Electrodynamic Suspension - Principles

Principles

When a conductive loop experiences a changing magnetic field, from Lenz's law and Faraday's law, the changing magnetic field generates an Electromotive Force (EMF) around the circuit. For a sinusoidal excitation, this EMF is 90 degrees phased ahead of the field, peaking where the changes are most rapid (rather than when it is strongest):

where N is the number of turns of wire (for a simple loop this is 1) and Φ_B is the magnetic flux in webers through a single loop.

Since the field and potentials are out of phase, both attractive and repulsive forces are produced, and it might be expected that no net lift would be generated. However, although the EMF is at 90 degrees to the applied magnetic field, the loop inevitably has inductance. This inductive impedance tends to delay the peak current, by a phase angle dependent on the frequency (since the inductive impedance of any loop increases with frequency).

where K is impedance of the coil, L is the inductance and R is the resistance, the actual phase lead being derivable as the inverse tangent of the product ωL/R, viz., the standard phase lead evidence in a single-loop RL circuit.

But:

where I is the current.

Thus at low frequencies, the phases are largely orthogonal and the currents lower, and no significant lift is generated. But at sufficiently high frequency, the inductive impedance dominates and the current and the applied field are virtually in line, and this current generates a magnetic field that is opposed to the applied one, and this permits levitation.

However, since the inductive impedance increases proportionally with frequency, so does the EMF, so the current tends to a limit when the resistance is small relative to the inductive impedance. This also limits the lift force. Power used for levitation is therefore largely constant with frequency. However there are also eddy currents due to the finite size of conductors used in the coils, and these continue to grow with frequency.

Since the energy stored in the air gap can be calculated from HB/2 (or μ₀H2/2) times air-gap volume, the force applied across the air gap in the direction perpendicular to the load (viz., the force that directly counteracts gravity) is given by the spatial derivative (= gradient) of that energy. The air-gap volume equals the cross-sectional area multiplied by the width of the air gap, so the width cancels out and we are left with a suspensive force of μ₀H2/2 times air-gap cross-sectional area, which means that maximum bearable load varies as the square of the magnetic field density of the magnet, permanent or otherwise and varies directly as the cross-sectional area.