Crosstalk compensation in analysis of energy storage devices
Estimating impedance of energy storage devices includes generating input signals at various frequencies with a frequency step factor therebetween. An excitation time record (ETR) is generated to include a summation of the input signals and a deviation matrix of coefficients is generated relative to the excitation time record to determine crosstalk between the input signals. An energy storage device is stimulated with the ETR and simultaneously a response time record (RTR) is captured that is indicative of a response of the energy storage device to the ETR. The deviation matrix is applied to the RTR to determine an in-phase component and a quadrature component of an impedance of the energy storage device at each of the different frequencies with the crosstalk between the input signals substantially removed. This approach enables rapid impedance spectra measurements that can be completed within one period of the lowest frequency or less.
1. A method of estimating an impedance of an energy storage device, comprising:
generating two or more input signals at different frequencies with a frequency step factor therebetween;
generating an excitation time record comprising a summation of the two or more input signals;
generating a deviation matrix of coefficients relative to parameters of the excitation time record to determine crosstalk between the two or more input signals;
stimulating an energy storage device with a signal generator applying the excitation time record and capturing a response time record indicative of a response of the energy storage device to the excitation time record with a response measurement device; and
applying the deviation matrix of coefficients to the response time record to remove the crosstalk and determine response signals including an in-phase component and a quadrature component of the impedance of the energy storage device at each of the different frequencies.
2. The method of claim 1 , wherein generating two or more input signals comprises generating sinusoidal signals.
3. The method of claim 1 , wherein generating two or more input signals comprises generating non-harmonic signals.
4. The method of claim 1 , wherein generating two or more input signals comprises generating harmonic signals.
5. The method of claim 1 , further comprising generating a magnitude and a phase for each of the different frequencies to determine a frequency response of the energy storage device.
6. The method of claim 1 , wherein generating the deviation matrix of coefficients further comprises performing a time domain analysis by developing a least squares solution for each frequency in the summation of the two or more input signals with a pseudoinverse matrix for N time steps wherein there are N+1 samples in the response time record.
7. The method of claim 1 , wherein generating the deviation matrix of coefficients further comprises performing a frequency domain analysis by:
generating a synchronously detected response time record for each frequency in the summation of the two or more input signals;
generating a discrete Fourier transform of the response time record for each frequency in the summation of the two or more input signals;
convolving the response time record in the frequency domain for each frequency in the summation of the two or more input signals;
normalizing the discrete Fourier transform of the convolved response time record to one-half the number of samples in the response time record; and
evaluating the normalized discrete Fourier transform of the convolved response time record at zero frequency;
wherein generating the deviation matrix of coefficients relative to the response time record further comprises combining the normalized discrete Fourier transform of the convolved response and the synchronously detected response time record for each frequency in the summation of the two or more input signals to form the deviation matrix of coefficients.
8. The method of claim 1 , further comprising conditioning the excitation time record to be compatible with measurement conditions of the energy storage device.
9. The method of claim 1 , wherein the two or more input signals are a series of frequencies comprising a lowest frequency and each additional frequency in the series of frequencies is the frequency step factor higher than a previous frequency in the series of frequencies.
10. The method of claim 9 , wherein generating the excitation time record comprises generating the excitation time record to have a duration of about a period of the lowest frequency.
11. The method of claim 9 , wherein generating the excitation time record comprises generating the excitation time record to have a duration of about one-half the period of the lowest frequency.
12. The method of claim 9 , wherein generating the excitation time record comprises generating the excitation time record to have a duration less than the period of the lowest frequency.
13. The method of claim 1 , further comprising augmenting the deviation matrix of coefficients with noise information within the energy storage device, a system coupled to the energy storage device, or a combination thereof, and wherein applying the deviation matrix of coefficients to the response time record also removes the noise information from the response signals.
14. An apparatus for characterizing an energy storage device, comprising:
a signal generator configured for stimulating an energy storage device with a stimulus signal generated responsive to an excitation time record;
a response measurement device configured for measuring a response signal when the stimulus signal is applied to the energy storage device, the response signal indicative of a response of the energy storage device to the stimulus signal; and
an analyzer operably coupled to the stimulus signal and the response signal, the analyzer configured for:
generating two or more input signals at different frequencies with a frequency step factor therebetween;
generating the excitation time record comprising a summation of the two or more input signals;
generating a deviation matrix of coefficients relative to parameters of the excitation time record to determine a crosstalk between the two or more input signals;
sampling the response signal to generate a response time record; and
applying the deviation matrix of coefficients to the response time record to remove the crosstalk and determine response signals including an in-phase component and a quadrature component of the impedance of the energy storage device at each of the different frequencies.
15. The apparatus of claim 14 , wherein the analyzer is further configured for generating the two or more input signals as sinusoidal signals.
16. The apparatus of claim 14 , wherein the analyzer is further configured for generating the two or more input signals as non-harmonic signals.
17. The apparatus of claim 14 , wherein the analyzer is further configured for generating the two or more input signals as harmonic signals.
18. The apparatus of claim 14 , wherein the analyzer is further configured for generating a magnitude and a phase for each of the different frequencies to determine a frequency response of the energy storage device.
