Practical Circuits: Internal Resistance of a Source
What is a Source of e.m.f.?
A source of electromotive force (e.m.f.) is like a battery or generator that pushes electric charge through a circuit. It has two key parts:
- Ideal e.m.f. (E) – the maximum voltage it can provide if no current flows.
- Internal resistance (r) – the resistance inside the source that causes a voltage drop when current flows.
Internal Resistance Analogy 🚗
Think of a battery as a water pump that pushes water through a pipe. The pump’s power is like the e.m.f. (E). The pipe’s friction is like the internal resistance (r). When the pipe is narrow (high r), more water (current) is lost as heat, so less water reaches the end (terminal voltage).
Terminal Potential Difference
The voltage you actually see across the battery terminals when a current I flows is:
\$V_{\text{t}} = E - I\,r\$
So, the higher the current or the internal resistance, the lower the terminal voltage.
Example Calculation
Given a battery with:
- \$E = 12\,\text{V}\$
- \$r = 0.5\,\Omega\$
- \$I = 2\,\text{A}\$
Find the terminal voltage:
\$V_{\text{t}} = 12\,\text{V} - (2\,\text{A})(0.5\,\Omega) = 12\,\text{V} - 1\,\text{V} = 11\,\text{V}\$
So the terminals drop to 11 V when 2 A flows.
Current vs. Terminal Voltage Table
| Current (A) | Terminal Voltage (V) |
|---|
| 0 | 12 |
| 1 | 11.5 |
| 2 | 11 |
| 3 | 10.5 |
Practical Implications
- Power Delivered: \$P = V_{\text{t}}\,I = (E - I\,r)I = EI - I^2 r\$ – the internal resistance reduces the power that reaches the load.
- Maximum Power Transfer: Occurs when the load resistance equals the internal resistance (\$R_{\text{load}} = r\$). At this point, half the source’s power is lost inside the source.
- Battery Life: Higher internal resistance means more heat and faster depletion.
Exam Tips 📚
- Always write the formula \$V_{\text{t}} = E - I\,r\$ before plugging in numbers.
- Check units: volts for voltage, amperes for current, ohms for resistance.
- Remember that internal resistance is always positive; it never increases the terminal voltage.
- For maximum power transfer, set \$R_{\text{load}} = r\$ and calculate the resulting current and voltage.
- When given a graph of \$V_{\text{t}}\$ vs. \$I\$, the slope equals \$-r\$ and the intercept equals \$E\$.