= Thévenin's theorem
{c}
{title2=$V=V_{\mathrm{oc}}-IR_{\mathrm{th}}$}
{wiki}
= Thevenin theorem
{c}
{synonym}
Thévenin's theorem replaces the terminal behavior of a <linear resistive circuit> by an ideal <voltage source> in series with a <Thévenin resistance>, whenever its terminal <electric potential difference> is a well-defined <affine function> of the delivered <electric current>. Fix terminals $a,b$, write $V=\phi(a)-\phi(b)$, and take $I$ positive \i[out of the network at $a$] into the <electrical load>, returning through $b$. Then
$$
V=V_{\mathrm{oc}}-IR_{\mathrm{th}}.
$$
The <open-circuit voltage> $V_{\mathrm{oc}}$ is the intercept at $I=0$; $-R_{\mathrm{th}}$ is the slope. Thus two parameters predict what happens when many different <electrical loads> are connected. The <Wheatstone bridge detector loading> calculation uses precisely this replacement to find the detector reading even when the bridge's two <voltage dividers> interact through its detector.
For example, a $12\,\mathrm V$ ideal <voltage source> with $2\,\mathrm{k\Omega}$ above and $4\,\mathrm{k\Omega}$ below a <voltage divider> junction supplies $V_{\mathrm{oc}}=8\,\mathrm V$ and $R_{\mathrm{th}}=\tfrac43\,\mathrm{k\Omega}$ at that junction relative to the lower terminal. Attaching a $4\,\mathrm{k\Omega}$ <electrical load> reduces its voltage to $6\,\mathrm V$ and draws $1.5\,\mathrm{mA}$. The equivalent explains why an unloaded divider ratio alone cannot predict a loaded output.
\Image[/thevenin-divider-and-equivalent-circuits.png]
{title=A 12 V voltage divider and its 8 V Thévenin equivalent feed the same load}
{id=thevenin-divider-and-equivalent-circuits}
{description=Original calculated example: the left circuit has $2\,\mathrm{k\Omega}$ and $4\,\mathrm{k\Omega}$ divider arms; the right circuit has an $8\,\mathrm V$ ideal <voltage source> and $\tfrac43\,\mathrm{k\Omega}$ series <Thévenin resistance>. Both use $V=\phi(a)-\phi(b)$ and delivered <electric current> $I$ from $a$ into the same <electrical load>. Dashed boundaries identify what is replaced.}
{source=thevenin-divider-and-equivalent-circuits.py}
The equivalence concerns the chosen terminals, including the <power> $VI$ delivered there. It does not identify internal branch <electric currents> or internal <Joule heating>. At open circuit the original divider still draws $2\,\mathrm{mA}$ from its $12\,\mathrm V$ supply, while the equivalent's series branch carries no <electric current>. <Thevenin and Norton Equivalents, University of Houston teaching notes>[Thevenin and Norton Equivalents, pp. 1–2, “The Idea” and “Important Notes”: equivalence at the specified terminals]{ref}
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