Transverse Resonance MethodThis is a general method that can be used to help us calculatevarious important quantities:2Wavenumbers for complicated waveguiding structures (dielectric-loaded waveguides, surface waves, etc.)Resonance frequencies of resonant cavitiesWe do this by deriving a “Transverse Resonance Equation (TRE).”
Transverse Resonance Equation (TRE)R = reference plane at arbitraryx = x0To illustrate the method, consider a lossless resonator formed by atransmission line with reactive loads at the ends.We wish to find the resonance frequency of this transmission‐line resonator.3R0Z0Zxx = x022LLZjX11LLZjXx = L
Examine the voltages and currents at the reference plane:R+V r-+V l-I rI lx = x0 4TRE (cont.)R0Z0Zxx = x022LLZjX11LLZjXx = L
TRE (cont.)Hence:Define impedances:rinrlinlVZIVZIininZZ rlrlVVIIBoundary conditions:R+V r-+V l-I rI lx = x0 inZinZx5
ininininZZYY TRE (cont.)RinZinZorTRENote about the reference plane: Although the location of the reference plane is arbitrary, a “good” choice will keep the algebra to a minimum.6
Example00,rZkkx22LLZjX11LLZjXLDerive a transcendental equation for the resonance frequency of thistransmission‐line resonator.