Although direct-current systems and calculations are still indispensable to the electrical engineer, virtually all public supplies are now alternating-current mains. The reasons for the changeover from DC to AC supplies will be considered in Section 10.2, our purpose here being to indicate how the two systems differ. The easiest method of portraying an alternating quantity is to draw a graph showing how it varies with time, as in Figure 10.1. Any part of the graph which lies above the horizontal (or zero) axis represents current or voltage in one direction, and values below it represent current or voltage in the other direction. The pattern given by the graph is known as the waveform of the AC system, and this usually repeats itself. There is no need for the waveform above the zero axis to have the same shape as that below it although, in most AC systems derived from mains supplies, this is the case.
An alternating current is thus one which rises in one direction to a maximum value, before falling to zero and repeating in the opposite direction. Instead of drifting steadily in one direction, the electrons forming the current move backwards and forwards in the conductor.
The time taken for an alternating quantity to complete its pattern (to flow in both directions and then return to zero) is called the periodic time (symbol T) for the
positive peak + i 0 T negative peak 5 10 15 20 25 30 time, ms positive halfcycle – negative halfcycle zero current
system, which is said to complete one cycle in this time. The complete cycle is split into the positive half-cycle above the axis, and the negative half-cycle below it.
The number of complete cycles traced out in a given time is called the frequency (symbol f ), usually expressed in hertz (Hz), which are cycles per second (c/s). If there are f cycles in one second, each cycle takes 1/f seconds, so that
T = 1
f and f =
1
T
Example 10.1
Calculate the frequency of the AC system shown in Figure 10.1. From Figure 10.1, T = 20 ms = 0.02 s, therefore
f = 1 T
= 0.021 hertz = 50 Hz
A frequency of 50 Hz is the standard for the supply system in many parts of the world, including the UK, but 60 Hz systems are also common for mains supplies.
10.2 Advantages of AC systems
There are certain complications which occur when using AC supplies, which are absent with DC supplies; these complications are explained later in this chapter. However, the advantages of AC supplies have led to their general use, some of the more important being as follows.
(a) An alternating-current generator (often called an alternator) is more robust, less expensive, requires less maintenance, and can deliver higher voltages than its DC counterpart.
(b) The power loss in a transmission line depends on the square of the current carried (P = I2R). If the voltage used is increased, the current is decreased, and losses
can be made very small. The simplest way of stepping up the voltage at the sending end of a line, and stepping it down again at the receiving end, is to use transformers, which will only operate efficiently from AC supplies.
(c) Three-phase AC induction motors are cheap, robust and easily maintained. (d) Energy meters, to record the amount of electrical energy used, are much simpler
for AC supplies than for DC supplies.
(e) Discharge lamps (fluorescent, sodium, mercury vapour etc.) operate more effi- ciently from AC supplies, although filament lamps are equally effective on either type of supply.
(f) Direct-current systems are subject to severe corrosion, which is hardly present with AC supplies.
Basic alternating-current theory 177
10.3 Values for AC supplies
The alternating current or voltage changes continuously, so that it is not possible to state its value in the same simple terms that can be used for a direct current.
Instantaneous values are the values at particular instants of time, and will be
different for different instants. Symbols for instantaneous values are small symbols,
v for voltage, i for current and so on.
Maximum or peak values are the greatest values reached during alternation,
usually occurring once in each half-cycle. Maximum values are indicated by Umfor voltage, Imfor current and so on.
Average or mean value is the average value of the current or voltage. If an average
value is found over a full cycle, the positive and negative half-cycles will cancel out to give a zero result if they are identical. In such cases it is customary to take the average value over a half-cycle. The average value of this kind of waveform can be found as shown in Example 10.2. Symbols used are Uav for voltage Iavfor current and so on.
Example 10.2
Table 10.1 gives the waveform of a half-cycle of alternating voltage. Find the fre- quency of the supply, its instantaneous values after 1.8 ms and 2.4 ms, the maximum value and the mean value of voltage.
f = T1
=8 ms1
= 1
0.008hertz = 125 Hz
The next step in the solution is to draw the half-cycle as a graph (Figure 10.2), reading off the instantaneous values (195 V at 1.8 ms, 287 V at 2.4 ms) and its maximum value (300 V).
To find the average or mean value, the base line (time axis) is divided into any number of equal parts. For clarity, eight parts have been chosen, although more would
Table 10.1 Waveform for Example 10.2
Time, ms 0 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0
Volts, V 0 45 72 91 104 118 142 185 240
Time, ms 2.25 2.5 2.75 3.0 3.25 3.5 3.75 4.0 Volts, V 278 295 300 280 248 195 85 0
300 250 200 150 v 100 50 0 0 1 2 3 maximum value = 300 V at 2·4 ms, v = 287 V at 1·8 ms, v = 195 V RMS value mean value 4 time, ms
Figure 10.2 Graph for Examples 10.2–10.4
give greater accuracy. At the centre of each part, a broken line has been drawn up to the curve.
The average value of voltage will be the average length of these lines (expressed in volts). To find this, we add the voltage represented by each line and divide by the number of lines. Uav= 45 + 91 + 118 + 185 + 278 + 300 + 248 + 858 volts = 1350 8 volts = 169 V
Effective value
Since the heat dissipated by a current is proportional to its square (P = I2R), the
average value of an alternating current is not the same as the direct current which produces the same heat or does the same work in the same time. The equivalent to a direct current is the value we use most in describing and calculating AC systems, and is called the effective or root mean square (RMS) value of the system. The RMS value is the square root of the average value of the squares of the instantaneous values. The symbols used for RMS values are U, I and so on. The method of find- ing the effective or RMS value of a given waveform is illustrated in the following example.
Basic alternating-current theory 179
Example 10.3
Find the RMS value of the voltage waveform of Example 10.2 To find the RMS value
(a) divide the base into equal parts and erect a vertical line to the curve from the centre of each part (as for finding the average value)
(b) square the value of each vertical line
(c) take the mean of the squared values (add them and divide by the number of lines)
(d) take the square root of the result – this is the root of the mean of the squared value. The graph has already been drawn and vertical lines have been erected for Example 10.2, and need not be repeated in this case. The sum of the squared values will be
452+ 912+ 1182+ 1852+ 2782+ 3002+ 2482+ 852= 294 468 V2 mean of the square values = 294 468
8 = 36 809 V2 root mean square value =√36 809 V = 191.9 V
It will be seen that the RMS value is greater than the mean value, and this is always the case, except for a direct current, for which they are equal.
Form factor for a particular waveform is the ratio of the RMS and mean values:
form factor = mean valueRMS value
Example 10.4
Find the form factor of the waveform of Example 10.2 form factor = mean valueRMS value
=191.9 V 169 V = 1.136
Form factor is an indication of the shape of a waveform; the higher its value the more ‘peaky’ the waveshape.
10.4 Sinusoidal waveforms
In Chapter 9, we considered a simple rectangular loop of wire rotating on an axis between the poles of a permanent magnet (Figures 9.5 and 9.6). The EMF induced in the loop is shown again in Figure 10.3, one cycle being induced for each revolution