Abstract
This paper describes a comprehensive analysis of the stability of continuous-time low-pass or band-pass ΔΣ modulators. This analysis is based on a fully S-domain approach which considers the modulator as a feedback control system with gain and phase margins. Theoretical responses for both the optimal filter and a sub-optimal one are derived using a mathematical formalism close to well-known boundary elements methods. A very accurate empirical model has also been developed and, for each case (optimal and sub-optimal), normalized sets of curves allowing to extract the maximum achievable loop gain for a given loop configuration. High achievable gains were found even in the case of ΔΣ modulators involving large delays (several clock times).
Résumé
Cet article décrit une méthode complète d’analyse de la stabilité des modulateurs ΔΣ passe-bande et passe-bas en temps continu. Cette analyse, entièrement en temps continu, modélise le modulateur comme un système asservi avec marges de gain et de phase. Une méthode mathématique proche de méthodes classiques de moments est utilisée pour calculer les réponses théoriques du filtre optimal et d’un filtre sous-optimal. Un modèle empirique très précis a été développé et dans chaque cas (optimal et sous-optimal) sont donnés des jeux de courbes normalisées permettant d’extraire le gain de boucle maximal pour une configuration de boucle donnée. Ces courbes montrent que des gains très élevés sont accessibles, même pour des modulateurs présentant de grands retards (plusieurs périodes d’horloge).
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Hode, JM., Flouzat, C. & De Gouy, JL. Optimal and sub-optimal Loop Filter Responses for Continuous-Time ΔΣ Modulators. Ann. Télécommun. 57, 1186–1225 (2002). https://doi.org/10.1007/BF02999459
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DOI: https://doi.org/10.1007/BF02999459
Key words
- ad Converter
- Sigma delta modulation
- Feedback control system
- Feedback loop
- Linear filtering
- Optimization
- Stability
- Nyquist criterion
- Transfer function