Dipole Antenna - Half-wave Antenna

Half-wave Antenna

Typically a dipole antenna is formed by two quarter wavelength conductors or elements placed back to back for a total length of . A standing wave on an element of a length ~ yields the greatest voltage differential, as one end of the element is at a node while the other is at an antinode of the wave. The larger the differential voltage, the greater the current between the elements.

Assuming a sinusoidal distribution, the current impressed by this voltage differential is given by:

For the far-field case, the formula for the electric field of a radiating electromagnetic wave is somewhat more complex:

But the fraction is not very different from .

The resulting emission diagram is a slightly flattened torus.

The image on the right shows the section of the emission pattern. We have drawn, in dotted lines, the emission pattern of a short dipole. We can see that the two patterns are very similar. The image on the left shows the perspective view of the same emission pattern.

This time it is not possible to compute analytically the total power emitted by the antenna (the last formula does not allow), though a simple numerical integration or series expansion leads to the more precise, actual value of the half-wave resistance:

$\begin{align}R_{\frac{\lambda}{2}} &= \frac{Z_0}{2\pi}\left\approx 60\operatorname{Cin}(2\pi)= 60\left=120\int_{0}^{\frac{\pi}{2}}\frac{\cos\left(\frac{\pi}{2}\cos\theta\right)^2}{\sin\theta}d\theta,\\ &=15\left,\\ &\approx 73.1296 \ \Omega; \end{align}\,\!$

This leads to the gain of a dipole antenna, :

$\begin{align}G_{\frac{\lambda}{2}} &=\frac{60^2}{30R_{\frac{\lambda}{2}}}=\frac{3600}{30R_{\frac{\lambda}{2}}}=\frac{120}{R_{\frac{\lambda}{2}}}=\frac{1}{{}^{\int_{0}^{\frac{\pi}{2}}\frac{\cos\left(\frac{\pi}{2}\cos\theta\right)^2}{\sin\theta}d\theta}},\\ &\approx\frac{120}{73.1296}\approx 1.64\approx 2.15\,\mathrm{dBi};\end{align}\,\!$

The resistance, however, is not enough to characterize the dipole impedance, as there is also an imaginary part——it is better to measure the impedance.

In the image below, the real and imaginary parts of a dipole's impedance are drawn for lengths going from to, accompanied by a chart comparing the gains of dipole antennas of other lengths, both as a number and in dBi: