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Table 3. Qualitative changes in the impedance spectrum in different state-of-charge areas: R resistance of the high-frequency arc, X reactance at 0.1 Hz, C capacitance at 0.1 Hz.

Table 3 Qualitative changes in the impedance spectrum in different state-of-charge areas: R resistance of the high-frequency arc, X reactance at 0.1 Hz, C capacitance at 0.1 Hz.

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Figure 1. Lithium iron phosphate battery (LithiumWerks 2.6 Ah): (a) Charge at constant current and constant voltage. (b) Flat discharge profile. EIS measurements were performed from 100% to 30% state-of-charge (SOC) in 2% steps. Impedance spectroscopy [17], coulomb counting [18], bookkeeping methods [19,20], and look-up tables [21] have been established since the mid-1970s, and have been supple- mented in the past decade by fuzzy logic, Kalman filters, learning algorithms, predictive methods [15], and the analysis of relaxation times [22]. At present, there is still no univer- sal method of precisely determining the usable capacity of a battery without completely discharging the battery.
Figure 1. Lithium iron phosphate battery (LithiumWerks 2.6 Ah): (a) Charge at constant current and constant voltage. (b) Flat discharge profile. EIS measurements were performed from 100% to 30% state-of-charge (SOC) in 2% steps. Impedance spectroscopy [17], coulomb counting [18], bookkeeping methods [19,20], and look-up tables [21] have been established since the mid-1970s, and have been supple- mented in the past decade by fuzzy logic, Kalman filters, learning algorithms, predictive methods [15], and the analysis of relaxation times [22]. At present, there is still no univer- sal method of precisely determining the usable capacity of a battery without completely discharging the battery.
Figure 2. (a) Schematic measurement setup: Impedance spectrometer BIM2 (BRS GmbH, Stuttgart, Germany), data logger (Agilent 34972A), electronic load (ET Systems ELP/DCM 9712C), DC power source (Elektro-Automatik EA-PS 2342-10B), and relay box reside in a climatic chamber (V6tsch VT7021). Data acquisition: Labview. (b) Basic model assumption and data evaluation.
Figure 2. (a) Schematic measurement setup: Impedance spectrometer BIM2 (BRS GmbH, Stuttgart, Germany), data logger (Agilent 34972A), electronic load (ET Systems ELP/DCM 9712C), DC power source (Elektro-Automatik EA-PS 2342-10B), and relay box reside in a climatic chamber (V6tsch VT7021). Data acquisition: Labview. (b) Basic model assumption and data evaluation.
Figure 3. Frequency response of impedance of the cell C LithiumWerks (2.6 Ah). The electrolyte resistance Rg = R(333 Hz) was subtracted. Capacitance versus frequency: Cs = —1/(wX), including leakage: Cp = —X/|w(R? + X)], and corrected by the electrolyte resis- tance: Cpp = —X/ [eo ((R - Rs)* + x?) according to Equation (4). Imaginary part of impedance: X = Im Z(w).
Figure 3. Frequency response of impedance of the cell C LithiumWerks (2.6 Ah). The electrolyte resistance Rg = R(333 Hz) was subtracted. Capacitance versus frequency: Cs = —1/(wX), including leakage: Cp = —X/|w(R? + X)], and corrected by the electrolyte resis- tance: Cpp = —X/ [eo ((R - Rs)* + x?) according to Equation (4). Imaginary part of impedance: X = Im Z(w).
Figure 5. (a) State-of-charge monitoring of a VoltSolar LFP cell (1.5 Ah) using capacitance at selected frequencies. Example: C/F = 0.1925-SOC + 11.64 at 0.15 Hz. SOC = Q/Qp is based on Ah counting. EIS measurements were performed from 100% to 50% SOC in 2% steps. (b) Linear relationship between capacitance and battery capacity: C = 14.65-Q + 11.64 (fit quality: 95.3%). In the following the significance of the pseudocapacitance C(w) for the actual electric charge Q and energy content of the battery is shown. The linear relationship between the pseudocapacitance and the SOC in Figure 5 is best at a frequency around 0.15 Hz. This means that the capacitance does not necessarily have to be measured down to extremely low frequencies. The chosen frequency of 0.15 Hz covers a part of the mass transport limited charge transfer reaction that is proportional to the genuine charge storage process by lithium ion intercalation at very low frequencies.
