Magnetic Torque Calculation
Magnetic Torque Calculation: Theoretical Foundations
Fundamental Principles of Torque
Professor, how is the torque of a rotating machine formulated?
Magnetic torque is derived from the angular component of the Maxwell stress tensor. Torque in the air gap:
$r$: Air gap radius, $L_{stk}$: Stack length, $B_r$: Radial magnetic flux density, $B_\theta$: Tangential magnetic flux density.
So torque is the product of $B_r$ and $B_\theta$.
Correct. In a Permanent Magnet Synchronous Motor (PMSM), torque can be separated into magnet torque and reluctance torque:
The first term is magnet torque, the second term is reluctance torque. In IPM motors, $L_d \neq L_q$, allowing utilization of reluctance torque as well.
Summary
- Maxwell Stress — $T \propto \int B_r B_\theta d\theta$
- Magnet Torque + Reluctance Torque — The two components of IPM motor torque
- dq-axis Theory — Correspondence with control parameters
Maxwell Stress Tensor—The Physics of "Force Exerted by a Magnetic Field on a Surface"
When calculating electromagnetic torque with FEM, the "Maxwell Stress Tensor Method" is most commonly used. It calculates torque and force by integrating the mechanical stress (normal component: attractive force, tangential component: shear force) that the magnetic field exerts on a boundary surface. Theoretically, it is derived directly from Maxwell's electromagnetic field theory (1865) and has high versatility applicable to objects of arbitrary shape. However, the accuracy of torque calculation degrades with coarse meshes because the integration surface position depends on the mesh density in the air. JMAG and ANSYS adopt improved implementations that combine this method with "Averaged Maxwell Method" or "Virtual Work Method" to address this issue.
Computational Methods for Magnetic Torque Calculation
Torque Calculation in FEM
Could you compare the methods for calculating torque in FEM?
| Method | Accuracy | Mesh Dependency | Remarks |
|---|---|---|---|
| Maxwell Stress Method | Medium | High | Depends on integration surface position |
| Arkkio Method | High | Medium | Averaging via air gap volume integration |
| Virtual Work Method | Highest | Low | Requires two calculations |
| Nodal Force Method | High | Medium | Convenient for structural coupling |
Is the Arkkio method most common in practice?
In JMAG, the Arkkio method is standard. In Ansys Maxwell, Virtual Work is the default. It's also important to separate torque spatial harmonic components (cogging torque, torque ripple orders) using FFT analysis.
Summary
- Arkkio Method — Practical standard for rotating machine torque
- Virtual Work Method — Highest accuracy but high computational cost
- FFT Analysis — Identification of torque harmonic components
Virtual Work Method—An Alternative Torque Calculation via "Energy Differentiation"
The Virtual Work Method calculates torque as the "displacement derivative of magnetic field energy" and is used as an alternative to the Maxwell Stress Method. Torque = δW/δθ is calculated from the energy change δW when a small displacement δ is applied. Its advantage is lower sensitivity to mesh quality compared to the Maxwell method; its drawback is the need for two calculations (before and after displacement). An implementation note is that if the displacement δ is too large, nonlinear error occurs, and if too small, numerical error increases. The optimal δ is generally around 0.01 to 0.1 times one electrical degree, established as a guideline for motor analysis.