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528 ⏐⏐⏐ SINUSOIDAL ALTERNATING WAVEFORMS
i
t
Sine wave
R, L, or C
FIG. 13.11
The sine wave is the only alternating
waveform whose shape is not altered by the
response characteristics of a pure resistor,
inductor, or capacitor.
E m
0° 90° 180° 270° 360°
Cosine wave
E m
0° 90° 180° 270° 360°
FIG. 13.12
Sine wave and cosine wave with the
horizontal axis in degrees.
57.296°
r
v
+
–
r
FIG. 13.13
Defining the radian.
1 radian
t
α
α
The need for defining polarities and current direction will become quite
obvious when we consider multisource ac networks. Note in the last
sentence the absence of the term sinusoidal before the phrase ac networks.
This phrase will be used to an increasing degree as we progress;
sinusoidal is to be understood unless otherwise indicated.
13.3 THE SINE WAVE
The terms defined in the previous section can be applied to any type of
periodic waveform, whether smooth or discontinuous. The sinusoidal
waveform is of particular importance, however, since it lends itself
readily to the mathematics and the physical phenomena associated with
electric circuits. Consider the power of the following statement:
The sinusoidal waveform is the only alternating waveform whose
shape is unaffected by the response characteristics of R, L, and C
elements.
In other words, if the voltage across (or current through) a resistor,
coil, or capacitor is sinusoidal in nature, the resulting current (or voltage,
respectively) for each will also have sinusoidal characteristics, as
shown in Fig. 13.11. If a square wave or a triangular wave were
applied, such would not be the case.
The unit of measurement for the horizontal axis of Fig. 13.12 is the
degree. A second unit of measurement frequently used is the radian
(rad). It is defined by a quadrant of a circle such as in Fig. 13.13 where
the distance subtended on the circumference equals the radius of the
circle.
If we define x as the number of intervals of r (the radius) around the
circumference of the circle, then
and we find
π radians
(3.14 radians)
C � 2pr � x ⋅ r
x � 2p
Therefore, there are 2p rad around a 360° circle, as shown in Fig.
13.14, and
2p rad � 360°
3
2
1 radian
4
5 6
0.28
2π π radians
(6.28 radians)
FIG. 13.14
There are 2p radians in one full circle of 360°.
(13.4)