Inside the ECM — Where the Equations Come From

Three physical processes, three timescales

When current flows through a lithium-ion cell, three processes activate simultaneously:

Ohmic drop (microseconds) Electrons through metal current collectors and ions through the electrolyte separator encounter resistive friction. The voltage response is instantaneous on any engineering timescale. Modelled by $R_0$.

Double-layer polarisation (seconds) At each electrode–electrolyte interface, ions rearrange into a charged double layer — physically a capacitor. It charges over seconds and dissipates over seconds when current stops. Modelled by the $R_1$–$C_1$ parallel branch.

Solid-state diffusion (minutes) Lithium must diffuse through solid electrode particles to the reaction sites. Concentration gradients take minutes to build and minutes to relax. Modelled by a second RC branch ($R_2$, $C_2$) in a 2-RC model.

When none of these are active — after a sufficiently long rest — the terminal voltage equals the Open-Circuit Voltage: a thermodynamic quantity that depends only on lithium stoichiometry, i.e., on SOC.

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The Thévenin (1-RC) circuit

The 1-RC model captures ohmic drop and double-layer polarisation:

         R₀              R₁
(+)────/\/\/────────┬────/\/\/────┐────(+)
                    │              │
         OCV(SOC)  C₁    V₁      ═══   V_t
                    │              │
(−)─────────────────┴──────────────┘────(−)

• $R_0$ — series resistance; causes the instantaneous voltage step when current switches on or off
• $R_1$, $C_1$ in parallel — polarisation branch; the voltage $V_1$ across it evolves continuously
• $V_t$ — terminal voltage; what the voltmeter measures

Sign convention: current $I$ is positive on discharge throughout. Fix this and never change it.

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Governing equations from KVL and KCL

Eq. A — Terminal voltage (KVL)

Kirchhoff's Voltage Law around the outer loop gives the terminal voltage directly:

$$V_t = \mathrm{OCV}(\mathrm{SOC}) - I\,R_0 - V_1 \tag{A}$$

This is algebraic — it holds at every instant given OCV, $I$, and $V_1$. The only quantity with its own dynamics is $V_1$.

Eq. B — Polarisation voltage (KCL)

Current entering the RC branch splits between $R_1$ and $C_1$:

$$I = \frac{V_1}{R_1} + C_1\,\frac{dV_1}{dt}$$

Solving for $\dot{V}_1$ and substituting $\tau_1 = R_1 C_1$:

$$\boxed{\frac{dV_1}{dt} = -\frac{V_1}{\tau_1} + \frac{I}{C_1}} \tag{B}$$

Equations A and B together fully describe the 1-RC ECM. There is nothing else.

Intuition for Eq. B:

• The $-V_1/\tau_1$ term: $R_1$ continuously discharges $C_1$ toward zero — the system forgets its past.
• The $+I/C_1$ term: current charges $C_1$ — the system responds to the present input.
• The steady state ($\dot{V}_1 = 0$) is $V_1 = I R_1$, reached asymptotically at rate $1/\tau_1$.

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Analytical solution

Eq. B is a first-order linear ODE with constant coefficients. For piecewise-constant current — constant over any interval $[t_0,\, t]$ — the exact solution is:

$$\boxed{V_1(t) = V_1(t_0)\,e^{-(t-t_0)/\tau_1} + I\,R_1\!\left[1 - e^{-(t-t_0)/\tau_1}\right]}$$

Term 1 — initial condition memory: decays exponentially at rate $1/\tau_1$. At $t = t_0$ it equals $V_1(t_0)$; as $t \to \infty$ it vanishes.

Term 2 — forced response: zero at $t = t_0$; approaches the steady-state value $I R_1$ as $t \to \infty$.

The solution interpolates smoothly from $V_1(t_0)$ toward $I R_1$, with the transition speed set by $\tau_1$. This is exactly the exponential shape visible on every battery voltage trace.

Verification: differentiating the master solution and substituting into Eq. B confirms it satisfies the ODE for all $t \geq t_0$. ✓

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Two exact special cases

Current applied from rest — $V_1(t_0) = 0$

If the cell is fully relaxed before current is applied, the first term vanishes:

$$V_1(t) = I\,R_1\!\left[1 - e^{-t/\tau_1}\right]$$ $$V_t(t) = \mathrm{OCV} - I\,R_0 - I\,R_1\!\left[1 - e^{-t/\tau_1}\right]$$

At $t = 0^+$: $V_t$ drops (or rises, for charge) by exactly $I R_0$ — instantaneously. The RC branch has not yet responded. $V_t$ then curves further as $V_1$ builds toward $I R_1$. The long-time asymptote, if current were held forever, would be $\mathrm{OCV} - I(R_0 + R_1)$.

Current removed — $I = 0$

With $t' = t - t_0$ measured from the moment current stops:

$$V_1(t') = V_1(t_0)\,e^{-t'/\tau_1}$$ $$V_t(t') = \mathrm{OCV} - V_1(t_0)\,e^{-t'/\tau_1}$$

$V_t$ jumps immediately by $I R_0$ when current switches off (R₀ drop disappears instantly), then recovers exponentially toward OCV as $V_1$ decays. In practice, $V_1(t_0)$ is unknown — it is what we are trying to find. So the equation is fitted to measured data with two free parameters:

$$V_t(t') = \mathrm{OCV} - \underbrace{A}_{\displaystyle =\,V_1(t_0)}\, e^{-t'/\tau_1}$$

The fit returns $A$ and $\tau_1$ directly. If the pulse was long enough to saturate the RC branch ($t_p \gg \tau_1$), then $V_1(t_0) \approx I R_1$, so $R_1 = A / I$

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References

1. G. L. Plett, Battery Management Systems, Vol. I: Battery Modeling, Artech House, 2015. — The standard graduate-level reference for ECM derivation, ODE solutions, and Kalman filter SOC estimation.

2. G. L. Plett, "Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs — Part 2: Modeling and identification," Journal of Power Sources, vol. 134, no. 2, pp. 262–276, 2004. — Original paper connecting the ECM discrete update to the EKF prediction step.

3. M. Chen and G. A. Rincon-Mora, "Accurate electrical battery model capable of predicting runtime and I–V performance," IEEE Transactions on Energy Conversion, vol. 21, no. 2, pp. 504–511, 2006. — Widely cited ECM paper; good comparison of topologies.

4. X. Hu, S. Li, and H. Peng, "A comparative study of equivalent circuit models for Li-ion batteries," Journal of Power Sources, vol. 198, pp. 359–367, 2012. — Systematic comparison of Rint, 1-RC, and 2-RC accuracy across SOC and temperature.