Analysis and Design of VCO-Based Phase-Domain $\Sigma \Delta $ Modulators

VCO-based phase-domain <inline-formula> <tex-math notation="LaTeX">$\Sigma \Delta $ </tex-math></inline-formula> modulators employ the combination of a voltage-controlled-oscillator (VCO) and an up/down counter to replace the analog loop filter used in conventional <inline-formula> <tex-math notation="LaTeX">$\Sigma \Delta $ </tex-math></inline-formula> modulators. Thanks to this highly digital architecture, they can be quite compact, and are expected to shrink even further with CMOS scaling. This paper describes the analysis and design of such converters. Trade-offs between design parameters and the impact of non-idealities, such as finite counter length and VCO non-linearity, are assessed through both theoretical analysis and behavioral simulations. The proposed design methodology is applied to the design of a phase-to-digital converter in a 40-nm CMOS process, which is used to digitize the output of a thermal-diffusivity temperature sensor, achieving ± 0.2° (<inline-formula> <tex-math notation="LaTeX">$3\sigma $ </tex-math></inline-formula>) phase inaccuracy from -40 to 125 °C and a sensor-limited resolution of 57 m° (RMS) within a 500-Hz bandwidth. Measurements on the prototype agree quite well with theoretical predictions, thus demonstrating the validity of the proposed design methodology.

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