Digital Signal Processing Reference
In-Depth Information
The voltage is evaluated with equation (2-58), where
dl
=
x(a)
−
x(b)
=
d
,
which is the distance between the plates:
b
Q
εA
E
dl
v
=
·
=
d
a
Therefore, the capacitance between two parallel plates is
Q
v
=
Q
(Q/εA) d
=
εA
d
C
=
farads
where
A
is the area of the parallel plates,
v
the voltage, and
d
the distance
between them.
2.4.4 Energy Stored in a Capacitor
The process of storing energy in a capacitor involves electric charges of equal
magnitude, but opposite polarity, building up on each plate. As long as the
capacitor holds a charge, it is storing energy. To calculate how much energy is
stored in a capacitor, consider how much energy it would take to transport a
single charge from the positive plate to the negative plate. From equation (2-58)
we know that voltage is the work done to move a charge from point
a
to
b
,
W
q
a
→
b
=
v
and capacitance is defined as
Q
v
C
≡
farads
(2-76)
Therefore, the amount of work needed to move one charge
q
of the total charge
Q
from plate
a
to plate
b
is
q
C
dW
=
vdq
=
dq
To calculate the total work done to charge up the capacitor to a value of
Q
, all
charges must be moved:
Q
Q
2
C
q
C
1
2
W
=
dq
=
0
However, from (2-69),
Q
=
Cv
. Therefore, the energy stored in a capacitor with
a final potential of
v
is
1
2
Cv
2
W
=
joules
(2-77)
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