User:Graeme Bartlett/tri-polarized antenna

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A tri-polarized antenna is an antenna with three orthogonally arranged dipoles or loops. Terms for the triple dipole antenna include three crossed dipoles, three orthogonal dipoles, tri-axis design, orthogonal triad or tripole.

There are 20 different ways to group electic dipoles and magnetic loops together to form such an antenna.

Capability
A tri-polarized antenna can measure the total energy level of an electromagnetic field, and thus can be considered isotropic. It can measure the direction of arrival of a wave in three dimensions. It removes the front-back ambiguity of direction that a simple loop or dipole antenna has. It can measure the full polarization of the wave, both for linear and circularly polarized signals.

Kinds
An electric dipole oriented in three directions can be given by ex, ey, ez and the magnetic loop oriented in three orthogonal directions hx, hy and hz. The twenty possible combinations of these elements is then: {ex, ey, ez}; {ex, ey, hx}; {ex, ey, hy}; {ex, ey, hz}; {ex, ez, hx}; {ex, ez, hy}; {ex, ez, hz}; {ex, hx, hy}; {ex, hx, hz}; {ex, hy, hz}; {ey, ez, hx}; {ey, ez, hy}; {ey, ez, hz}; {ey, hx, hy}; {ey, hx, hz}; {ey, hy, hz}; {ez, hx, hy}; {ez, hx, hz}; {ez, hy, hz}; {hx, hy, hz}

Response to three dimensional field
The power density of an arriving wave can be easily calculated by summing the power from each element of the antenna. The power is proportional to the square of the electric or magnetic field strength.
 * Power density W = (Ex2 + Ey2 + Ez2)/377 where the 377, is Z0 intrinsic impedance of free space.

The normalised fields are divided by this power so that the total of the field is 1 of its units. Lower case e is used for the normalised E, and h for normalised H.

The parameters for an electromagnetic wave is its arrival from direction φ in azimuth and θ in elevation, and polarization direction γ and phase angle η Each of these unknown values have a limited range: θ from 0 to π; φ from 0 to 2π; γ from 0 to $π/2$; and η from −π to +π. Elevation is measured from the nadir, the -z direction.

the normalised fields in spherical coordinates are given by:

eφ = sin γejη eθ = cos γ hφ = Z0cos γ hθ = Z0sin γejη

In Cartesian coordinates by

ex = sin γ cos θ cos φejη − cos γ sin θ ey = sin γ cos θ sin φejη + cos γ cos θ (check + or -) ez = − sin γ sin θ φejη hx = − cos γ cos θ cos φ − sin γ sin φejη hy = − cos γ cos θ sin φ − sin γ cos φejη hz = cos γ sin θ

The tripole elements can be aligned with the Cartesian coordinates. Then the dipole will be proportional to the electric field e, and a loop will respond linearly to the magnetic field h.

Solving for unknown direction and polarization
There are four unknown values to establish φ in azimuth and θ in elevation, γ polarization direction and η phase angle. With three antenna inputs, amplitude and phase for each are available. This gives enough input variables to solve for the unknown values, and also the unknown amplitude and phase of the arriving waves.

The Poynting vector p = [sin θ cos φ, sin θ sin φ , cos θ] = e × h* ÷ (‖e‖•‖h*‖)

If there are multiple waves from different directions impinging on the antenna with different frequencies, then this method of calculation results in a time varying direction. However by sampling over a longer period of time, a method called Unitary Matrix Pencil combined with the least squared method can result in a set of linear equations that can be solved, even when noise is present. 0002149.pdf

Use
Tri-polarized antennas are used in sensor probes, radio astronomy, omnidirectional transmitting antennas, and receiving antennas that can simultaneously detect polarization and direction. Examples included LOPES used to detect radio emissions from cosmic ray showers.

Advantage
Calculation of arrival directs can be done without knowing the exact size geometry of the antenna or calibration. There is no phase ambiguity as would happen with a spaced array antenna. The common centre ensures that the dipoles all measure the same region of space, but yet do not cast shadows on each other.