Conical Antenna

Conical antennas are useful for many applications due to their broadband characteristics and relative simplicity. This example includes an analysis of the antenna impedance and the radiation pattern as functions of the frequency for a monoconical antenna with a finite ground plane and a 50 Ω coaxial feed. The rotational symmetry makes it possible to model this in axially symmetric 2D. When modeling in 2D, you can use a dense mesh, giving an excellent accuracy for a wide range of frequencies.

Model Definition

The antenna geometry consists of a 0.2 m tall metallic cone with a top angle of 90 degrees on a finite ground plane of a 0.282 m radius. The coaxial feed has a central conductor of 1.5 mm radius and an outer conductor (screen) of 4.916 mm radius separated by a Teflon dielectric of relative permittivity of 2.07. The central conductor of the coaxial cable is connected to the cone, and the screen is connected to the ground plane.

Figure 1: The geometry of the antenna. The central conductor of the coaxial cable is connected to the metallic cone, and the cable screen is connected to the finite ground plane.

The model takes advantage of the rotational symmetry of the problem, which allows modeling in 2D using cylindrical coordinates. You can then use a very fine mesh to achieve an excellent accuracy.

Domain Equations

An electromagnetic wave propagating in a coaxial cable is characterized by transverse electromagnetic (TEM) fields. Assuming time-harmonic fields with complex amplitudes containing the phase information, you have:

where z is the direction of propagation and r,

, and z are cylindrical coordinates centered on axis of the coaxial cable. Z is the wave impedance in the dielectric of the cable, and C is an arbitrary constant. The angular frequency is denoted by ω. The propagation constant, k, relates to the wavelength in the medium λ as

In the air, the electric field also has a finite axial component whereas the magnetic field is purely azimuthal.

Boundary Conditions

The boundary conditions for the metallic surfaces are:

At the feed point, a matched coaxial port boundary condition is used to make the boundary transparent to the wave. The antenna is radiating into free space, but you can only discretize a finite region. Therefore, truncate the geometry some distance from the antenna using a scattering boundary condition allowing for outgoing spherical waves to pass with very little reflections. A symmetry boundary condition is automatically applied on boundaries at r = 0.

Results and Discussion

Figure 2 shows the antenna impedance as a function of frequency. Ideally, the antenna impedance should be matched to the characteristic impedance of the feed, 50 Ω, to obtain maximum transmission into free space. This is quite well fulfilled in the high-frequency range.

Figure 2: The antenna impedance in Ω as a function of frequency from 200 MHz to 1.5 GHz. The solid line shows the radiation resistance and the dotted line represents the reactance.

Figure 3 shows the antenna radiation pattern in the near field for three different frequencies. The effect of the finite diameter of the ground plane is to lift the main lobe from the horizontal plane. For an infinite ground plane or in the high-frequency limit, the radiation pattern is symmetric around zero elevation. This is easy to understand, as an infinite ground plane can be replaced by a mirror image of the monocone below the plane. Such a biconical antenna is symmetric around zero elevation and has its main lobe in the horizontal direction. The decreased lobe lifting at higher frequencies is just about visible in Figure 3.

Figure 4 shows the antenna radiation pattern in the far field for the same frequencies as the radiation pattern at the boundary in Figure 3.

Figure 3: Polar plot of the antenna radiation pattern in the near field versus the elevation angle for 200 MHz, 800 MHz, and 1.5 GHz. The scale is logarithmic.

Figure 4: Polar plot of the antenna radiation pattern for the far field versus the elevation angle for 200 MHz, 800 MHz, and 1.5 GHz. This plot is normalized differently but has a shape resembling the near field.

As the frequency increases the antenna impedance gets closer to 50 Ω, which means that a voltage generator connected to the input of the antenna should have an output impedance of 50 Ω.

RF_Module/Antennas/conical_antenna

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

Geometry 1

Import 1 (imp1)

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 conical_antenna.mphbin.

Click

The imported geometry is effectively a half circle with the metal areas removed. You model the electromagnetic waves in the air and the dielectric material inside the coaxial cable. There is no need to include the metal as a domain in the model because the fields in it are essentially zero except on its surface.

Global Definitions

Prepare for the impedance computation by making a few 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
Z_tl	50[ohm]	50 Ω	Characteristic transmission line impedance

Definitions

Variables 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Z	Z_tl*(1+emw.S11)/(1-emw.S11)		Antenna impedance

emw.S11 is the name of the automatically computed reflection S-parameter.

Define the following selections in order to get easy access to some frequently used domains and boundaries.

Air

In the toolbar, click

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

Select Domain 1 only.

