Closed Analytical Formulae for Multi-Pole Magnetic Rings

Multi-pole permanent magnetic encoders are used for wheel speed sensing in automotive systems. This paper discusses rings and discs magnetized along axial direction. The vector field is calculated analytically in 3D as sums over all poles. For the case of discs with vanishing inner and infinite outer diameter the summation is done in closed form with a new summation formula. The results are discussed and several plots of field patterns are given in normalized form: At very small air-gap the field shows an overshoot. At moderate and large air-gaps it is a sine-function with exponentially decaying amplitude. The amplitude versus air gap, reading radius, thickness of the magnetic layer, and number of poles is studied and excellent agreement with measurements is found. The effect of a steel-back on the field is explained. It is also shown how to maximize the torque transmitted in axially magnetized couplings.