Lithium-ion battery polarization voltage is not determined by current alone. During characterization, the measured polarization changes nonlinearly with current rate, initial SOC, initial resting condition and history, and aging state. A practical equivalent-circuit model integrates these effects by scaling the RC polarization response with current- and SOC-dependent factors, then adding history- and SOH-dependent offsets.
Core takeaway: A fixed linear RC model captures only the nominal dynamic response. A more realistic time-domain model treats current rate and SOC as multiplicative distortions of the RC response, while representing initial polarization history and aging as additive terms.
Why a Linear RC Model Is Insufficient
The nominal polarization response
In a standard multi-order RC equivalent circuit, each polarization branch consists of a resistance (R_{Pj}) and capacitance (C_{Pj}). For a current step, the polarization contribution of branch (j) evolves approximately as
[ U_{P,j}(T)=I R_{Pj}\left(1-\exp\left[-\frac{T}{R_{Pj}C_{Pj}}\right]\right). ]
The total response is therefore often written as
[ U_P(T)=U_P(0)+I\sum_{j=1}^{N}A_j(T), ]
where
[ A_j(T)=R_{Pj}\left(1-\exp\left[-\frac{T}{R_{Pj}C_{Pj}}\right]\right). ]
Here, (T) is the elapsed time after the current change, and (A_j(T)) has resistance units.
The limitation of constant polarization resistance
This formulation assumes that the polarization-voltage increment is proportional to current. In effect, it assumes that the slope
[ \frac{\Delta U_P}{\Delta I} ]
is constant under the tested conditions.
Real cells do not generally satisfy that assumption. Charge-transfer kinetics, ionic transport, concentration gradients, and electrode acceptance change with operating point, so the apparent polarization resistance depends on the test conditions.
The Four Non-Linear Influences
1. Current-rate distortion
The factor (K_I) represents the fact that the polarization response does not scale linearly with current magnitude.
At higher current rates, the cell can experience stronger activation losses, larger concentration gradients, and greater transport limitations. Consequently, doubling current does not necessarily double the polarization voltage.
A current-rate correction can be applied to the nominal RC response as
[ U_{P,I}(T)=K_I(I),I\sum_{j=1}^{N}A_j(T). ]
The notation (K_I(I)) emphasizes that this factor may itself depend on current. If (K_I=1), the model reduces to the nominal linear-current assumption.
2. Initial-SOC distortion
The factor (K_{SOC}) captures the dependence of polarization on the SOC at the beginning of the test.
Polarization is often lower in a cell’s mid-SOC range and higher near low- and high-SOC limits, where lithium-ion transport and charge acceptance become more restricted. Thus, the same charging current can produce different polarization voltages depending on the initial SOC.
The SOC-adjusted dynamic term becomes
[ U_{P,SOC}(T)=K_{SOC}(SOC_0),I\sum_{j=1}^{N}A_j(T), ]
where (SOC_0) is the initial SOC. This factor can be obtained from characterization data at several initial SOC levels.
3. Initial resting state and hysteresis
The term (B_{P0-}) represents the effect of the cell’s initial polarization state and previous operating history.
A cell that begins testing after charging, complete relaxation, or discharging may exhibit different voltage responses even when current, SOC, and temperature are nominally identical. This is a hysteresis or memory effect that a simple instantaneous RC response does not fully describe.
The state-dependent contribution is represented additively:
[ B_{P0-}=B_{P0-}(s_0,t_{\mathrm{rest}},\text{history}), ]
where (s_0) identifies the initial condition, such as a charging-related (+B) state, a fully rested (0) state, or a discharging-related (-B) state.
The additive form is useful because it shifts the polarization response according to the starting condition rather than changing only its current-dependent slope.
4. Battery aging and SOH distortion
The term (B_{SOH}) accounts for the increase in polarization as the cell ages.
Loss of capacity, rising internal resistance, changes in electrode structure, and degradation of transport and reaction pathways can all increase the voltage required to sustain a given charging current. Therefore, two cells with the same nominal SOC and current can show different polarization if their SOH differs.
The aging contribution can be represented as
[ B_{SOH}=B_{SOH}(SOH,\text{operating condition}), ]
or as a fitted function of cycle life, capacity retention, resistance growth, or another aging indicator.
How the Factors Are Integrated Mathematically
Unified time-domain expression
Combining the four effects gives the phenomenological model
[ \boxed{ U_P(T)=U_P(0) +K_{SOC}(SOC_0),K_I(I),I \sum_{j=1}^{N} R_{Pj} \left( 1-\exp\left[-\frac{T}{R_{Pj}C_{Pj}}\right] \right) +B_{P0-} +B_{SOH} } ]
or, using (A_j(T)),
[ \boxed{ U_P(T)=U_P(0) +K_{SOC},K_I,I\sum_{j=1}^{N}A_j(T) +B_{P0-} +B_{SOH}. } ]
The multiplicative terms (K_{SOC}) and (K_I) modify the amplitude of the current-driven RC response. The additive terms (B_{P0-}) and (B_{SOH}) represent shifts caused by initial history and aging.
Physical interpretation of the RC terms
For each RC branch, the time constant is
[ \tau_j=R_{Pj}C_{Pj}. ]
At short times, the branch response is still developing. At long times, it approaches (R_{Pj}), so the contribution approaches
[ I R_{Pj}. ]
Multiple branches allow the model to represent fast and slow polarization processes, such as rapid interfacial response and slower concentration or diffusion-related relaxation.
What “nonlinear” means in this model
The model is nonlinear in an operational sense because (K_I), (K_{SOC}), (B_{P0-}), and (B_{SOH}) vary with test conditions and cell state.
