# Brushless Motors and Torque

When it comes to brushless motors there are a number of constants inherent to a given motor that define how that motor behaves, both electrically and mechanically. Let’s take a closer look at two of these constants: the velocity constant (K_{v}) and the torque constant (K_{t}.)

## Velocity Constant (K_{v})

Probably the most familiar constant is the Velocity Constant or K_{v}. Technically K_{v} is the back voltage or Back Electro-Motive Force (BEMF) that is generated in the non-active coils by the motor when it is spun at a given RPMs. The constant is called the velocity constant, hence the notation K ( scientific notation for constant) and v for velocity. At a given velocity it generates a given voltage on the inactive phases. You can quite easily drive a motor at a known RPM with an external drive like a drill press, measure the RMS voltage across two of the phases and run it through a formula to get the velocity constant. The no-load RPMs of a motor, or the steady state RPMs at full voltage with no propeller attached, are a bi-product of the K_{v} that are commonly used as a reference point. The back voltage is the true measure if K_{v}, but the velocity constant can be fairly accurately simplified as the number of rotations per minute a motor will spin under no load per volt of applied electricity. Remember that as we’ll come back to it later.

The K_{v} of a motor is constrained by both the magnetic field strength at the stators and the number of turns on the windings. The air gap and magnet strength in turn constrains the magnetic field strength at the stator and the windings are tuned to match that field strength to achieve a specific K_{v}. The higher the field strength the lower the K_{v} with an equivalent number of turns, so as magnetic field strength increases the number of turns on the windings must be decreased to raise the K_{v} back to the initial value.

The number of turns used to achieve a specific K_{v} also impacts the internal resistance of the windings, which plays a role in a number of other performance related aspects. The diameter of the wire, the length of the wire, and the quality of the copper or any other materials used to augment it (like silver coated wire for instance) all impact the functional resistance of the coil which in turn impacts the voltage drop across the coils.

The rise time, or the time it takes for the coils to reach the current and voltage steady state, is impacted by the total circuit resistance and the inductance of the coil. The inductance changes based on the number of turns. The higher the inductance, the longer it takes to reach a full charge, and the lower the inductance the faster it will charge. The more turns that are used in an inductor, the higher the inductance, and the fewer turns, the lower the inductance. In our case this is inversely proportional to the velocity constant (K_{v}). As the current begins flowing through the coil, it creates an equal BEMF reaction in the inductor which resists rapid changes in current flow. This time factor interacting with the back flow of current in the non-active phases is what defines the actual RPMs the motor will hit.

The peak measured BEMF level in this situation is always a percentage of the total supply voltage. At zero RPMs the back EMF is 0v, as the speed of the motor increases, the back EMF gets closer and closer to the forward supply voltage. Since torque is a function of current in a brushless motor, and current is equal to the voltage drop in the circuit divided by the circuit resistance, when the back EMF is equal to the forward applied voltage minus the circuit losses the functional torque of the motor will reach a point where it runs out of the torque required to accelerate the bell and RPMs can't rise further. This is also why loaded RPMs are less than the unloaded value. The motor runs into the torque limit sooner because the required torque to spin the prop is higher than the required torque to spin the bell alone.

The stronger the magnetic field at the stator, the stronger the field created in the coils and the higher the BEMF level. This is why going to a stronger magnetic field (stronger/thicker magnets, tighter air-gap, etc) lowers the functional K_{v} of the motor even with the same number of turns on the coil. You can actually take different bells from different motors and put them on the same windings to tell which magnets and air-gap are stronger and tighter. The lower the K_{v} reads, the stronger the field at the stator.

## Torque Constant (K_{t})

So let's talk a bit about one of the other important constants, the torque constant, or K_{t}. One of the biggest myths that has been floating around since the beginning of RC use of brushless motors is related to the relationship between K_{v} and torque. Let’s set the record straight, ** a change in K_{v} doesn't necessarily predict a consistent change torque**. K

_{v}only changes the current it takes to produce the torque, or inversely, how much torque is produced by certain amount of current. The ability of a motor to actually produce torque is a factor of things like magnet strength, air-gap, cross-sectional area of the windings, and stator size. What K

_{v}does change is K

_{t}.

