A rectangular patch antenna is a type of microstrip antenna consisting of a rectangular conductive patch placed on a dielectric substrate with a ground plane on the opposite side. These antennas are widely used in wireless communication applications due to their simple design, ease of fabrication, and low profile.

In this notebook, we will demonstrate how to use Tidy3D to simulate a rectangular patch antenna [1] and compute key performance metrics. These include S-parameters using the TerminalComponentModeler, as well as directivity, axial ratio, and polarized far-field components using the DirectivityMonitor.

Units and Frequency Settings¶

In Tidy3D, the default units are micrometers (µm) and Hertz (Hz). In the following, we introduce a scaling factor to convert the default unit of micrometers to millimeters (mm), which is more commonly used in antenna simulations.

Materials¶

We use a simple dielectric substrate with a relative permittivity of 2.2 and PEC2D, a thickness-free perfect electric conductor (PEC) medium to represent the conducting materials in the antenna.

Building Geometry and Structures¶

We follow the parameters outlined in [1] to create the corresponding patch antenna step-by-step. Boolean operations on geometries can be applied using ClipOperation.

Define Monitors¶

A FieldMonitor is used to observe the field distribution on the antenna patch. To obtain radiation characteristics such as directivity, axial ratio, and polarized far fields, use a DirectivityMonitor. The DirectivityMonitor captures far-field information through a near-to-far field transformation.

Boundaries¶

Perfectly Matched Layers (PMLs) are applied in all directions to simulate an infinitely large domain.

Create Simulation Object¶

Create a simulation object that includes all the modules we have defined. Note that the excitation (source) is not defined here and will be introduced by as a lumped port.

Define Lumped Port (Excitation) and TerminalComponentModeler¶

A LumpedPort is a commonly used excitation port in RF simulations. It consists of a lumped resistor at the port and a uniform current source to excite the structure. The TerminalComponentModeler is used to combine the defined simulation object with the lumped port to compute S-parameters.

Examine Structures and Mesh¶

Before running the simulation, it is advisable to inspect the created structure and mesh. Here, we examine the structure along its propagation direction and the cross-section at the terminated end of the feedline, which includes the location of the lumped element.

Run simulation¶

Post processing: S-parameters¶

Post processing: Field distributions¶

Post processing: Directivity¶

Directivity is a dimensionless quantity defined as the ratio of the radiation intensity in a given direction to the average radiation intensity over all directions. Mathematically, directivity $D(\theta, \phi)$ in a given direction $(\theta, \phi)$ is defined as:

$$ D(\theta, \phi) = \frac{4\pi}U(\theta, \phi)}{P_{\text{rad}} $$

where:

• $U(\theta, \phi)$: Radiation intensity in direction $(\theta, \phi)$
• $P_{\text{rad}}$: Total radiated power

Computation¶

1. Radiation Intensity:
   Radiation intensity is given by:

   $$ U(\theta, \phi) = r^2 \cdot S_{\text{rad}}(\theta, \phi) $$

   where $S_{\text{rad}}(\theta, \phi)$ is the far-field Poynting vector magnitude, computed as:

   $$ S_{\text{rad}}(\theta, \phi) = \frac{1}{2} \cdot \text{Re}(E_\theta H_\phi^* - E_\phi H_\theta^*) $$

   Here, $E_\theta$ and $E_\phi$ are the electric field components in the far-field, $H_\phi^*$ and $H_\theta^*$ are the complex conjugates of the magnetic field components, and $r$ is the distance from the source.

2. Total Radiated Power:
   The total radiated power is computed by integrating the Poynting vector over the sphere:

   $$ P_{\text{rad}} = \int_{0}^{2\pi} \int_{0}^{\pi} S_{\text{rad}}(\theta, \phi) r^2 \sin\theta \, d\theta \, d\phi $$

3. Directivity:
   Directivity is determined by substituting $U(\theta, \phi)$ and $P_{\text{rad}}$ into the formula:

   $$ D(\theta, \phi) = \frac{4\pi}U(\theta, \phi)}{P_{\text{rad}} $$

Post processing: Axial Ratio¶

AR is a dimensionless quantity defined as the ratio of the major axis to the minor axis of the polarization ellipse $$ AR = \frac{E_{\text{major}}}{E_{\text{minor}}} $$

For a perfectly circularly polarized wave, $ AR = 1 $, corresponding to 0 dB.

Computation:¶

The complex far field components $E_\theta$ and $E_\phi$ are used to determine the AR:

$$AR = \sqrt{\frac{|E_\theta|^2 + |E_\phi|^2 + |E_\theta^2 + E_\phi^2|}{|E_\theta|^2 + |E_\phi|^2 - |E_\theta ^2 + E_\phi^2|}}$$

Post processing: Left and Right polarized far field¶

Left polarization represents the counter-clockwise component of circularly polarized electric fields, calculated as:

$$ E_{\text{left}} = \frac{E_\theta - j E_\phi}{\sqrt{2}} $$

while right polarization represents the clockwise component, calculated as:

$$ E_{\text{right}} = \frac{E_\theta + j E_\phi}{\sqrt{2}} $$

Reference¶

[1] Sheen, D.M., Ali, S.M., Abouzahra, M.D. and Kong, J.A., 1990. Application of the three-dimensional finite-difference time-domain method to the analysis of planar microstrip circuits. IEEE Transactions on microwave theory and techniques, 38(7), pp.849-857.