torque in a DC motor

[CPL.Luke]

alright so if I apply a voltage to a loop of wire in a constant magnetic field, then it should feel both a torque and after some period of time it will have a (presumed) constant angular velocity. my question is what that final angular velocity is going to be and or, the function of time that gives us the angular velocity at that time. also, what would the torque on the motor be as a function of time?

 

 

I've been working on this problem on and off for the past couple days after seeing a simple electric motor demonstration, I ended up with 4 equations that I thought I could use, namely an equation for the torque on the loop as a function of I,angular velocity, and t; an application of kirchoff's first law that contains the variables I,angular velocity, and t; and an equation for the power in the circuit containing the variables I,torque, and angular velocity; also an equation for the magnetic flux, and then its derivative.

 

note: for simplicity the loop of wire is square with the dimensions X x Y

 

V=IR+d(flux)/dt

power=IV=(torque)(angular velocity)+R(I^2)

flux=XYB cos((angularvelocity)(time))

torque=IXYB sin((angularvelocity)(time))

 

 

now the big problem is that when you plug the flux into the equation for V and then you plug V into the equation for power, then plug the equation for torque into the equation for power, you end up with both sides of the equation being the exact same thing. This then prohibits you or I from solving anything at this point. Yet the problem has to be solvable, as by merely plugging an electric motor into a battery you will get a torque, and an angular velocity.

[swansont]

T = IABsin(theta), equivalent to your last equation. If you want to solve this you would need to know the moment of inertia of the system, since T=I(alpha). Then it's just a rotational kinematics problem.

[CPL.Luke]

but I don't know the angular acceleration or the current, in the problem we only get the voltage. although we could have the rotational inertia.

[swansont]

I don't think that's enough to solve the problem. The torque depends on B, too, and you can't get I without R.

[CPL.Luke]

were assuming a known B/R/size of the sides of the loop etc.

 

we could even know the rotational inertia of the loop, but we can't take any measurements once the motor is turned on. The reason I'm interested in a problem like this is that it seems very difficult to solve but a solution must exist (otherwise how would the universe know what the torque/angular velocity should be)

[TonyMcC]

Hang on a minute! If you apply a voltage to a loop of wire that is free to rotate in a constant magnetic field it will move to a position at right angles to the magnetic field and then stop. You need a commutator to keep reversing the applied voltage at the correct positioning of the loop. Or is everyone assuming there is a commutator?

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Another aspect not considered is the phenomenon known as back e.m.f.. As the loop rotates it generates a voltage in opposition to the applied voltage. Thus the effective voltage is Applied Voltage minus Back e.m.f.. Since back e.m.f. increases with speed of rotation it has a lot to do with final maximum speed. As mentioned before, if you don't have flux density and loop resistance you can't really begin to consider making calculations.

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I forgot to mention that if coil bearings are frictionless and there is no "windage" then maximum speed of rotation will be when back e.m.f. equals applied voltage giving zero current in the loop.