Multisine Excitation Research Articles

Electrochemical Impedance Spectroscopy Beyond Linearity and Stationarity

Electrochemical impedance spectroscopy (EIS) is a widely used experimental technique for characterising materials and electrochemical devices. By measuring the impedance at selected frequencies, one can distinguish between electrochemical processes happening at different time-scales (such as diffusion and charge transfer).Classical EIS measurements [1,2] require the electrochemical processes under investigation to behave as a linear time-invariant (LTI) system. However, electrochemical processes do not naturally satisfy this requirement: their dynamics are inherently nonlinear, and evolve depending on several parameters. For lithium-ion batteries, the dynamics change with temperature, state-of-charge, and state-of-health [3]. Due to the time-invariance constraint of EIS, the technique can only reveal information about electrochemical processes at operating points, that is, at fixed temperature and state-of-charge, while in steady-state. Moreover, due to the linearity constraint, EIS can only reveal information on linear dynamics about these operating points. Hence, classical EIS cannot be used to characterise electrochemical processes in operating conditions such as charge and discharge, or relaxation. These are severe limitations!In this contribution, we demonstrate how to measure impedance data beyond the limiting constraints of linearity and stationarity [4]. As a case-study, we measure the impedance of commercial lithium-ion batteries during charge and discharge using a recently-developed operando EIS technique [5,6] (leveraging the use of the frequency domain and multisine excitations). The measurements show that impedance data during operation is different from ‘classical’ impedance data (at rest). More specifically, the charge transfer resistance is shown to decrease during operation, which can be explained via physics-based battery models. This showcases that operando EIS is a promising experimental tool enabling the characterisation of electrochemical processes during operation (which is not possible with classical EIS). Examples of applications where operando EIS is promising include monitoring fast-charging [7] and parametrising physics-based models.[1] Orazem, M.E. and Tribollet, B., 2008. Electrochemical impedance spectroscopy. Wiley.[2] Wang, S., Zhang, J., Gharbi, O., Vivier, V., Gao, M. and Orazem, M.E., 2021. Electrochemical impedance spectroscopy. Nature Reviews Methods Primers, 1(1), p.41.[3] Doyle, M., Fuller, T.F. and Newman, J., 1993. Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. Journal of the Electrochemical society, 140(6), p.1526.[4] Hallemans, N., Howey, D., Battistel, A., Saniee, N.F., Scarpioni, F., Wouters, B., La Mantia, F., Hubin, A., Widanage, W.D. and Lataire, J., 2023. Electrochemical impedance spectroscopy beyond linearity and stationarity—A critical review. Electrochimica Acta, p.142939.[5] Hallemans, N., Pintelon, R., Van Gheem, E., Collet, T., Claessens, R., Wouters, B., Ramharter, K., Hubin, A. and Lataire, J., 2021. Best linear time-varying approximation of a general class of nonlinear time-varying systems. IEEE Transactions on Instrumentation and Measurement, 70, pp.1-14.[6] Hallemans, N., Widanage, W.D., Zhu, X., Moharana, S., Rashid, M., Hubin, A. and Lataire, J., 2022. Operando electrochemical impedance spectroscopy and its application to commercial Li-ion batteries. Journal of Power Sources, 547, p.232005.[7] Zhu, X., Hallemans, N., Wouters, B., Claessens, R., Lataire, J. and Hubin, A., 2022. Operando odd random phase electrochemical impedance spectroscopy as a promising tool for monitoring lithium-ion batteries during fast charging. Journal of Power Sources, 544, p.231852. Figure 1

Constructing nonlinear data-driven models from pitching wing experiments using multisine excitation signals

