Arkkio's Method for Torque Computation

Hi folks – it’s been a while! But today, I’m finally back from vacation with a load of awesome! Namely, the torque computation method developed by and named after my professor, is now available in the commercial FEM software Comsol Multiphysics as well! This is a really nice acknowledgement in my opinion, and a long time in waiting.

Enter the Arkkio’s method.

Background

Arkkio’s method is used in finite element analysis, to obtain the torque produced by a rotating electrical machine. Well, to be exact, it can just as easily yield the total force acting on the machine rotor, but I’ll be focusing on torque here.

The method is based on the Maxwell stress tensor – a second-rank electromagnetic tensor representing the interaction between electromagnetic forces and (mechanical) momentum. See its Wikipedia page for some mathematical nightmare fuel.

Long (and horrible) story short, the tensor is very useful in analysis of electrical machines. Indeed, it can be used to calculate the total torque acting on the machine rotor, by means of integrating the magnetic flux density along any surface completely enclosing the rotor.

In other words, the electric field $\mathbf{E}$ terms seen on the Wiki page disappear, since their contribution is negligible compared to the flux density $\mathbf{B}$ .

Things get even simpler when we move to 2D cylindrical coordinates. Which we can, since rotating machines are typically analysed by 2D FEM, and the rotor is surrounded by a cylindrical air-gap after all.

In this case, the surface integral can be replaced by a simple line integral along a circle, multiplied by the length of the machine in the z-direction. This yields a very compact expression for the torque $T$

$T = \frac{l}{\mu_0} \int\limits_0^{2\pi} B_r B_\theta r^2 \mathrm{d}\theta$ ,

where $B_r$ and $B_\theta$ are the radial and circumferential components of the flux density in the air-gap, $l$ is the machine length, and $\mu_0$ is the air permeability. Finally, $r$ represents the radius of the circle that we are integrating on. Obviously, $r$ must be selected so that the circle lies wholly in the air-gap of the machine – no crossing the iron parts allowed!

Also, note that $B_r$ and $B_\theta$ are obviously dependent on both $r$ and $\theta$ .

Torque computation with the Maxwell stress tensor. Taken from the course slides for the Numerical Methods in Electromechanics.

Why the stress tensor sometimes fails (kinda)

The stress tensor method is mathematically correct, and will yield the correct torque – if the flux density $\mathbf{B}$ is correct.

Which it usually isn’t.

Remember, we are talking about numerical analysis here. Hence, we are dealing with approximate quantities most of the time.

And typically, what is actually solved in the procedure is the so-called vector potential $\mathbf{A}$ . The flux density is then obtained from the curl

$\mathbf{B} = \nabla \times \mathbf{A}$ .

That there is a differentiation, folks. And if you have any experience with computers and mathematics, you know that when you take the derivative of a somewhat-inaccurate quantity, you end up with something quite a bit more inaccurate.

In other words, we get larger errors in $\mathbf{B}$ than we have in $\mathbf{A}$ , resulting in larger inaccuracies in the torque $T$ .

Not nice.

Could this situation be remedied somehow?

Arkkio’s method

This is where the Arkkio’s method comes in. (Note: the following is my and only my interpretation about the method. For the original derivation, see prof. Arkkio’s thesis from page 55 on.)

Suppose we try to improve the accuracy by computing the torque with several different integration radii $r_i$ (that all still fit in the air-gap, mind you) equally spaced between $r_\mathrm{in}$ and $r_\mathrm{out}$ , and then taking their average. This, ideally, should cancel a significant portion of the numerical noise we have, caused by that nasty differentiation.

So, we try and write

$T \approx \sum\limits_{i=1}^{N} \frac{1}{N} \frac{l}{\mu_0} \int\limits_0^{2\pi} B_r B_\theta r_i^2 \mathrm{d}\theta$ .

Next, we multiply both the nominator and denominator with the small constant $\Delta r = \frac{r_\mathrm{out} - r_\mathrm{in}}{N}$ , getting

$T \approx \sum\limits_{i=1}^{N} \frac{1}{N \Delta r} \frac{l}{\mu_0} \int\limits_0^{2\pi} B_r B_\theta r_i r_i \Delta r\mathrm{d}\theta$ .

You see where I’m going with this?

Now we simply let $N$ approach infinity, so that $\Delta r$ can be replaced by $\mathrm{d}$ , and the sum by an integral. By definition, $N \Delta r$ can obviously be written as $r_\mathrm{out} - r_\mathrm{in}$ .

With this, we finally get the Arkkio’s method for torque computation

$T = \frac{l}{\mu_0 (r_\mathrm{out} - r_\mathrm{in})} \int\limits_o^{2\pi} \int\limits_{r_\mathrm{in}}^{r_\mathrm{out}} B_r B_\theta r r \mathrm{d} r \mathrm{d}\theta$ .

By also noting that $r \mathrm{d} r \mathrm{d}\theta$ represents the differential surface area $\mathrm{d}S$ in cylindrical coordinates, the method can be equivalently expressed in the more compact surface integral form

$T = \frac{l}{\mu_0 (r_\mathrm{out} - r_\mathrm{in})} \int\limits_S B_r B_\theta r \mathrm{d}S$ ,

which is indeed the canonical, most common expression for the Arkkio’s method.

Nice and simple!

Torque computation with Arkkio's method. — Torque computation with Arkkio’s method. From the same source.

Results

Indeed, Arkkio’s method does exactly what it is supposed to – improving the accuracy of torque computation by cancelling out some numerical noise due to the differentiation. Countless hours of applications examples have demonstrated this quite nicely.

The method is also very nice to implement – we are anyways evaluating surface integrals within the finite element method. Changing the integrand into $B_r B_\theta r$ is basically no work at all!

Hope you liked this knowledge bomb. You should.

-Antti

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Arkkio’s Method for Torque Computation

Antti Lehikoinen

Arkkio’s Method for Torque Computation

Background

Why the stress tensor sometimes fails (kinda)

Arkkio’s method

Results

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