Digital Signal Processing Reference
In-Depth Information
By following a similar approach, we can also derive the equation for modal
capacitance:
d
dz
(
T
i
i
m
)
=−
jω
CT
v
v
m
(4-71)
d
i
m
dz
=−
jω
T
−
i
CT
v
v
m
(4-72)
T
−
i
CT
v
C
m
=
(4-73)
L
m
and
C
m
are diagonal,
L
m
11
0
0
0
0
0
L
m
22
0
0
0
L
m
=
0
0
···
0
0
(4-74)
0
0
0
L
m(n
−
1
)(n
−
1
)
0
0
0
0
0
L
mnn
C
m
11
0
0
0
0
0
C
m
22
0
0
0
C
m
=
0
0
···
0
0
(4-75)
0
0
0
C
m(n
−
1
)(n
−
1
)
0
0
0
0
0
C
mnn
where the diagonal elements are defined by (4-70) and (4-71).
Another useful relationship that we will find useful is [Paul, 1994]
=
(
T
−
v
)
T
T
i
(4-76)
To carry out the modal transmission line analysis, we need to put the modal
quantities into the wave equation, starting with (4-33):
d
2
T
v
v
m
dz
2
ω
2
T
v
L
m
T
−
i
T
i
C
m
T
−
v
T
v
v
m
ω
2
T
v
L
m
C
m
v
m
=
=
Multiplying through by
T
−
v
gives the wave equation expressed in terms of the
modal voltage, inductance, and capacitance:
d
2
v
m
dz
2
=
ω
2
L
m
C
m
v
m
(4-77)
Following the same process with (4-15) gives
d
2
i
m
dz
2
=
ω
2
C
m
L
m
i
m
(4-78)
Again, since the modal quantities are orthogonal, we can use them to simulate
the system as
n
isolated lines rather than
n
coupled lines.
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