Thevenin's theorem

Thevenin's theorem is one of the most liberating ideas in circuit theory. It says that no matter how complicated a linear two-terminal network is inside, however many resistors and sources are crammed in there, when viewed from its two output terminals it behaves exactly like a single voltage source in series with a single resistor. The whole messy box reduces to just two numbers: a Thevenin voltage and a Thevenin resistance.

You find those two numbers with two measurements or calculations. The Thevenin voltage Vth is the open-circuit voltage, what you read across the terminals with nothing connected. The Thevenin resistance Rth is the resistance looking back into the terminals after you switch off all the internal sources, which means replacing each voltage source with a wire (a short) and each current source with a gap (an open). For example, a 9 V supply feeding a 10 kohm and 5 kohm divider has Vth = 3 V at the tap and Rth = 10 kohm in parallel with 5 kohm, which is about 3.3 kohm.

This is enormously powerful in practice. Once you have collapsed a complicated source network to its Thevenin pair, predicting what happens when you attach any load is trivial: it is just a voltage divider between Rth and the load. It is the natural language for talking about a source's strength, its output resistance, and loading, and you can swap loads in and out without re-solving the whole circuit. The one boundary is that the network must be linear, made of resistors and ideal sources, not diodes or transistors in their bending regions.