Accurate modelling of unsteady nonlinear aerodynamic loads is crucial for the effective design and control of aerodynamic systems. Data-driven modelling approaches have emerged as powerful tools for capturing intricate system dynamics without prior knowledge of the underlying physical principles. This paper focuses on the construction of nonlinear data-driven models for pitching wing systems using multisine excitation signals and their relative merits compared to swept sine signals. The polynomial nonlinear state-space (PNLSS) modelling framework is used, which has been proven in the system identification community to offer flexibility and richness in representing diverse nonlinear phenomena. Data are gathered with a dedicated experimental wind tunnel setup. A NACA 0018 wing is made to oscillate in pitch according to a swept sine and a multisine with a frequency up to 2 Hz. The experiments are repeated to cover the attached flow, light stall, and deep stall regimes. PNLSS models are trained on multisines and benchmarked against models trained on swept sines to assess their comparative performance. Then, all models are successfully cross-validated using single sine data at different angles of attack. We conclude that one single data-driven model can accurately represent the lift force across different linear (attached flow) and nonlinear (light and deep stall) operating regimes.

High-performance efficient embedded systems for impedance spectroscopy: Challenges and potentials

Impedance spectroscopy is nowadays in use in a wide range of applications where there is a particular need for efficient embedded measurement systems. Several embedded impedance spectrometers based on Field Programmable Gate Array (FPGA) have been proposed but have several drawbacks, notably high power consumption, high cost, and complex development frameworks. Meanwhile, microcontrollers have become more powerful, enabling the efficient design of embedded impedance spectrometers. A major challenge, however, is to achieve high measurement performance in terms of accuracy while complying with the conditions for impedance spectroscopy. This paper addresses the potential of microcontroller-based system design for embedded impedance spectrometers and proposes a methodology to comprehensively design compact and efficient microcontroller-based impedance measurement systems. After a review of possible measurement methods, we focus on the optimization of multi-sine excitation signals,the acquisition of current and voltage signals, and the efficient signal analysis, which are key elements in realizing embedded impedance measurement systems that achieve high quality measurements while maintaining linearity and system stability conditions. We demonstrate the efficiency of the proposed design methodology in the two application scenarios of battery diagnosis and bioimpedance spectroscopy.

Using multi-sine excitation and rigid body motion compensation in randomly sampled camera-based experimental modal analysis to improve SNR

Camera-based Experimental Modal Analysis (EMA) is able to measure full-field vibration modes contactlessly. However, camera measurements still suffer from a lower sampling frequency and lower Signal-to-Noise Ratio (SNR) compared to accelerometers and laser vibrometers. Since regular sampling confines the results to the Nyquist frequency (i.e., half the frame rate of a camera), in a previous paper we have proposed an approach that exploits images randomly sampled in time, allowing to measure modes beyond the Nyquist frequency. In this paper we employ such an approach and we propose a novel excitation scheme to increase the SNR, i.e., the specimen under investigation is excited at selected resonance frequencies, which results in high amplitudes of the structural responses. These frequencies are measured by accelerometers during a pre-test, and a multi-sine signal containing these frequency components is generated as structural excitation through a shaker. Furthermore, as Rigid Body Motion (RBM) due to pseudo-free suspension is not desired for EMA, in this paper we also propose a novel optimization workflow for RBM compensation. The accuracy of the proposed approach with multi-sine excitation is first evaluated numerically by reconstructing steady-state response signals. Secondly, the whole workflow (i.e., multi-sine excitation and RBM compensation) is validated on an experimental case with a 3D component and a stereo camera setup. From a sequence of randomly sampled images equivalent to a frame rate below 50fps (corresponding to an equivalent Nyquist frequency below 25Hz), four modes up to 250Hz (i.e., ten times higher than the Nyquist frequency) are extracted, which correlate well with the modes predicted through a finite element simulation.

