Reluctance networks 102

Hello there, and welcome back to my post series about reluctance networks. They are kind an expertise of mine. However, they do not seem to be the expertise of quite many other researchers. Indeed, today’s post is about where almost everybody goes wrong.

Before we start, a sidenote: I’ve been quiet a few days, I know. Well, at least blogging-wise – some people might argue I would do well to be a bit more quiet in the conversational sense. But I digress. What I mean we organized a week-long intensive course about permanent magnet machines, so spent quite a bit of time sitting at lectures. Moreover, I had/decided to organize a farewell Sitsit party for one of our interns, on a very short notice. (Mental todo: write a post about our parties. They rule).

Almost like electric circuits

But now I’m squarely back writing. Last time, I described some very elementary basics about reluctance networks. As you hopefully remember, the idea is to describe the magnetic field behaviour in an electrical machine with a network consisting of reluctance elements. Almost exactly like an electric circuit.

For each element, we have the magnetic Ohm’s law

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Like an electric circuit, remember?

Indeed, the well-known Kirchhoff’s circuit laws nicely describe the conservation of the flux and the mmf in the circuit. The only difference are the names of the quantities, really.

However, this is where things differ slightly.

You see, magnetic fields are almost always rotational. You know how the field will circle around a single conductor in air? Hence, rotational.

This can be seen from the Ampere’s law

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stating that the line integral of the magnetic field strength along any closed loop must equal the total current enclosed by that same loop.

In the context of reluctance network, this means that the sum of mmf drops (over individual reluctance elements) in any closed loop must equal the total current inside.

And this is a very fundamental difference from electric circuits. In those, a voltage source (or a current source, for that matter) can be presented by a single block in the circuit. That’s it. Nice and easy.

Where things differ

By contrast (a huge one), in a reluctance network the mmf source of a current will be circling around it. Thus, its influence can often be felt in every single reluctance element in the network. Remember the rotationality.

Admittedly, often the mmf sources can be expressed as simple “magnetic voltage sources”, connected in series with one or more of the reluctance elements. Even more often, they are expressed so – violating the underlying Ampere’s law!

You see, if you put the sources in the network like that, you have to be careful in doing so. You have to make sure that any possible closed loop satisfies the right-hand side of the Ampere’s law. That is, the sum of the mmf source-elements in that loop has to equal the current enclosed by that path. Or the current enclosed by the corresponding path in the actual physical device you are modelling, to be exact.

In simple networks, this can be done quite easily. However, it gets increasingly difficult as the size and complexity of the network are increased. In the extreme, we could have a particular current that needs to have a separate mmf source for every reluctance element in the network, to represent it correctly.

And this is where many people go wrong. Indeed, I once reviewed a reluctance network paper, where the authors were modelling the slot-leakage flux of a flux-switching machine with an extra reluctance element. That element passed through part of the stator winding, and was connected to the stator tooth and slot bottom. Thus, a flux loop was formed, linking part of the stator current. But alas, no sources could be found in the loop. A clear violation of the Ampere’s law.

What to do instead

In my opinion, in general, the mmf sources due to currents should not be presented with traditional circuit elements. They occasionally can, yes, but this approach can go wrong just as easily.

What should be done, instead, is forming the flux loops and finding the total current inside them. And the next post is going to be about that.

Reluctance networks 102