  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.
