Wideband RCS Calculation Using Time-Domain Simulation and FFT

This model shows how to calculate the radar cross section (RCS) of a scatterer in a wide frequency range with the Electromagnetic Waves, Time Explicit physics interface. The problem is solved in the scattered-field formulation, where the background field is a temporally modulated Gaussian pulse. The simulated results present the scattered field in the frequency domain and time domain, and the RCS per unit length of a circle in the frequency domain.

Figure 1: A PEC circle is enclosed by a circular vacuum domain. The outermost layer is finished with an absorbing layer to absorb outgoing waves from the modulated Gaussian background pulse and the scattered field from the circular scatterer.

Model Definition

The model is solved using the scattered field formulation, that is automatically activated by adding a background field domain. The built-in Gaussian pulse, modulated with a sinusoidal function at 200 MHz, is used for background field. The wave vector of the background field is pointing toward the +x direction with z-directional polarization and 120π Ω wave impedance. The distance from the coordinate system origin to the wave launching plane is set to 1.5 m (See Figure 1). So, the wave is launched at the boundary of the vacuum domain.

The metallic circle is modeled using the perfect electric conductor boundary condition and its inner part is removed from the model domain. This is done because the skin depth is much smaller than the size of the circle. The circle's scattered field is computed on the boundaries configured by the Far-Field Domain node and its Far-Field Calculation subfeature, by performing a near-field to far-field transformation in the frequency domain after a fast Fourier transform (FFT). The PEC circle is surrounded by vacuum. The simulations are performed with two different absorbing features:

A scattering boundary condition (first order absorbing boundary condition) applied on the exterior boundary of the vacuum domain.

An absorbing layer domain on the outside of the surrounding vacuum domain acting as an absorber of the scattered field. The absorbing layer is placed at more than one wavelength away from the scatterer. The effect of the absorbing layer is to make the outermost layer appear as much larger for the outgoing waves. At the outermost boundary, the waves will reach the boundary propagating in the normal direction. Thus, the scattering boundary condition, added to the outermost boundary, will effectively absorb the outgoing waves.

The entire simulation consists of two study steps and one study extension step. The first step is a Time Dependent study for computing a temporal solution. The second step is a Time to Frequency FFT study, where an FFT is performed to obtain a frequency spectrum of the temporal solution. The last step is a Combine Solutions study extension step. This study extension step removes the unwanted parts of the frequency spectrum after the FFT (the first and the last 5% of the frequency spectrum), using a user-defined expression such as freq<0.1*fb0 || freq>2*fb0-0.1*fb0.

Results and Discussion

In Figure 2, the dB-scaled RCS per unit length is plotted for 100 MHz, 200 MHz, and 300 MHz when using either an absorbing layer or a scattering boundary condition to absorb the outgoing wave from the background field and scattered field from the PEC circle. The RCS difference between two cases is less than 0.2 dB at the backward scattering direction and around 0.5 dB at the forward scattering direction.

Figure 2: The dB-scaled RCS per unit length with the absorbing layer (solid) and scattering boundary condition (dashed).

Notes About the COMSOL Implementation

In the scattered field formulation, the total field is defined as Etotal = Erel + Eb where Erel is the relative field and Eb is the background field. The relative field is the difference between the total field caused by the presence of the scatterer and the background field. After performing the time dependent study and the FFT, only the relative field and postprocessing variables related to far-field analysis are available in the frequency domain, since the FFT takes only dependent variables. Other postprocessing variables are valid only in the time domain and can be accessed via stored solutions.

Table 1: Availability of variables for postprocessing.
Variable	Description	Availability
comp1.Ez or Ez	Dependent variable, Relative (scattered) field	Frequency domain
ewte.Ez	Total field	Time domain
ewte.Ebz	Background field	Time domain
ewte.bRCS2D	RCS per unit length	Frequency domain

RF_Module/Scattering_and_RCS/rcs_time_explicit

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
fb0	200[MHz]	2E8 Hz	Modulated Gaussian center frequency
lda0	c_const/fb0	1.499 m	Wavelength
k0	2*pi/lda0	4.1917 1/m	Wave number
T0	1/fb0	5E-9 s	Modulation period

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

Geometry 1

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2.5.

Circle 2 (c2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.7.

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 object only.

Click

Electromagnetic Waves, Time Explicit (ewte)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Background Field 1

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

In the f0 text field, type fb0.

In the doffset text field, type 2.5.

Scattering Boundary Condition 1

In the toolbar, click

and choose .

Select Boundaries 1, 2, 5, and 8 only.

Far-Field Domain 1

In the toolbar, click

and choose .

Far-Field Calculation 1

In the window, expand the node, then click .

Select Boundaries 3, 4, 6, and 7 only.

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

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Size

In the window, under click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type lda0/5.

This sets the maximum mesh element size to 0.2 wavelengths.

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*fb0),10*T0). The Sampling rate 4*fb0 satisfies the Nyquist condition for the time to frequency fast Fourier transform (FFT) where its bandwidth is 2*fb0 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 20*T0. 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.