On the modeling and the stability of continuous-time ΣΔ-Modulators

In this paper an exact state equivalency including the input signal between continuous-time (CT) ΣΔ-modulators and their discrete-time (DT) counterparts is derived. The equivalency is then generalized to include the very important non-ideal effect of excess loop-delay, that plays a crucial role in continuous-time modulators. With this new state-space equivalency, an efficient tool for stability analysis based on continuous-time describing functions is introduced. In the end, it is explained how these findings can be used for a MATLAB toolbox for continuous-time modulators that complements the well-known Schreier-toolbox for discrete-time modulators.