Terminology & Standards

Battery modeling terminology varies significantly across contexts. This page defines how terms are used throughout this guide, along with our chosen conventions where multiple standards exist.

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Terminology

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Standards

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Direction of Current

Following the PyBaMM convention, positive current corresponds to discharge and negative current corresponds to charge. The discharge capacity is given by: $$q_{\text{dchg}}(t) = Q_0 + \int I(t) \, dt$$ where $Q_0$ is the starting capacity (equal to 0 if the cell is at 100% SOC and $Q_{\text{cell}}$ if the cell is at 0% SOC). The charging capacity is defined as: $$q_{\text{chg}}(t) = -q_{\text{dchg}}(t)$$

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Single Electrode

For a single electrode, we say that the electrode is “charged” when its lithiation/stoichiometry/capacity increases. The instantaneous capacity of the electrode is defined by $q^{\text{elec}}(t)$. The mathematical definition depends on whether the electrode is the anode or cathode of a full cell (see below), but in general $q^{\text{elec}}$ equals $q_{\text{chg}}$ (or $q_{\text{dchg}}$) plus an offset. We can express this in terms of electrode lithiation/stoichiometry: $$\theta(t) = \frac{q^{\text{elec}}(t)}{Q^{\text{tot}}}$$ where $Q^{\text{tot}}$ is the total capacity of the electrode.

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Whole Cell

The negative and positive electrodes behave differently when combined in a full cell. During discharge:

• Negative electrode: Lithiation decreases → open-circuit potential increases
• Positive electrode: Lithiation increases → open-circuit potential decreases

The open-circuit voltage of the full cell is: $$U = U_p - U_n$$ which is monotonically decreasing since $U_p$ is decreasing and $U_n$ is increasing.

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Capacity Relationships

Defining min/max electrode capacities by $Q^{\min/\max}$, and electrode capacities at 0%/100% cell SOC by $Q^{0/100}$: $$\begin{aligned} Q_n^{\min} &= Q_n^0 \\ Q_n^{\max} &= Q_n^{100} \\ Q_p^{\max} &= Q_p^0 \\ Q_p^{\min} &= Q_p^{100} \end{aligned}$$ The electrode instantaneous capacity is: $$q_n^{\text{elec}}(t) = Q_n^{100} - q_{\text{dchg}}$$ for the negative electrode and: $$q_p^{\text{elec}}(t) = Q_p^{100} + q_{\text{dchg}}$$ for the positive electrode. For each electrode, $Q^{\max} = Q^{\min} + Q^{\text{cell}}$, where $Q^{\text{cell}}$ is the theoretical capacity of the cell.

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Stoichiometry Relationships

In terms of stoichiometries: $$\theta_n^{\text{elec}}(t) = \theta_n^{100} - \frac{q_{\text{chg}}}{Q_n^{\text{tot}}}$$ $$\theta_p^{\text{elec}}(t) = \theta_p^{100} + \frac{q_{\text{chg}}}{Q_p^{\text{tot}}}$$ The cell’s state of charge is: $$z = \frac{q_{\text{chg}}}{Q_{\text{cell}}}$$

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Naming Standards