The force density in electrical machines modeled as tension and pressure gradients of magnetic field lines

This paper shows how to model the force density in electrical machines based on the field lines of the magnetic flux density. The force density is written as two vector components: the magnetic tension force and the magnetic pressure gradient force. This approach has been applied in physics but never to forces in engineering problems. The magnetic tension force acts to straighten bent field lines, based on the curvature of the flux density. The magnetic pressure gradient force acts from regions of high flux density to regions of low flux density. Both force densities are derived from the Lorentz force using the tnb-frame of Frenet–Serret formulas and shown to be equivalent to the divergence of the Maxwell stress tensor. It is shown how the force density could describe the forces in a synchronous machine, including both the angular torque of the load and the radial forces between the rotor and the stator. It could also be linked to the power flow and thereby to the energy flux of Poynting’s vector. The force densities could be used to improve the understanding of the Maxwell stress tensor, since they are easier to illustrate as vectors compared to the matrix form of the Maxwell stress tensor. It also shows the location of the force density, which could improve the use of enclosing volumes when calculating the force based on the divergence theorem with the Maxwell stress tensor.