Measurement method

A measurement method describes the general logical arrangement of operations used to make a measurement. A measurement procedure supplies the detailed instructions for implementing that arrangement with a particular measuring system. The VIM lists substitution measurement methods, null measurement methods, differential methods, and direct or indirect methods among the possible descriptions.^[ref]

For an original substitution measurement method, drive an unknown resistor with a fixed electric current $I$ and record a voltmeter indication. Replace it by an adjustable reference resistor without changing $I$ or the other relevant conditions, and adjust the reference until the indication agrees. In an ideal measurement model,so matching $v_x=v_s$ implies $R_x=R_s$. The common additive readout offset $v_0$ cancels. This cancellation depends on the current, connections and offset remaining common; the name of the method alone does not guarantee those conditions.

A null measurement method instead adjusts a reference to make a difference signal vanish, as in a balanced Wheatstone bridge. A resistance inferred from separately measured electric potential difference and electric current uses the relation $R=V/I$. These are descriptions of how the information is obtained; the operational record must still specify apparatus, settings, timing, reference values and the measurement uncertainty. The measurement principle concerns the physical phenomenon used, while the measurement method concerns the organization of the measurement. Different methods can exploit the same physical principle.

Polarity reversal measurement separates a signal that changes sign on reversal from an additive contribution that does not. If readings in two polarities obeythen the half-difference estimates the signal, $\hat{s}=(y_{+}-y_{-})/2$, while the half-sum estimates the unchanged offset, $\hat{b}=(y_{+}+y_{-})/2$. The procedure depends on reversing the intended signal while preserving the relevant offset.

A primary electrical application is cancellation of thermoelectric offset in low-level measurements. The Keithley handbook describes reversing sources when comparing small electric potential differences, and reversing test current for low-electrical resistance measurements. A thermal offset can arise at dissimilar-metal junctions and remain approximately unchanged during a sufficiently short cycle.^[ref]

For an original resistance example, a bipolar current source applies $+I$ and $-I$ to a resistor while a voltmeter measuresLet $I=10\mathrm{mA}$ and readings be $103$ and $-97\mu \mathrm{V}$. Their half-difference is $100\mu \mathrm{V}$, giving $R=10\mathrm{m\Omega }$; their half-sum reveals a $3\mu \mathrm{V}$ offset. Using only the positive reading would instead give $10.3\mathrm{m\Omega }$.

An offset changing from $b_{+}$ to $b_{-}$ leaves resistance error $(b_{+}-b_{-})/(2I)$. A $2\mu \mathrm{V}$ change in this example therefore leaves $0.1\mathrm{m\Omega }$ error. Allow electrical settling after reversal while keeping the cycle short relative to material temperature changes. The handbook explicitly notes this time-scale requirement.^[ref] Reversal does not automatically remove lead resistance, current-magnitude errors, self-heating or every offset in the circuit. Record the actual reversal operation and timing with the measurement procedure.

A substitution measurement method compares an unknown with a reference by placing them successively in the same relevant position of a measuring system. The comparison seeks a match or measures a difference while retaining as much of the system response as possible. The VIM lists substitution among measurement methods; the detailed sequence belongs to the measurement procedure.^[ref]

For a physical application, NIST's double substitution weighing procedure compares a standard and an unknown of the same nominal mass twice, using a weighing instrument as a comparator. It also determines response with a known sensitivity mass and includes measurement-assurance checks.^[ref] The measurement standard supplies the reference value; the comparator need not establish the whole mass independently from its own zero.

An original simplified drift model shows why sequence matters. Let a comparator indication at equally spaced time index $t$ bewith fixed response coefficient $a$, additive offset $b$ and linear instrumental drift $d$. Observe standard S at $t=0$, unknown X at $t=1,2$, and S again at $t=3$. The mean X indication and mean S indication have the same mean time $3/2$, givingBoth the constant offset and the stipulated linear drift cancel. For $m_S=m_0=100\mathrm{g}$, $a=500{\mathrm{g}}^{-1}$, $b=10$ and $d=2$, an unknown $m_X=100.004\mathrm{g}$ gives indications $10,14,16,16$. Their paired means differ by two, recovering $0.004\mathrm{g}$.

This teaching sequence isolates one cancellation mechanism; it is not the full NIST procedure. Nonlinear drift, unequal timing, changing response or different positioning can defeat it. NIST's procedure separately addresses buoyancy and uncertainty contributions.^[ref] A close match does not remove the reference's measurement uncertainty or establish that the substitution left every relevant condition unchanged.

Double substitution weighing determines the difference between an unknown mass and a reference through repeated substitutions on the same comparator, with a small known sensitivity mass used to determine response under load. NIST's SOP4 compares standard and unknown twice, rather than relying on the balance to establish their entire masses from its zero.^[ref]

For the source's sequence A, let S denote the standard, X the unknown and sw the added sensitivity mass. With optional tare masses omitted, the four pan loads areproducing measurement indications $O_1,O_2,O_3,O_4$. The standard remains the reference; the small addition probes transducer sensitivity close to the actual working load.^[ref]

An original simplified model neglects buoyancy, tare differences and drift, and assumes a common linear response coefficient $a$. Write $\Delta m=m_X-m_S$ and $m_{\mathrm{sw}}$ for the added mass. Relative to the standard load, the readings areThe two comparison differences estimate the same quantity; averaging them gives $D=[(O_2-O_1)+(O_3-O_4)]/2=a\Delta m$. The sensitivity step is $O_3-O_2=a{m}_{\mathrm{sw}}$, soFor original $m_{\mathrm{sw}}=10\mathrm{mg}$ and readings $(100,106,126,120)$, this gives $\Delta m=3\mathrm{mg}$. The offset $b$ and the unknown response coefficient cancel under the model.

