Design of a Bandpass Filter with Mixed Electromagnetic Coupling Paths Using Vertical Split-Ring Resonators

We propose bandpass filters (BPFs) with mixed electromagnetic coupling paths (MEMCPs) that comprise two coupled vertical split-ring resonators (VSRRs) and present the equivalent circuit models of the coupled VSRRs. We demonstrate that the dominant coupling modes for the top and bottom layers of the VSRRs are magnetic (M) and electric (E), respectively, and that M-dominant coupling is required for high-selectivity BPFs. BPFs with narrow and wide bandwidths were designed based on the generalized coupling matrix. The proposed BPFs were fabricated and measured, and it is verified that the proposed BPFs have high selectivity due to the transmission zeros and a small circuit footprint due to the vertical structure of resonators. The fabricated narrow- and wide-band BPFs have the fractional bandwidths of 3.62% and 5.81%, respectively, with the overall size of 0.29λg×0.043λg.