19. The apparatus of claim 14 , wherein the analyzer is further configured for generating the deviation matrix of coefficients by performing a time domain analysis to develop a least squares solution for equations for each frequency in the summation of the two or more input signals with a pseudoinverse matrix for N time steps wherein there are N+1 samples in the response time record.
20. The apparatus of claim 14 , wherein the analyzer is further configured for generating the deviation matrix of coefficients by performing a frequency domain analysis, comprising:
generating a synchronously detected response time record for each frequency in the summation of the two or more input signals;
generating a discrete Fourier transform of the response time record for each frequency in the summation of the two or more input signals;
convolving the response time record in the frequency domain for each frequency in the summation of the two or more input signals;
normalizing the discrete Fourier transform of the convolved response time record to one-half the number of samples in the response time record; and
evaluating the normalized discrete Fourier transform of the convolved response time record at zero frequency;
wherein generating the deviation matrix of coefficients relative to the response time record further comprises combining the normalized discrete Fourier transform of the convolved response and the synchronously detected response time record for each frequency in the summation of the two or more input signals to form the deviation matrix of coefficients.
21. The apparatus of claim 14 , wherein the analyzer is further configured for conditioning the excitation time record to be compatible with measurement conditions of the energy storage device.
22. The apparatus of claim 14 , wherein the two or more input signals are a series of frequencies comprising a lowest frequency and each additional frequency in the series of frequencies is the frequency step factor higher than a previous frequency in the series of frequencies.
23. The apparatus of claim 22 , wherein the analyzer is further configured for generating the excitation time record by generating the excitation time record to have a duration of about a period of the lowest frequency.
24. The apparatus of claim 22 , wherein the analyzer is further configured for generating the excitation time record by generating the excitation time record to have a duration less than the period of the lowest frequency.
25. The apparatus of claim 14 , wherein the analyzer is further configured for augmenting the deviation matrix of coefficients with noise information within the energy storage device, a system coupled to the energy storage device, or a combination thereof, and wherein applying the deviation matrix of coefficients to the response time record also removes the noise information from the response signals.
26. Non-transitory computer-readable media including computer-executable instructions, which when executed on one or more computers, cause the computers to:
generate two or more input signals at different frequencies with a frequency step factor therebetween;
generate an excitation time record comprising a summation of the two or more input signals;
generate a deviation matrix of coefficients relative to parameters of the excitation time record to determine a crosstalk between the two or more input signals;
stimulate an energy storage device with the excitation time record and capture a response time record indicative of a response of the energy storage device to the excitation time record; and
applying the deviation matrix of coefficients to the response time record to remove the crosstalk and determine response signals including an in-phase component of an impedance of the energy storage device and a quadrature component of the impedance of the energy storage device at each of the different frequencies.
27. The non-transitory computer-readable media of claim 26 , wherein the computer instructions are further configured to generate two or more input signals as non-harmonic sinusoidal signals.
28. The non-transitory computer-readable media of claim 26 , wherein the computer instructions are further configured to cause the computer to generate a magnitude and a phase for each of the different frequencies to determine a frequency response of the energy storage device.
29. The non-transitory computer-readable media of claim 26 , wherein the computer instructions configured to cause the computer to generate the deviation matrix of coefficients are further configured to cause the computer to perform a time domain analysis by developing a least squares solution for each frequency in the summation of the two or more input signals with a pseudoinverse matrix for N time steps wherein there are N+1 samples in the response time record.
30. The non-transitory computer-readable media of claim 26 , wherein the computer instructions configured to cause the computer to generate the deviation matrix of coefficients are further configured to cause the computer to perform a frequency domain analysis by:
generating a synchronously detected response time record for each frequency in the summation of the two or more input signals;
generating a discrete Fourier transform of the response time record for each frequency in the summation of the two or more input signals;
convolving the response time record in the frequency domain for each frequency in the summation of the two or more input signals;
normalizing the discrete Fourier transform of the convolved response time record to one-half the number of samples in the response time record; and
evaluating the normalized discrete Fourier transform of the convolved response time record at zero frequency;
wherein generating the deviation matrix of coefficients relative to the response time record further comprises combining the normalized discrete Fourier transform of the convolved response and the synchronously detected response time record for each frequency in the summation of the two or more input signals to form the deviation matrix of coefficients.
31. The non-transitory computer-readable media of claim 26 , wherein the two or more input signals comprise input signals at a series of frequencies comprising a lowest frequency and each additional frequency in the series of frequencies is the frequency step higher than the previous frequency in the series of frequencies.
32. The non-transitory computer-readable media of claim 31 , wherein the computer instructions configured to generate the excitation time record are further configured to generate the excitation time record to have a duration of about a period of the lowest frequency.
33. The non-transitory computer-readable media of claim 31 , wherein the computer instructions configured to generate the excitation time record are further configured to generate the excitation time record to have a duration less than the period of the lowest frequency.
34. The non-transitory computer-readable media of claim 26 , wherein the computer instructions are further configured to augment the deviation matrix of coefficients with noise information within the energy storage device, a system coupled to the energy storage device, or a combination thereof, and wherein applying the deviation matrix of coefficients to the response time record also removes the noise information from the response signals.