Internal Resistance

Inside the battery, the ions moving through the solution also encounter friction, due to collisions with the solvent and electrolyte molecules. This translates into an internal resistance $r$, in series with the ideal e.m.f. A real battery is therefore an ideal source $\mathcal{E}$ in series with $r$.

Equivalent model of a real battery: ideal e.m.f. $\mathcal{E}$ in series with the internal resistance $r$.

When the circuit is closed on an external resistor $R$ and a current $i$ flows in the circuit, the p.d. measured at the terminals (also called the terminal voltage) is no longer $\mathcal{E}$ but:

Key formula — Real battery

Terminal voltage: $\ev{\Delta V_{\text{morsetti}} = \mathcal{E} - r\,i}$ Current in the circuit closed on $R$: $\ev{i = \dfrac{\mathcal{E}}{R + r}}$

Applying Ohm’s laws to the whole circuit (total e.m.f. $=$ sum of the drops across $R$ and $r$):

$\mathcal{E} = R\,i + r\,i \quad\Longrightarrow\quad i = \frac{\mathcal{E}}{R + r}$

Example — Measuring the internal resistance

A battery with $\mathcal{E} = 1{,}5$ V reads $1{,}48$ V at the terminals on open circuit (there’s already a real voltmeter, which draws a trickle of current) and $1{,}35$ V when powering a resistor $R = 4{,}5\;\Omega$. In this latter case the current is $i = 1{,}35/4{,}5 = 0{,}30$ A. From the model $\Delta V = \mathcal{E} - ri$, with $\mathcal{E} \approx 1{,}5$ V and $\Delta V = 1{,}35$ V: $r = \frac{\mathcal{E} - \Delta V}{i} = \frac{1{,}5 - 1{,}35}{0{,}30} = \ev{0{,}5\;\Omega}$ At zero current the p.d. tends to $\mathcal{E}$: the idealised battery would only exist in the limit $i \to 0$.

Flat batteries

A battery runs down because the reactants are used up: the e.m.f. stays almost unchanged, but the internal resistance $r$ increases, until the current becomes too small to power the load. This is why an “old” battery can still light an LED but cannot drive a small motor.

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Topics: Circuiti elettrici Concepts: Forza elettromotrice

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