A Novel Single-Probe Setup for Multifrequency Simultaneous Measurement of In-Circuit Impedance

The single-probe setup (SPS) based on the inductive coupling approach is a promising candidate for in-circuit impedance measurement of an energized electrical system due to its non-contact characteristics and simple configuration. However, the conventional SPS performs the measurement with a frequency-domain measuring instrument via stepped swept-sine excitation. It measures in-circuit impedance at only single frequency at a time while sweeping across frequencies of interest, which is not only inefficient but also not fast enough to capture in-circuit impedance with time-variant characteristics. To overcome these limitations, this paper proposes a novel SPS with a time-domain measuring instrument via multi-sine excitation. The proposed SPS can perform multi-frequency simultaneous measurement of in-circuit impedance without direct electrical contact. By combining it with a short-time Fourier transform (STFT) algorithm, in-circuit impedance with time-variant characteristics can be captured. Experimental case studies validate the capability and accuracy of the proposed SPS.

Bioimpedance spectroscopy measurement method based on multisine excitation and integer-period undersampling

Bioimpedance spectroscopy (BIS) is a detection technology that uses the bioimpedance characteristics of human tissues and their changes to analyze their physiological and pathological status, and is widely used in clinical and scientific research applications. Traditional BIS measurement must satisfy the Nyquist sampling theorem so as to ensure that the measurement signal has no frequency aliasing, but at the same time the sampling frequency and the number of sampling points will be increased, which will increase the computation and hardware costs. This paper proposes a novel BIS measurement method based on multisine excitation and integer-period undersampling (IPUS) technology. Firstly, the multisine-based IPUS theory is deduced, and the BIS measurement principle based on multisine excitation and IPUS technology is introduced. Secondly, a BIS measurement system based on a field-programmable gate array + analog-to-digital converter + digital-to-analog converter architecture is designed, and multisine excitation with 32 pseudo-logarithmically distributed frequency components in the range of 2–997 kHz is generated. Comparative BIS measurement experiments on three RC three-element models are carried out under the Nyquist sampling condition (sampling frequency fs = 2.56 MHz) and under the IPUS condition (fs = 512 kHz), respectively. Experimental results show that the mean amplitude error of BIS measurement under the Nyquist sampling condition is 0.80% (±1.19% SD), while the mean amplitude error under the IPUS condition is 1.02% (±1.13% SD). Moreover, the signal-to-noise ratio (SNR z ) is calculated in 40 repeated BIS measurements, where the mean SNR z is 63.60 dB under the IPUS condition, similar to the value of 62.77 dB under the Nyquist sampling condition. The proposed multisine-based IPUS theory and its implementation method in this paper can complete a BIS measurement with only one fundamental period, and the sampling frequency and sampling point requirements are lower than for Nyquist sampling, laying a theoretical and technical foundation for a BIS measurement system with reduced hardware and computation requirements.

Frequency domain parametric estimation of fractional order impedance models for Li-ion batteries

The impedance of a Li-ion battery contains information about its state of charge (SOC), state of health (SOH) and remaining useful life (RUL). Commonly, electrochemical impedance spectroscopy (EIS) is used as a nonparametric data-driven technique for estimating this impedance from current and voltage measurements. In this article, however, we propose a consistent parametric estimation method based on a fractional order equivalent circuit model (ECM) of the battery impedance. Contrary to the nonparametric impedance estimate, which is only defined at the discrete set of excited frequencies, the parametric estimate can be evaluated in every frequency of the frequency band of interest. Moreover, we are not limited to a single sine or multisine excitation signal. Instead, any persistently exciting signal, like for example a noise excitation signal, will suffice. The parametric estimation method is first validated on simulations and then applied to measurements of commercial Samsung 48X cells. For now, only batteries in rest, i.e. at a constant SOC after relaxation, are considered.

On the design of multisine signals for maintaining stability condition in impedance spectroscopy measurements of batteries

For measuring the impedance spectra of batteries, several challenges should be overcome to maintain the stability condition, especially at low frequencies in the mHz range. Even if the sine sweep signals realize the highest signal power, they lead to a severe stability problem at low frequencies in the mHz region. In this paper, we propose a novel design method for multisine excitation signals, which reduces measurement time and maintains battery stability. The novel excitation signals are structured in a dummy interval for considering the transient behavior of the battery and three folds of a multisine signal with an optimized crest factor, frequency bins, and measurement time. The system stability is evaluated by a consistency check based on the Linear Kramers–Kronig (LKK) method. The results show that the novel excitation signals outperform the often-used sinusoidal sweep signals by maintaining the stability condition during impedance measurements even at low frequencies in the mHz range. For example, for a signal with six frequencies/decade from 10 mHz to 1 kHz, a root means squares error of the LKK of 31.70 ppm could be realized instead of 201.21 ppm for the sine sweep, and the measurement time could be reduced to 315 s from 1440 s.

