What a load really sees

Take any network made of linear elements and independent or dependent sources, and hide it inside a box. Leave only two terminals accessible. From the outside, the internal drawing is no longer observable directly. A load can only establish a current at the port and observe the resulting terminal voltage.

This shift in viewpoint is the useful content of Thévenin's theorem. The theorem is not primarily a recipe for replacing three resistors by one resistor. It says that, for a linear one-port in a given regime, the external world can be made insensitive to the internal complexity if we preserve the port relation.

Historically, equivalent-source ideas appeared before electronic engineering existed as a discipline. Helmholtz obtained a voltage-source equivalent in 1853; Thévenin published an equivalent result independently in 1883 in the context of telegraph networks. The recurring engineering problem was already familiar: how can a large network be replaced by the smallest model that predicts what happens at its terminals?

Define the port voltage as V = V_A − V_B. Let I be positive when the hidden network delivers current to the external load. Because the network is linear, superposition lets us separate two contributions to the port voltage.

First set I = 0. The port is open, so the voltage that remains is the open-circuit voltage V_OC. Now zero the independent sources and impose a current at the port. In a linear network, the resulting voltage change must be proportional to that imposed current. With our current convention, that proportionality appears as a negative slope.

The familiar drawing — an ideal voltage source in series with a resistor — is therefore not the theorem itself. It is simply the smallest circuit that has the same intercept and the same slope in the V–I plane.

Two operating points are especially useful. With an open circuit, I = 0, so the terminal voltage is V_TH. With an ideal short circuit, V = 0, so the delivered current is V_TH/R_TH whenever R_TH is finite and non-zero.

This is the cleanest way to see why Thévenin and Norton are the same model. One emphasizes the voltage intercept; the other emphasizes the current intercept. Both reproduce the same straight terminal characteristic.

Suppose a sensor interface is represented by V_TH = 5.0 V and R_TH = 100 Ω, and it drives R_L = 900 Ω. Calling it merely a “5 V source” hides the fact that the source itself participates in the voltage division.

The load receives only 90% of the open-circuit voltage. The important quantity is therefore the ratio R_L/R_TH, not the label “high impedance” by itself.

The theorem survives far beyond elementary resistor circuits, but the object that replaces R_TH changes. In sinusoidal steady state, it becomes a complex impedance Z_TH(ω). Around a nonlinear device bias point, a local small-signal linearization can create a useful incremental equivalent.

A single fixed V_TH and R_TH, however, cannot reproduce a strongly nonlinear V–I curve over a wide operating range. The model is useful because it preserves the behaviour needed for a specific question, not because it reproduces every internal physical detail.