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How is linearity checked in EIS with Lissajous plots and the Fourier transform? (CVSim example)

Using the free-access CVSim simulator in EC-Lab, 10 mV and 50 mV sine perturbations are applied to a simple redox reaction (E mechanism); linear and non-linear responses are compared in the time, Lissajous and frequency domains.

Key points

  • Simulation: E mechanism, both redox species initially in solution (10 mM), k° = 10⁻⁴ cm/s, αf = 0.5, D = 10⁻⁶ cm²/s, 1 mHz sine excitation.
  • At 10 mV the current is a pure sine after the first period, the Lissajous plot (I vs E) is an ellipse and the Fourier transform shows a single line at 1 mHz: the system behaves linearly.
  • At 50 mV the current response is distorted, the Lissajous plot departs from an ellipse and a third harmonic appears at 3 mHz in the Fourier spectrum, definite proof of non-linearity.
  • Only the third harmonic appears because the symmetry factor is αf = 0.5; with αf = 0.1 a second harmonic appears at 2 mHz as well. More harmonics appear as the amplitude increases.
  • The Fourier transform is far more sensitive and quantitative than the shape-based, qualitative Lissajous method. The THD, NSD and NSR indicators in EC-Lab are calculated from this discrete Fourier transform.

Figures

CVSim settings: E mechanism, both redox species initially in solution, 1 mHz sine excitation, E1 = +10 mV and E2 = -10 mV.
Figure 1. CVSim settings: E mechanism, both redox species initially in solution, 1 mHz sine excitation, E1 = +10 mV and E2 = -10 mV.
Potential modulation with 10 mV amplitude.
Figure 2. Potential modulation with 10 mV amplitude.
With 10 mV excitation the current response is a pure sine after the first period.
Figure 3. With 10 mV excitation the current response is a pure sine after the first period.
Lissajous plot (I vs E) for 10 mV is an ellipse: the response is linear.
Figure 4. Lissajous plot (I vs E) for 10 mV is an ellipse: the response is linear.
Fourier transform of the current for 10 mV: a single line at the 1 mHz excitation frequency.
Figure 5. Fourier transform of the current for 10 mV: a single line at the 1 mHz excitation frequency.
Potential modulation with 50 mV amplitude.
Figure 6. Potential modulation with 50 mV amplitude.
With 50 mV excitation the current response is distorted; it is no longer a single sine.
Figure 7. With 50 mV excitation the current response is distorted; it is no longer a single sine.
The Lissajous plot for 50 mV departs from an ellipse.
Figure 8. The Lissajous plot for 50 mV departs from an ellipse.
Fourier transform for 50 mV and αf = 0.5: a third harmonic at 3 mHz, definite proof of non-linearity.
Figure 9. Fourier transform for 50 mV and αf = 0.5: a third harmonic at 3 mHz, definite proof of non-linearity.
Same conditions with αf = 0.1: because symmetry is broken a second harmonic also appears at 2 mHz.
Figure 10. Same conditions with αf = 0.1: because symmetry is broken a second harmonic also appears at 2 mHz.

Images: Bio-Logic Science Instruments; taken from the source article.

Related application notes

Related curriculum module: Electrochemical Impedance Spectroscopy (EIS)

Related products:EC-Lab®SP-150eSP-300
Source: Bio-Logic Learning Center, “CVSim as a learning tool III: Linearity in EIS” · Source updated: November 18, 2024
Copyright of the images and the original text belongs to Bio-Logic SAS.

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