Computing Q-Factors and Resonant Frequencies of Cavity Resonators

A classic benchmark example in computational electromagnetics is to find the resonant frequency and Q-factor of a cavity with lossy walls. Here, models of rectangular, cylindrical, and spherical cavities are shown to be in agreement with analytic solutions.

Model Definition

This example considers three geometries:

•

a rectangular cavity of dimensions 0.9 in-by-0.9 in-by-0.4 in;

•

a cylindrical cavity of radius 0.48 in and height 0.4 in; and

•

a spherical cavity of radius 1.35 cm.

The cavity walls are assumed to be a good conductor, such as copper, with an electric conductivity of 5.7·107 S/m, and relative permeability and permittivity of unity. The interior of the cavity is assumed to be vacuum, with zero electric conductivity, and unit permeability and permittivity. The analytic solutions to these three cases are given in Ref. 1.

The lossy walls of the cavity are represented via the impedance boundary condition. This boundary condition accounts for the frequency dependent losses on the walls of a cavity due to the nonzero electric conductivity, which makes the eigenvalue problem nonlinear. When solving any eigenvalue problem, it is necessary to provide a frequency around which to search for modes. In addition, when solving a nonlinear eigenvalue problem, it is also necessary to provide a frequency at which to initially evaluate the frequency-dependent surface losses. Although the guesses for these frequencies do not need to be very close, solution time is less the closer they are.

It is usually possible to estimate the resonant frequency of interest, and to use this as an initial guess. It is also possible to quickly estimate the resonant frequency by building a second model that uses the perfect electrical conductor (PEC) boundary condition instead of the impedance boundary condition. A model that uses only PEC boundaries results in a linear eigenvalue problem, and is less computationally intensive to solve. Such a model only requires a rough guess at the frequency of the mode, and does not require a frequency at which to evaluate the surface losses. Therefore, it is often convenient to also solve a version of a model without losses.

Q-Factor and Resonant Frequency in Cavity Structures

Q-factor is one of important parameters characterizing a resonant structure and defined as Q = ω (average energy stored/dissipated power). The average energy stored can be evaluated as a volume integral of Energy density time average (emw.Wav) and the dissipated power can be evaluated as a surface integral of Surface losses (emw.Qsh).

Another way to calculate Q-factor at the dominant mode is via equations in Ref. 1. For a rectangular cavity, the dominant mode is TE101, at which the cavity provides the lowest resonant frequency. The Q-factor and resonant frequency at this mode is

There are two dominant modes for a cylindrical cavity. One dominant mode of the cylindrical cavity is TE111 when the ratio between the height and radius is more than 2.03. The other dominant mode is TM010 when the ratio is less than 2.03. For this case, the Q-factor and resonant frequency are given as

For a spherical cavity, TM mode provides the lowest resonant frequency.

In the above equations, Rs is surface resistance defined as

and η is the characteristic impedance of free space,

These two analytical approaches are compared with the Q-factor obtained from Eigenfrequency analysis.

Results and Discussion

The analytic resonant frequencies and Q-factors for these three cases, and the results of the COMSOL model for various levels of mesh refinement, are shown below. These show that the solutions agree. As the mesh is refined, the polynomial basis functions used by the finite element method better approximate the analytic solutions, which are described by sinusoidal functions for the rectangular cavity and Bessel functions for the cylindrical and spherical cavities. This difference between the numerical results and the analytic solution is discretization error, and is always reduced with mesh refinement.

Table 1: Results for the TE101 mode of a rectangular cavity.
maximum mesh size	Resonant frequency, GHz (Analytic=9.273)	Q-factor (Analytic=7770)
h_max	9.706	7039
h_max/2	9.283	7687
h_max/4	9.273	7765
h_max/8	9.273	7770

Table 2: Results for the TM010 mode of a cylindrical cavity.
maximum mesh size	Resonant frequency, GHz (Analytic=9.412)	Q-factor (Analytic=8065)
h_max	9.458	7891
h_max/2	9.419	8004
h_max/4	9.411	8056
h_max/8	9.411	8065

Table 3: Results for the TM011 mode of a spherical cavity.
maximum mesh size	Resonant frequency, GHz (Analytic=9.698)	Q-factor (Analytic=14594)
h_max	9.752	14121
h_max/2	9.723	14430
h_max/4	9.701	14616
h_max/8	9.697	14641

Note that convergence with respect to the mesh is fastest for the rectangular cavity and slowest for the spherical cavity. This is because the isoparametric finite-element mesh represents curved surfaces approximately, via second order polynomials by default. This introduces some small geometric discretization error that is always reduced with mesh refinement. Although it is possible to use different element orders, the default second-order curl element (also known as a vector or Nedelec element) is the best compromise between accuracy and memory requirements. Because memory requirements for three-dimensional models and direct solvers increase close to quadratically with increasing number of unknowns, there is strong motivation to use as coarse a mesh as reasonable. For larger models, one may use an iterative solver that scales more favorably with the number of unknowns but then solution time typically goes up substantially. Figure 1 shows the fields within the cavities, as well as the surface currents and surface losses.

Figure 1: Arrow plots of electric and magnetic fields. Slice plot of electric field.

Figure 2: Arrow plots of surface currents. Surface plot of surface losses.

Notes About the COMSOL Implementation

Solve this example using an Eigenfrequency study. Search for a single eigenfrequency around 9·109 Hz. Because of the impedance boundary condition with a finite conductivity value, the model becomes a nonlinear eigenvalue problem and it is necessary to provide a frequency at which to initially evaluate the frequency-dependent surface losses. In the Eigenvalue Solver settings window you can see the linearization point is automatically specified to the value in “Search for eigenfrequencies around” in the study settings.

Reference

1. C.A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, 1989.

RF_Module/Verification_Examples/cavity_resonators

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file cavity_resonators_parameters.txt.

Here, mu0_const and epsilon0_const in the imported table are predefined COMSOL constants for the permeability and permittivity in free space. From the Value column you can read off the values f_TE101_analytic_r = 9.273 GHz, Q_TE101_analytic_r = 7770 for the rectangular cavity, f_TM010_analytic_c = 9.412 GHz, Q_TM010_analytic_c = 8065 for the cylindrical cavity, f_TM011_analytic_s = 9.698 GHz, and Q_TM011_analytic_s = 14594 for the spherical cavity.

Since air and lossy wall materials will be used on multiple components, add them on the global material node. They will be linked to each individual component later on.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click

In the toolbar, click

to close the window.

Global Definitions

Lossy Wall

In the window, under right-click and choose .

In the window for , type Lossy Wall in the text field.

Click to expand the section. In the tree, select .

Click

In the tree, select .

Click

In the tree, select .

Click

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	sigma_wall	S/m	Basic

Geometry 1

Create a block for the rectangular cavity.

Block 1 (blk1)

In the toolbar, click

In the window for , locate the section.

In the text field, type a_r.

In the text field, type b_r.

Click

Electromagnetic Waves, Frequency Domain (emw)

Now set up the physics. Override the default perfect electric conductor condition on the exterior boundaries by an impedance condition.

Impedance Boundary Condition 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Materials

Assign material properties on the model by linking the global material already created. First, apply air to all domains.

Material Link 1 (matlnk1)

In the window, under right-click and choose .

Material Link 2 (matlnk2)

Right-click and choose .

Define a lossy conductive material for all exterior boundaries.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Definitions

Add variables for Q-factor calculation and visualization. For this Q-factor calculation, add two nonlocal integration couplings: one for volume and the other for surface integration.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , type int_v in the text field.

Locate the section. From the list, choose .

Integration 2 (intop2)

In the toolbar, click

and choose .

In the window for , type int_s in the text field.

Locate the section. From the list, choose .

From the list, choose .

Variables 1

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file cavity_resonators_model1_variables.txt.

The emw. prefix is for the Electromagnetic Waves, Frequency Domain interface in the first model. Wav and Qsh are Energy density time average and Surface losses, respectively. Qfactor included in this text file shows up in orange indicating an unknown variable. It will be known after solving the model.

Mesh 1

The maximum mesh size is one dimension of the cavity scaled inversely by d_f, a discretization factor defined in Parameters. The discretization factor is also used as a parametric sweep variable to see the effect of the mesh refinement.

Free Tetrahedral 1

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type h_max_r/d_f.

In the text field, type 2.

In the text field, type 1.

In the text field, type 0.1.

Click

Provide the number of modes and a frequency around which to search for modes.

Study 1

Step 1: Eigenfrequency

In the window, under click .

In the window for , locate the section.

Select the check box.

In the associated text field, type 1.

In the text field, type 9[GHz].

Add a Parametric Sweep over the discretization factor, d_f.

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
d_f (Discretization factor)	1 2 4 8

In the toolbar, click

Results

Electric Field (emw)

The default plot shows the distribution of the norm of the electric field. Add arrow plots of the electric and magnetic fields.

Arrow Volume 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

In the toolbar, click

Arrow Volume 2

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. Find the subsection. In the text field, type 1.

Locate the section. From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

Compare the resulting plot with that shown in Figure 1, top. The exact numbers that you get may differ slightly.

