# Maximum Angular Velocity of an Electric Motor

1. Aug 18, 2015

### Physicist97

Hello! I was looking to find out about an equation that would tell you the maximum angular velocity an electric motor can put out in terms of the geometry of the motor (area of the rotating coil, number of turns, etc), the EMF applied to the coil, magnetic field surrounding the coil, and so on. When I say electric motor, the setup I have in mind is a coil of wire in a uniform magnetic field. When a current is applied, a force is generated which will rotate the coil and when half a turn is reached a commutator will switch the direction of the current to keep the coil rotating. When the coil rotates another EMF is induced because you have a changing magnetic flux. This EMF will oppose the input EMF and thus the motor will reach maximum angular velocity when the input EMF and induced EMF balance. Here is how far I got.

$V=V_{ind}=-N{\frac{d{\Phi}}{dt}}=-NAB{\frac{dcos{\theta}}{dt}}$
So here $N$ is the number of turns of the coil, $A$ is the area the coil encloses, $B$ is the magnetic field surrounding the coil, ${\theta}$ is the angle between the vector normal to the area and the magnetic field. (${\Phi}$ was the magnetic flux through the coil). From this you get a differential equation.

${\frac{V}{NAB}}{\int{dt}}={\int_{0}^{{\theta}}{sin{\theta}}{d{\theta}}}=-cos{\theta}$
I do not now what bounds to put on the time integral. Would it go from $0$ to $T/2$ , where $T$ is the period of rotation, since the equation is only good for half a rotation (due to the need to change the direction of current)? But since $T=(2{\pi})/{\omega}$ , where ${\omega}$ is the angular velocity, how can I have a dependent variable as a bound on the integral? If anyone can point me in the right direction, or fix an error I made, that would be appreciated.

2. Aug 18, 2015

### Nidum

3. Aug 23, 2015

### CWatters

Perhaps this helps..

Most DC permanent magnet motors can also be used as a dynamo. The faster you spin them the more voltage or (back EMF) they generate. When used as a motor they typically accelerate until the back EMF roughly equals the applied voltage. At that point they stop accelerating.

The stronger the magnets (B) the more back EMF they generate so the point at which the back EMF equals the applied voltage occurs at a lower RPM. This leads to the somewhat surprising conclusion that to make a faster motor you should use weaker magnets :-)

So why are powerful rare earth magnets used in motors at all. The answer is that if you want a motor to run at a particular speed a rare earth magnet motor could have fewer turns. Fewer turns means you can fit fatter wire with a lower resistance into the armature. So for a motor with a given power input the I2R losses in the windings are lower and the motor is more efficient.

I can't claim to be a motor designer but I don't think there is a simple answer to the question "What is the Maximum Angular Velocity of an Electric Motor?" without getting into a lot of detail about the design constraints for a particular motor.