Dielectric

In the toolbar, click

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

The dielectric domain is inside the coaxial cable just below the cone. It is easier to select it if you zoom in a little.

Select Domain 2 only.

Outer Air Boundaries

In the toolbar, click

In the window for , type Outer Air Boundaries in the text field.

Locate the section. From the list, choose .

Select Boundaries 14 and 15 only.

With all selections and expressions now defined, it is time to set up the materials and the physics of the model.

Add Material

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.

Materials

Air (mat1)

In the window for , locate the section.

From the list, choose .

Material 2 (mat2)

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

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	2.07	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

Electromagnetic Waves, Frequency Domain (emw)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Port 1

In the toolbar, click

and choose .

Set up Boundary 6, at the bottom of the coaxial cable, to be a Port. You can zoom in on this part of the geometry to easier find and select this boundary.

Select Boundary 6 only.

In the window for , locate the section.

From the list, choose .

For the first port, wave excitation is by default.

Scattering Boundary Condition 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

The Scattering boundary condition is a simple way of letting the waves undergo only minor artificial reflections as they leave the computational domain through the exterior boundaries. To minimize these reflections, but at a greater computational cost, you can use Perfectly Matched Layers).

Perfect Electric Conductor 1

As you can see if you click the Perfect Electric Conductor node under Electromagnetic Waves, the physical boundaries to which you have not assigned any boundary condition will by default be considered perfect electric conductors. This is a good approximation for most metals throughout the frequency range considered in this model.

Far-Field Domain 1

In the toolbar, click

and choose .

Select Domain 1 only.

Far-Field Calculation 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Adding a Far Field Calculation feature does not affect the physics of the model, but makes it possible to study the far field generated by the antenna. Select the boundaries to use for this computation so that, in the physical (3D) geometry, they surround all sources and reflecting objects. The outer air boundaries are a convenient choice.

Mesh 1

In the window, under right-click and choose .

Study 1

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the text field, type range(200[MHz],25[MHz],1.5[GHz]).

The frequency range you just entered runs from 200 MHz to 1.5 GHz in steps of 25 MHz.

In the toolbar, click

Results

Electric Field (emw)

The plot that appears once the solution process is finished shows the norm of the electric field at 1.5 GHz. The reason it is mostly dark blue is because the range is dominated by the high values in and near the coaxial cable. To better see how the wave propagates, try plotting the instantaneous value of the H-field using a manual range.

Surface

In the window, expand the node, then click .

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

Click to expand the section. Select the check box.

In the text field, type -0.5.

In the text field, type 0.5.

Locate the section. Click

In the dialog box, select in the tree.

Click .

In the toolbar, click

Click the

button in the toolbar.

Smith Plot (emw)

Radiation Pattern 1

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. In the text field, type 100.

Find the subsection. In the text field, type -1.

In the text field, type 0.

In the toolbar, click

To plot the impedance as a function of the frequency, set up a 1D plot.

1D Plot Group 6

In the toolbar, click

and choose .

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
real(Z)	Ω	Resistance
imag(Z)	Ω	Reactance

Click to expand the section. Find the subsection. From the list, choose .

In the toolbar, click

You have now reproduced Figure 2. Next, visualize the near-field and far-field radiation patterns using polar plots.

Polar Plot Group 7

In the toolbar, click

and choose .

Select a few of the frequencies from the list of parameter values. Showing the radiation pattern for all of them would take a bit of time and lead to a cluttered plot.

In the window for , locate the section.

From the list, choose .

In the list, choose , , and .

Use Ctrl-key to select multiple frequencies.

Line Graph 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Click in the upper-right corner of the section. From the menu, choose .

Locate the section. In the text field, type 10*log10(emw.nPoav).

The variable emw.nPoav represents the outgoing power flow through the boundaries where it is evaluated. The expression you just entered gives you the same in a logarithmic scale.

Select the check box. In the associated text field, type Near-field radiation pattern.

Locate the θ section. From the list, choose .

In the text field, type atan2(z,r).

Click to expand the section. Find the subsection. From the list, choose .

Click to collapse the section. Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
200 MHz
800 MHz
1.5 GHz

Click to collapse the section. In the toolbar, click

Click the

button in the toolbar.

Your near-field radiation plot should look like that in Figure 3.

Finally, visualize the far-field radiation pattern.

Polar Plot Group 8

In the window, under right-click and choose .

Line Graph 1

In the window, expand the node, then click .

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

Locate the section. In the text field, type Far-field radiation pattern.

In the toolbar, click

The plot should look like that in Figure 4.