Source Models, Internal Resistance & Loading

A source is useful only through the terminal behavior it presents to a load. A voltage reading with nothing connected identifies one point on that behavior, not the whole source. Internal resistance, current limiting, regulation, and nonlinear protection determine what happens once energy is drawn.

Linear source models #

The Thevenin form places an ideal voltage source $V_{th}$ in series with $R_{th}$:

$$V=V_{th}-IR_{th}.$$

The Norton form places $I_N=V_{th}/R_{th}$ in parallel with the same resistance:

$$I=I_N-\frac{V}{R_{th}}.$$

They are identical straight lines in the terminal $V$-$I$ plane. The intercepts are:

For a safely testable linear source, two distinct operating points determine the line and hence $R_{th}=-\Delta V/\Delta I$. A short-circuit test is not required and may be destructive.

Loading error #

A voltmeter with input resistance $R_m$ is itself a load. Connected to a source with $V_{th}$ and $R_{th}$, it reads

$$V_m=V_{th}\frac{R_m}{R_{th}+R_m}.$$

The familiar rule "use a high-impedance voltmeter" is therefore quantitative: the relative loading error is approximately $R_{th}/R_m$ when $R_m\gg R_{th}$. An ammeter has the dual problem; its burden resistance changes the current being measured.

The source model must match the question. Use a constant-resistance equivalent for a narrow linear range; use a piecewise or dynamic model when the load crosses operating modes.

At AC, replace $R_m$ by the probe input impedance $R_m\parallel 1/(j\omega C_m)$. The capacitive branch can dominate at high frequency even when the quoted DC input resistance is very large; this is the loading mechanism emphasized in the Tektronix probe guide.