In the pursuit of enhancing the accuracy and robustness of battery state estimation, this study investigates the coupled mechanical and electrochemical behavior of commercial lithium iron phosphate (LiFePO4) batteries. The core objective is to explore the potential of an additional, highly sensitive physical signal—the in-situ surface expansion force—as a complementary input for state of charge (SOC) estimation algorithms. Traditional methods often rely heavily on voltage and current, which can exhibit flat characteristics in certain SOC ranges for LiFePO4 chemistry. This work systematically characterizes the LiFePO4 battery’s responses across a spectrum of temperatures and electrical loading conditions, providing a foundational dataset and analysis for future multi-physics battery management strategies.
The experimental foundation of this research rests on a custom-designed multi-physics signal acquisition platform. A critical component is the mechanical fixture engineered to accurately measure the surface expansion force of a prismatic LiFePO4 battery. The fixture comprises upper and lower clamping plates made of thermally conductive aluminum alloy, which provide rigid mechanical constraint in the through-plane direction while facilitating heat dissipation. A uniaxial pressure sensor is mounted centrally on the upper plate, ensuring direct and consistent contact with the battery surface. A 3D-printed positioning frame prevents in-plane movement of the LiFePO4 battery cell while allowing for electrical tab connections. This entire assembly is housed within a thermal chamber for precise environmental temperature control. Electrical cycling is performed using a high-precision battery test system, synchronously logging voltage, current, and the force sensor data. The repeatability of force measurements was rigorously validated, with consecutive tests under identical conditions showing a maximum deviation of less than 1.8%, confirming the reliability of the platform for characterizing the LiFePO4 battery’s mechanical signature.
Electrochemical Impedance and Open-Circuit Voltage Profiling
A comprehensive understanding of the LiFePO4 battery’s electrochemical parameters under varying temperatures is essential. Hybrid Pulse Power Characterization (HPPC) tests were conducted at temperatures ranging from 5°C to 30°C. The total internal resistance ($R_{total}$) is decomposed into its ohmic ($R_{ohm}$) and polarization ($R_{pol}$) components. The results, summarized for a representative temperature, reveal distinct trends.
| SOC | Charge $R_{ohm}$ (mΩ) | Charge $R_{pol}$ (mΩ) | Charge $R_{total}$ (mΩ) | Discharge $R_{ohm}$ (mΩ) | Discharge $R_{pol}$ (mΩ) | Discharge $R_{total}$ (mΩ) |
|---|---|---|---|---|---|---|
| 0.1 | 7.2 | 4.1 | 11.3 | 6.8 | 3.8 | 10.6 |
| 0.3 | 7.3 | 4.5 | 11.8 | 6.9 | 4.0 | 10.9 |
| 0.5 | 7.3 | 5.0 | 12.3 | 6.9 | 4.5 | 11.4 |
| 0.7 | 7.4 | 5.8 | 13.2 | 7.0 | 5.0 | 12.0 |
| 0.9 | 7.5 | 6.5 | 14.0 | 7.1 | 5.5 | 12.6 |
The data indicates that for the LiFePO4 battery, the ohmic resistance remains relatively stable across the SOC window, while the polarization resistance shows a gradual increase during both charge and discharge. Furthermore, the charge resistance is consistently higher than the discharge resistance, a phenomenon attributed to greater polarization and possible SEI layer effects during lithium intercalation into the graphite anode.
The temperature dependence of the LiFePO4 battery’s internal resistance is pronounced and follows an Arrhenius-type relationship. The governing equation for the temperature-dependent resistance can be expressed as:
$$R(T) = R_{ref} \cdot \exp\left[\frac{E_a}{R_g}\left(\frac{1}{T} – \frac{1}{T_{ref}}\right)\right]$$
where $R(T)$ is the resistance at temperature $T$, $R_{ref}$ is the resistance at reference temperature $T_{ref}$, $E_a$ is the activation energy, and $R_g$ is the universal gas constant. Analysis shows the ohmic resistance is more sensitive to temperature changes, with a reduction of up to 57% observed between 5°C and 30°C, compared to a 53% reduction in polarization resistance.
