Student Myshko V., Ph.D. Kovalenko M.

National Technical University of Ukraine «Kyiv Polytechnic Institute», Ukraine

Theoretical foundations for numerical calculation of electromagnetic field of the Electric Machines

Simulation of magnetic fields is of interest when studying magnets, motors, transformers, and conductors carrying static or alternating currents.

Quasi-static analysis of magnetic and electric fields is valid under the assumption that:

(1)

It is mean that there are no dielectric displacement currents in quasistatic mode simulation of electrical machines. This implies that it is possible to rewrite Maxwell’s equations in the following manner:

(2)

Here is an externally generated current density from the power supply of the electric motor and is the velocity of the conductor. The crucial criterion for the quasi-static approximation to be valid is that the currents and the electromagnetic fields vary slowly. This means that the dimensions of the structure in the problem need to be small compared to the wavelength.

Using the definitions of the potentials,

(3)

(4)

and the constitutive relation , Ampère’s law can be rewritten as:

(5)

Equation (5) it is a basis for calculating the electromagnetic field in a cross-section area of calculation of electric the machines or electromechanical energy converter (with any configuration or design. The equation of continuity, which is obtained by taking the divergence of the above equation, adds the following equation:

(6)

The electric and magnetic potentials are not uniquely defined from the electric and magnetic fields.

The variable transformation of the potentials is called a gauge transformation. To obtain a unique solution, choose the gauge, that is, put constraints on that make the solution unique. Another way of expressing this additional condition is to put a constraint on . This is called Helmholtz’s theorem.

Important observations are that in the dynamic case A and V are coupled via the selected gauge. For a dynamic formulation, it is also possible to select a such that the scalar electric potential vanishes and only the magnetic vector potential has to be considered. The dynamic formulations (frequency domain and time dependent study types) of the Magnetic Fields interface are operated in this gauge as it involves only A. The Magnetic and Electric Fields interface involves both A and V and is inherently ungauged for all study types. In the static limit, A and V are not coupled via the gauge selection and thus any gauge can be chosen for A when performing magnetostatic modeling.

After eliminating the electric potential by choosing the appropriate gauge and disregarding the velocity term. The equation of continuity obtained by taking the divergence of Ampère’s law reads:

(7)

It is clear that unless the electrical conductivity is uniform, the particular gauge used to eliminate V cannot be the Coulomb gauge as that would violate the equation of continuity and would thereby also violate Ampère’s law.

The field is used to impose a divergence constraint. In the most simple case, that is for magnetostatics, Ampère’s law for the magnetic vector potential reads:

(8)

The equation for is used to impose the Coulomb gauge: However, to get a closed set of equations, must be able to affect the first equation and this is obtained by modifying the first equation to:

(9)

In the time-harmonic case, there is no computational cost for including the displacement current in Ampère’s law (then called Maxwell-Ampère’s law):

(9)

In the transient case the inclusion of this term leads to a second-order equation in time, but in the harmonic case there are no such complications. Using the definition of the electric and magnetic potentials, the system of equations becomes:

(10)

(11)

Current conservation is inherent in Ampère’s law and it is known that if current is conserved, explicit gauge fixing is not necessary as iterative solvers converge towards a valid solution. However, it is generally not sufficient for the source currents to be divergence free in an analytical sense as when interpolated on the finite element functional basis, this property is not conserved.

References:

1.R.K. Wangsness, Electromagnetic Fields, 2nd ed., John Wiley & Sons, 1986.2. O. Wilson, Introduction to Theory and Design of Sonar Transducers, Peninsula Publishing, 1988.

2. O.C. Zienkiewicz, C. Emson, and P. Bettess, “A Novel Boundary Infinite Element,”

International Journal for Numerical Methods in Engineering, vol. 19, no. 3, pp. 393–404, 1983

3. B.D. Popovic, Introductory Engineering Electromagnetics, Addison-Wesley, Reading, Massachusetts, 1971 .