The relentless pursuit of a green, low-carbon society is inextricably linked to the advancement of new energy industries. Within this landscape, battery energy storage systems, with the lithium-ion battery as its quintessential representative, play a pivotal role. Statistical data indicates that in the first half of 2024, China’s cumulative installed capacity of new energy storage reached 48.18 GW, with lithium-ion battery storage accounting for over 95% of this share. Renowned for their high energy density, long cycle life, high operating voltage, wide operational temperature range, and capability for fast charging, lithium-ion batteries have become indispensable across diverse sectors including consumer electronics, electric vehicles, and aerospace. Critical to ensuring the safe and reliable operation of these systems are internal state parameters, primarily the State of Charge (SOC) and State of Health (SOH). SOC, defined as the ratio of remaining capacity to fully charged capacity, is a fundamental metric for battery management. However, SOC cannot be measured directly; it must be estimated indirectly through parameters like terminal voltage, charge/discharge current, and internal resistance. These parameters are susceptible to various uncertainties such as battery aging and ambient temperature fluctuations, making accurate SOC estimation, especially under complex operating conditions, a significant challenge. Traditional methods, including coulomb counting and open-circuit voltage measurement, often struggle with localized state anomalies within a single cell, potentially overlooking weak performance areas or incipient failure points that could lead to safety hazards.
To address these limitations, ultrasonic non-destructive testing (NDT) has emerged as a promising complementary technique. The fundamental principle lies in the mechanical changes within a lithium-ion battery during operation. During charge and discharge cycles, lithium ions shuttle between the cathode and anode through the electrolyte and separator. This intercalation and de-intercalation process alters the density and elastic modulus of the electrode materials. Since ultrasonic waves are mechanical vibrations propagating through a medium, their characteristics—such as velocity and amplitude—are highly sensitive to the material’s mechanical properties. Consequently, ultrasonic signals can serve as a powerful probe for internal battery states. Early pioneering work by Pecht’s team at the University of Maryland demonstrated the feasibility of using ultrasound to monitor electrode wrinkling and expansion. Subsequent research by Hsieh’s group at Princeton University established clear correlations between ultrasonic signal amplitude, Time-of-Flight (TOF), and the battery’s SOC during cycling. This body of work underscores the potential of acoustics for battery diagnostics.
This article delves into the application of ultrasonic guided waves, enabled by flexible Macro Fiber Composite (MFC) sensor arrays, for the spatial characterization and quantitative assessment of SOC in lithium-ion batteries. The inherent flexibility and compact profile of MFC sensors make them ideally suited for integration into tightly packed battery modules where conventional rigid transducers are impractical. We first establish the proof-of-concept using a “pitch-catch” configuration with a single MFC transmitter-receiver pair, validating the correlation between guided wave features and SOC. We then scale the approach to a multi-element array configuration to investigate potential spatial variations in SOC across different regions of a commercial pouch cell. Finally, we integrate the extracted acoustic features with conventional electrical parameters to construct a comprehensive dataset. A Backpropagation (BP) neural network model is trained on this multimodal data to achieve high-precision, quantitative SOC inversion, demonstrating the synergistic power of combining acoustic and electrical sensing for advanced battery state estimation.
Theoretical Background: Ultrasonic Guided Waves in Multilayer Structures
A lithium-ion battery pouch cell is essentially a multilayer structure, typically comprising a cathode current collector, cathode active material, separator, anode active material, and anode current collector, all encapsulated within a flexible laminate pouch. When an ultrasonic wave is excited in such a thin, layered medium, it propagates as a guided wave, or Lamb wave, which is constrained between the two parallel surfaces of the cell. Lamb waves exist in two fundamental families of modes: symmetric (S) and antisymmetric (A) modes. For thin plates, the fundamental antisymmetric mode (A0) and symmetric mode (S0) are most readily excited. The A0 mode is characterized by out-of-plane displacement, making it particularly sensitive to changes in the mechanical properties of the bulk electrode materials, which are directly influenced by lithium ion concentration (and thus SOC).
