Signal conditioning, calibration and error budgets

Prepare a signal for the next stage

Conditioning can supply excitation to a passive sensor, amplify a small difference, remove a known offset or limit the frequency content reaching a converter. Each function changes the measurement model. A filter may reduce unwanted variation while also delaying the signal or attenuating a real peak. Its effect belongs in the interpretation.

For a separate ideal force channel, let sensor voltage be Vin=.50+.020F. A supplied conditioner Vout=1.5(Vin−.50)+.10 gives Vout=.10+.030F. Over the assumed 0–100 N range, output spans .10–3.10 V. This is a model of range use, not a component circuit or an assurance of available input/output headroom.

A calibration relation is more than a zero adjustment

Calibration compares indications with appropriate references under stated conditions and establishes a relationship with uncertainty. Adjusting a zero changes the instrument; it is not the whole calibration. A nonzero offset and an incorrect slope can coexist. Reference pairs across a range can distinguish their patterns better than one unloaded observation.

Our supplied observed voltages differ from the nominal line by 2, 5 and 8 mV as reference force increases. These three constructed points fit V=.102+.03006F. The exact straight fit is a choice of teaching data. Real residuals, reference uncertainty, loading direction and environmental changes determine whether a linear model is adequate.

Convert each input contribution before combining it

Write the inverse channel model as F=(V−b)/s, where V is the measured output voltage, b its offset in volts and nonzero s the sensitivity in V/N. Standard uncertainty u(x) describes a supplied standard-deviation-equivalent contribution associated with an input estimate x; it is not a known signed error to subtract. To see how it reaches force, first ask how a small change of one input changes F while the others are fixed.

At fixed b and s, a voltage change ΔV gives ΔF=ΔV/s exactly. The corresponding standard-uncertainty contribution is uV(F)=u(V)/|s|: volts divided by volts per newton gives newtons. Raising b by Δb instead changes F by −Δb/s, so its uncertainty contribution has the same magnitude rule u(b)/|s|. The opposite signs matter if the inputs are correlated, even though independent variances use squared magnitudes.

Changing s at fixed V−b requires more care because it is a denominator. Replacing s by s+Δs gives Fnew=F/[1+Δs/s]. For a small fractional change, ΔF≈−F Δs/s; thus the slope contribution is approximately |F|u(s)/|s|. This is a local first-order approximation, suitable only when the sensitivity estimate stays well away from zero and its relative uncertainty is small enough for the model. Large nonlinear effects require a more complete evaluation.

The table supplies one possible origin of the later common-unit budget at the separate nominal state V=1.600 V, b=0.100 V, s=0.030 V/N, F=50 N. Treat the three listed inputs as uncorrelated for this constructed example. Do not count a physical contribution twice by including it both inside the voltage term and again as a separate component. Coefficients obtained from the same calibration can be correlated; the actual budget must preserve that information.

Express every contribution in compatible units

A hypothetical 12-bit converter with 3.3 V span has 4096 ideal code intervals, giving a step 3.3/4096≈0.806 mV. Using the nominal .030 V/N sensitivity, that is about .0269 N per interval. It describes ideal resolution. Noise, reference variation, gain, nonlinearity, sensor effects and drift can make the measurement uncertainty much larger.

For an uncertainty budget, convert contributions through the measurement equation before combining them. Standard uncertainty terms expressed in newtons may combine by the square root of summed squares when the relevant inputs are uncorrelated and the linear approximation is suitable. Shared references or fitted coefficients can be correlated; simply listing independent-looking components does not establish that condition.

Correct a slope while keeping uncertainty visible

At nominal 50 N, the constructed observed voltage is 1.605 V. The nominal conversion gives (1.605−.100)/.030≈50.167 N, while the fitted relation gives (1.605−.102)/.03006=50 N. A model matching its own three reference points does not independently validate unseen conditions.

Separately assume standard uncertainty contributions .10, .20 and .05 N after propagation, with zero correlations. Their combined standard uncertainty is √(.10²+.20²+.05²)≈.229 N. An explicitly chosen coverage factor 2 gives expanded uncertainty about .458 N. That factor alone does not guarantee a particular coverage probability or acceptance decision.

The new table connects input estimates to the original 0.10, 0.20 and 0.05 N contributions: output-voltage uncertainty becomes 0.10 N, offset uncertainty 0.20 N and the local relative-sensitivity term 0.001×50=0.05 N. Their root-sum-square value is therefore the same approximately 0.229 N under the stated independence assumption. This nominal illustration is separate from the three-point fitted example; it does not claim that a calibration fit makes b and s independent.

For a changed-gain case, replace the ideal conditioner gain 1.5 by 3 while retaining Vout=G(Vin−0.50)+0.10 and Vin=0.50+0.020F. Sensitivity doubles from 0.030 to 0.060 V/N. A contribution introduced after the gain, fixed at u(Vout)=0.003 V, now maps to 0.050 N instead of 0.100 N. But a separate upstream input-voltage uncertainty 0.002 V is amplified to 0.003 V at gain 1.5 and 0.006 V at gain 3; divided by their corresponding sensitivities, both give 0.100 N. Gain changes where a voltage contribution sits in the account; it does not automatically improve the underlying physical measurement.

The range also changes. Over a restricted illustrative 0–40 N interval, gain 3 gives 0.10–2.50 V. At 100 N it would demand 6.10 V, beyond the stated 3.3 V converter span; the original 0–100 N range cannot simply be carried across. Available conditioner headroom, noise, drift and actual converter behaviour still need evidence. These ideal comparisons introduce no component choice or operating recommendation.

Try the reasoning

1. Can adding amplifier gain recover a signal already clipped in an earlier stage?

   Show the answer and reasoning

   No. Information lost by clipping is not restored by later scaling.

2. Why is one ideal code step not the complete uncertainty?

   Show the answer and reasoning

   It describes quantisation alone; the other physical, reference and conversion contributions remain.

3. When is root-sum-square combination incomplete?

   Show the answer and reasoning

   When contributions have correlations, inappropriate units or an inadequate measurement model.

4. Keep s=0.030 V/N but change the supplied output-voltage standard uncertainty to 0.0015 V. What force contribution follows, and can it replace the entire uncertainty budget?

   Show the answer and reasoning

   The single contribution is 0.0015/0.030=0.050 N. It does not replace offset, sensitivity, sensor, reference or other contributions. Combine only the relevant separate terms with their correlations and model limits; a smaller voltage term alone does not establish 0.050 N total uncertainty.

5. At gain 3 and sensitivity 0.060 V/N, a downstream voltage term stays 0.003 V while an upstream term is 0.002 V before amplification. Which contribution improves relative to gain 1.5?

   Show the answer and reasoning

   The fixed downstream term becomes 0.003/0.060=0.050 N rather than 0.100 N. The upstream term becomes 3×0.002/0.060=0.100 N, the same as 1.5×0.002/0.030. Only the first contribution improves in this stipulated comparison; the larger gain also requires a new range/headroom check.

What to carry forward

Track the complete conversion and its uncertainty, rather than equating numerical detail with trustworthy evidence.