10.2 Halfwavelength dipoleBasic theoryA dipole antenna, developed by Heinrich Rudolph Hertz around 1886 is an antenna with a centerfed driven element for transmitting or receiving radio frequency energy. These antennas are the simplest practical antennas
from a theoretical point of view. Typically, a dipole antenna is formed by two quarter wavelength conductors or elements placed back to back for a total length of λ/2.

Fig. 10.2A.1  Wire model of dipole 

A standing wave on an element of a length ~ λ/4 yields the greatest voltage differential, as one end of the element is at a node while the other element is
at an antinode of the wave. Assuming a sinusoidal distribution, the current impressed by this voltage differential is given by

$I={I}_{0}{e}^{j\omega t}\mathrm{cos}\left(kl\right)$.

( 10.2A.1 )

Here, I_{0} is the current in the maximum, denotes angular frequency and t time, k is wavenumber and l is the length of the dipole arm.
In the farfield zone, the electric field of a radiating electromagnetic wave is given by:

${E}_{\theta}=\frac{j{I}_{0}}{2\pi {\epsilon}_{0}cr}\frac{\mathrm{cos}\left(\frac{\pi}{2}\mathrm{cos}\theta \right)}{\mathrm{sin}\theta}{e}^{j\left(\omega tkr\right)}$.

( 10.2A.2 )

Here, E_{} is the respective component of the electric field, ε_{0} is the permitivity of vacuum, c is the speed of light in vacuum, r is the distance from
the observation point, is the angle measured from the dipole axis, and other symbols were already explained.
In figures below, frequency responses of standing wave ration and input impedance of a dipole (l = 19.64 m) is depicted. The halfwavelength resonance appears at the
frequency f = 3.68 MHz.

Fig. 10.2A.2  Frequency response of standing wave ratio of halfwavelength dipole (l = 19.64 m) 


Fig. 10.2A.3  Frequency response of input impedance of halfwavelength dipole (l = 19.64 m) 

In fig. 10.2A.4, far field radiation pattern of a dipole in the halfwavelength resonance is depicted. The maximum radiation appears in the direction perpendicular to the axis of the dipole. The
antenna does not radiate in the axis of the dipole. Since the dipole is placed above the conducting plane, the antenna radiates into a halfspace only.

Fig. 10.2A.4  Far field radiation pattern of halfwavelength dipole above the conducting plane in the height h = 20 m (l = 19.64 m, f = 3.68 MHz) 

