Environmental Engineering Reference
In-Depth Information
1.0
i
I
0.5
v
V
0.0
ωt
a
R
b
0
5
1
5
2
2
3
-0.5
-1.0
Time
1.0
i
0.5
0.0
0
5
1
5
2
5
3
v
V
ωt
-0.5
a
b
C
-1.0
Time
1.0
V
v
ωt
0.5
0.0
0
5
0
1
2
2
3
-0.5
a
b
L
I
-1.0
i
Time
FIGURE 7.4
Current and voltage across a resistor, capacitor, and inductor, showing the phase relationship
between the voltage and current. Voltage, dashed line; current, solid line.
where
V
p
and
I
p
are the peak values;
q
, rad/s, is the angular velocity (which is 2
.
times the fre-
quency); and the angle,
b
, is the difference in degrees between the instantaneous voltage and current
(the sine wave for voltage and the sine wave for current). For a resistor the voltage and current are in
phase and the average power over one cycle is
P
V
I
V
p
sin (
qt
)
I
p
sin (
qt
) 0.5
V
p
I
p
(7.12)
For capacitors and inductors, the voltage and current are 90° out of phase and the average power
is zero:
P
V
p
sin (
qt
)
I
p
sin (
qt
90)
V
p
sin (
qt
)
I
p
cos (
qt
) 0
In all real circuits there is inductance, capacitance, and resistance, so the current and voltage will
not be completely in phase (
Figure 7.5
). The instantaneous power to an arbitrary AC circuit oscil-
lates because both the voltage and current oscillate:
p
vi
V
p
sin (
qt
)
I
p
sin (
qt
f
)
(7.13)
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