The torque constant (K_{t}) describes how much current the motor will consume to produce a certain amount of torque. As the RPMs increase the current rises dramatically primarily due to the non-linear relationship between energy and RPMs. Eventually the torque limit defined by the stators, magnets, and air-gap will be reached and the RPMs won’t be able to continue, but until that point the current will continue to rise at an exponential rate.

Torque itself is simply a definition of how much rotational force can be applied, which does not have a time component or a current component inherently. However, in terms of its relationship with a brushless motor torque can be calculated as *mechanical power = torque * angular velocity*. To calculate angular velocity we need to convert RPMs to radians per second, so multiply RPMs by *(2*pi)/60 *or 0.1047.

Looking at the formulas:

RPMs = K_{v}*V

P = V*I

We can make substitutions in the original formula:

V*I = T * (K_{v}*V*0.1047)

(V*I) / (K_{v}*V*0.1047) = T

T = I/(0.1047*K_{v})

We can arrive at the same spot by using the formulas for kT.

K_{t} = 1/(0.1047*K_{v})

K_{t} = T/I

So setting these formulas equal to each other creates:

T/I = 1/(0.1047*K_{v})

T = I/(0.1047*K_{v})

A cursory look at this formula might make one assume that torque and kV have an inverse relationship, but that doesn’t account for current. As K_{v} increases the torque constant changes, which means current increases as well. Assuming current and K_{v} increase proportionally, that means torque stays relatively constant.

Obviously this is based on an ideal theoretical model, and in the real world we have friction, heat, and other efficiency losses that make it harder to maintain torque the higher we go in K_{v}. What this boils down to is that a motor of a given K_{v} will attempt to generate the same amount of torque **per amp of current **regardless of other physical properties. This also explains why larger stator motors of equivalent K_{v} end up drawing much more current than their smaller stator counterparts. The larger motors inherently have more torque, and since the torque per amp of current is constant for a constant K_{v}, that motor will draw more current when it delivers the increased torque.

In motors with weaker magnets and larger air-gaps there was a much larger disparity between K_{v} and torque due to the inherently low torque capacity of the designs themselves. The lower torque limits were more readily apparent as K_{v} increased, because they simply couldn’t push the props hard enough to establish the current required to deliver torque at higher K_{v}. At lower K_{v} the lower torque was not as much of a problem due to the lower RPM target, but if you look at the response times and load capacity even at the lower k_{v} the lack of torque is readily apparent.

On the very high torque motors that are the core of the mini-quad world right now, the minimal torque change is much more readily apparent. It is also important to note that on high torque motors the change in K_{t} doesn't necessarily impact efficiency at equivalent output in a practical way, it just means that there is a lot higher current spikes during rapid changes in RPMs, and sustained increases at the top end of the throttle as the RPMs exceed the RPMs of the lower K_{v} motor. At equivalent constant RPMs, the difference in current draw is fairly insignificant. If you were to limit a high K_{v} motor to similar RPMs though you would see very little difference in overall efficiency, but you would get some small gains in the rate of RPM change due to the higher K_{v}.

## Conclusions

The end results of all of this discussion is that we are limited primarily by the current delivery capacity of our batteries on high torque motors. Yes we can probably push the heavy 5" props and even 6" props on high torque high K_{v} motors, but the current demands will very quickly exceed the capacity of our batteries to deliver them. High K_{v} motors will often have faster response times and more power than their lower kV counterparts, but they will dramatically change both the throttle curve and the power requirements. That being said, often a low kV motor on a heavy prop and a high K_{v} motor on a lighter prop will have very similar power requirements.

