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20.2Generators and alternating current

The ac generator

One very important application of electromagnetic induction is the ac generator—the method used universally to produce electricity (Figure 20.38). A coil is made to rotate in a region of magnetic field. This can be accomplished in a variety of ways: by a diesel engine burning oil, by falling water in a hydroelectric power station, by wind power and so on. The ends of the coil are firmly attached to two slip rings that rotate along with the coil. The slip rings touch carbon brushes that transfer the current into an external circuit.

Figure 20.38: a A coil that is forced to turn in a region of magnetic field will produce an emf. b Generators at the Hoover hydroelectric power plant in the United States.

The flux in the coil changes as the coil rotates and so an emf is produced in it. We assume that the coil has N = 10 turns of wire around it, the magnetic field is B = 0.21 T, the coil has an area of 0.50 m2 and the coil rotates with frequency f = 50 revolutions per second. The flux linkage in the coil changes as time goes on according to a cosine function as shown in Figure 20.39.

Figure 20.39: The flux linkage in the coil is changing with time.

The red, white and blue bar that is superposed on the graph indicates the position of the coil as we look at it along the axis of rotation: at t = 0 for example the coil is vertical with the part painted red on top. The equation of the flux (linkage) is, in general:

Φ=N B A cos θ

where θ is the angle between the magnetic field and the normal to the coil and N is the number of turns in the coil.

The coil rotates with angular speed of rotation ω; it follows that θ = ω t.

But, since ω = 2πf, we also have θ = 2πf t. So the flux linkage becomes:

Φ=N B A cos(2 π f t)

By Faraday’s law, the emf induced in the coil is (minus) the rate of change of the flux linkage and is given by:

V=−d Φ d t

=2 π f N B A sin(2 π f t)

The important thing here is that V ∝ f.

The quantity V 0 = 2πf N B A is the peak voltage produced by the generator. The variation of the induced emf with time is given by the graph in Figure 20.40. The peak voltage in this example is 325 V.

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Figure 20.40: The emf induced in the loop as a function of time. The peak voltage is 325 V.

Note that the induced emf is zero whenever the flux assumes its maximum or minimum values and, conversely, it is a maximum or minimum whenever the flux is zero. The noteworthy thing here is that the voltage can be negative as well as positive. This is what is called alternating voltage voltage which varies between positive and negative values and the current that flows in the coil is alternating current (ac)current which varies between positive and negative values. This means that, unlike the ordinary direct current (dc) that flows in a circuit connected to a battery, the electrons do not drift in the same direction but oscillate back and forth with the same frequency as that of the voltage. The flux and the emf are out of phase by π 2 or 90°.

The current in a circuit of resistance R can be found from:

I=V R=V 0 sin(2 π f t)R=I 0 sin(2 π f t)

where I 0=V 0 R is the peak current. Notice that since V ∝ f it follows that also I ∝ f. For the emf of Figure 20.40 and a resistance of 16 Ω, the current is shown in Figure 20.41.

Figure 20.41: The induced current in the rotating loop. Note that the current is in phase with the emf. The peak current is found from peak voltage divided by resistance, i.e. 325 16, which is about 20 A.

CHECK YOURSELF 6

The average current produced by a generator is zero. How then can it produce a heating effect in a resistor?

Power in ac circuits

The power P generated in an ac circuit is given by:

P=V I

Because both the current I and voltage V vary with time, the expression for power becomes:

P=V 0 I 0 sin 2⁡(2 π f t)

This means that, just like the current and the voltage, power is not constant in time. It has a peak value P max given by the product of the peak voltage and peak current:

P max=V 0 I 0

The power as a function of time is shown in Figure 20.42. The average power dissipated is half the peak value.

Figure 20.42: The power dissipated in a resistor as a function of time. Note that the period of one rotation of the coil is 20 ms. The power becomes zero with every half rotation of the coil. The horizontal dotted line indicates the average power, which is half the peak value.

The relationship between voltage, current and power in an ac circuit is shown in Figure 20.43.

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Figure 20.43: Power (orange), voltage (red) and current (blue) in an ac circuit resistor.

Since P = V I, and V ∝ f as well as I ∝ f it follows that P ∝ f 2.

