# 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, ${\displaystyle L}$, as measured in units of wavelength, ${\displaystyle \lambda }$. Two common regimes are:

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

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

${\displaystyle E_{\theta }={\frac {-iI_{0}}{2\pi \epsilon _{0}cr}}{\frac {\cos({\frac {\pi }{2}}\cos \theta )}{\sin \theta }}e^{2\pi i(\nu t-{\frac {r}{\lambda }})},\,\!}$

where ${\displaystyle \theta }$ 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 ${\displaystyle Z_{0}=73.13\Omega }$, 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.