Step 2: The Brushless DC Motor

At the heart of most hub motors is a brushless DC motor. To build a hub motor right, you need to understand some basics of brushless DC motors. To understand brushless DC motors, you should understand brushed DC motors. If you've taken a controls class, chances are that you've used brushed DC motors as a "plant" to test your controls on.

I've highlighted and bolded the juicy stuff that you'll need, but for the sake of continuity it's probably good to grunge through all of it anyway.

Brushed DC Motor Physics

Perhaps the best DC motor primer I have seen (I'm not biased at all, I promise guys! Pinky promise! ) is the MIT OpenCourseware notes for 2.004: Dynamics and Control II. Take a read through it at your own leisure, but the basic rundown is that a brushed DC motor is a bidirectional transducer between electrical power and mechanical power that is characterized by a motor constantKm , and an internal resistanceRm. For simplicity, motor inductance L will not be considered. Essentially if you know Km and Rm, and a few details about your power source, you can more or less characterize your entire motor.\

Update10/06/2010: The original 2.004 document link is dead, but here's one that's roughly the same content-wise. Also from MIT OCW.

The motor constant Km contains information about how much torque your motor will produce per ampere of current draw (Nm / A) as well as how many volts your motor will generate across its terminals per unit speed that you spin it at (V / rad / s, or Vs / rad, or simply V*s). This "back-EMF constant" is numerically equal to Km, but some times called Kv.

In a DC motor, Km is given by the expression

Km = 2 * N * B * L * R

where N is the number of complete loops of wire interacting with your permanent magnetic field of strength B (measured in Tesla). This interaction occurs across a certain length L which is generally the length of your magnets, and a radius R which is the radius of your motor armature. The 2 comes from the fact that your loop of wire must go across then back across the area of magnetic influence in order to close on itself. This R has nothing to do with Rm, by the way.

As an aside, I will be using only SI (metric!!!!!) units here because they are just so much easier to work with for physics.

Let's look at the expression for Km again. We know from the last page that

Pe = V * I and Pm = T * ω

In the ideal motor of 100% efficiency (the perfect transducer), Pe = Pm, because power in equals power out.  So

V * I = T * ω

Where have we seen this before? Swap some values:

V / ω = T / I

Kv = Km

Oh snap.

The takeaway fact of this is that knowing a few key dimensions of your motor: The magnetic field strength, the length of the magnetic interaction, the number of turns, and the radius of the armature, you can actually ballpark your motor performance figures usually to within a factor of 2.

Now it's time for...

The Brushless DC Motor

BLDC motors lie in the Awkward Gray Zone between DC motor and AC motors. There is substantial disagreement in the EE and motor engineering community about how a machine which relies on three phase alternating current can be called a DC motor. The differentiating factor for me personally is:

In a brushless DC motor, electronic switches replace the mechanical brush-and-copper switch that route current to the correct windings at the correct time to generate a rotating magnetic field. The only duty of the electronics is to emulate the commutator as if the machine were a DC motor. No attempt is made to use AC motor control methods to compensate for the AC characteristics of the machine.

This gives me an excuse to use DC motor analysis methods to rudimentarily design BLDC motors.

I will admit that I do not have in depth knowledge of BLDC or AC machines. In another daring act of outsourcing, I will encourage you to peruse James Mevey's Incredible 350-something-page Thesis about Anything and Everything you Ever Wanted to Know about Brushless Motors Ever. Like, Seriously Ever

There's alot of things you don't need to know in that, though, such as how field-oriented control works. What is extremely helpful in understand BLDC motors is the derivation of their torque characteristics from pages 37 to 46. The short rundown of how things work in a BLDC motor is that an electronic controller sends current through two out of three phases of the motor in an order that generates a rotating magnetic field, a really trippy-ass thing that looks like this.

The reason that we consider two out of three phases is because a 3 phase motor has, fundemantally, 3 connections, two of which are used at any one time. Here's a good illustration of the possible configurations of 3 phase wiring. Current must come in one connection, and out the other.

