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550 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
40
0
–40
1600
0
v (V)
v 2 (V)
10 20 t (ms)
1 cycle
FIG. 13.59
Example 13.23.
10
20
t (ms)
FIG. 13.60
The squared waveform of Fig. 13.59.
EXAMPLE 13.23 Determine the average and rms values of the square
wave of Fig. 13.59.
Solution: By inspection, the average value is zero.
v 2 (Fig. 13.60):
(1600)(10 � 10 �3 ) � (1600)(10 � 10 �3 )
�����
20 � 10 �3
V rms � ����
32,000 � 10
� � �1�6�0�0� �� �3
���3
20 � 10
Vrms � 40 V
(the maximum value of the waveform of Fig. 13.60)
The waveforms appearing in these examples are the same as those
used in the examples on the average value. It might prove interesting to
compare the rms and average values of these waveforms.
The rms values of sinusoidal quantities such as voltage or current
will be represented by E and I. These symbols are the same as those
used for dc voltages and currents. To avoid confusion, the peak value
of a waveform will always have a subscript m associated with it: I m
sin qt. Caution: When finding the rms value of the positive pulse of a
sine wave, note that the squared area is not simply (2A m) 2 � 4A 2 m;it
must be found by a completely new integration. This will always be
the case for any waveform that is not rectangular.
A unique situation arises if a waveform has both a dc and an ac component
that may be due to a source such as the one in Fig. 13.61. The
combination appears frequently in the analysis of electronic networks
where both dc and ac levels are present in the same system.
+
3 sin ωt
–
6 V
+
v T
–
7.5 V
6 V
4.5 V
v T
0 t
FIG. 13.61
Generation and display of a waveform having a dc and an ac component.
The question arises, What is the rms value of the voltage vT? One
might be tempted to simply assume that it is the sum of the rms values
of each component of the waveform; that is, VT � 0.7071(1.5 V) �
rms
6 V � 1.06 V � 6 V � 7.06 V. However, the rms value is actually
determined by
Vrms � �V� 2 �dc �� V� 2 ac� (rm� s) � (13.37)
which for the above example is
Vrms � �(6� V�) 2 � �� (�1�.0�6� V�) 2 �
� �3�7�.1�2�4� V
� 6.1 V