Electromagnetic Force
Electromagnetic Force: Theoretical Foundations
Fundamentals of Electromagnetic Force
Professor, calculating electromagnetic force is at the core of motor design, right?
The force experienced by an electric current in a magnetic field, namely the Lorentz force, is the starting point.
For current in a conductor, the volume force density is:
How is the Maxwell stress tensor used?
The stress tensor derived from the magnetic field energy:
Force is calculated via surface integration: $F_i = \oint_S T_{ij} n_j \, dS$. Another method is the principle of virtual work: $F = -\partial W_m / \partial x$.
Summary
- Lorentz Force — $\mathbf{f} = \mathbf{J} \times \mathbf{B}$
- Maxwell Stress Tensor — Calculates force via surface integration
- Virtual Work Method — High-precision method to obtain force from energy change
The Discovery of Lorentz Force—The Force Was Measured Before the Electron's Mass
When we say "electromagnetic force," we think of the Lorentz force F = q(E + v × B). This force law was experimentally established in the 1890s, thanks to Hendrik Lorentz's theoretical organization. Interestingly, the law of electromagnetic force was measured before the existence of the electron (discovered by J.J. Thomson in 1897). At a stage when neither the mass nor the charge of charged particles was known, the phenomenon of "a current experiencing a force in a magnetic field" could be described quantitatively. This is a classic example in the history of physics where theory construction preceded the elucidation of the physical entity.
Computational Methods for Electromagnetic Force
Force Calculation Methods in FEM
Are there multiple methods to calculate electromagnetic force in FEM?
There are mainly three methods.
1. Maxwell Stress Tensor Method — Calculates force on an integration surface in the air gap. Results depend on surface position, so average over multiple surfaces
2. Virtual Work Method — Calculates force from energy change when an object is infinitesimally displaced. Highest accuracy
3. Arkkio's Method — Volume integration over the air gap. Standard method for torque calculation in rotating machines
Which method is used in JMAG?
JMAG also implements the Nodal Force Method. It directly calculates the electromagnetic force acting on each node, which can be passed as a load for structural analysis. In Ansys Maxwell, the Virtual Work Method is the default.
Summary
- Maxwell Stress — Simple but depends on integration surface
- Virtual Work Method — High accuracy but high computational cost
- Arkkio's Method — Standard for rotating machine torque
Maxwell Stress Method or Lorentz Force Method?—The Mystery of Mismatched Answers
There are two types of methods for determining electromagnetic force: the "Maxwell Stress Tensor Method" and the "Lorentz Force Method (Virtual Work Method)." In theory, the total force should match regardless of which method is used. However, in practical FEA calculations, results can sometimes slightly differ. The cause is mesh coarseness; the stress tensor method is sensitive to the accuracy of magnetic field values on the integration surface, so errors can occur if the mesh on the air gap surface is coarse. The standard countermeasure is to "set up a fine mesh layer on the air gap." Mastering just this one point makes the results from the two methods almost coincide.