Shaft encoder characterization via theoretical model of differentiator with both differential and integral nonlinearities

A model of an incremental shaft encoder is developed to facilitate sensor characterization. The model is obtained through derivation of a new mathematical formula for the spectral characteristics of the error which accrues when a sampled, nominally constant-rate signal is uniformly quantized, having been subject to both differential and integral nonlinearities. The spectrum of the error in the rate estimate generated when a digital differentiator is applied to such a signal is shown to be of particular importance. Subsequent sensor characterization involves some basic signal processing of a set of sampled sensor outputs, obtained when the encoder rotates at an almost uniform rate, followed by a simple curve-fitting procedure using the formula for estimated rate error. Both computer-generated, finite-length data sets and experimental data derived from encoder-based shaft velocity measurements are utilized to verify the theoretical model. The methodology of the mathematical analysis is applicable to other digital sensors and to a more general class of systems, such as data converters, which involve the digital differentiation of quantized, noise-affected signals. The paper illustrates how the combined influence of quantization error and of additional sources of noise can be described in an analytical, but applicable, manner.