## 4 Design and Electrical Performance Characterization
### Input, Inter-Resonator, and Output Coupling Characterization
#### 4.4.1 Input and Output Coupling  
The generation of the TE\({}_{0.6}\) has been studied through various feeding techniques [16; 35; 36]. In this work, two established coupling methods are investigated as follows:  
The first approach, illustrated in Figure 11a, employs parallel transmission line coupling similar to classic microstrip coupled-line configurations as presented in [26]. However, this method exhibits limited coupling enhancement with increasing line length due to diminishing interaction with the resonator's induced currents. The second approach, shown in Figure 12a, implements a contour-following parallel transmission line or curved transmission line (C-TL), building upon prior coaxial implementations [36] but adapted to planar technology. Here, the rotation angle \(\theta\) corresponds to the physical arc length along the DR circumference (top view). In Figure 12a, \(\theta=90^{\circ}\) serves as an example, while this work also analyzes the angle in terms of wavelength to characterize the C-TL's EM behavior. Figure 10: Aspect ratio variation for TE\({}_{0.6}\) mode. The analysis results shown in Figures (b)b and (b)b--expressed using the notation in (2)--validate the bandwidth-limiting constraints of the first coupling approach. While the second method exhibits more favorable characteristics, it displays an unexpected behavior, as follows: the coupling factor \(k\) (inversely proportional to quality factor) initially grows with increasing arc-shaped C-TL length (\(\theta\)), reaches a maximum, and then decreases to complete cancellation (Figure 13). This phenomenon occurs when the induced current vectors along the coupled line transition from constructive to destructive interference with respect to the resonator mode. Coupling restoration occurs only when the currents re-establish phase coherence for the desired mode. This behavior, intrinsic to both the mode excitation mechanism and the \(TE_{01\delta}\) mode characteristics, reveals inherent bandwidth limitations for \(TE_{01\delta}\)-based implementations while simultaneously suggesting potential approaches for innovative bandwidth extension techniques. The maximum achievable bandwidth coupling, calculated using expression (2), is presented in Figure (b)b. This configuration demonstrates a coupling \(k\cdot f_{0}\) exceeding 200 MHz with gradual variation, representing a significant improvement over the rectangular transmission line results shown in Figure (b)b. The smoother response characteristic enables reliable filter tuning through mechanical adjustments while providing enhanced absolute input coupling compared to conventional methods. Figure 11: \(TE_{01\delta}\) feeding methods—straight transmission line. Figure 12: \(TE_{01\delta}\) feeding methods—curved transmission lines (C-TL). The simulation results in Figure 13 demonstrate that although the transmission line C-TL terminates in an open circuit, the majority of the electromagnetic field transfers from the transmission line to the DR, exciting the TE\({}_{01\delta}\) mode. The simulations reveal anomalous behavior where the resonator response varies with the excitation's electrical length. Under certain conditions, the transmission line mode couples efficiently to the resonator, while at other lengths the coupling coefficient becomes negative, with periodic recoupling occurring at specific intervals. This undesirable effect limits resonator capabilities and introduces heightened design sensitivity. The peak coupling condition coincides with the minimum \(Q_{u}\). The electrical spacing between consecutive minima remains constant at \(\lambda_{\text{eff}}/2\), while the maxima spacing follows an irregular pattern. Specifically, subsequent maxima occur at approximately \(\lambda_{\text{eff}}\) intervals. To elucidate this coupling mechanism and explain the periodicity disruption, a mathematical model characterizes the \(Q_{u}\) dependence on wavelength. Figure 14 presents the comparison between simulated results and model predictions.