SS-V: 9003 Resonator with Two Materials

Test No. VE04Analysis of modes in resonator with two materials

Definition

The details of the model are:

Figure 1: a1 = 5mm, a2 = 15mm, a = a1 + a2, b = 10 mm, L = 100 mm
All walls are under PEC (Perfect Electric Conductor) conditions

The material properties are:

Properties: Value
Dielectric relative permittivity ( $ε_{r}$ ): 4.0
Dielectric relative permeability ( $μ_{r}$ ): 1.0
Air relative permeability ( $ε_{r}$ ): 1.0
Air relative permeability ( $μ_{r}$ ): 1.0

Reference Solution

Figure 2. Cross section of the resonator

We are looking for the

T E_{10 l}

modes (the one with l=1 is the fundamental mode), which means that there is no dependence on y^¹.

Equations for longitudinal magnetic field become:

(\frac{\partial^{2}}{\partial x^{2}} + {k_{D}}^{2}) H_{z} = 0 for 0 \leq x \leq a_{1}

(\frac{\partial^{2}}{\partial x^{2}} + {k_{A}}^{2}) H_{z} = 0 for a_{1} \leq x \leq a

Where,

{k_{A}}^{2} = {k_{0}}^{2} - {(\frac{πl}{L})}^{2}

and

{k_{D}}^{2} = {ε_{r} k_{0}}^{2} - {(\frac{πl}{L})}^{2}

Depending on the combination of the model parameters both ${k_{A}}^{2}$ and ${k_{D}}^{2}$ are positive or ${k_{D}}^{2}$ is positive and ${k_{A}}^{2}$ is negative.

In the first case the value of $k_{0}$ is obtained from the solution of the following equation:

k_{A} \tan (k_{D} a_{1}) + k_{D} \tan (k_{A} a_{2}) = 0

This is after substituting ${k_{A}}^{2}$ and ${k_{D}}^{2}$ from their definitions above^¹.

In the case of the negative value of ${k_{A}}^{2}$ in a similar way the value of $β$ is obtained from the solution of the equation (this case is complementary to the one in Pozar^¹):

k_{A} \tan (k_{D} a_{1}) + k_{D} \tan h (k_{A} a_{2}) = 0

Where,

{k_{A}}^{2} = - {k_{A}}^{2} = - {k_{0}}^{2} + {(\frac{πl}{L})}^{2}

and it is positive.

The roots of those transcendental equations can be found numerically or graphically depending on required accuracy.

In the first case the electric field distribution is:

In the second case the electric field distribution is:

Results

Comparison of the theoretical resonant frequencies for the

T E_{10 l}

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

Figure 5. Comparison of theoretical and model resonant frequencies

The z dependence of the field is simple and was followed in the solution with high accuracy. It is interesting to check, how accurate is the x dependence reproduced in the model. Electric field magnitude was output at fixed z corresponding to one of its maximums and normalized to that maximum. Below, this is illustrated for the case of l=3.

Figure 6. Electric field magnitude $E (X, Z)$ distribution for the $T E_{103}$ mode

As it is described above for some values of l the solution on both sides of the boundary is described by the sin function, for other values (in our case – starting from l=7) the sin function in the area with air switches to the hyperbolic sin. This can be observed in the figure below.