The a.c. generator Revision Notes

An a.c. generator primarily consists of a coil of wire that can rotate within a magnetic field. This coil is usually rectangular and mounted on an axle so it can spin. The magnetic field is provided by either permanent magnets or electromagnets arranged so that the coil rotates between their poles.

The coil is connected to two slip rings, which are circular metal rings attached to the ends of the coil wire. These slip rings rotate with the coil.

Stationary carbon or metal brushes press against the slip rings, providing an electrical connection between the rotating coil and the external circuit.

• Rotating coil: rectangular wire loop that spins in a magnetic field.
• Magnetic field: created by permanent magnets or electromagnets.
• Slip rings: two metal rings connected to coil ends, rotate with coil.
• Brushes: stationary contacts that press on slip rings to transfer current.

When the coil rotates in the magnetic field, the magnetic flux linking the coil changes continuously. According to Faraday's law of electromagnetic induction, a changing magnetic flux induces an electromotive force (e.m.f.) in the coil.

As the coil turns, the angle between the coil plane and magnetic field changes, causing the magnetic flux through the coil to vary sinusoidally. This induces an alternating voltage across the coil terminals.

The induced e.m.f. causes an alternating current to flow if the coil is connected to an external circuit. The current changes direction every half turn of the coil, producing alternating current (a.c.).

• Magnetic flux changes as coil rotates, inducing e.m.f.
• Faraday's law: induced e.m.f. ∝ rate of change of magnetic flux.
• Alternating voltage and current produced due to continuous rotation.
• Current reverses direction every half turn of the coil.

Slip rings are essential for transferring the induced alternating voltage from the rotating coil to the external circuit without twisting the connecting wires.

Each end of the coil wire is connected to a slip ring. As the coil rotates, the slip rings rotate with it, maintaining electrical contact with the stationary brushes.

Brushes are made of carbon or metal and press gently against the slip rings, allowing current to flow through them to the external circuit while the coil spins freely without tangling wires.

• Slip rings rotate with coil, providing continuous electrical contact.
• Brushes remain stationary, pressing on slip rings to conduct current.
• This arrangement prevents wire twisting and allows continuous current flow.

Mechanical energy, supplied by turning the coil (e.g., by a turbine or hand crank), is converted into electrical energy by electromagnetic induction.

The output voltage is alternating because the direction of the induced e.m.f. reverses every half rotation of the coil.

The frequency of the alternating current depends on the speed of rotation and the number of coil turns.

• Mechanical energy input causes coil rotation.
• Induced voltage and current alternate in direction.
• Frequency of output a.c. depends on rotation speed.

To draw a labelled diagram of an a.c. generator: start with a rectangular coil mounted on an axle.

Draw two slip rings attached to the coil ends, shown as two separate rings around the axle.

Add stationary brushes in contact with the slip rings, connected to an external circuit or load (e.g., a resistor). Include arrows to indicate coil rotation direction and magnetic field lines between the poles of permanent magnets around the coil.

• Rectangular coil on axle, labelled 'rotating coil'.
• Two slip rings on axle ends, labelled 'slip rings'.
• Stationary brushes touching slip rings, labelled 'brushes'.
• Magnetic field lines between magnets, labelled 'magnetic field'.
• Arrow showing coil rotation direction.

An alternating current (a.c.) generator produces an electromotive force (e.m.f.) by rotating a coil within a magnetic field. The coil is connected to an external circuit via slip rings and brushes, allowing current to flow.

As the coil rotates, the magnetic flux linkage through it changes continuously, inducing an e.m.f. according to Faraday's law of electromagnetic induction:

The magnitude of the induced e.m.f. depends on the rate of change of magnetic flux linkage through the coil, which varies with the coil's orientation relative to the magnetic field lines.

• Faraday's law: e.m.f. = - dΦ/dt, where Φ is magnetic flux linkage (Wb).
• Magnetic flux linkage Φ = NBA cos θ, where N = number of turns, B = magnetic flux density (T), A = coil area (m²), θ = angle between coil normal and magnetic field.
• The negative sign indicates the direction of induced e.m.f. (Lenz's law).

The coil rotates at a constant angular velocity ω (radians per second). The angle θ changes with time t as θ = ωt.

The magnetic flux linkage varies as Φ = NBA cos(ωt). Differentiating, the induced e.m.f. is:

e.m.f. = - dΦ/dt = NBAω sin(ωt). This shows the e.m.f. varies sinusoidally with time, producing alternating voltage and current.

• Maximum e.m.f. (peak) occurs when sin(ωt) = ±1, i.e., when the coil plane is perpendicular to the magnetic field lines (coil normal parallel to field).
• Zero e.m.f. occurs when sin(ωt) = 0, i.e., when the coil plane is parallel to the magnetic field lines (coil normal perpendicular to field).
• The e.m.f. changes sign every half rotation, causing alternating current direction.

The graph of e.m.f. (vertical axis, volts) against time (horizontal axis, seconds) is a sine wave.

Key features to plot: peaks (maximum positive e.m.f.), troughs (maximum negative e.m.f.), and zero crossings (points where e.m.f. = 0).

The period T of the wave is the time for one complete rotation of the coil, related to angular velocity by ω = 2π/T.

• Label axes: vertical axis as e.m.f. (V), horizontal axis as time (s).
• Mark zero e.m.f. points where coil is parallel to magnetic field.
• Mark peaks where coil is perpendicular to magnetic field.
• The graph is symmetrical about the time axis, showing alternating polarity.

At zero e.m.f., the coil is aligned so that its plane is parallel to the magnetic field lines; no magnetic flux is being cut, so no e.m.f. is induced.

At peak e.m.f., the coil is perpendicular to the magnetic field lines, cutting the maximum amount of flux per unit time, inducing maximum voltage.

The sinusoidal shape reflects the continuous and smooth rotation of the coil, causing the rate of change of flux linkage to vary sinusoidally with time.

• Zero e.m.f. corresponds to coil edges aligned with field lines.
• Peak e.m.f. corresponds to coil faces cutting field lines at maximum rate.
• The alternating sign of e.m.f. corresponds to the coil rotating through 180° intervals, reversing the direction of induced current.

Given: A coil with N = 100 turns, area A = 0.05 m², rotating in a magnetic field B = 0.3 T at 50 revolutions per second (rps). Calculate the peak e.m.f.

Step 1: Calculate angular velocity ω = 2π × frequency = 2π × 50 = 314 rad/s.

Step 2: Use formula for peak e.m.f.: E₀ = NBAω = 100 × 0.05 × 0.3 × 314 = 471 V (peak).

• Formula: E₀ = NBAω
• Substitution: E₀ = 100 × 0.05 × 0.3 × 314
• Answer: E₀ = 471 V (peak e.m.f.)
• Common error: forgetting to convert frequency to angular velocity (multiply by 2π).