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The airfoil thickness-to-chord ratio is the dimensionless maximum section thickness divided by the airfoil chord:Its percentage form is $100\eta$. The airfoil thickness convention must specify how upper and lower surface points are paired and in which direction their separation is measured. For the standard NACA four-digit airfoil construction, the prescribed thickness envelope is placed normal to the airfoil mean camber line, not simply added vertically everywhere.^[ref]

For an original scale example, $t_{\max }=24\mathrm{mm}$ and $c=200\mathrm{mm}$ give $\eta =0.12$, or twelve percent. Scaling every coordinate by three gives $72\mathrm{mm}$ thickness and $600\mathrm{mm}$ chord, with the same ratio. Holding the chord fixed while adding $0.2\mathrm{mm}$ thickness instead gives $\eta =0.121$: the percentage thickness rises by $0.1$ percentage point.

The differential relationshows why an identical fractional scale error in both lengths cancels to first order, while an independent thickness error survives. Exact common scaling cancels in the quotient itself. This is the geometric counterpart of a ratiometric measurement.

The scalar $\eta$ does not specify where the thickest station lies, the airfoil leading-edge radius, or the airfoil camber. Two sections can therefore share $\eta$ while having different pressure distributions and structural envelopes. The early NACA investigation varied thickness and mean-line form independently in a variable-density wind tunnel, making those distinctions experimentally useful.^[ref]

A geometric check of a manufactured section should report the chord reference, the thickness direction and the coordinate scale. Measuring a projected thickness in a tilted image without correcting its direction can change the reported ratio even when the physical section is unchanged.

Correlated threshold and amplitude errors in RC timing must be propagated together. An input level and a comparator threshold can share a reference, so treating their measurement uncertainties as independent can overestimate or underestimate the uncertainty of the RC threshold-crossing time. The correct cancellation depends on which levels share an error, including the initial baseline.

First hold the time constant $\tau$ fixed and take a known zero initial output. Write $H>0$ for the step's high level and $0<\theta <H$ for the threshold. The RC threshold-crossing time and its two sensitivity coefficients areThe first-order law of propagation of uncertainty retains the covariance term. If the two input standard uncertainties are $u_{\theta },u_H$, and their correlation coefficient is $\rho$, thenThe negative cross term follows from the opposite signs of the two sensitivities: raising the threshold delays a crossing, whereas raising the input makes it earlier. Setting covariance to zero is an assumption about the measurement process.^[ref]

For an original example, let $H=1\mathrm{V}$, $\theta =0.50\mathrm{V}$, $\tau =1\mu \mathrm{s}$, $u_H=20\mathrm{mV}$ and $u_{\theta }=10\mathrm{mV}$. The nominal crossing is $0.69315\mu \mathrm{s}$, andIf the two errors are independent random variables, the result is $28.28\mathrm{ns}$; $\rho =0.9$ gives $8.94\mathrm{ns}$; perfect anticorrelation gives $40.00\mathrm{ns}$. These are standard uncertainties from a local approximation, not automatically coverage intervals.

There is an exact cancellation under a more specific model. With zero baseline, if $H=G{H}_0$ and $\theta =G{\theta }_0$ with the same positive gain $G$, the ratio $a$ is fixed andThis ratiometric cancellation of RC timing gain errors is a ratiometric measurement principle. With $u_{\theta }=a{u}_H$, the zero at $\rho =1$ reflects exact common scaling, rather than a claim that every source of timing uncertainty has vanished. Uncertainty in $\tau$, for example, still gives $u_t/t\approx u_{\tau }/\tau$.

A common offset in RC timing needs the baseline to be specified. If the initial output remains fixed at zero while only $H$ and $\theta$ receive the same offset $B$, thenIn the numerical example, a small shared offset with standard uncertainty $10\mathrm{mV}$ gives about $10\mathrm{ns}$ of timing uncertainty. Perfect positive correlation between these two levels alone does not ensure cancellation.

