Optical Encoder Sine/Cosine Interpolation and Subdivision: How One Scale Pitch Becomes Nanometer Resolution

Why sine and cosine: two signals that encode position twice

A transmitted-light optical readhead illuminates a periodic scale grating and detects the resulting light modulation with photodiodes arranged to produce two signals 90 degrees apart: sine and cosine. Over one scale pitch, the pair completes one full cycle. The 90-degree shift is what makes fine position recoverable, because it turns the pair into a two-dimensional coordinate. Plotting cosine against sine traces a circle, and every position within the pitch corresponds to exactly one angle on that circle.

Direction and coarse travel come free from the same pair. As the readhead moves forward the point rotates one way; reversed motion rotates it back. The controller can count full periods for coarse position while the angle within the period supplies the fine position. This dual role is why the sine/cosine pair remains the backbone of incremental optical encoding even as electronics have pushed subdivision factors from tens to tens of thousands.

How subdivision turns one period into thousands of steps

Subdivision is the measurement of the phase angle within the sine/cosine cycle. The electronics compute the angle, typically through an arctangent function or equivalent analog phase method, and divide the full 360 degrees of the cycle into equal steps. A subdivision factor of 4096 on a 20 µm pitch yields a theoretical increment of about 4.9 nm. The published resolution is simply pitch divided by the subdivision factor, rounded to the output format the interface transmits.

The angle computation is only as good as the signals feeding it. The arctangent assumes a circle: equal amplitudes, zero DC offset, no harmonic distortion and correct 90-degree phasing. Each violation deforms the circle into an ellipse or off-center spiral, and the deformation returns as a periodic position error that repeats once or twice per pitch. This is why interpolated resolution is quoted from a clean bench signal, and why installation, gain setup and contamination control determine whether the machine achieves anything close to it.

• Subdivision factor divides each scale period into equal angle steps
• Interpolated resolution equals scale pitch divided by subdivision factor
• The method assumes a clean circle: balanced amplitudes, zero offsets, 90-degree phase
• Signal defects reappear as periodic error repeating once or twice per pitch

What limits the subdivision factor

Signal quality sets the practical ceiling. Noise on the sine and cosine inputs translates through the angle computation into position jitter, so the signal-to-noise ratio bounds how finely the period can be divided before the extra digits report noise rather than position. Amplitude imbalance between channels, DC offsets and residual harmonics each contribute systematic periodic error components with distinct signatures, which is why interpolation error testing measures the error over one pitch and compares it against the signal parameters.

Speed and bandwidth impose a second ceiling. At maximum velocity the signal frequency equals velocity divided by pitch, and the electronics must sample, compute and transmit position within the control-loop period. Higher subdivision means more computation per period and finer output timing requirements. This is where pitch selection interacts with interpolation: a coarser pitch lowers the signal frequency at speed and buys bandwidth headroom, while a fine pitch raises signal frequency and demands faster electronics for the same velocity.

Choosing and specifying interpolated resolution responsibly

Start the selection from the error budget, not the finest number on the datasheet. Ask what position uncertainty the process tolerates, then allocate shares to scale accuracy, interpolation error, mechanics, thermal behavior and environment. An interpolated resolution several times finer than the sum of the other error sources contributes little except cost and bandwidth pressure; conversely, resolution coarser than the servo loop can use wastes control performance. The right setting is the one that keeps the output increment below what the loop can correct, with interpolation error small against the budget.

For specification and purchase, require the configuration-specific data: subdivision factor, interpolated resolution, maximum output frequency for that setting, the signal amplitude window and the interpolation error figure over one pitch. Confirm that the gain and offset setup procedure is defined for installation, because a subdividing interpolator commissioned by signal level rather than by guesswork keeps its periodic error specification in service. SENFU documents resolution options and signal behavior per configuration, and the application review can map them to your travel, speed, controller and environment before order.