Antenna Parameters

The radiation pattern of an antenna is the normalised spatial distribution of radiated (or received) power as a function of direction $(\theta, \phi)$ in spherical coordinates:

$$F(\theta, \phi) = \frac{U(\theta, \phi)}{U_{\max}}$$

where $U(\theta, \phi)$ is the radiation intensity (power per unit solid angle, in W/sr). The pattern is typically plotted in polar or Cartesian form on a dB scale:

$$F_{\text{dB}}(\theta, \phi) = 10\log_{10} F(\theta, \phi)$$

Key features of the pattern include:

• Main lobe: the direction of maximum radiation
• Side lobes: secondary maxima away from the main beam
• Nulls: directions of zero radiation
• Back lobe: radiation in the hemisphere opposite the main lobe

An isotropic radiator is a hypothetical antenna that radiates equally in all directions:

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

where $P_{\text{rad}}$ is the total radiated power. No physical antenna is truly isotropic, but it serves as the reference for defining directivity and gain (hence units of dBi — decibels relative to isotropic).

The directivity $D$ of an antenna is the ratio of its maximum radiation intensity to that of an isotropic radiator with the same total radiated power:

$$D = \frac{U_{\max}}{U_{\text{iso}}} = \frac{4\pi\, U_{\max}}{P_{\text{rad}}} = \frac{4\pi}{\int_0^{2\pi}\!\int_0^{\pi} F(\theta,\phi)\,\sin\theta\,d\theta\,d\phi}$$

Directivity is dimensionless (or expressed in dBi). A higher directivity means the antenna concentrates energy more tightly into a narrower beam.

The antenna gain $G$ accounts for both the directional focusing (directivity) and ohmic/mismatch losses:

$$G = \eta\, D$$

where $\eta \in (0, 1]$ is the radiation efficiency:

$$\eta = \frac{P_{\text{rad}}}{P_{\text{in}}}$$

In practice, $G$ is the quantity used in link budgets. For well-designed antennas $\eta > 0.9$, so $G \approx D$. Gain is expressed in dBi (relative to isotropic).

The effective aperture (or effective area) $A_e$ of an antenna characterises its ability to capture power from an incident electromagnetic wave:

$$P_r = A_e \cdot S_i$$

where $S_i$ is the incident power flux density (W/m$^2$) and $P_r$ is the received power. For any antenna, $A_e$ is related to gain by the fundamental relation (Theorem 7.1.1).

The half-power beamwidth (HPBW) is the angular width of the main lobe between the $-3$ dB points:

$$\text{HPBW} = \theta_2 - \theta_1 \quad \text{where } F(\theta_1) = F(\theta_2) = 0.5$$

For a uniformly illuminated aperture of length $L$:

$$\text{HPBW} \approx 0.886\,\frac{\lambda}{L}$$

Narrower beamwidth implies higher directivity. The approximate relation is $D \approx \frac{4\pi}{\Theta_E\,\Theta_H}$, where $\Theta_E$ and $\Theta_H$ are the HPBW in the E-plane and H-plane (in radians).

The polarization of an antenna describes the orientation of the electric field vector of the radiated wave. Common polarization states:

• Linear (vertical or horizontal): $\mathbf{E}$ oscillates along a fixed direction
• Circular (RHCP or LHCP): $\mathbf{E}$ rotates at the carrier frequency, tracing a circle
• Elliptical: general case; $\mathbf{E}$ traces an ellipse

Polarization mismatch between transmit and receive antennas causes a power loss quantified by the polarization loss factor:

$$\text{PLF} = |\hat{\mathbf{e}}_t \cdot \hat{\mathbf{e}}_r|^2$$

where $\hat{\mathbf{e}}_t$ and $\hat{\mathbf{e}}_r$ are the polarization unit vectors. Cross-polarized antennas have $\text{PLF} = 0$ ($-\infty$ dB loss).