Digital Signal Processing Reference
In-Depth Information
capacitance is simply the series combination of the capacitance between equal
potential lines as shown in Figure 3-24a. If the equal potential lines are chosen
so that each capacitance is the same and
n
s
denotes the number of elements in
series, the total capacitance is
C
0
n
s
C
=
(3-92)
Furthermore, each of the series capacitors in Figure 3-24a can be subdivided
into parallel capacitance values with a value of
C
for each cell, as shown
in Figure 3-24b. Assuming
n
p
parallel elements,
C
0
=
n
p
C
, yielding a total
capacitance
n
p
n
s
C
C
=
(3-93)
In order to use (3-93), we need to calculate the cell capacitance
C
. Assuming
that the charge
q
and
−
q
are present on the top and bottom of each cell wall,
we can write the cell capacitance in terms of the potential difference between the
boundaries using (2-76) and (3-1):
q
C
=
(3-94)
E
·
dl
If the problem is simplified by drawing the field lines such that each cell is
approximately the same size, the capacitance can be written in terms of the
average electric fields,
q
E
ave
h
ave
C
=
(3-95)
where
h
ave
is the average height of a cell and
E
ave
is the average electric field
for a cell.
Equation (3-95) can be simplified to eliminate the electric field using the
integral form of Gauss's law, (2-59),
S
ε E
·
ds
=
S
D
·
ds
=
q
:
q
E
ave
h
ave
εq
D
ave
h
ave
εq
(q/ lw
ave
)h
ave
εw
ave
h
ave
C
=
=
=
=
l
(3-96)
where
w
ave
is the average cell width and
ds
=
lw
ave
is the area of the cell
surface for the transmission line length
l
. If the cells can be drawn so that
w
ave
≈
h
ave
, the total capacitance is calculated by substituting (3-96) into
(3-93):
C
l
=
ε
n
p
F
/
m
(3-97)
n
s
Example 3-2
Use field mapping techniques to calculate the impedance of a coax-
ial transmission line shown in Figure 3-25, where
b/a
=
2 and the permittivity
of the dielectric is
ε
r
=
2
.
3.
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