Battery Internal Resistance Tool

Physics Electricity and Magnetism • Current and Circuits

Written by STEM Calculators Team Published January 14, 2026 Updated February 24, 2026

8. Battery Internal Resistance Tool

Solve a battery’s internal resistance using the Thevenin model: $V_{\text{terminal}}=\varepsilon - I\,r$ . Also compute the operating point with a load $$R_L$$ and the maximum power-transfer case $$R_L=r$$ .

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication.

Mode + Inputs

Mode: EMF $\varepsilon$ (V): Terminal voltage $V_{\text{t}}$ (V): Current $I$ (A): Internal resistance $r$ (Ω): Load $R_L$ (Ω): Graph: Plot type: Plot span factor: Cursor fraction $f\in[0,1]$ (dot on plot): Circuit schematic:

The cursor fraction $f$ only moves the dot along the plotted range; it does not change your input values.

Ready

Steps

Enter values and click Solve.

Graph (pan/zoom)

Pan/zoom • hover to probe

Hover: (x, y)=…

Double-click the plot to reset the view.

Circuit schematic (for the computed operating point)

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Theory — Battery internal resistance

1) Thevenin model and terminal voltage

A real battery can be modeled as an ideal voltage source (EMF) $\varepsilon$ in series with an internal resistance $$ r $$ . With a load $$R_L$$ , the circuit current is

\[ I = \frac{\varepsilon}{R_L + r}. \]

The terminal voltage (voltage delivered to the load) is

\[ V = I R_L = \varepsilon - I r. \]

Interpretation: $Ir$ is the internal “voltage drop” inside the battery.

2) Solving for internal resistance from a measurement

If you measure terminal voltage $V$ while a known current $I$ is drawn and the EMF is known (or assumed), rearrange $V=\varepsilon-Ir$ :

\[ r = \frac{\varepsilon - V}{I}. \]

Be consistent with signs: if $I$ is defined as current leaving the + terminal, then $V$ should be the + terminal voltage across the load.

3) Open-circuit and short-circuit limits

These limits are useful sanity checks:

\[ \text{Open circuit: } I=0 \Rightarrow V_{oc}=\varepsilon, \qquad \text{Short circuit: } R_L=0 \Rightarrow I_{sc}=\frac{\varepsilon}{r}. \]

Large $r$ ⇒ “dead battery” behavior: voltage sags and short-circuit current is small.

4) Power to the load and maximum power transfer

Power delivered to the load is

\[ P_L = I^2 R_L = \frac{\varepsilon^2 R_L}{(R_L+r)^2}. \]

This is maximized when

\[ R_L = r, \qquad P_{max} = \frac{\varepsilon^2}{4r}. \]

At maximum power, efficiency is only 50% because $P_L = P_{int}$ (equal power lost inside $r$).

5) Multi-cell packs (quick approximation)

For a pack with $N_s$ cells in series and $N_p$ parallel strings (identical cells):

\[ \varepsilon_{pack} = N_s \varepsilon_{cell}, \qquad r_{pack} = \frac{N_s\,r_{cell}}{N_p}. \]

Real packs have balancing, temperature effects, and non-identical cells; this is a first-pass model.

Steps

Graph (pan/zoom)

Circuit schematic (for the computed operating point)

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