Experimental Circuit Validation & Measurement Uncertainty

The purpose of a circuit lab is not to "prove Kirchhoff's laws" by producing a small remainder. It is to test whether a particular lumped model, with measured component values and instrument models, predicts observations within stated uncertainty. Conservation-law residuals are powerful diagnostics, but only after sign conventions and measurement loading are correct.

Build the prediction from measured inputs #

Measure the quantities the model actually uses: resistor values rather than color-code nominals, source open-circuit voltage and output resistance rather than the front-panel setting, and meter burden where it is not negligible. Record units, ranges, and reference directions before energizing the circuit.

For a predicted quantity $y_m$ and observation $y_o$, useful residuals include

$$r=y_o-y_m,\qquad r_{rel}=\frac{y_o-y_m}{\max(|y_m|,y_{floor})}.$$

The floor prevents meaningless relative errors near zero. Compare $r$ with a combined uncertainty, not with an arbitrary percentage.

Conservation residuals #

At a measured node, define

$$r_I=\sum_k s_k I_k,$$

where $s_k$ follows the declared arrows. Around a loop,

$$r_V=\sum_k s_k V_k.$$

For a whole network under the passive sign convention,

$$r_P=\sum_k V_kI_k.$$

Each residual should be near zero relative to its propagated uncertainty. A shared calibration error can move several readings together, so uncertainties are not always independent.

Failure classification #

A valid conclusion names the model, operating point, residuals, uncertainty budget, and any systematic deviation. "Theory confirmed" without those items is not a reproducible result.

A compact uncertainty budget #

Suppose a calculated divider output is $V_o=V_sR_2/(R_1+R_2)$. Treat $V_s$, $R_1$, and $R_2$ as input quantities with standard uncertainties and propagate them through the same measurement model. For uncorrelated inputs,

$$u_c^2(V_o)=\sum_i\left(\frac{\partial V_o}{\partial x_i}u(x_i)\right)^2.$$

NIST TN 1297 separates statistically evaluated Type A components from Type B components evaluated by other information, then combines standard uncertainties. It also makes the reporting rule explicit: a result without a quantitative uncertainty statement is incomplete.