Open a brushless motor datasheet and two numbers sit near the top: the torque constant, in newton-metres per amp, and the back-EMF constant, in volt-seconds per radian. Different names, different units, different sections of the page. In SI units they are the same number, and the reason why tells you almost everything about how the motor behaves.
The two constants
The first is easy to accept. Push current through the windings and the motor produces torque in proportion: T = KtI. Double the amps, double the twist. The constant of proportionality is fixed by the magnets, the number of turns in the winding and the geometry of the air gap — things decided when the motor was wound and unchangeable afterwards.
The second is less intuitive. Spin the motor and it generates a voltage that opposes whatever is driving it: V = Keω. The magnets sweeping past the windings induce it, and it grows with speed. This is back-EMF, and it is the reason a motor draws a huge current at stall and a small one at full speed — at stall there is no back-EMF to oppose the supply, so only the winding resistance limits the current.
Why they are the same number
Set the two constants against each other with nothing but conservation of energy. Electrical power going in is VI. Mechanical power coming out is Tω. In an ideal motor — no resistance, no friction, no iron losses — those are equal:
VI = Tω
Substitute both definitions. The voltage the motor generates is Keω, and the torque it produces is KtI:
(Keω)I = (KtI)ω
The current cancels. The speed cancels. What is left is Ke = Kt. Not approximately, not by coincidence of unit choice — it falls straight out of the requirement that the motor cannot invent energy. A motor with a torque constant of 0.05 N·m/A has a back-EMF constant of 0.05 V·s/rad, and no amount of clever winding can separate them.
What the identity is good for
Practically, it means one measurement gives you both numbers. Spin an unpowered motor at a known speed, read the voltage across its terminals, divide: that is Ke, and therefore also Kt. No dynamometer, no load cell.
It also fixes the motor's no-load speed. At no load the motor accelerates until its back-EMF nearly equals the supply, at which point almost no current flows and it stops accelerating. So ω at no load is roughly the supply voltage divided by Ke — and the same constant tells you the stall torque, since stall current is the supply voltage divided by winding resistance. Two constants and a resistance, and the entire speed-torque line is determined.
The part that replaced the brushes
A brushed motor keeps its windings correctly energised mechanically: the commutator is a rotary switch that the rotor operates on itself as it turns. It works without any electronics at all, at the price of sliding carbon contacts that wear, spark and drop a volt or so every time current crosses them.
Brushless motors move that switching into a controller, which then has to answer a question the brushes answered automatically: where is the rotor right now? There are two ways to find out. Hall-effect sensors embedded in the stator detect the magnet poles going past and report position directly — reliable from a standstill, at the cost of three extra wires and three more things to fail. Or the controller can watch the back-EMF induced in whichever winding it is not currently driving, and infer position from where that waveform crosses zero.
Sensorless control is cheaper and has nothing to break inside a sealed motor, and it is why a drone or a cordless drill can do without sensors. Its weakness is exactly the constant this article started with: back-EMF is proportional to speed, so at zero speed there is no signal at all. A sensorless controller has to start the motor blind, energising the windings in a fixed sequence and hoping the rotor follows, then switch to closed-loop control once the motor is turning fast enough to be heard from.
The rotating magnetic field those controllers are synthesising is worth seeing in motion — the rotating field demo builds it up one phase at a time.
Go deeper: the thermal feedback loopfor engineers
Copper's resistivity rises with temperature at roughly 0.4% per kelvin. That number turns the motor's losses into a feedback loop rather than a fixed penalty: winding losses go as I²R, so a hot motor dissipates more for the same torque, which makes it hotter, which raises R again. A 100 K rise adds about 40% to the resistance and therefore to the loss at constant current.
The loop is stable in any well-cooled motor — heat escapes faster than the resistance climbs — but it explains why continuous ratings are so much lower than peak ratings, and why the honest way to size a motor is around the heat it must reject rather than the torque it must produce. Torque is limited by current; current is limited by what the windings can survive.