Replace the whole network with two components

Thevenin's theorem is the most useful result in DC circuit analysis: everything to one side of a pair of terminals can be reduced to one source and one resistor.

• Any linear two-terminal circuit is equivalent to a voltage source V_Th in series with a resistor R_Th.
• V_Th is the open-circuit voltage at the terminals, measured with the load removed.
• R_Th is the resistance seen at those terminals with all independent sources turned off, voltage sources shorted and current sources opened.
• The point is reuse: swap the load a hundred times and you only solve the network once.

When dependent sources are present

Killing the sources no longer works, because a dependent source is controlled by the circuit itself.

• Turn off the independent sources only, and leave the dependent ones alive.
• Apply a 1 V test source across the terminals, find the resulting current io, and take R_Th = 1/io.
• Equivalently, inject a 1 A test current and read the terminal voltage.

Norton, and maximum power

Norton is the same equivalence viewed as a current source, and the two forms convert directly.

• Norton's equivalent is a current source I_N in parallel with R_N, where I_N is the short-circuit current at the terminals.
• R_N = R_Th, and the two forms are linked by V_Th = I_N × R_Th, which is exactly a source transformation.
• Maximum power transfer: a load draws the most power when R_L = R_Th, and that maximum is P_max = V_Th²/(4R_Th).