If the baseline shares the offset too, it must be included. The nonzero-baseline model from residual baseline in RC pulse-width discrimination isAdding $B$ to all three levels leaves both differences, and hence the time, unchanged. This is a change of common zero reference. Holding the baseline fixed while shifting only the two other levels is a different experiment. The extra sensitivity is $c_b=-\tau /(H-V_b)$; at $V_b=0$, the three voltage sensitivities sum to zero, as required by the exact common-offset invariance.

An exact numerical check can use bounded errors with prescribed correlation. Take two independent random variables $Z,W$ with the Rademacher distribution and setThe four equally likely states have the specified variances and covariance. At the chosen error scales, all states retain $0<\theta <H$, so the exact logarithmic crossing exists in every state. Evaluating those four times checks the local approximation without silently assigning a crossing to an invalid input pair. Near a threshold or pulse-duration boundary, missing crossings and nonlinear effects require separate treatment.

For a full measurement uncertainty calculation, include the baseline and time constant when their uncertainties matter. With input vector $(\theta ,H,V_b,\tau )$, Jacobian matrix $J$ and covariance matrix $\Sigma$, the first-order result is $u_t^2\approx J\Sigma J^{\mathsf{T}}$. The measurement model must represent significant corrections and uncertainty sources, rather than only a physical law.^[ref] A bench check can vary a common amplitude reference and then an independent threshold offset while recording actual levels and crossings on an oscilloscope; the baseline should be recorded in both tests.

Joseph Wu is the named author of Texas Instruments' 2018 guide to resistance thermometer measurements, revised in 2023. The guide connects sensor wiring, excitation electric current, reference resistors and analog-to-digital converters rather than treating the converter as a complete thermometer by itself.^[ref]

One concrete design distinction is between two-wire resistance measurement and four-terminal sensing. In the two-wire arrangement, the same leads carry excitation electric current and contribute to the measured electric potential difference. Their electrical resistance is therefore counted with the sensor. Separate sense leads with high input impedance can exclude that series drop, although input leakage and other nonidealities still require attention.^[ref]

For an original ideal example, let a sensor have electrical resistance $R_T=100\Omega$, with two $1\Omega$ leads carrying $I=1\mathrm{mA}$. Two-wire resistance measurement givesThe $2\Omega$ lead error remains even when a shared reference resistor cancels excitation variations in the ratiometric measurement. Ideal four-terminal sensing instead measures $I{R}_T=100\mathrm{mV}$ directly across the sensor. This illustrates the platinum resistance thermometer lead resistance error mechanism without assuming a particular resistance-to-temperature calibration.

A proposed bench comparison replaces the sensor with a stable calibrated resistor and adds known series resistors to its current leads. Comparing two-wire and four-terminal sensing readings then tests lead-drop rejection without changing sensor temperature. The calculation distinguishes the wiring error from sensor self-heating in a ratiometric resistance measurement, which changes the sensor's actual electrical resistance.

Luis Chioye writes about precision analog-to-digital converters and their measurement circuits. The author biography in a 2013 Texas Instruments reference design identifies him as an applications engineer in its Precision Analog group, with degrees in electrical engineering from the University of Arizona and Walden University/NTU School of Engineering and Applied Science.^[ref]

His work includes ratiometric measurement with resistance thermometers: sharing an excitation electric current can cancel its variation, but the input and reference filters must preserve that shared variation consistently.^[ref] This is the engineering distinction captured by ratiometric cancellation with matched signal and reference filters.

With Mike Beckman, Chioye also published a thermocouple reference design. Its reference-junction temperature sweep in thermocouple testing keeps the measurement junction in a stable bath while changing the temperature of the connector and converter board; the acquisition computer stays outside the temperature-controlled chamber. This isolates reference-side effects from changing the temperature being measured.^[ref]

An original ideal calculation explains the test. Let $E(T)$ be the thermocouple electromotive force response from the Seebeck effect relative to $0{}^{\circ }\mathrm{C}$. With isothermal connector junctions at $T_c$, the measured electric potential difference is $V=E(T_h)-E(T_c)$. Cold-junction compensation reconstructs $E(T_h)=V+E(T_c)$ before applying the inverse function of the response. In the stipulated linear model $E(T)=ST$ with $S=40\mu V/K$, a fixed $T_h=100{}^{\circ }\mathrm{C}$ gives $V=3.2\mathrm{mV}$ at $T_c=20{}^{\circ }\mathrm{C}$ and $2.8\mathrm{mV}$ at $30{}^{\circ }\mathrm{C}$. Correct compensation gives the same $100{}^{\circ }\mathrm{C}$ in both cases. These are constructed values, not measured performance or a calibration table for a particular thermocouple.

