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Apr 21, 2013, 07:56 AM
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# The relationship between Kv, Resistance, RPM, and torque in BLDC motors - explained!

Short answer, the motor tries to run at Kv*V, but it never can get there, due to losses. A perfect motor would actually run at Kv*V.

Long (more interesting!) answer follows. I warn you, I am going to be super verbose with this:

First, some quick core concepts:

1) Whenever the motor is turning, it generates a voltage (called back EMF, BEMF) of RPM divided by Kv.

2) The torque generated by the motor is proportional to the current running through it, and inversely proportional to the Kv.

Now, step by step, what is happening to a motor (The motor is lets say 1000 Kv with 10 volts across it) at zero load (obviously doesn't happen in reality) with a voltage across it, and then what happens when you add a load:

Basically, we enter this example some time after the zero-load motor has been started and it has spun up to speed and reached equilibrium (it would generally take only a fraction of a second to spin our motors up to like 99.99% speed by the way). It is therefore running at 10k RPM (10v*1000Kv). Why doesn't it go faster? Well, at 10K Rpm, we know from concept 1 it is generating that voltage (back EMF) of 10000/1000Kv = 10V. This voltage opposes the battery voltage of 10V, and therefore the total voltage across the windings is zero. Current does not flow when voltage is zero. And concept 2 says torque generated by the motor is proportional to current. No current flow, no torque. So the motor has no torque to accelerate further.

So what happens when a wizard appears and adds a prop to the motor while it is running?

Suddenly there is a big torque load trying to slow the motor. But currently the motor is at 10K RPM and thus still can't generate any of its own torque to oppose this. So, since the slowing torque is unopposed, the motor starts slowing down.

Suddenly we are at some amount of RPM that is less than 10,000! That means the back EMF opposing the battery EMF is now [something less than 10,000]/1000= Something less than 10! That means the back EMF no longer perfectly matches the battery voltage - suddenly, a net voltage appears across the windings! And with that voltage, a current starts to flow.

Concept 2! Torque is proportional to current!

That means the motor has started resisting the slowing-torque of the prop. But is it enough? Well, if it isn't, then the motor will keep slowing down! And the difference between battery voltage and the motors back EMF will keep growing, thus the voltage pushing current across the windings will keep growing - and so will the torque. At some stage, the back EMF will drop low enough, the current will get high enough, and with it the torque, and the prop load torque will be perfectly counteracted. This is our new equilibrium! The motor seeks a new equilibrium every time the load changes, that balances its output torque with the load torque, by changing its back EMF.

Some relationships between parameters:

-The actual amount of current from a given drop in back EMF is given, as with all resistive circuits, by the voltage divided by the winding resistance. If you have a lot of winding resistance, you can't get a lot of current without lots of voltage - as a result, your motor has to slow down a *lot* more than you might hope. By contrast, if you have very low current, even a very small drop in back EMF will allow a vast current to flow, and your motor will operate at a very constant speed. Taken to extremes, if you have a superconducting (0 resistance) motor, the speed change would be infinitesimal, since it would need only an infinitesimal voltage, and thus an infinitesimal speed drop, to generate ridiculously huge currents and resist any load torque.

- The current required to resist a given torque: Conveniently, thanks to the magic of conservation of energy, this is given by the inverse of Kv. You have to use SI units for that though. That means your Kv has to be expressed in radians per second, then when you invert it, your torque constant units are in units of newton-meters per amp.

Interesting observation of the realities of life in motor design:

-One key reason larger motors can handle more power: If you want a motor with the same Kv but you have for example a way wider diameter to fit the stuff in, you can then fit way thicker windings in. Thicker windings have less resistance. Less resistive windings mean they can pass more current (and generate more torque) without as much resistive losses. There are other loss factors as well but they are not to worry about just yet

-The actual magnitude of the speed drop, in typical BLDC outrunner setups in our hobby here, tends to be in the area of 20-25%, I.e. the motor runs at around 75-80% of its Kv*V value. It's a good indication of how much relative stress you are putting your motor under for its size - how efficiently it is running and whether it is over or undersized.
Last edited by Nereth; May 16, 2013 at 11:13 PM.