Far-Field Results | XFdtd

XF's far zone sensor requests steady-state far-field results for an antenna. Realized gain, gain, and directivity are ratios relative to available power, input power, and radiated power, respectively. The following relationship applies.

\begin{equation} P_{av}(1-|\Gamma|^2) = P_{in} = P_d + P_l + P_r \end{equation}

where,

• $P_{av}$ is the power available at the antenna's port.
• $\Gamma$ is the reflection coefficient due to the impedance mismatch at the port. It is also expressed in terms of S-parameters as $S_{11}$.
• $P_{in}$ is the power input into the antenna's port.
• $P_d$ is the power dissipated in the device due to lossy materials like PCB substrates or plastic.
• $P_l$ is the power lost to passive components like lossy RLCs or inactive ports.
• $P_r$ is the power radiated into free space. Also expressed as $P_r = P_{in} - P_d - P_l$

The following results utilized the theta, phi coordinate system, however, XF supports other coordinate systems when users substitute the polarization unit vectors.

Electric Field

The electric field vector is expressed

\begin{equation} \tilde E (\theta, \phi) = \hat a_{\theta} \tilde E_{\theta} (\theta, \phi) + \hat a_{\phi} \tilde E_{\phi} (\theta, \phi)\,\,\,\,\,\text{(V/m)} \end{equation}

where,

• $(\theta, \phi)$ is the observation angle.
• $\hat a_{\theta}$ and $\hat a_{\phi}$ are the theta and phi unit vectors, respectively.
• $\tilde E_{\theta}$ is the complex theta-polarized electric field value.
• $\tilde E_{\phi}$ is the complex phi-polarized electric field value.

XF utilizes the near- to far-field transform to compute far-field $\tilde E_{\theta}$ and $\tilde E_{\phi}$ from a near-field FDTD simulation.

Realized Gain

The realized gain of an antenna accounts for reflection, or mismatch, losses at the port. It is proportional to the ratio of electric field strength in a given direction to the available power at the port. For a given observation angle, $(\theta, \phi)$, realized gain is expressed

\begin{equation} R_\theta = \frac{2\pi|\tilde E_\theta|^2}{\eta P_{av}}\,\,\,\,\,\text{(dimensionless)} \end{equation} \begin{equation} R_\phi = \frac{2\pi|\tilde E_\phi|^2}{\eta P_{av}}\,\,\,\,\,\text{(dimensionless)} \end{equation}

where,

• $R_{\theta}$ and $R_{\phi}$ are the theta- and phi-polarized realized gain values at $(\theta, \phi)$, respectively.
• $|\tilde E_\theta|$ and $|\tilde E_\phi|$ are the theta- and phi-polarized electric field magnitudes at $(\theta, \phi)$, respectively.
• $\eta$ is the impedance of free space.
• $P_{av}$ is the port's available power provided by the system sensor.

Gain

The gain is the ratio of the radiation intensity, in a given direction, to the radiation intensity that would be obtained if the power accepted by the antenna were radiated isotropically [1]. It is proportional to the ratio of electric field strength in a given direction to the power input at the port. For a given observation angle, $(\theta, \phi)$, gain is expressed

\begin{equation} G_\theta = \frac{2\pi|\tilde E_\theta|^2}{\eta P_{in}}\,\,\,\,\,\text{(dimensionless)} \end{equation} \begin{equation} G_\phi = \frac{2\pi|\tilde E_\phi|^2}{\eta P_{in}}\,\,\,\,\,\text{(dimensionless)} \end{equation}

where,

• $G_{\theta}$ and $G_{\phi}$ are the theta- and phi-polarized gain values at $(\theta, \phi)$, respectively.
• $|\tilde E_\theta|$ and $|\tilde E_\phi|$ are the theta- and phi-polarized electric field magnitudes at $(\theta, \phi)$, respectively.
• $\eta$ is the impedance of free space.
• $P_{in}$ is the power input from the port provided by the system sensor.

Directivity

Directivity is the ability of an antenna to radiate power in a particular direction. It is proportional to the ratio of electric field strength in a given direction to the radiated power. For a given observation angle, $(\theta, \phi)$, directivity is expressed

\begin{equation} D_\theta = \frac{2\pi|\tilde E_\theta|^2}{\eta P_r}\,\,\,\,\,\text{(dimensionless)} \end{equation} \begin{equation} D_\phi = \frac{2\pi|\tilde E_\phi|^2}{\eta P_r}\,\,\,\,\,\text{(dimensionless)} \end{equation}

where,

• $D_{\theta}$ and $D_{\phi}$ are the theta- and phi-polarized directivity values at $(\theta, \phi)$, respectively.
• $|\tilde E_\theta|$ and $|\tilde E_\phi|$ are the theta- and phi-polarized electric field magnitudes at $(\theta, \phi)$, respectively.
• $\eta$ is the impedance of free space.
• $P_r$ is the power radiated from the device provided by the system sensor.

Identities

The following equation converts the scalar, dimensionless realized gain, gain, and directivity values to dBi.

\begin{equation} G_{dBi} = 10 \log_{10} G\,\,\,\,\,\text{(dBi)} \end{equation}

The total gain of an antenna, in a specified direction, is the sum of the partial gains for any two orthogonal polarizations [1]

\begin{equation} G_{total} = G_\theta + G_\phi\,\,\,\,\,\text{(dimensionless)} \end{equation}

Reference

1. Antenna Standards Committee of the IEEE Antennas and Propagation Society, IEEE Standard Definitions of Terms for Antennas, IEEE Std 145-1993.