Method of detecting system function by measuring frequency response
Methods of rapidly measuring an impedance spectrum of an energy storage device in-situ over a limited number of logarithmically distributed frequencies are described. An energy storage device is excited with a known input signal, and a response is measured to ascertain the impedance spectrum. An excitation signal is a limited time duration sum-of-sines consisting of a select number of frequencies. In one embodiment, magnitude and phase of each frequency of interest within the sum-of-sines is identified when the selected frequencies and sample rate are logarithmic integer steps greater than two. This technique requires a measurement with a duration of one period of the lowest frequency. In another embodiment, where selected frequencies are distributed in octave steps, the impedance spectrum can be determined using a captured time record that is reduced to a half-period of the lowest frequency.
1. A method for detecting function of an energy storage device under test by measuring frequency response, the method comprising the acts of:
generating a vector of frequencies by defining a lowest frequency for the vector of frequencies, defining an integer step factor, multiplying the lowest frequency by the integer step factor to obtain a second frequency for the vector of frequencies, determining subsequent frequencies for the vector of frequencies by multiplying a previous frequency in the vector of frequencies by the integer step factor;
defining a vector of amplitudes and phases corresponding to the frequencies in the vector of frequencies;
assembling an excitation signal using the vector of frequencies and the vector of amplitudes and phases;
conditioning the excitation signal to be compatible with the energy storage device under test;
exciting the energy storage device under test with the conditioned excitation signal for one period of the lowest frequency;
capturing a response time record with a data acquisition system, the response time record comprising a response of the energy storage device to the act of exciting; and
processing the response time record to the frequency domain to obtain the frequency response.
2. The method of claim 1 , wherein the integer step factor is greater than 2.
3. The method of claim 1 , wherein the excitation signal is a sum-of-sines excitation signal.
4. The method of claim 3 , wherein the response time record is captured at a sampling rate that is a step-factor harmonic to the sum-of-sines excitation signal and is at least a step factor squared times a highest frequency in the sum-of-sines excitation signal.
5. The method of claim 1 , wherein the response time record is processed by rectifying the response time record relative to various forms.
6. The method of claim 5 , wherein the response time record is rectified with r forms, where r is the integer step factor, and N is the number of samples over the period (i.e., n=n 1 , n 2 , . . . , N), and the first form of the rectification is:
R
1
(
n
)
=
[
1
,
0
≤
n
<
N
r
-
1
,
N
r
≤
n
<
N
]
For the i th rectification, where 1<i<r, the form is as follows:
R
i
(
n
)
=
[
-
1
,
0
≤
n
<
(
i
-
1
)
N
r
1
,
(
i
-
1
)
N
r
≤
n
<
iN
r
-
1
,
iN
r
≤
n
<
N
]
for the i th rectification, the form is as follows:
R
r
(
n
)
=
[
-
1
,
0
≤
n
<
(
r
-
1
)
N
r
1
,
(
r
-
1
)
N
r
≤
n
<
N
]
to obtain a rectified signal.
7. The method of claim 6 , wherein each of the r forms are summed up to obtain r different sums including m 1 . . . m r .
8. The method of claim 7 , wherein the summation terms (m 1 . . . m r ) are normalized to the response time record:
S 1 =Qm 1 , S 2 =Qm 2 , . . . S r =Qm r
Where
m 1 . . . m r are summation terms of the rectified signal
Q is the number of periods in the response time record for a frequency being detected
S 1 , S 2 , . . . S r are corrected summation terms.
9. The method of claim 8 , wherein the rectified signal is used to calculate magnitude and phase of the response time record with:
[
V
P
cos
ϕ
P
V
P
sin
ϕ
P
]
=
{
[
K
11
K
21
⋯
K
r
1
K
12
K
22
⋯
K
r
2
]
[
K
11
K
12
K
21
K
22
⋮
⋮
K
r
1
K
r
2
]
}
-
1
[
K
11
K
21
⋯
K
r
1
K
12
K
22
⋯
K
r
2
]
[
m
1
m
2
⋮
m
r
]
Where
V P is a desired magnitude
φ P is a desired phase
m 1 . . . m r are summation terms of the rectified signal
K 11 . . . K r1 are summations of the rectified sine term
K 12 . . . K r2 are summations of the rectified cosine term.
10. The method of claim 9 , wherein the magnitude and phase of the response time record are calculated at each frequency of interest of the vector of frequencies to generate an impedance spectrum of the frequency response.
