Analysis processing method using impedance spectrum data, impedance spectrum data analysis processing system, and impedance spectral analysis processing program
This method for analysis processing by using measured impedance spectrum data includes: a step for determining whether to reset the maximum value and/or the minimum value of logarithmic relaxation time corresponding to a measured logarithmic frequency on the basis of the measured impedance spectrum data; a step for setting the number of equal-interval divisions within the range of the logarithmic relaxation time set in the determination step; and a step for analyzing parameters R∞, T, L, and Rl for the range of the set logarithmic relaxation time so as to satisfy a prescribed expression (A) by applying a regularized least-squares method.
1. A method of analysis processing which is performed in a system including a processor and a non-transitory memory coupled to the processor and storing software instructions executed by the processor, the method comprising:
(i) acquiring a measured impedance spectrum data of a sample;
(ii) determining whether to reset a maximum value and/or a minimum value of a logarithmic relaxation time corresponding to a measured logarithmic frequency based on the measured impedance spectrum data;
(iii) resetting the maximum value and/or the minimum value of the logarithmic relaxation time when it is determined to reset in the determining;
(iv) setting a number of divisions to be divided at equal intervals in a range of the logarithmic relaxation time corresponding to the measured logarithmic frequency when it is determined not to reset in the determining, or a range of the logarithmic relaxation time reset when it is determined to reset in the determining;
(v) applying regularized least squares to analyze parameters R ∞ , T, L, and R l with respect to the range of the logarithmic relaxation time and the number of divisions set so as to satisfy the following formula (A):
Z
(
f
p
)
≈
R
∞
+
T
2
π
f
p
j
+
2
π
f
p
L
+
∑
l
=
1
M
R
l
1
+
2
π
j
exp
(
ln
τ
l
)
exp
(
ln
f
p
)
❘
"\[LeftBracketingBar]"
Δ
ln
τ
❘
"\[RightBracketingBar]"
(
A
)
(p is a sequence number of a measured data point, l is a sequence number of the logarithmic relaxation time, f p is a p-th frequency, j is the unit imaginary number, M is the number of divisions of the logarithmic relaxation time, T is a reciprocal of a capacitance (C), L is an inductance, R ∞ is a high frequency limiting resistance, τ l is an l-th relaxation time, R l is a resistance value at τ l , and |Δ lnτ| is an absolute value of an interval of a natural logarithmic relaxation time);
(vi) substituting the analyzed parameters into the formula (A) to obtain a theoretical impedance spectrum of the sample;
(vii) displaying the measured impedance spectrum and the theoretical impedance spectrum such that they are comparable from user's view on a display device, and
(viii) comparing the obtained theoretical impedance spectrum and the measured impedance spectrum, and determining whether or not the spectra match,
wherein, when it is determined that the spectra do not match, the method further comprises repeating the steps of (iii) to (viii), and
wherein, when it is determined that the spectra do match, the method further comprises determining an impedance property of the sample using the theoretical impedance spectrum of the sample.
2. The method according to claim 1 , wherein
the determining determines to reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing when, in the measured impedance spectrum data,
(1) in a relationship between an impedance real term and the logarithmic frequency, the impedance real term has no plateau on a low frequency side and a high frequency side of a measured logarithmic frequency range,
(2) in a relationship between an impedance imaginary term and the logarithmic frequency, the impedance imaginary term has no peak on the low frequency side and the high frequency side in the measured logarithmic frequency range, or
(3) there is no arc in an impedance plot, and
the resetting
(1′) resets the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing so that, in the relationship between the impedance real term and the logarithmic frequency, the impedance real term becomes a plateau on the low frequency side and the high frequency side beyond the measured logarithmic frequency range,
(2′) resets the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing so that, in the relationship between the impedance imaginary term and the logarithmic frequency, the impedance imaginary term has a peak on the low frequency side and the high frequency side beyond the measured logarithmic frequency range, or
(3′) resets the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing so that there is an arc in the impedance plot.
