Computing Electromagnetic Forces and Torques
Two methods are available to compute electromagnetic forces and torques:
•
The most general method is to use the Maxwell stress tensor.
•
Another method that works for the special case of computation of magnetic forces on nonmagnetic, current-carrying domains uses a predefined physics interface variable for the Lorentz force distribution in a magnetic flux density B.
Maxwell Stress Tensor
In most AC/DC interfaces, the Force Calculation feature can be used to compute forces and torques using Maxwell stress tensor. In electric field and current interfaces, the force is calculated by integrating
(3-8)
on the surface of the object that the force acts on,
In magnetic field interfaces, the expression
(3-9)
is integrated on the surface to obtain the force.
In the Magnetic and Electric Fields interface, both expressions (Equation 3-8 and Equation 3-9) are included. E is the electric field, D the electric displacement, H the magnetic field, B the magnetic flux density, and n1 the outward normal from the object (Material 1).
With the Maxwell stress tensor, the force is expressed as
(3-10)
where ∂Ω represents the domain boundary. The torque is computed as
(3-11)
where r0 is a point on the axis of rotation. For a theoretical discussion about the stress tensor, see Electromagnetic Forces.
Lorentz Forces
The Lorentz force is defined as F = J × B. The Lorentz force is very accurate for electromagnetic force calculations in electrically conducting domains. The Lorentz force variables <name>.FLtz<component> are available both in domains and on boundaries (in the case of surface currents).
For moving objects containing magnetic materials such as a magnetizable armature, the Lorentz force is no longer the only force contribution. In this case, the force contribution from the magnetic polarization might need to be considered. This can be done by adding the Force Calculation feature which computes the force using Maxwell stress tensor. Note that the Force Calculation feature includes the force contribution not only from magnetic polarization but also from the Lorentz force, that is, the total electromagnetic force.
Cycle-Averaged Quantities
In Frequency Domain studies, the formulas presented above can be used to compute the instantaneous force and torque by taking the real part of all the variables involved. It is often more interesting to compute the average of these quantities over a cycle to, for example, couple the electromagnetic forces to another physics. The cycle-averaged Maxwell’s stress tensor (including both electric and magnetic forces) is computed as:
Arkkio’s Method
In the Rotating Machinery, Magnetic interface, in addition to the Force Calculation feature, the Arkkio Torque Calculation feature based on Arkkio’s method is available. Arkkio’s method is a variant of the Maxwell stress tensor method, specifically useful in modeling rotating machines where the thin air gap separates rotating parts. In this case, the normal n1 of the air gap can be defined as the radial outward unit vector. Arkkio’s method computes the torque as
(3-12)
where Fφ is the azimuthal force density; Br, Bφ are the radial and azimuthal magnetic flux density, respectively; rA is the distance from the rotation center; nsec is the number of sectors. Different from surface integral Equation 3-11 used in Maxwell stress tensor method, the torque expression for Arkkio’s method (Equation 3-12) is a volume integral on the domain Ω, yielding a less mesh-dependent result.