Measurement resolution

Measurement resolution is the smallest change in an instrument indication that its reading method can distinguish. It contributes to measurement uncertainty, but does not by itself establish accuracy or remove systematic error. The manometer height-to-pressure calibration shows how a height scale affects the inferred pressure.

Input-referred noise expresses observed or modeled output noise as an equivalent disturbance at a system's input. For a locally linear response $y=Gx$ with constant nonzero gain over the relevant measurement band, an output-noise root mean square $n_{\mathrm{out}}$ corresponds toThis is a way to compare noise with the input signal; it does not locate every physical noise source at the input. Walt Kester's ADC tutorial explicitly models converter noise as an equivalent series input source and describes its measurement through repeated codes under a fixed input.^[ref]

For an original amplifier with voltage gain $100$, output noise $1\mathrm{mV}$ rms is equivalent to $10\mu \mathrm{V}$ rms at the input. Changing gain can change this quotient without removing the device's own physical noise. State the measurement bandwidth and processing, since a longer averaging interval or different filter changes the observed rms value.

An original two-stage example shows why the position of noise matters. Let gains be $G_1,G_2$ and independent input-equivalent noise contributions be $e_1,e_2$ for the respective stages. The output-noise variance isso referring it to the original input givesWith $e_1=2\mu \mathrm{V}$, $e_2=30\mu \mathrm{V}$ and $G_1=10$, this is $\sqrt{13}\simeq 3.61\mu \mathrm{V}$. Increasing $G_1$ to $100$ reduces the equivalent total to about $2.02\mu \mathrm{V}$, approaching the first-stage contribution. The independent-noise assumption matters; correlated contributions require covariance terms.

A quiet output code does not by itself establish a small total error, and rms noise is not the same criterion as input-referred resolution. Kester distinguishes noise-based resolution from dynamic converter measures and notes the conditions needed for averaging to help.^[ref] Retain offsets, gain uncertainty and instrumental drift separately in the measurement model.

Minimum incremental motion is the smallest displacement increment that a positioning system can execute repeatedly for the specified input and test conditions. It is an operational motion property, whereas design resolution is derived from the drive geometry and electronic increments. Friction, stiction, elasticity and backlash can make them differ.^[ref]

For an original illustrative test, independently measure the displacement produced by repeated commands from equivalent starting conditions. A nominal 10 nm command gives moves $0,20,0,20\mathrm{nm}$. Their mean is 10 nm, but that mean does not show repeatable 10 nm increments. A nominal 40 nm command instead gives $39,41,40,40\mathrm{nm}$.

Define an example acceptance rule before testing: every measured move must lie within 5 nm of the command, and the readout must have sufficiently smaller uncertainty, stipulated here as 1 nm standard uncertainty. The 10 nm trials fail this rule; the 40 nm trials pass. Among these two candidate increments, 40 nm is the smallest accepted one. It is not a proof of the globally smallest achievable increment because intermediate candidates and other conditions were not tested.

Report the command convention, load, motion direction, starting procedure, displacement sensor, measurement uncertainty and acceptance rule. A rotary encoder on the motor does not automatically measure the final load motion. Independent readout, for example with an interferometer, tests the assembled stage rather than only the commanded count. Positioning repeatability at a target and the ability to make a small move are related but distinct properties.

Design resolution is the theoretical smallest commanded movement implied by a positioning mechanism's drive components. For a screw drive, it can be calculated from screw pitch, reduction gearing and motor or encoder increments. It is not automatically the minimum incremental motion that the final load executes repeatedly.^[ref]

For an original ideal drive, let a lead screw advance $p=0.50\mathrm{mm}$ per revolution, with 50 motor revolutions per screw revolution and 2000 resolved counts per motor revolution. One count corresponds toDoubling the count resolution to 4000 per motor revolution gives theoretical $q=2.5\mathrm{nm}$ without changing the screw. That calculation does not halve friction or remove backlash. A controller may display both increments even if the load moves only after several such commands have accumulated.

Keep the location and counting convention of the feedback sensor explicit. Counts at a motor shaft, counts at a screw and directly measured load position are different observations. A gearbox can introduce compliance between them. Pair the design calculation with an independent motion test to establish operational performance. A smaller theoretical increment also does not by itself establish absolute positioning accuracy or a smaller measurement uncertainty.