Biomedical Engineering Reference
In-Depth Information
harmonics, with the modulus dependent solely on the location within the transmis-
sion line (
x
).
φ
P
and
φ
Q
are the pressure and flow phase angles at
t
0. The voltage
difference between two points on the transmission line denoted (
x
) and (
x
+
dx
)is
due to losses over the resistance and inductance:
=
l
x
dx
∂q
∂t
+
−
=−
·
−
p(x
dx)
p(x)
r
x
dx
q
(4.40)
and the current difference between the same points is due to leakage losses and
storage in the capacitor:
c
x
dx
∂p
∂t
+
−
=−
·
−
q(x
dx)
q(x)
g
x
dx
p
(4.41)
After division with
dx
, knowing that in the limit
dx
−→
0, and introducing
(
4.39
) in the first and second derivation gives, respectively:
∂P
∂x
=−
(r
x
+
=−
jωl
x
)Q
Z
l
Q
∂Q
∂x
=−
(g
x
+
jωc
x
)P
=−
P/Z
t
∂
2
P
∂x
2
(4.42)
=−
(r
x
+
jωl
x
)
∂Q
∂x
=−
Z
l
∂Q
∂x
Z
t
∂
2
Q
∂x
2
=−
(g
x
+
jωc
x
)
∂P
∂P
∂x
∂x
=−
with
∂P
∂x
Z
l
=−
Q
=
r
x
+
jωl
x
(4.43)
the longitudinal impedance and
P
−
1
g
x
+
jωc
x
Z
t
=
=
(4.44)
∂Q
∂x
the transversal impedance.
From (
4.42
) we obtain the system equations for
P(x)
and
Q(x)
:
∂
2
P
∂x
2
−
Z
l
P/Z
t
=
0
(4.45)
∂
2
Q
∂x
2
−
Z
l
Q/Z
t
=
0
Introducing the notation
Z
l
Z
t
(r
x
+
γ
=
jωl
x
)(g
x
+
jωc
x
)
=
(4.46)
