A new impedance spectroscopy measurement procedure for automotive battery cells is presented, which is based on waveform shaping. The method is optimized towards a short measurement duration, high excitation power and increased frequency resolution and overcomes limitations of established methods. For a given spectral magnitude profile, a corresponding time domain waveform is derived from the inverse discrete Fourier transform. Applying an identical initial phase angle for each frequency component, the resulting signal exhibits a high peak-to-peak amplitude at relatively low total excitation power. This limits the maximum allowed power for quasi-linear excitation. Altering the phase angles randomly spreads the excitation power across the complete measurement duration. Thereby, linearity is preserved at higher excitation power. A large set of phase patterns is evaluated statistically in order to obtain a phase pattern with a significant peak-to-peak amplitude decrease. By means of numerical optimization, even further peak-to-peak amplitude reduction is achieved. Including window functions in the synthesis concept minimizes spectral leakage without compromising the spectral signal magnitude in the frequency range of interest. A time domain waveform optimized for impedance spectroscopy on lithium ion cells is synthesized based on the proposed approach and evaluated on real automotive cells. The resulting impedance data show good concordance with established standard measurement procedures at significantly reduced measurement duration and charge throughput. Additionally, increased frequency resolution is achieved, enhancing the level of detail of the obtained impedance data. The method is used for improved localization of aging effects in the cells, without further stress of the cells by the measurement procedure.

Lithium ion cells for use in electric vehicles are subject to ongoing research, aiming to reach higher energy
densities and longer lifetimes while maintaining safe operation

Electrochemical impedance spectroscopy (EIS) is an established method in the
characterization of electrochemical systems. By analyzing measurement data of
current and voltage of a lithium ion cell, the frequency-dependent complex
impedance

The application of EIS during an ongoing long-term aging test on lithium ion
cells for automotive energy storage is a challenging task. EIS is usually
performed on small cells with low capacity or on dedicated test structures
with low-power test equipment

An established test approach is based on sinusoidal excitation with a given
amplitude, which ensures a quasi-linear response. The impedance is measured
at a discrete number of frequencies, one at a time. This method is robust
against influences of noise, as the spectral excitation power is concentrated
on a single frequency. However, especially for low frequencies in the

In impedance measurements, a dense frequency resolution is desirable. For the
single-frequency approach, several periods at each frequency need to be
analyzed sequentially. The complete measurement procedure is very
time consuming, especially at low frequencies. The measurement requires a
duration of several hours. Therefore, time invariance of the device under test (DUT) cannot be
guaranteed

An alternative approach optimized towards short measurement duration is based
on pulse excitation. As opposed to the single-frequency approach, the impedance
is simultaneously evaluated at multiple frequencies. The frequency domain
characteristic results from the time domain pulse shape. Although rectangular
pulses can be synthesized on almost any test equipment, their spectra are
less useful due to the non-uniform power density. Using sophisticated test
equipment capable of arbitrary waveform generation, spectrally optimized
pulse shapes like sinc (

Broadband excitation based on band-limited noise signals avoids temporal
energy concentration. However, spectral magnitude is randomly distributed
across the observed frequency range for practical noise signals. Therefore,
suitable excitation is not guaranteed for all frequencies of interest

In this work, a novel excitation signal generation method is presented,
aiming to combine the advantageous spectral magnitude properties of pulse
signals with widespread time domain characteristics. This is achieved using
a waveform-shaping approach based on the inverse discrete Fourier transform (IDFT)
in the digital signal synthesis

This paper is based on a previous conference publication

The impedance of any electrochemical system can be defined as the voltage
response following a current excitation

Nyquist plot for lithium
ion cell impedance from 10

Impedance measurements on a DUT are performed by applying
an excitation signal and measuring the system response. In theory, current
and voltage excitation can be applied equivalently. In the event of low
internal impedance in the

For the interpretation of EIS results, the measured impedance is often fitted
to an electrochemically motivated equivalent circuit. This allows a link
between the measured electrical characteristics and the underlying
electrochemical processes inside the cell. The effects observable in
automotive lithium ion cells can roughly be separated according to their
characteristic frequency range. In the

In Fig.

