# Transverse resonance method by malj

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```									    ECE 5317-6351
Microwave Engineering
Fall 2011
Fall 2011
Prof. David R. Jackson
Dept. of ECE

Notes 13
Transverse Resonance
Method

1
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-
§ Resonance frequencies of resonant cavities

We do this by deriving a “Transverse Resonance Equation (TRE).”

2
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.

3
TRE (cont.)
R

x            x = x0    x=L

Examine the voltages and currents at the reference plane:

Il       R            Ir

+                      +
Vl                     Vr
-                      -

x = x0
4
TRE (cont.)
R         Ir
Il

+                +
Vl               Vr
-                -
x
x = x0

Define impedances:    Boundary conditions:   Hence:

5
TRE (cont.)
R

TRE

plane: Although the
location of the reference
or            plane is arbitrary, a
“good” choice will keep
the algebra to a minimum.

6
Example
Derive a transcendental equation for the resonance frequency of this
transmission-line resonator.

L

x

We choose a reference plane at x = 0+.

7
Example (cont.)
L

R

x

Apply TRE:

8
Example (cont.)

9
Example (cont.)

After simplifying, we have

Special cases:

10
Rectangular Resonator
Derive a transcendental equation for the resonance frequency of a
rectangular resonator.
z

PEC boundary
Orient so that
b < a <h                 h
y
a
x
b

The structure is thought of as supporting RWG modes bouncing back
and forth in the z direction.
The index p describes the
We have TMmnp and TEmnp modes.       variation in the z direction.
11
Rectangular Resonator (cont.)

We use a Transverse Equivalent Network (TEN):

h

z                                  We choose a reference
plane at z = 0+.

Hence                                           12
Rectangular Resonator (cont.)
Hence

h

z

13
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.                   14
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.
15
Rectangular Resonator (cont.)
z
Practical excitation
by a coaxial probe
PEC boundary

h
y
a
x
b
Lp (Probe inductance)

R    L C        Circuit model
Tank (RLC) circuit

16
Rectangular Resonator (cont.)

Q = quality factor of resonator

Lp (Probe inductance)

R   L C             Circuit model
Tank (RLC) circuit

17
Rectangular Resonator (cont.)

18
Grounded Dielectric Slab
Derive a transcendental equation for wavenumber of the TMx surface
waves by using the TRE.

x

h
z

Assumption: There is no variation of the fields in the y direction,
and propagation is along the z direction.

19
Grounded Dielectric Slab

x

TMx    H
E
z

20
TMx Surface-Wave Solution
h    R

The reference plane is
TEN:                     chosen at the interface.

x

21
TMx Surface-Wave Solution (cont.)
TRE:

22
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!
23

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