Sub-Divisional Error (SDE) is cyclic and non-accumulative. It results from interpolation errors (not count loss), typically caused by imperfections in the analogue sine and cosine signals fed to the interpolator by the encoder’s read head. To understand SDE it is necessary to understand what a Lissajous figure is and how it is constructed.
The term “Lissajous” is named after Jules Antoine Lissajous (1822-1880), a French mathematician who pioneered work on the optical observation of vibration. A Lissajous figure is a way of displaying two analogue signals relative to each other. If the magnitude of a sine wave is plotted against the magnitude of a cosine wave at any time a circular plot can be created.
As the analogue sine and cosine signals change due to movement of the encoder relative to the scale, the point on the Lissajous moves in a circular path one turn per period of the signals. This circular plot can be displayed on an oscilloscope. The direction of movement can be deduced by the direction of rotation of this circle.
Because SDE results from cyclic Lissajous effects, it has a pattern repeat related to the signal period, typically one cycle of SDE per signal period (i.e. 20μm distance for RGS-S scale). SDE can cycle with harmonics (e.g., 2 x per Lissajous, hence 10μm distance) but not subharmonics – hence never 40μm.
SDE is measured in μm, mean to peak and can be thought of as the maximum error from the true position in any one cycle, not from the scale origin. The convention for specifying SDE in terms of mean to peak value comes from the convention for describing sine waves being also in terms of amplitude mean to peak. This is a non-accumulative cyclic error, i.e. it is independent of scale length.