For robust model fitting, appropriate impedance data quality and integrity are required. Data quality refers to frequency resolution and the signal-to-noise ratio (SNR), which is limited by the excitation signal amplitude and the performance of the test equipment.

Furthermore, the model fitting approach implicitly requires that the underlying system has to be linear and time invariant. These conditions are not necessarily satisfied by a lithium ion cell. Therefore, verification of the EIS results' integrity is required.

The Kramers–Kronig relations describe the interdependency of the real and
imaginary part of causal LTI (linear time invariant) system transfer
functions in system theory. A Kramers–Kronig compliance test method
applicable to EIS results is presented in

As the impedance is defined in the frequency domain, spectra calculation usually involves a time–frequency domain transform like the discrete Fourier transform (DFT). The inverse DFT (IDFT) is suitable for the synthesis of waveforms featuring well-defined frequency domain properties. The main characteristics of the (I)DFT are briefly summarized in this section.

Time–frequency relations

For an input time domain signal consisting of

In theory,

Properties of real valued time domain signals

The IDFT of a frequency domain signal

Signal power

The power

Flow chart of an impedance spectroscopy measurement scenario. The solid arrows describe a generic EIS measurement procedure, while the dashed red arrows indicate typical error contributions.

An overview of a typical EIS measurement scenario is depicted in
Fig.

Noise effects induced by non-ideal components in the measurement unit
compromise the resulting data quality. Besides noise influence, additional
measurement errors not related to noise may occur, because the DUT's
impedance is interpreted as the transfer function of a causal, linear and
time-invariant system for most applications. Only if these conditions are
satisfied, the EIS results can be considered valid. The aforementioned linear
Kramers–Kronig test (LinKK) is a suitable method to verify data integrity

In an ideal measurement scenario, the mean residual

Time-variant behavior due to variation of the cell's state of charge or temperature

The impedance of a lithium ion cell is dependent on various influence
factors, such as temperature, state of charge and state of health. Therefore,
the impedance changes over time (

Linearity violation caused by excessive peak-to-peak voltage amplitudes

Non-linear effects in the DUT are stimulated by high values of the
peak-to-peak voltage amplitude

Noise contributions

Noise influences induced by non-ideal components of the measurement unit also
lead to increased values of the residual

The excitation source and the measurement unit are encapsulated into an electrochemical workstation and cannot be replaced by alternative components. As a consequence, noise performance improvements achieved by hardware modification are generally not available. Instead, this work focuses on defining an optimized measurement procedure based on improved waveform shapes to enhance the data quality without compromising data integrity, regardless of the test equipment used.

Arithmetic mean of the linear
Kramers–Kronig residual

This section provides a short overview of established EIS measurement procedures, discussing the main advantages and disadvantages of each method.

In a conventional EIS approach, the DUT is stimulated at only one frequency
at a time using sinusoidal excitation signals. In literature, this procedure
is referred to as stepped-sine EIS

The complete measurement duration can be calculated using the following
equation, assuming

In order to reduce measurement duration while preserving the same frequency
resolution, broadband excitation using pulse waveforms can be applied.
As opposed to sinusoidal signals, multiple frequency components are present in
the excitation signal

The spectral density characteristic determine the performance of a pulse waveform for EIS. For most commercial test equipment, only rectangular pulses are available, suitable for measurements with low demand on duration and charge throughput. Such an excitation waveform provides a non-uniform power density across the frequency range of interest, which results in a poor signal-to-noise ratio.

Alternative pulse forms like sinc pulses enable a homogeneous broadband
excitation

For pulse signals covering the complete frequency range of interest, the
required measurement duration is calculated as a multiple of the period of
the minimum frequency of interest:

Even spectrally optimized pulse shapes like sinc pulses show low average signal power at high peak-to-peak amplitude in time domain, limiting the achievable quality of the measured impedance spectra due to linearity constraints.