In Mevey 38, equation 2.30, the torque of one BLDC motor phase is given by

T = 2 * N * B * Y * i * D/2

where Y has replaced L in my previous DC motor equation and D/2 (half the rotor diameter) replaces R.

If you do it my way, it becomes

T = 2 * N * B * L * R * i , replacing D/2 with R.

Remember now that two phases of the motor has current i flowing in it. Hence,

T = 4 * N * B * L * R * i

This is the Equations to Know for simple estimation of BLDC torque. Peak torque production is (modestly) equal to 4 times the:

� number of turns per phase
� strength of the permanent magnetic field
� length of the stator / core (or the magnet too, if they are equal)
� radius of the stator
� current in the motor windings

As expected, this scales linearly with current. In real life, this will probably get you within a factor of two. That is, your actual torque production might be between this theoretical T and T/2

Wait, 4? Does that mean if I turn my brushed DC motor into a brushless motor, it will suddenly have twice the torque? Not necessarily. This is a mathematical construct - a DC motor's windings are considered in a different fashion which causes the definition of N and L to change.

Next, we will see how to use this equation to size your motor.

28  July 2010 Updateto the definition of T

In the equation T = 4* N * L * B * R i, the constant 4 comes from the derivation of a motor with only one tooth per phase, assuming N is the number of turns of wire per tooth on the stator.

The full derivation of this constant involves each loop of wire actually being two sections of wire, each of length L. This is due the fact that a loop involves going across the stator, then back again. Next, in a BLDC motor, two phases are always powered, therefore contributing torque.

We can observe that in a motor with only 1 tooth per phase (a 3-toothed stator), there are no more multiplicative factors. However, for each tooth you add per phase (2 teeth per phase in a 6-tooth stator, 3 teeth per phase in a 9-tooth stator, etc.) the above constant must be multiplied accordingly. The constant in front of the equation essentially accounts for the number of active passes of wire, which is 2 passes per loop times 2 phases active times number of teeth per phase.

So, what I actually mean is that T = 4 * m * N * B * L * R * i where

m = the newly defined teeth per phase count.

As the windings themselves have yet to be introduced, keep in mind the number of teeth per phasein the dLRK winding is 4.

