6.10 AC Generator

The phenomenon of electromagnetic induction has been technologically exploited in many ways. An exceptionally important application is the generation of alternating currents (ac). The modern ac generator with a typical output capacity of 100 MW is a highly evolved machine. In this section, we shall describe the basic principles behind this machine. The Yugoslav inventor Nicola Tesla is credited with the development of the machine. As was pointed out in Section 6.3, one method to induce an emf or current in a loop is through a change in the loop’s orientation or a change in its effective area.

As the coil rotates in a magnetic field $\mathbf{B}$, the effective area of the loop (the face perpendicular to the field) is $Acos\theta$ , where $\theta$ is the angle between $\mathbf{A}$ and $\mathbf{B}$. This method of producing a flux change is the principle of operation of a simple ac generator. An ac generator converts mechanical energy into electrical energy.

The basic elements of an ac generator are shown in Fig. 6.16. It consists of a coil mounted on a rotor shaft. The axis of rotation of the coil is perpendicular to the direction of the magnetic field. The coil (called armature) is mechanically rotated in the uniform magnetic field by some external means. The rotation of the coil causes the magnetic flux through it to change, so an emf is induced in the coil. The ends of the coil are connected to an external circuit by means of slip rings and brushes.

When the coil is rotated with a constant angular speed $\omega$ , the angle $\theta$ between the magnetic field vector B and the area vector A of the coil at any instant t is $\theta =\omega t$ (assuming θ = 0° at t = 0). As a result, the effective area of the coil exposed to the magnetic field lines changes with time, and from Eq. (6.1), the flux at any time t is

$${\mathrm{\Phi }}_{\mathrm{B}}=BA\cos \theta =BA\cos \omega t$$

From Faraday’s law, the induced emf for the rotating coil of N turns is then,

$$\epsilon =-N\frac{\mathrm{d}{\mathrm{\Phi }}_B}{\mathrm{dt}}=-\mathrm{NBA}\frac{\mathrm{d}}{\mathrm{d}t}(\cos \omega t)$$

Thus, the instantaneous value of the emf is

$$\epsilon =NBA\omega \sin \omega t$$

where $NBA\omega$ is the maximum value of the emf, which occurs when $sin\omega t=\pm 1$ . If we denote $NBA\omega$ as ${\epsilon }_0$ , then

$$\epsilon ={\epsilon }_0\sin \omega t$$

Since the value of the sine fuction varies between +1 and –1, the sign, or polarity of the emf changes with time. Note from Fig. 6.17 that the emf has its extremum value when $\theta =90°$ or $\theta =270°$, as the change of flux is greatest at these points.

The direction of the current changes periodically and therefore the current is called alternating current (ac). Since $\omega =2\pi \nu$ , Eq (6.22) can be written as

$$\epsilon ={\epsilon }_0\sin 2\pi vt$$

where ν is the frequency of revolution of the generator’s coil.

Note that Eq. (6.22) and (6.23) give the instantaneous value of the emf and $\epsilon$ varies between $+{\epsilon }_0$ and $–{\epsilon }_0$ periodically. We shall learn how to determine the time-averaged value for the alternating voltage and current in the next chapter.

In commercial generators, the mechanical energy required for rotation of the armature is provided by water falling from a height, for example, from dams. These are called hydro-electric generators. Alternatively, water is heated to produce steam using coal or other sources. The steam at high pressure produces the rotation of the armature. These are called thermal generators. Instead of coal, if a nuclear fuel is used, we get nuclear power generators. Modern day generators produce electric power as high as 500 MW, i.e., one can light up 5 million 100 W bulbs! In most generators, the coils are held stationary and it is the electromagnets which are rotated. The frequency of rotation is 50 Hz in India. In certain countries such as USA, it is 60 Hz.