Biomedical Engineering Reference
In-Depth Information
If we calculate the total admittance until
m
=
1, we have
1
/Zl
1
(s)
Y
2
(s)
=
(5.13)
Zt
1
(s)/Zl
1
(s)
1
+
Zt
1
(s)/Zl
2
(s)
1
+
1
+
U
2
(s)
I
1
(s)Zl
2
(s)
or, equivalently,
1
/Zl
1
(s)
Y
2
(s)
=
(5.14)
Zt
1
(s)/Zl
1
(s)
1
+
Zt
1
(s)/Zl
2
(s)
1
+
Zt
2
(s)/Zl
2
(s)
1
1
+
Zt
2
(s)
I
2
(s)
U
2
(s)
+
From (
5.12
)-(
5.14
) one may generalize via recurrence the form of the total admit-
tance with
m
=
N
cells, for
N
→∞
:
1
/Zl
1
(s)
Y
N
(s)
=
(5.15)
Zt
1
(s)/Zl
1
(s)
1
+
Zt
1
(s)/Zl
2
(s)
1
+
Zt
2
(s)/Zl
2
(s)
1
+
...
...
Zt
N
−
1
(s)/Zl
N
(s)
1
1
+
+
Zt
N
(s)/Zl
N
(s)
which is, in fact, a continued fraction expansion [
118
]. Re-writing (
5.15
)usingthe
explicit form of the longitudinal and transversal impedances gives
1
/(R
e
1
+
L
e
1
s)
Y
N
(s)
=
(5.16)
1
/
[
C
e
1
s(R
e
1
+
L
e
1
s)
]
1
+
1
/
[
C
e
1
s(R
e
2
+
L
e
2
s)
]
1
+
1
/
[
C
e
2
s(R
e
2
+
L
e
2
s)
]
1
+
1
/
[
C
e
2
s(R
e
3
+
L
e
3
s)
]
1
+
...
...
1
/
[
C
e(N
−
1
)
s(R
eN
+
L
eN
s)
]
1
1
+
+
1
/
[
C
eN
s(R
eN
+
L
eN
s)
]
which, in terms of the recursive ratios from (
5.4
) can be re-written as
1
/(R
e
1
+
L
e
1
s)
Y
N
(s)
=
(5.17)
1
/
[
C
e
1
s(R
e
1
+
L
e
1
s)
]
1
+
L
e
1
s
α
1
/
[
C
e
1
s(λR
e
1
+
)
]
1
+
L
e
1
s
α
1
/
[
χC
e
1
s(λR
e
1
+
)
]
1
+
L
e
1
s
α
2
)
]
1
/
[
χC
e
1
s(λ
2
R
e
1
+
1
+
...
...
L
e
1
s
α
N
−
1
)
χ
N
−
2
C
e
1
s(λ
N
−
1
R
e
1
+
1
/
[
]
1
+
L
e
1
s
α
N
−
1
)
]
1
+
1
/
[
χ
N
−
1
C
e
1
s(λ
N
−
1
R
e
1
+
For the set of conditions:
1
R
e
1
·
R
e
1
L
e
1
L
e
1
R
e
1
|
s
|
<
and
|
s
|
(5.18)
C
e
1
and
α
·
χ>
1
,α
·
λ>
1
,λ>
1 and
χ>
1
,
(5.19)
