Idealised Nyquist Plot of a Lithium Ion Battery

It is useful to look at an idealised Nyquist Plot of a lithium ion battery. Starting with a detailed, feature-by-feature description of the idealised Electrochemical Impedance Spectroscopy (EIS) Nyquist plot, tracking from the highest frequencies on the left to the lowest on the right.

1. The Axes and Flow

• X-axis (Z’): Represents the Real Impedance (Ω). This captures the resistive (Ω) elements of the cell where energy is dissipated as heat.
• Y-axis (-Z”): Represents the Negative Imaginary Impedance (Ω). This captures the reactive elements of the cell, primarily capacitance, which stores energy. Plotting it as negative ensures the dominant capacitive behavior of the battery appears in the positive (upper) quadrant.
• Frequency Sweep: The measurement sweeps from high frequency to low frequency, moving from the left side of the plot to the right.

2. The Inductive Artefact (Highest Frequency)

• Visual: The small tail dipping below the x-axis into the negative region (where -Z” becomes negative, meaning true imaginary impedance is positive).
• Physical Meaning: This represents the inductance of the system. In a lithium-ion cell, this primarily comes from the physical geometry of the cell (such as the metallic jelly-roll windings) as well as the metallic cables and tabs used by the testing equipment. It is generally not indicative of the electrochemical health of the active materials.

3. Internal Ohmic Resistance (R_ohmic)

• Visual: The exact point where the curve crosses the x-axis (Z’=0).
• Physical Meaning: Also known as Equivalent Series Resistance (R_s), this is the pure electrical resistance of the cell. It is the sum of the resistance from the bulk liquid electrolyte, the current collectors, the electrode terminals, and the contact resistance between these components. An increase in R_ohmic over a cell’s life is a primary indicator of electrolyte dry-out or mechanical delamination.

4. Contact / Interfacial Resistance (Mid-High Frequency)

• Visual: The first, smaller semicircle.
• Physical Meaning: This represents contact resistance at interfaces such as the current collector foil/active material interface and particle-to-particle contacts within the electrode. It is modelled as a parallel resistor-capacitor (Rcontact and Ccontact) circuit. In SEI-forming systems (graphite anodes, NMC cathodes), a growing insulating layer from electrolyte decomposition adds to this resistance over life, which is why the semicircle is often loosely labelled “SEI.” But the feature isn’t caused by SEI itself: it shows up in LFP and LTO systems too, which don’t form significant SEI (LTO’s ~1.55V vs Li/Li+ operating potential sits above the reduction window of most electrolyte solvents, and stabilized LFP/electrolyte pairings behave similarly), because the foil/particle contact resistance exists independently of any insulating film growth.

Note: In practice the time constants for contact resistance and charge transfer often overlap, which is why real EIS data typically shows one stretched or “depressed” semicircle rather than two distinct ones (see Real-World Caveats below).

5. Charge Transfer Resistance (Mid Frequency)

• Visual: The second, larger semicircle.
• Physical Meaning: This captures the charge transfer kinetics (R_ct) at the interface between the electrode active material and the electrolyte. It represents the energy barrier the lithium ion must overcome to shed its solvation shell and accept an electron to intercalate into the solid electrode. The capacitance associated with this is the electrochemical double-layer capacitance (C_dl). Cold temperatures drastically slow down these kinetics, which would visually result in a massive expansion of this semicircle.

6. Warburg Diffusion (Low Frequency)

• Visual: The straight, diagonal line extending to the right at approximately a 45° angle.
• Physical Meaning: This represents solid-state diffusion (Warburg impedance). At these very low alternating frequencies, the rate-limiting step of the battery is no longer surface reactions, but the physical time it takes for lithium ions to diffuse deep into the bulk crystalline lattice of the active electrode particles.

In a real-world teardown or structural review, analysing how these specific features shift under different states of charge or temperatures reveals exactly which component of the battery architecture: the electrolyte, the interfaces, or the bulk material is limiting performance.

Real-World Caveats (vs. Idealized Data)

While this is a perfect educational schematic, measured EIS data from a physical cell will look a bit messier:

1. Merged Semicircles: In reality, the time constants for the SEI layer and charge transfer often overlap. Instead of two distinct, perfectly round semicircles, you will typically see a single, somewhat stretched or “depressed” semicircle.
2. Imperfect Angles: The Warburg tail is rarely exactly 45° in real cells due to the porous nature of battery electrodes and surface roughness.
3. Constant Phase Elements (CPE): Because physical capacitors in batteries are not mathematically perfect, modelers use Constant Phase Elements instead of pure capacitors to account for the depressed semicircles.

