## 4 Design and Electrical Performance Characterization
### Input, Inter-Resonator, and Output Coupling Characterization
#### 4.4.1 Input and Output Coupling
The analysis reveals a periodic behavior (though not identical between maxima and minima) in the simulated results, from which we extract the unlo  
Figure 14: Comparison of simulated and modeled \(Q_{u}\) behavior, w/o stub configuration. Figure 13: Unloaded quality factor \(Q_{u}\) performance—curved TL. characteristics showing distinct properties in two separate regions, as mathematically represented by the piecewise model in (11). \[Q_{u}(\theta)= \begin{cases}Q_{\text{low-region}}(\theta),&\theta\leq\theta_{c}\\ Q_{\text{upper-region}}(\theta),&\theta>\theta_{c}\end{cases}  \tag{11}\]  
where \(\theta_{c}\approx 0.55\lambda_{\text{eff}}\) represents the transition point between regions. The short-range coupling (\(\theta\leq\theta_{c}\))--low-region--is described by the following:  
\[Q_{\text{low-region}}(\theta)=\underbrace{\left[A_{c}\cos^{2}(\beta\theta+ \phi_{c})+A_{s}\sin^{2}(\beta\theta+\phi_{s})\right]}_{\text{Standing wave coupling}}\underbrace{\underline{\underline{\underline{\epsilon}}^{-(\  
On the other hand, progressive stub extension showed the secondary peak merging with the primary coupling peak at \(\ell_{s}=40^{\circ}\), beyond which coupling efficiency degraded rapidly. This merged condition represents the optimal operating point, combining the original local maxima observed at 80\({}^{\circ}\) and 150\({}^{\circ}\) in shorter and stub-less configurations while effectively suppressing unwanted C-TL resonances. A key advantage of this configuration lies in the differential phase variation between the main C-TL and the open-ended stub. For equivalent electrical length changes, the stub characteristics vary more rapidly, facilitating fine-tuning, while the main line responds more gradually to coarse adjustments. The optimized \(\ell_{s}\) = 40\({}^{\circ}\) configuration achieves an ideal balance between coupling efficiency and bandwidth preservation, resulting in direct improvements to input-output port performance. The optimized system employing a higher-impedance open-circuit stub demonstrates improved coupling behavior, as described by the expressions below, which were obtained from a predicted behavior model derived through genetic algorithm optimizations of simulation data as follows:  
\[k_{e}(\theta)= \begin{cases}k_{\text{low-region}}(\theta),&\theta\leq\theta_{c} \\ k_{\text{upper-region}}(\theta),&\theta>\theta_{c}\end{cases}  \tag{14}\]  
where the low-region--rising edge (\(\theta\leq\theta_{c}\approx 0.32\lambda_{\text{eff}}\))--follows a modified logistic growth as follows:  
\[k_{\text{low-region}}(\theta)=\frac{k_{\text{max}}}{1+e^{-\kappa(\theta-\theta _{0})}}+\eta\theta+k_{0} \tag{15}\]  
with \(\kappa\) representing the coupling growth rate (dependent on stub tuning), \(\theta_{0}\) indicating the position of maximum coupling gradient, and \(\eta\) denoting the linear coupling coefficient from the main line. The upper-region--falling edge (\(\theta>\theta_{c}\))--exhibits a dual-exponential decay as follows:  
\[k_{\text{upper-region}}(\theta)=k_{1}e^{-\gamma(\theta-\theta_{t})}+k_{2}e^{ -[(\theta-\theta_{c})/\sigma]^{2}}+k_{b} \tag{16}\]  
characterized by \(\gamma\) (dominant energy decay rate), \(\sigma\) (coupling resonance width), and \(k_{b}\) (background coupling level). In Figure 16, the correlation between the model and the simulated behavior is shown. Figure 15: Input–output coupling characterization. The stub implementation delivers three primary performance enhancements. First, it maintains phase synchronization and constructive interference up to \(\theta=80^{\circ}\) through precise impedance matching. Second, it effectively combines the primary (around \(0.25\lambda_{\text{eff}}\)) and secondary (approximately \(0.5\lambda_{\text{eff}}\)) coupling peaks--case with stub length \(\ell_{s}\approx 10^{\circ}\), via optimized stub length \(\ell_{s}\). Finally, the design achieves substantial bandwidth enhancement.