Q.Give the construction and theory of an a.c. generator. OR Derive an expression for average power in $L$-$C$-$R$ circuit connected to an a.c. supply. What is wattless current?

An a.c. generator converts mechanical rotational energy into electrical energy by rotating a coil in a magnetic field, using electromagnetic induction; as the coil rotates uniformly, the changing flux produces a sinusoidal e.m.f.

Construction

An a.c. generator (alternator) essentially consists of:

• Armature: a coil of many turns of insulated copper wire, wound on a soft-iron core (to increase flux linkage), mounted on a rotating shaft.
• Field magnet: strong permanent magnets or electromagnets providing a uniform magnetic field $\vec B$ between whose poles the armature rotates.
• Slip rings: two rings fixed to the rotating shaft, rotating along with the coil, each connected to one end of the armature coil.
• Brushes: stationary carbon/graphite contacts that press against the slip rings, providing the electrical connection between the rotating coil and the stationary external circuit.
• A mechanical energy source (turbine, engine, etc.) rotates the shaft (and hence the coil) at a constant angular speed $\omega$.

Theory

Let the coil (area $A$, $N$ turns) rotate with constant angular velocity $\omega$ in a uniform field $B$, with the angle between the field and the normal to the coil being $\theta=\omega t$ at time $t$ (assuming $\theta=0$ at $t=0$). Flux linked with the coil:

$\Phi(t) = NBA\cos(\omega t)$

By Faraday's law of electromagnetic induction, the induced e.m.f. is:

$e = -\frac{d\Phi}{dt} = NBA\,\omega\sin(\omega t) = e_0\sin(\omega t)$

where $e_0=NBA\omega$ is the peak (maximum) e.m.f. This sinusoidally varying e.m.f. is delivered to the external circuit through the slip rings and brushes — the basic principle of the a.c. generator.

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