Alternating Current Generators

An electric generator, or dynamo, is a device that converts mechanical energy into electrical energy. The simplest practical generator consists of a rectangular coil rotating in a uniform magnetic field. The magnetic field is usually supplied by a permanent magnet. This setup is illustrated in Figure 2.32.

Figure 2.32: An alternating current generator.

Let $l$ be the length of the coil along its axis of rotation, and $w$ the width of the coil perpendicular to this axis. Suppose that the coil rotates at constant angular velocity $\omega$ in a uniform magnetic field of strength $B$. The velocity $v$ with which the two long sides of the coil (i.e., sides $ab$ and $cd$) move through the magnetic field is simply the product of the angular velocity of rotation $\omega$ and the distance $w/2$ of each side from the axis of rotation, so $v = \omega\,w/2$. The motional emf induced in each side is given by $V = B_\perp\,l\,v$, where $B_\perp$ is the component of the magnetic field perpendicular to instantaneous direction of motion of the side in question. If the direction of the magnetic field subtends an angle $\theta$ with the normal direction to the coil, as shown in the figure, then $B_\perp = B\,\sin\theta$. Thus, the magnitude of the motional emf generated in sides $ab$ and $cd$ is

$$V_{ab} = \frac{B\, w\, l\, \omega\,\sin\theta}{2} = \frac{B\,A\, \omega\,\sin\theta}{2},$$ (2.402)

where $A=w\,l$ is the area of the coil. The emf is zero when $\theta = 0^\circ$ or $180^\circ$, because the direction of motion of sides $ab$ and $cd$ is parallel to the direction of the magnetic field in these cases. The emf attains its maximum value when $\theta = 90^\circ$ or $270^\circ$, because the direction of motion of sides $ab$ and $cd$ is perpendicular to the direction of the magnetic field in these cases. Incidentally, it is clear, from symmetry, that no net motional emf is generated in sides $bc$ and $da$ of the coil.

Suppose that the direction of rotation of the coil is such that side $ab$ is moving into the page in Figure 2.32 (side view), whereas side $cd$ is moving out of the page. The motional emf induced in side $ab$ acts from $a$ to $b$. Likewise, the motional emf induce in side $cd$ acts from $c$ to $d$. It can be seen that both emfs act in the clockwise direction around the coil. [The direction of the emf is the same as the direction of the electric field seen in the rest frame of the sides. See Equation (2.401).] Thus, the net emf $V$ acting around the coil is $2\,V_{ab}$. If the coil has $N$ turns then the net emf becomes $2\,N\,V_{ab}$. Hence, the general expression for the emf generated around a steadily-rotating, multi-turn coil in a uniform magnetic field is

$$V = N\,B\,A\,\omega\,\sin( \omega\, t),$$ (2.403)

where we have written $\theta = \omega \,t$ for a steadily rotating coil (assuming that $\theta=0$ at $t=0$). This expression can also be written

$$V = V_{\rm max}\,\sin (2\pi\, f\, t),$$ (2.404)

where

$$V_{\rm max}= 2\pi\,N\,B\,A\,f$$ (2.405)

is the peak emf produced by the generator, and $f=\omega/2\pi$ is the number of complete rotations the coils executes per second. Thus, the peak emf is directly proportional to the area of the coil, the number of turns in the coil, the rotation frequency of the coil, and the magnetic field-strength.

Figure 2.33 shows the emf specified in Equation (2.404) plotted as a function of time. It can be seen that the variation of the emf with time is sinusoidal in nature. The emf attains its peak values when the plane of the coil is parallel to the plane of the magnetic field, passes through zero when the plane of the coil is perpendicular to the magnetic field, and reverses sign every half period of revolution of the coil. The emf is periodic (i.e., it continually repeats the same pattern in time), with period $T= 1/f$ (which is, of course, the rotation period of the coil).

Figure 2.33: Emf generated by a steadily rotating AC generator.

