Biomedical Engineering Reference
In-Depth Information
1
β
β
β
t
−
t
025
1
τ=
d
31
(9.20)
Vt
Vt
()
(
Tt
Tt
11
1 025
ln
)
025
025
025 1
Vt
1
(
1
)
K
=
(9.21)
31
β
t
d
1
t
T
−
τ
F
1
1
e
1
0
Hou et al. [4] then indicated that the ideal fitting result is to find a specific
value of β that makes
τ
d
= τ
d
12
= τ
d
23
= τ
d
31
and
K
=
K
12
=
K
23
=
K
31
Due to the presence of measurement errors, the fitting criterion is used to
find a β(0 < β < 1) which minimizes the error function
Error(
=τ −τ +τ −τ +τ −τ
)
d
12
d
23
d
23
d
31
d
31
d
12
(9.22)
× −+−+−
(
KK KK KK
)
12
23
23
31
31
12
under the six restrictive conditions |τ
d
12
− τ
d
23
|≤ ∆ τ, |τ
d
23
− τ
d
31
|≤ ∆ τ,
|τ
d
31
− τ
d
12
|≤ ∆ τ, |
K
12
−
K
23
|≤ ∆
K
, |
K
23
−
K
31
|≤ ∆
K,
and |
K
31
−
K
12
|≤ ∆
K
, where Δτ
and Δ
K
are error thresholds of time constants and proportional coefficients,
respectively. Then the values of τ
d
and
K
are defined as their average value:
τ
d
= (τ
d2
1
+ τ
d2
3
+ τ
d
31
)/3
and
K
= (
K
12
+
K
23
+
K
31
)/3. Because τ
d
and
K
are not of the
same order of magnitude, the error function is configured by multiplying the
sum of the three time constant differences and the sum of the three propor-
tional coefficient differences.
Equations (9.16)-(9.21) imply that if a specific β is fo
u
nd
K,
τ
d
ca
n be deter-
m
ined based on the three piezovoltage values of
(
025
,
()
)
Vt
025
Vt
05
05
, and
()
V
1
1
at a
specific tim
e
t.
For the curve fitting, the three piezovoltages of
(
025
,
()
)
Vt
025
V
11
at
t
025
= 0.25 s,
t
05
= 0.5 s, and
t
1
= 1 s, respectively,
are used, which correspond to the three peak values of the three piezovolt-
age time curves.
To find suitable fitting results under this criterion, Hou et al. [4] carried
out circular calculations for the preceding equations when β increased from
0 to 1 step by step. The parameters
K,
τ
d
,
and β obtained are listed in Table 9.3.
Hou et al. [4] found from their experimental results that if the value of Δτ
or Δ
K
was set to be less than 5%-8% of the final fitting value of τ
d
or
K,
the
Vt
05
05
, and
()
