Power & Resistance Measurement: a Metrology Walkthrough

This tutorial is a full metrological worked example of measuring the electrical power dissipated in a precision resistor and assessing conformity of the resistor against its nominal value. It ties GUM methodology (JCGM 100:2008) to VIM vocabulary (JCGM 200:2012) and shows how the SymbolicUncertainties.jl API fits the calibration-certificate workflow end-to-end.

Unlike the single-section summaries in Worked Examples, this page walks through a complete uncertainty budget, coverage-interval construction, conformity assessment, and reporting — the sequence a calibration lab would follow in practice.

VIM terms used in this tutorial

For full VIM definitions see JCGM 200:2012; this tutorial paraphrases only.

Scenario

A calibration lab receives a precision resistor rated at R₀ = 100 Ω (an E96 series value) with a manufacturer tolerance of ±0.1 %. The lab applies a stable voltage source and measures both voltage across and current through the resistor using calibrated instruments:

• Voltmeter across the resistor — reading V = 10.000 V with σV = 0.001 V (1 mV standard uncertainty).
• Ammeter in series — reading I = 0.100 02 A with σI = 1.0 × 10⁻⁵ A (10 μA standard uncertainty).

The lab needs to deliver:

1. P — power dissipated, with uncertainty u(P).
2. R_meas — resistance inferred from Ohm's law, with uncertainty u(R).
3. A conformity assessment — is R_meas consistent with R₀ = 100 Ω at 95 % coverage?

Inputs are treated as independent (no correlated drift between the voltmeter and ammeter readings) — a standard JCGM 100:2008 §5.1.2 eq (10) setup.

Build the measurement models

using SymbolicUncertainties
using SymbolicUncertainties: ±
using Symbolics
using DynamicQuantities

# Symbolic inputs — values and standard uncertainties
@variables V I σV σI

# Model 1: power P = V · I
P = propagate((v, i) -> v * i, [V ± σV, I ± σI])
# P.val = V * I
# P.err = sqrt((I·σV)² + (V·σI)²)   — uncorrelated-inputs form

# Model 2: resistance R = V / I
R = propagate((v, i) -> v / i, [V ± σV, I ± σI])
# R.val = V / I
# R.err = sqrt((σV/I)² + (V·σI/I²)²)

\[V / I \pm sqrt(((1 / I)^2)*(σV^2) + (((-V) / (I^2))^2)*(σI^2))\]

propagate is used (rather than the binary * / / operators) because the underlying sensitivity analysis is explicit and the result is guaranteed to collapse repeated symbolic-variable occurrences — see Sensitivity Analysis.

Power budget

P_rows = uncertainty_budget(P, [V, I], [σV, σI])
# An UncertaintyBudget of 2 rows — one per input.
# Columns: variable, sigma, sensitivity, contribution,
#          relative (fraction of variance).

Uncertainty budget (EA-4/02 §7.3) for y = I*V
  V:  u = σV   c = I   |c|u = abs(I)*σV
  I:  u = σI   c = V   |c|u = abs(V)*σI
  u_c = sqrt((I^2)*(σV^2) + (V^2)*(σI^2))

Each row follows the GUM §5.1.6 / EA-4/02 §7.3 schema. Substituting numerics:

readings = Dict(
    V => 10.000us"V", σV => 1e-3us"V",
    I => 0.10002us"A", σI => 1e-5us"A",
)

# Several calls below rank or substitute plain numbers, so the same
# readings are also kept stripped.
dict = Dict(k => ustrip(v) for (k, v) in readings)

# The unit of the answer is derived from the model, not asserted:
# volts times amperes is watts, and nothing here had to say so.
evaluate(P, readings)

(val = 1.0002 A V, err = 0.0001414354990799693 A V)

Dominant source:

dominant = dominant_source(P, [V, I], [σV, σI];
                           values = dict)
# → returns `V` or `I`; in this setup they contribute
#   nearly equally (1.00e-8 each), so the dominant source
#   depends on tie-breaking.

(index = 1, variable = V, contribution = abs(I)*σV, ranked_by = :numeric)

This is a balanced budget — a deliberate choice in the scenario design. Real calibration work usually reveals one dominant source on which to focus improvements.

Resistance budget

R_rows = uncertainty_budget(R, [V, I], [σV, σI])

# Volts over amperes is ohms — again derived, not stated:
evaluate(R, readings)

(val = 99.98000399920016 A⁻¹ V, err = 0.014137894184807242 A⁻¹ V)

Coverage interval (VIM §2.36)

Expand u(R) to a coverage interval with k = 2 (p ≈ 0.95, assuming approximate normality per GUM §6.3.3):

R_expanded = expanded_uncertainty(R, 2)
# R_expanded.val = R.val        (unchanged)
# R_expanded.U = 2 · R.err

R_exp_num = Symbolics.substitute(R_expanded, dict)
# R_exp_num.val ≈ 99.98  Ω
# R_exp_num.U ≈ 2 · 14 mΩ ≈ 28 mΩ

V / I ± 2sqrt(((1 / I)^2)*(σV^2) + (((-V) / (I^2))^2)*(σI^2)) (k = 2)

The coverage interval is therefore [99.98 − 0.028, 99.98 + 0.028] Ω = [99.952, 100.008] Ω.

