# Dipole Antennas

## 1 Dipole Antennas

Dipole antennas are a type of radio antenna that is very common at lower frequencies, where wavelengths are long enough that such elements are reasonable to build. The performance of a dipole antenna depends on its tip-to-tip length, $L$ , as measured in units of wavelength, $\lambda$ . Two common regimes are:

• Short ($L\ll \lambda$ )
• Half-wave ($L=\lambda /2$ )

The most efficient of these is the half-wave dipole, which we will consider in more detail.

### 1.1 Deriving Far-Field Beam Patterns

The far-field $E$ -field generated by currents flowing in an antenna can by calculated using the Fourier transform of the current density flowing in the aperture (modified by a directionality component that reflects that currents flowing in a direction emit most strongly perpendicular to that direction). And by the reciprocity theorem, the reverse is true: the Fourier transform of the beam pattern show the currents that are excited in the aperture. So when deriving the beam response of a dipole, it is important to note that current does not flow at the tips of the dipole, and that the current excited increases toward the center.

### 1.2 Half-wave Dipole

The formula for the $E$ -field at a distance $r\gg L$ for a half-wave dipole being driven with a current $I=I_{0}e^{2\pi i\nu t}$ is given by:

$E_{\theta }={\frac {-iI_{0}}{2\pi \epsilon _{0}cr}}{\frac {\cos({\frac {\pi }{2}}\cos \theta )}{\sin \theta }}e^{2\pi i(\nu t-{\frac {r}{\lambda }})},\,\!$ where $\theta$ is the direction angle, measured from the axis of the dipole.

Half-wave dipoles have, at the wavelength that they are tuned to, a resistance of $Z_{0}=73.13\Omega$ , and a gain of 2.15 dBi. This means that the peak response of the dipole beam is a factor of 1.64 higher than an (ideal) isotropic beam would have.