# 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, λ. Two common regimes are:

• Short ($L\ll \lambda$)
• Half-wave (L = λ / 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 = I0eiνt 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 θ 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 Z0 = 73.13Ω, 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.