This post has been built based on the support of and sponsorship from: illumion, Custom Power, Quarto Technical Services, Eatron Technologies and About:Energy

Why the Semicircles Can’t Overlap

A reader flagged an issue with the idealised plot above: if you draw the two semicircles so they physically overlap, converting that same curve to a Bode plot (Z’ against log frequency) forces the real impedance to drop, then rise, then drop again as frequency sweeps through that region. That’s worth walking through step by step, because it isn’t a stylistic nit-pick, it falls straight out of the equivalent circuit the plot is supposed to represent.

1. Each semicircle comes from one R‖C block, active over its own frequency window. The circuit behind the plot is a resistor in series with two parallel resistor-capacitor blocks: R_ohmic + (R₁ ‖ C₁) + (R₂ ‖ C₂), one block per physical process (contact/interfacial resistance, charge transfer). Each block has a time constant τ = R×C, and it only dominates the impedance over the narrow band of frequencies where ωτ is close to 1. Well above that band the capacitor short-circuits the block and it contributes almost nothing; well below it, the block contributes its full resistance.

2. The blocks add in series, and each one moves in only one direction. At any given frequency, the total real impedance is R_ohmic plus whatever block 1 is contributing plus whatever block 2 is contributing. Each of those two contributions shrinks as frequency rises and grows as frequency falls, there’s no frequency at which either one reverses.

3. So the sum can only move in one direction too. Adding two quantities that are each always shrinking as frequency rises can never produce a sum that rises. That’s true regardless of how close together the two time constants are. Plotted against log frequency, Z’ has to trace a single, smooth, non-increasing curve, a staircase down from R_ohmic+R₁+R₂ at low frequency to R_ohmic alone at high frequency.

4. The Nyquist plot is just that same curve, re-plotted. Every point on the Nyquist plot is Z'(frequency) paired with -Z”(frequency) at that frequency. Since Z’ can only move in one direction as frequency sweeps, the plot can only visit each real-impedance value once. Two arcs generated this way can be sequential, meeting at a single point on the real axis, but they can’t cross or double back over each other. A drawing that shows the semicircles overlapping is implicitly claiming a Bode curve where Z’ reverses direction mid-sweep, which this circuit has no way to produce.

The figure below rebuilds the idealised plot from one set of circuit values, R_ohmic = 0.5 Ω, contact/interfacial block = 1.7 Ω, charge-transfer block = 3.6 Ω. Run through the same calculation for both panels, so the transition points (0.5 → 2.2 → 5.8 Ω) land in exactly the same place on the Nyquist arcs and the Bode steps.

One caveat carried over from the Real-World Caveats section above: this doesn’t mean measured semicircles always look this crisp. If the two time constants sit close together, the arcs partially merge into one flattened, “depressed” shape rather than two clean circles, that’s normal and already covered above. What the maths rules out is specifically a crossing, where Z’ would have to reverse direction; a merged or overlapping-looking arc without an actual reversal is still physically consistent. Genuinely non-monotonic Z’ behaviour does show up in some cells, but from a different mechanism. Pseudo-inductive loops linked to consecutive reaction steps with a potential-dependent adsorbed intermediate, sometimes reported around lithium plating or in fuel-cell electrodes, not from two ordinary capacitive semicircles.

References

1. Wenxuan Hu, Yufan Peng, Yimin Wei, Yong Yang, Application of Electrochemical Impedance Spectroscopy to Degradation and Aging Research of Lithium-Ion Batteries, The Journal of Physical Chemistry C, ol 127/Issue 9
2. Yuhao Zhang, Jiayuan Xu, Xinyi Xie, Qiu-An Huang, Shijie Cheng, Jia Xie; Measurement, interpretation, and application of electrochemical impedance spectroscopy to lithium-ion batteries. EES Batteries 2026; 2 (1): 80–102. https://doi.org/10.1039/d5eb00153f
3. Lujun Wang, Ziang Song, Lijun Zhu, Jiuchun Jiang, Fast electrochemical impedance spectroscopy of lithium-ion batteries based on the large square wave excitation signal, iScience, Volume 26, Issue 4, 2023