Thévenin, Norton and Superposition

A load connected to two terminals responds only to the voltage across it and the current through it. Anything behind those terminals producing the same relationship is indistinguishable to it.

For a linear network that relationship is a straight line, fixed by two numbers: its intercept — the open-circuit voltage — and its slope, the internal resistance. That is why the equivalent has exactly two components.

Deactivating a source means setting its value to zero — so a voltage source becomes a short and a current source becomes an open, because that is what a zero-valued source looks like. V_th = I_N·R_th converts between the two forms.

Superposition says the response to several sources is the sum of the responses to each acting alone. It is what linearity means, and it is the licence under which Thévenin and Norton are derived.

It fails for power, because P = I²R is quadratic. The power from source A plus the power from source B is not the power from both together — a reliable trap.

Power delivered to a load peaks when R_L = R_th. Too small and the terminal voltage collapses; too large and the current does.

At that point efficiency is exactly 50% — as much power is dissipated inside the source as reaches the load. A power system would never do this. Matching belongs to signal work, where the available power is tiny and getting the most of it matters more than wasting half.

For AC the theorem holds with complex impedances, and maximum transfer requires the conjugate match Z_L = Z_th* — so the reactances cancel and only the resistances match.

• Internal power — R_th dissipates nothing like what the real network dissipates.
• Internal voltages — you cannot ask what a node inside the original is doing.
• Non-linear networks — a diode or transistor rules the theorem out entirely.

The equivalence is exact at the terminals and meaningless anywhere else. That is acceptable because the load only ever sees the terminals, which was the whole premise.

It earns its keep when one network faces many loads: reduce once, then each load is a two-component calculation. R_th is the output impedance quoted on a datasheet, and a battery's internal resistance is the same quantity. Both numbers are measurable directly — open-circuit volts and short-circuit amps.