3. The method according to claim 1 , wherein
in relation to the measured logarithmic frequency range, the determining includes obtaining logarithmic relaxation time dependence of a natural logarithmic relaxation time distribution resistor R k using the following formula (a) and the following formula (b):
Z
(
f
p
)
≈
R
∞
+
∑
k
=
1
N
R
k
1
+
2
π
j
exp
(
ln
τ
k
)
exp
(
ln
f
p
)
❘
"\[LeftBracketingBar]"
Δlnτ
❘
"\[RightBracketingBar]"
(
a
)
ln
τ
k
=
-
ln
(
f
k
2
π
)
(
b
)
(k is a sequence number of the logarithmic relaxation time, p is a sequence number of the measured data point, f k and f p are k-th and p-th frequencies, j is the unit imaginary number, N is the number of measurement data points, R ∞ is a high frequency limiting resistance, τ k is a k-th relaxation time, R k is a resistance value at τ k , and |Δ lnτ| is the absolute value of the interval of the natural logarithmic relaxation time), to thereby determine to reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing when R k has no peak in the range of the logarithmic relaxation time corresponding to the measured logarithmic frequency, and
the resetting resets the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing so that R k has a peak beyond the range of the logarithmic relaxation time corresponding to the measured logarithmic frequency.
4. The method according to claim 1 , wherein
in the analyzing, an impedance of the capacitor in the formula (A) is given by the following formula (B):
Z ( f )= T /( j 2π f ) (B).
5. The method according to claim 1 , wherein the analyzing is calculated using regularized nonlinear optimization.
6. The method according to claim 5 , wherein the regularized nonlinear optimization employed is regularization selected from the group consisting of Tikhonov regularization, Lasso regularization, and Elastic Net regularization.
7. The method according to claim 5 , wherein the regularized nonlinear optimization employed is nonlinear optimization selected from the group consisting of a Levenberg-Marquardt method, a simplex method, and a conjugate gradient method.
8. The method according to claim 1 , the measured impedance spectrum data is impedance spectrum data of a sample selected from the group consisting of primary batteries, secondary batteries, fuel cells, capacitor materials, anticorrosion-treated metals, and ceramic materials.
9. An impedance spectrum data analysis processing system, comprising:
a control device including a processor and a memory; and
a display device,
wherein the processor is configured to:
(i) acquire a measured impedance spectrum data of a sample,
(ii) determine whether to reset a maximum value and/or a minimum value of a logarithmic relaxation time corresponding to a measured logarithmic frequency based on the measured impedance spectrum data,
(iii) reset the maximum value and/or the minimum value of the logarithmic relaxation time corresponding to the measured logarithmic frequency when the processor determines to reset,
(iv) set a number of divisions to be divided at equal intervals in a range of the logarithmic relaxation time corresponding to the measured logarithmic frequency when the processor determines not to reset, or a range of the logarithm relaxation time reset by the processor when the processor determines to reset,
(v) apply regularized least squares to analyze parameters R ∞ , T, L, and R l of the sample with respect to the range of the logarithmic relaxation time and the number of divisions set so as to satisfy the following formula (A):
Z
(
f
p
)
≈
R
∞
+
T
2
π
f
p
j
+
2
π
f
p
L
+
∑
l
=
1
M
R
l
1
+
2
π
j
exp
(
ln
τ
l
)
exp
(
ln
f
p
)
❘
"\[LeftBracketingBar]"
Δ
ln
τ
❘
"\[RightBracketingBar]"
(
A
)
(p is a sequence number of a measured data point, l is a sequence number of the logarithmic relaxation time, f p is a p-th frequency, j is the unit imaginary number, M is the number of divisions of the logarithmic relaxation time, T is a reciprocal of a capacitance (C), L is an inductance, R ∞ is a high frequency limiting resistance, τ l is an l-th relaxation time, R l is a resistance value at τ l , and |Δ lnτ| is an absolute value of an interval of a natural logarithmic relaxation time),
(vi) substitute the analyzed parameters into the formula (A) to obtain a theoretical impedance spectrum of the sample,
(vii) control the display device to display the measured impedance spectrum and the theoretical impedance spectrum such that they are comparable from user's view in the control device, and
(viii) compare the obtained theoretical impedance spectrum and the measured impedance spectrum, and determine whether or not the spectra match,
wherein, when it is determined that the spectra do not match, the processor is further configured to repeat the steps of (iii) to (viii), and
wherein, when it is determined that the spectra do match, the processor is further configured to determine an impedance property of the sample using the theoretical impedance spectrum of the sample.
