Question

Briefly describe the construction of a simple ac generator. Obtain an expression for the emf induced in a coil rotating with a uniform angular velocity in a uniform magnetic field. Show graphically the variation of the emf with time (t).

OR 

Describe the construction of a simple ac generator and explain its working.

Answer 1

Construction : A simplified diagram of an ac generator is shown in below It consists of many loops of wire wound on an armature that can rotate in a magnetic field. When the armature is turned by some mechanical means, an emf is generated in the rotating coil.

Consider the coil to have N turns, each of area A, and rotated with a constant angular speed ω – about an axis in the plane of the coil and perpendicular to a uniform magnetic field \(\vec B\), as shown in the figure. The frequency of rotation of the coil is f = ω / 2π.

Working : The angle 9 between the magnetic field \(\vec B\) and the area of the coil \(\vec A\) at any instant t is θ = ωt (assuming θ = 0° at t = 0). At this position, the magnetic flux through the coil is Φ_m = N\(\vec B\). \(\vec A\) = NBA cos θ = NBA cos ωt

Coil pf an generator rotating in a uniform magnetic field

As the coil rotates, the changing magnetic flux induces an emf the coil gives by

∴ e = e₀ sin ωt, where e₀ = NBAω.

Therefore the induced emf varies as sin cot and is called sinusoidally alternating emf. In one rotation of the coil, sin cot varies between +1 and – 1 and hence the induced emf varies between +e₀ and -e₀. The maximum value e₀ of an alternating emf is called the peak value or amplitude of the emf.

Variation of emf with time

The sinusoidal variation of emf with time t is shown in above figure. The emf changes direction at the end of every half rotation of the coil. The frequency of the alternating emf is equal to the frequency/of rotation of the coil. The period of the alternating emf is T = \(\cfrac 1f\)

magine looking at the coil of the ac generator from the slip rings along the rotation axis in.The magnetic flux, rate of change of flux and sign of the induced emf are shown in the table below for the different orientations of the coil as in below figure.

Coil orientation	Flux Φm	dΦm/dt	Induced emf
1	Positive maximum	Momentarily zero (constant flux)	Zero
2	Positive	Decreasing (negative)	Positive
3	Zero	Decreasing (negative)	Positive
4	Negative	Decreasing (negative)	Positive
5	Negative maximum	Momentarily zero (constant flux)	Zero
6	Negative	Increasing (positive)	Negative
7	Zero	Increasing (positive)	Negative
8	Positive	Increasing (positive)	Negative
9	Return to positive maximum	Momentarily zero (constant flux)	Zero

Coil orientations at selected instants.