Resistive Sensor Output Curves: Breakpoints and Interpolation Error

Specify what happens between two listed points

A resistance table may be interpreted as exact points joined by straight lines, a smooth fitted function or acceptance windows at selected positions. These interpretations are different. Define the intended rule, whether endpoints are included and how values outside the specified travel are handled.

Keep mechanical travel in one coordinate system with a named origin and direction. If a table mixes degrees from one datum with percentage travel from another, interpolation can be numerically correct but physically wrong. Include the electrical measurement boundary and whether the listed quantity is resistance, voltage or normalized ratio. The same card can produce different loaded outputs from the same intrinsic resistance function.

Use a simple interpolation formula transparently

Between two points, linear interpolation weights each endpoint according to the fractional position within the interval. It preserves both endpoint values and provides a straightforward function that can be reproduced in inspection software or a controller. The method does not imply that the underlying physical curve is actually straight.

For a hypothetical interval from 20 degrees at 100 ohms to 30 degrees at 160 ohms, the interpolated value at 24 degrees is 124 ohms. If an independently measured or specified target at 25 degrees is 132 ohms, the straight-line prediction there is 130 ohms, revealing a two-ohm intermediate deviation. Endpoint agreement alone would not expose it.

Place more points where the curve bends

For a smooth function, linear-interpolation error is governed by curvature as well as interval width. A steep straight segment can be interpolated exactly from its endpoints, while a shallow curved segment may need more points. Slope and curvature therefore serve different roles in the point-selection process.

The standard interpolation remainder gives a bound proportional to the maximum second derivative multiplied by interval width squared, divided by eight for a linear interval. If the curvature bound is four ohms per square degree and the interval is two degrees, the corresponding error bound is two ohms. This hypothetical bound is useful only when the curvature estimate itself is justified and the function is smooth over the interval.

Use local slope for the mechanical contribution

A position error moves the contact along the curve. Its first-order output effect is local slope multiplied by position error. This contribution can dominate in a steep region even when interpolation error is small. Allocate it separately so adding more breakpoints does not create a false impression that the total error has been reduced.

For an assumed slope of eight ohms per millimeter, a 0.05-millimeter position uncertainty contributes approximately 0.4 ohm. If the output budget is 0.5 ohm, little remains for interpolation, card variation and electrical measurement. The correct response may be improved positioning or a revised system requirement rather than a denser resistance table.

Choose points according to the local decision

Begin with endpoints and known function transitions, then add points where a quantified check shows that the current interval is inadequate. Retain the reason for each added breakpoint. This produces a useful curve definition without turning every acquisition sample into a contractual target.

Preserve the required direction of the output

If the application expects output to increase or decrease consistently with travel, verify that the breakpoint sequence and interpolation preserve that behavior. A higher-order spline can overshoot between otherwise monotonic points. Noise can make an unconstrained spline unsuitable when monotonic output is required.

A smooth-looking curve is not automatically a better engineering representation. Choose the method based on the target function and its allowed behavior. For piecewise linear data, check duplicate coordinates, reversed ordering and sign changes in successive slopes. If a physical plateau is intended, define its extent and consequence for position inference rather than disguising it with a fitted slope.

Validate using positions that did not define the table

Measure or evaluate additional positions inside each interval, with denser checks where curvature or local features are suspected. Compare the interpolated prediction with the independent target or measurement and retain signed residuals. Do not choose verification points only after looking for locations that pass.

Repeat selected checks in both travel directions and after remounting where fixture variation matters. A residual pattern that shifts with mounting may arise from coordinate alignment rather than insufficient breakpoint density. Keep the target-curve error, actual-card error and measurement uncertainty distinct. Otherwise, refining the table can unintentionally compensate one fixture or specimen instead of defining the intended product function.

Deliver a curve that software and inspection can reproduce

The final table should include coordinate units, output units, interpolation rule, valid domain and acceptance limits. State whether tolerances apply at points, over complete intervals or both. Use enough numeric precision for implementation, while avoiding digits unsupported by the source data and measurement method.

For quotation, provide the original engineering relationship or underlying target data as well as the proposed reduced table. Include the allowed interpolation contribution and the mechanical-position budget. This lets the card curve be reviewed with its actual purpose. Breakpoint selection defines the function efficiently; complete calibration still requires the agreed fixture, electrical interface and verification sequence.