10. The analysis processing system according to claim 9 , wherein
the processor is configured to determine to reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the processor when, in the measured impedance spectrum data,
(1) in a relationship between an impedance real term and the logarithmic frequency, the impedance real term has no plateau on a low frequency side and a high frequency side of a measured logarithmic frequency range,
(2) in a relationship between an impedance imaginary term and the logarithmic frequency, the impedance imaginary term has no peak on the low frequency side and the high frequency side in the measured logarithmic frequency range, or
(3) there is no arc in an impedance plot, and
the processor is configured to:
(1′) reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the processor so that, in the relationship between the impedance real term and the logarithmic frequency, the impedance real term becomes a plateau on the low frequency side and the high frequency side beyond the measured logarithmic frequency range,
(2′) reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the processor so that, in the relationship between the impedance imaginary term and the logarithmic frequency, the impedance imaginary term has a peak on the low frequency side and the high frequency side beyond the measured logarithmic frequency range, or
(3′) reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the processor so that there is an arc in the impedance plot.
11. The analysis processing system according to claim 9 , wherein
in relation to the measured logarithmic frequency range, the processor is configured to obtain logarithmic relaxation time dependence of a natural logarithmic relaxation time distribution resistor R k using the following formula (a) and the following formula (b):
Z
(
f
p
)
≈
R
∞
+
∑
k
=
1
N
R
k
1
+
2
π
j
exp
(
ln
τ
k
)
exp
(
ln
f
p
)
❘
"\[LeftBracketingBar]"
Δlnτ
❘
"\[RightBracketingBar]"
(
a
)
ln
τ
k
=
-
ln
(
f
k
2
π
)
(
b
)
(k is a sequence number of the logarithmic relaxation time, p is a sequence number of the measured data point, f k and f p are k-th and p-th frequencies, j is the unit imaginary number, N is the number of measurement data points, R ∞ is a high frequency limiting resistance, τ k is a k-th relaxation time, R k is a resistance value at τ k , and |Δ lnτ| is the absolute value of the interval of the natural logarithmic relaxation time), to thereby determine to reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the processor when R k has no peak in the range of the measured logarithmic frequency, and
the processor is configured to reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the processor so that R k has a peak beyond the range of the logarithmic relaxation time corresponding to the measured logarithmic frequency.
12. The analysis processing system according to claim 10 , wherein the control device includes an input terminal with which an operator visually determines the (1) to (3) from the measured impedance spectrum data displayed on the display device, and manually resets the (1′) to (3′).
13. The analysis processing system according to claim 11 , wherein the control device includes an input terminal with which an operator visually determines whether or not the natural logarithmic relaxation time distribution resistor R k has a peak in the measured logarithmic frequency range from the logarithmic relaxation time dependence of R k displayed on the display device, and manually resets the maximum value and/or the minimum value of the logarithmic relaxation time used in the processor so that R k has a peak beyond the measured logarithmic frequency range.
14. The analysis processing system according to claim 9 , wherein
in the formula (A), an impedance of the capacitor is given by the following formula (B):
Z ( f )= T /( j 2π f ) (B).
