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# DIPOLE

A dipole antenna is a radio antenna that can be made of a simple wire, with a center-fed driven element. It consists of two metal conductors of rod or wire, in line with each other, with a small space between them. The radio frequency voltage is applied to the antenna at the center, between the two conductors. These antennas are the simplest practical antennas from a theoretical point of view. They are used alone as antennas, notably in traditional “rabbit ears” television antennas, and as the driven element in many other types of antennas, such as the Yagi. Dipole antennas were invented by German physicist Heinrich Hertz around 1886 in his pioneering experiments with radio waves.

An elementary doublet is a small length of conductor δℓ (small compared to the wavelength λ) carrying an alternating current: I=I_0 e^{iomega t}. Here ω = 2πf is the angular frequency (and f the frequency), and i = √−1 is the imaginary unit, so that I is a phasor.

Note that this dipole cannot be physically constructed because the current needs somewhere to come from and somewhere to go to. In reality, this small length of conductor will be just one of the multiple segments into which we must divide a real antenna, in order to calculate its properties.

In the case of the elementary doublet it is possible to find exact, closed-form expressions for its electric field, E, and its magnetic field, H. In spherical coordinates, they are

E_r=frac{Z ,I_0 delta ell}{2pi}left(frac{1}{r^2}-frac{i}{kr^3} right) e^{i(omega t-k,r)},cos(theta) E_theta= ifrac{Z ,I_0 delta ell}{4pi} left(frac{k}{r} – frac{i}{r^2} – frac{1}{k r^3}right) e^{i(omega t-k,r)},sin(theta) H_phi= i frac{I_0delta ell}{4pi} left(frac{k}{r} – frac{i}{r^2} right) e^{i(omega t-k,r)},sin(theta) E_phi = H_r = H_theta = 0,

where r is the distance from the doublet to the point where the fields are evaluated, k =2π/λ is the wavenumber, and Z = √μ/ε = 1/εc = μc is the wave impedance of the surrounding medium (usually air or vacuum).

The energy associated with the term of the near field flows back and forth out and into the antenna. The exponent of e accounts for the phase dependence of the electric field on time and the distance from the dipole.

Often one is interested in the antenna’s radiation pattern only in the far field, when r ≫ λ/2π. In this regime, only the 1/r term contributes, and hence E_theta= ifrac{Z ,I_0 delta ell, k}{4pi r} e^{i(omega t-k,r)},sin(theta) H_phi= i frac{I_0delta ell, k}{4pi r} e^{i(omega t-k,r)},sin(theta) E_r = E_phi = H_r = H_theta = 0.

The far electric field Eθ of the electromagnetic wave is coplanar with the conductor and perpendicular with the line joining the dipole to the point where the field is evaluated. If the dipole is placed in the center of a sphere in the axis south-north, the electric field would be parallel to geographic meridians and the magnetic field of the electromagnetic wave would be parallel to geographic parallels.

A short dipole is a physically feasible dipole formed by two conductors with a total length L very small compared with the wavelength λ. The two conducting wires are fed at the centre of the dipole. We assume the hypothesis that the current is maximal at the centre (where the dipole is fed) and that it decreases linearly to be zero at the ends of the wires. Note that the direction of the current is the same in both the dipole branches: to the right in both or to the left in both. The far field Eθ of the electromagnetic wave radiated by this dipole is

Emission is maximal in the plane perpendicular to the dipole and zero in the direction of wires which is the direction of the current. The emission diagram is circular section torus shaped (right image) with zero inner diameter. In the left image the doublet is vertical in the torus centre.

Knowing this electric field, we can compute the total emitted power and then compute the resistive part of the series impedance of this dipole due to the radiated field, known as the radiation resistance: R_text{series}={piover6}Z_0 left({Loverlambda}right)^2 qquad text{ for } L ll lambda, where Z_0 is the impedance of free space. Using a common approximation of Z_0 approx 120 pi ohms, we get R_text{series}approx 20pi^2left({Loverlambda}right)^2 qquad text{(in ohms)}.

Antenna gain, G, is the ratio of surface power radiated by the antenna to the surface power radiated by a hypothetical isotropic antenna: G=frac{(P/S)_text{ant}}{(P/S)_text{iso}}.  The surface power carried by an electromagnetic wave is left(frac{P}{S}right)_text{ant} = frac{1}{2}c varepsilon_0 E_theta^{,2} simeq frac{E_theta^{,2}}{120pi}, while the surface power radiated by an isotropic antenna feed with the same power is left(frac{P}{S}right)_text{iso} = frac{tfrac{1}{2} R_text{series} I_0^{,2}}{4pi r^2}.

Combining these expressions with the far-field expression for Eθ for a short dipole gives G = frac{3}{2} = mathrm{1.76 dBi}, where dBi means decibels gain relative to an isotropic antenna.

Dipoles that are much smaller than the wavelength of the signal are called Hertzian, short, or infinitesimal dipoles. These have a very low radiation resistance and a high reactance, making them inefficient, but they are often the only available antennas at very long wavelengths. Dipoles whose length is half the wavelength of the signal are called half-wave dipoles, and are more efficient. In general radio engineering, the term dipole usually means a half-wave dipole (center-fed).

A half-wave dipole is cut to length l for frequency f MHz according to the formula l=frac{143}{f} where l is in metres or l=frac{468}{f} where l is in feet.  This is because the impedance of the dipole is purely resistive at about this length. The length of the dipole antenna is approximately 95% of half a wavelength at the speed of light in free space. The exact value depends on the ratio of the wire radius to wavelength, as the reactance of a dipole depends on the ratio of wire radius to wavelength. For thin wires (radius = 0.000001 wavelengths), this is approximately 98.1%, dropping to 91.5% for thick wires (radius = 0.01 wavelengths).

The constants above are derived from a one Hz wavelength which is the distance that light radio travels in one second. Speed of light in vacuum is 299,792,458 m/s, which is divided by 1 million to account for MHz rather than Hz, which is then divided by 2 for a half-wave dipole antenna. An adjustment of 0.95 is multiplied to account for the fact the input reactance of a dipole is inductive when exactly 0.5 wavelengths long, but the reactance reduces to 0 (i.e. the dipole becomes resonant) at a length of approximately 0.475 wavelengths, which results in the constants of 143 m·MHz or 468 ft·MHz. But as noted above, a more accurate calculation would need to take into account the wire radius. Such a calculation is quite complex.

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