## 4 Results and Discussion
Usually, this third harmonic is a result of variation in the optical-scanning process. The following conclusions could be drawn from the obtained results:  
* The initial overview of the Lissajous curves showed that the SDE of the tested encoder depended on the scanning speed. More detailed analysis at different velocities is necessary to figure out the dependency. * Statistical investigation showed a strong linear relationship between SDE and scanning speed. In this case, it is important to know the maximal traversing velocity at which the encoder runs in a specific application. Different maximal speed gives a different maximal SDE value and sets a distinct limitation for the highest resolution. In another case, when the SDE retains the same meaning in the whole speed range, or its relationship is nonlinear, the maximal SDE value should be determined. * The maximal recommended traversing velocity of the encoder was 1 m/s. Working in this range, maximal SDE was \(\pm\)0.49 \(\upmu\)m, reached at 900 mm/s. After interpolation of these encoder signals, the resolution should be more than 0.5 \(\upmu\)m. Otherwise, the interpolation error is bigger than the measuring step. * At each speed, the biggest part of the SDE budget formed the second harmonic. This directly correlates to increasing velocity. This means that the difference of the signal amplitudes or the phase shift increased with speed. Amplitude, offset, or phase errors could be caused by the physics of the optical-scanning principle, and the dynamical behavior and improper adjustment of the electric components (like photodiodes, used processing chips, or analog amplifiers), or the  
Figure 4:0 Three first harmonics of determined SDEs at different scanning speeds. quality of the used cable and other effects. It is necessary to pay attention to these aspects not only during the design process, but also while choosing a suitable encoder for a specific application. For example, the tested encoder's working principle was based on the four-field scanning method. If the application requires more stability to scanning speed, or if there is an increased possibility of measuring scale contamination, the optical encoder based on the single-field scanning principle must be selected. * The calculated total SDE values at 900 and 1000 mm/s velocities were distinct from the determined linear relationship. The magnitudes of the first and third harmonics reached their maximum values at 900 mm/s. When analyzing the first harmonic graph, in speeds above 500 mm/s the offset error of the encoder signals started to greatly vary. It is very likely that the dynamic behavior of the encoder was affected in this range of speed. The reading head is a complex mechanical part containing optical and electronic components, and flexible spring-based suspension, so even the smallest translation, distance variation, tilt, or other change in a relative position between scanning reticle and measuring scale could be generated by resonant frequency, friction, or other forces. For more detailed analysis, the generated frequencies at these speeds should be determined and compared with the natural frequencies of the encoder. * FFT analysis showed that the major part of the SDE was made up of only a few first harmonics. That means that the trend of the SDE could be quite accurately approximated by using a simple equation that contains information of only the three first harmonics (\(n=3\)). Figure 4:1 (3)  
After some extra processing of the experimental-investigation data, the multivariable function of the SDE value could be derived. This function could tell the approximated SDE value at a relative position inside the period at any scanning speed. \[\delta(x,v)=f(x,v)\ +\ \varepsilon,\ for\left\{ \begin{array}{l}0\leq x\leq p \\ 0<v\leq v_{Max}\end{array} \right. ;\ x,v\in Z, \tag{4}\]  
where \(f(x)\) is a multivariable approximation function of the SDE value, of which the arguments are relative position \(x\) and scanning speed \(v\).