Dipole antenna#Hertzian dipole .28current element.29

The Hertzian dipole ( after Heinrich Hertz ), also called Elementardipol, is the idealization of an electric radiator and used to calculate the radiation of real antennas, as well as a reference antenna to capture the directivity of an antenna numerically as profit. A generalization of the results ( mitbehandelte ) multipole.

• 2.1 Definitions
• 2.2 formula
• 2.3 Near and far field

The Hertzian dipole as a model

The Hertzian dipole as a model is based on a sinusoidally varying ( with angular frequency ) electric dipole moment, in complex notation

Such a pure dipole moment without spatial extension arises in the limit of oscillating charge carriers with vanishing oscillation amplitude and diverging amount of charge.

Exact equations

For the magnetic and electric field at the given distance and direction by Location applies:

It is

• The velocity of light
• The wavelength of the radiation.
• The absolute permittivity in vacuum so. Thus, it is used at this point the SI-system, even though the equivalent cgs system simplifies some formulas.

From these equations for the Hertzian dipole can be, in contrast to all other types of antennas, calculate the velocities of propagation of the wave fronts analytically. Overall, a radiation field at any point has closed field lines, with a re- passed in all textbooks characteristic kidney shape ( see, eg, the outer box in Figure 1). Emphasis is also the time dependence, we obtain animation above, which among other things in a realistic way the phase velocity of, the group velocity and the propagation speed of the energy as a function of the distance to the source results in terms of the velocity of light in units of the circular wave number. For large distances, all these speeds approaching the speed of light. In the near field only the speed of signal propagation is properly again.

• Short range dominates because of the term, the electric field, while the magnetic field is negligible: it is for example in the ratio (r / λ ) less and in anti-phase to the electric field ( that is, when one field is at a maximum, the other is a minimum ). this acts as a quasi-static (ie, slowly oscillating ) dipole field and the magnetic field is analogous to a low impedance in relation to the strong inductive ohmic resistance is negligible.

• In the far field, are the radius vector, electric field and magnetic field pairwise orthogonal to each other and the fields in the in-phase, in the cgs system even of identical strength. In this system quantitatively valid (or radiation intensity.

Consequences

The last formula has many consequences, among others, for the entire radio and television technology, for the blue color of the sky, which arises that in the spectral range "green" dominant radiation of the sun ( light spectrum Hertz) excites the air molecules to dipole radiation, in the " Blue " dominated ( frequencies around the higher hertz, the approximate ratio, 4 x 6.5 / 5.5 corresponds to almost a doubling of the radiation intensity in the transition from a green to a blue frequency with a given dipole moment ). Furthermore, the formula given is also relevant for today has become almost commonplace mobile telephony. The communication happens via the hazards arising from the mobile phone to the nearest switching node dipole radiation whose frequency range (~ 109 Hertz) is sufficiently high that, despite minimal energy consumption of the mobile phones, the signal intensity for the transmission of information is sufficient. At the same time the frequencies of the mobile telephony are currently in biologically harmless or area, in contrast to X-rays.

From the far-field approximation to the antenna pattern

In the far field are negligible and the terms. If we write only the dominant terms on, it follows:

The amount of common factor contains the directional dependence of the field strength. It varies as the angle to the equatorial plane and is independent of the azimuth ( see adjacent antenna diagram).

The Poynting vector gives the energy flux density. Its amount, averaged over time, is in the far field

And up to a factor equal to the radiation intensity In this case, the pressure measured by polar angle of the vector from the azimuthal angle, however, the result does not depend on.

Thus, the broadcasting that is at its maximum in the directions perpendicular to perpendicular to the antenna. In antenna direction even it disappears.

By integrating over all directions, we obtain for the total power radiated into the far field, as valid and the full solid angle. In isotropic distribution instead would result in a radiation intensity of Referred to as antenna gain ratio is thus 1.5 in vacuum ( about 1.76 dBi ).

Generalization: multipole

Definitions

Supplying an alternating current to the angular frequency of an antenna of length L so produces a periodic oscillating electric dipole vector of the antenna with the direction (z-direction ) as the dipole. ( Where Q is the electric dipole moment ( t) is the periodic oscillating electric charge. )

Well layer is in the (x, y ) is generated on a circle of radius circulating particles with a constant charge Q0, a magnetic dipole vector, which also has the z- direction by convention, and is circularly polarized in accordance with the sense of rotation. ( The magnetic dipole moment is the angular frequency of the circulation

Magnetic dipole radiation from the outset is therefore an order of magnitude lower than electric dipole radiation because of the quadratic dependence of the torque of the (compared to λ ) of small length. However, the already known linear Beziehunng applies to these.

Two slightly mutually shifted opposes - same dipole vectors yield a so-called " quadrupole tensor ," two slightly mutually shifted opposite - same quadrupoles an " octupole ", etc. The number of degrees of freedom thereby increases each time by two, not three, because in the direction the displacement of only the two angular coordinates are involved perpendicular to the z -axis.

Spherical coordinates are used in place of the Cartesian coordinates in (x, y, z) in the following, are associated in the usual manner.

Formula

The corresponding generalization of the Hertzian dipole radiation is the so-called multipole. Instead of Dipolvektors occur electric plus magnetic multipole moments and, with the indices l and m on the polar brw. azimuthal angle variables and spherical coordinates refer. The general formula is by John David Jackson

This roughly corresponds to the interchange of E and H, taking into account the sign ( iZ0 -> -i/Z0 ), analogous to the formal Vertauschungssymmetrie the free Maxwell 's equations in the cgs system (vacuum, B = H, D = E):

The term, the " real part " is often omitted for simplicity. is the impedance of the vacuum, the spherical radius of the torque vector. The weight factors and describe for l = 1 electric and magnetic dipole radiation and for l = 2 quadrupole, each with 2l 1 different m values ​​. So you have three or five m values ​​for the successive l - values ​​. In the far field, the radial function can be a spherical Bessel function simplifies to. , In accordance with the above formulas The size of k is equal to finally ω / c.

Near and far field

In the vicinity of the field components are now - with complicated directional dependence is given by the so-called spherical harmonics - proportional to In the far field, however, are by - as - before all of the components and electric and magnetic fields, and the radius vector is as planar electromagnetic waves mutually orthogonal to each other.

Monopole radiation would correspond to l = 0. That this may not occur is intuitively clear, for example, because the external field of a small charged sphere is given regardless of the oscillating sphere radius only by the constant total charge united in the center of the sphere.