Magnetic Torque Calculation
Theory and Physics
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.
Physical Meaning of Each Term
- Electric Field Term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. A time-varying magnetic flux density generates an electromotive force. [Everyday Example] A bicycle dynamo (generator) produces voltage in a nearby coil by rotating a magnet—a direct application of this law that a time-varying magnetic field induces an electric field. Induction heating (IH) cookers also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heat.
- Magnetic Field Term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère-Maxwell's law. Current and displacement current generate a magnetic field. [Everyday Example] When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, passing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ cannot be ignored, describing electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Indicates that electric charge is the divergence source of electric flux. [Everyday Example] Rubbing hair with a plastic sheet creates static electricity, making hair stand up—the charged sheet (electric charge) radiates electric field lines outward, exerting force on the light hair. Capacitor design calculates the electric field distribution between electrodes using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis following Gauss's law.
- Magnetic Flux Conservation $\nabla \cdot \mathbf{B} = 0$: Expresses the non-existence of magnetic monopoles. [Everyday Example] Cutting a bar magnet in half does not create a magnet with only an N pole or only an S pole—N and S poles always exist as a pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, the formulation using vector potential $\mathbf{B} = \nabla \times \mathbf{A}$ is used to satisfy this condition, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity are independent of magnetic/electric field strength (nonlinear B-H curve needed in saturation region)
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis
- 2D assumption (cross-section analysis): Effective when current direction is uniform and end effects can be ignored
- Isotropic assumption: Direction-specific property definitions needed for anisotropic materials (e.g., rolling direction of silicon steel sheets)
- Non-applicable cases: Additional constitutive laws needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Magnetic Flux Density $B$ | T (Tesla) | 1T = 1 Wb/m². Permanent magnets: 0.2 to 1.4T |
| Magnetic Field Strength $H$ | A/m | Horizontal axis of B-H curve. Conversion with CGS unit Oe (Oersted): 1 Oe = 79.577 A/m |
| Current Density $J$ | A/m² | Calculated from conductor cross-sectional area and total current. Note non-uniform distribution due to skin effect |
| Permeability $\mu$ | H/m | $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m |
| Electrical Conductivity $\sigma$ | S/m | Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise |
Numerical Methods and Implementation
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.
Edge Elements (Nedelec Elements)
Elements specialized for electromagnetic field analysis. Automatically guarantee continuity of tangential components and eliminate spurious modes. Standard for 3D high-frequency analysis.
Nodal Elements
Used for scalar potential formulation. Effective for scalar potential methods in magnetostatics and electrostatic field analysis.
FEM vs BEM (Boundary Element Method)
FEM: Handles nonlinear materials and inhomogeneous media. BEM: Naturally handles infinite domains (open boundary problems). Hybrid FEM-BEM is also effective.
Nonlinear Convergence (Related Topics
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