Crest Factor Optimization for Multisine Excitation Signals with Logarithmic Frequency Distribution Based on a Hybrid Stochastic-Deterministic Optimization Algorithm

For diagnosis of batteries and fuel cells based on impedance spectroscopy, excitation signals are required, including low frequencies down to the mHz range. This leads to a long measurement time and compromises the stability condition for impedance spectroscopy. Multisine excitation signals with logarithmic frequency distribution can significantly reduce the measurement time but need optimization of the crest factor to realize a high signal-to-noise ratio at all excitation frequencies and maintain at the same time the linearity and stability conditions of impedance spectroscopy. Crest factor optimization is challenging, as the obtained results strongly depend on the initial phase values and many trials are necessary. It takes a very long time and can not be easily performed automatically up to now. In this paper, we propose a time-efficient hybrid stochastic-deterministic crest factor optimization method for multisine signals with logarithmic frequency distribution. A sigmoid transform on the multisine signal gradually transforms the multi-frequency signal into a binary-alike signal. The crest factor is significantly decreased, but the phases of the singular frequency signals remain sub-optimal. Further optimization based on the Gauss-Newton algorithm can determine the final phases, realizing a lower crest factor. The proposed method is less sensitive to initial phase values and provides more reasonable results in a reasonable time. The validation on a Samsung INR-18650-25R Lithium-ion battery cell shows that the crest factor of the optimized multisine signals has a median of 3.62 ± 0.7 within 6 min of run time, which is significantly better than the best previous work in the state-of-the-art of 3.85 ± 0.11 for the same run time.

Improved crest factor minimization of multisine excitation signals using nonlinear optimization

In frequency-domain system identification, multisine waveforms are frequently used as an excitation signal. Supplying a low Crest Factor (CF) input signal to the actuator of the system is essential for measuring transfer functions with a high signal-to-noise ratio. In this paper, we present a method for achieving a lower CF compared to state-of-the-art methods. Our two-step method starts with Guillaume’s method, followed by solving a slack reformulation of the ∞-norm minimization problem for additional improvement. The proposed method is compared to the existing ones in terms of the achievable CF. The performance of the CF reduction methods has been analyzed using statistical methods.

Lumped Parametric Model for Skin Impedance Data in Patients with Postoperative Pain.

The societal and economic burden of unassessed and unmodeled postoperative pain is high and predicted to rise over the next decade, leading to over-dosing as a result of subjective (NRS-based) over-estimation by the patient. This study identifies how post-surgical trauma alters the parameters of impedance models, to detect and examine acute pain variability. Model identification is performed on clinical data captured from post-anesthetized patients, using Anspec-PRO prototype apriori validated for clinical pain assessment. The multisine excitation of this in-house developed device enables utilizing the complex skin impedance frequency response in data-driven electrical models. The single-dispersion Cole model is proposed to fit the clinical curve in the given frequency range. Changes in identified parameters are analyzed for correlation with the patient's reported pain for the same time moment. The results suggest a significant correlation for the capacitor component. Clinical Relevance- Individual model parameters validated on patients in the post-anesthesia care unit extend the knowledge for objective pain detection to positively influence the outcome of clinical analgesia management.

Improving Wireless Power Transfer Efficiency With DC/DC Boost Charger by Multi-Sine Excitation at 5.8 GHz

This work deals with new design techniques of RF to dc converter coupled with commercial dc/dc converters, for efficient wireless power transfer. The dc/dc boost converter operates as current sinks, thus imposing challenging design constraints. We investigate the impact of applying a multi-sine RF input signal on power transfer efficiency, involving a source-pull optimization to maximize RF to dc energy conversion and maximizing the current throughput. This letter provides an experimental demonstration of 1.8 improvements with respect to conventional design at the operative frequency of 5.8 GHz.

