Linear vs. non-linear systems in impedance measurements (EIS linearity) Battery – Application Note 9

Electrochemical Impedance Spectroscopy (EIS) is an interesting tool devoted to the study of linear systems. However, electrochemical systems are often non-linear. Before explaining the different ways to deal with this issue, we will point out the main differences between linear and non-linear systems. These differences of behavior are shown in Table I. The impedance measurement was performed using the potentiostatic mode. The potential is defined by:

$$E(t) = E_\text {WE} + V_\text a\ \mathrm{sin}(2\pi ft) \tag{1}$$

where E_WE is the stationary potential for the impedance measurement, V_a is the potential amplitude of the sine signal, f is the frequency, and t is the time.

Table I: Differences between linear and non-linear systems.

Three test circuits have been designed in order to highlight the differences in behavior between linear and non-linear systems.

The experiments described on this paper were performed with a test box specifically designed for the user to learn how to achieve impedance measurements on linear and non-linear electrochemical systems.

There are three different electrical circuits inside the test box simulating real electrochemical systems. With Test Box-3, it is possible to study general electrochemistry protocols like Cyclic Voltammetry or corrosion protocols such as Linear Polarization and Generalized Corrosion.

Figure 1 shows the electrical circuit of the test circuit #1, the corresponding steady-state curve (I vs. E_WE), and the Nyquist diagram of the impedance measured at points a and b of the steady-state curve. The impedance diagrams do not depend on the steady-state potential E_WE or on the amplitude V_a. The Nyquist impedance diagram displays two semi-circles. Only one impedance measurement is needed to characterize the equivalent test circuit #1. It is a worthwhile exercise to determine the different frequencies characterizing circuit test #1 and compare them with the experimental values shown on the impedance diagram of Fig. 3.

Impedance is measured at steady-state points a and b (Fig. 2, Fig. 3). . R₀= 501 Ω, R₁= 1 kΩ, R₂= 3.56 kΩ, C₁= 10 nF, C₂= 2.3 µF.

Figure 1: Test circuit #1

Figure 2: I vs. E_WE steady-state curve.

Figure 3: Nyquist diagram for the impedance.

Values of the various parameters of circuit #1 are obtained using ZFit, available in EC-Lab^® and EC-Lab^® Express software. The result of the fit is given in Figure 4 .

Figure 4: ZFit result on circuit #1 of Test Box-3.

To verify these results with test circuit #1, you can perform the following experiments:

1. Plot the steady-state I vs. E_WE curve using cyclic voltammetry technique (in EC‑Lab^® software, load the CV_circuit1.mps file, accept, and run the experiment).
2. Select the PEIS technique. Apply a constant potential E_WE = 0 V and perform an impedance measurement from f_i = 200 kHz to f_f = 1 Hz frequency with a low amplitude V_a = 10 mV (it is also possible to load the PEIS_circuit1.mps file).
3. In the PEIS technique, apply a constant potential E_WE = 0.4 V and perform an impedance measurement from f_i = 200 kHz to f_f = 1 Hz frequency with a low amplitude V_a = 10 mV (it is also possible to load the PEIS_0-4V_circuit1.mps file).
4. In the PEIS technique, apply a constant potential E_WE = 0.4 V and perform an impedance measurement from f_i = 200 kHz to f_f = 1 Hz frequency with a high amplitude V_a = 100 mV (it is also possible to load the PEIS_ampl100mV_circuit1.mps file).
5. Overlay the three impedance measurement curves in the Nyquist plot mode.

As test circuit #1 is a linear system, the three impedance diagrams should be identical.

Test circuit #2 is made mainly of two semi-conductor diodes. This is a model for exponential non-linearity. Test circuit #2 results from a circuit studied in [1].

The I vs. E_WE steady-state curve is not a straight line for test circuit #2. Therefore the test circuit #2 is a non-linear circuit. The impedance of this circuit depends on the steady-state value of the working electrode potential E_WE and on the amplitude V_a.

The Nyquist diagram of the impedance measurement performed on test circuit #2 is a semi-circle whose diameter changes along with the electrode potential E_WE. (Fig. 6). The impedance of a non-linear system is potential dependent.

Figure 5: I vs. E_WE steady-state curve.

Figure 6: Nyquist diagram for the impedance measured at point a and b.

Figure 8 shows the impedance change versus the amplitude V_a for a given E_WE value. We can clearly see that the semi-circle diameter decreases when V_a values increase. The impedance does not depend on the amplitude of the excitation signal for low values of amplitude (Fig. 8). In that case, this system’s behavior is similar to that of a linear system.
Therefore, one impedance measurement is not sufficient to characterize a non-linear system. It can be a good exercise to try and find an equivalent electrical circuit for test circuit #2. The user can also determine the electrical components’ values when its behavior can be compared to one of a linear system.

