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 constant*

*, and an*

**Km***internal resistance*

**. For simplicity, motor inductance**

*Rm**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.\*

**L****Update**

**10/06/2010**: 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

**contains information about how much torque your motor will produce per ampere of current draw (**

*Km**) as well as how many*

**Nm / A***volts*your motor will generate across its terminals per unit speed that you spin it at (

**or**

*V / rad / s,***or**

*Vs / rad,***). This "back-EMF constant" is numerically equal to Km, but some times called Kv.**

*simply V*s*In a DC motor,

**is given by the expression**

*Km***where**

*Km = 2 * N * B * L * R*

*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

**and**

*Pe = V * I*

*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 ***Where have we seen this before? Swap some values:*

**ω**

*V /*

**ω = T / I****Oh snap.**

*Kv = Km*

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

*, replacing D/2 with R.*

**T = 2 * N * B * L * R * i**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 Update**

**to the definition of T**

In the equation

*, the constant*

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

*where*

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