15. A non-transitory computer-readable recording medium storing an impedance spectrum data analysis processing program that causes a computer to execute a process, the process comprising:
(i) acquiring a measured impedance spectrum data of a sample;
(ii) determining whether to reset a maximum value and/or a minimum value of a logarithmic relaxation time corresponding to a measured logarithmic frequency based on the measured impedance spectrum data;
(iii) resetting the maximum value and/or the minimum value of the logarithmic relaxation time when it is determined to reset in the determining;
(iv) setting a number to be divided at equal intervals in a range of the logarithmic relaxation time corresponding to the measured logarithmic frequency when it is determined not to reset in the determining, or a range of the logarithmic relaxation time reset when it is determined to reset in the determining;
(v) applying regularized least squares to analyze parameters R ∞ , T, L, and R l of the sample with respect to the range of the logarithmic relaxation time and the number of divisions set so as to satisfy the following formula (A):
Z
(
f
p
)
≈
R
∞
+
T
2
π
f
p
j
+
2
π
f
p
L
+
∑
l
=
1
M
R
l
1
+
2
π
j
exp
(
ln
τ
l
)
exp
(
ln
f
p
)
❘
"\[LeftBracketingBar]"
Δ
ln
τ
❘
"\[RightBracketingBar]"
(
A
)
(p is a sequence number of a measured data point, l is a sequence number of the logarithmic relaxation time, f p is a p-th frequency, j is the unit imaginary number, M is the number of divisions of the logarithmic relaxation time, T is a reciprocal of a capacitance (C), L is an inductance, R ∞ is a high frequency limiting resistance, τ l is an l-th relaxation time, R l is a resistance value at τ l , and |Δ lnτ| is an absolute value of an interval of a natural logarithmic relaxation time);
(vi) substituting the analyzed parameters into the formula (A) to obtain a theoretical impedance spectrum of the sample;
(vii) displaying the measured impedance spectrum and the theoretical impedance spectrum such that they are comparable from user's view on a display device, and
(viii) comparing the obtained theoretical impedance spectrum and the measured impedance spectrum, and determining whether or not the spectra match,
wherein, when it is determined that the spectra do not match, the process further comprises repeating the steps of (iii) to (viii), and
wherein, when it is determined that the spectra do match, the process further comprises determining an impedance property of the sample using the theoretical impedance spectrum of the sample.
16. The non-transitory computer-readable recording medium according to claim 15 , wherein
the determining determines to reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing when, in the measured impedance spectrum data,
(1) in a relationship between an impedance real term and the logarithmic frequency, the impedance real term has no plateau on a low frequency side and a high frequency side of a measured logarithmic frequency range,
(2) in a relationship between an impedance imaginary term and the logarithmic frequency, the impedance imaginary term has no peak on the low frequency side and the high frequency side in the measured logarithmic frequency range, or
(3) there is no arc in an impedance plot, and
the resetting
(1′) resets the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing so that, in the relationship between the impedance real term and the logarithmic frequency, the impedance real term becomes a plateau on the low frequency side and the high frequency side beyond the measured logarithmic frequency range,
(2′) resets the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing so that, in the relationship between the impedance imaginary term and the logarithmic frequency, the impedance imaginary term has a peak on the low frequency side and the high frequency side beyond the measured logarithmic frequency range, or
(3′) resets the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing so that there is an arc in the impedance plot.
17. The non-transitory computer-readable recording medium according to claim 15 , wherein
in relation to the measured logarithmic frequency range, the determining includes obtaining logarithmic relaxation time dependence of a natural logarithmic relaxation time distribution resistor R k using the following formula (a) and the following formula (b):
Z
(
f
p
)
≈
R
∞
+
∑
k
=
1
N
R
k
1
+
2
π
j
exp
(
ln
τ
k
)
exp
(
ln
f
p
)
❘
"\[LeftBracketingBar]"
Δlnτ
❘
"\[RightBracketingBar]"
(
a
)
ln
τ
k
=
-
ln
(
f
k
2
π
)
(
b
)
(k is a sequence number of the logarithmic relaxation time, p is a sequence number of the measured data point, f k and f p are k-th and p-th frequencies, j is the unit imaginary number, N is the number of measurement data points, R ∞ is a high frequency limiting resistance, τ k is a k-th relaxation time, R k is a resistance value at τ k , and |Δ lnτ| is the absolute value of the interval of the natural logarithmic relaxation time), to thereby determine to reset the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing when R k has no peak in the range of the measured logarithmic frequency, and
the resetting resets the maximum value and/or the minimum value of the logarithmic relaxation time used in the analyzing so that R k has a peak beyond the range of the logarithmic relaxation time corresponding to the measured logarithmic frequency.