- #1
Physicist97
- 31
- 4
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.
##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.