Electrochemical impedance spectroscopy (EIS) modeling is a powerful tool for battery diagnostics, providing insights into internal processes, degradation mechanisms, and performance characteristics. By analyzing a battery's frequency-dependent impedance response, EIS models can extract key parameters related to charge transfer, diffusion, and interfacial phenomena. Three primary modeling approaches are used: equivalent circuit models, distribution of relaxation times analysis, and physics-based impedance models. Each offers unique advantages for battery state-of-health estimation and fault detection.

Equivalent circuit models (ECMs) are the most widely used approach for interpreting EIS data due to their simplicity and intuitive connection to electrochemical processes. These models represent physical processes within a battery using electrical components such as resistors, capacitors, and Warburg elements. A typical ECM for lithium-ion batteries includes elements representing ohmic resistance (RΩ), charge transfer resistance (Rct), double-layer capacitance (Cdl), and diffusion impedance (Zw). The ohmic resistance accounts for electronic and ionic resistances in current collectors, electrodes, and electrolyte. The parallel Rct-Cdl combination models the electrode-electrolyte interface, while the Warburg element represents diffusion limitations. More advanced ECMs may include constant phase elements (CPEs) to account for non-ideal capacitive behavior caused by surface inhomogeneity or porous electrode effects. ECM parameterization involves fitting experimental EIS spectra using non-linear least squares optimization, with the quality of fit often evaluated through chi-squared values or residual analysis. While ECMs provide easily interpretable parameters for battery diagnostics, their empirical nature limits physical interpretability and predictive capability.

Distribution of relaxation times (DRT) analysis has emerged as a powerful alternative to ECMs, offering a model-free approach to impedance data interpretation. DRT transforms the frequency-domain impedance into a time-domain representation, revealing distinct electrochemical processes without requiring a priori assumptions about equivalent circuits. The method is based on the principle that any linear electrochemical system can be represented as a superposition of parallel resistor-capacitor circuits with a continuous distribution of time constants. Mathematically, the impedance Z(ω) is expressed as an integral equation involving the DRT function γ(τ), where τ represents relaxation time. Solving this ill-posed inverse problem requires regularization techniques to obtain physically meaningful solutions. DRT analysis excels at separating overlapping processes in the impedance spectrum, enabling identification of individual contributions from charge transfer, solid-electrolyte interphase (SEI) growth, and other interfacial phenomena. For battery diagnostics, the evolution of DRT peaks provides quantitative indicators of aging mechanisms such as lithium inventory loss, active material degradation, or electrolyte decomposition. The method's resolution depends on the frequency range and quality of experimental data, with typical lithium-ion battery processes exhibiting relaxation times from microseconds to thousands of seconds.

Physics-based impedance models offer the highest level of mechanistic insight by directly coupling electrochemical theory with impedance response. These models derive impedance expressions from fundamental governing equations including Butler-Volmer kinetics, concentrated solution theory, and porous electrode theory. A comprehensive physics-based model accounts for charge conservation in solid and electrolyte phases, species conservation in electrodes and electrolyte, and electrochemical reactions at interfaces. The linearized form of these equations under small-signal perturbation yields analytical or numerical solutions for impedance. Physics-based models explicitly incorporate design parameters such as electrode porosity, particle size distribution, and electrolyte composition, enabling direct correlation between impedance features and material properties. For example, the low-frequency Warburg impedance slope relates to lithium diffusion coefficients in active materials, while mid-frequency semicircles reflect charge transfer kinetics and double-layer effects. Multi-particle models further extend this approach to account for heterogeneous electrode architectures common in commercial batteries. While computationally intensive compared to ECMs or DRT, physics-based models provide unparalleled ability to predict impedance changes under varying operating conditions or aging states.

Applications of EIS modeling in battery diagnostics focus primarily on state-of-health (SOH) estimation and fault detection. For SOH estimation, ECM parameters such as RΩ and Rct serve as degradation indicators, with numerous studies demonstrating strong correlation between charge transfer resistance increase and capacity fade. Physics-based models enable more sophisticated SOH estimation by quantifying loss of active material, lithium inventory, and kinetic degradation separately. DRT analysis provides complementary information through time-constant distributions that evolve characteristically with different aging modes. Combined approaches that integrate multiple modeling techniques often achieve highest accuracy in SOH prediction, particularly when accounting for temperature and state-of-charge dependencies.

Fault detection using EIS modeling relies on identifying abnormal impedance signatures associated with specific failure modes. Internal short circuits manifest as depressed semicircles in Nyquist plots due to additional parallel conduction paths. Lithium plating produces distinct low-frequency features detectable through DRT analysis or physics-based models. Electrolyte drying increases ohmic resistance disproportionately at high frequencies, while electrode delamination alters both charge transfer and diffusion characteristics. Advanced fault detection algorithms compare real-time impedance parameters against baseline models to identify deviations indicative of incipient failures. Model-based approaches prove particularly valuable for early detection of thermal runaway precursors through subtle changes in interfacial impedance before bulk temperature rise occurs.

The choice between modeling approaches depends on diagnostic requirements and available computational resources. ECMs offer rapid implementation suitable for embedded applications but may lack resolution for complex degradation analysis. DRT provides detailed process separation without requiring physical assumptions but demands careful data preprocessing. Physics-based models deliver deepest mechanistic understanding but require extensive parameterization and computational power. Hybrid approaches that combine strengths of multiple methods are increasingly common in advanced battery management systems and laboratory diagnostics.

Future developments in EIS modeling will likely focus on improving parameter identifiability, extending models to extreme operating conditions, and integrating machine learning for pattern recognition in complex impedance data. Challenges remain in modeling non-linear impedance behavior at high currents or deep states of charge, as well as accounting for coupled aging mechanisms in multi-component systems. Nevertheless, continued advances in computational power and electrochemical theory will further enhance the diagnostic capabilities of EIS modeling for next-generation battery systems.