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er of cycles shown. For a fixed time interval, the greater the angular
velocity, the greater the number of cycles generated.
Due to Eq. (13.10), the general format of a sine wave can also be
written
GENERAL FORMAT FOR THE SINUSOIDAL VOLTAGE OR CURRENT ⏐⏐⏐ 533
(13.14)
with qt as the horizontal unit of measure.
For electrical quantities such as current and voltage, the general format
is
i � Im sin qt � Im sin a
e � Em sin qt � Em sin a
where the capital letters with the subscript m represent the amplitude,
and the lowercase letters i and e represent the instantaneous value of
current or voltage, respectively, at any time t. This format is particularly
important since it presents the sinusoidal voltage or current as a function
of time, which is the horizontal scale for the oscilloscope. Recall
that the horizontal sensitivity of a scope is in time per division and not
degrees per centimeter.
EXAMPLE 13.8 Given e � 5 sin a, determine e at a � 40° and a �
0.8p .
Solution: For a � 40°,
e � 5 sin 40° � 5(0.6428) � 3.214 V
For a � 0.8p,
180°
a (°) � � (0.8p) � 144°
p
and e � 5 sin 144° � 5(0.5878) � 2.939 V
The conversion to degrees will not be required for most modern-day
scientific calculators since they can perform the function directly. First, be
sure that the calculator is in the RAD mode. Then simply enter the radian
measure and use the appropriate trigonometric key (sin, cos, tan, etc.).
The angle at which a particular voltage level is attained can be
determined by rearranging the equation
e � Em sin a
in the following manner:
which can be written
A m sin qt
sin a �
a � sin �1 e
�� E
Similarly, for a particular current level,
e
� Em
a � sin �1 i
�� I
The function sin �1 is available on all scientific calculators.
m
m
(13.15)
(13.16)