Biomedical Engineering Reference
In-Depth Information
Fig. 5.1
Recursive tree representation of the respiratory tree in its electrical equivalent
Fig. 5.2
Equivalent ladder network for the symmetric recursive tree
Ta b l e 5 . 1
Ratios of the mechanical parameters between consecutive levels. Values are presented
as mean
±
standard deviation values for the 1-15 levels, respectively, for the 16-24 levels
Nominal
Levels 1-15
Levels 16-24
λ
0
.
81
±
0
.
32
0
.
68
±
0
.
16
1
/α
0
.
56
±
0
.
17
0
.
56
±
0
.
08
χ
1
.
71
±
0
.
77
1
.
55
±
0
.
29
Zl
m
/
2
m
−
1
,asinFig.
5.2
. In the same figure, the capaci-
tance is denoted using (
5.3
).
We introduce the following notations for the ratios between the levels:
R
em
+
1
R
em
representation
Zl
m
(s)
=
L
em
+
1
L
em
1
α
,
C
em
+
1
C
em
=
λ,
=
=
χ
(5.4)
with the ratios including both morphological and geometrical properties, as in
Fig.
5.2
. Hence, using the morphological values from Table
2.1
in (
5.1
), (
5.2
), and
(
5.3
), the 'nominal' ratios from (
5.4
) are calculated and given in Table
5.1
.
The total input impedance
Z
N
(s)
of the ladder network from Fig.
5.2
can be writ-
ten as a continuous fraction expansion [
118
]. For the sake of mathematical clarity,
we shall derive the analysis in terms of the admittance, which is the inverse of the
impedance
Y
N
(s)
=
1
/Z
N
(s)
.
