Biomedical Engineering Reference
In-Depth Information
it follows that (
4.45
) can be re-written as
∂
2
P
∂x
2
−
γ
2
P
=
0
and
(4.47)
∂
2
Q
∂x
2
−
γ
2
Q
=
0
to which the solution is given by
Ae
−
γx
Be
γx
P(x)
=
+
and
(4.48)
Ce
−
γx
De
γx
=
+
Q(x)
with complex coefficients
A, B, C, D
;using(
4.48
) in the first two relations from
(
4.42
), the system can be reduced to
Z
0
Ae
−
γx
Be
+
γx
,
1
Q(x)
=
−
with
(4.49)
r
x
+
Z
l
Z
t
jωl
x
Z
0
=
jωc
x
=
(4.50)
g
x
+
in which
Z
0
is the characteristic impedance of the transmission line cell.
Using the trigonometric relations
e
γx
−
e
−
γx
2
sinh
(γ x)
=
(4.51)
e
γx
+
e
−
γx
2
cosh
(γ x)
=
we can write the relationship between the input
x
=−
and the output
x
=
0as
cosh
(γ )
Z
0
sinh
(γ )
P
1
Q
1
=
=
P
2
Q
2
(4.52)
1
Z
0
sinh
(γ )
cosh
(γ )
with
r
x
+
Z
l
Z
t
jωl
x
Z
0
=
jωc
x
=
(4.53)
g
x
+
the characteristic impedance and
Z
l
=
r
x
+
jωl
x
=
γZ
0
(4.54)
the longitudinal impedance, respectively,
Z
t
=
1
/(g
x
+
jωc
x
)
=
Z
0
/γ
(4.55)
the transversal impedance.
