Energy Balance of the Battery

Starting from the real-battery relation $\mathcal{E} = R\,i + r\,i$ and multiplying by the current $i$:

$\mathcal{E}\,i = R\,i^2 + r\,i^2$

Each term is a power and has a precise physical meaning.

The three terms of the balance

• $\mathcal{E}\,i$ : power supplied by the chemical reaction (conversion of chemical energy into electrical energy).
• $R\,i^2$ : power dissipated in the external load (the useful work).
• $r\,i^2$ : power dissipated inside the battery (heats the battery!).

The efficiency of the battery is thus the ratio of useful power to supplied power:

$\ev{\eta = \dfrac{R}{R + r}}$

The efficiency tends to 1 when $R \gg r$ (almost all the power goes to the load), and collapses when $R \sim r$.

Short circuit — caution

When $R \sim r$ (or worse, $R \to 0$) the efficiency collapses: leaving a copper wire connected directly from one terminal to the other heats the battery and destroys it, because the whole chemical power $\mathcal{E}\,i$ ends up dissipated in $r$. A short circuit is the condition that sends the entire power of the source into internal dissipation.

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Topics: Circuiti elettrici Concepts: Potenza · Forza elettromotrice

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