Figure 7: Nyquist impedance diagram measured for different values of potential (E_WE = 0.15 V).

Figure 8: diagram displaying the in-phase impedance (evaluated for a limiting value in low frequencies) versus log (V_a).

It is possible to verify these results with test circuit #2:

1. Plot the steady-state I vs. E_WE curve using cyclic voltammetry technique (in EC-Lab^® software, load the CV_circuit2.mps file, accept, and run the experiment).
2. Select PEIS technique. Apply a constant potential E_WE = 0.2 V and perform an impedance measurement from frequency f_i = 200 kHz to f_f = 1 Hz with an amplitude V_a = 1 mV (it is also possible to load the PEIS_0-2V_circuit2.mps file).
3. In the PEIS technique, apply a constant potential E_WE = 0.25 V and perform an impedance measurement from frequency f_i = 200 kHz to f_f = 1 Hz with an amplitude V_a = 1 mV (it is also possible to load the PEIS_0-25V_circuit2.mps file).
4. In the PEIS technique, apply a constant potential E_WE = 0.15 V and perform several impedance measurements from frequency f_i = 200 kHz to f_f = 1 Hz with an amplitude V_a/mV= 2.5, 5, 10, 25, 50, 100 (it is also possible to load the PEIS_0‑15V_xxmV_circuit2.mps files for the different values of V_pp).

Test circuit #3 mainly consists of two transistors. It is a model for metal passivation which has been extracted from [2]. This circuit has also been studied in [1-4].

The shape of the impedance diagram evolves along with the electrode potential E_WE as it can be observed in Figs. 9, 10 and 11. Let us consider the steady-state I vs. E_WE curve showing a peak (as in the case of metal passivation, Fig. 9). For low frequencies, the in-phase impedance is negative. Therefore, a part of the impedance diagram is on the left side of the complex plot, i.e. Re(Z(w)) < 0.

Figure 9: I vs. E_WE steady-state curve.

Figure 10: Nyquist diagram for the impedance which has been measured at point b.

Figure 11: Nyquist diagram for the impedance which has been measured at point a.

It is possible to verify these results with test circuit #3 by performing the following experiments:

1. Plot the steady-state I vs. E_WE curve using cyclic voltammetry technique (in EC-Lab^® software, load the CV_circuit3.mps file, accept, and run the experiment).
2. To plot diagram a on Fig. 11: select the PEIS technique and apply a constant potential E_WE = 0.8 V and perform an impedance measurement from frequency f_i = 200 kHz to f_f = 1 Hz with an amplitude V_a = 10 mV (it is also possible to load the PEIS_0-8V_circuit3.mps file).
3. To plot the b diagram (Fig. 10): in the PEIS technique apply a constant potential E_WE = 1.3 V and perform an impedance measurement from frequency f_i = 200 kHz to f_f = 1 Hz with an amplitude V_a = 10 mV (it is also possible to load the PEIS_1‑3V_circuit3.mps file).

Circuit test #1: the characteristic frequency, i.e., the frequency at the top of the semi-circle of an RC circuit, is given by

$$f_\text c = \frac{1}{2\pi \text{RC}} \tag{2}$$

The two characteristic frequencies for circuit test #1 are:

$$f_\text {c1} = \frac{1}{2\pi \times 10^3 \times 10^{-8}} ≈ 16\ \text {Hz} \tag{3}$$
$$f_\text {c2} = \frac{1}{2\pi \times 3.57\text {x}10^3 \times  2.2\text {x}10^{-6}} ≈ 20\ \text {Hz} \tag{4}$$

These values are close to the frequency values given at the top of the semi-circles present on Fig. 3.

Circuit test #2: the impedance diagram represents a semi-circle. The equivalent electrical circuit can be assimilated to an RC circuit. It is possible to determine the circuit resistance values on the Im(Z) vs. RE(Z) plot. They correspond to the in-phase impedance for the lowest frequencies (-Im(Z) ≈ 0). R value varies with E_WE.
For point a of the steady-state curve, we find: R_a= 420 Ω
and for point b: R_b= 70 Ω.

With the resistance values determined above, the user can determine the capacitance using the following equation:

$$C = \frac{1}{2\pi f_\text cR} \tag{5}$$

where f_c is the frequency at the top of the semi-circle. The results should be similar for R_a and R_b. We find C~490 nF.

Revised in 07/2018