Recent Developments of Electrical Drives - Part 22. The book stating the recent developments of electrical drives, can be useful for engineers and researchers investigating and designing electrical and electronic devices as well as for students and young researchers dealing with electrical and electronic engineering, computer sciences (advanced computer modelling, sophisticated control systems with artificial intelligence tools applied, optimal design bye use of classical and genetic algorithms employed), applied mathematics and all the topics where electromagnetic, thermal, mechanical phenomena occur | 202 Belahcen Methods Magnetic and elastic fields The A formulation of the magnetic field in two dimensions and the displacement based formulation of the elastic field are used. -S Ak 1 uk 1 d S Ak 1 Uk 1 . dA Ak 1 Dr Dr1T Cr LDs1T 0 d S Ak 1 Uk 1 . dU Ak 1 0 L Ds 0 Gs 0 d Fk 1 L dA S Ak Ak Dr T 0 0 uk Ds1t k Ti k K d Fk 1 K d U A Ak 1 A k 1 Ai k 1 AUk 1 An x An i r i r nsi rL sn x S Ak 1 Ak 1 D1 Uk 1 ID 1 K 1 k 1 LDS Ak H ik Cs Vk i Vk LD Ak i Hi k 1 C s Vk i Vk TV a six r z r r D Ak C Uk G i k r n r rn r rn D Ak 1 C Uk 1 G i k 1 F 1 K u k 1 1 The 2D finite element FE equations for the magnetic and elastic field are coupled through the displacements and the forces. These equations are solved together with the circuit equations of the windings of the machine as described in 9 . The system of equations to be solved at each iteration is written as 1 where A is the magnetic vector potential C D G and H are the coupling matrices between the magnetic vector potential and the electric parameters in the windings parts of the machine L is a matrix for the connections of the stator windings S and K are the magnetic and mechanical stiffness matrices u and i are respectively voltages and currents and U is the nodal displacements vector. The superscripts r and s refer respectively to rotor and stator. The subscripts k and k 1 refer respectively to previous and present steps in the time stepping method. The sign over the matrix K and the force F means that they are replaced by their dynamic counterparts. A full description of the above matrices is given in 9 11 . Magnetic and magnetostrictive forces The forces which are the load for the elastic field are separated into magnetic force also called reluctance forces in some works 4 and magnetostriction forces. The magnetic forces are calculated at any iteration from the calculated magnetic vector potential based on the local application of the virtual work principle d w lu Ft constant 2 II-6. Effect of Stress-Dependent .
