A simple loop in a uniform magnetic field
The figure below shows a simple rotating loop in a uniform magnetic field:
(a) is the front view and;
(b) is the view of the coil;
The rotating part is called the rotor, and the stationary part is called the stator.
This case in not representative of real ac machines (flux in real ac machines is not constant in either magnitude or direction).
However, the factors that control the voltage and torque on the loop are the same as the factors that control the voltage and torque in real ac machines.
The voltage induced in a simple rotating loop.
If the rotor (loop) is rotated, a voltage will be induced in the wire loop. To determine the magnitude and shape, examine the phasors below:
To determine the total voltage induced etot on the loop, examine each segment of the loop separately and sum all the resulting voltages.
The voltage on each segment is given by equation
Segment ab: The velocity of the wire is tangential to the path of rotation, while the magnetic field B points to the right.
The quantity vxB points into the page, which is the same direction as segment ab. Thus, the induced voltage on this segment is:
Segment bc: In the first half of this segment, the quantity vxB points into the page, and in the second half of this segment,
the quantity vxB points out of the page. Since the length l is in the plane of the page, vxB is perpendicular to l for
both portions of the segment. Thus, ecb = 0.
Segment cd: The velocity of the wire is tangential to the path of rotation, while B points to the right.
The quantity vxB points into the page, which is the same direction as segment cd. Thus,
Segment da:same as segment bc, vxB is perpendicular to l. Thus, eda = 0
Total induced voltage on the loop
since ab = 180o -
cd and sin
sin (180o - )
If the loop is rotating at a constant angular velocity ,
then the angle of the loop will increase linearly with time.
also, the tangential velocity v of the edges of the loop is:
where r is the radius from axis of rotation out to the edge of the loop and is the angular velocity of the loop. Hence,
since area, A = 2rl
Finally, since maximum flux through the loop occurs when the loop is perpendicular to the magnetic flux density lines, so
From here we may conclude that the induced voltage is dependent upon:
Flux level (the B component)
Speed of Rotation (the v component)
Machine Constants (the l component and machine materials)
The Torque Induced in a Current-Carrying Loop
Assume that the rotor loop is at some arbitrary angle
wrt the magnetic field, and that current is flowing in the loop.
To determine the magnitude and direction of the torque, examine the phasors below:
The force on each segment of the loop is given by: