The physical grating pitch of an encoder scale (typically 20 µm for optical, or the equivalent capacitive period) represents the coarsest resolution the sensor can provide without interpolation. At a 20 µm grating pitch on a rotary scale with 2,000 lines, the uninterpolated resolution would be 1/2,000 of a revolution = 0.18°.
Modern servo systems require very high resolutions of typically 0.001° or better. Interpolation (applying the arctangent function to the differential sinusoidal encoder signals) subdivides each grating period by a factor of 100× to 100,000×, enabling theoretical sub-arc-second resolution from a physical grating that the unaided eye can barely resolve.
What Interpolation Requires: The Sin/Cos Signal
Interpolation requires a sinusoidal signal pair — sine and cosine channels — from the encoder sensor. For an optical encoder, the sensor generates two sinusoidal signals in 90° quadrature as the scale moves through each grating period:
- sin channel: V_sin = A × sin(2πx / d) + offset
- cos channel: V_cos = A × cos(2πx / d) + offset
Where:
- A = signal amplitude (typically 1 Vpp for standard analog output, or proportional for digital processing)
- x = position within the grating period
- d = grating pitch (e.g., 20 µm)
- offset = DC bias (typically removed by differential signal processing)
Position within the grating period:
θ_interp = arctan(sin / cos)
This arctangent computation divides each 360° electrical cycle of the sin/cos signal into as many angular increments as the electronics can resolve — limited by the signal-to-noise ratio of the sin and cos channels.
Interpolation Factor and Achievable Resolution
The interpolation factor (N) is the number of position counts generated per grating period:
Count resolution = grating pitch / N
For typical optical encoder parameters:
| Grating Pitch | Interpolation Factor | Position Resolution |
|---|---|---|
| 20 µm | ×100 | 200 nm |
| 20 µm | ×1,000 | 20 nm |
| 20 µm | ×4,000 | 5 nm |
| 20 µm | ×40,000 | 0.5 nm |
At ×40,000 interpolation on a 20 µm grating, the output produces 0.5 nm resolution counts — approximately the size of 2 silicon atoms placed side by side.
In rotary encoder terms:
For a rotary scale with 256 periods per revolution and ×4,000 interpolation:
- Total counts per revolution: 256 × 4,000 = 1,024,000
- Angular resolution: 360° / 1,024,000 = 0.00035° = 1.26 arc-seconds
Signal Quality Requirements for High Interpolation Factors
As the interpolation factor increases, the sensitivity to signal imperfections increases proportionally. The following signal quality parameters must offer tighter specifications at higher interpolation:
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1. Signal Amplitude (A)
If sin and cos amplitudes are not equal, the Lissajous figure (sin plotted vs. cos) is an ellipse rather than a circle. The arctangent of an ellipse is not linearly proportional to position — it introduces a periodic position error at twice the grating frequency.
Typical amplitude imbalance specification for high interpolation:
| Interpolation Factor | Max Amplitude Imbalance tolerance |
|---|---|
| ×100 | ±5% |
| ×1,000 | ±2% |
| ×4,000 | ±1% |
| ×40,000 | ±0.2% |
2. Phase Offset (Quadrature Error)
The sin and cos channels must be in exactly 90° phase quadrature. A phase error φ produces a Lissajous ellipse rotated relative to the horizontal axis, introducing a different form of periodic position error.
Effect: Phase error of φ produces a position error of approximately φ/2 arc-second at the output per arc-second of phase error input (for small φ). At ×4,000 interpolation, a 1° phase error contributes approximately 0.5° / 4,000 = 0.00013° position error — typically acceptable. At ×40,000 interpolation, the same 1° phase error contributes 10× more — potentially significant.
3. Signal-to-Noise Ratio (SNR)
Random noise on the sin/cos channels produces random noise on the computed position. The rms position noise:
Position noise ≈ noise_amplitude / (A × N)
Where A is the signal amplitude and N is the interpolation factor. Higher signal amplitude and lower noise floor improve position noise performance at any given interpolation factor.
For A = 1 Vpp and 1 mV rms noise at the interpolation input:
- At ×100: position noise ≈ 0.001 / (0.5 × 100) = 0.00002 per-period fraction = 0.4 nm at 20 µm pitch
- At ×4,000: position noise ≈ 0.001 / (0.5 × 4,000) = 0.0000005 per-period → 0.01 nm — dominated by other error sources
SNR is typically not the limiting factor at standard interpolation levels; signal quality (amplitude balance, phase) usually dominates.
4. Harmonic Distortion
Real encoder signals are not perfect sinusoids. Higher harmonics (3rd, 5th) in the sin/cos signals produce systematic position errors at those harmonic frequencies. Total Harmonic Distortion (THD) must be specified and controlled:
- At ×100: THD < 5% acceptable
- At ×4,000: THD < 1% required
Interferential optical encoders typically achieve THD < 0.5%, supporting interpolation factors up to ×40,000 or higher.
Digital Interpolation vs. Analog Interpolation
Analog Interpolation
Early interpolation electronics used an analog arctangent circuit. The sin and cos signals were sampled continuously by analog circuits, and the arctangent function was computed by an analog approximation circuit (typically a ratio circuit using diodes and op-amps).
Advantages: No quantization noise at the interpolation stage; continuous output. Disadvantages: Drift with temperature; calibration required; limited interpolation factor (typically ×100).
Digital Interpolation
Modern interpolators sample the sin and cos signals with high-resolution ADCs (typically 14–16 bits) and compute the arctangent digitally using lookup tables or CORDIC algorithms.
Advantages: No drift; calibration stored in non-volatile memory; interpolation factors up to ×1,000,000 in software. Disadvantages: Quantization noise from the ADC at the interpolation input (mitigated by oversampling); finite computation time (latency).
Digital interpolation dominates modern encoder designs. The interpolation electronics may be:
- In the sensor head: The encoder outputs a digital position word (BiSS-C, SSI) directly
- In a remote interpolator module: The encoder outputs a sin/cos analog signal; an external interpolator board performs the ADC and arctangent computation
Remote interpolators allow the interpolation factor to be adjusted externally, enabling flexible resolution configuration without changing the encoder.
Inductive Encoders: Robust Sine/Cos Generation in Confined Spaces
While interferential optical encoders offer ultra-high interpolation factors through precise physical masks, FLUX inductive encoders achieve outstanding position feedback by scanning the variable electrical impedance of a passive metallic rotor loop.
Instead of utilizing fragile glass scales or active permanent magnets, the stator features a multi-layer printed circuit board (PCB) that generates a high-frequency, oscillating electromagnetic field. As the rotor turns, its continuous 360° holographic pattern modulates this field , generating raw sinusoidal and cosinusoidal signals inherently averaged across the entire circumference.
This holistic sensing geometry completely eliminates the typical periodic errors caused by mechanical eccentricity, which normally distort the Lissajous circle into a modulated wave on single-point scanning systems. Because the output relies entirely on stable PCB trace geometries rather than optical clarity, the resulting sin/cos signal pair maintains incredibly low total harmonic distortion (THD) and zero backlash.
This pristine signal quality enables real-time digital interpolation up to 20-bit or 22-bit resolutions within ultra-flat, bearingless profiles under 6 mm in thickness.
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