Terminology and standards¶

Battery modeling terminology can vary significantly across different contexts. This page defines how terms are used in this package and the broader PyBaMM ecosystem. It also outlines our chosen conventions where multiple standards exist.

Terminology¶

Name	Meaning
Anode	The electrode with the lower open-circuit potential, often graphite, silicon, or lithium, commonly referred to as the “negative electrode”. In some definitions, which electrode is the “anode” switches based on whether the battery is charging or discharging.
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	Can be used to mean either the total available capacity of an electrode or whole cell, denoted by capital \(Q\), or the instantaneous capacity of the electrode or whole cell (i.e. integral of the current in amps with respect to time in hours), denoted by lower case \(q\).
Lithiation	The amount of lithium intercalated relative to the minimum and maximum possible lithium content. Goes from 0 to 1.
Nominal capacity	The rated capacity of the cell.
Theoretical capacity	The total capacity that can be extracted from the cell 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.
Stoichiometry	Used interchangeably with “lithiation”.

Standards¶

Direction of current¶

In PyBaMM, and therefore in this package, we follow the convention that positive current (increasing capacity) corresponds to discharge, and negative current (decreasing capacity) corresponds to charge. The discharge capacity is given by

\[ q_{dchg}(t) = Q_0 + \int I(t) d\mathrm{t}, \]

where \(Q_0\) is the starting capacity (equal to 0 if the cell is at 100% SOC and \(Q_{cell}\) if the cell is at 0% SOC). We also define the charging capacity

\[ q_{chg}(t) = -q_{dchg}(t), \]

where we use lower case \(q\) to indicate that this varies during operation, as opposed to capital \(Q\) which is used to represent 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. In terms of capacity, the instantaneous capacity of the electrode is defined by \(q^{elec}(t)\). The mathematical definition of \(q^{elec}\) depends on whether the electrode is the anode or cathode of a full cell (see below), but in general \(q^{elec}\) is equal to \(q_{chg}\) (or \(q_{dchg}\)) plus an offset.

We can think in terms of electrode lithiation/stoichiometry by defining

\[ \theta(t) = q^{elec}(t) / Q^{tot}, \]

where \(Q^{tot}\) is the total capacity of the electrode (often just referred to as “the electrode capacity”).

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

Whole cell¶

The negative and positive electrodes behave differently when put together in a full cell, which means we need to think carefully about what is happening to the lithiation of each electrode and its corresponding open-circuit potential.

During discharge, the negative electrode lithiation decreases, which causes its open-circuit potential to increase. The positive electrode lithiation increases, which causes its open-circuit potential to decrease. The open-circuit voltage of the full cell is given by

\[U = U_p - U_n,\]

which is monotonically decreasing since \(U_p\) is decreasing and \(U_n\) is increasing.

Defining min/max electrode capacities by \(Q^{min/max}\), and electrode capacities at 0%/100% cell SOC by \(Q^{0/100}\), we have the following relationships:

\[\begin{split} \begin{align*} Q_n^{min} &= Q_n^0, \\ Q_n^{max} &= Q_n^{100}, \\ Q_p^{max} &= Q_p^0, \\ Q_p^{min} &= Q_p^{100}. \end{align*} \end{split}\]

The electrode instantaneous capacity is given by

\[q_n^{elec}(t) = Q_n^{100} - q_{dchg} \]

for the negative electrode and

\[q_p^{elec}(t) = Q_p^{100} + q_{dchg} \]

for the positive electrode. For each electrode, \(Q^{max} = Q^{min} + Q^{cell}\), where \(Q^{cell}\) is the theoretical capacity of the cell (the capacity that can be extracted from the electrode between its voltage limits).

In terms of stoichiometries,

\[\theta_n^{elec}(t) = \theta_n^{100} - \frac{q_{chg}}{Q_n^{tot}}, \]

and

\[\theta_p^{elec}(t) = \theta_p^{100} + \frac{q_{chg}}{Q_p^{tot}}, \]

The cell’s SOC is given by \(z = q_{chg}/Q_{cell}\).

Variable	Meaning
\(Q^{cell}\)	Useable capacity of the electrode
\(Q^{min}\)	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¶

Historically, we have used “anode”/”negative electrode” and “cathode”/”positive electrode” interchangeably throughout PyBaMM projects. For consistency, always use “negative” and “positive” to refer to the electrodes, instead of “anode” and “cathode”.
The concept of “charge” and “discharge” can be confusing for an electrode since the two electrodes gain or lose lithium in opposite directions depending on whether they are the anode or cathode. To avoid confusion, we use “charge” and “discharge” to refer to the full cell, and “lithiation” and “delithiation” to refer to the individual electrodes. 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.