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

2. 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).”

3. 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.

4. 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

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

6. 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

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

8. Example (cont.) R x L
Apply TRE:

9. Example (cont.)

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

11. 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.

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

13. Rectangular Resonator (cont.)
Hence z h

14. 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.

15. 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.

16. 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

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

18. Rectangular Resonator (cont.)

19. 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.

20. Grounded Dielectric Slabx z H E
TMx

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

22. TMx Surface-Wave Solution (cont.)
TRE:

23. 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!