Introduction

Electrochemical impedance spectroscopy (EIS) is a non-destructive method that characterizes electrochemical processes inside the battery. It can be used both in battery steady state during idling, and dynamically during battery cycling [1, 2]. Subsequently, various dependencies such as state-of-charge, temperature, and C-rates can be covered. During EIS measurements, an AC signal over a wide frequency range is applied to the battery, and the cell response is measured [3].

With the wider spread of electric cars and the use of traction lithium-ion batteries, there is increasing pressure to diagnose them as accurately as possible and monitor their State of Health (SOH) while they are in use. For this reason, the speed of measurement using the conventional single-sine method is no longer sufficient [4].

Newly developed faster methods include the excitation of lithium-ion batteries using a multi-sine signal, whose response is then processed using fast-Fourier transform (FFT) or excitation of the battery using a Pseudo Random Sequence (PRS) [5].

This paper provides an overview and a description of various EIS methods and highlights their advantages and disadvantages.

Methods

The paper compares the single-sine method, currently the most widely used method for lithium-ion battery diagnostics, with innovative methods that use, for example, multi-sine signal processing using fast-Fourier transform or battery excitation using pseudo-random sequence. Impedance measurement methods can be divided into two essential subgroups, the first category consists of single-sine frequency methods, and the second category includes broadband methods. A more detailed classification of the perspective methods is given in Fig. 1.

Fig. 1
figure 1

Classification of EIS methods

Full size image

Single-sine battery excitation

A sinusoidal voltage or current excitation signal is applied at different frequencies, and the respective response of the current or voltage is examined. Impedance \(\widehat{Z}\) is from current phasor (\(\widehat{I})\) and current phasor (\(\widehat{U})\) calculated as:

$$\widehat{Z}=\frac{\widehat{U}}{\widehat{I}}.$$

Usually, the battery response to the input signal is measured from higher frequencies to lower frequencies. This is done mainly since transient response subsides earlier at higher frequencies and thus steady state occurs earlier at higher frequencies. The frequencies are in the range from 10 kHz to 0.01 Hz. The single-sine method is currently the most accurate.

Multi-sinusoidal excitation and data evaluation based on FT

The aim of the multi-sinusoidal excitation method is to decrease the duration of the measurement. The lithium-ion battery is not a truly linear system and to consider its linear approximation, it must be excited with a signal of specific amplitude. Superposing of multi-sinusoidal excitation leads to higher amplitudes of the multi-sinusoidal signal compared to a single sinusoidal signal. It means that when assuring the maximum amplitude of the excitation signal, amplitudes of sinusoidal signals have to be lower.

Assuming N sinusoidal signals in a linear circuit, the maximum amplitude of a single sinusoidal signal has to be equal to the maximum acceptable amplitude divided by √N. Fourier transform is used to transform a continuous signal from time domain to frequency domain. The Fourier transform S(ω) of the function s(t) is defined by the integral formula:

$$S\left(\omega \right)=\underset{-\infty }{\overset{\infty }{\int }}s\left(t\right){{\text{e}}}^{-{\text{j}}\omega t}{\text{d}}t.$$

The function s(t) can then be calculated from S(ω) using the inverse Fourier transform:

$$s\left(t\right)=\frac{1}{2\uppi }\underset{-\infty }{\overset{\infty }{\int }}S\left(\omega \right){{\text{e}}}^{{\text{j}}\omega t}{\text{d}}\omega .$$

A common way to write the complex spectrum is also the phasor notation with the expression S(jω), which corresponds to a phasor with magnitude \(\left|S(\omega )\right|\) and angle φ(ω).

