How to read EIS accuracy contour plots Electrochemistry & Battery – Application Note 54

Introduction

The impedance measurement accuracy of an impedance analyzer depends on the impedance magnitude and on the measurement frequency. Potentiostats/galvanostats/Impedancemeters specifications are given in graphs called accuracy contour plots¹. The aim of this note is to help the user understand and use EIS accuracy contour plots.

Contour Plot

Figure 1: 3D graph of a two-variable function, level curve (blue curve) and projection in the x, y plan (red curve).

Contour plots are isovalue curves of a function depending on two variables. As an example 3D graph of a two variables function f(x, y) = x² + y² and a level curve of this function for f(x, y) = 10 are shown in Fig. 1. The level curve is a cross-section of the 3D graph parallel to the (x, y) plan at f(x, y) = 10. The projection of this curve on the x, y plane is a contour plot. For any point (x, y) inside of this contour, we have f(x, y) < 10 and for any point outside of this contour we have f(x, y) ≥ 10.

EIS Accuracy Contour Plot

SP-300 EIS Accuracy Contour Plot

The scheme of an EIS accuracy contour plot is shown in Fig. 2.

Figure 2: Scheme of an EIS accuracy contour plot.

The two horizontal limits (high (5) and low (1) Z) are related to the ability of the instrument to accurately measure/control the current. To increase the high Z portion of accuracy contour plot, the instrument has to be able to measure low current (often related to the electrometer impedance in the specifications).

To increase the low part of the accuracy contour plot, the instrument has to be able to manage high current. A current booster may be added to improve the measurement in this area. The vertical limit (3) depends on the regulation speed of the instrument. This is the maximum frequency that can be reached by the instrument with an acceptable error. It is related to the bandwidth of the instrument. The two other limits (4) and (2) are related to stray capacitor and stray inductor, respectively. Inductive effects are very sensitive to current levels and the cell connections.

The contour plot of an SP-300 potentiostat is shown in Fig. 3. The shaded areas show different ranges of impedances that can be measured at various frequencies within a specified error in magnitude and in phase. These errors refer to the borders and not to the areas as explained below.

Figure 3: SP300 EIS accuracy contour plot. Black dot: f = 106 Hz, |Z| = 104 Ω, white dot: f = 104 Hz, |Z| = 102 Ω.

Example

Let us consider the black dot in Fig. 3. This dot corresponds to the measurement of an impedance |Z| = 10⁴ Ω measured at a frequency f = 10⁶ Hz. As this measurement is located between the 0.3 %, 0.3^◦ and 1 %, 1^◦ contour plots, the accuracy is 0.3 % < |Z|/|Z| < 1 %, for the modulus (relative error) and 0.3^◦ < φ_Z < 1^◦ for the phase (absolute error). The phase is given in absolute error because the relative error could be indeterminate in the case where φ = 0. An impedance whose modulus is 100 Ω at 10⁴ Hz is measured with a relative uncertainty lower than 0.3 % for the modulus and lower than 0.3^◦ for the phase (Fig. 3, white dot).

How to Use an EIS Contour Plot

Impedance of the R1+R2/C2 Circuit

The impedance for the R1+R2/C2 is given by

$$Z(f)=R_1+ \frac{R_2}{1+R_2C_2j2\pi f}\tag{1}$$

The modulus of the impedance as a function of the frequency is shown in Fig. 4 for R₁ = 0.1 Ω, R₂ = 3 Ω and C₂ = 10⁻⁶ F.

Figure 4: Modulus Bode diagram of the impedance for the R₁+R₂ /C₂ circuit. R₁ = 0.1 Ω, R₂ = 3 Ω and C₂ = 10^– 6 F.

Example #1

The modulus Bode impedance diagram of this electrical circuit R₁+R₂/C₂ and the SP-300 contour plot are superimposed in Fig. 5. This figure allows us to predict the precision of the impedance measurements as a function of frequency. One has to determine the frequency values where the modulus diagram moves into a different precision domain. For instance, the modulus diagram moves from the high precision domain (|Z|/|Z| < 0.3 % and φ_Z < 0.3^◦) to the mid-range precision domain (0.3 < |Z|/|Z| <1 % and 0.3 < φ_Z < 1^◦) for a frequency moving from f₁ ≈ 10 kHz to f₂ ≈ 60 kHz (Fig. 5).

Figure 5: Superimposition of the modulus Bode impedance diagram of the electrical circuit R₁+R₂/C₂ and the SP-300 contour plot.

The Nyquist impedance diagram of the R₁+R₂/C₂ circuit is shown in Fig. 6 with the impedances measured at f₁, f₂ and f₃.

Figure 6: Nyquist diagram of the impedance for the R1+R2 /C2 circuit, f1 ≈ 10 kHz, f2 ≈ 60 kHz, f3 ≈ 100 kHz.

Tab. I shows how the precision of the impedance modulus and phase evolves with the frequency for the R₁+R₂/C₂ electrical circuit with intermediate Rs and C values.
Table I:  Impedance measurement accuracy as a function of the frequency. R₁ = 0.1 Ω, R₂ = 3 Ω and C₂ = 10⁻⁶ F.

Example #2

Another situation can be observed for high values of R₂ and low values of C₂, for instance R₁ = 10⁴ Ω, R₂ = 0.2 × 10⁷ Ω and C₂ = 5 × 10⁻⁹ F (Figs. 7 and 8).

Figure 7: Superimposition of the modulus Bode impedance diagram of the electrical circuit R₁+R₂/C₂ (high Rs values and low C value) and the SP-300 contour plot.

Figure 8: Nyquist diagram of the R1+R2/C2 circuit. R1= R₁ = 10⁴ Ω, R₂ = 0.2.10⁷ Ω and C₂ = 5.10^-10 F.

As the frequency reaches lower values, measurement precision moves outside of the best precision domain. Tab. II shows how the precision of the impedance modulus and phase, evolves with the frequency for an R+R/C electrical circuit with high R values and low C values.
Tab. II: Change of impedance measurement accuracy with frequency. R₁ = 10⁴ Ω, R₂ = 0.2 × 10⁷ Ω and C₂ = 5 × 10−¹⁰ F.