SS-V: 9004 Resonator with Conducting Material

Test No. VE05Modal analysis in resonator filled with conducting material

Definition

The details of the model are:

Figure 1: a = 20mm, b = 10 mm, L = 40 mm
All walls are under PEC (Perfect Electric Conductor) conditions

The material properties are:

Properties: Value
Dielectric relative permittivity ( $ε_{r}$ ): 1.0
Dielectric relative permeability ( $μ_{r}$ ): 1.0
Conductivity ( $σ$ ): 1.0 [S/m]

Reference Solution

Equation for the electric field in this case is^¹:

\nabla \times (\frac{1}{μ} \nabla \times E) + ε \frac{\partial^{2} E}{\partial t^{2}} + σ \frac{\partial E}{\partial t} = 0

For harmonic waves (with the time dependence $e^{j ω t}$ ) under the conditions of this problem it takes the following form:

\nabla^{2} E_{ω} + {(\frac{ω}{c})}^{2} E_{ω} - j σ η (\frac{ω}{c}) E_{ω} = 0

Where,

$μ = μ_{r} μ_{0}$: Magnetic permeability
$ε = ε_{r} ε_{0}$: Dielectric permittivity
$η = \sqrt{\frac{μ}{ε}}$: Intrinsic impedance of the material
$c = \sqrt{\frac{1}{ε μ}}$: Speed of light in the material

Spatial dependence of the standing waves in the above resonator is described by a product of sin or cos functions for each coordinate representing fixed number of half wavelengths in those directions. So, the action of the Laplacian in the above equation reduces to multiplication by the following number with the negative sign^²:

{k_{m n l}}^{2} = {(\frac{πm}{a})}^{2} + {(\frac{πn}{b})}^{2} + {(\frac{πl}{L})}^{2}

Finding the eigen frequencies reduces to solving the quadratic equation:

{(\frac{ω}{c})}^{2} - j σ η (\frac{ω}{c}) - {k_{m n l}}^{2} = 0

γ = α + j β = j {(\frac{ω}{c})}_{m n l} = \frac{(- σ η \pm j \sqrt{- {(σ η)}^{2} + 4 {k_{m n l}}^{2}})}{2}

Where,

$α$: Attenuation constant
$β$: Phase constant (wave number)

If ${(σ η)}^{2} \geq 4 {k_{m n l}}^{2}$ , the phase constant is zero and no oscillations are observed. If such mode were excited it would be decaying to zero. For ${(σ η)}^{2} < 4 {k_{m n l}}^{2}$ , the mode is oscillating and decaying at a rate defined by time constant $τ = \frac{1}{(c α)} = \frac{2}{c σ η}$ that is independent of frequency $f = \frac{c β}{2 π}$ .

Results

Comparison of the theoretical values f and

\frac{1}{τ}

for the

T E_{10 l}

modes with those obtained in the modeling is presented in the picture below.

Figure 2. Comparison of theoretical and model resonant parameters (complex conjugate values are not shown)

The figure below demonstrates electric field distribution in one of the modes.

Figure 3. Real part of the electric field $E_{y}$ distribution for the $T E_{103}$ mode

¹ Jianming, J., The finite element method in electromagnetics, 3rd Edition, John Wiley & Sons, Inc., 2014, Ex.12.8.

² Pozar, D.M., Microwave Engineering, 4th Edition, John Wiley & Sons, Inc., 2012, §6.3.