Generators

Principle and construction

• The basic construction of an electric generator is the same as that of an electric motor. The difference is that in a generator the armature is turned by some external agency or torque, instead of being driven by a current.
• As the conductor wires of the armature cut across the magnetic lines of force, an induced emf $e = Blv$ is produced across the terminals of the commutator (or slip rings). The induced emf is found to be proportional to the speed of rotation $\omega$ of the armature.
• Diagram in words: a rectangular coil (the armature) mounted on an axle between the north and south poles of a magnet; the ends of the coil go to two rings on the axle, and brushes press on the rings to carry the current out to an external lamp; an arrow shows the external torque turning the axle.

Waveform generation, followed on one conductor

• Position (i): the conductor is moving upward across the lines of force — the induced emf is maximum.
• Position (ii): the conductor is moving parallel to the field — no lines are cut, so $e = 0$.
• Position (iii): the same conductor is now moving down across the lines of force — the induced emf and current are directed opposite to those in position (i).
• Position (iv): the graph of the current in the lamp against time is a sinusoid.

Expression for the generated emf

• Let the coil of $N$ turns and area $A$ rotate with constant angular velocity $\omega$ in a uniform field $\vec{B}$. The angle between $\vec{B}$ and the area vector $\vec{A}$ of the coil at any instant $t$ is $\theta = \omega t$ (taking $\theta = 0$ at $t = 0$).
• Because the effective area of the coil normal to the field keeps changing as it rotates, the flux at any time is

$$\phi_B = \vec{B}\cdot\vec{A} = BA\cos\theta = BA\cos\omega t$$

• From Faraday's law, the induced emf in a coil of $N$ turns is

$$e = -N\dfrac{d\phi_B}{dt} = -N\dfrac{d}{dt}\left(BA\cos\omega t\right) = NBA\omega\sin\omega t$$

• Writing $e_0 = NBA\omega$ for the value when $\sin\omega t = 1$,

$$e = e_0\sin\omega t = e_0\sin 2\pi f t$$

• where $f$ is the frequency of revolution of the coil, and

$$e_0 = NBA\omega = 2\pi f NBA$$

Alternating current and pulsating DC

• Since $\sin\omega t$ varies between $+1$ and $-1$, the polarity of the emf changes with time. The emf has its extremum values at $\theta = 90^\circ$ and $\theta = 270^\circ$, because the change in flux is greatest at those positions.
• As the direction of the induced current changes periodically, it is called alternating current (AC). The frequency of AC is the number of times per second the current goes from positive to negative and back again. Domestic electrical current in India varies at a frequency of 50 cycles per second.
• For charging a storage battery a steady or direct current is needed. The reversing action of a split-ring commutator can be used to generate pulsating DC: the commutator acts like a rapid switch which reverses the connections to the armature at just the right times to match the reversals in the current, so the output never goes negative.
• Modern AC motors are more compact and rugged than DC motors.

Standard results

• $\phi_B = BA\cos\omega t$; $e = NBA\omega\sin\omega t = e_0\sin\omega t$; $e_0 = NBA\omega = 2\pi f NBA$.
• Emf is zero when the plane of the coil is perpendicular to $\vec{B}$ (flux maximum) and maximum when the plane of the coil contains $\vec{B}$ (flux zero, rate of change maximum).
• Semicircular wire of radius $R$ rotated about the axis through its ends at frequency $f$: area $\dfrac{\pi R^2}{2}$, so $e_0 = B\left(\dfrac{\pi R^2}{2}\right)(2\pi f) = \pi^2BR^2f$.
• Convert r.p.m. to hertz before using $\omega = 2\pi f$: 6000 r.p.m. is 100 Hz.