*2.3.2 Quarter-mode SIW (QMSIW) filters*

The authors of Ref. [15] proposed the bandpass filter based on QMSIW for the first time. The geometries of the two filters are shown in **Figure 9**. The bandpass filter consists of cascaded QMSIW cavities, and its overall size is greatly reduced. For verification, two filters are designed. The prototype I is designed at the center frequency of 5.85 GHz with a fractional bandwidth of about 14%. Prototype II is designed at the center frequency of 5.5 GHz with a fractional bandwidth of about 26%.

The authors of Ref. [16] reported a QMSIW for designing a series of bandpass filters. Quarter-mode SIW cavities can be connected in various ways, such as sharing a side or sharing a corner, with some metal vias removed to allow cavity coupling. The authors introduce novel filter topologies based on the proposed two different techniques, side-coupling, and corner-coupling. Taking the filter shown in **Figure 10** as an

**Figure 9.** *The geometries of the two filters. (a) The prototype I, (b) the prototype II [15].*

**Figure 10.** *Configuration of the side-coupled four-pole filter [16].*

**Figure 11.** *Configuration of the corner-coupled four-pole filter [16].*

example, the resonator is laterally coupled by removing some through holes in the common wall. By using a plus-single topology, it can be used to implement filters with any number of poles. **Figure 11** shows the filter composed of the new coupling. By using a coplanar resonator and quarter-mode cavity together, a three-pole filter with a center frequency of 4 GHz is designed, and its relative bandwidth is 16%.
