Important dimensions of SRMs and SynRMs

Hi there, and welcome back to our post series about switched and synchronous reluctance motors!

In case you missed them, the first post introduced both motor types and how they work.

The second post, over here, then described some very basic, elementary tools for calculating their performance, based primarily on the airgap radius and a ‘current-in-slot’ (or electrical loading, for those with a little more experience in motor design).

Well, today we’ll look a little deeper into modelling aspects. Namely, what other important *dimensions* we have to consider, and exactly how much current we can ram into the aforementioned slot. Let’s begin with the dimensions.

Sidenote: Before everything, lemme point out that most of the stuff in this post is by no means specific to SRMs or SynRMs. By contrast, it applies to pretty much all rotating machinery. But that just makes it better, I’d say.

### Important dimensions

Last time, we saw that the torque of the machine is pretty much proportional to the square of the airgap radius and the currents, times the airgap flux density* (That’s directly evident from the linear current-density approach we used with the SynRM. It also follows from the single-coil approach we used with SRM, once we realize we can fit moar coilz around a larger airgap.)

*times length, but length’s boring.

On the other hand, we often have a maximum limit for the *outer *radius of the machine. After all, it has to fit within our application.

So, between the airgap radius and the outer radius, we have to fit our slots. And slots, very roughly speaking have a width and a height.

##### Slot width

Make ’em wider, and more current fits in. This *would* increase the torque, if not for one caveat. The airgap flux has to go somewhere, and that somewhere is the *stator teeth*. And a wider slot means a narrower teeth. And a narrower tooth means less space for the flux, meaning a higher flux density. And since the tooth flux density can’t be too high, there’s an upper limit for the relative slot width.

Therefore, if we decide to fix the tooth flux density (and we often do!), the airgap flux density ends up being negatively proportional to the slot width.

One goes up, the other comes down.

On the other hand, a wider slot *does* permit more current, leading to parabolic (downward-opening) relationship between tooth width and torque.

Two points are easy to deduce:

- Zero slot width –> no current, no torque.
- Zero tooth width –> no flux, no torque.

Indeed, under simple assumptions, the optimum is midway, meaning equally wide slots and teeth (on the airgap side).

In practice, you tend to see some variation. A slightly different number may be good from the losses’ point of view, and of course all the nonidealities in this simplified approach also influence the results once you actually take them into account. But nevertheless, 0.5 is a good starting point.

##### Slot height

Slot height is quite similar in behaviour to the width. Meaning, more seems to be better for torque (higher slot = more current), but flux density constraints set an upper limit.

Meaning, the entire flux of a half-pole – meaning the flux you get by integrating the airgap flux over half the pole pitch – has to fit inside the stator yoke. That’s the area between the slot bottom and the outer radius of the stator, you know.

And like in the teeth, the yoke flux density can only get so high. Meaning, the yoke simple needs be at least *this *high, and the slot only gets whatever is left. Or, alternatively, you choose the slot height and then adjust the airgap flux density accordingly.

Which kinda brings us to the third important dimension.

##### Split ratio

Earlier, we saw how the torque is dependent on the square of the airgap radius and current*. Furthermore, we also saw how the slot height – and thus current – is basically defined by what’s available once we have accounted for the yoke height needed for the flux (assuming a fixed outer diameter).

*and length. Suuuuuper boring.

In other words, have too large an airgap radius, and there won’t be any room for the current. Correspondingly, have too small a radius, and the ‘lever arm’ for the tangential force is too low, again killing the torque. Clearly, the optimum lies somewhere in the middle.

Luckily, the optimal split ratio – the ratio between airgap and outer radius – is covered in pretty much all textbooks and other study material. And *un*luckily, it is usually covered without any justification at all.

Bruh, it’s around 0.6 – trust me.

Except for the fact that if you actually write open the torque formula, using a fixed current density and a fixed yoke (and tooth) flux density, the optimum comes at around 0.3 to 0.4. And for anyone who’s actually seen an electrical machine, that doesn’t seem right at all, apart from maybe some high-speed machines.

So where’d the difference come from?