Figure 5. (a) State-of-charge monitoring of a VoltSolar LFP cell (1.5 Ah) using capacitance at selected frequencies. Example: C/F = 0.1925-SOC + 11.64 at 0.15 Hz. SOC = Q/Qp is based on Ah counting. EIS measurements were performed from 100% to 50% SOC in 2% steps. (b) Linear relationship between capacitance and battery capacity: C = 14.65-Q + 11.64 (fit quality: 95.3%). In the following the significance of the pseudocapacitance C(w) for the actual electric charge Q and energy content of the battery is shown. The linear relationship between the pseudocapacitance and the SOC in Figure 5 is best at a frequency around 0.15 Hz. This means that the capacitance does not necessarily have to be measured down to extremely low frequencies. The chosen frequency of 0.15 Hz covers a part of the mass transport limited charge transfer reaction that is proportional to the genuine charge storage process by lithium ion intercalation at very low frequencies.
Table 2. Validation of the linear correlation between state-of-charge and pseudocapacitance for three lithium-iron phosphate batteries of the same manufacturer.
Table 2. Validation of the linear correlation between state-of-charge and pseudocapacitance for three lithium-iron phosphate batteries of the same manufacturer.
Figure 6. VoltSolar LFP battery. (a) Pseudocharge Q(0.15 Hz) = C(0.15 Hz)-U and cell voltage U versus the “true” state- of-charge Q measured by Ah counting. (b) Linear correlation between pseudocharge Q(w) and “true” battery capacity at different discharge states. Example: Q/ As = 66.92-Q(0.15 Hz) — 2226. LPF batteries from different manufacturers show qualitatively the same results.
Figure 6. VoltSolar LFP battery. (a) Pseudocharge Q(0.15 Hz) = C(0.15 Hz)-U and cell voltage U versus the “true” state- of-charge Q measured by Ah counting. (b) Linear correlation between pseudocharge Q(w) and “true” battery capacity at different discharge states. Example: Q/ As = 66.92-Q(0.15 Hz) — 2226. LPF batteries from different manufacturers show qualitatively the same results.
Figure 7. (a) Energy W of three types of LFP cells determined by the pseudocharge Q = C U at 0.15 Hz against the real part of impedance (ohmic resistance) at 0.15 Hz. A VoltSolar 3.2 V/1.4 Ah, B Sony 3.2 V/1.1 Ah, C LithiumWerks 3.3 V/2.6 Ah. (b) For comparison: Peukert diagram of true energy versus C rate, (c) Ragone plot: true energy versus power (measured by current and voltage). Pseudocharge Q(0.15 Hz) allows to calculate an equivalent for the stored energy V| and the specific energy of the battery (energy divided by mass), as shown in Figure 7a.
Figure 7. (a) Energy W of three types of LFP cells determined by the pseudocharge Q = C U at 0.15 Hz against the real part of impedance (ohmic resistance) at 0.15 Hz. A VoltSolar 3.2 V/1.4 Ah, B Sony 3.2 V/1.1 Ah, C LithiumWerks 3.3 V/2.6 Ah. (b) For comparison: Peukert diagram of true energy versus C rate, (c) Ragone plot: true energy versus power (measured by current and voltage). Pseudocharge Q(0.15 Hz) allows to calculate an equivalent for the stored energy V| and the specific energy of the battery (energy divided by mass), as shown in Figure 7a.
Figure 8. Impedance spectra of different lithium-ion chemistries at rest in defined charge states. NMC as measured to show the influence of SOC on electrolyte resistance. LCO and NCA are shifted on the real axis to the electrolyte resistance of SOC = 1. Frequency range 0.1 Hz to 1 kHz.
Figure 8. Impedance spectra of different lithium-ion chemistries at rest in defined charge states. NMC as measured to show the influence of SOC on electrolyte resistance. LCO and NCA are shifted on the real axis to the electrolyte resistance of SOC = 1. Frequency range 0.1 Hz to 1 kHz.
