The most common systematic error in rotary encoder installations is eccentricity, a mismatch between the center of the encoder disc and the geometric center of the rotation axis.
Unlike grating uniformity errors (which are inherent to the encoder) or interpolation errors (which are minimized by electronics design), eccentricity is an installation-dependent error. Its magnitude can range from negligible to many times larger than the encoder’s inherent accuracy. Understanding and controlling eccentricity is essential for high-precision rotary applications.
Definition and Mathematical Model
Eccentricity (e) is the lateral displacement between the center of the encoder scale disc and the center of the rotation axis. It is typically expressed in micrometers.
When a disc with eccentricity e rotates about an axis, the sensor (which is fixed relative to the rotation axis) reads different positions on the disc at different rotation angles. The position error introduced by eccentricity follows:
Angular error (radians) = arctan(e / R)
Where R is the radius of the optical diameter (the radius at which the sensor reads the grating).
For small eccentricity values (e << R):
Angular error (radians) ≈ e / R
Converting to arc-seconds: 1 radian = 206,265 arc-seconds
Angular error (arc-seconds) = (e / R) × 206,265
Numerical Examples: Eccentricity Error vs. Scale Diameter
| Scale Diameter (D) | Optical Radius (R) | Eccentricity (e) | Angular Error |
|---|---|---|---|
| 100 mm | 50 mm | 10 µm | 41 arc-seconds |
| 100 mm | 50 mm | 25 µm | 103 arc-seconds |
| 100 mm | 50 mm | 100 µm | 412 arc-seconds (0.11°) |
| 50 mm | 25 mm | 10 µm | 83 arc-seconds |
| 50 mm | 25 mm | 25 µm | 206 arc-seconds (0.057°) |
| 25 mm | 12.5 mm | 10 µm | 165 arc-seconds |
| 25 mm | 12.5 mm | 5 µm | 83 arc-seconds |
Interpretation: An encoder with inherent accuracy of ±2 arc-seconds (interferential optical) has its effective system accuracy dominated by eccentricity at any eccentricity greater than approximately 0.5 µm at 50 mm radius. Achieving the encoder’s inherent accuracy requires sub-micron eccentricity control.
Practical implications:
- At typical machined hub tolerances (h6/H7 fits): eccentricity ≈ 5–25 µm → dominant error for precision encoders.
- At precision-lapped hub seats: eccentricity < 2 µm → encoder inherent error becomes relevant.
- At standard pressed-on hubs with tolerances: eccentricity may exceed 50 µm → eccentricity dominates for all encoder technologies.
Sources of Eccentricity
Manufacturing Tolerance in Hub-Disc Assembly
The disc is mounted to a hub that is pressed or shrunk onto the shaft. The fit tolerance between the hub bore and the shaft determines the maximum displacement between the shaft center and the hub center. For a standard H7/h6 press fit at 20 mm diameter:
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- Fundamental deviation for H7: +21 µm / 0 µm.
- Fundamental deviation for h6: 0 / -13 µm.
- Maximum clearance (or interference at tightest): combines to potentially 21 µm maximum radial deviation in worst case.
This worst-case eccentricity (21 µm at 50 mm optical radius) produces a 87 arc-second error, completely dominating an interferential optical encoder’s ±2 arc-second inherent accuracy.
Bearing Runout
Bearing inner race runout displaces the shaft relative to the outer race. For precision ABEC-7 bearings:
- Maximum radial runout: 2.5 µm (inner race, shaft).
- This is a 10 arc-second error at 50 mm radius — still significant for interferential encoders.
For angular contact bearing pairs (common in precision motor shafts): 1–2 µm runout achievable.
Disc Balancing Error
If the disc material has density variations (inclusions, voids), the geometric center of the disc does not coincide with the mass center. During high-speed rotation, unbalance forces deflect the shaft, introducing speed-dependent eccentricity.
Compensation Method 1: Two-Readhead Averaging
Principle: If two readheads are positioned exactly 180° apart on the same scale, the eccentricity error they each measure is equal in magnitude and opposite in sign. The average of the two readings cancels the eccentricity error:
Corrected position = (position₁ + position₂) / 2
Why this works mathematically:
At any position θ, eccentricity introduces an error ε(θ) = (e/R) × sin(θ). When the second readhead is at position θ + 180°:
Error at readhead 2: ε(θ + 180°) = (e/R) × sin(θ + 180°) = -(e/R) × sin(θ) = -ε(θ)
Average: [θ + ε(θ) + θ + 180° + (-ε(θ))] / 2 = θ + 90°
After accounting for the 180° offset, the error cancels exactly (for pure first-harmonic eccentricity).
Higher-order errors (from bearing runout harmonics or disc non-circularity) are only partially cancelled by two readheads.
Four-readhead configuration (readheads at 0°, 90°, 180°, 270°) averages out both first and second harmonic eccentricity errors — suitable for the highest accuracy requirements.
Compensation Method 2: Electronic Calibration
Principle: For each encoder unit, the eccentricity error is measured during a calibration run and stored as a correction table. During operation, the stored correction is subtracted from the measured position.
Procedure:
- Measure the encoder output at N equally spaced angular positions using a reference standard (e.g., a calibrated polygon mirror or a more accurate reference encoder).
- Compute the error at each calibration point: error(θ_i) = reference(θ_i) – encoder(θ_i).
- Fit the error profile to a Fourier series or store as a lookup table.
- Apply the correction in the encoder electronics or controller.
This method corrects all systematic errors (not just eccentricity) including grating uniformity errors and interpolation bias.
Limitation: Calibration is performed at one temperature. Differential thermal expansion between the disc and hub introduces temperature-dependent eccentricity change that the static calibration table cannot correct.
Hub Design Best Practices for Minimum Eccentricity
Precision fits: Use P5/P4 bearing precision classes and precision ground hub bores. Target hub bore/shaft eccentricity < 2 µm.
Kinematic mounting: Some high-precision encoder disc designs use a three-point kinematic mount with adjustable radial eccentric pins. The disc is positioned relative to the axis by adjusting the pin positions under interferometric monitoring.
Adhesive mounting with alignment fixture: Bond the disc to the hub with structural adhesive while the disc is held in a precision centering fixture. Achieves eccentricity < 5 µm without precision grinding.
Do not rely on encoder disc OD as a centering reference: The optical diameter (where the grating lines are read) is not necessarily concentric with the disc OD. Use the optical track itself as the centering reference whenever possible.
Before you go, you might want to dive deeper into:
- Rotary Encoder Selection for CNC Machine Tools: Resolution, Accuracy, and Interface Requirements,
- discover more about Multi-Turn Absolute Encoders: Architecture and Application in Multi-Revolution Actuation Systems,
- or check out our guide on Slip Ring Assemblies for Offshore Oil and Gas: ATEX Certification and Hazardous Area Requirements.
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