CHECK YOURSELF 7

The frequency of rotation of the coil of a generator is doubled. By what factor do a the peak voltage and b the peak power change?

Worked Example 20.5

The graph of Figure 20.44 shows the variation with time of the power delivered by an ac generator.

aState the frequency of rotation of the generator.

bOn a copy of the graph, sketch the graph of the power delivered when the frequency of rotation is halved.

Figure 20.44: The variation with time of the power delivered by an ac generator.

Answer

aThe period T is found by looking at two loops, i.e. it is 20 ms.

So the frequency is found from f=1 T

f=1 0.020=50 Hz

bSince P ∝ f 2, if you halve the frequency, the power is reduced by a factor of 4. So the peak of the graph will be at 4 W.

Changing the frequency also changes the period. If the frequency is halved, then the period is doubled to 40 m s.

So we get the graph in red shown in Figure 20.45.

Figure 20.45: The power delivered when the frequency of rotation is halved.

Exam tip

It is important to understand how to get the period from a graph of power against time. The interval between two consecutive times when the power is zero is half a period.

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Links

Maxwell’s intuition about the symmetry between electric and magnetic fields led to electromagnetic waves and a unification of electric and magnetic phenomena. Einstein carried this connection further. Imagine a charged particle that moves parallel to a straight wire that carries current; there will be a magnetic force on this particle. But what does an observer who travels along with the charged particle see? For this observer, the particle is at rest and so there can be no magnetic force. For this observer, the wire is moving and so suffers length contraction. A detailed analysis shows that as a result the density of positive charges in the wire increases and so there is now an electric force on the particle. So not only are electric and magnetic phenomena related, but what is called ‘electric’ and what is called ‘magnetic’ also depends on the frame of reference one uses to describe the situation.

Science in context

The discovery of EM waves and EM induction has brought endless applications and uses. A very small selection includes all modern communications with cell phones and similar devices as well as ordinary radio and television which involve EM waves; so do antitheft devices and countless other types of radio frequency identification devices (RFIDs, for example the transponder you use to raise the bar at toll stations on a highway); levitating trains; electric cars; and most of our electromechanical technology; we owe much to what we know about the early universe by studying the cosmic microwave background radiation (CMB); the exploitation of solar energy and the greenhouse effect that sustains life on earth are both based on understanding EM waves; X rays are used in medical imaging, radar and navigation, microwave ovens and so forth.

NATURE OF SCIENCE

Much of the electromechanical technology we use today is due to the discoveries made by Michael Faraday. In 1831, using very simple equipment, Faraday observed a tiny pulse of current in one coil of wire when the current in a second coil was switched on or off, but nothing while a constant current was flowing. In further experiments he found these transient currents when he slid a magnet quickly in and out of a coil of wire. Faraday explained this electromagnetic induction using the idea of lines of force but did not provide a mathematical relationship. The mathematical description of these phenomena was given much later by the Scottish physicist James Clerk Maxwell (1831–1879).

TEST YOUR UNDERSTANDING

22A coil is rotated in a magnetic field. The graphs show the variation of the flux with rotation angle and the variation of the induced emf with time, Figure 20.46.

Figure 20.46: For question 22.

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aDraw a graph to show the variation of the induced emf with angle.

The same coil is now rotated at double the speed in the same magnetic field. Draw graphs to show:

bthe variation of the flux with angle

cthe variation of the induced emf with angle

dthe variation of the induced emf with time.

23A transformer consists of two coils wrapped around an iron core (Figure 20.47a). When alternating current is established in the primary coil, a varying magnetic field is created in the primary coil. The iron core ensures that all the magnetic field lines stay within the core and so also go through the secondary coil.

Figure 20.47: For question 23.

aExplain why a current will be established in the secondary coil.

The input current in the primary coil varies with time, as shown in Figure 20.47b.

bDraw a sketch graph (no numbers required) to show how the current in the secondary coil varies with time.

24Figure 20.48 shows the variation with time of the power dissipated in a resistor when an alternating voltage from a generator is established at its ends.

Figure 20.48: For question 24.

aFind the period of rotation of the coil.

bThe coil is now rotated at double the speed. Draw a graph to show the variation with time of the power dissipated in the resistor.