Magnetic Torque Report¶

The torque is calculated using the Maxwell stress tensor $\overline{\overline{T}}$ [Pa] given by

\[ \overline{\overline{T}} = \mathbf{B} \otimes \mathbf{H} - \frac{1}{2} \left( \mathbf{B} \cdot \mathbf{H} \right) \overline{\overline{I}} , \]

where

$\overline{\overline{T}}$ — Maxwell stress tensor [Pa],
$\mathbf{B}$ — magnetic flux density [T],
$\mathbf{H}$ — magnetic field [A/m].

Here $\otimes$ denotes the outer (dyadic) product, $\left( \mathbf{B} \otimes \mathbf{H} \right)_{ij} = B_i H_j$, and $\overline{\overline{I}}$ is the identity tensor.

The torque $\boldsymbol{\tau}$ [N $\cdot$ m] is obtained by integrating over a surface $S$ enclosing the body of interest:

\[ \boldsymbol{\tau} = \int_S \mathbf{r} \times \left( \overline{\overline{T}} \cdot \mathbf{n} \right) \, dS, \]

where $\mathbf{r}$ is the position vector [m] and $\mathbf{n}$ the outward unit normal.

In practice the surface integral is evaluated as a volume integral of the stress tensor over the layer of mesh elements immediately surrounding the selected body.

Torque from the Maxwell stress on the layer around the body

Figure 1: The traction $\overline{\overline{T}}\cdot\mathbf{n}$ on the air layer enclosing the body, weighted by the lever arm $\mathbf{r}$ about the axis $O$, integrates to the net torque $\boldsymbol{\tau}$.

Note

The surrounding region must be force-free. The elements adjacent to the selected body — the layer over which the stress tensor is integrated — must carry no body force: the region must be non-conductive (no eddy currents, so no $\mathbf{J}\times\mathbf{B}$ force) and have vacuum permeability ($\mu = \mu_0$, no magnetization force). Surround the body with at least one layer of air/vacuum elements; do not place the body directly against another magnetic or conductive region, or the reported torque will be wrong.

Usage¶

from mufem.electromagnetics.timedomainmagnetic import MagneticTorqueReport

rotor_torque_report = MagneticTorqueReport(
    name="Rotor Torque",
    marker="Rotor" @ Vol,
)

rotor_torque = rotor_torque_report.evaluate()
print(f"Rotor torque: {rotor_torque}")

The evaluate() method returns the torque vector about the origin. The marker selects the body on which the torque is summed.

When to use this¶

Rotating electric machines — average and ripple torque on a motor / generator rotor; cogging-torque curves vs. rotor angle.
Stepper and reluctance motors — detent and holding torque characterisation.
Galvanometers and rotary actuators — torque vs. coil current.
Magnetic couplings — transmitted torque vs. angular misalignment.
Halbach-array motors — torque density studies as a function of magnetisation pattern.