Cold-junction thermal coupling matters because the temperature sensor must represent the connector temperature. A precise converter reading cannot by itself remove an unmeasured temperature difference between those locations.^[ref]

Ratiometric cancellation of RC timing gain errors follows when the same positive gain multiplies every voltage level relevant to a threshold ratio. With zero initial baseline, fixed time constant $\tau$, high level $H>0$ and threshold $0<\theta <H$,Replacing $H,\theta$ by $GH,G\theta$ with $G>0$ leaves $a$ and the time unchanged. This is an exact invariance, not merely the first-order cancellation of two uncertainty terms. It is a ratiometric measurement property of the specified model.

For an original example, choose $H=2\mathrm{V}$, $\theta =0.6\mathrm{V}$ and $\tau =0.8\mu \mathrm{s}$. The crossing is approximately $0.285340\mu \mathrm{s}$. Gains $G=0.8$ and $G=1.2$ both give the same result when they scale high level and threshold together. A shared readout gain can cancel from the calculated time in the same way; whether a reference also changes the physical circuit must be stated separately.

The mismatch remaining after common RC gain cancellation depends on the relative gains, not their common magnitude. Let the high level use gain $G_H$ and the threshold gain $G_{\theta }$, with $G_{\theta }/G_H=1+ϵ$. Provided $0<a(1+ϵ)<1$,In the example, a one-percent relative mismatch delays the crossing by about $3.436\mathrm{ns}$, compared with $3.429\mathrm{ns}$ from the first-order expression. Thresholds nearer the high level amplify the residual.

A nonzero initial baseline must share the gain too: $t=\tau \log [(H-V_b)/(H-\theta )]$ is unchanged by multiplying all three levels by $G$. Scaling only the high level and threshold against a fixed nonzero baseline is another experiment. Likewise, additive errors follow the conditions in common offset in RC timing, not the two-level scaling rule.

Uncertainty in the time constant, independent threshold error, baseline memory, and failure to reach the threshold remain even when one common gain cancels. Their covariance structure must follow the complete measurement model, as in correlated threshold and amplitude errors in RC timing and NIST's propagation law.^[ref]

A ratiometric measurement infers a quantity from a ratio of two measured signals chosen to share an unwanted multiplicative variation. The useful question is which inputs actually share that variation: a reference resistor error or a change in the sensor itself generally survives the cancellation.

In a shared-excitation resistance ratio measurement, a current source passes the same electric current $I$ through a sensor of electrical resistance $R_T$ and a reference resistor $R_{\mathrm{ref}}$. Their electric potential differences are $V_T=I{R}_T$ and $V_{\mathrm{ref}}=I{R}_{\mathrm{ref}}$. An ideal normalized analog-to-digital converter reading is thereforeThe current source need not have a perfectly constant output for this cancellation, provided both measurements sample the same electric current, the denominator is nonzero, and the converter stays within its operating ranges. This is a standard arrangement for a resistance thermometer.^[ref]

For an original ideal example, take $R_T=100\Omega$, $R_{\mathrm{ref}}=2000\Omega$ and $I=1\mathrm{mA}$. Then $V_T=0.100\mathrm{V}$, $V_{\mathrm{ref}}=2.00\mathrm{V}$ and $r=0.05$. Increasing $I$ by $5\%$ gives $0.105\mathrm{V}$ and $2.10\mathrm{V}$: the ratio remains $0.05$. If the converter instead uses an independent fixed $2\mathrm{V}$ reference and the calculation assumes the original $1\mathrm{mA}$, the inferred electrical resistance becomes $105\Omega$.