Add a surface plot of the surface losses and an arrow plot of the surface current (Figure 2, top).

3D Plot Group 2

In the toolbar, click

and choose .

Surface 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

Surface Losses (emw)

In the window, under click .

In the window for , type Surface Losses (emw) in the text field.

Arrow Surface 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

In the toolbar, click

Root

Next, set up a model for the cylindrical cavity.

Add Component

In the window, right-click the root node and choose .

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

You will copy the settings from the existing study later on.

Click in the window toolbar.

In the toolbar, click

to close the window.

Geometry 2

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type a_c.

In the text field, type height_c.

Click

Electromagnetic Waves, Frequency Domain 2 (emw2)

Set up the second physics interface. The steps are same as for the first model.

In the window, under click .

Impedance Boundary Condition 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Materials

Assign material properties on the second model. Apply air to all domains.

Material Link 3 (matlnk3)

In the window, under right-click and choose .

Material Link 4 (matlnk4)

Right-click and choose .

Define a lossy conductive material for all exterior boundaries.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Definitions (comp2)

Add variables and two nonlocal integration couplings. The purpose of these are the same as in the first model.

Integration 3 (intop3)

In the toolbar, click

and choose .

In the window for , type int_v in the text field.

Locate the section. From the list, choose .

Integration 4 (intop4)

In the toolbar, click

and choose .

In the window for , type int_s in the text field.

Locate the section. From the list, choose .

From the list, choose .

Variables 2

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file cavity_resonators_model2_variables.txt.

The emw2. prefix refers to the Electromagnetic Waves, Frequency Domain interface for the second model.

Mesh 2

Apply the same logic in the mesh set up as you have done in the first model.

Free Tetrahedral 1

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type h_max_c/d_f.

In the text field, type 2.

In the text field, type 1.

In the text field, type 0.1.

Click

Study 1

Parametric Sweep, Step 1: Eigenfrequency

In the window, under , Ctrl-click to select and .

Right-click and choose .

Study 2

Parametric Sweep

In the window, right-click and choose .

Study 2

Step 1: Eigenfrequency

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Physics interface	Solve for	Discretization
Electromagnetic Waves, Frequency Domain (emw)		Physics settings
Electromagnetic Waves, Frequency Domain 2 (emw2)	√	Physics settings

In the toolbar, click

Results

Arrow Volume 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

In the toolbar, click

Arrow Volume 2

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. Find the subsection. In the text field, type 1.

Locate the section. From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

The plot should now look like that in Figure 1, middle.

Again, add a surface plot of the surface losses and an arrow plot of the surface current (Figure 2, middle).

3D Plot Group 4

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Surface 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

Surface Losses (emw2)

In the window, under click .

In the window for , type Surface Losses (emw2) in the text field.

Arrow Surface 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

In the toolbar, click

Root

Now add a model for the spherical cavity.

Add Component

In the window, right-click the root node and choose .

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Geometry 3

Sphere 1 (sph1)

In the toolbar, click

In the window for , locate the section.

In the text field, type a_s.

Click

Electromagnetic Waves, Frequency Domain 3 (emw3)

Set up the third physics interface.

In the window, under click .

Impedance Boundary Condition 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Materials

Assign material properties on the third model. Apply air to all domains.

Material Link 5 (matlnk5)

In the window, under right-click and choose .

Material Link 6 (matlnk6)

Right-click and choose .

Define a lossy conductive material for all exterior boundaries.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Definitions (comp3)

Add variables and two nonlocal integration couplings.

Integration 5 (intop5)

In the toolbar, click

and choose .

In the window for , type int_v in the text field.

Locate the section. From the list, choose .

Integration 6 (intop6)

In the toolbar, click

and choose .

In the window for , type int_s in the text field.

Locate the section. From the list, choose .

From the list, choose .

Variables 3

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file cavity_resonators_model3_variables.txt.

The emw3. prefix in the imported table is for the physics interface, , in the third model.

Mesh 3

Free Tetrahedral 1

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type h_max_s/d_f.

In the text field, type 2.

In the text field, type 1.

In the text field, type 0.1.

Click

Study 2

Parametric Sweep, Step 1: Eigenfrequency

In the window, under , Ctrl-click to select and .

Right-click and choose .

Study 3

Parametric Sweep

In the window, right-click and choose .

Study 3

Step 1: Eigenfrequency

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Physics interface	Solve for	Discretization
Electromagnetic Waves, Frequency Domain (emw)		Physics settings
Electromagnetic Waves, Frequency Domain 2 (emw2)		Physics settings
Electromagnetic Waves, Frequency Domain 3 (emw3)	√	Physics settings