1. Microwave Engineering
ECE
Microwave Engineering
Adapted from notes by Prof. Jeffery T. Williams
Fall 2011
Fall 2018
Prof. David R. Jackson
Dept. of ECE
Notes 15
Transverse Resonance Method

2. This leads to a “Transverse Resonance Equation (TRE).”
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 (resonators)
The transverse resonance method involves establishing a reference plane and enforcing the KVL and KCL.
This leads to a “Transverse Resonance Equation (TRE).”

3. Transverse Resonance Method
To illustrate the method, consider a lossless resonator formed by a lossless transmission line with reactive loads at the ends.
We wish to find the resonance frequencies of this transmission-line resonator.
(Here we develop the method. We will do the actual algebra for this structure a little later.)
A resonator can have nonzero fields at a resonance frequency,
when there is no source.

4. Transverse Resonance Method (cont.)
We start by selecting an (arbitrary) reference plane R.
R = reference plane at arbitrary x = x0
Note:
Although the location of the reference plane is arbitrary, a good choice will often simplify the derivation of the TRE and the complexity of the final TRE.

5. Transverse Resonance Method (cont.)
Examine the voltages and currents at the reference plane:

6. Transverse Resonance Method (cont.)
Define impedances:
Boundary conditions:
Hence:
TRE

7. Summary
TRE: or

8. Derive the resonance frequency of a parallel RLC resonator.
Example:
Derive the resonance frequency of a parallel RLC resonator.
Lossless:
At the resonance frequency, voltages and currents exist with no sources.

9. A reference plane is chosen.
RLC Resonator (cont.)
A reference plane is chosen.

10. RLC Resonator (cont.)
The TRE is obtained.

11. Solve the quadratic equation.
RLC Resonator (cont.)
Solve the quadratic equation.

12. RLC Resonator (cont.) Hence, the plus sign is the correct choice.
Factor out 4LC from the square root.
For the lossless limit, G  0:
(must be a positive real number)
Hence, the plus sign is the correct choice.

13. complex resonance frequency
RLC Resonator (cont.)
Hence, we have
complex resonance frequency
We can write this as
where

14. quality factor of RLC resonator
RLC Resonator (cont.)
Ratio of imaginary and real parts of complex frequency: so or
where
quality factor of RLC resonator

15. In the time domain we have:
RLC Resonator (cont.)
In the time domain we have:
In the phasor domain:

16. RLC Resonator (cont.)

17. Q of a General Resonator
The quality factor (Q) for a general resonator is defined as:
Note:
is often denoted simply as 0 in this equation.

18. Q for RLC Resonator
For the RLC resonator we have:
Phasor domain
Hence

19. Q for RLC Resonator (cont.)
For the RLC resonator we have:
Similarly:
Phasor domain
Hence so

20. General Q Formulas
These formulas hold for any resonator:

21. Transmission Line Resonator
Example:
Derive a transcendental equation for the resonance frequency of this lossless transmission-line resonator.
We choose a reference plane at x = 0+.
Note: The load reactances may be functions of frequency.

22. Transmission Line Resonator (cont.)
Apply TRE:

23. Transmission Line Resonator (cont.)

24. Transmission Line Resonator (cont.)
After simplifying, we have
Special cases:

25. Transmission Line Resonator (cont.)
For the resonance frequencies, we have
We then have

26. Rectangular Resonator
Example:
Derive a transcendental equation for the resonance frequencies of a rectangular resonator.
Orient the structure so that
b < a < h
The structure is thought of as supporting rectangular waveguide 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.

27. (Choose Z0 to be the wave impedance.)
Rectangular Resonator (cont.)
We use a Transverse Equivalent Network (TEN) to model any one of the waveguide modes:
We choose a reference plane at z = 0+:
(Choose Z0 to be the wave impedance.)
Hence

28. Rectangular Resonator (cont.)
Hence

29. Rectangular Resonator (cont.)
Also, we have
Hence, we have

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

31. The other field components, Ey and Hx, can be found from Hz.
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.

32. Rectangular Resonator (cont.)
Here we examine the Q of the resonator:
(dielectric and conductor loss)
We now allow for dielectric and conductor loss in the resonator.

33. Rectangular Resonator (cont.)
Q of TE101 mode

34. (This holds for any mode.)
Rectangular Resonator (cont.)
Q of TE101 mode
Results (from Pozar book):
This gives:
(This holds for any mode.)

35. Excitation of Resonatorh
Practical excitation by a coaxial probe
Circuit model:
(probe inductance)
Tank (RLC) circuit
Note:
The value of the circuit elements will depend on where the resonator is fed, and also the size of the probe. The values of Q and 0 do not depend on the feed.

36. Excitation of Resonator (cont.)
(from circuit theory – see next slide)
where
Approximate form (see slide 35):

37. Excitation of Resonator (cont.)
Derivation of ZRLC formula:
Note:
Red color highlights where equivalent substitutions have been made. Blue color highlights where terms combine to unity.

38. Excitation of Resonator (cont.)
Derivation of approximate form:

39. Excitation of Resonator (cont.)
Note:
A larger Q of the resonator (less loss),
means a more sharply peaked response, and a larger R value.
Lossless resonator:

40. Excitation of Resonator (cont.)
High Q resonator
Low Q resonator
Note:
A larger Q of the resonator (less loss),
means a more sharply peaked response, and a larger R value.

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

42. Grounded Dielectric Slab (cont.)
We think of the transmission lines in the TEN running in the x direction.

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

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

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

46. The modes are hybrid in the z direction (not TEz or TMz).
Waveguide With Slab
Choose this representation:
TExmn modes
TMxmn modes
Note:
The modes are hybrid in the z direction (not TEz or TMz).
TEN:

47. Waveguide With Slab (cont.)
TEN:
TRE:

48. Waveguide With Slab (cont.)
Choose TEx:
Separation equations: so

49. Waveguide With Slab (cont.)
Final transcendental equation for the unknown wavenumber kz:
TEx:
with
Note:
The integer n is arbitrary but fixed.
The equation has an infinite number of solutions for kz for a given n: m = 1, 2, 3, …

50. Waveguide With Slab (cont.)
Limiting case: w  0:
This mode becomes the usual TE10 mode of the hollow waveguide.
Hence, we have

51. Waveguide With Slab (cont.)
Longitudinal section plane (yz plane)
Alternative notation:
Another notation that is common in the literature for slab-loaded waveguides is the designation of “LSE” and “LSM” modes.
LSE mode: “longitudinal section electric” mode
LSM mode: “longitudinal section magnetic” mode
LSE: TExmn (The electric field vector stays in the “longitudinal section”.)
LSM: TMxmn (The magnetic field vector stays in the “longitudinal section”.)