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 régime, only the 1/r term contributes,[1] 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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