The pairs in the Fourier transform are known as original s(t) and transformation S(jω). Symbol \(\mathcal{F}\) represents operator of Fourier transform. Relationship between s(t) and S(jω) is represented by notation:

$$S\left(j\omega \right)=\mathcal{F}\left[s\left(t\right)\right] \mathrm{ \,{and} } \,{s\left(t\right)}={\mathcal{F}}^{-1}\left[S\left(j\omega \right)\right].$$

For engineer’s purposes is ω used for angular frequency, and S(jω) represents spectrum of signal s(t). Spectrum is complex variable which can be expressed as follows:

$$S\left({\text{j}}\omega \right)= \left|S({\text{j}}\omega )\right|{e}^{\mathrm{j arg} S({\text{j}}\omega )}.$$

The parameter \(\left|S({\text{j}}\omega )\right|\) is called the amplitude spectrum of signal and angle \(\varphi \left({\text{j}}\omega \right)={\text{arg}}S({\text{j}}\omega )\) is signal phase spectrum.

Considering all the above assumptions, we can then write for signals u(t), i(t) in frequency domain:

$$Z\left({\text{j}}\omega \right)=\frac{U({\text{j}}\omega )}{I({\text{j}}\omega )}.$$

Two signals—current signal and voltage signal—are obtained during measurement. The sensed signal is processed using Fourier transform which gives the magnitude and phase of the impedance of the measured system for each frequency. In actual applications, the Fast Fourier Transform is often used instead of the Fourier Transform, which is a specific modification of the above mentioned approach in discrete time, with the advantage of shorter computational time.

Measuring techniques of electrochemical impedance spectroscopy based on PRS

An alternative approach to applying sinusoidal signal injection is to use broadband excitation, such as a pseudo-random sequence (PRS) (sometimes also referred to as pseudo-random binary sequence—PRBS), which enables fast and low-complexity broadband battery impedance measurements.

Linear feedback shift register (LFSR) is usually used for PRS generation. LFSR is most often a shift register whose input bit is driven by the XOR of some bits of the overall shift register value. This means that the generated value is always determined by the previous state and the register has a finite number N of possible states defined by the equation: N = 2n–1, where n = 1, 2, 3 … is the order of the shift register.

The main advantages of this method are the significant reduction in measurement time, its low complexity, and its online real-time application.

A common problem with most measurements based on this principle is the negative impact of noise on the measured values because of the wide bandwidth. For this reason, there is pressure to modify the sequence to achieve a higher signal-to-noise ratio. Table 1 shows the waveforms of variously modified PRS excitation signals.

Table 1 Comparison of PRS methods; adapted from [6,7,8]
Full size table

An example of a basic pseudo-random frequency signal is shown in Fig. 2. A common problem with most measurements based on this principle is the very significant negative impact of noise on the resulting measured values. For this reason, individual research teams try to slightly modify the generated sequence to achieve a higher signal-to-noise ratio.

Fig. 2
figure 2

PRS sequence

Full size image

The following subsections describe three techniques used to improve the general PRS method and their advantages.

Measurement method based on dual PRS

A method using the PRS basis is presented by Du et al. [8]. They aimed to reduce noise as much as possible by extending the frequency band that has a sufficient power spectrum.

In this case, the proposed dual DPRS signal is composed of two serially sorted partial sequences, which together form the signal that excites the battery. They achieved the required result by folding individual PRS into a dual signal.

Although in the time domain, the proposed DPRS signal has the same amplitude as the standard PRS signal, in the frequency domain the authors are able to obtain 8.46 times higher usable power than the PRS signal. This increase in power can significantly improve the interference immunity and makes it more suitable for broadband measurements of lithium-ion batteries. Another advantage of this modification is the significant reduction in the volume of measured data, which then requires less computational power to process the signal [8].

The functionality of the proposed method was verified by an experimental study. This verification was performed on three cells 18,650 which had different SOH, State of Charge (SOC) and temperatures. To reduce the required measurement time, the frequency range was 0.1–720 Hz and the resulting measurement time was 10.16 s. The amplitude of the applied excitation current was 1 A, and the measurement achieved a Normalized Root Mean Square Error of 0.1%.