The reference-resistor error in ratiometric resistance measurement remains even with perfect shared excitation. If the actual reference resistor is $R_{\mathrm{ref},\mathrm{nom}}(1+e)$ but the calculation uses its nominal value, thenThus a $+0.1\%$ reference error makes the nominal estimate of a true $100\Omega$ sensor about $99.9001\Omega$. Reference accuracy and its variation with temperature still matter.^[ref]

Nor does the ratio remove sensor self-heating in a ratiometric resistance measurement. Joule heating dissipates $P=I^2{R}_T$, which is $0.1\mathrm{mW}$ in the example. A temperature rise changes the actual electrical resistance of a resistance thermometer; the ratio then measures that changed resistance. Reducing the electric current or comparing several currents can reveal this physical effect.^[ref]

The law of propagation of uncertainty makes the role of covariance explicit. For small perturbations of nonzero mean signals, the first-order measurement uncertainty of $r=V_T/V_{\mathrm{ref}}$ obeysThe contribution from a shared excitation fluctuation cancels through the covariance term. Treating the two signals as independent random variables would discard that cancellation. Other errors need their own inputs in the complete model.^[ref]

For time-varying excitation, ratiometric cancellation with matched signal and reference filters requires matching dynamics as well as shared wiring. With constant resistances and identical linear time-invariant systems of impulse response $h$, the filtered signals are $R_T(h\ast I)$ and $R_{\mathrm{ref}}(h\ast I)$, so their ratio is still $R_T/R_{\mathrm{ref}}$ wherever defined. Here $\ast$ denotes convolution. Different filters or different sampling times can leave an excitation-dependent error.^[ref]

A simple proposed bench test first replaces the sensor by a stable calibrated resistor, sweeps the electric current within the instrument limits, and records both electric potential differences and their ratio. That separates electrical cancellation from changing sensor temperature. Repeating with a resistance thermometer in a stable bath then tests Joule heating; changing the reference resistance or one filter tests the predicted failure modes. These are proposed checks of the model, rather than reported experimental results.

Sampling skew after matched ratiometric filters is the timing difference between the signal and reference samples in a ratiometric measurement, after their analog paths have the same response. Shared wiring and matched filters cancel excitation at the same effective time; they do not make two different times interchangeable.

For constant sensor and reference resistances, let the common filtered excitation be $F(t)=(h\ast I)(t)$, where $h$ is the common impulse response and $\ast$ denotes convolution. With consistent initial conditions, the two filtered electric potential differences are $R_{T}F(t)$ and $R_{\mathrm{ref}}F(t)$. If the sensor is sampled at $t$ but the reference at $t+\delta t$, the measured ratio isBoth denominator factors must be nonzero. The sampling instants here mean the effective acquisition times; conversion-result arrival times need not be those instants. Ratiometric cancellation with matched signal and reference filters treats the equal-time cancellation, supported by the primary circuit report's matching requirement.^[ref]

For positive differentiable $F$ and small skew, expanding the logarithm givesA later reference sample during a rising excitation therefore makes the inferred resistance low. The relevant rate is the fractional slope $F^{\prime }/F$, not the absolute slope alone. A timing-skew bound for a ratiometric measurement can express a worst-case requirement using a bound on this logarithmic slope.

For an original local example, suppose $F(t)=F_0{e}^{\kappa t}$ over the acquisition window, with $\kappa =20{\mathrm{s}}^{-1}$ and $\delta t=100\mu \mathrm{s}$. Then the exact scale error is $e^{-0.002}-1\approx -0.199800\%$. The small-skew estimate is $-0.200000\%$. Doubling the common amplitude $F_0$ leaves both errors unchanged; reversing the sample order changes their sign to first order. This is a stipulated short-window excitation model, not an instrument-performance claim.

A focused bench check replaces the sensor with a stable resistor, imposes a controlled excitation change, and compares simultaneous acquisition with deliberately delayed reference sampling. Recording the excitation on an oscilloscope supplies its fractional slope. Repeating on a settled plateau separates timing error from reference-resistor error in ratiometric resistance measurement and other steady scale errors.

Sensor self-heating in a ratiometric resistance measurement occurs when the sensing electric current changes the sensor's own temperature through Joule heating. A ratiometric measurement can remove the direct dependence of the electrical reading on that electric current while still reading the heated sensor correctly. For a resistance thermometer, that means the indicated temperature can differ from the unperturbed surroundings.

