How batteries work Lab 5 EIS: analysis

Lab handbook · Lab 5

Electrochemical impedance spectroscopy (EIS) II: equivalent circuits and the distribution of relaxation times

What does a spectrum mean, and what does a good fit not prove?

Figure L5.1 Two equivalent circuits with their element values: a resistor followed by two resistor-capacitor pairs in series, and a nested ladder of a capacitor in parallel with a resistor followed by a resistor-capacitor pair. Below, their Nyquist plots lie exactly on top of each other: a line for the first and open circles for the second.
circuit 1circuit 2
Two circuits, one spectrum: a perfect fit proves nothing about the circuit. Try this: change R2 and watch the ladder’s values change while the two curves stay on top of each other.
SourceMathematically identical circuits, a series chain of R‖C pairs and its nested ladder, after Lazanas and Prodromidis 2023, section 8 and Fig. 7 (they show a three-pair version); the hierarchical ladder for the SEI, non-uniqueness, and minimal, physics-based circuits after Meddings et al. 2020, section 3.2.2. The conversion between the two-pair versions here (matching the admittance as a continued fraction) is this page’s working, checked numerically. R0 = 20 mΩ, R1 = 4 mΩ, C1 = 0.25 F, C2 = 2 F; illustrative.

From spectrum to meaning

With a valid spectrum in hand, we can ask what it means. The usual analysis fits an equivalent circuit to the spectrum by complex nonlinear least squares and judges the fit by its χ2.1 The trap is that circuits are not unique: different circuits can give exactly the same impedance at every frequency, and adding elements always improves the fit.1 The same model can also return different values depending on the starting guesses and the weighting.2

So the circuit has to come from the physics of the cell, each element tied to a process, with as few elements as possible.1,2 Physics-based models go further: the transmission line of figure L4.4 with spherical diffusion in each grain, or full electrochemical models solved for their impedance.2,3

Figure L5.2 Left, a Nyquist plot of two resistor-capacitor pairs in series after an ohmic resistance: two semicircles whose diameters are the resistances, with the apex frequency written above each. Right, the negative imaginary part against frequency on a log axis, one peak per arc. Sliders set the second resistance and the ratio of the two time constants; below a ratio of a hundred the arcs merge into one.
Reading an arc: diameter, apex frequency, time constant; and when two arcs resolve. Try this: bring the ratio of time constants down from 300 to 10 and watch the two arcs become one.
SourceResistance as the diameter, the apex at ωRC = 1 and τ = RC, and the hundredfold rule for resolving two arcs, after Lazanas and Prodromidis 2023, sections 9 and 10. R0 = 10 mΩ, R1 = 20 mΩ, C1 = 100 µF, ideal capacitors; illustrative.

Reading before fitting

Before any fit, a spectrum can be read by hand, and the reading is what the fit must later agree with. A resistor in parallel with a capacitor draws a semicircle whose diameter is the resistance: the curve leaves the real axis at R0 and returns to it at R0 + R1. At its top, where the imaginary part is largest, ωRC = 1, so the frequency at the top gives the time constant τ = RC, and the capacitance follows from R and τ.1 Two pairs in series draw two arcs only when their time constants differ by about a hundredfold; closer than that they overlap, and at equal time constants a single arc appears.1 A Bode plot of −Z″ against frequency shows each arc as a peak and keeps the frequencies, which a Nyquist plot discards; the two plots are read together.

These readings are approximate for real arcs, which are depressed below the axis and are drawn by constant phase elements rather than capacitors (L4.4); but a fit that returns a resistance far from the diameter read by eye, or a time constant far from the apex, is a fit to be distrusted.

Figure L5.3 Left above, a Nyquist plot of noisy data made from two depressed arcs, with the fitted curve for one, two or three arcs chosen by buttons. Left below, the residuals of the fit, real and imaginary, against frequency: a pattern for one arc, noise for two and three. Right, the fitted values and the chi-square, which falls at every step, with the true values for comparison.
How many elements? The χ2 always falls; the residuals say when to stop. Try this: go from two arcs to three and read what happened to the second resistance.
SourceComplex nonlinear least squares, modulus weighting, χ2 as the measure of a fit, the improvement with every added element and the demand that each element stand for a process, after Lazanas and Prodromidis 2023, sections 8 and 16; the dependence of fitted values on starting guesses and weighting after Meddings et al. 2020, section 3.2.2. The fitter is this page’s own Levenberg-Marquardt (assets/js/physics.js, m11.cnls); data from arcs of 4 mΩ (n = 0.90) and 10 mΩ (n = 0.85) after 20 mΩ, with 0.4 % noise.

