Battery Equivalent Circuit Model

This tool computes the terminal voltage of a battery under dynamic load using a 1st-order Thévenin equivalent circuit. It calculates the instantaneous voltage drop across R₀, the voltage across the RC parallel branch (R₁–C₁), and the resulting terminal voltage at a specified time after load application. The model accounts for both ohmic loss and polarization effects.

The Thévenin model balances simplicity and accuracy for real-time applications like BMS design and SOC estimation. Its single RC pair captures dominant voltage relaxation behavior—faster than electrochemical models but more realistic than pure resistive models. It’s widely adopted in automotive and portable electronics due to its low computational overhead and good empirical fit.

R₀ is the instantaneous ohmic resistance (electrolyte, contacts, current collectors). R₁ represents charge-transfer resistance at electrode interfaces, while C₁ models double-layer capacitance and surface charge storage. Together, R₁–C₁ approximates the battery’s transient voltage recovery after load changes.

For a standard 18650 Li-ion cell: R₀ ranges from 10–100 mΩ; R₁ from 5–50 mΩ; and C₁ from 0.1–10 kF (100–10,000 F), depending on capacity and chemistry. Higher-capacity or prismatic cells often exhibit larger C₁ values due to greater interfacial area.

The terminal voltage evolves exponentially over time as the RC branch charges/discharges. At t = 0, only R₀ contributes to voltage drop. As t increases, the R₁–C₁ voltage decays toward steady state (I × R₀). Longer times yield lower dynamic deviation—useful for estimating steady-state vs. transient performance.

Not directly—the model assumes fixed parameters and doesn’t include SOC-dependent OCV or aging effects. However, it can be embedded in larger SOC/SOH estimators (e.g., Kalman filters) when combined with OCV–SOC lookup tables and parameter adaptation routines.

OCV serves as the ideal no-load voltage source in the Thévenin model and is strongly correlated with SOC. Accurate OCV improves terminal voltage prediction, especially during relaxation phases. For best results, use measured or interpolated OCV values from manufacturer datasheets or lab characterization.

First verify units: R₀ and R₁ must be in milliohms (mΩ), not ohms; C₁ in kilofarads (kF), not farads. Also ensure load current is positive for discharge (negative for charge, if supported). If still inconsistent, check whether your R₁–C₁ time constant (τ = R₁×C₁) is physically plausible—typical τ for Li-ion is 1–100 seconds.

The time constant τ = R₁ × C₁ governs voltage relaxation speed. With R₁ in ohms and C₁ in farads, τ is in seconds—but here R₁ is entered in mΩ and C₁ in kF, so τ (s) = (R₁ / 1000) × (C₁ × 1000) = R₁ × C₁. A value of R₁ = 20 mΩ and C₁ = 1 kF yields τ = 20 s, matching typical mid-frequency polarization response.

Yes—with caution. For series packs, sum OCVs and resistances (R₀, R₁), and divide total C₁ by the number of series cells. For parallel packs, sum capacities and capacitances, but keep R₀ and R₁ per parallel branch. Always validate against pack-level test data, as cell imbalance and thermal gradients introduce additional complexity.