964 IEEE TRANSACTIONS ON MAGNETICS, zyxwv VOL. 26, NO. 2, MARCH 1990 A FINITE ELEMENT SIMULATION OF AN ALTERNATOR CONNECTED TO A NON LINEAR EXTERNAL CIRCUIT F. Heclit, A. Marrocco INRIA Domaine de Voluceau- Rocquencourt BP 105 78153 LE CHESNAY FRANCE Abstract We consider a squirrel cage alternator (with laminated rotor and sta- tor), connected to an external circuit whose components may be resis- tances, inductances, diodes, tension or current generators. The rota- tion speed of the rotor given, we compute the electromagnetic behavior of the system. The field approximation inside the alternator is achieved by finite el- ement technique and the relative rotor movement with respect to the stator is taken into account using a rotation line (or slipping line) lo- cated in the air-gap. The external circuit is implemented in the model via an admittance matrix and in this way an integro-differential equ& tion for vector potential A only is obtained. Numerical results have been compared with experimental measurements and found to be in good agreement with them. This work has been supported by VALEO. Introduction When modelling and analysing electro-magnetic systems, two main strategies can be followed. Firstly, electric part and magnetic part of the system can be decoupled and parameters from one subsystem are used for the simulation of the other one [I]. Such an approach will simplify the problem to be solved, and is often used, but cannot always be applied in practical situations. When the interactions be- tween magnetic and electric tircuits become larger, such a strategy will certainly induce erroneous computational results. Secondly, equations governing electric and magnetic states can be taken into account si- multaneously. As the coupling between external circuit and alternator is very strong the second strategy will be used in the following. The laminated rotor and stator imply, zyxwvutsrq as usual, that a 2D magnetic simulation will be a valid approximation. The field approximation inside the alternator is achieved using finite element technique; the rotor movement is taken into account in the model by using zyxwvutsrq time stepping technique in connection with tiine discretization in Maxwell’s equations and using a rotation line (or slipping line) with equally dis- tributed nodes, located in the air-gap [2,3]. Other techniques such as macro-element [4,5], movement stripe {GI have been used when mod- elling movement in electromagnetic devices. Two coordinate systems are used implicitely, one is fixed and is attached to the stator, the other one is mobile and is in synchronous rotation with the rotor; by this way no first order terms due to displacement appear explicitely in the equations [6,7,8]. Within each time step the external circuit (the state of external cir- cuit) is considered to be fixed zyxwvutsrqpo ( a diode is modelled -as usual in electrotechnics- as a small resistance r when the current in the cor- responding branch is in the right direction and as a large resistance R when not, a threshold voltage is also included in diode simulation), the commutation between the two states for a diode occur only at the ends of the time step interval. The external circuit is taken into account in the Maxwell’s equations via an admittance matrix (generalization of the concept developped and used in [9]). At each time step a non linear problem with a symmetric positive operator has to be solved. A software including a language for an easy description of external circuit connected to the machine under consideration has been devel- opped and validated (during 1986-1987) by comparison between nu- merical results and experimental measurements. We want to thank the Direction des Etudes Auance‘es of VALEO Group for the financial support of these studies, for the experimental measurements and for the fruitful discussions with their engineers and technicians. Problem formulation and solution When modelling electromagnetic systems, it is usual [3,10,11,12,13] to derive the following equation dA V zyxwv x (vV x A) = zyxw -U-+ zyxwvu at zyxwv J on the set R c zyxwv E’, with appropriate boundary and initial conditions. U is the electric conductivity, A=(O,O,A) is the vector potential, Y is the reluctivity and J=(O,O,J) is the current density. The terms in the right hand side of (1) are not given quantities. The movement is taken into account in the simulation by solving equation (1) within two different coordinate systems. In this way, first order terms like a(v x V x A) are not present in the formulation (see [7,8]). The current density J in (1) can be written as: k=l where JO represents a given current density (prescribed), ik is the current in the phase ( or branch) k, np is the number of phases and $k is a L2(n) function defined as a current density distribution (in 0) when the current is 1A in the phase number k. (obviously $Jk E 0 outside the support of phase t). The electric equations for branches k, 1 5 k 5 np, with ends IC1 and kz may be written and for branches q,nP < q 5 nb (3-4 where nb represents the total number of branches of electric circuit, (vkl - vkz) or - Vq2) the potential drop in the branch k or q, Rk or R, the resistance , [k or 1, are the inductances not included in magnetic equation, @k is the flux through a coil and -y(i,) the potential drop for a diode. As the time constant for a diode is negligeable with respect to the time constant of the system, the diode is modelled as an equivalent circuit with a resistance Rd and a threshold voltage E,, More precisely, if a and b are the extremities of a diode, we have 7(iq) = V, - & = &(i,)i, + (3%) Rd(i,) = R (large value for the resistance) if i, < 0 { r (small value for the resistance) otherwise The relations (1-3) show how magnetic and electric equations are cou- pled ( the flux @k can be expressed as a function of vector potential A). 0018-9464/90/0300-0964$01.00 zyxwvut 0 1990 IEEE