1. Notes 13 ECE 5317-6351 Microwave Engineering
Fall 2011
Fall 2015
Prof. David R. Jackson
Dept. of ECE
Notes 13
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 KCL 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 transmission line with reactive loads at the ends. x
x = L L
We wish to find the resonance frequencies of this transmission-line resonator.
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 x
x = x0
x = L
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: R x
x = x0
x = L R +
V r -
V l -
I r
I l
x = x0

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

7. Summary R
TRE: or

8. Derive the resonance frequency of a simple 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. R L C

10. RLC Resonator (cont.)
The TRE is obtained. R L C

11. RLC Resonator (cont.)

12. Hence, the plus sign is the correct choice.
RLC Resonator (cont.)
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. RLC Resonator (cont.) Hence, we have We can write this as where
complex resonance frequency
We can write this as
where

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

15. is often denoted simply as 0 in this equation.
RLC Resonator (cont.)
The Q for a resonator is in general defined as:
Note:
is often denoted simply as 0 in this equation.

16. RLC Resonator (cont.) For the RLC resonator we have: Hence so V0 + R L- V0
Phasor domain
Hence so

17. In the time domain we have:
RLC Resonator (cont.)
In the time domain we have: R L C + - V0
In the phasor domain:

18. RLC Resonator (cont.) R L C + - V0

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

20. Transmission Line Resonator (cont.)x L
Apply TRE:

21. Transmission Line Resonator (cont.)

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

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

24. Rectangular Resonator
Derive a transcendental equation for the resonance frequencies of a rectangular resonator. y z x
PEC boundary a b h
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.

25. (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: z h R
We choose a reference plane at z = 0+.
(Choose Z0 to be the wave impedance.)
Hence

26. Rectangular Resonator (cont.)
Hence z h R

27. Rectangular Resonator (cont.)
Also, we have z h R
Hence, we have

28. Note: 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.
TEmnp mode: or
The lowest mode is the TE101 mode.

29. 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.

30. Excitation of Resonatory z x
PEC boundary a b h
Practical excitation by a coaxial probe Lp
(Probe inductance)
Tank (RLC) circuit R L C
Circuit model

31. Excitation of Resonator (cont.)
(from circuit theory – see next slide)
where R L C
Approximate form (see next slide):

32. Excitation of Resonator (cont.)
Derivation of ZRLC formula:

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

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

35. 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.

36. 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.

37. Grounded Dielectric Slab (cont.)x z H E
TMx

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

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

40. 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!

41. Note: The modes are hybrid in the z direction (not TEz or TMz).
Waveguide With Slab a x b w y
Choose this representation:
TExmn modes
TMxmn modes
Note: The modes are hybrid in the z direction (not TEz or TMz).
TEN:
L = (a-w) w

42. Waveguide With Slab (cont.)
TEN:
L = (a-w) w x R
TRE:

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

44. Waveguide With Slab (cont.)
Final transcendental equation for the unknown wavenumber kz:
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, …

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