Antenna Polarization

The polarization of an antenna describes the orientation of the electric field of the wave it radiates. For a given position along the propagation axis \(z\), the electric field vector \(\mathbf{E}\) has components along the axes of the diagram:

Horizontal axis → \(E_x(t,z) = a \cos(\omega t - \beta z)\)
Vertical axis → \(E_y(t,z) = b \cos(\omega t - \beta z + \phi)\)

Here \(a\) and \(b\) are the amplitudes, \(\phi\) is the phase difference between the components, \(\omega\) is the angular frequency, and \(\beta = 2\pi/\lambda\) is the propagation constant.

The tip of the electric field vector \(\mathbf{E}(t) = (E_x, E_y)\) traces a curve in the xy-plane (horizontal x, vertical y). Using trigonometry, we can derive the equation of the curve:

\[ \left(\frac{E_x}{a}\right)^2 + \left(\frac{E_y}{b}\right)^2 - 2 \frac{E_x E_y}{ab} \cos \phi = \sin^2 \phi \]

This equation represents:

**Linear polarization** when \(\phi = n\pi\) (\(n\) integer \( = 0 , \pm 1 , \pm 2 , ... ) \)
**Circular polarization** when \(a = b\) and \(\phi = (n+1) \pi/2 \) (\(n\) integer \( = 0 , \pm 1 , \pm 2 , ... ) \)
**Elliptical polarization** in the general case

The canvas below shows the tip of \(\mathbf{E}\) as a function of time. The horizontal axis corresponds to \(E_x\) and the vertical axis to \(E_y\). The shape of the trace directly illustrates the type of polarization.

By varying \(a\), \(b\), and \(\phi\), different polarizations can be obtained: linear, circular, or elliptical.

Antenna Polarization

Interactive Polarization Simulator