The phase velocity ($c_p$) of a specific Lamb wave mode is a function of the frequency-thickness product ($f \cdot d$) and the material properties (density $\rho$ and elastic constants) of the layered medium. The relationship is governed by the Rayleigh-Lamb frequency equations. For a single, isotropic plate, the simplified equation for the A0 mode at low frequency-thickness values is given by:
$$ c_p \approx \sqrt[4]{\frac{E}{3\rho(1-\nu^2)}} \cdot \sqrt{\omega d} $$
where $E$ is Young’s modulus, $\nu$ is Poisson’s ratio, $\rho$ is density, $\omega$ is angular frequency, and $d$ is plate thickness. Although a lithium-ion battery is anisotropic and multilayered, this equation illustrates the principle: changes in the effective modulus $E$ and density $\rho$ of the electrodes will alter the wave velocity. As lithium ions intercalate into the anode during charging, the anode material expands and its modulus changes, affecting the overall effective properties of the cell stack. Therefore, monitoring the Time-of-Flight (TOF) of a specific wave mode (e.g., A0) over a fixed propagation path provides a direct measure of these property changes, which are correlated with SOC.
The signal amplitude is attenuated due to material damping and scattering. The attenuation coefficient $\alpha$ can also be sensitive to the microstructure and viscoelastic properties of the electrodes, which may change with lithiation level. Thus, both velocity (derived from TOF) and signal amplitude serve as potential acoustic feature vectors for SOC estimation:
$$ \text{TOF} = \frac{L}{c_p(SOC)} $$
$$ \text{Amplitude} = A_0 \cdot e^{-\alpha(SOC) \cdot L} $$
where $L$ is the propagation path length and $A_0$ is the initial excitation amplitude.
Sensor Technology: Macro Fiber Composite (MFC)
The practical deployment of ultrasonic monitoring in real-world battery modules demands sensors that are compact, durable, and capable of conforming to curved or confined surfaces. The traditional piezoelectric ceramic disc transducers are brittle and rigid, making them unsuitable for this application. The Macro Fiber Composite (MFC) sensor, developed by NASA, provides an elegant solution. An MFC consists of rectangular piezoceramic (typically PZT) fibers aligned in one direction and embedded in an epoxy matrix, sandwiched between layers of polyimide film and equipped with interdigitated electrodes (IDEs) on the surface.
The MFC used in this study is the P2 type, where the IDE pattern and the poling direction of the PZT fibers are configured to primarily exploit the $d_{33}$ piezoelectric coefficient. When a voltage is applied across the IDEs, the resulting electric field causes the PZT fibers to expand or contract along their length, generating in-plane strain. Conversely, when subjected to mechanical strain, the fibers generate a charge collected by the IDEs. For exciting the out-of-plane A0 mode in a thin plate, the in-plane contraction/expansion of the MFC bonded to the surface creates a local bending moment, efficiently coupling energy into the A0 mode. The flexibility of the polyimide substrate allows the MFC to be easily bonded onto the flexible pouch cell surface without inhibiting its natural expansion/contraction during cycling. Key specifications of the MFC-2814-P2 used are summarized in Table 1.
| Parameter | Value | Description |
|---|---|---|
| Model | M2814-P2 | Smart Material Corp. |
| Overall Dimensions (a x b) | 37 mm x 18 mm | Length x Width |
| Active Area Dimensions (c x d) | 28 mm x 14 mm | Length x Width |
| Operating Frequency Range | Up to ~500 kHz | Used at 60 kHz center freq. |
| Piezoelectric Type | $d_{33}$ mode | In-plane strain actuation/sensing |
Experimental Methodology: From Single Path to Array Sensing
1. Proof-of-Concept: Single Pitch-Catch Configuration
The initial experiment established the baseline correlation between ultrasonic features and SOC. A custom-made soft-pouch lithium-ion battery (theoretical capacity: 2325 mAh, dimensions: 118 mm × 88 mm × 1.992 mm) was used. Two MFC-2814-P2 sensors were bonded to the surface with a center-to-center distance of 47 mm, configured in a pitch-catch mode. One MFC acted as the transmitter, the other as the receiver.
The experimental setup comprised a function generator, a digital oscilloscope, a battery cycler (Neware CT-4008T-5V6A), and a PC for data acquisition. The function generator excited the transmitter MFC with a 5-cycle Hanning-windowed tone burst with a center frequency of 60 kHz and a peak-to-peak voltage of 10 V. The cycler performed a constant-current constant-voltage (CC-CV) charge (1C rate, cutoff at 4.2V, taper current 2A) followed by a constant-current (CC) discharge (1C rate, cutoff at 3.0V). During the discharge phase, ultrasonic signals were captured by the oscilloscope every 5 minutes. The received signals were band-pass filtered around 60 kHz to improve the signal-to-noise ratio.
