Time-to-Frequency Fast Fourier Transform of a Coaxial Low-Pass Filter

A very wideband coaxial low-pass filter is designed using a 2D axisymmetric model. To address the wideband frequency response with a fine frequency resolution, the model is built with a transient physics interface first and then S-parameters are calculated using a time-to-frequency fast Fourier transform (FFT). The computed S-parameters show a low-pass frequency response with a cutoff frequency around 24.5 GHz.

Figure 1: Coaxial structure with irises that is symmetric around the axis of the center conductor. The example is efficiently modeled using a 2D axisymmetric configuration.

Model Definition

To achieve a low-pass frequency response, an air-filled coaxial cable is tuned with five annular rings (irises) that are added to the outer conductor wall in this example model. The coaxial cable walls and iris parts are modeled as perfect electric conductor (PEC). The volume inside the center conductor of the coaxial cable is removed and only the conducting metal surfaces are modeled. The space between the center and outer conductor is set to vacuum. Each end of the coaxial cable is terminated with a 41.56 Ω lumped port. The characteristic impedance for a lossless coaxial cable is calculated using the radius of the center and outer conductor parts:

where Z0 is the characteristic impedance of free space, εr is the relative permittivity of the material between the center and outer conductors, Rcoax is the radius of the outer conductor, and rcoax is the radius of the inner conductor.

In the lumped port setting window, by clicking the check box “Calculate S-parameter” on the excitation port, the voltage excitation type is set to the modulated Gaussian and the center frequency (f0) of the modulating sinusoidal function can be specified. The excitation voltage is defined as:

where σ is the standard deviation 1/2f0, f0 is the center frequency and ηf is the modulating frequency shift ratio.

The frequency here has to be matched to the center frequency of the S-parameter calculation used in the Time to Frequency FFT study step (see below).

The end time of the Time Dependent study step is set to 100 times of the period of the modulating sinusoidal function, which is long enough in this model to ensure that the input energy is fully decayed. This would work for a typical passive circuit except for closed cavity type devices, where the energy decay time can be much longer. The stop condition is automatically added under the Time-Dependent solver (the check box activates this stop condition in the solver settings). When the sum of total electric and magnetic energy in the modeling domain is below 70 dB compared to the input energy, the Time Dependent study is terminated by the stop condition and all time-domain data will be passed to the FFT step. To generate the frequency-domain data without significant distortion in the frequency range between 0 and 2f0, the time step, satisfying the Nyquist criterion, is set to 1/4f0 = 1/2B, where B is the bandwidth 2f0.

To provide a fine frequency resolution, the end time of the FFT study step is much longer than that of the Time Dependent study. Zero-padding is applied between the end time of the Time Dependent study and that of the FFT study step.

Results and Discussion

The computed S-parameters (Figure 2) shows the low-pass frequency response of the coaxial filter. The -3 dB cut-off is observed at 24.5 GHz. The last 15% close to the highest frequency is noisy, so the current simulation settings are not appropriate for the analysis close to the high-frequency limit of the FFT. Figure 3 shows the contour plot of the electric field norm distribution and the arrow plot of the time-averaged power flow at 10 GHz, which is within the passband where the electric field is confined on the irises. The power flow is straight toward the observation port from the excitation port.

Figure 2: Time-to-frequency fast Fourier transform of a transient simulation calculates frequency responses. The -3 dB cutoff is observed at 24.5 GHz.

Figure 3: A 3D result plot from a 2D axisymmetric model. The electric field norm distribution and time-averaged power flow at 10 GHz are visualized.

RF_Module/Filters/coaxial_low_pass_filter_transient

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.

In the table, enter the following settings:


Name	Expression	Value	Description
r_coax	1[mm]	0.001 m	Inner radius, coaxial cable
R_coax	2[mm]	0.002 m	Outer radius, coaxial cable
f0	30[GHz]	3E10 Hz	Center frequency
L	c_const/f0	0.0099931 m	Wavelength, free space
T0	1/f0	3.3333E-11 s	Period
h_max	L/10	9.9931E-4 m	Maximum mesh element size
Tend	100*T0	3.3333E-9 s	End time
Z0	(Z0_const/2/pi)*log(R_coax/r_coax)	41.56 Ω	Characteristic impedance

Here, c_const used in the free space wavelength is a predefined COMSOL constant for the speed of light in vacuum.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Create the air-filled part of a coaxial cable.

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type R_coax-r_coax.

In the text field, type 40.

Locate the section. In the text field, type r_coax.

Add a few irises inside the coaxial cable to convert the cable to a low-pass filter.

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.82.

In the text field, type 0.2.

Locate the section. In the text field, type R_coax-0.82.

In the text field, type 23.9.

Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.75.

In the text field, type 0.2.

Locate the section. In the text field, type R_coax-0.75.

In the text field, type 27.9.

Mirror 1 (mir1)

In the toolbar, click

and choose .

Select the objects and only.

In the window for , locate the section.

Select the check box.

Locate the section. In the text field, type 20.

Locate the section. In the text field, type 0.

In the text field, type 1.

Rectangle 4 (r4)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.86.

In the text field, type 0.2.

Locate the section. In the text field, type R_coax-0.86.

In the text field, type 19.9.

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Select the

toggle button.

Select the objects , , , , and only.

Click

Next, set up the physics using appropriate boundary conditions.

Electromagnetic Waves, Transient (temw)

Lumped Port 1

In the window, under right-click and choose .

Select Boundary 2 only.

This is the input port of the filter.

In the window for , locate the section.

Select the check box.

In the f0 text field, type f0.

Locate the section. In the Zref text field, type Z0.

Lumped Port 2

In the toolbar, click

and choose .

Select Boundary 3 only.

This is the output port of the filter.

In the window for , locate the section.

In the Zref text field, type Z0.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , 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	0	S/m	Basic

Set the maximum mesh size smaller than 0.2 wavelengths.

Mesh 1

Size

In the window, under right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type h_max.

Click

Study 1

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,1/(4*f0),Tend). The Sampling rate 4*f0 satisfies the Nyquist condition for the time to frequency fast Fourier transform (FFT) where its bandwidth is 2*f0 excluding negative frequencies.

Step 2: Time to Frequency FFT

In the window, click .

In the window for , locate the section.

In the text field, type Tend*3. This makes sure that the FFT end time is longer than the simulation time so zero-padding can be applied during the time to frequency FFT. This will generate a finer frequency resolution in the resulting frequency response.