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# How to make reliable EIS measurements

Latest updated: February 12, 2020

Three quality indicators are available in EC-Lab® to ensure the reliability of EIS measurements.

**1 ^{st} Quality Indicator: the Total Harmonic Distortion (THD)**

THD indicates if the amplitude of the current or potential modulation applied to the system is small enough to consider that the system behaves linearly. If it behaves non-linearly, the output signal will contain some harmonics. THD quantifies the non-linearity by evaluating the amplitudes of the N harmonics:

$$THD_N=\frac{1}{\text{|}Y_f\text{|}} \sqrt{\sum_{k=2}^N\text{|}Y_k\text{|}^2}$$ (1)

where |*Y*_{f}| is the amplitude of the signal at the fundamental frequency *f* (or first harmonic) and |*Y*_{k}| is the amplitude of the k^{th} harmonic number.

THD is expressed as a percentage. Generally, it is considered that a THD below 5 % is acceptable. In EC-Lab®, it is calculated on the potential and on the current and over 7 harmonics including the fundamental (N = 7 in Eq. 1).

**2 ^{nd} Quality Indicator: the Non-Stationary Distortion (NSD)**

We can distinguish two causes for the non-stationarity of a system: i) the response of the system has not reached its permanent regime; ii) the parameters defining the system are changing with time.

The response of a non-stationary system will contain, in addition to the fundamental frequency, some adjacent frequencies.

Hence the NSD indicator is defined as:

$$NSD_{\Delta f} = \frac{1}{\text{|}Y_f\text{|}} \sqrt{\text{|}Y_{f-\Delta f}\text{|}^2 + \text{|}Y_{f+\Delta f}\text{|}^2}$$ (2)

where |Y* _{f}*| is the amplitude of the signal at the fundamental frequency f (or first harmonic), |Y

*-Δ*

_{f}*f*| et |Y

*+Δ*

_{f}*f*| are the amplitudes of the peaks right next to the fundamental frequency and Δ

*f*is the frequency resolution , equal to the inverse of the total measurement time.

NSD is expressed in percentage and calculated on the potential and on the current.

**3 ^{rd} Quality Indicator: the Noise to Signal Ratio (NSR)**

In an ideal EIS measurement, all the signal energy is contained in the fundamental frequency, but because of various factors such as the accuracy and precision of the measuring device or external perturbations, there might be some energy in other frequencies than the fundamental one, the harmonics and the adjacent frequencies. In this document, this additional signal is called noise.

NSR quantifies the extent of noise in the measurement. It is expressed using the following formula:

$$NSR_f=\frac{1}{\text{|}Y_f\text{|}}\sqrt{\sum_k^{}\text{|}Y_{k\Delta f}\text{|}^2}$$ (3)

with: $k\Delta f \notin \{f;2f;3f;4f;5f;6f;7f;f- \Delta f; f+ \Delta f \}$

It represents all the signals not contained in:

- The fundamental frequency,
- The 7 harmonics used to calculate THD
- The signal at frequencies adjacent to the fundamental frequency used to calculate NSD.

**How do I use them?**

*Finding the right amplitude*: NSR and NSD can be used to find the right AC amplitude of the input signal in your EIS measurements. The NSR needs to be minimized but the THD should not exceed 5% at all frequencies. 5% is an arbitrary value and depends on your system and on the frequency. 5% can also be used as a threshold value above which the EIS data can be considered as non-reliable.

*Finding the right frequency range*: similarly, NSD can also be used to establish at which frequencies can your data be considered reliable. Usually, NSD will increase at lower frequencies unless your system changes in a faster way than the input frequency, which is never really the case. Time-variance usually leads to slow changes, only visible below 1 Hz. Therefore, if your system time-variance cannot be adapted, the data obtained at a frequency for which the NSD is above 5% should be left out.

**For further information:**

Application notes

EIS Quality Indicators: THD, NSD & NSR – Electrochemistry, Battery & Corrosion – Application Note 64

EIS Quality Indicators THD – Electrochemistry, Battery & Corrosion – Application Note 65

White paper

Systems and EIS quality indicators – Electrochemistry

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