Environmental Engineering Reference
In-Depth Information
v
i
p
Time
Figure A.7
Power in an inductive component
The instantaneous power is
==
cos
p iVI
ωω
t
sin
t
and through Equation (A.9),
2
p
=
(
ω
LI
)
sin
ω
t
cos
ω
t
=
(
ω
LI
)
2
2
sin
ω
t
cos
ω
t
=
(
ω
LI
)
2
sin
2
ω
t
(A.12)
where
I
is the RMS value of the current.
Equation (A.12) is plotted in Figure A.7. Here the power is of double frequency, as in the
case of the resistor, but it is wholly symmetrical with respect to the time axis. This means
that a perfect inductor (one that possesses no winding resistance) is not associated with a net
power transfer over the long run. During periods in the cycle when the power is positive, the
inductor absorbs energy from the network to which it is connected and stores it in its magnetic
fi eld. During periods when the power is negative, the stored energy is returned to the rest of
the system. It can be concluded that an inductor acts as a consumer or sink of electrical energy
during part of the cycle and a generator of energy during the remaining cycle.
The purpose of power systems is to transfer electrical power from generators to
consumers so that the latter could convert this energy usefully into heat, light, mechanical
or chemical power or use it to drive IT systems. The power associated with an inductor
serves none of these purposes. It consists of power oscillating between power system
components.
A.5.3 Capacitance
Capacitors oppose changes in voltage by drawing or supplying current as they charge or dis-
charge to the new voltage level (Figure A.8). The fl ow of electrons through a capacitor is
directly proportional to the rate of change of voltage across the capacitor. This opposition to
voltage change is a mirror image to the kind exhibited by inductors.
Ohm's law for a capacitance
C
in farads is the differential relationship
iC
v
t
d
d
=
(A.13)
If a sinusoidal voltage
=
sin
(A.14)
vV
ω
t
is applied to a capacitor, the resulting current from Equation (A.13) is
