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540 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
Height
Height
Sand
Distance
(a)
Average height
Sand
Same
distance
(b)
FIG. 13.34
Effect of distance (length) on average value.
Height
Sand
Height
Distance
(a)
Average height
Sand
Same
distance
(b)
Ground level
FIG. 13.35
Effect of depressions (negative excursions) on
average value.
If the distance parameter includes a depression, as shown in Fig.
13.35(a), some of the sand will be used to fill the depression, resulting
in an even lower average value for the landscaper, as shown in Fig.
13.35(b). For a sinusoidal waveform, the depression would have the
same shape as the mound of sand (over one full cycle), resulting in an
average value at ground level (or zero volts for a sinusoidal voltage over
one full period).
After traveling a considerable distance by car, some drivers like to
calculate their average speed for the entire trip. This is usually done by
dividing the miles traveled by the hours required to drive that distance.
For example, if a person traveled 225 mi in 5 h, the average speed was
225 mi/5 h, or 45 mi/h. This same distance may have been traveled at
various speeds for various intervals of time, as shown in Fig. 13.36.
By finding the total area under the curve for the 5 h and then dividing
the area by 5 h (the total time for the trip), we obtain the same result
of 45 mi/h; that is,
area under curve
Average speed � �� (13.25)
length of curve
A1 � A2 Average speed ��
5 h
(60 mi/h)(2 h) � (50 mi/h)(2.5 h)
�����
5 h
225
� � mi/h
5
� 45 mi/h
Equation (13.25) can be extended to include any variable quantity, such
as current or voltage, if we let G denote the average value, as follows:
algebraic sum of areas
G (average value) � ��� (13.26)
length of curve
Speed (mi/h)
70
60
50
40
30
20
10
A1 A2 Average speed
0 1 2 3
Lunch break
4 5 6 t (h)
FIG. 13.36
Plotting speed versus time for an automobile excursion.
The algebraic sum of the areas must be determined, since some area
contributions will be from below the horizontal axis. Areas above the
axis will be assigned a positive sign, and those below, a negative sign.
A positive average value will then be above the axis, and a negative
value, below.
The average value of any current or voltage is the value indicated on
a dc meter. In other words, over a complete cycle, the average value is