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534 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
4
0
e
0° 30° 90°
e
v (V)
�
1
t1 10
10
10
90°
FIG. 13.19
Example 13.9.
180° 270° 360°
— 3
π
π 2 2π
α α (°)
0
— π — π
α (rad)
6 2
10
10
�2
t2 180° �
FIG. 13.20
Example 13.10, horizontal axis in degrees.
FIG. 13.21
Example 13.10, horizontal axis in radians.
EXAMPLE 13.9
a. Determine the angle at which the magnitude of the sinusoidal function
v � 10 sin 377t is 4 V.
b. Determine the time at which the magnitude is attained.
Solutions:
a. Eq. (13.15):
a 1 � sin �1
v
� Em
� sin �1
4 V
� 10 V
� sin �1 0.4 � 23.578°
However, Figure 13.19 reveals that the magnitude of 4 V (positive)
will be attained at two points between 0° and 180°. The second
intersection is determined by
a2 � 180° � 23.578° � 156.422°
In general, therefore, keep in mind that Equations (13.15) and
(13.16) will provide an angle with a magnitude between 0° and 90°.
b. Eq. (13.10): a � qt, and so t � a/q. However, a must be in radians.
Thus,
p
a (rad) � �(23.578°) � 0.411 rad
180°
a 0.411 rad
and t1 � � ��� �1.09 ms
q 377 rad/s
For the second intersection,
p
a (rad) � �(156.422°) � 2.73 rad
180°
a 2.73 rad
t2 � � ���7.24 ms
q 377 rad/s
The sine wave can also be plotted against time on the horizontal
axis. The time period for each interval can be determined from t � a/q,
but the most direct route is simply to find the period T from T � 1/f and
break it up into the required intervals. This latter technique will be
demonstrated in Example 13.10.
Before reviewing the example, take special note of the relative simplicity
of the mathematical equation that can represent a sinusoidal
waveform. Any alternating waveform whose characteristics differ from
those of the sine wave cannot be represented by a single term, but may
require two, four, six, or perhaps an infinite number of terms to be represented
accurately. Additional description of nonsinusoidal waveforms
can be found in Chapter 25.
EXAMPLE 13.10 Sketch e � 10 sin 314t with the abscissa
a. angle (a) in degrees.
b. angle (a) in radians.
c. time (t) in seconds.
Solutions:
a. See Fig 13.20. (Note that no calculations are required.)
b. See Fig. 13.21. (Once the relationship between degrees and radians
is understood, there is again no need for calculations.)