A field theoretic foundation for the representation of quality factor in terms of reactance-slope parameters for electrically large 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. This paper provides a field-theoretic foundation for reactance-slope parameters based upon the treatment of arbitrarily shaped electrically large resonators comprised of electric and magnetic conducting enclosures. Traditional representations for these parameters in terms of the reactance for series-connected network elements and susceptance for parallel-connected network elements are related to electromagnetic fields. The resulting generalized expressions for reactance-slope parameters with utility for evaluating quality (Q) factor are developed for both electric and magnetic conducting resonators using strictly surface electromagnetic fields and sources. The generalized expressions are used to evaluate the unloaded Q for TE₁₀₁-mode rectangular resonators comprised of silver, copper, aluminum, and brass based upon two different standard rectangular waveguides with enclosed ends. Q results are also presented for TM₀₁₁-mode circular cylindrical resonators comprised of the same conducting materials. Results from the method here are compared to well-established results published in the literature. These expressions are useful for either analytically or numerically evaluating the reactance-slope parameter and Q factor.