Exploration of Second-Order Effects in High-Performance Continuous-Time $\Sigma \Delta$ Modulators Using Discrete-Time Models

This paper proposes a method for the discretization of continuous-time sigma-delta modulators (CT-ΣΔMs) with various circuit nonidealities. Recurrence equations for the sampled states of a CT-ΣΔM are derived to find the equivalent discrete-time (DT) transfer functions of CT loop filters along with several second-order effects, such as finite DC gain, finite unity-gain bandwidth (GBW) of an amplifier, and excess loop delay (ELD), etc. This allows a synthesis flow that considers these nonidealities at the system level. The proposed approach generalizes existing works relating to the DT modeling of a CT-ΣΔM, and is applicable to arbitrary-order loop filters with unconstrained digital-to-analog converter (DAC) output waveforms. According to the DT model, a third-order low-pass modulator design based on a numerical optimization shows that the second-order effects of a CT-ΣΔM can be noticeably mitigated by the appropriate coefficient scaling.