However, the RC state equation remains exponential for fixed parameter values. This is best understood as a state-dependent or piecewise nonlinear equivalent-circuit model, rather than a complete first-principles description of Butler–Volmer reaction kinetics and spatial diffusion.
Connecting the Model to Physical Polarization
Ohmic contributions
Ohmic polarization is associated with electronic and ionic resistance in components such as current collectors, electrodes, separator, electrolyte, contacts, and leads.
It responds rapidly to a current change and is often identified from the instantaneous voltage step when current is interrupted.
Activation contributions
Activation polarization arises from the energy barrier for charge-transfer reactions at the electrode–electrolyte interfaces.
Its dependence on current is fundamentally nonlinear and is commonly associated with Butler–Volmer behavior. In an equivalent circuit, it is approximated through fitted polarization resistance and time constants rather than modeled explicitly at the reaction-kinetics level.
Concentration contributions
Concentration polarization results from lithium-ion concentration gradients and transport limitations in the electrolyte and porous electrodes.
These effects become more prominent at high current and near SOC extremes. Slow RC branches are often used to approximate their time-dependent relaxation, while (K_I) and (K_{SOC}) adjust the amplitude under different operating conditions.
How Characterization Data Identify the Terms
Current-rate sweeps
Test the cell at several current magnitudes while controlling initial SOC, temperature, resting time, and SOH as closely as possible.
The change in polarization amplitude with current is used to estimate (K_I). If the fitted current-to-voltage relationship is not proportional, a constant polarization resistance is inadequate.
SOC-dependent tests
Repeat the same current profile from multiple initial SOC values.
The relative polarization amplitude can be used to determine (K_{SOC}(SOC_0)), commonly through a lookup table or fitted curve rather than a single constant.
Resting and relaxation tests
A current-interrupt or rest test separates the immediate voltage drop from the subsequent relaxation curve.
The instantaneous change provides information about rapid resistance, while the relaxation trajectory helps fit (R_{Pj}), (C_{Pj}), and the initial-state term (B_{P0-}). A fully rested condition can serve as a reference state, but the chosen rest criterion must be kept consistent across tests.
Aging campaigns
Repeat the same characterization protocol at different cycle-life or capacity-retention points.
The resulting change in polarization is used to estimate (B_{SOH}), or to update the RC parameters themselves if aging changes the time constants and resistance values rather than merely adding a voltage offset.
Understanding the Trade-offs
What the unified model does well
The model is compact and practical for battery testing systems, control development, and real-time estimation.
It captures the most important observed distortions without requiring a full electrochemical model with spatially resolved reaction and transport equations.
Where the additive aging term can be limiting
Representing aging only through (B_{SOH}) assumes that degradation primarily shifts the polarization voltage.
In practice, aging may also change (R_{Pj}), (C_{Pj}), the time constants (\tau_j), open-circuit voltage behavior, hysteresis, and temperature sensitivity. If relaxation shapes change substantially with age, the model should use SOH-dependent RC parameters as well as an aging offset.
Parameter identifiability
The four effects can be difficult to separate because they may produce similar voltage changes.
For example, a high-SOC test at high current on an aged cell combines SOC, current-rate, and SOH effects. Independent test matrices and controlled conditions are necessary to avoid attributing one factor to another.
Temperature must not be ignored
The stated expression does not explicitly include temperature, although temperature strongly affects resistance, reaction kinetics, and ion transport.
If testing spans a meaningful temperature range, the correction factors and RC parameters should be treated as temperature-dependent or the data should be restricted to a controlled temperature condition.
Avoid over-interpreting fitted terms
The factors are effective model parameters, not direct one-to-one measurements of individual electrochemical mechanisms.
A fitted (K_I), for example, may collectively reflect activation and concentration effects. The equivalent circuit is therefore most reliable within the SOC, current, temperature, and SOH range used for parameter identification.
Applying the Model to Cell Characterization
The model should be implemented as a calibrated, state-dependent equivalent circuit rather than as a single fixed resistance-capacitance network.
- If your primary focus is current-rate behavior: Identify (K_I(I)) from controlled current sweeps and retain the RC time constants that reproduce the measured transient response.
- If your primary focus is SOC-dependent fast charging: Map (K_{SOC}(SOC_0)) across the usable SOC range and reduce current where the correction factor and measured polarization become large.
- If your primary focus is hysteresis and relaxation: Use controlled rest and current-interrupt tests to estimate (B_{P0-}) and the slow RC branches.
- If your primary focus is lifetime modeling: Recharacterize cells at multiple SOH levels and update (B_{SOH}), while checking whether (R_{Pj}) and (C_{Pj}) also evolve.
- If your primary focus is physical diagnosis: Combine equivalent-circuit fitting with current-interrupt or impedance methods, because the fitted terms alone do not uniquely separate ohmic, activation, and concentration polarization.
A calibrated state-dependent RC model turns measured polarization distortions into usable parameters for more accurate characterization, control, and lifetime-aware charging decisions.
Summary Table:
| Factor | Effect on Polarization | Mathematical Representation |
|---|---|---|
| Current Rate | Higher current increases polarization but not linearly | Multiplicative factor K_I(I) scales RC response: K_I(I)Isum(A_j(T)) |
| Initial SOC | Polarization varies with SOC, higher at extremes | Multiplicative factor K_SOC(SOC_0) scales RC response: K_SOCIsum(A_j(T)) |
| Initial Resting State | History and rest time cause offsets | Additive term B_P0-: shifts U_P(T) by B_P0- |
| Aging/SOH | Degradation increases polarization | Additive term B_SOH: shifts U_P(T) by B_SOH |
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