Electronic – How the Kv rating relates to power generation with BLDC motors

My experience with motors is that they are usually specified as K_e = RMS voltage line-to-line / 1000RPM. The reason RMS voltage line-to-line is used, is because it is very easy to measure with a multimeter + doesn't need an oscilloscope.

K_v would be proportional to the reciprocal, and I'd expect it to be RPM / RMS line-to-line voltage.

In your case: 2193RPM / 0.738Vrms l-l = 2972RPM / Vrms,l-l, that's 15% higher than expected. There is bound to be some error in measurement (voltage in scopes is usually 1-2% or so off worstcase), and in part-to-part variation of the motor, but it sounds a little different.

The torque output of an electric motor is directly proportional to the motor current (not voltage!), and the current (I) is roughly equal to

$$ I=\dfrac{V-\varepsilon}{R} $$

Where V is the motor supply voltage, R is the winding resistance and ε is the back-electromotive force (back EMF).

KV and back EMF

The back EMF is the voltage that would be present at the motor terminals as the motor spins without anything being connected to it. This voltage is produced by the motor acting as an alternator, if you will, and it is directly proportional to the rotation speed. The KV rating is nothing but another way to state the relationship between rotation speed and back EMF (KV ≈ RPM / ε). It limits the maximum motor speed at any given battery voltage, because at some KV-dependent speed the back-EMF will "cancel out" the battery voltage. This prevents any more current from flowing to the motor and thus reduces the torque to zero.

When you first power your motor on, the speed is zero. This means that the back EMF is also zero, so the only things limiting the motor current are the winding resistance and the supply voltage. If the motor controller (ESC) was to output the full battery voltage to the motor at low speeds, the motor and/or ESC would just melt down.

Voltage, frequency, throttle and speed

In closed loop brushless motor control schemes the motor speed (which the output frequency is a function of) is not directly controlled. The throttle instead controls the output voltage and the ESC continously adjusts the output frequency in response to the phase shift between the angle of the rotor and the drive waveform. The phase of the back EMF tells sensorless ESCs directly the current angle of the rotor, while sensored ESCs use hall effect sensors for the same purpose.

Doing things the other way around (setting the frequency directly and controlling the voltage in response to the measured phase shift) would become a fine balancing act:

• Setting the voltage too low would allow too little current to flow, limiting the torque. If the torque falls but the load remains constant the motor must slow down, leading to immediate loss of sync.

• Too much voltage would cause excessive current to flow, wasting power and heating the motor and ESC unnecessarily.

Thus the optimum efficiency point is unstable with "frequency first" control. A control loop could keep it close, but if the ESC can't react fast enough to a load transient loss of sync will occur. This is not true for "voltage first" control, where a load transient will just cause a momentary reduction in speed with no ill effects.

ESCs used in collective pitch RC helicopters often have a "governor" function, which maintains a fixed motor speed proportional to the throttle setting. Even these ESCs don't actually control the frequency directly, instead implementing a PID controller which sets the voltage in response to the difference between the desired and actual frequency.

ESC "timing"

The motor timing setting of ESCs adjusts the setpoint of this mechanical-electrical phase shift: High timing means that the ESC output leads the sensed rotor position by e.g. 25 degrees, while with low timing this phase shift is kept much closer to zero. A high timing setting produces more power less efficiently.

Torque

Normal RC ESCs cannot do constant torque control or torque limiting, since they lack current sensing circuitry as a cost and weight saving measure. The torque output is not controlled in any way; the motor just produces as much torque (and draws proportionally as much current) as the load requires at a given speed. In order to prevent rapid throttle punches from overloading the ESC, battery and/or motor (as overcoming inertia produces potentially unlimited torque), ESCs usually have limits to acceleration and the voltage at a given frequency.

Braking

If the motor is kept spinning by external means while the voltage is reduced, eventually the back EMF will become larger than the level the ESC tries to drive. This causes negative current and brakes the motor. The electricity thus produced is either dissipated in the motor coils or fed back into the power supply/battery, depending on the PWM decay mode used.