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When learning about regenerative braking, I began to ponder as to what the relationship between the number of turns in the coil of a DC motor is to the efficiency of kinetic energy retrieval from the rotating wheel.

In math terms, what is the relationship between N and eta where: $$\eta = \frac{E_E}{E_K}$$

If you can cite some references as well relating to your answer for further reading, that would be extremely helpful.

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  • \$\begingroup\$ What factors should you consider? Bearing friction? Drive train losses? What else? \$\endgroup\$ – Solar Mike Mar 3 '19 at 6:36
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The number of turns is irrelevant.

All changing the number of turns does, assuming you also change the wire diameter so that the resulting winding fills the space available, is to change the impedance of the winding, the voltage/current ratio, not the efficiency.

Consider the space filled with two identical windings, running at the same current, voltage, field, dissipation etc. If you connect them in series, it runs at 2V and I, you have twice as many turns as each coil. If you connect them in parallel, it runs at V and 2I, you have double the copper area but the same number of turns. All of the important parameters, field, weight, torque, power lost, cost of the copper, are unaltered, even as you've doubled or halved the number of turns.

That's all to first order of course. There are many small second order things going on. The wires to the motor can be thinner if the motor impedance is higher. Thin wire tends to have a greater part of its cross sectional area occupied by insulation than thick wire, so it's not quite as efficient as medium weight wire. Very thick wire also doesn't pack as well due to its stiffness. A very high voltage motor may well dedicate more space to insulation than a lower voltage motor. There may be other considerations for voltage, for instance availability of 12v batteries, or staying below 40v so you can use 'low voltage' insulation rules.

That's enough about number of turns not affecting efficiency, so what does?

The main loss is \$I^2R\$ loss in the copper. Unfortunately there's little that can be done about the resistivity of copper. Silver is too expensive (and barely any better), cryogenics is very complicated, though superconductor is worth doing if the machine is really, really big (>> 100MW).

High field magnets produce more volts per turn, and more torque per current than low field magnets, so you should use the highest field you can afford. More volts per turn means a higher ratio of 'useful' volts (coupled to the speed of the motor) to 'useless' \$IR\$ drop volts that only generate heat, so better efficiency. A small airgap helps produce a higher field with the same magnets, but requires better machining tolerance and produces more air viscosity losses (aka windage), so it can't be reduced too much.

A high speed motor produces more power than a low speed one at the same output torque, which is another way of saying generates a higher EMF than a slow one. Again, a higher ratio of useful volts to useless volts. However, high speed requires strong materials, and careful balancing, and produces losses that increase rapidly with speed, so it can't be increased too much. The higher rotor frequency means more hysteresis loss in the magnetic materials, and the windage losses increase as the speed cubed, so will rapidly come to dominate at very high speeds.

Because of the way some power and loss terms scale, a big motor, other things being equal, will have lower losses than a small motor. But let's say you want four driven wheels. Is it better to use a small motor per wheel, or a big motor and a mechanical drive train? Almost certainly the former, for lower weight and complexity, even if it's slightly less efficient.

Motor/generator design is a compromise between efficiency and, well, cost really, once you have a motor powerful enough for your specifications. Unfortunately, even the simple things like cost, weight and volume are a multidimensional space you can't simply optimise over. When you add things you can't easily put numbers on, like time to design, maintainability, reliability, the rest of the vehicle efficiency, convenience, use of strategic materials (in strong magnets), there is no one size fits all, no one equation describes all.

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  • \$\begingroup\$ Neil_UK, would it be too much trouble for you to also list some citations in your answer as well for further reading? I am finding it really difficult to find relevant papers on similar topics to this. \$\endgroup\$ – James Balajan Mar 7 '19 at 5:38
  • \$\begingroup\$ @JamesBalajan There isn't the literature available, because people don't write about turns versus efficiency. Most motor design application notes I've read talk about turns, cross sectional area, total flux, resistance, current, volts etc and rely on you to understand what varies with what. Efficiency is assumed to be 100% for a first order design, then when you take account of winding resistance, friction, hysteresis losses etc etc in a more detailed analysis, you get a better figure. But it varies with so many things, and number of turns cancels out of the final efficiency equation. \$\endgroup\$ – Neil_UK Mar 7 '19 at 6:46
  • \$\begingroup\$ @JamesBalajan To put it another way. When you design a motor, you can do all the significant design assuming each winding has a single, very thick turn. You can do power, efficiency, speed, torque, flux, hysteresis loss, copper loss. Right at the end, you then decide whether you want a 1v 1000A motor, or a 10V 100A motor, or a 100V 10A motor, or a 1kV 1A motor, and choose a wire thickness and a number of turns for the winding that delivers that. BTW, what are \$E_E\$ and \$E_K\$? \$\endgroup\$ – Neil_UK Mar 7 '19 at 6:52
  • \$\begingroup\$ Thanks for the reply. E_E refers to electrical energy. E_K refers to kinetic energy. Apologies for my lack of knowledge however I am still in high school. \$\endgroup\$ – James Balajan Mar 7 '19 at 7:06
  • \$\begingroup\$ If number of turns does not affect the efficiency of the motor's generation, what factors do? \$\endgroup\$ – James Balajan Mar 7 '19 at 7:08
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In a word: complicated.

For a given motor physical design, if it's at all well done, the coils fill the available space. You cannot increase the number of turns without reducing the size of the wire, and if you just remove turns you reduce efficiency because of increased \$I^2R\$ losses at a given torque.

If you change the wire size, and wind the motor to fill the available space for wires, then for a given \$I^2R\$ loss, the torque produced (or absorbed) by the motor is a constant across different coil windings. If you look at manufacturers like Pittmon Motors or Maxon, you'll see that for a given physical size of motor, they have a number of different voltage ratings, all of which have roughly the same torque vs. dissipation rating.

So unless the motor is poorly designed, increasing the turns of the same wire isn't possible (because they know the space needs to be filled), and decreasing the wires just makes things worse.

Once you get past that, there's a whole lot of factors that affect how a motor is built which, I admit, I can't put my finger on in detail. But as near as I can tell it mostly depends on how strong of magnets you can buy (if it's a permanent magnet motor), the conductivity of copper, and at what point the cost of maintaining higher mechanical tolerances exceeds the gain over the lifetime of the generator.

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