4.1 ALTERNATING QUANTITIES
An alternating current or voltage is
‘periodic’ - that is to say, it repeats itself exactly after each period or
‘cycle’. Ideally an alternating quantity
can be represented by a pure sine-wave (see Figure 4.1(a)). In practice such waves are seldom quite pure
and suffer some distortion. With
rectifier equipments, which are dealt with later in the manual ‘Fundamentals of
Electricity 3’, the distortion can be considerable (see Figure 4.1(b)). But with this exception most alternating
currents and voltages are reasonably pure, and for the purposes of this and the
following paragraphs they will be assumed to have a pure sine-wave form.
FIGURE 4.1
ALTERNATING QUANTITIES
The vertical distance from the
centreline to the peaks on either side is called the ‘amplitude’. Sometimes the expression ‘peak-to-peak value’
may be seen; this is double the amplitude and is a term which should be avoided,
as it could lead to confusion.
If an alternating quantity repeats
itself f times per second, f is called the ‘frequency’ of the
quantity and is measured in ‘hertz’ (Hz) (formerly in ‘cycles per second’ or
c/s). The time of one period or ‘cycle’
is then 1/f seconds. For example, if the frequency of an
electrical system is 50Hz, then the time for one cycle is one-fiftieth of a
second.
4.2 SIMPLE HARMONIC MOTION
An alternating quantity can be
developed as shown in Figure 4.2. Here
there is a bar OP of length ‘A’ rotating with a constant angular velocity about
O.
FIGURE 4.2
SIMPLE HARMONIC MOTION
If the angular displacement q (radians) of the bar OP from its starting horizontal position at
t = 0 occupies a time t
(seconds), then the angular velocity of OP is:
If the point N on the vertical line
through O is the projection (or ‘shadow’) of P, then N moves vertically up and
down about O in what is called ‘simple harmonic motion’, and the length ON is
in fact a pure alternating quantity. Its
maximum length is equal to OP either side of O; this is its amplitude A.
The value of ON at any instant is OP sin q,
or ON = A sin q ….
(i)
The point N will complete one full
cycle in the time it takes for P to complete one full circle: that is, in the
time that the bar OP takes to rotate through 2p radians (360°) at its constant angular velocity of q /t
radians per second.
The time for
one complete cycle is therefore 
seconds. Put another way, the number of complete
cycles in one second is 
. But the number of
cycles per second has already been defined as the ‘frequency’ f.
seconds. Put another way, the number of complete
cycles in one second is . But the number of
cycles per second has already been defined as the ‘frequency’ f. 
The general expression for simple harmonic
motion can now be written from expression (i):
ON
= A sin 2pft (ii)
Suppose N were the pen of a chart
recorder, and the paper were moving from left to right at a steady speed. Then N would leave a trace-as shown in Figure
4.2 which is a pure sine-wave. It could
represent any alternating quantity of amplitude A and frequency f. The value of that quantity at any instant of
time is given by A sin 2pft.
Any alternating quantity can thus be
completely specified by its amplitude and its frequency, as in the expression
(ii) above. If the amplitude is used in
this manner, it represents the ‘peak’ value of the quantity.
The expression (ii) above now
contains variable quantities which can be measured by practical instruments:
t: the time in seconds
f: the frequency in hertz
A: the amplitude of the wave (volts, amperes,
etc.)
In electrical engineering practice
the actual amplitude A - that is, the peak value - is not what is
usually measured. It is however a
function of a particular value of the wave which can be measured, the
so-called ‘root mean square’ or ‘rms’ value.
This concept is discussed in Chapter 5.
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