Design of multi-frequency electrical impedance tomography system based on multisine excitation and integer-period sampling

Starting from the principle that the integer-period sampling (IPS) of periodic signals is free of spectrum leakage, in this paper we propose the multisine-IPS theory, deduce theoretically the sampling rate setting formula of multisine-IPS condition for the first time, and build its realization method based on field-programmable gate array (FPGA) plus digital-to-analog converter (DAC) plus analog-to-digital converter (ADC). A new multi-frequency electrical impedance tomography (mfEIT) system based on multisine excitation and its IPS theory is developed, and a dual-target imaging model including a carrot stick and a cucumber stick is designed. The experiments of multi-frequency time-difference imaging and frequency-difference imaging are carried out on the mfEIT system. The experimental results show that the newly-designed mfEIT system can achieve full-band impedance measurements on multiple objective tissue boundary at 20 frequency points (2–997 kHz) within one fundamental period (1 ms), and the structure and position of biological tissues with different electrical properties can also be distinguished from the resulting images. The proposed multisine-IPS theory and its implementation method can complete a full-band impedance measurement within one multisine fundamental period, which lays a theoretical and technical foundation for developing high-speed mfEIT system.

Time-Frequency Design of a Multi-Sine Excitation With Random Phase and Controllable Amplitude for (Bio) Impedance Measurements

Impedance spectroscopy has become a standard electroanalytical technique to study (bio)electrochemical and physiological systems. From an instrumentation point of view, the measurement of impedance can be carried out either in the frequency domain using the classical frequency sweep method or in the time domain using a variety of broadband signals. While time-domain techniques can be implemented with relatively simple hardware and can achieve faster acquisition time, they are still not that popular because of their lower accuracy and modularity. In this work we present a method and an algorithm for computing the impedance spectrum of samples using an arbitrary time-domain signal excitation initially engineered from a multi-sine summation. The signal is designed in such a way that its spectral phase is derived from a discrete uniform distribution (obtained from a physically-true random number generator), but its time-domain maximum amplitude remains controllable. As such, samples can be excited with adjustable amplitude signals (as low as 50mV to 2V in the current setup) in order to avoid exciting the nonlinearities in the samples under test, which is critical for the very definition of the impedance. Furthermore, the method is proved to combine both speed of measurement, given that it permits to excite at once a large number of frequency components of the sample, as well as accuracy given that the signal’s power spectrum is relatively flat. We verified our method and algorithm for monitoring yogurt quality using (bio)impedance measurements over a period of 48 hours.

Uncertainty Characterization of a Practical System for Broadband Measurement of Battery EIS

Electrochemical impedance spectroscopy (EIS) is of fundamental importance for characterization and monitoring of rechargeable batteries in automotive, energy storage, and electronics sectors. In this article, a low-complexity practical system for EIS of batteries is presented, and the uncertainty associated with its measurement results is analyzed. The system is based on a voltage-controlled current generator and the acquisition of current and voltage signals from the battery under test. Arbitrary current waveforms can be generated, and thus the system is flexible and reconfigurable. In particular, a broadband multisine excitation signal is used, enabling faster measurements with respect to the conventional stepped sine approach. The proposed system is characterized by repeated measurements on a 18 650 lithium-ion battery. Moreover, it is validated by comparison with a commercial benchtop instrument, and with the ground truth provided by a discrete component Randles circuit. Results show that the system is able to accurately and repeatably measure the impedance of a battery in a wide frequency range from 0.05 Hz to 1 kHz. Furthermore, it provides results that are consistent with the commercial instrument and with the ground truth of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> Randles circuit. Therefore, the proposed system using the broadband multisine method is suitable for implementing fast and inexpensive monitoring capabilities in smart battery management systems.