Practical weighing records timing and environmental conditions, keeps settings fixed through the comparison, and controls handling and equilibration. The full NIST procedure supplies buoyancy corrections, uncertainty contributions and a comparison using a check standard.^[ref] For equally spaced readings, a linear additive drift cancels from $D$, but can still affect the sensitivity step $O_3-O_2$. Neither the simplified equation nor the method's name guarantees immunity to drift, contamination, nonlinear response or uncertainty in the measurement standard.

A sensitivity mass is a small calibrated mass added to a loaded weighing instrument to determine its response to a known mass increment. NIST's double-substitution procedure calls this a sensitivity weight and uses it to establish sensitivity under the conditions of the comparison.^[ref] It supplies a local input step rather than a new determination of the entire load from zero.

Let an original comparator give indications $r_0$ before and $r_1$ after adding mass $m_{\mathrm{sw}}$. For a stable linear response over that increment,where $\Delta r$ is another indication difference near that working load. A constant additive zero cancels from each difference; instrumental drift or load-dependent response need not cancel.

For original indications $r_0=860$, $r_1=885$ and assigned sensitivity mass $10.020\mathrm{mg}$, the local response is about $2.495$ indication increments per milligram. A difference $\Delta r=7.5$ increments gives $\Delta m=3.006\mathrm{mg}$. Using only a nominal $10\mathrm{mg}$ label instead would give $3.000\mathrm{mg}$. The assigned mass and its measurement uncertainty, rather than the nominal label alone, enter the comparison.

The added step must be large enough to resolve reliably but remain within the response region relevant to the unknown/reference comparison. NIST specifies selection criteria and regularly spaced observations; it also requires the standards to have valid calibration information.^[ref] Keep handling, settling and environmental conditions consistent. A sensitivity mass does not by itself cancel buoyancy, temperature effects or uncertainty in the main measurement standard. Those contributions remain in the full double substitution weighing model.

A null measurement method adjusts a reference or a system setting until a difference signal is brought to zero. A null detector tests the residual mismatch; the final measured quantity value follows from the reference and the balance condition. The VIM lists null measurement among measurement methods.^[ref]

A Wheatstone bridge provides a concrete electrical apparatus. The PHYWE manual connects a source across two resistor dividers and a current detector between their middle junctions; balance occurs when those junction potentials agree and detector current vanishes.^[ref] A sliding-wire implementation adjusts the divider ratio continuously.

For an original notation, let the left branch contain upper resistance $R_a$ and lower $R_b$, while the right contains upper unknown $R_x$ and lower reference $R_s$. With source electric potential difference $V_s$ and negligible loading at balance, the middle-junction potentials above the lower supply terminal areSetting $V_L-V_R=0$ yieldsFor $R_a=2000\Omega$, $R_b=1000\Omega$ and $R_s=500\Omega$, the balance identifies $R_x=1000\Omega$ independently of the nonzero source magnitude in this ideal model.

Zero on a display has finite meaning. With $V_s=3\mathrm{V}$, the derivative of residual $V_L-V_R$ with respect to $R_x$ at this balance is $1/1500\mathrm{V}{\mathrm{\Omega }}^{-1}$. A residual magnitude below $10\mu \mathrm{V}$ therefore corresponds locally to about $0.015\Omega$ of unresolved mismatch. A detector zero offset can also shift the apparent balance. The null measurement method reduces reliance on calibrated detector gain at exact balance, but reference uncertainties, lead effects, detector sensitivity and instrumental drift remain relevant to the complete measurement result.

A null detector indicates whether a residual signal has been brought sufficiently close to zero in a null measurement method. A galvanometer between the middle junctions of a Wheatstone bridge is a concrete example: equal junction potentials give zero detector current.^[ref] Its useful property is the ability to distinguish small residuals and their sign near balance.

For an original electronic model, let the detector output bewhere $ϵ$ is the residual electric potential difference, $G$ is gain, $z$ an output zero offset and $n$ an output noise term. With known zero offset and negligible noise, an output threshold $|d|\le 1\mathrm{mV}$ at $G=1000$ corresponds to $|ϵ|\le 1\mu \mathrm{V}$. It is a resolution criterion for this model, not a total measurement uncertainty.

A false zero exposes another limit. If $z=2\mathrm{mV}$, a displayed $d=0$ can occur at $ϵ=-2\mu \mathrm{V}$ for the same gain. Increasing gain improves the output response to a given residual, but measurement calibration must still address offset and noise. Polarity reversal measurement can help separate signal from an unchanged internal offset: ifthen $ϵ=(d_{+}-d_{-})/(2G)$ and $z=(d_{+}+d_{-})/2$. The separation assumes reversal changes the signal sign while leaving the offset unchanged.

Detector loading also matters away from null. For a residual source with series electrical resistance $R_s$ connected to detector input resistance $R_d$, the detector receives $ϵR_d/(R_s+R_d)$. The ideal zero stays at $ϵ=0$, but a low input resistance reduces the response to a small departure from it. Verify the response with known small positive and negative signals, allow settling, and investigate persistent offsets. A quiet zero may otherwise reflect a broken connection or insufficient sensitivity rather than actual balance. Check the working range when approaching null because detector saturation invalidates the assumed linear response.