= Thévenin measurement experiment
{c}
An authored teaching experiment uses an adjustable steady-voltage supply, a digital multimeter, fixed <resistors>, and a $10\,\mathrm{k\Omega}$ <potentiometer> or resistance decade box. Its original circuit uses a $10\,\mathrm V$ supply and $3.3$, $6.8$, and $4.7\,\mathrm{k\Omega}$ <resistors>; the student compares $8.2$ and $2.2\,\mathrm{k\Omega}$ <electrical loads>. The readout is load voltage, recorded against prediction and deviation. The student then removes the <electrical load> to measure the <open-circuit voltage>, disconnects the supply and substitutes a wire to measure the <Thévenin resistance> with the multimeter's resistance mode, and restores the supply for an independent method: adjust the variable <electrical load> until the <voltmeter> reads half the unloaded voltage. Finally the student builds the equivalent and repeats both load measurements. Its tables are blank recording forms, not reported observations. <Laboratory Manual for DC Electrical Circuit Analysis by James M. Fiore>[Laboratory Manual for DC Electrical Circuit Analysis by James M. Fiore, version 1.4.3, experiment 11, printed pp. 52–55: Equipment, Procedure 1–10, Tables 11.1–11.3]{ref}
The half-voltage inference assumes negligible <voltmeter loading> and positive $R_{\mathrm{th}}$. A finite <voltmeter> input <electrical resistance> $R_m$ actually gives an apparent unloaded reading
$$
V_m=V_{\mathrm{oc}}\frac{R_m}{R_{\mathrm{th}}+R_m}.
$$
When the same <voltmeter> remains connected across a variable <electrical load> $R_L$, the network sees $R_p=R_LR_m/(R_L+R_m)$. Setting its reading to $V_m/2$, rather than to the true $V_{\mathrm{oc}}/2$, gives
$$
R_L=\frac{R_{\mathrm{th}}R_m}{R_{\mathrm{th}}+R_m}.
$$
Thus the uncorrected half-reading method underestimates positive $R_{\mathrm{th}}$. In the original $12\,\mathrm V$ example, a stipulated $R_m=10\,\mathrm{k\Omega}$ gives $V_m=120/17\,\mathrm V\approx7.059\,\mathrm V$ and half-reading load $R_L=20/17\,\mathrm{k\Omega}\approx1.176\,\mathrm{k\Omega}$, instead of $\tfrac43\,\mathrm{k\Omega}$. These are model predictions for a deliberately loading meter, not the specification or observed readout of the source experiment's instrument.
Finite readings can also estimate the terminal line without a physical short. For two distinct load settings with corrected delivered <electric currents> $I_1,I_2$ and terminal voltages $V_1,V_2$,
$$
R_{\mathrm{th}}=\frac{V_1-V_2}{I_2-I_1},\qquad
V_{\mathrm{oc}}=V_1+I_1R_{\mathrm{th}}.
$$
If the <voltmeter> remains across each <electrical load>, the delivered <electric current> is $I_k=V_k(1/R_{L,k}+1/R_m)$, including the meter branch. Omitting it biases the inferred slope.
Useful additional controls are to measure each <resistor> separately before assembly, verify terminal polarity, repeat an earlier load setting to detect supply drift, and compare a third setting with the inferred <Thévenin terminal response>. These are proposed controls, not reported outcomes of the source experiment. Closely spaced <electric currents> make the slope estimate sensitive to voltage resolution and <measurement error>; widely separated settings may trigger source current limiting or <Joule heating> that changes <electrical resistance>. Lead and contact <electrical resistance>, meter input uncertainty and rounding limit the inference. An <ohmmeter> injects its own test <electric current>; its reading on a powered network is not the suppressed-network ratio. Measurements with independent sources physically removed do not characterize an active <dependent electrical source> whose control or supply has also been disconnected.
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