Notes 13 ECE 5317-6351 Microwave Engineering Fall 2011 Fall 2011 Prof. David R. Jackson Dept. of ECE Notes 13 Transverse Resonance Method

Transverse Resonance Method This is a general method that can be used to help us calculate various important quantities: Wavenumbers for complicated waveguiding structures (dielectric- loaded waveguides, surface waves, etc.) Resonance frequencies of resonant cavities We do this by deriving a “Transverse Resonance Equation (TRE).”

Transverse Resonance Equation (TRE) To illustrate the method, consider a lossless resonator formed by a transmission line with reactive loads at the ends. R x x = x0 x = L R = reference plane at arbitrary x = x0 We wish to find the resonance frequency of this transmission-line resonator.

TRE (cont.) R x x = x0 x = L Examine the voltages and currents at the reference plane: I r I l R + V l - + V r - x = x0

TRE (cont.) R I r I l + V l - V r - x Define impedances: x = x0 x Define impedances: Boundary conditions: Hence:

TRE (cont.) R TRE Note about the reference plane: Although the location of the reference plane is arbitrary, a “good” choice will keep the algebra to a minimum. or

Example Derive a transcendental equation for the resonance frequency of this transmission-line resonator. x L We choose a reference plane at x = 0+.

Example (cont.) R x L Apply TRE:

Example (cont.)

Example (cont.) After simplifying, we have Special cases:

Rectangular Resonator Derive a transcendental equation for the resonance frequency of a rectangular resonator. y z x PEC boundary a b h Orient so that b < a < h The structure is thought of as supporting RWG modes bouncing back and forth in the z direction. The index p describes the variation in the z direction. We have TMmnp and TEmnp modes.

Rectangular Resonator (cont.) We use a Transverse Equivalent Network (TEN): z h We choose a reference plane at z = 0+. Hence

Rectangular Resonator (cont.) Hence z h

Rectangular Resonator (cont.) Solving for the wavenumber we have Hence Note: The TMz and TEz modes have the same resonance frequency. TEmnp mode: or The lowest mode is the TE101 mode.

Rectangular Resonator (cont.) TE101 mode: Note: The sin is used to ensure the boundary condition on the PEC top and bottom plates: The other field components, Ey and Hx, can be found from Hz.

Rectangular Resonator (cont.) y z x PEC boundary a b h Practical excitation by a coaxial probe Lp (Probe inductance) Tank (RLC) circuit R L C Circuit model

Rectangular Resonator (cont.) Q = quality factor of resonator Lp (Probe inductance) Tank (RLC) circuit R L C Circuit model

Rectangular Resonator (cont.)

Grounded Dielectric Slab Derive a transcendental equation for wavenumber of the TMx surface waves by using the TRE. x z h Assumption: There is no variation of the fields in the y direction, and propagation is along the z direction.

Grounded Dielectric Slab x z H E TMx

TMx Surface-Wave Solution h The reference plane is chosen at the interface. TEN: x

TMx Surface-Wave Solution (cont.) TRE:

TMx Surface-Wave Solution (cont.) Letting We have or Note: This method was a lot simpler than doing the EM analysis and applying the boundary conditions!