Closed-Form Formulas for Designing Ultra-Low Phase-Noise Cross-Coupled Dynamically Body-Biased Only-NMOS LCVCOs

This paper presents a system-level analytical framework for modeling and minimizing phase noise in body-biased cross-coupled LC-tank voltage-controlled oscillators (LC-VCOs). Building upon Impulse Sensitivity Function (ISF) theory, the impulse sensitivity and noise modulation mechanisms associated with both flicker and thermal noise sources are systematically characterized. By modeling the oscillator as a nonlinear dynamical system and incorporating transistor operation across multiple regions, analytical expressions for device-level noise power spectral densities (PSDs) are derived as functions of transconductance parameters under symmetric body excitation. Using these results, effective ISF representations corresponding to dominant noise sources are formulated, enabling a unified description of noise-to-phase conversion dynamics. The phase noise minimization problem is then cast as an optimization over system parameters, where both DC and RMS components of the effective ISF are analytically evaluated and minimized. This leads to the derivation of three closed-form expressions that explicitly capture the interaction between circuit parameters and the applied body-bias signals. The proposed framework provides insight into parameter sensitivity and design trade-offs in nonlinear oscillator systems and offers generalizable analytical tools for guiding the design of ultra-low phase noise LC-VCOs, as well as for exploring new oscillator architectures.