With actual sensor electrical resistance $R_T(I)$ and a stable reference resistor $R_{\mathrm{ref}}$ carrying the same electric current,The cancellation leaves the dependence of $R_T$ on $I$ intact. In a small, steady perturbation, a model with thermal resistance $R_{\mathrm{th}}$ gives $\Delta T\simeq R_{\mathrm{th}}{I}^2{R}_T(0)$ andwhere ${\alpha }_R$ is the temperature coefficient of resistance. These approximations require small heating and thermal conditions that remain comparable between readings. The BIPM guide distinguishes heat flow inside the probe from heat flow into its surroundings; changing the surrounding medium can change the effect.^[ref]

A practical diagnostic is a two-current extrapolation. Model $R_T(I)\simeq R_0+c{I}^2$ over the tested range. Measurements $R_1,R_2$ at distinct nonzero current magnitudes $I_1,I_2$ imply$R_0$ is an inferred zero-power electrical resistance, not an electrical reading taken with zero excitation. The guide describes this squared-current method and requires a stable surrounding temperature and settled readings.^[ref]

For an original numerical example, suppose a $1000\Omega$ reference resistor gives ratios $0.120006$ at $0.5\mathrm{mA}$ and $0.120024$ at $1.0\mathrm{mA}$. The inferred resistances are $120.006\Omega$ and $120.024\Omega$. Since $I_2^2=4I_1^2$, the estimate is $(4R_1-R_2)/3=120.000\Omega$. Both ratios canceled their excitation exactly, yet they disagree because the sensor changed.

A bath experiment should alternate the two excitation settings, allow thermal settling and monitor the bath temperature. Repeated low/high/low readings help reveal drift rather than attributing every difference to heating. Resistance thermometer self-heating gives the wider physical setting; reference-resistor error in ratiometric resistance measurement and mismatched filters are separate contributions.

A shared-excitation resistance ratio measurement compares two electrical resistances by passing the same electric current through them and comparing their electric potential differences. It is the circuit realization of a ratiometric measurement: the unknown excitation becomes a common multiplicative input.

For a sensor $R_T$ in series with a reference resistor $R_{\mathrm{ref}}$, ideal high-input-impedance sensing givesThe electric current must be nonzero and shared at the time relevant to each sample. One implementation feeds $V_T$ to an analog-to-digital converter input and $V_{\mathrm{ref}}$ to its external reference pins; the conversion is then scaled by their ratio. The primary circuit guide shows this wiring and includes the programmed gain when translating a digital code to resistance.^[ref]

For an original check, choose $R_T=180\Omega$ and $R_{\mathrm{ref}}=1200\Omega$. Excitation at $0.4\mathrm{mA}$ gives $0.072\mathrm{V}$ and $0.480\mathrm{V}$; excitation at $1.2\mathrm{mA}$ gives $0.216\mathrm{V}$ and $1.440\mathrm{V}$. Both give $r=0.15$ and recover $180\Omega$. This calculation assumes unchanged resistances; the larger excitation dissipates nine times the sensor electric power, making sensor self-heating in a ratiometric resistance measurement an independent test.

A common supply is insufficient if different branch currents pass through the two resistances. With currents $I_T,I_{\mathrm{ref}}$,This is branch-current mismatch in a resistance ratio measurement. Sequential sampling of a changing excitation has the same problem unless the two paths retain the same effective excitation history. Ratiometric cancellation with matched signal and reference filters specifies the dynamic condition.

Wiring matters even with an ideal shared current source. A two-wire sensor reading includes lead electrical resistance; four-terminal sensing separates the current leads from high-impedance potential-sensing leads. The reference itself needs the same care if lead drops would otherwise enter its measured denominator.^[ref]

A bench check can replace the sensor with a stable known resistor, sweep excitation, and compare each measured potential with their ratio. A constant ratio accompanied by changing individual potentials tests shared excitation. It does not establish reference accuracy, eliminate sensor heating, or prove operation outside the converter's input and reference ranges.