High kV benefits more from a lighter weight build where it can make use of the extra RPMs at the top of the throttle, where mid K_{v} gains ground on heavier quads where the motors have to do more work low in the throttle to get the quad moving and where the high K_{v} motors can't unwind enough to take advantage. For instance on a heavy rig, the difference felt in the air between a high K_{v} motor on a lighter loaded prop and a mid kV motor on a heavier loaded prop might be fairly minimal, where the same two motors and props on a light rig may feel dramatically different in speed and handling. The high K_{v} motor may be able to push the heavier props and see some gains, but they will be dramatically limited by the ability of the battery to deliver the required current.

### Comments (7)

- 1

kartupeli:

Sep 14, 2017 at 12:00 PM

Amazing write up. How about a mid-heavy 5" quad with 2205-2206 1500-1900kV motors? Would it result in a less power hungry quad (albeit without the higher end throttle), or is the current then going way up?

quadmcfly:

Sep 18, 2017 at 10:12 AM

Sure, reducing kV will reduce top end current by a good bit, but as I mentioned you can get almost the same effect by reducing max throttle. I would be wary about going down too much in stator size though, as you'll need that torque in order to make the quad responsive enough with the weight. One of my favorite freestyle setups back in the day was a larger stator 2100kv motor on quad props, so not much top end, but good response,good control, and good flight times.

Brian Bailey:

Sep 14, 2017 at 05:24 PM

I think you just eloquently described what we've known for awhile now. New motors are cool, but we're really just limited by our batteries. I'd like to think we're finally done with the bigger motors = better motors debacle, and we're now left to explore the phase space of Motor, battery (weight), and prop combinations.

Dean:

Sep 15, 2017 at 12:30 AM

I'm more interested these days on the "electrical noisyness" of the motors and the ease, or difficulty in tuning with different motors. We are also starting to get limited by the amount of motor electrical noise that the gyro's on our flight controllers can tolerate. On my latest build, I have fitted 5 extra capacitors with a total of 14g to my quad just to control the amount of electrical noise.

quadmcfly:

Sep 18, 2017 at 10:22 AM

If you're having to add that much capacitance to your system then my guess is you are having other issues rather than just noisy motors. I'm running some of the most powerful (and traditionally considered noisy) motors out there with minimal damping on the FC and almost no capacitance without issues. A lot of the noise issues are a result of inductance in the wiring and gyro issues rather than the motors themselves. For instance, any flight controller with a built in PDB is much more likely to have noise issues due to proximity of the high inductance portions of the board to the low voltage electronics.

Kevin:

Sep 18, 2017 at 08:27 PM

Not sure if this is what Dean meant, but I've run into issues of cogging-resonance - very narrow RPM bands where all motors of a particular model induce fairly severe z-axis gyro noise. The motors otherwise seem fine - very smooth bearings, and very little gyro noise outside of the narrow RPM band where they seem to hit resonance. I haven't tried that many different motors, but it's been the ones with extremely sharp magnet detent (strong magnets, small air gap)that had this problem. I don't think there's any easy way to measure this on your current test setup. You would really need a torque transducer, and logging quick enough to pick up the high RPMs. Of course, a torque transducer would also let you isolate motor and propeller efficiency, which would be an awesome addition to your test results (although maybe more than most people want to know).

I've never run into EMI issue from a motor or ESC, but have from a low-cost VTx that was in very close proximity to the IMU chip on a FC. It was in a Kraken, with not too much room in the pod for physical separation. An inch was sufficient separation, but alas impossible in the tiny pod.

quadmcfly:

Sep 19, 2017 at 08:48 AM

Yes, I've seen what you're talking about too, but believe it or not, it is an artifact not purely of the motors, but of how we drive them. One of the causes it is actually torque ripple, which is a combination of cogging torque and specific flux harmonics. This is accentuated by the trapezoidal commutation methodology used in most of our ESCs, as well as some harmonic interference between the ESC drive frequency and the commutation rate. BLHeli_32 solves some of that by making the drive frequency adjustable, so you can move it out of the range where it causes harmonic interference, and it also uses a slightly tweaked commutation method. I think more gains are possible given some looks at alternative commutation methods like sinusiodal or FOC, or maybe some sort of hybrid. There's a lot of stuff that goes in to the dynamics between motor and ESC and we have a lot of room for growth still.