Figure 9. (a) NMC battery #8: Normalized pseudocapacitance and pseudocharge at 0.1 Hz (reference: 589 F and 2470 As at SOC = 100%) versus “true” SOC from Ah counting. (b) Cell voltage of different cell chemistries versus state-of-charge from Ah counting. Batteries under test see Table 1. a Figure 9 demonstrates that pseudocapacitance reflects the state-of-charge in the correct order of magnitude. Unfortunately, the connex is not linear. Rather two linear regions can be distinguished from SOC = 60% to 100% for all battery chemistries, which can be fitted by a 3rd order polynomial (coefficients see Table 1). The linearity between impedance and SOC holds as long as the slope of the discharge curve AU/ASOC does not change. This is not surprising, because the impedance represents the slope of the current-voltage curve, |Z| = dU/dI. The steeper the voltage-charge curve, the more resistance and reactance increase when the battery is depleted.
Figure 9. (a) NMC battery #8: Normalized pseudocapacitance and pseudocharge at 0.1 Hz (reference: 589 F and 2470 As at SOC = 100%) versus “true” SOC from Ah counting. (b) Cell voltage of different cell chemistries versus state-of-charge from Ah counting. Batteries under test see Table 1. a Figure 9 demonstrates that pseudocapacitance reflects the state-of-charge in the correct order of magnitude. Unfortunately, the connex is not linear. Rather two linear regions can be distinguished from SOC = 60% to 100% for all battery chemistries, which can be fitted by a 3rd order polynomial (coefficients see Table 1). The linearity between impedance and SOC holds as long as the slope of the discharge curve AU/ASOC does not change. This is not surprising, because the impedance represents the slope of the current-voltage curve, |Z| = dU/dI. The steeper the voltage-charge curve, the more resistance and reactance increase when the battery is depleted.
This section is intended to show briefly that the proposed pseudocapacitance can correctly reflect the state-of-health of a battery even if the shape of the impedance spectra changes significantly during cycle aging. With aging and progressive SEI formation, the electrolyte resistance at high frequencies becomes larger and larger until two arcs become visible in the impedance spectrum (Figure 10). The internal resistance increases until the end of service life is reached. The electrolyte resistance is not a good SOH indicator, because the passive layer formation in some new batteries leads to an apparent improvement of conductivity during the first charge-discharge cycles until the gradual deterioration follows. For combined SOC and SOH measurements, the pseudocapacitance is still appropriate. As shown above, it is useful to normalize C(w) to the fully charged battery (SOC = 1) or the new battery (SOH = 1).
This section is intended to show briefly that the proposed pseudocapacitance can correctly reflect the state-of-health of a battery even if the shape of the impedance spectra changes significantly during cycle aging. With aging and progressive SEI formation, the electrolyte resistance at high frequencies becomes larger and larger until two arcs become visible in the impedance spectrum (Figure 10). The internal resistance increases until the end of service life is reached. The electrolyte resistance is not a good SOH indicator, because the passive layer formation in some new batteries leads to an apparent improvement of conductivity during the first charge-discharge cycles until the gradual deterioration follows. For combined SOC and SOH measurements, the pseudocapacitance is still appropriate. As shown above, it is useful to normalize C(w) to the fully charged battery (SOC = 1) or the new battery (SOH = 1).
Figure 10. Aging of an LCO battery at the same state-of-charge (Panasonic UR18650 FK). (a) Impedance spectra of fully charged batteries during aging by a thousand charge-discharge cycles. (b) Pseudocapacitance correlates linearly with the actual battery capacity determined by Ah counting: C = 141-Q + 191. Fit quality 0.92. (c) Increase of electrolyte resistance in time.
Figure 10. Aging of an LCO battery at the same state-of-charge (Panasonic UR18650 FK). (a) Impedance spectra of fully charged batteries during aging by a thousand charge-discharge cycles. (b) Pseudocapacitance correlates linearly with the actual battery capacity determined by Ah counting: C = 141-Q + 191. Fit quality 0.92. (c) Increase of electrolyte resistance in time.
Related topics:
Materials ScienceState of ChargeElectrical Impedance
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