From least action in electrodynamics to magnetomechanical energy -- a review

The equations of motion for electromechanical systems are traced back to the fundamental Lagrangian of particles and electromagnetic fields, via the Darwin Lagrangian. When dissipative forces can be neglected the systems are conservative and one can study them in a Hamiltonian formalism. The central concepts of generalized capacitance and inductance coefficients are introduced and explained. The problem of gauge independence of self-inductance is considered. Our main interest is in magnetomechanics, i.e. the study of systems where there is exchange between mechanical and magnetic energy. This throws light on the concept of magnetic energy, which according to the literature has confusing and peculiar properties. We apply the theory to a few simple examples: the extension of a circular current loop, the force between parallel wires, interacting circular current loops, and the rail gun. These show that the Hamiltonian, phase space, form of magnetic energy has the usual property that an equilibrium configuration corresponds to an energy minimum.Comment: 29 pages, 9 figures, 65 reference

Modeling magnetic saturation and saliency effects via Euler-Lagrange models with complex currents for three-phase permanent magnet machines

International audiencePermanent magnet machines with both magnetic saturation and saliency effects can be directly described via Euler-Lagrangian formulation with complex currents. The Lagrangian is the sum of a mechanical kinetic energy and a magnetic Lagrangian. This second term is expressed in terms of rotor angle, complex stator and rotor magnetizing currents. Via simple modification of magnetic Lagrangian we derive a non-trivial dynamical model describing permanent-magnet machines with both saturation and saliency. We propose an experimental validation of such models on a customized torque machine of 1.2 kW. This first validation relies on injections of high frequency oscillations on the stator voltage. According to the proposed saturation model, the resulting amplitudes of the current-ripples is an increasing function of the current offset. Such dependance is effectively observed experimentally and confirmed by simulations