<p>this is one of the absolute best instructables I've ever read : mainly because it is actually instructive !!</p><p>great work</p><p>for my part I'm interested in the topic because I'd like to make an electric skateboard with hub motors : options are cheap Chinese ones I'm afraid to waste money on, custom made ones that are quite expensive (250$ apiece+shipping) and now I think it might be worth going the DIY way if I can get my hands on an acceptable stator...</p><p>in fact I think the best would probably be to make 2 with lower power to make them easier to wind and especially to heat less...</p><p>thanks again for the inspiration</p>
<p>Hey there I just recently found this instructable and first I want to say, nice work! I did want to <br>point out that in step 2 Kv and Km are inversely proportional. I.E. Km has units <br>[V/(rad/s)] and Kv should be [(rad/s)/V], which means when you multiply <br>Kv by your battery voltage you get a rough estimate for top speed. What you calculated by substituting your equations is that the back-emf constant Ke is equal to the torque constant Kt, which is true. You refer to both of those as Km.</p>
<p>End-Turn also contributes to Torque. Anything that increases magnetic flux desity in the cores also increases the torque.<br></p>
<p>Hi</p><p>I wanna make an hub motor inside an longboard wheel.</p><p>How should I build it?</p>
<p>great write up, thank you!</p>
<p>great job</p><p>can i control this motor using arduino ? and if i can do that how could i do it ?</p>
<p>Hi, great job.</p><p>what if regenerative braking concept. in the sense do we get back emf in BLDC motor.</p><p>if yes what if you harness the power from it</p>
<p>I am not good in electricity, but you explain very well, thanks!</p>
<p>I learned something A=heat that's why ev are high voltage.</p>
<p>This looks so cool, but after I read the why not to build list it got a little stale. </p>
<p>how large a motor would I have to build to power a bike light consisting of approx 50 led lights?</p>
<p>Could you just clarify for me the &quot;AC&quot; part of this whole thing.</p><p>Does the controller simply emulate the commutator of a DC motor, energizing the coils in such a way as to create a rotating magnetic field or are you saying it also converts the supplied DC to a true alternating current which it distributes to the coils in such a way as to produce a rotating AC field?</p><p>Thanks</p><p>Doug</p>
hi. i'm from tunisia. good job man !. can you contact me on jakefouly@gmail.com i need some help. thank you
<p>hello guys. i am new member. hope to could find good things for better future for succeses. tx all</p>
<p>Issues with your formula:</p><p>You pick Mevey's 2.30 equation:</p><p><strong>T = 2 * N * B * Y * i * D/2</strong></p><p>Where N is the number of turns per pahse, and '2' is the number of active phases. Then you redefine N as the number of turns per tooth, and define m number of teeth per phase. So (previous definition of N) = (new definition of N) * m. The result equation:</p><p><strong>T = 2 * </strong><strong>m * </strong><strong>N * B * Y * i * D/2</strong></p><p>Not four but two!!!</p><p>Next, in Mevey's 2.30 equation D is the diameter of the coil centers. Not stator's outer diameter, but diameter of the coil centers which is obviously smaller than stator's outer diameter.</p><p>P.S. Thanks for the article anyway. It is really motivating!</p>
<p>the most in depth view about brushless motors awesome saving this page for future guidence cheers</p>
<p>Hi, </p><p>good math derivation, T = 4 * m * N * B0 * (t / (t + g)) * L * R * i, <br>but this is exactly the double the torque you calculate using what reported on <br>LRK Motor Analysis Worksheet<br><a href="http://www.femm.info/examples/lrk40/lrk-bldc.pdf" rel="nofollow">http://www.femm.info/examples/lrk40/lrk-bldc.pdf<br></a>T = 4 * (rr+rs)/(rs-rr) * Br * N * L * t * ( I - I0)</p><p>where this formula is calculated taking into account that just two phase current are active on a trapezioidal drive, and that the current to be used should be just the active current (total current less free running current).</p><p>Which formula is the correct one???</p>
Torque should definitely depend on current. So, it should be T = 4 * m * N * B0 * (t / t + g) * L * R * i instead of T = 4 * m * N * B0 * (t / t + g) * L * R
<p>Or better, using correct parenthesis position:<br>T = 4 * m * N * B0 * (t / (t + g)) * L * R * i<br>but furthermore:<br>- for &quot;i&quot; you have use the useful portion of total current, i.e. &quot;i&quot; - free running i zero (taking into account the eddy current and friction losses);<br>- for &quot;R&quot; you have to intend the medium radius between stator and rotor (where the magnetic force act) which is in the middle of (t+g)=magnet thickness + air gap zone.</p>
