Biomedical Engineering Reference
In-Depth Information
The relation for the longitudinal impedance in function of aerodynamic variables
is obtained from (
4.43
), and gives
μδ
2
πR
4
M
1
jωρ
πR
2
M
1
e
−
j(
2
−
ε
1
)
e
−
jε
1
Z
l
=
=
πR
4
M
1
sin
(ε
1
)
+
j
cos
(ε
1
)
μδ
2
=
(4.56)
respectively, in terms of transmission line parameters, the longitudinal impedance is
given by
Z
l
=
r
x
+
jωl
x
.
By equivalence of the two relations we find that the resistance per unit distance
is
μδ
2
πR
4
M
1
r
x
=
sin
(ε
1
)
(4.57)
R
ω
μ
μδ
2
πR
4
M
1
It follows that
ωl
x
=
cos
(ε
1
)
and recalling that
δ
=
, the inductance
per unit distance is
ρ
πR
2
cos
(ε
1
)
M
1
l
x
=
(4.58)
4.2.1 Elastic Tube Walls
In case of an
elastic pipeline
, the characteristic impedance is obtained using rela-
tions (
4.43
), (
4.44
), and (
4.50
), leading to
Eh
2
ρR
ρ
πR
2
1
1
√
M
1
e
−
j
ε
2
Z
0
=
(4.59)
−
ν
P
1
and for a lossless line (no air losses trough the airway walls, thus
conductance g
x
is
zero
), the transversal impedance is
Z
0
1
jωc
x
=
Eh
(j ω(
2
πR
3
(
1
Z
t
=
Z
l
=
(4.60)
−
ν
P
))
from where the capacity per unit distance can be extracted:
2
πR
3
(
1
−
ν
P
)
Eh
c
x
=
(4.61)
Thus, from the geometrical
(R, h)
and mechanical
(E, ν
P
)
characteristics of the
airway tube, and from the air properties
(μ, ρ)
one can express the
r
x
,
l
x
and
c
x
parameters. In this way, the dynamic model can be expressed in an equivalent loss-
less transmission line by Eqs. (
4.57
)-(
4.61
). Notice that the compliance parameter