11. A method for detecting function of an energy storage device under test by measuring frequency response, the method comprising the acts of:
generating a vector of frequencies by defining a lowest frequency for the vector of frequencies, defining an integer step factor, multiplying the lowest frequency by the integer step factor to obtain a second frequency for the vector of frequencies, determining subsequent frequencies for the vector of frequencies by multiplying a previous frequency in the vector of frequencies by the integer step factor, and removing the second frequency from the vector of frequencies;
defining a vector of amplitudes and phases corresponding to the frequencies in the vector of frequencies;
assembling an excitation signal using the vector of frequencies and the vector of amplitudes and phases;
conditioning the excitation signal to be compatible with the energy storage device under test;
exciting the energy storage device under test with the conditioned excitation signal for one-half period of the lowest frequency;
capturing a response time record with a data acquisition system, the response time record comprising a response of the energy storage device to the act of exciting; and
processing the response time record to the frequency domain to obtain the frequency response.
12. The method of claim 11 , wherein the integer step factor is a factor of two (i.e., r=2, 4, 8, 16, etc.).
13. The method of claim 11 , wherein the excitation signal is a sum-of-sines excitation signal.
14. The method of claim 13 , wherein the response time record is captured at a sampling rate that is an octave harmonic to the sum-of-sines excitation signal and is at least four times a highest frequency in the sum-of-sines excitation signal.
15. The method of claim 11 , wherein the captured response time record is inverted and concatenated to the captured response time record.
16. The method of claim 11 , wherein the response time record is processed by rectifying the response time record relative to sine and cosine.
17. The method of claim 16 , wherein the response time record is rectified using a method selected from the group consisting of non-zero crossing, with N as the number of samples over the period, the sine waveform of the rectification is:
Rs
(
n
)
=
[
1
,
0
≤
n
<
N
2
-
1
,
N
2
≤
n
<
N
]
The cosine form is:
Rc
(
n
)
=
[
1
,
0
≤
n
<
N
4
-
1
,
N
4
≤
n
<
3
N
4
1
,
3
N
4
≤
n
<
N
]
and, zero crossing, with N as the number of samples over the period, the sine waveform of the rectification is:
Rs
(
n
)
=
[
0
,
n
=
0
1
,
0
<
n
<
N
2
0
,
n
=
n
2
-
1
,
N
2
<
n
<
N
]
The cosine form is:
Rc
(
n
)
=
[
1
,
0
≤
n
<
N
4
0
,
n
=
N
4
-
1
,
N
4
<
n
<
3
N
4
0
,
n
=
3
N
4
1
,
3
N
4
<
n
<
N
]
to obtain a rectified signal.
18. The method of claim 17 , wherein the rectified signal is used to calculate magnitude and phase of the response time record with:
[
V
P
cos
ϕ
V
P
sin
ϕ
]
=
=
[
K
4
K
1
K
4
-
K
2
K
3
,
K
2
K
2
K
3
-
K
1
K
4
K
3
K
2
K
3
-
K
1
K
4
,
K
1
K
1
K
4
-
K
2
K
3
]
[
m
1
m
2
]
Where
V P is a desired magnitude
φ P is a desired phase
m 1 , m 2 are summation terms of the rectified signal
K 1 , K 2 are summations of the rectified sine term
K 3 , K 4 are summations of the rectified cosine term.
19. The method of claim 18 , wherein the calculated magnitude and phase of the lowest frequency are used to generate a half-period time record that is subtracted from the response time record.
20. The method of claim 17 , wherein the summation terms (m 1 . . . m r ) are normalized to the response time record:
S 1 =Qm 1 , S 2 =Qm 2 , . . . S r =Qm r
Where
m 1 . . . m r are summation terms of the rectified signal
Q is the number of periods in the response time record for a frequency being detected
S 1 , S 2 , . . . S r are corrected summation terms.
21. The method of claim 20 , wherein a magnitude and phase response are calculated at each frequency in the vector of frequencies to generate an impedance spectrum of the frequency response.
22. The method of claim 11 , further comprising: repeating the acts of claim 11 , except:
rather than removing the second frequency, the method includes removing the lowest frequency and a third frequency from the vector of frequencies; and
the energy storage device under test is excited for one-quarter period of the lowest frequency rather than one-half period of the lowest frequency; and
combining the frequency response and a frequency response obtained from the repeated acts to obtain a frequency response with a duration of three-quarters of a period of the lowest frequency.