Machine Riddle No.30 - Flux Vs Induced EMF
In an alternator, why does the induced EMF lags the main field flux by 90°?

#1
Mon, November 8th, 2010 - 12:02
In static system such transformer, the difference time angle between induced voltage and its origin sinusoidal flux base of faraday's law (e = - dø/dt) is 90˚. But the condition of motion induced voltage which involve to rotary electrical machines is different.
The Figures below show a rotating rotor with a sinusoidally distributed magnetic field in the centre of a stationary coil.

Assume that the magnetic of the flux density vector B in the air gap between the rotor and the stator varies sinusoidally with mechanical angle, while the direction of B is always radially outward. The magnitude of the flux density vector B at a point around the rotor is given by:

B = BM cos α

Note that α is the angle between the maximum flux density (Bm) and the current magnetic flux density phasor B. Since the rotor is itself rotating within the stator at an angular velocity ωm the magnitude of the flux density vector B at any angle αaround the stator is given by:

B = BM cos (ωt - α)

Induced voltage in a wire is  e = (v x B) l

However, this equation was derived for the case of a moving wire in a stationary magnetic field. In this case, the wire is stationary and the magnetic field is moving, so the equation for induced voltage does not directly apply. Hence, we need to assume that we are “sitting on the magnetic field” so that the magnetic field appears to be stationary, and the sides of the coil will appear to go by at an apparent velocity vrel and the equation can be applied.

The total voltage induced in the coil will be the sum of the voltages induced in each of its four sides.

1.  Segment ab
α= 180º. Assume that B is directed radially outward from the rotor, the angle between v and B in segment ab is 90º, while v x B is in the direction of l, so

Where the minus sign comes from the fact that the voltage is built up with a polarity opposite to the assumed polarity.

2.  Segment bc
The voltage is zero, since the vector quantity v x B is perpendicular to l.

3.  Segment cd
α= 0º
Assume that B is directed radially outward from the rotor, the angle between v and B in segment cd is 90º, while v x B is in the direction of l, so

4.  Segment da
The voltage is zero, since the vector quantity v x B is perpendicular to l.

Therefore total induced voltage: