## 4 Design and Electrical Performance Characterization
### Input, Inter-Resonator, and Output Coupling Characterization
#### 4.4.2 Inter-Resonator Coupling  
Two distinct inter-resonator coupling methods have been analyzed as fikkiws: conventional distance-based coupling (Figure 17a) and C-TL coupling (Figure 17b).  
The C-TL coupling approach exhibits both advantages and disadvantages compared to distance-based coupling. While enabling larger physical separation between resonators reduces unwanted cross-coupling parasitics, it simultaneously increases those from the C-TL extension.  
Figure 16: Comparison of simulated and modeled \(\Delta f\) behavior, w/ stub.  
Figure 17: Even–odd analysis.  
This method also facilitates more compact layouts through meandered line geometries. However, this approach suffers from three key limitations. First, the maximum achievable coupling factor remains lower than with distance-based coupling. Second, the C-TL configuration excites quasi-TEM mode resonances within the cavity that can interfere with the desired passband response. Third, as characterized in Figure 13, the curved transmission line exhibits an inherent length limitation where coupling reaches a maximum before decreasing as the coupling coefficient changes sign beyond a certain electrical length.  
Figure 17 compares both coupling configurations between DR2 and DR3 using even-odd mode analysis. The full filter structure is considered to ensure representative characterization due to housing size, and to define minimum distance to minimize parasitic inter-resonator coupling non desired.  
For the distance-based coupling setup (Figure 17a), the minimum center-to-center spacing is 6 mm (DR radius = 3 mm), corresponding to tangential resonator contact--an impractical scenario due to required tuning mechanisms. Nevertheless, the results confirm that increasing coupling with reduced resonator separation (Figure 18a) follows an exponential coupling behavior as distance increase.  
The expected behavior is observed in Figure 13 through quality factor analysis. The key distinction lies in the coupling mechanisms as follows: Figure 13 demonstrates input-output coupling via (8), while Figure 18b shows inter-resonator coupling through (9) and (10). This results in different coupling rate dynamics due to the interactions between two dielectric resonators.  
Figure 18b reveals that the transmission line limits maximum coupling, as modeled in (11)-(13). Peak coupling occurs at \(\theta=100^{\circ}\), beyond which coupling decreases--an effect amplified by adjacent resonator field interactions. The secondary peak remains a local maximum, never reaching the principal peak's magnitude.  
Based on the advantages and drawbacks outlined in Table 3, distance-based coupling emerges as the preferred approach for inter-resonator coupling.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Coupling-Based Distance** & **Coupling-Based TL** \\ \hline Behavior & Increasing w/frequency & Phase-dependent coupling \\ Max. Coupling (C) & High & Medium \\ Parasitic Influence & Higher & Lower \\ Spurious Modes & N/A & TL mode (Q-TEM) \\ Size & Compact & Compact \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of inter-resonator coupling methods.  
Figure 18: Inter-resonator coupling characterization.