### Thermal stuff

Electrical machines generate heat. And that heat has to be lead somewhere else, and fast enough to keep the machine cool enough. And overheated motor won’t last as long, or might fail right away in severe cases.

The heat comes from three different sources, mainly.

**Friction losses** and other mechanical losses are usually not that significant from the thermal point of view, unless you go into the 10+ krpm range.

**Iron losses** likewise depend on the frequency, so rpm and number of poles both. They also depend on the amplitude of the flux density in the iron parts, but that is anyways “somewhere near” saturation. There IS some variation – high-speed machines typically have nowhere near the 1.6 to 1.8 T that 50 Hz machines do – but that’s not the main explanation to our split-ratio problem.

Which leaves us with the resistive, or **copper losses**. As you probably know, they are proportional to the current density squared, times the conductor volume. However, it also turns out that the resistive loss *density *– in Watts per cubic meter – is significantly higher than the iron loss density. Meaning, however much we play with the split ratio, just considering the effect on resistive losses is good enough.

##### But what effect should we consider?

Like mentioned, the heat generated in the winding (plus elsewhere) has to go somewhere. And in a traditional machine (not extremely short, no spray-cooling end-winding), that *somewhere *is into the machine frame through the stator yoke.

And this path for the heat flow is not really influenced by the split ratio. Meaning, unless we change the outer dimensions or cooling system, our total losses must stay fixed to stay at a certain temperature rise.

Meaning, our total winding losses must stay the same. Roughly speaking – making the slots higher (decreasing split ratio) will also lengthen the teeth and thus increase the iron losses. But thanks to the differences in the loss densities, the copper losses will increase so much more.

Meaning, we get the simple relationship

from which we see that the current density must be inversely proportional to the winding area.

In other words, lowering the split ratio, and thus increasing the winding area (due to increased slot height), must be accompanied by a lowered current density to avoid overheating the motor.

Now, taking this result and plugging it all the way back to our earlier results, will indeed yield an optimal split ratio in the 0.5-0.7 neighbourhood, depending on the number of poles (which primarily affects the required yoke height).

Sidenote: slot size also plays a role – several small slots cool better than a single large one. However, this effect is often neglected in many sources.

Yay!

### Number of poles

Which brings us to the final dimension to consider – the number of poles.

In my experience, the number of pole pairs is perhaps the most misunderstood dimension. After all, it *does *act as a speed divider for direct-on-line motors. Combine that with the fact that it appears as a *multiplier* in many textbook torque equations, and it’s no wonder many people think of it as a magnetic gear ratio of sorts.

Which it, sadly, isn’t.

Increasing the number of poles *does *reduce the rpm with fixed supply frequency. Unfortunately, it *does not* (directly) do anything to the torque. It may often appear as a multiplier in many a torque equation, but just as often it is hidden inside another parameter as a divisor.

Let me stress it out.

Doubling the number of poles Does. Not. Double. The torque.

However, that’s not to say you *can’t* get more torque out of a multi-pole motor.

Quite often, you can. Just maybe not double. And definitely not due to some magnetic gearbox magic.

For instance, increasing the number of poles decreases the pole flux. This, in turn, leads to a thinner yoke, leading to a higher slot and increased airgap radius – so double yay for torque!

Furthermore, in those cases where saturation due to stator currents is the limiting factor, increasing the number of poles can allow higher linear current densities (total current in a slot, essentially) to be used with the same airgap flux density. As the torque is proportional to the product of these two, it is of course increased. (Up to a limit, leakage flux paths can become a serious issue in reluctance machines especially.)

### Conclusion – Relevance to S(yn)RMs

Finally, what does all of this have to do with switched and synchronous reluctance motors?

Specifically and uniquely, nothing.

Generally, *everything*.

Meaning, what was presented here applies to *all *machines, more or less. SRMs and SynRMs included – they are machines after all.

In other words, to be able to design good SRMs or SynRMs, these factors *have *to be considered. Their exact values will depend on, well, each other and the application in question. Therefore, a combination of design skills, modern optimization, and finite element analysis (thanks to all the nonidealities already discussed) is usually needed for getting the best result.

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