Fitting, and knowing when to stop

The fit itself is complex nonlinear least squares: the parameters of the circuit are adjusted until the weighted sum of the squared differences between the model and the data, real and imaginary parts together, is as small as it can be, and that sum, the χ2, measures the fit.1 The weighting matters, because the impedance spans decades: dividing each residual by the modulus of its data point gives every frequency the same say, where an unweighted fit would see only the largest arc.2 And the answer can depend on where the fit starts, so a fit is run from several starting guesses and the one with the smallest χ2 kept, with suspicion if the others land far away.2

L5.3 shows the trap and the way out. One arc cannot follow two: the residuals, the differences between fit and data plotted against frequency, swing systematically. Two arcs leave residuals that are noise, and return the values the data were made with. Three arcs lower the χ2 a little further, because adding an element always does,1 but the third arc stands for nothing: depending on where the fit starts, it collapses to a fraction of a milliohm or takes a share of the second arc, and the data cannot decide between the two. The rule is the one the tutorial gives: as many elements as there are processes, each one tied to a process by the physics of the cell, and no more.1 The residual plot, not the look of the Nyquist curve, is the judge.

Stop at the smallest circuit whose residuals are noise and whose every element names a process.

Figure L5.4 Left, an Arrhenius plot: the charge-transfer resistance relative to its value at 25 degrees against 1000 over T, two straight lines for the two electrodes with activation energies of 17.8 and 35.0 kilojoules per mole, the two measured NMC points at 25 and 60 degrees, and a moving dot at the chosen temperature. Right, the charge-transfer arc of the NMC electrode shrinking as the temperature rises.
Temperature tells the arcs apart: charge transfer follows an Arrhenius law. Try this: warm the cell from 25 to 60 °C and read how much the NMC arc shrinks; then compare with what the graphite line would do.
SourceCharge-transfer resistances of the NMC 811 electrode (1.80 Ω at 25 °C, 0.94 Ω at 60 °C), the activation energies of 17.8 kJ/mol (NMC 811) and 35.0 kJ/mol (graphite-SiOx), the measurement conditions (50 % state of charge, 10 mV, 10 mHz to 500 kHz, 25 to 60 °C in a chamber) and the circuit of a series resistance, two R-CPE arcs and a CPE tail, from Chen et al. 2020, Table VI, eq. 18 and the surrounding text. The exchange current from Rct = RT/(F i0) is the linear limit of Butler-Volmer of chapter 5. The arc on the right is schematic.

Which arc is which: let the temperature decide

A fit returns resistances, not names. The name comes from how each resistance responds to what we change, and temperature is the most telling change. Charge transfer is an activated process: its resistance falls with temperature along an Arrhenius line, and the slope of that line is an activation energy. On an NMC 811 electrode the charge-transfer resistance fell from 1.80 Ω at 25 °C to 0.94 Ω at 60 °C, an activation energy of 17.8 kJ/mol; on the graphite-SiOx electrode of the same cell the activation energy was 35.0 kJ/mol, twice as large.4 The spectra were taken at 50 % state of charge with a 10 mV excitation from 10 mHz to 500 kHz, in a temperature chamber, and fitted with a series resistance, two R-CPE arcs and a CPE for the low-frequency tail.4

Once an arc has been named charge transfer, its resistance becomes kinetics: in the linear limit of the Butler-Volmer equation (chapter 5), Rct = RT/(F i0), so the exchange current follows, and its temperature dependence gives the activation energy of the reaction itself.4 The same logic names the other arcs: change the state of charge, the cycle count or the cell itself, in symmetric cells of two harvested electrodes, and see which resistance follows (L5.6). A spectrum measured once, at one temperature and one state of charge, cannot be named at all.

A resistance is named by what moves it: temperature for charge transfer, cycling for films, state of charge for the electrode.

Figure L5.5 Top: a spectrum with two overlapping depressed arcs and slight noise, as dots, with the fit from the distribution of relaxation times as a line. Bottom: the distribution against relaxation time on a log axis from 1 microsecond to 10 seconds, with the exact distribution dashed and peaks labelled; shaded bands mark typical time ranges for bulk conduction, surface layers and charge transfer. A slider sets the regularisation strength.
data; recovered distributionfitexact distribution
The distribution of relaxation times: overlapping arcs become separate peaks, if the regularization is right. Try this: find the range of λ that gives exactly two peaks; then go one step either side.
SourceThe DRT relation Z = R0 + Rpol∫g(τ)/(1 + jωτ)dτ with τ = 1/(2πf), peaks as processes, the need for regularisation and its narrow useful range, sensitivity to bad points, and the restriction to the SEI and charge-transfer arcs, after Meddings et al. 2020, eq. 5 and section 3.2.1, and Bakenhaster and Dewald 2025, eqs. 9 and 10 and section 4.3; time ranges of the bands read from their Fig. 12 (approximate). Here: non-negative least squares with a ridge penalty λ on 61 log-spaced time constants (this page’s method); data from two R‖CPE arcs (4 mΩ, n = 0.9; 10 mΩ, n = 0.85) with 0.2 % noise; the exact distribution of such arcs is the Cole–Cole form (this page’s working).

