Equivalent lumped elements G, L, C, and unloaded Q's of closed- and open-loop ring resonators

A transmission-line model is used to extract the equivalent lumped-element circuits for the closed- and open,loop ring resonators. The unloaded Q's of the ring resonators can be calculated from the equivalent lumped elements G, L, and C. Four different configurations of microstrip ring resonators fabricated on low and high dielectric-constant substrates are used to investigate the lumped elements and unloaded Q's. The largest difference between the measured and calculated unloaded Q is 5.7%, which is due to measurement uncertainties and accuracies of the calculation. These simple expressions introduce an easy method for analyzing ring resonators in filters and provide, for the first time, a means of predicting their unloaded Q.