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Two resistive channels can observe the same motion while using different slopes, offsets or directions. Their raw voltages should not be subtracted as though they were interchangeable. First map one channel into the coordinate of the other using released coefficients and explicit normalization. Then calculate a signed residual at synchronized positions. The residual contains channel mismatch, measurement uncertainty and possibly real mechanical separation; it is not automatically a fault flag. A disciplined record distinguishes the mapping used by design from coefficients fitted to one convenient specimen.
Key design decisions
- Choose a comparison coordinate and define each channel's valid interval.
- Control gain and offset coefficients independently from the specimens being assessed.
- Budget residual uncertainty before an application owner sets a plausibility threshold.
Define what each channel reports before comparing them
Document excitation, return, wiper, receiver and mechanical reference for both channels. One track may rise while the second falls, or both may rise with different usable spans. State whether voltages are absolute, divided by a common supply, or normalized by separate references. A shared excitation creates correlation that must be preserved. If channels use different supplies or returns, a common-mode assumption is unsafe. Also identify electrical start and end positions because one channel may intentionally enter its inactive region earlier than the other.
Select channel 1 voltage, normalized ratio or physical position as the comparison coordinate. Mapping into physical position is attractive but requires validated inverse curves for both channels. Mapping channel 2 directly into channel 1 units can be simpler for an online plausibility calculation. Neither choice proves redundancy: independence of conductors, contacts, supplies, mechanics and diagnostics belongs to the surrounding architecture. This method owns only the numerical residual after a declared mapping.
Use gain and offset only where an affine relationship is valid
For channels designed as straight and unequal ramps, represent channel 2 in channel 1 coordinates with a gain and offset. The sign of gain handles opposite directions. Determine coefficients from design-controlled endpoints or a released calibration procedure, not by minimizing residual on every tested unit. Per-unit fitting can erase the very mismatch the residual is intended to reveal. When either curve is nonlinear, use a controlled piecewise or polynomial mapping and evaluate its separate interpolation error rather than stretching a straight-line formula across the full travel.
The signed residual retains useful direction. An absolute value may be used later for a limit comparison, but storing only magnitude hides whether a channel is consistently high or low.
ŷ_2→1(x) = g y_2(x) + o; r(x) = y_1(x) - ŷ_2→1(x)
- y_1 and y_2 are synchronized connected outputs at the same mechanical coordinate x.
- g is the released scale factor, including sign where directions differ.
- o is the released offset in channel 1 units.
- r is the signed mapped residual and ŷ_2→1 is channel 2 expressed in channel 1 coordinates.
The affine mapping applies only within the stated travel interval and under the excitation and loading conditions used to define g and o.
Calculate a residual for oppositely directed channels
Take an illustrative synchronized point where channel 1 reads 2.620 V and channel 2 reads 1.410 V. The released mapping from channel 2 to channel 1 is g = −0.800 and o = 3.760 V. Mapped channel 2 is therefore 2.632 V, giving r = −12 mV. At a neighboring point, channel 1 is 2.780 V and channel 2 is 1.190 V; the same coefficients produce 2.808 V and a residual of −28 mV. These values are calculation examples, not product limits or measured capability.
The change from −12 to −28 mV matters more than a single pass or fail statement. It can reflect channel shape mismatch, position skew, loading or coefficient error. If a local channel-1 slope is 16 mV per mechanical degree, the second residual corresponds numerically to −1.75 degrees in that coordinate. That translation does not prove the shaft is displaced because electrical errors can create the same residual. It merely provides a common scale for allocating the subsequent investigation.
Keep mapping coefficients independent of the evaluation data
A design mapping may come from nominal curve definitions, while production calibration may use a specified set of stations. Record which one controls the plausibility calculation. If coefficients are estimated from two endpoints, endpoint noise propagates through every interior residual. If least-squares fitting is permitted, define the positions, weighting and specimen scope in advance. Do not refit after seeing an unexpected interior feature. That practice turns a diagnostic residual into a self-correcting curve and can conceal a genuine channel difference.