The key acoustic features, Time-of-Flight (TOF) and signal amplitude, were extracted. TOF was calculated using a cross-correlation method between a reference signal (at SOC=100%) and the signal at a given SOC. The amplitude was measured as the peak-to-peak voltage of the first arrived A0 mode packet. The results clearly demonstrated a monotonic relationship: as SOC decreased from 100% to 0%, the TOF increased (indicating decreasing wave velocity), and the signal amplitude decreased. This is consistent with the expected increase in electrode stiffness and density upon lithiation (during charge), which would increase wave speed and potentially alter attenuation. The discharge process (delithiation) produces the opposite trend.
2. Spatial Characterization: MFC Array Configuration
To investigate potential spatial inhomogeneity in SOC distribution—a critical aspect for large-format cells—an array-based “one-actuate, multiple-receive” strategy was employed. A larger commercial pouch cell (145 mm × 100 mm × 2.2 mm) was instrumented with six MFC sensors, as shown in Figure 4 of the original text. Two MFCs served as actuators (Actuator 1, Actuator 2), and four served as receivers (Rx1, Rx2, Rx3, Rx4). The actuators were placed 15 mm from the center of the long edge. Receivers were placed 60 mm on either side of each actuator, effectively dividing the cell into four interrogation zones: Zone A (Rx1), Zone B (Rx2), Zone C (Rx3), and Zone D (Rx4).
The test protocol involved a sequential actuation: first, Actuator 1 fired, and signals were recorded from Rx1 and Rx2. After a 10-second delay (during which SOC change was negligible ~0.0028), Actuator 2 fired, and signals were recorded from Rx3 and Rx4. This process was repeated every 5 minutes during a 1C constant-current discharge. The setup required two synchronized function generators and oscilloscopes. The collected time-domain signals for all four zones over the entire discharge were processed similarly to the single-path experiment.
The 3D time-domain cloud plots for each zone (see original text) visually confirmed the consistent trend across all regions: decreasing amplitude and increasing TOF with decreasing SOC. This confirms the global nature of the SOC-acoustic relationship. However, a quantitative comparison of the extracted feature ranges revealed interesting details, as summarized in Table 2.
| Zone | TOF Range (μs) | Amplitude Change (mV) (from SOC 100% to 0%) |
Relative Amplitude Level |
|---|---|---|---|
| A (Rx1) | ~76 – 86 | 15.92 | Lowest |
| B (Rx2) | ~76 – 86 | 17.64 | High |
| C (Rx3) | ~76 – 86 | 17.28 | High |
| D (Rx4) | ~76 – 86 | 13.25 | Low |
The table shows that while the TOF variation range was remarkably consistent across all zones (~10 μs), the total amplitude change and the absolute signal levels differed. Zones B and C exhibited larger amplitude changes and higher signal levels compared to Zones A and D. This asymmetry could be attributed to minor variations in sensor bonding quality, local material properties, or edge effects in wave propagation. Crucially, the core acoustic-SOC trend remained intact in every zone, proving the feasibility of multi-point monitoring for detecting localized anomalies. If a region were to have a significantly different SOC (e.g., due to internal short or uneven current distribution), its local acoustic features would deviate from the trend observed in other zones.
Data-Driven SOC Inversion Using a BP Neural Network
To move from qualitative correlation to quantitative estimation, a machine learning model was developed. The goal was to fuse acoustic and electrical features to create a robust, high-accuracy SOC estimator. A fresh commercial lithium-ion battery underwent 30 complete charge-discharge cycles (1C CC-CV charge, 1.5C CC discharge) under controlled temperature (25°C). Throughout these cycles, ultrasonic signals (using a single pitch-catch path) and electrical data were sampled every 5 minutes.
Feature Extraction: From the ultrasonic signal, two acoustic features were extracted: 1) Time-of-Flight (TOF), and 2) Signal Amplitude (Amp). From the battery cycler, five electrical features were logged: 1) Current (I), 2) Voltage (V), 3) Contact Resistance (R_c), 4) Charging Capacity (C_chg), and 5) Discharging Capacity (C_dis). The instantaneous SOC was calculated in real-time by the coulomb counting method, using the actual discharge capacity of each cycle as the reference (SOH was stable over 30 cycles).
