Antenna Polarization: Theory, Equations & Interactive Simulator

This page explores the concept of antenna polarization, describing how the orientation of the electric field vector of a radiated wave behaves over time. Review foundational equations and use the interactive simulator below to test linear, circular, and elliptical polarizations in real-time.

Theory & Mathematical Formulation

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:

\[ E_x(t,z) = a \cos(\omega t - \beta z) \quad \text{(Horizontal axis)} \] \[ E_y(t,z) = b \cos(\omega t - \beta z + \phi) \quad \text{(Vertical axis)} \]

Here \(a\) and \(b\) represent amplitudes, \(\phi\) is the phase difference between components, \(\omega\) is the angular frequency, and \(\beta = \dfrac{2\pi}{\lambda}\) 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. Trigonometric derivation yields the standard curve equation:

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

This general formula classifies polarization types into:

Linear polarization: when \(\phi = n\pi\) (\(n\) integer \( = 0, \pm 1, \pm 2, \dots \))
Circular polarization: when \(a = b\) and \(\phi = (n+\frac{1}{2}) \pi \) or similar quarter-wave conditions (\(n\) integer \( = 0, \pm 1, \pm 2, \dots \))
Elliptical polarization: in the general case

Interactive Polarization Simulator

Varying amplitudes \(a\), \(b\), and phase difference \(\phi\) changes the polarization trace dynamically.

Amplitude a (x-component)

Value: 1

Amplitude b (y-component)

Value: 1

Phase difference \(\phi\)

Value: 0

0π/2π3π/22π

The canvas displays the tip of vector \(\mathbf{E}\) evolving over time. The horizontal axis represents \(E_x\) and the vertical axis represents \(E_y\).

Antenna Polarization

Theory & Mathematical Formulation

Interactive Polarization Simulator

Related Resources