Thevenin's and Norton's Theorems

Reducing a whole network to two numbers

Thevenin's theorem: any linear circuit, viewed from just two terminals, behaves exactly like a single voltage source $V_{th}$ in series with a single resistor $R_{th}$, no matter how complicated the actual network behind those terminals is.

Norton's theorem is the same idea with a current source instead: a current source $I_N$ in parallel with $R_N$. The two are directly related, same resistance, and:

$V_{th}$ is the open-circuit voltage across the terminals (nothing connected). $R_{th}$ is the resistance seen from the terminals with every independent source turned off (voltage sources shorted, current sources opened).

Worked example

A 12V source feeds a 4Ω resistor in series, which then splits into a 6Ω resistor to the return path. Find the Thevenin and Norton equivalents seen from the terminals of that 6Ω resistor (with the 6Ω itself removed, as the external load being analyzed).

$V_{th}$: with the load removed, no current flows anywhere except through the 4Ω and 6Ω in series, so the terminal voltage is just the voltage-divider result across the 6Ω:

$R_{th}$: shorting the 12V source and looking back into the terminals, the 4Ω and 6Ω are now simply in parallel with each other:

Norton equivalent:

Cross-check: the Norton current should also equal the short-circuit current at the terminals. Shorting the terminals shorts out the 6Ω branch entirely, leaving only the 4Ω resistor across the 12V source:

Matches $I_N$ exactly, confirming both equivalents independently.