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Euler-Lagrange models with complex currents of three-phase electrical machines.pdf

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Euler-Lagrange models with complex currents of three-phase electrical machines

a r X i v : 0 8 0 6 .0 3 8 7 v 2 [ m a t h .O C ] 2 4 J u n 2 0 0 8 Euler-Lagrange models with complex currents of three-phase electrical machines D. Basic F. Malrait P. Rouchon ?? June 24, 2008 Abstract A Lagrangian formulation with complex currents is developed and yields a direct and simple method for modeling three-phases permanent-magnet and induc- tion machines. The Lagrangian is the sum of the mechanical kinetic energy and of the magnetic en- ergy. This magnetic energy is expressed in terms of rotor angle, complex stator and rotor currents. Such Lagrangian setting is a precious guide for modeling space-harmonics and saturation effects. A complex- ification procedure is applied here in order to derive the Euler-Lagrange equations with complex stator and rotor currents. Such complexification process avoids the usual separation into real and imaginary parts and simplifies notably the calculations. Via simple modification of magnetic energies we derive non-trivial dynamical models describing permanent- magnet machines with both saturation and saliency, and induction machines with both saturation and space harmonics. Keywords: Lagrangian with complex coordinates, space-harmonics, magnetic saturation, induction ma- chine, permanent-magnet machine. ?Duro Basic and Franc?ois Malrait are with Schnei- der Electric, STIE, 33, rue Andre? Blanchet, 27120 Pacy- sur Eure. E-mail: duro.basic@, francois.malrait@ ?Pierre Rouchon is with Mines ParisTech, Centre Automa- tique et Syste?mes, 60 Bd Saint-Michel, 75272 Paris cedex 06, France. E-mail: pierre.rouchon@ensmp.fr 1 Introduction Modeling electrical machines with magnitude satu- ration and space-harmonics effects is not a straight- forward task and could lead to complicated devel- opments when a detailed physical description is in- cluded (see, e.g.,[3, 1]). Even if such effects are not dominant they play an important role for sensor-less control. For a permanent magnet machine, the rotor position will be unobservab