Simultaneously, the open-circuit voltage (OCV) of the LiFePO4 battery was characterized using a rest-period method. The SOC-OCV relationship is critical for model-based estimators. The results demonstrate the signature flat plateau regions of the LiFePO4 battery, centered around approximately 3.29 V and 3.33 V.
| SOC | 5°C OCV (V) | 15°C OCV (V) | 30°C OCV (V) | Remarks |
|---|---|---|---|---|
| 0.05 | 3.157 | 3.159 | 3.162 | Lower Plateau Start |
| 0.20 | 3.273 | 3.275 | 3.277 | Rising Edge |
| 0.35 | 3.291 | 3.292 | 3.293 | Mid-Lower Plateau |
| 0.50 | 3.295 | 3.296 | 3.297 | Upper Lower Plateau |
| 0.65 | 3.321 | 3.322 | 3.324 | Rising to Upper Plateau |
| 0.80 | 3.331 | 3.332 | 3.333 | Mid-Upper Plateau |
| 0.95 | 3.338 | 3.339 | 3.341 | Upper Plateau End |
A key finding is the high consistency of the OCV curve for the LiFePO4 battery across the tested temperature range, with maximum deviations less than 0.7%. This temperature-insensitive OCV-SOC map is advantageous for SOC estimation but highlights the inherent challenge: the flat voltage plateaus offer low sensitivity to SOC changes, necessitating the exploration of supplementary signals like the mechanical response of the LiFePO4 battery.
In-Situ Surface Expansion Force: Dynamic and Static Relationships with SOC
The mechanical signature of the LiFePO4 battery, namely its surface expansion force ($F_{exp}$), was measured during galvanostatic cycling. When measured with a low current (e.g., C/20) to minimize polarization effects, the $F_{exp}$-SOC relationship exhibits a highly characteristic and sensitive non-monotonic profile.
During charging, the force initially increases, reaches a local maximum, then decreases to a local minimum, before rising again towards the fully charged state. A symmetric trend is observed during discharge. This behavior is directly linked to the phase transformations within the graphite anode of the LiFePO4 battery. The initial force rise corresponds to lithium intercalation into graphite forming LiC18, causing lattice expansion. The subsequent force dip coincides with the transition from LiC18 to LiC12, where layer sliding mitigates volume increase while the LiFePO4 cathode contracts. The final force increase is associated with the formation of LiC6, which induces significant anisotropic expansion. The dynamic force signal for the LiFePO4 battery thus provides a mechanical “fingerprint” of the underlying electrochemical staging reactions.
The sensitivity of this mechanical signal is particularly valuable in the mid-SOC ranges where the OCV of the LiFePO4 battery is flat. The force gradient $dF_{exp}/dSOC$ in these regions is significantly higher than the voltage gradient $dOCV/dSOC$. This characteristic can be formalized by defining a sensitivity ratio $S_R$ for SOC estimation:
$$S_R = \frac{|\partial F_{exp} / \partial SOC|}{|\partial OCV / \partial SOC| + \epsilon}$$
where $\epsilon$ is a small constant to avoid division by zero. For the LiFePO4 battery, $S_R >> 1$ in the plateau regions, underscoring the potential benefit of integrating the force signal.
The dynamic force profile of the LiFePO4 battery is influenced by operational parameters. Increasing the C-rate amplifies the initial force magnitude but dampens and shifts the characteristic peaks and valleys. The local extremum points move to lower SOCs during charge and higher SOCs during discharge as current increases, eventually smoothing out at very high rates. Notably, a convergence point was observed where discharge force curves at different C-rates intersected at approximately SOC = 0.5, providing a potential invariant feature for diagnostics.
| Current (A) [C-rate] | 1st Peak SOC | Valley SOC | 2nd Peak SOC | Observation |
|---|---|---|---|---|
| 1 [~C/20] | 0.26 | 0.56 | 1.00 (Endpoint) | Clear features |
| 11.5 [0.5C] | 0.23 | 0.51 | 0.94 | Features shifted earlier |
| 23 [1C] | 0.18 | 0.47 | Feature flattened | Pronounced damping |
The equilibrium or “rested” expansion force ($F_{sta}$) was also measured after prolonged relaxation at various SOC points. The resulting $F_{sta}$-SOC curve retains the same non-monotonic shape as the dynamic curve but represents the stress state at thermodynamic equilibrium. The positions of its extremum points (near SOC~0.3 and SOC~0.6) were found to be highly stable, making them robust candidates for feature-based SOC calibration or SOH indication in a LiFePO4 battery.
