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A rotary contact can follow a repeatable circle around its shaft while moving across a different circle on the resistor card. When the shaft and printed arc centers do not coincide, the contact's angle about the printed center differs from the shaft angle, and its radial position changes through the sweep. Evaluate both effects before attempting an electrical calibration: correction of a stable curve cannot restore contact area that has moved beyond the usable track.
Key design decisions
- Locate the actual pivot and printed arc center in the same assembly coordinate system.
- Calculate both angle mapping and radial footprint for the permitted travel sector.
- Allocate angular residual and physical edge margin separately; satisfying one does not establish the other.
Define the motion and track circles separately
Place the printed track center at the coordinate origin. Let the pivot lie a distance e along the positive horizontal axis, and let the contact center remain a fixed distance R from that pivot. Shaft angle theta is measured from the same positive axis. The contact position is then x equal to e plus R cosine theta and y equal to R sine theta.
This is a planar fixed-arm model. Axial motion, arm bending, bearing movement and contact-finger compliance are excluded initially and must be added when they matter. The printed track's nominal centerline radius is also taken as R for the numerical comparison. If the intended arm and track radii differ, include that nominal difference instead of assigning every radial deviation to eccentricity.
Convert the moving contact into track-centered coordinates
The distance from the printed center is the square root of R squared plus e squared plus twice e R cosine theta. Its angle about that center is the two-argument arctangent of R sine theta and e plus R cosine theta. Use a continuous angle convention over the active sector; a software jump between plus and minus pi is not a physical discontinuity.
Subtracting shaft angle from track-centered angle gives delta phi equal to atan2 of minus e sine theta and R plus e cosine theta. For e smaller than R, the denominator of this last expression stays positive. At zero eccentricity both the angular difference and the radial variation vanish, which is a useful implementation check.
r(theta) = sqrt(R² + e² + 2eR cos(theta)); delta phi = atan2(-e sin(theta), R + e cos(theta))
- R is fixed pivot-to-contact-center distance and e is pivot-to-track-center offset, in matching length units.
- r is contact-center radius about the printed center.
- theta is shaft angle and delta phi is track-centered angle minus shaft angle, in radians before conversion to degrees.
Planar fixed geometry with 0 <= e < R and a point representing the contact center. The finite footprint and actual electrical transfer are evaluated separately.
Find the angular extrema within the permitted sector
Over a complete rotation, the largest magnitude of the coordinate difference is arcsine of e divided by R. The stationary positions satisfy cosine theta equal to minus e divided by R. For small offsets, the error is approximately minus e divided by R times sine theta in radians, but use the exact relation when allocating a tight geometric budget.
For partial travel, evaluate the sector endpoints and every stationary position that lies inside that sector. A full-circle bound is conservative if the sector excludes the worst position. Do not automatically place the maximum at exactly ninety degrees: that is the small-offset approximation, while the exact extremum shifts slightly with eccentricity.
Calculate one offset before assigning a calibration correction
Assume a 10.0 mm arm length and 0.20 mm offset. The full-rotation angular-error magnitude reaches approximately 1.1460 degrees, with one extremum near 91.1460 degrees. The radial contact-center position ranges from 9.80 to 10.20 mm, a peak-to-peak excursion of 0.40 mm.
At shaft angle zero, the track-centered angle error is zero and the radial position is 10.20 mm. At ninety degrees the angular error is approximately minus 1.1458 degrees while the radius is approximately 10.0020 mm. Thus the point with nearly maximum angular error is not the point with maximum radial displacement. A single observation chosen to inspect one error can miss the worst condition of the other.
| Shaft angle | Track-centered angle difference | Contact-center radius |
|---|---|---|
| 0 degrees | 0 degrees | 10.2000 mm |
| 90 degrees | Approximately -1.1458 degrees | Approximately10.0020 mm |
| 180 degrees | 0 degrees | 9.8000 mm |
Reserve track width for the finite contact footprint
The contact center is not the complete contact region. For a simple annular usable track band with radial half-width w, and a footprint enclosed within a radial half-extent b about its center, require the magnitude of r minus the track radius plus b to remain no greater than w. Add a positive reserved edge margin when setting the actual acceptance condition.
In the numerical example, a usable half-width of 0.30 mm and footprint half-extent of 0.10 mm leave exactly 0.20 mm for center displacement, matching the assumed eccentricity but leaving no remaining margin. Reserving another 0.05 mm reduces the allowed displacement to 0.15 mm. This geometric containment test does not establish contact pressure, wear life or electrical stability, and real irregular footprints need their actual envelope.
Intersect the angle and contact constraints
If the full-circle angular-coordinate allowance is 0.5 degree for a 10 mm arm, the geometric bound requires e no greater than R times sine of 0.5 degree, approximately 0.08727 mm. The preceding contact-width allowance with a 0.05 mm reserve permits 0.15 mm. In this comparison, angular mapping controls the offset allocation even though the contact remains inside the usable track at larger offsets.
A narrower track or wider contact could reverse that decision. Preserve both inequalities rather than converting everything into one generic alignment tolerance. For a limited travel sector, use the actual calculated maximum of each quantity. Also include nominal radius mismatch and other allocated displacement contributors before assigning the entire remaining budget to pivot location.
Do not equate geometric angle with an unverified output law
The coordinate calculation identifies where the contact moves. Predicting electrical output additionally requires the actual track potential as a function of position, including any radial dependence, segmentation, collector connections and receiver loading. A broad resistive band is not automatically equipotential along every radial line. Use the measured or justified electrical map for the specific construction.
Where a stable track-centered angle-to-output function is established, substitute the calculated contact angle into that function to estimate the eccentricity contribution. A zero adjustment removes a constant indexing error but not the sinusoidal coordinate difference. A calibrated full-curve correction may address repeatable mapping error, but cannot correct unstable bearing movement, intermittent edge contact or a changed assembly offset after remounting.
Verify position, footprint and output on the same assembly
Measure pivot and printed-center locations from shared datums, then compare actual contact travel against the calculated radial envelope. Sweep through the specified sector in both directions using an independent shaft-angle reference. Retain contact observations and raw electrical output together so a geometric residual can be distinguished from backlash or a local track irregularity.
For the drawing review, provide the nominal centers, arm length, usable band and contact envelope with separate angular and edge-margin allocations. Recheck after any change to the carrier, locating features or contact construction. This makes eccentricity an explicit mechanical-to-electrical interface requirement instead of a curve defect that is discovered only after the track artwork has been finalized.
Send the pivot, printed center and contact envelope
Provide both motion and track geometry so angular error and physical contact margin can be evaluated together.
- Shared-datum drawing locating pivot,printed arc center,nominal radius and active angular sector.
- Arm length and its tolerance,radial/axial runout and relevant carrier compliance.
- Usable track band,contact footprint envelope and required reserved edge margin.
- Target electrical transfer,allowed angular contribution and any existing calibration map.
- Independent angle sweeps,raw output,contact observations and remounting comparison records.
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