Thévenin's Theorem

Any linear two-terminal network collapses into one source and one resistor.

Circuit AnalysisElectrical Engineering Year 1
⏱️ About 17 min

Any two-terminal linear network — no matter how many sources or resistors — can be replaced by a single voltage source in series with a single resistor.

💡
The big idea: Any linear two-terminal resistive network can be replaced by an equivalent circuit consisting of V_th in series with R_th, preserving all terminal behavior.
🎯 By the end, you'll be able to
  • State Thévenin's theorem and define V_th and R_th
  • Calculate the open-circuit voltage V_th at a pair of terminals
  • Calculate R_th by deactivating all independent sources
  • Use the Thévenin equivalent to find load current or voltage
📎 Helpful to know first

The Thévenin Equivalent

Thévenin's theorem states that any linear two-terminal network can be replaced by a single voltage source V_th in series with a single resistor R_th. From the perspective of whatever is connected to the two terminals — a load, a meter, another circuit — the Thévenin equivalent is indistinguishable from the original network.

🔑 Two Quantities Define the Equivalent

V_th is the open-circuit voltage at the terminals — the voltage measured when nothing is connected. R_th is the equivalent resistance looking into the terminals when all independent sources are turned off (voltage sources shorted, current sources opened).

Finding V_th

Remove the load from the terminals and calculate the voltage that appears across them. Any method works — nodal analysis, mesh analysis, voltage dividers, or superposition. The key is that no current flows through the terminals in this open-circuit condition.

Finding R_th

Turn off all independent sources: replace voltage sources with short circuits and current sources with open circuits. Then calculate the equivalent resistance seen looking into the two terminals. If the network contains only resistors and independent sources, this is a straightforward series/parallel simplification.

\[ I_L = \frac{V_{th}}{R_{th} + R_L} \]
Once the Thévenin equivalent is known, load current reduces to a simple series-circuit calculation.
📝 Worked example: A voltage-divider network has a 12 V source in series with R1 = 4 Ω, feeding a node that also connects to ground through R2 = 4 Ω. Find the Thévenin equivalent as seen at that node (relative to ground), then calculate the current through a load R_L = 2 Ω connected at that node.
  1. Find V_th (open-circuit voltage at the node, with R1 and R2 forming the source-side voltage divider): V_th = 12 V × (R2/(R1+R2)) = 12 × (4/8) = 6 V.
  2. Find R_th: turn off the 12 V source (replace with a short). Looking into the node, R1 = 4 Ω and R2 = 4 Ω are now in parallel: R_th = (4 × 4)/(4 + 4) = 2 Ω.
  3. The Thévenin equivalent is a 6 V source in series with 2 Ω.
  4. Connect R_L = 2 Ω: I_L = V_th/(R_th + R_L) = 6/(2 + 2) = 1.5 A.
✓ V_th = 6 V, R_th = 2 Ω, and the current through R_L = 2 Ω is 1.5 A.
✏️ Practice: A voltage-divider network has a 9 V source in series with R1 = 3 Ω, feeding a node that also connects to ground through R2 = 6 Ω. Find the current (in A) through a load R_L = 4 Ω connected at that node, using the Thévenin equivalent.
A
Solution
  1. V_th = 9 × (6/(3+6)) = 6 V (open-circuit voltage via voltage divider).
  2. R_th = 3 ∥ 6 = (3 × 6)/(3+6) = 2 Ω (turn off source, resistors in parallel).
  3. I_L = V_th/(R_th + R_L) = 6/(2+4) = 1 A.

Check your understanding

1. How is V_th determined for a two-terminal network?
V_th is the open-circuit voltage — the voltage at the terminals when the load is removed and no current flows out.
2. To find R_th, what do you do with the independent sources in the network?
R_th is the equivalent resistance seen looking into the terminals with all independent sources deactivated.
🔎 Related lessons
Norton's Theorem · soon
✅ Key takeaways
  • V_th is the open-circuit voltage at the terminals of interest.
  • R_th is the equivalent resistance looking into the terminals with all independent sources turned off.
  • The Thévenin equivalent is V_th in series with R_th.
  • Once found, the equivalent simplifies any load analysis to a single series loop.
➡️ If a voltage source in series has a dual, so does it — that's Norton's theorem, up next.
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