CHAPTER 4 SIMPLE HARMONIC MOTION

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.

                                                                              

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.

It should be noted that the reason for the term ‘sine-wave’ being given to the waveform generated in this unique manner is simply that its value at any instant is a function of the displacement angle q and is, in fact, A sin q from equation (i).