Thevenin Resistance from an Open-Circuit and Loaded Measurement

Léon Thévenin's 1883 theorem says something extravagant: any tangle of sources and resistors, seen from two terminals, behaves exactly like one voltage source behind one resistor. You never have to know what is inside. And you can measure both numbers with a voltmeter and one known resistor — read the open-circuit voltage, hang the load, read the sagged voltage, and the internal resistance falls out of the droop.

Work an example the long way and the formula becomes obvious. A 9 V cell reads 9.00 V open and 8.10 V across 100 Ω. That load is drawing $8.10/100 = 0.081$ A, and the 0.90 V that went missing was dropped inside the cell, so $R_{Th} = 0.90/0.081 = 11.1$ Ω. A fresh alkaline should be well under 2 Ω, so this one is tired — which is exactly how a battery tester works.

The mistake to avoid is shorting the terminals to find the internal resistance directly. On a cell that is merely rude; on a car battery or a lithium pack it is a welding accident. Use a load that draws a sensible current and let the droop tell you. One caution on the theorem itself: it holds for linear circuits only. Put a diode, a lamp filament or a switching supply behind the terminals and R_Th becomes a number that changes with the load you used to measure it.