# Why do brushless motors have a kv rating?

I am wondering why brushless motors, such as ones used for quadrotors, have a kv rating, which supposedly means RPM per voltage across the motor. So a 2300 kv motor spins at 2300 rpm if "1 volt is applied" to it.

The part in parenthesis does not make sense to me. An ESC generates 3 phase AC current. And from what I understand the frequency of the AC waveform completely determines motor speed, and the amplitude (peak voltage minus trough voltage) of the waveform is more or less constant. To me this seems like voltage really has nothing to do with determining the speed of a brushless motor.

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.

• Thanks for the detailed explanation @jms. So if I understand correctly increasing the throttle increases the amplitude of the AC signal on the 3 motor wires, which momentarily creates a phase shift, which the esc detects (with back emf?) and then changes its frequency output accordingly? – Thomas Kirven Oct 7 '16 at 3:50
• @ThomasKirven That's correct, and a very nice way to put it. – jms Oct 7 '16 at 4:11
• It should be made clear "kv rating" has nothing to do with torque – Tony Stewart Sunnyskyguy EE75 Oct 7 '16 at 15:41
• @TonyStewart.EEsince'75 While they are completely distinct parameters, there is a tradeoff between the two: When you buy two motors of the same size, mass and design, but one is wound to a higher KV than the other, the high KV motor will spin faster and generate less torque at the same power input. – jms Oct 7 '16 at 16:48
• yes of course like gears on a bike , torque vs speed but unrelated to HP or actual power – Tony Stewart Sunnyskyguy EE75 Oct 7 '16 at 18:04

An ESC generates 3 phase AC current. And from what I understand the frequency of the AC waveform completely determines motor speed, and the amplitude (peak voltage minus trough voltage) of the waveform is more or less constant. To me this seems like voltage really has nothing to do with determining the speed of a brushless motor.

Sorry, but this is all wrong. The motors used in quadcopters are Brushless DC motors (BLDC), which are equivalent to a brushed DC motor but with electronic commutation.

Motor speed is determined by the voltage ('back-emf') the motor generates as it is spinning, not the commutation frequency (which has to follow in lock step with the motor's rotation or it won't spin). BLDC motors have permanent magnets so back-emf is directly proportional to rpm. Back-emf equals applied voltage minus voltage drop across the winding resistance and inductance, and the motor will speed up or slow down as it draws the current required to produce the torque absorbed by the load - exactly the same as a brushed DC motor.

The ESC controls motor speed by varying the voltage applied to it. Usually this is done with PWM so the peak voltage is always equal to the battery voltage, but the average voltage (which the motor responds to) varies according to the PWM on/off ratio. The ESC produces whatever commutation frequency the motor demands of it, similar to how the armature in a brushed motor causes the commutator to switch at the frequency it demands.

So the applied voltage has everything to do with motor speed. This is why these motors have a Kv rating - it is an essential parameter for determining what rpm can be achieved with a particular voltage. Since the power absorbed by a propeller is proportional to the 3rd power of rpm and the 4th power of prop diameter, Kv is a critical parameter when matching the components of a quadcopter.

The specified Kv value should be the theoretical rpm at 1V when the motor is not drawing any current. However it is commonly calculated by simply dividing measured no-load rpm by applied voltage, which gives a slightly lower (incorrect) value. And just as the speed of a brushed motor can be increased by advancing the brushes, so a brushless ESC can increase the effective Kv of a BLDC motor by advancing the commutation timing. Add in manufacturing tolerances and poor quality control, and it is not usual for a motor to have an actual Kv 20% higher or lower than its specification.

Motors designed for other uses often don't have a Kv rating because it is not considered so important. However the no-load rpm at nominal voltage is usually provided, from which Kv can be derived. The motor's torque constant (Kt) may also be specified. Kv is the inverse of Kt.

