Frequency-differential representation of losses to express quality factor in terms of reactance-slope parameters for three-dimensional, arbitrarily-shaped resonators

The use of reactance slope parameters for the design of RF and microwave filters was drawn from lumped-element circuit representations. In recent work, a field-theoretic framework was established to represent the quality factor in terms of reactance-slope parameters for electrically-large, arbitrarily-shaped, three dimensional resonators. The work presented in this paper develops a relationship for use in the denominator of the field-theoretic representation for modal Q that involves strictly surface-based field and source quantities, even for losses distributed throughout the resonator volume. By doing so the field-theoretic representation for Q becomes a strictly surface field and source dependent representation with the attendant reduced level of dimensionality (two-dimensional instead of three-dimensional integrals) and with a potentially corresponding reduction in computational complexity. The relationships presented here are used to evaluate the effect of dielectric losses on the unloaded Q for TE₁₀₁-mode rectangular resonators comprised of silver, copper, aluminum, and brass and are used to compare unloaded Q values for air- and fused-quartz-filled resonators. Results are also presented for circular cylindrical resonators comprised of the same lossy metals and lossy dielectrics.