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The voltage on a resistive track is not necessarily the voltage delivered at its wiper terminal. Current drawn by the receiver crosses the wiper interface and develops an additional drop. That contact resistance belongs in series with the output path, not simply in the lower track segment. This calculation separates those locations, predicts the resulting transfer curve and identifies which measurements can distinguish track loading from contact loss. It applies to a continuous potentiometric track with a defined contact position.
Key design decisions
- Place contact resistance between the local track potential and the receiver terminal.
- Evaluate the full active travel because relative loading and full-span error peak at different positions.
- Validate the contact model under the actual load instead of inferring it from a nearly unloaded curve.
Define electrical position independently of commanded travel
Let u represent the fraction of total track resistance between the low terminal and the local contact point. For a uniform track with ideal end regions, u equals normalized active travel. If the track is tapered or includes conductive transitions, obtain u from the actual unloaded resistance distribution instead. A geometric midpoint is not automatically the electrical midpoint.
The model uses one effective contact location. A wiper footprint bridging several conductive regions may change the track network itself and needs a distributed contact model. That is a different question from resistance in the output contact path. Keep mechanical alignment, direction and active endpoints fixed during this calculation so a shifted contact coordinate is not absorbed into an artificially large fitted contact resistance.
Move the wiper contact outside the track equivalent
With an ideal excitation supply, the unloaded track voltage at u is u times Vexc. The resistance seen looking back into the track is the parallel combination of its upper and lower portions, equal to RT u(1−u). Replace the track by this Thevenin source and resistance, then add contact resistance Rc in series between that source and the output terminal.
The receiver RL connects from the output terminal to the low return. It draws current through both the track's Thevenin resistance and Rc. Putting Rc in parallel with the bottom track leg would instead describe leakage; adding it to the bottom leg before forming the source would change even the no-load voltage. Neither represents this contact topology. Collector and connector resistance may be included in Rc only if the record states that combined boundary.
Rth(u) = RT u(1−u); K(u) = Vout/Vexc = u RL/[RL + Rc + Rth(u)]; Iw = Vout/RL.
- RT: total continuous track resistance in ohms.
- u: electrical position from zero to one measured from the low terminal.
- Rc: positive effective series resistance from local track contact to output terminal.
- RL: linear receiver resistance to the low return; Iw: wiper output current.
Ideal excitation at track endpoints, settled ohmic behavior, one effective contact position, and negligible leakage or current injection. RT and Rc refer to the evaluated condition.
Calculate several positions for a declared contact resistance
Take illustrative values Vexc = 5 V, RT = 10 kΩ, Rc = 200 Ω and RL = 100 kΩ. At u = 0.5, Rth is 2500 Ω, giving Vout = 2.5 × 100000/102700, approximately 2.43427 V. The ideal midpoint output is 2.5 V. The total shortfall is therefore about 65.73 mV.
Wiper current at that point is approximately 24.34 µA, so only about 4.87 mV of the shortfall appears directly across the assumed contact resistance. The rest reflects the track's finite output resistance under load. These quantities are calculated from hypothetical inputs; they do not describe a tested sensor or a permissible contact current. Repeat the calculation across travel rather than using the midpoint as the complete transfer specification.
| Electrical position u | Track output resistance | Calculated output | Ideal output |
|---|---|---|---|
| 0.25 | 1875 Ω | 1.22459 V | 1.25000 V |
| 0.50 | 2500 Ω | 2.43427 V | 2.50000 V |
| 0.75 | 1875 Ω | 3.67377 V | 3.75000 V |
| 1.00 | 0 Ω | 4.99002 V | 5.00000 V |
Separate relative attenuation from full-span deviation
For positive u, the fractional attenuation relative to the local ideal output is [Rc + RT u(1−u)]/[RL + Rc + RT u(1−u)]. With constant Rc, this attenuation is largest at the electrical midpoint, where the track source resistance peaks. At u = 0, division by ideal output is undefined; report the absolute voltage or full-span error there.
Full-span deviation is u minus K(u), so the local attenuation is also multiplied by position. Its largest value need not occur at the midpoint. In the example the absolute shortfall at u = 0.75 exceeds that at u = 0.5 even though its relative attenuation is smaller. An endpoint-calibrated residual is a third quantity, with zero and span adjustments removed. State which definition the drawing limits before ranking sensor curves.
Distinguish contact change from a different track profile
Measure the same position with two defined receiver loads while holding excitation and mechanics constant. A change consistent with the Thevenin model supports an impedance explanation; a residual that persists with negligible output current suggests that the track profile or contact coordinate also needs review. Load substitution cannot by itself separate every possible series path, so retain connector and collector details.
At a known u and RT, the model can be inverted to estimate the combined series contact term from a loaded observation: Rc = u Vexc RL/Vout − RL − RT u(1−u). Avoid using this inversion near zero output, where small voltage or position uncertainty becomes disproportionately large. A negative fitted Rc is not a useful passive-contact result; investigate incorrect u, offsets, excitation definition or the assumed receiver model before interpreting it.
Check whether a constant contact resistance is adequate
Contact behavior can vary with position, force, speed and surface condition. Treat Rc(u) as a measured function when a single value fails to describe repeated data. An abrupt intermittent event cannot be represented faithfully by a smooth average resistance. Retain the acquisition bandwidth and unfiltered observations so a low-rate curve does not conceal short interruptions.
An amplifier may draw bias current rather than behave as a fixed resistor, and an ADC may draw charge during acquisition. Add those mechanisms to the output circuit explicitly. Reducing the static contact drop does not prove adequate settling after motion. Increasing receiver impedance makes the DC drop smaller, but a truly open contact can leave the input floating or governed by leakage and stored charge; the finite-Rc formula is not an open-contact diagnostic model.
Validate the transfer with an independent position reference
Check the algebra first at zero position, with Rc set to zero, and as RL becomes very large. The no-load limit should approach u times excitation for finite Rc. Compare the closed-form results with a direct circuit solution containing the upper track, lower track, contact and load as four distinct resistances. This is a useful check against placing the contact on the wrong node.
Then sweep a representative assembled track with an independent travel reference in both directions, using the actual receiver or controlled equivalents. Record excitation at the track terminals, wiper output, contact position and load. Repeat selected points with a substituted load and after the specified contact/environment sequence. Compare predicted and observed residual shapes without refitting every point, and preserve both raw and calibrated transfer curves.
Use residual shape to locate the next investigation
An unloaded curve that is correct while a loaded curve droops suggests output impedance or contact loss. A near-high-end error with little midpoint explanation can expose a series contact or terminal path that remains when track Rth becomes small. Direction-dependent shifts suggest mechanics or contact history. Abrupt spikes tied to motion require time-resolved inspection rather than another smooth polynomial calibration.
Provide the resulting transfer table with the contact boundary, receiver condition and explicit error definition. Keep the assumed values separate from measured functions and agreed acceptance limits. A two-point calibration may remove endpoint error yet leave interior load curvature and contact variability. The defensible output is a circuit-specific prediction to validate with the assembled sensor, not a universal claim of track linearity or contact life.
Send the wiper and receiver circuit
A loaded-transfer review needs the contact path and the electrical position definition.
- Track total resistance, unloaded resistance-versus-position data, active travel datums and endpoint connection drawing.
- Wiper/contact/collector construction, effective contact resistance observations, contact force and motion conditions.
- Excitation at the track terminals, receiver resistance or bias/acquisition model, wiring and measurement bandwidth.
- Required raw or calibrated transfer-error definition, forward/reverse data, environmental sequence and specimen acceptance criteria.
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