The most common description of a motor’s Kv constant is that it is the ratio of the unloaded speed to the applied voltage of a motor. Or RPM per volt, as many people say.
The main problem with that description is that it is wrong. The other problem with that description is that it seems right.
And when something seems right but is actually wrong, it tends to cause a lot of confusion.
Why is it wrong? That’s a long story (which I’m about to tell). But first, we need to know a little bit about brushed motors and back-emf.
Brushed Motor Model
I’ve explained the very basics of what a brushed motor is in another post. One concept I didn’t explain was the back-emf. In a motor, whenever there is relative motion between the magnets and the coils, a voltage is induced in the coils.
This voltage is called the back-emf. The back-emf is proportional to the rotational speed of the motor.
In this equation is the back-emf, is the back-emf constant and is the speed. is often given in units of Volts/RPM.
Given this, we can make a simple model of a DC brushed motor. It consists of a resistor, an inductor, and a back-emf in series with an applied voltage. If we assume that the current through the circuit is a constant DC current, we can ignore the inductor in the model and we are left with a resistor and a back-emf in series. This is shown in Figure 1.
Using basic circuit analysis, we can write:
From this simple model we can learn a lot about a motor. For example, if we know the applied voltage, V, the motor resistance, R, and the no-load current, , we can determine the maximum speed of the motor:
In general, we can say that the speed of a DC motor is equal to the applied voltage minus the voltage drop due to the resistance and the current in the motor, all divided by the back-emf constant.
So as the current in the motor increases, the speed of the motor decreases. And because current is proportional to torque, we can also say that as the torque of the motor increases, the speed decreases.
Another thing we can do is see what happens if the motor speed is 0 and full voltage is applied. This situation occurs the motor can’t rotate. For example, if you crash your quadcopter and the prop of one of the motors can’t spin because the ground is preventing it from turning.
In this case the back-emf reduces to zero and the current is equal to the applied voltage divided by the resistance: . Because the resistance can easily be measured and the applied voltage is known, the current when the rotor is locked can be found easily.
But that doesn’t really answer the question of what Kv is or why it is misunderstood.
What about Motor Kv?
As I mentioned above, Kv is often taken to be the RPM/Volt of an unloaded motor.
In other words, if you apply 1 Volt to an unloaded motor, the Kv constant tells you how fast the motor will rotate. Then if you apply 2 Volts, the motor will rotate twice the value of Kv.
And, to be honest, this will get you pretty close. It’s not a bad estimate.
But thinking about it this way will prevent you from understanding what is really going on in a motor.
So what is wrong?
Motor Kv has nothing to do with the applied voltage. Instead, Kv has to do with the back-emf I talked about above.
The motor Kv constant is the reciprocal of the back-emf constant:
So Kv tells us the relationship between motor speed and generated back-emf.
A 2300 Kv motor will generate a 1 V back-emf when the motor is rotating at 2300 RPM. At 23,000 RPM that motor will generate 10 V.
Brushless Motor Kv “Rating”
The model I described above was for a brushed DC motor. It turns out that everything above also applies to the type of brushless motors and ESC’s that are used in RC quadcopters.
Even though brushless motors are a type of AC motor, the assumptions made allow us to use Kv in a similar way as with brushed DC motors.
However, I want to stress that Kv is not a “rating,” but rather it is a motor constant. It tells you how the generated back-emf in the motor relates to motor speed. But it doesn’t tell you how powerful the motor is or how much current it can handle or how efficient the motor is.
You can have a large motor and a small motor with the same Kv constant. The large motor will be more powerful than the small power. So there is no way to use Kv to tell you how powerful your motor is.
However, Kv is related to another important motor constant – Kt, the torque constant.
Brushless Torque Constant, Kt
The motor torque constant for brushed DC motors and brushless motors tells you how current relates to torque.
In this equation, T is torque, Kt is the torque constant, and I is current. Kt has units of Newton-meters per Amp.
How does Kv relate to Kt? While it is beyond the scope of this post, Ke and Kt are actually the same constant. They are equivalent. This means that
Another way of writing this is:
All of these equations assume SI units (Nm/A and V/(rad/sec) for Kt and Ke, respectively).
Wait … What? How Can Kt and Ke be Equal?
They have different units, after all. Nm/A is different than V/(rad/sec). Right?
No, they are the same actually.
This tells us:
Which tells us that the units are at least equivalent.
How do you Measure Kv?
The typical way to measure Kv is to spin your motor at a known RPM and then measure the voltage generated on the motor leads. Many hobbyists use a drill press to spin the motor. You could also use another motor. You can measure the RPM with a tachometer or strobe. You can measure the voltage using a digital multimeter.
For brushed DC motors, you would measure the voltage between the two motor leads. For brushless motors, you would measure the RMS voltage between any two leads.
The formula to calculate Kv for brushless motors is:
This is just speed divided by voltage but the 1.414 term is to convert RMS voltage to peak voltage. The 0.95 term is “fudge factor” that has been found by hobbyists to give you the right answer.
The formula to calculate Kv for a brushed DC motor is to just divide the speed by the generated DC voltage that you measure using your multimeter.
Another option for brushless motors is to buy a BL motor tester that will calculate Kv for you. One example of this tester is here, but there are other available out there.
The main thing I want you to take away from this post is that when a motor rotates it generates a back-emf. That back-emf is proportional to the motor speed and the Kv constant tells you how they relate to each other.
For rough calculations, you can treat Kv as the RPM/Volt (meaning, the speed the motor will turn per applied volt) but that will always be an approximation.