The split ratio, defined as the rotor OD divided by the the stator OD, is one of the key dimensions defining the performance of an inrunner electrical machine. For a (relatively) fixed OD and stack length, it’s probably **the** most important one. But why do the split ratios – and by extension all the machines on the market – look the way they do? Here’s a brief, equation-free dive into the maths (yes) of the split ratio.

## The naiive

So, what makes a good split ratio?

We know that the torque of an electric motor is somewhat proportional to the airgap flux density times some definition of a ‘current’ – be it a linear current density, the phase current in an equivalent circuit, or the per-slot current in a Lorentz-force-like approach, times the airgap radius, times the stack length. Stack length we are not considering here, and airgap flux density we are commonly trying to maximize to a reasonable degree, leaving us with the airgap radius (hello split ratio) and the current.

For the current, a common but naiive assumption is to begin with a fixed number, based on a rule of thumb based again on the machine size and cooling type.

With this assumption, we are faced with two contradicting effects

- Increasing the split ratio increases the torque in a linear fashion – for a fixed current
- Increasing the split ratio decreases the slot area somewhat quadratically

The latter effect is more complex than a pure quadratic relationship, though. We have to account for the tooth width, and especially for the fact that increasing the split ratio linearly increases the flux per pole, and thus the required yoke height. Thus, the number of poles also enters the field here.

Anyways, the math here is relatively easy to work through.

The only caveat?

The resulting split ratios look **nothing **like what is commonly seen in production. For instance, the optimum for a 2-pole machine ends up around 0.3 if I remember right – a tiny rotor inside a very bulky stator.

## The simple

Obviously, we are missing something. And, in the traditional, textbook-way-of-things, that *something* are considering the thermals.

Indeed, the above approach is typically modified by taking into account a very simple thermal model. Traditional, and even today quite commonly, the main cooling path is through the stator yoke, into the housing, into whatever the heat finally gets rejected into.

On the level of our very simple conceptual-level mathematics here, we are assuming that the maximum amount of heat that can be rejected stays constant. In other words, we take the torque-maximizing approach above, but at the same time require the resistive losses to stay constant.

In other words, smaller split ratios, with deeper slots and larger conductor area, allow for a smaller current density, and vice versa. In other words, we want the slot area times current density squared to remain constant.

This change alone suddenly makes things much more realistic. Split ratios for 2-pole machines jump into the 0.5 neighbourhood, and slowly grow as the pole count is increased.

All good, on the pen-and-paper level.

## The realistic

Of course, the devil is in the details, and details on our field mean electromagnetic FEA, thermal analysis, mechanics, demagnetization analysis, manufacturability, and others.

For instance, one key factor neglected by our simple analysis is saturation, and leakage fluxes in general. Deeper slots generally mean a higher electric loading, and a careless approach here can lead into excessive tooth saturation. And, even without saturation, a higher loading can lead into an excessively poor power factor, due to the armature reaction field (the p-pole airgap field created by the stator winding, or something) and the slot leakage fluxes (for which deep slots offer an exquisite path).

Also, high electric loading does also spell a higher demagnetization risk in most cases. Whether or not that means a *realistic* case depends on many aspects. Industrial products usually utilize both less-exotic cooling *and* weaker magnet grades, while traction motors opt for the premium end of both categories. Meaning, both might potentially see demag-management as a limiting factor, especially if and when short-circuit scenarios are considered.

Also for thermal analysis, reality is typically more complex than the simple approach outlined earlier. Meaning, even if the total losses are kept constant, extracting 1 kW of losses is typically easier for 15-mm-deep slots compared to 45-mm-slots.

Mechanics can *also* spoil the party here. High-speed machines often have far smaller split ratios than their more-moderate cousins, for multiple reasons. For one, they literally **can’t** have rotors too large – the windage losses in the airgap would ruin the efficiency and thermal management both, if they could even get up to speed without blowing apart from centrifugal stresses. Secondly, they can *tolerate* smaller split ratios without much of an efficiency penalty – small split ratios do generally result in increased resistive losses, but with high surface speeds that generally does not hurt the efficiency too much.

## EMDtool

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Thanks again for a good post!

On this topic, I stumbled upon this short note by Soong, which I like a lot:

https://www.scribd.com/document/60016511/Soong-Sizing-of-Electrical-Machines-2008

Thank you, looks good!