论文标题
对叠加傅里叶序列引入的角度误差的谐波分析，用于在正弦/余弦角度上应用
Harmonic analysis of the arctangent function regarding the angular error introduced by superimposed Fourier series for application in sine/cosine angle encoders
论文作者
论文摘要
我们提出了一种严格的分析方法，用于谐波分析旋转和线性编码器的角度误差，其正弦/余弦输出信号在正交中被叠加的傅立叶级数扭曲。为了计算正交中测得的正弦和余弦编码器通道的角度，通常使用了arctangent函数。因此，原始信号和计算的角度之间的非线性关系（通常被认为是黑匣子）使角误差及其谐波分解的估计变得复杂。通过谐波幅度的Taylor系列扩展，我们的方法允许量化谐波信号扭曲对谐波顺序，大小和相位角度误差的影响，包括剩余误差术语的上限 - 无需对Arctangent函数的数值评估。通过在复杂平面中直观的几何近似，可以实现相同的近似值，从而验证了结果。另外，高阶泰勒膨胀考虑了信号中谐波之间的相互作用效应。近似值显示了与数值示例中的精确计算的绝佳一致性，即使发生较大的失真幅度，也会导致对角误差分解的可行估计。
We present a rigorous analytical method for harmonic analysis of the angular error of rotary and linear encoders with sine/cosine output signals in quadrature that are distorted by superimposed Fourier series. To calculate the angle from measured sine and cosine encoder channels in quadrature, the arctangent function is commonly used. The hence non-linear relation between raw signals and calculated angle -- often thought of as a black box -- complicates the estimation of the angular error and its harmonic decomposition. By means of a Taylor series expansion of the harmonic amplitudes, our method allows for quantification of the impact of harmonic signal distortions on the angular error in terms of harmonic order, magnitude and phase, including an upper bound on the remaining error term -- without numerical evaluation of the arctangent function. The same approximation is achieved with an intuitive geometric approximation in the complex plane, validating the results. Additionally, interaction effects between harmonics in the signals are considered by higher-order Taylor expansion. The approximations show an excellent agreement with the exact calculation in numerical examples even in case of large distortion amplitudes, leading to practicable estimates for the angular error decomposition.
