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Symmetrical transmission line ring resonators are studied using the generalised reflection coefficient concept and the odd- and even-mode analysis method. The mode distributions for closed, tunable and selectively damped rings are derived. A normlised tuning characteristic is generated for a capacitively tunable ring, showing agreement with derivations based on current continuity in the limit of loose coupling. The perturbation of resonant frequencies by nonzero coupling is also calculated. An optimum value for an even-mode damping resistance is derived. A circuit simulator is used to demonstrate that the features derived analytically for lossless lines persist when realistic values of loss and dispersion are introduced.
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Gardner et al. (1996) studied this question.
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