EM Tracker HFluxPerI Derivation

6DOF electromagnetic tracker electromagnetics

Most of the following can be found at https://en.wikipedia.org/wiki/Magnetic_dipole and in classical-electromagnetics textbooks.

We assume that, at the working frequency or frequencies, the wavelength is large compared to the distance between transmitter coil trio and receiver coil trio. This is the quasi-static approximation, which permits us to ignore radiation fields.

We assume that each coil is so small that its shape does not matter, only its size times its number of turns. This is the dipole approximation. Each coil has an effective-area vector, Aeff_vect (measured in square meters) which completely describes a dipole coil's quasi-static magnetic properties.

Consider a single transmitter coil, with effective-area vector Aeff_tmtr_vect. We pass a current I (measured in amperes) through the transmitter coil, which causes the coil to emit a vector magnetic field Hvect (measured in amperes per meter) which varies depending upon where we observe the magnetic field.

Let Rvect (measured in meters) be the vector from the transmitter coil to the position of the magnetic-field observer. The magnetic field at the observation point is Hvect(Rvect).

We can write Rvect as the product of its scalar magnitude, Rmag (measured in meters), and a unit vector Runit_vect (unitless) in the direction of Rvect:

Rvect = Rmag * Runit_vect

The magnetic field at the observation point is then:

Hvect(Rvect) = (I / (4 pi Rmag^3)) (3 Runit_vect (Runit_vect .dotproduct.) Aeff_tmtr_vect) - Aeff_tmtr_vect)

This can be written more compactly (being a little free with the notation) as:

Hvect(Rvect) = (I / (4 pi Rmag^3) ((3 Runit_vect Runit_vect .dotproduct.) -1) Aeff_tmtr_vect