Conformity assessment (VIM §4.17)

Question: is R_meas = 99.98 Ω consistent with R₀ = 100 Ω at 95 % coverage?

R₀ = 100.0
lower = R_exp_num.val - R_exp_num.U
upper = R_exp_num.val + R_exp_num.U
pass = lower ≤ R₀ ≤ upper
# → true : R₀ = 100 ∈ [99.952, 100.008]  ✓

Result: R₀ lies inside the 95 % coverage interval, so the resistor is consistent with its nominal value at the 95 % level. The calibration certificate reports R = 99.98 Ω ± 0.028 Ω (k = 2, p ≈ 0.95) and the conformity status PASS.

Counter-example — a failing conformity check

Suppose instead the ammeter reading had been I = 0.10050 A (a 0.48 % drift):

dict_fail = Dict(k => ustrip(v) for (k, v) in Dict(
    V => 10.000us"V", σV => 1e-3us"V",
    I => 0.10050us"A", σI => 1e-5us"A",
))

R_fail = Symbolics.substitute(R_expanded, dict_fail)
# R_fail.val ≈ 99.502 Ω
# R_fail.U ≈ 28 mΩ   (roughly unchanged)
# coverage interval: [99.474, 99.530]

lower_f = R_fail.val - R_fail.U
upper_f = R_fail.val + R_fail.U
pass_f = lower_f ≤ R₀ ≤ upper_f
# → false : R₀ = 100 ∉ [99.474, 99.530]  ✗

Result: the measured resistance is inconsistent with the nominal value at 95 % coverage. The certificate reports NON-CONFORM and the device is tagged for adjustment or rejection.

Reporting (GUM §7 + VIM §4.18)

A calibration-certificate-ready summary block:

function report(label, m, k)
    v = Symbolics.value(Symbolics.substitute(m.val, Dict()))
    u = Symbolics.value(Symbolics.substitute(m.err, Dict()))
    # (in practice m is already fully substituted — the
    # `substitute(_, Dict())` round-trip is just defensive)
    println(label, " = ", round(v, sigdigits = 5),
            " ± ", round(k*u, sigdigits = 3),
            "  (k = ", k, ", coverage ≈ ",
            k == 2 ? "95 %" : "…", ")")
end

report("P", P_num, 2)
# P = 1.0002 ± 2.83e-4  (k = 2, coverage ≈ 95 %)

report("R", R_num, 2)
# R = 99.98 ± 2.83e-2   (k = 2, coverage ≈ 95 %)

Certificate excerpt (human-readable):

Device under test: precision resistor, nominal R₀ = 100.000 Ω (E96 series).

Conditions: T = 23.0 °C, applied voltage V_applied ≈ 10 V.

Result:

• R = 99.98 Ω ± 0.028 Ω (k = 2, coverage ≈ 95 %).
• P = 1.0002 W ± 0.00028 W (k = 2, coverage ≈ 95 %).

Dominant uncertainty sources: voltmeter reading and ammeter reading contribute approximately equally (~50 % each).

Conformity: PASS — R₀ = 100 Ω lies within the 95 % coverage interval [99.952, 100.008] Ω.

Method: JCGM 100:2008 §5.1.2 eq (10), uncorrelated inputs; software implementation: SymbolicUncertainties.jl v0.10.0.

What this tutorial demonstrates

• Symbolic propagation is calibration-ready. Every intermediate expression in this walkthrough is symbolic until the final Symbolics.substitute, giving the lab a traceable derivation of the final number.
• VIM vocabulary maps cleanly to the API: measurand → SymbolicMeasurement.val, measurement uncertainty → .err, coverage interval → expanded_uncertainty(m, k), dominant source → dominant_source(...).
• Conformity assessment is a downstream consumer of the coverage interval — the library provides the interval; the decision logic (R₀ ∈ [y-U, y+U]) is two lines of user code.
• Paraphrase-only — no normative JCGM / VIM text is reproduced in the library or this tutorial. Section numbers are cited to anchor the methodology; the authoritative document is JCGM 100:2008 / JCGM 200:2012 itself.

References

• JCGM 100:2008 — Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM). BIPM. See CITATION.bib at the repository root for the BibTeX entry.
• JCGM 200:2012 — International vocabulary of metrology — Basic and general concepts and associated terms (VIM), 3rd edition. BIPM.
• EA-4/02 M:2022 — Evaluation of the Uncertainty of Measurement in Calibration. European co-operation for Accreditation.
• For the full methodology reference of every SymbolicUncertainties.jl export, see Methodology Reference.
• For the six quick worked examples (Ohm, voltage divider, RC time constant, dissipated power, RLC resonance, RC-charge ODE), see Worked Examples.