Hi mate I'm making a hub motor for a skateboard, 100kv 15turns in wye dlark with 6 strands 0.34mm wire (this is all I can fit)<br>80mm diameter wheel, 50mm diameter motor, 40.7mm x30mm stator 12t 14p, with 40SH magnets 30x7x3, air gap between 0.5-0.7 depending on tolorance.<br>Will be using 29v batterys 8ah 30c.<br><br>I want to get up a hill of about 10-15% grade at about 20km/h<br>70-90kg. How much power do I need, is my wire cross section ok?<br>Iv been struggling with the math, and worried about the new winding not having the current capabilities I need, but it's hard to fit the copper inside of the stator, but maybe I'm just not good at it!<br>I'm also using hall sensors or optical sensors soon to get better start up as I seem to get a lot of cogging! From even a small push!<br><br>Each skateboard is using 2 motors on the rear.<br><br>I really need help!<br>My email is jacob.bloy(at)gmail.com<br><br>My build page.<br>http://endless-sphere.com/forums/viewtopic.php?f=31&amp;t=65636&amp;sid=32c74875705d1d55d0801eeae1381c11
<p>So I have a question about the equation to measure the theoretical torque. T = 4 * N * B * L * R * i in my case would be 10 turns per phase, 52 for the neodymium magnet and assuming you measure things using the imperial system the length of the stator would be 19.8 inches and the stator radius 3 inches. Putting through 42 amps would theoretically give me 5189184.0 torque. Now this can't be accurate because at 1500 RPM that would give me 1.4821e+6 horse-powers just to put things into perspective, which is a insane amount of horsepower.<br>t = 4 * 10 * 52 * 19.8 * 3 * 42 - Where did I go wrong in the equation?</p>
<p>If we rewire series connection of coils to parallel i.a.w. <a href="http://www.thebackshed.com/windmill/FPRewire.asp" rel="nofollow">http://www.thebackshed.com/windmill/FPRewire.asp</a> without changing position of hall sensors, does it influence on steering algorithm? How to include this wiring difference in one equation(e.g. torque equation)?</p>
<p>Excellent instructable! Thanks for taking the time to document and share your work. I need clarification on (at least) one topic. In the section discussing Magnet Length the author states:</p><p>&quot;Optimally, the magnet length is equal to the stator length (<em>L</em>).&quot;</p><p>In the same section, magnet width is mentioned. Would someone clarify for me the magnet dimensions that should be used for a given stator? Specifically, what is the stator length (L)? A diagram would be especially useful.</p><p>Thank you.</p>
<p>Jah mahn, danke mahn, huge, this is monster info bro.</p><p>Thanks you!</p>
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<p>You mention that you used 2 x 22AWG wires instead of 1 x 18AWG wire because it was hard to wrap and bend for 25turns. You said &quot;Use the wire gauge table to compare diameters!&quot;, now 22 AWG wire is 0.644mm in diameter and 18AWG wire is 1.024mm in diameter. So 2 x 0.644mm is 1.288mm and thats well over the diameter of the 18AWG wire. Now 24AWG wire is 0.511mm in diameter, and 2 x 0.511mm is 1.022mm which is a lot closer to 18AWG. I don't want to be annoying i'm just confused. If we compare the surface areas though 18AWG wire is 0.823mm^2 and the closest pair that would measure similar is a pair of 21AWG wires, at 0.411mm^2 x 2 = 0.822mm^2, BUT neither of those are the wire you said you were using. Should the diameter of the multiple strands not add up to close to the diameter of the single wire? Thanks for any help, just confused.</p>
<p>Hi, I thought I'd jump in here and clarify a few things for you: When doing anything with electrical wire, especially when said wire is going to be carrying a significant percentage of its maximum safe current, you should be aware that the determining factor in current capacity is the cross-sectional area of the conductor, not the outside diameter. Since the area increases faster than the diameter or circumference, a wire of half the diameter will have one fourth the cross-section and thus one fourth the current capacity. A wire 1.024mm in diameter has a section of 0.82 mm^2, where a wire of 0.511mm diameter has a section of only 0.20mm^2. A 0.644mm wire has a section of 0.32, which means a pair are up to 0.64, close to the original 0.82. If I were doing this, I'd use three strands of 22, for a section of 0.96, better than the 18 gauge.</p><p>TL;DR don't go by diameter, go by cross section. They don't equate directly.</p>
<p>Thanks for the formulas to make my motor, but your math was off by .01 on 3.66/(4*4*10.9*0.03 *0.035)=19.98.</p><p>it is really 19.99 (19.986)</p>
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<p>Being one of the last electrical and electronic engineering graduates from my school, before they dropped the &quot;electrical&quot; part, electric machines have always been a favourite subject of mine. This 'ible is one of the best I've ever read. Excellent work. </p><p>Incidentally, you can get tyres made by the guys who can retread forklift truck wheels. They vulcanise the tyre onto your own hub. </p>
