Thévenin and Norton Formulas
Thévenin and Norton equivalent circuit formulas: open-circuit voltage, equivalent resistance, load current and the load for maximum power transfer.
The formulas
| To find | Formula |
|---|---|
| Thévenin voltage | Vth = open-circuit voltage at the terminals |
| Thévenin resistance | Rth = resistance seen at the terminals with sources set to zero |
| Norton current | IN = Vth / Rth (the short-circuit current) |
| Norton resistance | RN = Rth |
| Load current | IL = Vth / (Rth + RL) |
| Maximum power to the load | RL = Rth, and Pmax = Vth² / (4 × Rth) |
To set sources to zero, replace a voltage source with a short circuit and a current source with an open circuit. The Thévenin and Norton circuits give the same voltage and current at the terminals, but not the same power inside the circuit. At maximum power transfer only half of the source power reaches the load, so the efficiency is 50%.
Worked example: a divider with a load
A 24 V source feeds R1 = 100 Ω in series, with R2 = 200 Ω from the output to ground. The load connects across R2.
- Vth = 24 × 200 / (100 + 200) = 16 V
- Rth = 100 × 200 / (100 + 200) = 66.67 Ω, so IN = 16 / 66.67 = 0.24 A
- With RL = 100 Ω, IL = 16 / (66.67 + 100) = 0.096 A and VL = 9.6 V
- Maximum power is at RL = 66.67 Ω: Pmax = 16² / (4 × 66.67) = 0.96 W
Load power for different load resistances
| Load | Current | Load power |
|---|---|---|
| 25 Ω | 174.5 mA | 761.7 mW |
| 66.67 Ω | 120 mA | 960 mW |
| 100 Ω | 96 mA | 921.6 mW |
| 200 Ω | 60 mA | 720 mW |
Frequently asked questions
When is a Thévenin equivalent useful?
When one part of a circuit changes, for example the load, and you want to find its current or voltage without solving the whole network again.
What is the difference between the Thévenin and Norton forms?
They are the same circuit in two forms: a voltage source in series with Rth, or a current source IN in parallel with the same resistance.
Does maximum power transfer mean maximum efficiency?
No. At RL = Rth the efficiency is only 50%. Power systems are designed for a high efficiency with a load much larger than the source resistance.
Can I use this for AC circuits?
Yes, with complex impedances. The load impedance for maximum power is then the complex conjugate of the Thévenin impedance.
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