If choosing the circuit is the problem, we can try not to choose one in advance. That is what the distribution of relaxation times (DRT) does. Every R‖C pair relaxes with one time constant, so a spectrum can be written as a sum, in the limit a distribution, of such pairs:2,5

Z(ω)=R0+Rpol∫0∞g(τ)1+jωτdτ

Recovering g(τ) turns the spectrum into peaks along the time axis, one per process, even where the arcs overlap in a Nyquist plot. That is why DRT is used on commercial cells: to count the processes and then build a circuit with the right number of elements.2,5

The catch is that recovering g from noisy data is unstable and needs a regularization, a penalty of strength λ; only a narrow range of λ gives the right answer, and the number of peaks depends on it.2,5 Bad data points distort it strongly, so DRT needs a Kramers–Kronig check first, and because the inductive and diffusion parts distort it too, it is usually applied to the SEI and charge-transfer arcs.2

Figure L5.6 Top left: impedance spectra of one cell at check-ups every 100 cycles up to 400, shifting right with growing arcs. Top right: the fitted ohmic resistance and the sum of SEI and charge-transfer resistances against cycles, and the capacity in percent of new. Bottom: a six-step machine-learning workflow from impedance check-ups to features, a training and test split, training, prediction of state of charge or health, and error checks.
current spectrum; RSEI + RctR0capacity
Following a cell as it ages, and the data-driven route. Try this: play the ageing and compare how fast the interface resistance and the ohmic resistance grow.
SourceImpedance rising with age, check-ups that do not stress the cell, the two effects of degradation, and individual contributions that may correlate better with health or sudden death than total impedance, after Meddings et al. 2020, sections 1 and 3.3.2; the arc growing faster than the intercept moves follows the trend of their Fig. 4d (read from the figure). The workflow after Bakenhaster and Dewald 2025, Fig. 10 (from their ref. 81); Gaussian process regression, relaxation-time models near 1 % error and the limits of the training data after their section 4.2. All numbers in the panels are illustrative, not data.

Following ageing

We now have the tools to do what chapter 16 asked of us: follow a cell as it ages. Impedance generally rises as a cell ages, and because the measurement does not harm the cell, it can be repeated through its whole life. Ageing studies run check-ups: the same measurement at fixed conditions every so many cycles, often enough to follow the ageing, rarely enough not to add to it.2 Ageing has two visible effects: capacity fade, from losing active material, and power fade, from growing resistance through lost contacts, thickening surface films and delamination.2

A spectrum is reduced to a few features: fitted resistances, DRT peaks or impedance at chosen frequencies. Individual contributions may follow the state of health, and give early warning of sudden fade, better than the total impedance.2 The difficulty is that temperature, state of charge and ageing all move the same spectrum; for some chemistries, LiFePO4 in particular, estimating the state of charge from impedance is much harder than for NMC.2

Machine learning is the newest route: train a model on features from many cells, then predict the state of health or charge of a new one. Gaussian process regression is common, and models based on relaxation times have reached errors around 1 %.5 Their knowledge ends at their training data, and the studies so far cover few systems and conditions.5 Local impedance, measured on one spot of an electrode, is a further step, since a whole-cell spectrum averages over a structure that is not uniform.5

References for this page

Numbers in the text refer to this list. All references of the course, with the chapters that use each one.

  1. 1 A. Ch. Lazanas, M. I. Prodromidis, "Electrochemical Impedance Spectroscopy—A Tutorial", ACS Measurement Science Au 3, 2023, 162–193. doi:10.1021/acsmeasuresciau.2c00070
  2. 2 N. Meddings, M. Heinrich, F. Overney, J.-S. Lee, V. Ruiz, E. Napolitano, S. Seitz, G. Hinds, R. Raccichini, M. Gaberšček, J. Park, "Application of electrochemical impedance spectroscopy to commercial Li-ion cells: A review", Journal of Power Sources 480, 2020, 228742. doi:10.1016/j.jpowsour.2020.228742
  3. 3 I. Abbas, H. T. Tran, T. T. N. Tran, T. L. Pham, E.-C. Shin, C.-W. Park, S.-B. Yu, O. J. Lee, A.-G. Nguyen, D. Jeong, B. H. Ka, H.-H. Cho, J. Lim, N. Shin, M. Gaberšček, S.-M. Hur, C.-J. Park, J. Kim, J.-S. Lee, "GITT Limitations and EIS Insights into Kinetics of NMC622", Batteries 11, 2025, 234. doi:10.3390/batteries11060234
  4. 4 C.-H. Chen, F. Brosa Planella, K. O'Regan, D. Gastol, W. D. Widanage, E. Kendrick, "Development of Experimental Techniques for Parameterization of Multi-scale Lithium-ion Battery Models", Journal of The Electrochemical Society 167, 2020, 080534. doi:10.1149/1945-7111/ab9050
  5. 5 S. T. Bakenhaster, H. D. Dewald, "Electrochemical impedance spectroscopy and battery systems: past work, current research, and future opportunities", Journal of Applied Electrochemistry 55, 2025, 1657–1681. doi:10.1007/s10800-025-02273-6

Cite this page

Using this page in teaching, a thesis or an article? Cite it as follows; the date is the last update of the course.

Shyma Sajeevan, A. (2026). Electrochemical impedance spectroscopy (EIS) II: equivalent circuits and the distribution of relaxation times. How batteries work. https://asajeevan.github.io/batteries/lab/eis-analysis/

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