• It seems misleading to talk about voltage applied to the motor, since the signal is AC and always varying. If I measured the waveform output of an ESC while connected to an actual motor would I really see the waveform growing in amplitude at higher throttle values?? – Thomas Kirven Oct 7 '16 at 13:49
• The voltage must be converted from RMS to DC to get the equivalent commutated brush type DC motor or the actual average DC applied to the ESC that generates PWM modulated DC . Ignore the ESC commutation and 3 phases to understand it. It is not a variable frequency control drive. – Tony Stewart Sunnyskyguy EE75 Oct 7 '16 at 16:03
• "If I measured the waveform output of an ESC while connected to an actual motor would I really see the waveform growing in amplitude at higher throttle values?? " - Sort of. You would see a PWM square wave with duty cycle increasing at higher throttle values. The PWM wave is flat-topped for 1 commutation step, then ramps down/up linearly for 2 steps to get to the opposite polarity (the average voltage tracing out a trapezoid waveform). This is 'AC' at the commutation frequency, but 'averaged DC' at the PWM frequency. – Bruce Abbott Oct 7 '16 at 19:27
• @Tony Stewart this is where it gets complicated. With low frequency PWM the current switches fully on and off so the motor responds to the PWM wave's rms voltage, and the throttle curve is non-linear (more power than expected at low throttle, but also more heating and lower efficiency). When PWM frequency is increased the motor's inductance makes current flow more smoothly so voltage response shifts from rms to average, and the throttle curve becomes linear. – Bruce Abbott Oct 7 '16 at 19:46
• and if commutation rate is too fast for the ESC, it flips and falls out of the sky.. perhaps a low pressure spot causing rapid RPM increase. – Tony Stewart Sunnyskyguy EE75 Oct 7 '16 at 19:50

Why do brushless motors have a kv rating?

"kv Rating" has nothing to do with expected Torque, current, power, thrust , lift or drag

• The exception is relative torque can change with the number of magnets and the number of stator windings per revolution so like gears , this ratio can be modified. So in a sense, same size motors with relatively higher kv values are made for more speed and less lift.

It is based on number of magnets, number of stator windings per rotation, number of phases per pole and has no indication of power.

It is purely the rotational velocity which generates the back EMF voltage to match the applied voltage. This match only occurs at no load and drag reduces this ratio up to 10% with increase towards rated voltage depending on inherent losses. (e.g. eddy current, friction , generally small compared to power avail. Changing the winding stator pattern or changing the number of magnets will change the number of RPM ratio per volt for the same material used like the gear ratio on a bike.

• Example calculations with various magnets, Determine field rotation

• total magnets/2 = field rotation factor
• Field rotation factor * kV = magnetic cycle/V

• So with 14 magnets, field rotation factor = 7, thus field rotation = 7609 cycles/v

• For 2200 kv:

• 14 magnet - 2200 * 7 = 154000 cycles/V
• 10 magnet - 2200 * 5 = 11000 cycles/V
• 8 magnet - 2200 * 4 = 8800 cycles/V

Power is a function of current and load only is rated with EITHER a linear load or the nonlinear load of the aerodynamic prop. or an incremental linear load in terms of gm/W or gm/A where gm is the prop thrust.

Background thumbnail on theory (over simplified)

• It is based on the laws of Physics defined by Maxwell and in more depth by Heaviside, and Lorenz who proved that this Force on charge q is a product of the sum of the E field and the velocity of the B field.

So the vector equations says. F=q(E+vxB)

The Lorenz force, F acting on a particle of electric charge q with instantaneous velocity v, due to an external electric field E and magnetic field B. This force is what we call the Electromagnetic Force and is matched by the Back EMF at no load.

