Chapter 25

Methods of adding

alternating waveforms

25.1 Combination of two periodic functions

There are a number of instances in engineering and sci-
ence where waveforms have to be combined and where
it is required to determine the single phasor (called
the resultant) that could replace two or more separate
phasors. Uses are found in electrical alternating cur-
rent theory, in mechanical vibrations, in the addition of
forces and with sound waves.
There are a number of methods of determining the
resultant waveform. These include:

(a) by drawing the waveforms and adding graphically

(b) by drawing the phasors and measuring the

resultant

(c) by using the cosine and sine rules

(d) by using horizontal and vertical components

(e) by using complex numbers

25.2 Plotting periodic functions

This may be achieved by sketching the separate func-
tions on the same axes and then adding (or subtracting)
ordinates at regular intervals. This is demonstrated in
the following worked problems.

Problem 1. Plot the graph ofy 1 =3sinAfrom

A= 0 ◦toA= 360 ◦. On the same axes plot

y 2 =2cosA. By adding ordinates, plot

yR=3sinA+2cosAand obtain a sinusoidal

expression for this resultant waveform.

y 1 =3sinA and y 2 =2cosA are shown plotted

in Fig. 25.1. Ordinates may be added at, say, 15◦

intervals. For example,

at 0◦,y 1 +y 2 = 0 + 2 = 2

at 15◦,y 1 +y 2 = 0. 78 + 1. 93 = 2. 71

at 120◦,y 1 +y 2 = 2. 60 +− 1 = 1. 6

at 210◦, y 1 +y 2 =− 1. 50 − 1. 73 =− 3 .23, and

so on.

The resultant waveform, shown by the broken line,

has the same period, i.e. 360◦, and thus the same fre-

quency as the single phasors. The maximum value, or

y

y 15 3 sin A

y 25 2 cos A

yR 5 3.6 sin (A 1 34) 8

(^0) A
23
22
21
3
3.6
2
1
348
908 1808 2708 3608
Figure 25.1