Control coefficient precision in software and documentation. Rounding a negative gain or offset can create a position-dependent numerical residual even when ideal curves agree. Evaluate quantization with the implementation data type, not only with spreadsheet decimals. Reopen the analysis when ADC reference, channel loading, curve endpoints or mapping code changes. A version identifier linking coefficients to the circuit and track revision prevents cross-use of incompatible calibrations.
| Input | Control question | Residual consequence |
|---|---|---|
| Gain g | Design value or authorized calibration result? | Changes residual slope |
| Offset o | Which terminals and zero state define it? | Moves the residual baseline |
| Sample alignment | Are both channels captured at the same position and time? | Creates apparent mismatch on motion |
| Valid interval | Where are both tracks active and monotonic? | Prevents endpoint misuse |
Propagate channel and mapping uncertainty into the residual
Residual uncertainty includes both output measurements, gain, offset and any time or position alignment error. Shared excitation can cancel partly in ratiometric channels, so treating all terms as independent may overstate or misstate the result. Conversely, separate grounds can introduce a common-looking offset that does not cancel. Use partial derivatives of the chosen mapping and the applicable covariance information. When only bounded limits exist, calculate adverse directional combinations and label the result a bound rather than a statistical confidence interval.
Position skew is often dominant during a sweep. With output slopes s1 and s2, a sample delay Δt at motion rate dx/dt creates a residual contribution related to those slopes and the mapping gain. Synchronized acquisition or low-speed dwell points can isolate it. Quantization creates discrete steps whose effect should be evaluated in mapped units. Maintain separate budget rows for expected channel manufacturing variation and measurement uncertainty so evidence about one is not mistaken for proof of the other.
Read residual shape rather than relying on its maximum alone
A nearly constant residual points toward offset, reference or return difference. A residual that changes linearly through travel suggests gain mismatch. A localized peak can indicate curve geometry, contact disturbance or a mapping-knot problem. Opposite signs on forward and reverse approaches suggest mechanical separation or timing. A residual that appears only when one diagnostic load is enabled implicates the circuit state. These patterns guide controlled contrasts; none is sufficient as a stand-alone root-cause verdict.
Plot y1, raw y2, mapped y2 and signed residual against an independent coordinate. Preserve values beyond any proposed threshold so clipping does not hide shape. Compare repeated cycles and multiple specimens. If both channels move together relative to the independent reference while residual stays small, the paired comparison cannot see that common-mode error. This is why a low residual is evidence of agreement between channels, not proof that either channel is accurate in absolute position.
Validate nominal mapping and intentional perturbations separately
First acquire slow, synchronized forward and reverse sweeps under the released supply and receiver conditions. Confirm that each channel remains in its stated active range, apply the controlled coefficients, and compare residual with the uncertainty prediction. Then use permitted diagnostic perturbations one at a time: alter a receiver within an approved range, introduce a known reference offset in a simulation or test interface, or create a defined sample delay in analysis. The residual response should follow the model. A perturbation validates detectability, not field failure probability.
Record raw ADC values, physical position, excitation, returns, mapping revision, timing and filtering. Threshold selection must account for normal residual, uncertainty, transients, fault objectives and system response; it belongs to the responsible application and safety teams. Tests on moving machinery or safety-related controls require their authorized procedure. ChipSimple can support review of the paired printed networks against supplied drawings, but cannot authorize a system diagnostic threshold without complete architecture evidence.
Release the two curves and mapping as one controlled interface
The interface package should provide terminal maps, output direction, excitation, receiver loading, active intervals, mechanical datum and simultaneous curve points for both tracks. State g and o with units, precision, rounding rule and version ownership. Define whether residual is signed or absolute and how out-of-range values are handled. If a piecewise mapping replaces the affine relationship, provide knots and interpolation method. Avoid labeling two nominal curves as redundant without describing shared contacts, supplies or mechanical elements.
Quotation inputs should distinguish the tolerances allocated to each printed channel from the residual allowed by the complete system. Include any requirement for separate terminations, track spacing, glazing, contact paths or inspection access on the drawing. Revisit the mapping after artwork, substrate, contact geometry, connector, receiver, reference or software changes. That release discipline makes the residual reproducible across design, manufacture and validation.
Provide both channel definitions and the controlled mapping
A paired-track review requires simultaneous data and the system rule used to compare it.
- Two track schematics, terminal maps, excitation and connected receiver circuits.
- Mechanical datum, travel direction, active intervals and synchronized curve points.
- Released gain, offset or piecewise mapping with coefficient precision and ownership.
- Residual allocation, uncertainty assumptions, validation conditions and diagnostic objective.
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