<p>Being one of the last electrical and electronic engineering graduates from my school, before they dropped the &quot;electrical&quot; part, electric machines have always been a favourite subject of mine. This 'ible is one of the best I've ever read. Excellent work. </p>
<p>Oh my, that's a lot of work and thanks for putting it all up here.</p><p>I was looking for a motor I could pass my leg though instead of using a ring gear and a small motor to rotate it off to one side. The open motors would be used to rotate segments of a leg roughly depicted here: </p><p><a href="http://youtu.be/RV9fvg3C_fo" rel="nofollow">http://youtu.be/RV9fvg3C_fo</a></p><p>I'm still working out how many segments and at what angle and speed each segment should rotate at for the maximum comfort of the rider while still providing a good, natural leg motion. Seems making the motor would be beyond my capabilities and I'll have to settle on the ring gear driving by a motor or the like.</p>
very interesting very ( ty iv bine tring to find info on moters like this )( o and I Quote &quot; Their large outrunner motors are inexpensive enough to consider cannibalizing for stators. &quot; <br>well LOL!!! ) thank you for this it was very help full. :)
Can the stator core be plastic? Does it need to still be magnetic at all? I dont' understand why you wouldn't build it out of something completely non-magnetic
No not plastic. The material has to have a high permeability to concentrate the magnetic fields and at the same time reduce Eddy Currents.
I wonder if a motorcycle stator from the magneto would make a nice stator for a brushless motor? Used they are not too expensive.
Is it possible to melt down many cores in a foundry and then cast my own core? The core I need is huge and would cost a lot of money to have it machined and cast by a specialized group. Especially when I will need at least 3.
Stators are not cast. If you look at one, you'd notice they are many thin and fine layers. Each of those actually are insulated from each other. <br><br>A cast stator would basically be a big magnetic brake and would be extremely inefficient and heat up quickly due to eddy currents. I think you should look into motorcycle alternators and washing machines for large-ish (5&quot; - 6&quot; - 12&quot;) stators.
Where can we buy one of these motors (not the wheel) as a kit to put together and learn? It's easy to get the wire, but not the pieces the wires get wrapped around :-(
it looks to me that the torque should be proportional to the square of the radius. At constant magnet induction and current density the force per unit circumference length would be constant so the torque would be proportional to the radius and the length of the circumference, in turn proportional to the radius , hence the radius square.
Im an electrician, and house wiring is done in 14, 12 and 10 gauges mostly. Winding a motor in 18 gauge must be a chore! But im sure chris farley would say, &quot; It builds dexterity!&quot;
That's a lot to read but I read it anyway I can't make one of these. Because I don't have the tools nor the supplys to build it but awesome job
very nice and educative. learned a lot from this.
So there is probably something stupidly wrong with what I am about to write, but I am tired and can not get this idea out of my head, so on with it. <br> <br>What is to stop you from taking a standard dc motor, like the ones used in toy scooters, and reinforcing the !#$@% out of it, namely in the (casing? or is it a shell?) itself and the axel, welding a rim to the (reinforced) casing of the motor and using that as a hub motor with the motors axel acting like the axel of a bike wheel, with everything revolving around it? <br> <br>Would the motor just plain not have enough torque?, or is there some other blatantly obvious issue that I can't think of?
here is a mild example of your concept and a solotiuon using a standard dc brushed motor as an axle or pivot point. and having to add a gear reduction to it to get it to move. <br> <br>https://www.instructables.com/id/6-AXIS-ROBOTIC-ARM/ <br> <br>check it out. <br> <br>and vote for me <br>
You would not have enough torque, even on scooters with small diameter wheels the motor is usually geared down at least one to five, on a bike you will need a gearing of at least four times that! I hope this helps.
Sort of like this, or this.

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More by teamtestbot:How to Build your Everything Really Really Fast Chibikart: Rapid-Prototyping a Subminiature Electric Go-Kart Using Digital Fabrication and Hobby Components The New and Improved Brushless Electric Scooter Power System Guide 
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