Numerical solution of differential and integral equations
The availability of adequate accuracy characteristics of the electromagnetic field with consideration for the electric motor geometry, motion of its structural elements, eddy currents influence and saturation of the magnetic system is of great importance for designing an electric motor.
The method proposed in this report allows the calculation of the electromagnetic field in the motor volume at every instant of time and then according to its values the calculation of forces acting upon the rotor.
The calculated forces, in their turn, allows the calculation of the rotor displacement and thus in every following instant of time the calculation of the electromagnetic field in the motor volume with consideration for the changing of rotor position. Consequently, all the characteristics necessary for design (such as values of the winding current, power consumption, available capacity, heat and magnetic losses) are calculated in the modelling process and they at most represent the facts.
The method developed includes the procedures of solving the non-stationary non-linear differential equation system with partial derivative that obtained from Maxwell’s equations system. The approximation is fulfilled using the finite element method (FEM) on triangular and rectangular meshes depending on the type of a motor. In this case the finite element mesh during the simulation process is rebuilt automatically.
The possibilities of the proposed approach are demonstrated by the examples of work cycle calculation for two types of electric motors.
The first is an electromagnetic linear motor used as a drive for various press equipment. For such a motor the numerical modeling results were compared with the experimental results including the measurement of current oscillogram, voltage oscillogram, armature motion trajectory and armature speed at the moment of blow. The difference between calculated and experimental characteristics was not more than 2% that corresponded to experimental uncertainty.
The second type is an asynchronous motor. In contrast to the previous case, the geometry of such motors is three-dimensional. However, the results with not bad precision can be achieved using two-dimensional electromagnetic field model having assumed that the field does not change along motor axis, but the coils have finite size. In this case the developed software allows not only to take into consideration the dependence of magnetic characteristics of rotor and stator steel and speed of rotation from the electromagnetic field but also to set various coil communication circuits and connections of coils to the different phases. The electric circuit can include such elements as diodes and capacitors by means of including to the mathematical model not only differential equation with partial derivative that describes the electromagnetic field behavior, but also differential equations describing the interrelation between current and voltage on those elements.
Thus, the proposed mathematical modelling apparatus and the developed software based on it allow the optimizing of motor structural components during a design phase, the obtaining of information about motor work characteristics for various mode of operation with various loads as well as the analysis of its wake-up, acceleration and deceleration modes.
Note. Abstracts are published in author's edition
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