The Angular Velocity per Volt is a more complex one with the number of stator poles and rotor poles giving a ratiometric conversion and the commutation of the motor current is automatically reversed just an adequate number of arc seconds after the null magnetic field to ensure no dead stop. ( design/process failure)

Thus the magnetic charge velocity is proportional to the Field strength which is due to Voltage and is also referred to as Back EMF field strength

• Those that can find fault must prove it, those that recognize truth ought to approve it – Tony Stewart Sunnyskyguy EE75 Oct 7 '16 at 15:38
• This is technically correct so I zeroed out (+1) your votes, but there is definitely a much more comprehensible way to say this in the way you're trying to express it. – Daniel Oct 7 '16 at 15:59
• im adding a +1 as well, this covers where the constants come from. I have added an answer linking Kv, Kt and Ke – JonRB Oct 7 '16 at 16:01
• @Daniel agreed , but the theory is just thrown in to show roots of the magnetic principles which are far more complex than this reader's digest version. The BEMF can be interpreted different ways but always matches Vin with no load , regardless if a megawatt or a milliwatt motor. – Tony Stewart Sunnyskyguy EE75 Oct 7 '16 at 17:05
• Know of any sources that explain this in gritty detail, down to the time varying fields used in maxwells equations? – Thomas Kirven Oct 8 '16 at 14:53

The KV rating refers to the maximum RPM/volt that can be achieved with the motor - so a 2300 KV motor at 1 V would work at speeds up to 2300 RPM, regardless of the frequency. The lower the voltage, the lower the maximum torque the motor can produce. If you were to increase the frequency and attempt to run it at a higher speed, the motor would not have enough torque to overcome friction at that speed and stall.

• So is that the actual zero torque RPM for that voltage? ie is it the waveform peak voltage when you spin it with a drill at that RPM? – Daniel Oct 7 '16 at 2:45
• The zero torque RPM will usually be somewhere above the KV rating - the KV rating is just a point where the motor can provide a reasonable amount of torque and running it at a higher frequency may cause reduced torque, unreliable operation, or eventually stalling when it can no longer overcome friction. – hexagon Oct 7 '16 at 2:48
• Do you have any additional information you can add to your answer like where and why this rating was developed? It seems to be pretty limited to quad-copters and similar markets. – Daniel Oct 7 '16 at 2:49
• It's hard to tell, but it was probably developed by the RC industry as a way of rating motors for a safe maximum speed. I have never seen this on brushless motors intended for non-RC applications – hexagon Oct 7 '16 at 2:51
• So the amplitude of the signal that an ESC produces is actually not constant? – Thomas Kirven Oct 7 '16 at 2:58

For a BLDC machine there is two key constants

$K_t$ with units Nm/A

$K_e$ with units V/$\omega$ (peak line-line voltage)

For an ideal BLDC machine $K_t \equiv K_e$ but due to specifics as to where these two constants a defined ($K_e$ being open-terminal voltage & $K_t$ being torque production at rated current) $K_t$ tends to be lower due to saturation of the stator

What does this have todo with BLDC motors for quadrotors & $K_v$

Well $K_v$ is just the reciprocal of $K_e$ ONCE converted to rpm.

Because quadrotors and such RC devices are usually supply-voltage limited, this rpm constant will tell you the rotor velocity that can be achieved (unloaded) for a given battery. Likewise you can estimate the torque that can be produced due to the relationship between these constants.

The role of an ESC is to keep the stator flux at 90 degrees with respect to the rotor flux. This is done with the use of position sensor, like hall element or by using back EMF sensing - sensorless control.
Further the ESC can output sinewave three phase output, so called FOC (Field Oriented Control) or square voltage, where only two coils are connected at the same time, the third is left floating.
It is not the case, that rotor is following the stator field, rather the opposite - it's the stator filed that follows the rotor position. With FOC, the amplidute of the vector stator voltage is constant and rotating with respect of rotor position. The voltage has to be higher than back EMF generated voltage in order to spin the motor. This is where the Kv factor plays a role.

Not sure why this is miss quoted in this context.

It should be V/krpm. or volts /1000 revolutions/minute. I could possibly understand V/k short hand but kv is kilo volts.
Maybe volts between legs on the motor or a leg and neutral may be ambiguous but the convention is between 2 legs of the motor leads. I would guess it is because it is easier if no neutral wire exists.