Dataset Construction: Data was resampled at a 10-minute interval, resulting in 430 data points per feature across the 30 cycles. Thus, the input feature matrix $X$ had dimensions $430 \times 7$ (7 features), and the target output vector $Y$ (SOC) had dimensions $430 \times 1$. The dataset was split: the first 25 cycles (359 data points) were used for training, and the last 5 cycles (71 data points) were reserved for testing.
Model Architecture and Training: A standard three-layer Backpropagation (BP) neural network was implemented. The structure is defined as follows:
– Input Layer: 7 neurons, corresponding to the 7 input features (TOF, Amp, I, V, R_c, C_chg, C_dis).
– Hidden Layer: One hidden layer with 10 neurons (determined empirically). The hyperbolic tangent sigmoid (tansig) function was used as the activation function:
$$ \Phi(z) = \frac{e^z – e^{-z}}{e^z + e^{-z}} $$
– Output Layer: 1 neuron, outputting the estimated SOC value, using a linear activation function.
The weights ($\omega_{ij}$, $\omega_{jk}$) and biases were initialized randomly. The Levenberg-Marquardt algorithm was used for training to minimize the mean squared error (MSE) between the network output and the true SOC value.
Model Comparison: Two separate models were trained and compared:
1. Model E: Trained using only the 5 electrical features (I, V, R_c, C_chg, C_dis).
2. Model E+A: Trained using all 7 features, i.e., the 5 electrical features plus the 2 acoustic features (TOF, Amp).
The performance was evaluated on the unseen test data (last 5 cycles) using Absolute Error (AE):
$$ AE = |SOC_{true} – SOC_{predicted}| $$
The results are summarized in Table 3 and visualized in the prediction vs. truth plots.
| Model | Input Features | Max Absolute Error on Test Set | Typical Absolute Error Range | Notable Issues |
|---|---|---|---|---|
| Model E (Electrical Only) | I, V, R_c, C_chg, C_dis | 0.06 – 0.08 | 0.02 – 0.06 | Poor prediction near SOC=100%, multiple large errors. |
| Model E+A (Electrical + Acoustic) | I, V, R_c, C_chg, C_dis, TOF, Amp | < 0.0025 | < 0.0001 for most points | Significant error reduction, stable performance across full SOC range. |
The improvement afforded by the acoustic features is dramatic. Model E+A achieves sub-0.25% error across the entire test set, with the vast majority of estimates having errors below 0.01%. The acoustic features, particularly the TOF which is directly tied to the bulk mechanical state of the electrodes, provide physical information that is complementary to the electrical parameters. They effectively regularize the estimation problem, especially in regions like high SOC where electrical parameters like voltage plateau, leading to superior accuracy and robustness.
Conclusion and Outlook
This work has demonstrated a novel, non-destructive methodology for characterizing and estimating the State of Charge in lithium-ion batteries using flexible piezoelectric fiber composite sensor arrays and ultrasonic guided waves. The key conclusions are as follows:
1. Feasibility of MFC-based Acoustic Sensing: The use of flexible Macro Fiber Composite (MFC) sensors for exciting and receiving A0 mode Lamb waves in lithium-ion pouch cells has been successfully validated. The experimentally observed linear correlations between acoustic features (Time-of-Flight and signal amplitude) and battery SOC establish a solid physical foundation for this acoustic characterization technique.
2. Spatial Mapping Capability: By deploying MFCs in an array configuration, the methodology was extended from a single-point measurement to a multi-zone interrogation system. The results confirmed that the fundamental SOC-acoustic relationship holds across different regions of a commercial cell, while also revealing subtle differences in signal amplitude that could be indicative of local conditions. This lays the groundwork for detecting spatial inhomogeneities in SOC or early-stage localized degradation.
3. High-Accuracy Quantitative SOC Inversion: Fusing the extracted acoustic features with conventional electrical parameters within a data-driven BP neural network framework resulted in a highly accurate SOC estimation model. The model incorporating acoustic data (Model E+A) drastically outperformed the electrical-only model (Model E), achieving a maximum error below 0.25% and typical errors near 0.01%. This underscores the significant value-add of acoustic information for battery management systems, enabling precise, real-time state estimation.
The proposed approach offers a powerful new tool for the in-situ, non-invasive monitoring of lithium-ion batteries. Future work will focus on extending this technique to in-situ monitoring within operational battery modules, investigating the correlation between acoustic features and other states like State of Health (SOH) and detecting specific failure modes such as lithium plating. The integration of this acoustic sensing layer with battery management systems holds great promise for enhancing the safety, reliability, and longevity of energy storage systems and electric vehicles.