Terminology & Standards

Battery modeling terms and PyBaMM sign conventions: anode/cathode, lithiation, capacity, current direction, stoichiometry, and SOC

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.

Terminology

Term	Definition
Anode	The electrode with the lower open-circuit potential, often graphite, silicon, or lithium. Commonly referred to as the “negative electrode”.
Cathode	The electrode with the higher open-circuit potential, e.g., NMC or LFP. Commonly referred to as the “positive electrode”.
Negative electrode	Used interchangeably with “anode”.
Positive electrode	Used interchangeably with “cathode”.
Capacity	Either the total available capacity of an electrode or cell (denoted $Q$), or the instantaneous capacity during operation (denoted $q$).
Lithiation	The amount of lithium intercalated relative to the minimum and maximum possible lithium content. Bounded between 0 and 1.
Stoichiometry	Used interchangeably with “lithiation”.
Nominal capacity	The rated capacity of the cell.
Theoretical capacity	The total capacity extractable at open-circuit voltage (infinitely slow discharge) between voltage limits.
Potential	The electric potential of a single electrode relative to metallic lithium (0V).
Voltage	The difference between positive and negative electrode potentials.

Standards

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:

$$q_{\text{chg}}(t) = -q_{\text{dchg}}(t)$$

Lower case $q$ indicates a quantity that varies during operation, while capital $Q$ represents a scalar property of the electrodes or cell.

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.

Variable	Meaning
$q^{\text{elec}}$	Instantaneous capacity of an electrode
$Q^{\text{tot}}$	Total capacity of the electrode
$\theta(t)$	Instantaneous stoichiometry of an electrode

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.

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.

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:

$$z = \frac{q_{\text{chg}}}{Q_{\text{cell}}}$$

Variable	Meaning
$Q^{\text{cell}}$	Usable capacity of the cell
$Q^{\min}$	Capacity at the lower voltage cut-off
$Q^{\max}$	Capacity at the upper voltage cut-off
$\theta^{\min}$	Stoichiometry at the lower voltage cut-off
$\theta^{\max}$	Stoichiometry at the upper voltage cut-off

Naming Standards

Use 'Negative' and 'Positive'

Always use “negative” and “positive” to refer to the electrodes, instead of “anode” and “cathode”.

Use 'Lithiation' for Electrodes

Use “lithiation” and “delithiation” to refer to individual electrodes. Reserve “charge” and “discharge” for the full cell.

During a whole-cell discharge, the negative electrode delithiates and the positive electrode lithiates. During a whole-cell charge, the negative electrode lithiates and the positive electrode delithiates.