As an alternative to broadband pulses, signals with more than one discrete
frequency component can be used to simultaneously provide sparse broadband
excitation, which is also referred to as multi-sine excitation

In the following, a generic signal synthesis concept for broadband signals is presented, which does not require a priori information about the DUT.

In this section, a waveform-shaping method is developed. The design flow can
be used to synthesize time domain waveforms based on an arbitrary spectral
magnitude profile

The measurement duration cannot be reduced significantly below

The time domain waveform

For the unambiguous definition of frequency domain signals, both magnitude
information

Top: normalized frequency domain
magnitude profile for a relative bandwidth of one frequency decade. Bottom:
synthesized time domain waveform

As mentioned in Sect.

As stated in Eq. (

Time domain waveforms with equal spectral magnitude and different phase characteristics.

Suitable phase patterns can be employed in order to reduce the peak-to-peak
amplitude

In Fig.

Peak-to-peak amplitude decrease distribution for 10 000 random phase patterns

As a consequence of the random phase shift introduced in
Sect.

The signal power of

Based on the definition in Eq. (

A numerical optimization procedure is applied for this purpose. The window
function introduced in Sect.

Numerical optimization flow chart.

The optimization uses the well-known Levenberg–Marquardt algorithm

The subsidiary cost functions

The resulting waveform

Optimization parameters used for
the synthesis of the time domain signal

In the previous sections, a waveform-shaping flow is developed and improved
in three steps, each addressing the main disadvantages of the previous
version. The signal

By employing random phase patterns, a significant peak-to-peak amplitude
decrease of up to 66

However,

A comparison of all signal optimization stages presented in this work is
given in Table

Time domain waveforms resulting from

Comparison of all waveforms discussed in this work.

For the practical evaluation of the proposed measurement procedure, a
precision lithium ion single-cell test unit developed by our research group
was used

In order to verify the spectral characteristics of the proposed waveform,

Verification of the time (top)
and frequency domain (bottom) characteristics of the proposed waveform

To evaluate the suitability of the proposed waveform for EIS studies, the
impedance is measured in the frequency range from 10

Especially at low frequencies, the spectrum obtained from the optimized waveform is enriched by the increased frequency resolution achieved by the broadband excitation.

Impedance measured using stepped
sine (blue) and

To evaluate the data quality and integrity of the measured spectra, a
Kramers–Kronig compliance test is performed on the impedance data obtained
using both methods. The Kramers–Kronig residual distribution for the
impedance spectra is depicted in Fig.

Due to the increased available excitation power for each frequency in
stepped-sine measurements, a lower residual is achieved compared to
measurements based on the proposed waveform at the cost of substantially
higher charge throughput and excessively longer measurement duration. In
Table

Linear Kramers–Kronig residual for impedance
spectra measured using stepped-sine (blue) and

Quantitative comparison of stepped-sine and optimized time domain EIS measurements on an automotive lithium ion cell.

A waveform-shaping method to generate broadband time domain signals based on user-defined magnitude profiles was developed and optimized for high signal power at low peak-to-peak amplitudes. Compared to common pulse waveforms with equal spectral properties, a significant decrease in peak-to-peak amplitude is achieved by means of random phase variation. The signal enhancements introduced by employing window functions are inherently included in the optimization work flow. By combining statistical and numerical optimization approaches, even lower peak-to-peak amplitude values are achieved. The resulting waveform exhibits only negligible pass-band ripple. As a consequence, the overall excitation signal power in EIS can be increased significantly while ensuring quasi-linear excitation of the DUT.

The concept was verified in simulation and measurement. EIS results based on the optimized time domain waveform show good concordance with reference stepped sine EIS measurements, drastically reducing both measurement duration and charge throughput. The numerical optimization is based on a parametric cost function, which allows to emphasize the target waveform properties in accordance to application-specific constraints. Therefore, the waveform-shaping method is not restricted to the synthesis of impedance spectroscopy excitation waveforms, but can also supplement other system identification applications, including audio channel characterization and ultrasonic measurement systems.

The dataset used in this work is available in the Supplement.

The authors declare that they have no conflict of interest.