Measurement method based on ternary signal

A method based on a ternary signal instead of PRS or DPRS was proposed by Sihvo et al. [6] to reduce the measurement time. A ternary signal is a ternary sequence with quadratic residues. The sequence consists of two consecutive sequences and results in a generated signal that has three amplitude levels and zero power on all harmonics of even order [6].

The specificity of the ternary sequence in the time domain is one extra signal level, in the frequency domain it is mainly a significantly higher power spectrum for individual frequencies, which makes it possible to limit the influence of interference on the results.

The functionality of this method was verified on a lithium-ion cell with lithium iron phosphate cathode, which had a nominal voltage of 3.3 V and a capacity of 2.5 Ah. The range of frequencies tested was 0.21 Hz–3.5 kHz. The amplitude of the excitation ternary signal is 1.375 A, the time required for the measurement is 4.7 s. When compared with the conventional measurement method, an RMS error of 2.9% was achieved [6].

Measurement method using discrete-interval-binary-sequence injection

In contrast to the previous methods, this is not a randomly generated signal, but a signal composed of binary sequences that are specifically chosen to achieve the most accurate results.

A discrete interval binary sequence (DIBS) is a pseudo-random sequence that is software optimized to get as much energy as possible into the desired frequencies without increasing the signal amplitude. Compared to a conventional PRS signal, it is therefore possible to achieve much higher power densities for specified harmonic frequencies. Another advantage is the possibility to precisely select the target frequencies according to the specific processes to be monitored in the battery [7].

The suitability of the chosen method was experimentally verified on a cell type 21,700 with a nominal voltage value of 3.6 V and a nominal capacity of 5 Ah. The frequency of the excitation signal was in the range of 3.1 Hz–3 kHz and the amplitude of the injected signal was chosen to be 0.025 A. Unfortunately, the authors do not present specific measurement outputs for this method from which the error could be clearly determined.

Table 1 brings a comparison of the different methods described in the subsections above.

Results

Based on the analysis of above-introduced methods, Table 2 makes it possible to evaluate and compare the methods and to find out the positives and negatives of each diagnostic method. The presented parameters are extracted from the mentioned studies.

Table 2 Comparison of EIS methods
Full size table

When comparing the duration of different methods, frequency range has to be kept in mind. For instance, Dual PRBS method with 10.16 s duration is almost equal to reciprocal value of the lowest applied frequency 0.1 Hz, while for Single sinusoidal method 500 s is five times more than that.

It is important to note that generally, the impedance measurement accuracy is the highest around the nominal, often meaning middle, frequency and impedance measurement points of the measurement device. It decreases with increasing distance toward its frequency and impedance range limits, as illustrated in Fig. 3. Moreover, it is dependent on the limited phase angle of the impedance, and it is specified for the defined signal power level. Several EIS measuring instruments currently in use are not impedance adapted to the low impedances measured during Lithium-ion battery diagnostics. Because of this, the measuring device may be presented as highly accurate, but out of the range required for Lithium-ion battery measurements. This issue is reflected in Fig. 3, where f and \(\left|Z\right|\) are the measured values and f0 Z0 are the nominal values of the instrument, δ is the standard deviation.

Fig. 3
figure 3

Typical accuracy of an impedance meter

Full size image

Conclusion

Diagnostic of lithium-ion batteries using EIS is a perspective method that can significantly reduce the time required for testing compared to different methods, such as capacity tests. Nevertheless, single-sine methods, which are currently the most used EIS methods, are too slow to be used for online battery diagnostics in a battery management system. To reduce the required measurement time, it is possible to use one of the innovative methods described in this research, such as PRS or the use of a multi-sine signal and its post-processing by FFT, however, it is often for the cost of lower impedance measurement accuracy.

In any case, the quality of the measured data depends on the level of noise in the system, which seems to be a critical parameter determining its usability and it is necessary to properly assess it. Thus, a future study will focus on investigating the performance of the methods under normalized conditions and utilizing a consistent evaluation metric.