# A Novel Magnetic Coupling for Miniaturized Bandpass Filters in Embedded Coaxial SIW
## 2 Materials and Methods
### Synthesis  
As proposed in [9; 14] for the CSIW topology with a loading capacitance created by an air isolating gap, the resonator main parameters can be extracted straightforwardly, considering it as a piece of circular-square coaxial line of length \(h_{p}\) (where \(h_{p}=h-g_{p}\)) and characteristic admittance \(Y_{0}\) embedded  
Figure 1: 3D sketches of an embedded CSIW resonator including main dimensions: **(a)** top view; **(b)** detail of the inner conductor and capacitive plate; and **(c)** lateral view.  
Figure 2: 3D E-field distribution for the proposed embedded coaxial SIW resonator in the gap that separates the disk plate and the top ground plane.  
into the dielectric substrate. Then, such elements can be modeled as a TEM-mode transmission line short-circuited at one end and terminated with a capacitor on the other. Therefore, the TEM-mode resonant frequency is given by the condition \(B(\omega)=0\), being the susceptance \(B\) of the coaxial resonator defined as  
\[B(\omega)=\omega C_{l}-\frac{1}{Z_{0}}\cot\beta h_{p} \tag{1}\]  
where \(\beta\) is the TEM-mode propagation constant, and \(Z_{0}\) is the coaxial resonator characteristic impedance. It should be noted that the resonator electrical length \(\theta_{0}\) at the desired working frequency \(f_{0}\) corresponds to \(\theta_{0}=\beta h_{p}\).  
Next, the characteristic impedance \(Z_{0}\) of the CSIW resonator, and the proportion between the SIW cavity side \(l_{c}\) and \(d_{v}\) value are obtained from [14] as  
\[Z_{0}=\left(\frac{\cot\theta_{0}+\csc^{2}\theta_{0}}{2b}\right) \tag{2}\]  
\[\frac{l_{c}}{d_{v}}=0.9268\,e^{\frac{Z_{0}/\sqrt{\pi}}{\delta\theta}} \tag{3}\]  
Thus, the synthesis procedure proceeds by setting the resonator slope parameter \(b\) at center frequency \(\omega_{0}\) of the desired filtering response. As has been explained in [15], the susceptance slope parameter \(b\) for resonators with zero susceptance at \(\omega_{0}\) is well approximated by  
\[b=\frac{1}{2}\left[\omega_{0}C_{l}+\frac{\theta_{0}}{Z_{0}\sin^{2}\theta_{0}}\right] \tag{4}\]  
On the one hand, the coaxial line admittance \(Y_{0}\) can be calculated by using the approximation proposed in [16] for a circular inner conductor of diameter \(d_{v}\) and an external square contour of side \(l_{c}\), which is  
\[Y_{0}=\left[\frac{60}{\sqrt{\epsilon_{r}}}\ln\left(1.079\frac{l_{c}}{d_{v}} \right)\right]^{-1}\text{ for }l_{c}\gg d_{v} \tag{5}\]  
where \(\epsilon_{r}\) is the dielectric substrate permittivity of the LTCC dielectric tape.  
On the other hand, the loading capacitance \(C_{l}\) between the circular disk and the top metal layer may be written as  
\[C_{l}=C_{pp}+\frac{C_{ff}}{1.65}=\epsilon_{r}\epsilon_{0}\frac{\pi r_{p}^{2}} {g_{p}}+\frac{C_{ff}}{1.65} \tag{6}\]  
where \(\epsilon_{0}\) is the permittivity of the free space, \(C_{pp}\) is the associated parallel plate capacitance, and \(C_{ff}\) is the capacitance due to the fringing effect of the field between the edges of the plates, which are visible in the right part of Figure 2. A correcting factor has been taken into account, due to the specific configuration of the implemented parallel plate capacitor of the CSIW resonator that shows only one circular disk plate. Indeed, the other plate consists of the wider top ground plate. Thus, as demonstrated in [17], the fringing effect of a parallel plate capacitor composed of two identical circular disks can be well approximated as  
\[C_{ff}=\epsilon_{r}\epsilon_{0}r_{p}\log\left(16\pi\frac{r_{p}}{g_{p}}-1\right) \tag{7}\]  
Finally, to finely tune \(C_{l}\), a quick optimization of the circular disk plate radius \(r_{p}\) by means of 3D EM simulations is usually required, which includes the metal layer thickness \(t\) of the conducting layer that forms the plate.