Temperature-Dependent Multi-Physics Behavior
The interaction between temperature and the electromechanical response of the LiFePO4 battery is complex. As previously noted, internal resistance decreases with temperature. The open-circuit voltage shows minimal thermal dependency. The expansion force, however, exhibits a non-linear relationship with temperature.
For the LiFePO4 battery, the magnitude of $F_{exp}$ at a given SOC generally increased from 5°C to 25°C, then slightly decreased at 30°C. This can be modeled phenomenologically as a superposition of a thermally activated process and a high-temperature relaxation effect:
$$F_{exp}(T) \propto A \cdot \exp\left(-\frac{E_{a,F}}{k_B T}\right) – B \cdot (T – T_{ref}) \cdot H(T – T_{ref})$$
where $A$ and $B$ are coefficients, $E_{a,F}$ is an apparent activation energy for the force-generating process, $k_B$ is Boltzmann’s constant, and $H$ is the Heaviside step function accounting for the high-temperature force reduction potentially linked to graphite delithiation.
Critically, while the magnitude of the force changes with temperature, the SOC positions of the characteristic peaks and valleys in the $F_{exp}$-SOC curve for the LiFePO4 battery remain remarkably stable, with shifts less than 0.03 over the 5°C to 30°C range. This temperature-robust location of features is a significant finding, suggesting that a force-based SOC estimation algorithm for LiFePO4 batteries could require less intensive temperature compensation compared to purely voltage-based methods in certain ranges.
| Temperature (°C) | Avg. $R_{total}$ @ 0.5 SOC (mΩ) | OCV @ 0.5 SOC (V) | $F_{sta}$ @ 0.5 SOC (N) | 1st Peak SOC Position (Discharge) |
|---|---|---|---|---|
| 5 | 13.8 | 3.295 | 220 | 0.30 |
| 15 | 11.0 | 3.296 | 255 | 0.30 |
| 25 | 9.5 | 3.297 | 285 | 0.29 |
| 30 | 9.0 | 3.298 | 275 | 0.29 |
Synthesis and Implications for Battery State Estimation
This comprehensive experimental study on a commercial LiFePO4 battery elucidates the intricate coupling between its electrochemical and mechanical states under varied thermal and electrical conditions. The surface expansion force emerges as a rich information source, complementing traditional electrical signals. Its high sensitivity to SOC in the voltage-plateau regions of the LiFePO4 battery directly addresses a fundamental limitation in existing estimation techniques. The identified invariant features—such as the stable SOC locations of force extrema and the C-rate convergence point—provide tangible anchors for developing novel estimation algorithms.
The data supports the feasibility of a multi-physics sensor fusion approach for Battery Management Systems (BMS). By integrating real-time measurements of voltage ($V$), current ($I$), temperature ($T$), and expansion force ($F_{exp}$), the observability of the LiFePO4 battery’s internal state can be significantly enhanced. A conceptual framework for a joint estimator can be proposed:
$$\begin{aligned}
\text{State Vector: } & \mathbf{x} = [SOC, V_{polarization}, T_{core}, \sigma_{mech}]^T \\
\text{Measurement Vector: } & \mathbf{y} = [V_{terminal}, I, T_{surface}, F_{exp}]^T \\
\text{Model: } & \dot{\mathbf{x}} = f(\mathbf{x}, I, T_{ambient}) + \mathbf{w} \\
& \mathbf{y} = h(\mathbf{x}) + \mathbf{v}
\end{aligned}$$
where $f(\cdot)$ and $h(\cdot)$ represent coupled electrochemical-thermal-mechanical model functions, and $\mathbf{w}, \mathbf{v}$ are process and measurement noise. The function $h(\cdot)$ for the force output would directly incorporate the characterized $F_{exp}$-SOC-temperature-current relationships established for the LiFePO4 battery in this work.
Future work will focus on translating these characterization results into practical models. This includes refining empirical or physics-based models that describe the force generation in a LiFePO4 battery, conducting long-term cycling tests to correlate force evolution with state of health (SOH), and ultimately implementing and validating multi-physics SOC/SOH estimation algorithms in a BMS prototype. The insights gained from this detailed characterization of the LiFePO4 battery pave the way for more intelligent, reliable, and accurate management of lithium-ion battery systems.