Binary Sequences for Online Electrochemical Impedance Spectroscopy of Battery Cells

Online diagnostic of lithium-ion battery (LIB) cells may have significant impact on chemical energy storage systems. Electrochemical impedance spectroscopy (EIS) is widely used for the characterization of LIBs and could be the most appropriate technique for online diagnostic, but its response time should be shortened. This work investigates the usage of multisine excitation to shorten the measurement time and simplify the hardware implementation for EIS of battery cells. Two types of multisine binary sequences are considered: sigma–delta modulated multisine sequences (SDMSs) and maximum length binary sequences (MLBSs). Their applicability to online and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> EIS monitoring is evaluated by designing a measurement architecture also suitable to be implemented in a system-on-chip. The calibrated measurement system is compared with a benchtop reference instrument, reporting an RMSE deviation smaller than 5% in the frequency range of interest 1–200 Hz. The realized system is then used for online monitoring of the discharge process of a commercial 18650 LIB cell. The two proposed sequences are compared in terms of accuracy using a digital battery emulator circuit. Both the sequences demonstrated to be suitable for fast measurement and simple hardware integration, enabling online <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> EIS monitoring at cell level.

Identification of Parallel Wiener-Hammerstein Systems

This paper is focused on the problem of system identification for a class of nonlinear systems constituted of a Wiener subsystem and Hammerstein subsystem connected in parallel. Most of existing works on the identification of compound systems, involving Wiener and Hammerstein subsystems, have been focused on series connections. Presently, we present a frequency identification approach to determine estimates of the different components of the considered parallel Wiener-Hammerstein system. The approach comes in two versions. The first one consists in using repeatedly a one-frequency sine input experiment, with different frequencies in the various experiments. The second version consists in performing a signal experiment involving a sufficiently rich multi-sine excitation. The applicability of the approach and its relevance are highlighted by simulation.

Influence of friction bearings on the frequency response of a variable stiffness mechanism

This contribution examines a variable stiffness mechanism that is intended for the use as a boundary element in vibration tests. The mechanism consists of two antagonistic nonlinear springs guided along the central axis. The axis is axially guided via two bearings. This paper concerns itself with comparing the effects of friction and roller bearings on the frequency response. Further, a comparison of different measurement techniques for the dynamic characterization of the aforementioned mechanism is in focus. An experimental study utilizing a multisine, sine-sweep, and impulse excitation signal is conducted. A nonlinear numerical model as well as a linear approximation are presented and compared with the measurement results. The results indicate that friction bearings have a strong influence on the frequency response of the system, especially at low displacement amplitudes. The influence of the friction contact on the results of different measurement methods is shown. The presented variable stiffness mechanism with the roller bearing configuration yielded in a predictable vibration behavior even for high mechanism pretensions.

Design of a multi-frequency electrical impedance tomography system based on multisine excitation and integer-period sampling

本文从周期信号的整周期采样无频谱泄露这一原理出发，提出基于multisine信号的整周期采样理论，首次从理论上推导出满足multisine整周期采样的采样率设置条件，构建了基于FPGA+数模转换器+模数转换器的整周期采样实现方法，研制了一种基于multisine激励和整周期采样的新型多频电阻抗成像(EIT)系统；设计了胡萝卜棒+黄瓜棒的双目标成像模型，并进行了多频时差成像和频差成像实验。实验表明，本多频电阻抗成像系统能够一个基波周期（1 ms）内实现20个频率点（2 kHz -997 kHz）多目标组织边界的全频阻抗测量，成像结果可区分具有不同电特性生物组织的结构与位置。本文提出的基于multisine信号的整周期采样理论及其实现方法，只需一个multisine基波周期即可完成一次全频阻抗测量，为研制高速多频EIT系统奠定了理论和技术基础。

Shaping multisine excitation for closed-loop identification of a flexible transmission

Shaping multisine excitation for closed-loop identification of a flexible transmission