Design Method for High-Order Sigma–Delta Modulator Stabilized by Departure Angles Designed to Keep Root Loci in Unit Circle

In this brief, a systematic method to design stable single-stage high-order sigma–delta modulators (SDMs) without the need of stability-recovery mechanisms is proposed. It is shown that if the critical root loci of an SDM transfer function are all located inside a unit circle, then the SDM is guaranteed to be stable. Instead of basing the design on noise transfer function as in previous designs, the presented method bases the design on proper choice of open-loop transfer function departure angles so as to keep the critical root loci within the unit circle. A stable fourth-order SDM with the critical root loci inside the unit circle is given as a design example. Simulation results reveal that based on the presented design method, stable single-stage high-order SDMs can be free from overload instability even with full-scale input magnitude. Tradeoffs between stability and performance of the SDMs are presented. By allowing parts of the critical root loci to be located outside the unit circle, this method can be extended for SDM designs with higher signal-to-noise ratio but smaller stable input range.