## 4 Results and Discussion
Figure 4: Variations in cut-off frequencies w.r.t electrical length \(\theta_{1}\). Figure 5: Variations in cut-off frequencies w.r.t electrical length \(\theta_{4}\). Figure 8: Variation in third pass band w.r.t stub length L4. Figure 6: Realization of the third passband by embedded L-shaped resonator. Figure 7: Response of filter without L-shaped structure. The parameters frequently considered while designing bandpass filters are the coupling coefficient (K) and external quality factor (Q). The coupling coefficient describes the degree of energy transfer or closeness between two resonators, a physical quantity with no dimensions [46]. It was first incorporated into the microwave filter theory by [PERSON] [47]. He began with a bandpass network comprising a ladder chain of alternate series and parallel dissipative resonant circuits set to the same resonant frequency \(\omega_{0}=1/\surd\)L.C. The following design equation gives the exact values of coupling coefficients obtained in [48] for a certain asymmetrical pair of adjacent resonant circuits featuring Chebyshev frequency response. \[|\mathbf{k}|=(\omega_{2+}-\omega_{2-})\ \pm\ (\omega_{2+}+\omega_{2-}) \tag{7}\]  
where \(\omega_{-}\) and \(\omega_{+}\) are the lower and upper frequencies, and \"w\" is the relative bandwidth defined as  
\[\mathrm{w}=\frac{(\omega_{2}-\omega_{1})}{\surd\omega_{1}\omega_{2}}\]  
The next parameter is the external quality factor (Q), the ratio of the central resonance frequency to the 3 dB bandwidth. It can be determined by the outer resonator connecting to feeding lines and can be expressed as. \[\text{Quality factor }(\text{Q})=\frac{\text{Resonance frequency}}{3\ \text{dB bandwidth}} \tag{8}\]  
In Figure 3, the parameters G and S are responsible for K, while Q is achieved due to the parameter T\({}_{2}\). The coupling coefficient decreases when the values G and S increase while the quality factor increases, as depicted in Figures 10-12. Figure 13 shows the response of the three passbands' quality factors concerning T\({}_{2}\). The Q increases with varying values from 18 mm to 20 mm. Figure 14 illustrates the group delay of the proposed triband filter. In designing bandpass filters, group delay is a crucial factor since it impacts the filter's phase response. Group delay is a term that describes the delay that various frequency components of a signal endure as they pass through the filter [49]. To further validate the working principle, Figure 15 shows the current distribution of the proposed topology at the passbands. The current distribution indicates how the electrical current travels along the filter structure. It contributes to understanding signal propagation and the resonant behaviour of the  
Figure 9.0 Location of T.Z.s at different frequencies with sharpness factor. filter [50]. As seen in Figure 15a, the current density is maximum on the surface of the SIR at frequency ratio K (f\({}_{2}\)/f\({}_{1}\) = 2.2) because, as discussed, the first two passbands are obtained due to the ratio of the electrical lengths of the SIR. Figure 15b shows the current distribution at 4.23 GHz for the third passband, obtained by the embedded L-shaped resonator. As seen, the current is uniformly distributed only on the surface of the L-shaped resonator. Similarly, the magnitude of the electric field (E) intensity of the passbands is illustrated in Figure 16, respectively. Figure 9.1 Effect of G parameter external quality factor on three passbands. Figure 9.2 Effect of G parameter coupling coefficient on three passbands. Figure 9.3 Group delay response of the three passbands. Figure 9.4 Effect of S parameter coupling coefficient on three passbands. Figure 9.5 Effect of T\({}_{2}\) parameter external quality factor on three passbands.