Thévenin and Norton equivalent circuits

A circuit with a dozen resistors and three batteries can be genuinely tedious to analyse every time something downstream changes. Thévenin's theorem says you don't have to: from the point of view of anything connected to two particular terminals, the whole network — no matter how tangled — behaves exactly like one ideal voltage source V_th in series with one resistor R_th. Swap in the equivalent, and every measurement at those terminals comes out identical.

Finding the two numbers only takes two measurements (or two calculations). V_th is the open-circuit voltage — what a voltmeter reads across the terminals with nothing else connected. R_th is then the open-circuit voltage divided by the short-circuit current, R_th = V_oc/I_sc, where the short-circuit current is what flows if the terminals are joined by a plain wire (zero resistance), not whatever current happens to flow through some particular load.

Norton's equivalent is the same idea from the other side: one ideal current source I_N in parallel with the same resistance R_th, where I_N is exactly that short-circuit current. The two equivalents describe the same network and convert into each other directly — which one is more convenient just depends on whether the next thing you attach is easier to analyse against a voltage source or a current source.