Battery load power and efficiency ($P_L={\mathcal{E}}^{2}R/(R+r{)}^2$)

In the constant-parameter battery internal resistance model, the source's available electrical power splits between the electrical load and internal dissipation:The stipulated $3\mathrm{V}$, $1\Omega$, $4\Omega$ calculation gives $P_L=1.44\mathrm{W}$, $P_{\mathrm{int}}=0.36\mathrm{W}$, and $\mathcal{E}I=1.80\mathrm{W}$. A resistive electrical load converts its share to Joule heating; another electrical load might convert some of it to mechanical work or light. The source term is available electrical work per time, not a complete cell heat budget: reaction entropy can also matter.

Define the delivery efficiency relative to the source's available electrical work, rather than to all chemical energy or to a complete charging cycle:Here $\eta =0.80$. The largest load power occurs at $R=r$: since $(R+r{)}^2\ge 4Rr$,Thus maximum load power and maximum delivery efficiency ask different questions. As $R\to \infty$, $V_L\to \mathcal{E}$ and $\eta \to 1$, but $I\to 0$ and $P_L\to 0$: a high nearly unloaded voltage supplies almost no power. As $R\to 0$, the model gives $V_L\to 0$, $P_L\to 0$ and internal power tending to ${\mathcal{E}}^2/r$. Real high-current operation can invalidate constant $r$ and $\mathcal{E}$; the mathematical maximum does not establish a permissible operating point for an actual battery.