Suppose that some electrical load (e.g., a light-bulb, or an electric heating element) of resistance $R$ is connected across the terminals of the generator. In practice, this is achieved by connecting the two ends of the coil to rotating rings that are then connected to the external circuit by means of metal brushes. According to Ohm's law, the current $I$ that flows in the load is given by

$$I = \frac{V}{R} = \frac{ V_{\rm max}}{R}\, \sin (2\pi\, f\, t).$$ (2.406)

(See Section 2.1.11.) Note that this current is constantly changing direction, just like the emf of the generator. Hence, the type of generator described previously is usually termed an alternating current, or $AC$, generator.

The current $I$ that flows through the load must also flow around the coil. Because the coil is situated in a magnetic field, this current gives rise to a torque acting on the coil which, as is easily demonstrated, acts to slow down its rotation. Suppose, as before, that side $ab$ is moving into the page in Figure 2.32 (side view), whereas side $cd$ is moving out of the page, and the current is circulating in a clockwise sense. Side $ab$ experiences a magnetic force per unit length $F_{ab}= {\bf I}\times {\bf B}= I\,B\,\sin\theta$ that acts to oppose its motion. (See Section 2.2.2.) Hence, the braking force acting on the side is $f_{ab}=F_{ab}\,l= I\,B\,\sin\theta\,l$. Thus, the braking torque acting on the side is $\tau_{ab}=f_{ab}\,w/2= I\,B\,\sin\theta\,l\,w/2= I\,B\,\sin\theta\,A/2$, where $A=l\,w$ is the area of the coil. Side $cd$ experiences an equal torque. So, taking into account the fact that the coils has $N$ turns, the net braking torque $\tau$ acting on the coil is given by

$$\tau = N\,I\,B\,A\,\sin\theta.$$ (2.407)

It follows from Equation (2.403) that

$$\tau = \frac{V\, I}{\omega},$$ (2.408)

because $V= N\,B\,A\,\omega\,\sin\theta$. An external torque that is equal and opposite to the breaking torque must be applied to the coil if it is to rotate uniformly, as was initially assumed above. The rate $P$ at which this external torque does work is equal to the product of the torque $\tau$ and the angular velocity $\omega$ of the coil. (See Section 1.7.4.) Thus,

$$P= \tau\,\omega = V\, I.$$ (2.409)

Not surprisingly, the rate at which the external torque performs works exactly matches the rate $V\,I$ at which electrical energy is generated in the circuit comprising the rotating coil and the load.

Equations (2.403), (2.406), and (2.408) yield

$$\tau = \tau_{\rm max}\,\sin^2(2\pi \,f\,t),$$ (2.410)

where $\tau_{\rm max} = (V_{\rm max})^2/(2\pi\,f\,R)$. Figure 2.34 shows the braking torque $\tau$ plotted as a function of time $t$, according to Equation (2.410). It can be seen that the torque is always of the same sign (i.e., it always acts in the same direction, so as to continually oppose the rotation of the coil), but is not constant in time. Instead, it pulsates periodically with period $T$. The braking torque attains its maximum value whenever the plane of the coil is parallel to the plane of the magnetic field, and is zero whenever the plane of the coil is perpendicular to the magnetic field. It is clear that the external torque needed to keep the coil rotating at a constant angular velocity must also pulsate in time with period $T$. A constant external torque would give rise to a non-uniformly rotating coil, and, hence, to an alternating emf that varies with time in a more complicated manner than $\sin(2\pi\, f\, t)$.

Figure 2.34: The braking torque in a steadily rotating AC generator.

Virtually all commercial power stations generate electricity using AC generators. The external power needed to turn the generating coil is usually supplied by a steam turbine (steam blasting against fan-like blades that are forced into rotation). Water is vaporized to produce high pressure steam by burning coal, or by using the energy released inside a nuclear reactor. Of course, in hydroelectric power stations, the power needed to turn the generator coil is supplied by a water turbine (which is similar to a steam turbine, except that falling water plays the role of the steam). More recently, a new type of power station has been developed in which the power needed to rotate the generating coil is supplied by a gas turbine (basically, a large jet engine that burns natural gas). In the U.S. and Canada, the alternating electrical signal generated by power stations and fed into ordinary households, which is known as mains electricity, oscillates at $f=60$ Hz, which implies that the generator coils in power stations rotate exactly sixty times a second. In Europe and Asia, the oscillation frequency of mains electricity is $f=50$ Hz.