Lesson DC Circuits · Advanced analysis and integration
Thévenin, Norton, and superposition
Thévenin replaces a network with voltage (E_{Th}) in series with (R_{Th}). Norton uses current (I_N) in parallel with (R_N). Superposition solves multi-source circuits one source at a time, then adds effects algebraically. Shortcuts when the load changes or the diagram is heavy.
What these theorems are for
Imagine a power source and internal network feeding a load that changes often. Instead of re-analyzing the whole network each time, replace everything seen from the load terminals with a simple equivalent—then only recalculate the load.
That is the daily value of Thévenin and Norton.
Thévenin steps
At the two terminals where the load connects:
- Remove the load (open the terminals).
- \(E_{Th}\) = open-circuit voltage between terminals.
- \(R_{Th}\): deactivate independent sources (voltage source → short; current source → open); find resistance looking into terminals.
- Equivalent = \(E_{Th}\) in series with \(R_{Th}\). Reconnect load; use Ohm.
Thévenin's theorem reduces the network to one voltage source and one series resistance.
Norton steps
Same terminals:
- \(I_N\) = short-circuit current between load terminals (replace load with a short).
- \(R_N\) same as \(R_{Th}\) with sources deactivated.
- Equivalent = current source \(I_N\) in parallel with \(R_N\).
Useful when you think naturally in available short-circuit current.
Thévenin ↔ Norton conversion
\[ E_{Th} = I_N imes R_{Th},\quad R_N = R_{Th} \] Convert when the other form makes the next step easier.
Superposition theorem
For linear resistive networks with multiple sources:
- Keep one source active; deactivate others (V → short, I → open).
- Find the desired voltage or current contribution.
- Repeat for each source.
- Add algebraically (watch signs/directions).
Does not apply blindly to nonlinear devices. In DC resistive circuits, it works.
Choosing a tool
| Need | Use |
|---|---|
| Try many different loads | Thévenin or Norton |
| Think in Thevenin voltage | Thévenin |
| Think in short-circuit current | Norton |
| Several sources, one quantity | Superposition or Kirchhoff |
| Check equivalence | Convert Th ↔ N |
Toy Thévenin example
12 V source, 4 Ω series, 8 Ω to return from tap. Load \(R_L\) on tap.
- Open load: \(E_{Th} = 12 × 8/(4+8) = 8\,\mathrm{V}\) (divider view).
- Short source: \(R_{Th} = 4 \parallel 8 = 8/3\,\Omega\).
- With \(R_L = 8\,\Omega\): \(I_L = E_{Th}/(R_{Th}+R_L)\).
Faster than full re-analysis for each \(R_L\).
Field warnings
- \(R_{Th}\) is not always "the resistor you see"—deactivate sources and compute what the terminals see.
- Do not bolt a short across live power terminals to "get Norton" without a controlled procedure.
- Equivalent is valid only from the same two terminals.
Field case
Situation. A lab supply with internal resistance feeds changing test loads. Each time the team redraws mesh equations.
How to think with this lesson.
- Measure open-circuit V → \(E_{Th}\).
- Measure loaded V and I with known \(R_L\); estimate \(R_{Th} = (E_{Th}-V_L)/I_L\), or compute from schematic.
- For each new load: \(I_L = E_{Th}/(R_{Th}+R_L)\).
Conclusion: Thévenin turns "ugly network + variable load" into basic arithmetic.
In the field
Symptom
Load voltage sags more than expected
Where to look
High (R_{Th}): weak source, long leads, fuse resistance
Likely causes
- Ignored internal resistance
- current-limited supply
What to measure
- Open-circuit V ((E_{Th}))
- loaded V and I
- estimate (R_{Th})
What not to do
- Short power terminals for Norton current without control
Checklist
- I find (E_{Th}) with load removed (open terminals)
- I find (R_{Th}) with sources properly deactivated
- I relate Norton: (I_N), (R_N = R_{Th})
- I apply superposition one source at a time
- I know equivalent is for two specific terminals
- I estimate (R_{Th}) from loaded sag when needed