials and Methods
### Coupling Mechanisms  
Figure 5 depicts the topology of the novel magnetic coupling mechanism that is especially suited for obtaining strong coupling values in highly loaded CSIW resonators. Specifically, Figure 5a shows the layout of the magnetic external coupling mechanism, while the inter-resonator coupling mechanism is shown in Figure 5b. As reflected, the magnetic couplings are based on embedded stripline probes that are arranged at different layers of the LTCC stack-up, so that inner conductors of adjacent CSIW resonators can be short-circuited between them in a similar manner to conventional tapped-line couplings widely used in combine planar filters. Thus, this solution exploits the high flexibility of the LTCC multi-layer technology to improve the performance of CSIW filters.  
By changing the height of the interconnection between coupling probes and the plated via holes forming the inner conductor, the coupling magnitude can be coarsely tuned, as shown in Figure 6. In this context, let us name the height position in the stack-up of the external and the inter-resonator couplings as \(h_{e}\) and \(h_{k}\), respectively. In addition, let us set \(h_{i}=0\) mm at the external bottom level and \(h_{i}=2.24\) mm at the external top level, for \(i=(e,k)\).  
In Figure 6a, the external quality factor (\(Q_{e}\)) is represented as functions of \(h_{e}\), while the inter-resonator coupling coefficient (\(k_{1,2}\)) between two adjacent resonators versus \(h_{k}\) is shown in Figure 6b. Taking into account that the height position of the circular disk plate corresponds to \(h_{p}=1.792\) mm, the highest coupling value is achieved when the stripline probes are short-circuited to this capacitive plate element. By contrast, the coupling drops quickly as the probe is moved towards the bottom layer. Please note that such coupling values enable the design of wideband BPF with FBW easily higher than 15%.  
Nevertheless, a fine tuning of both couplings is achieved by varying the stripline probe width, which are named \(w_{e}\) for external coupling and \(w_{k}\) for inter-resonator coupling, as it can be seen in Figure 7a,b, respectively. Thus, once the height position of the coupling probes is defined in the LTCC stack-up, the probe widths allow us to finely control \(Q_{e}\) and \(k_{1,2}\) values, enabling a good optimization of the filtering response.  
Figure 5: Layout of the proposed magnetic (**a**) external and (**b**) inter-resonator coupling mechanisms based on stripline probes short-circuited to the inner posts of the CSIW resonators. Please note that bottom layer and via holes that form the SIW cavity walls are not shown for improving readability.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**SIW Topology** & **Area (mm\({}^{2}\))** & **MF (\%)** & \(Q_{u}\) & \(\Delta\)\(Q_{u}\) (\%)** & **Spurious-Free Band** \\ \hline TE\({}_{101}\) (\(h=2.25\) mm) & \(2704\) & \(REF\) & \(253\) & \(REF\) & \(1.58\times f_{0}\) \\ TE\({}_{101}\) (\(h=1.12\) mm) & \(2704\) & \(0\) & \(213\) & \(-15.8\) & \(1.58\times f_{0}\) \\ Coaxial \(b=0.1055\) & \(84.6\) & \(96.8\) & \(209\) & \(-17.4\) & \(7.48\times f_{0}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of coaxial SIW topology versus standard TE\({}_{101}\)-based SIW topology.