**3.1 Equivalent circuit model**

**Figure 7** depicts the equivalent lumped equivalent circuit for **Figure 6**. CSRRs etched in the center are excited by the electric field induced by the SIW. Therefore, this coupling can be labeled by connecting the SIW capacitance to the CSRR. In these models, L is the inductance of SIW vias, and C is the coupling capacitance between the CSRR and SIW. The resonator is represented by a parallel LC tank, where Lc and Cc represent the reactive elements, and R accounts for losses.

#### **Figure 6.**

*(a) Top and (b) bottom view of basic unit cell [11] [reproduced courtesy of the electromagnetics academy].*

**Figure 7.** *The equivalent circuit model.*

*Analysis and Design of Miniaturized Substrate Integrated Waveguide CSRR Bandpass Filters… DOI: http://dx.doi.org/10.5772/intechopen.104733*

#### **Figure 8.**

*(a) Dimensional layout of BPF and (b) fabricated BPF [11] [reproduced courtesy of the electromagnetics academy].*

**Figure 8a** shows the dimensional geometry of the proposed SIW-CSSRs bandpass filter [11], and **Figure 8b** shows the photograph of the fabricated design. The substrate used in the filter is RT/Duroid 5880, with a permittivity of 2.2 and a height of 0.254 mm.

**Figure 9** compares the simulated and measured results of the filter. The measured insertion and return losses are about 2.16 dB and 11.6 dB, respectively. The filter shows a wide bandwidth ranging from 6.2 to 8.6 GHz (FBW of 32.4%).
