Modeling and Stability Analysis for Multiphase Constant On-Time Control With Phase Overlapping

Abstract

Multiphase buck voltage regulators (VRs) powering today's high-performance processor loads must regulate output voltage within a small tolerance window and handle a large steady-state current. For output voltage regulation, multiphase current-mode and V² constant on-time (COT) control techniques are widely used because these control methods can help increase control bandwidth, minimize output impedance, and achieve fast transient response with less output capacitance. To supply large current demands, the phase count of buck VRs continues to increase. With increasing phase count, steady-state phase overlapping becomes inevitable in the practical duty range. However, currently, there is no good small-signal model to analyze stability and design high-bandwidth control loops for multiphase COT buck converters operating under phase-overlapping conditions. To address this issue, this dissertation presents a systematic small-signal modeling approach for multiphase COT buck converters operating with phase overlap, utilizing the describing function method. Prior works extended the small-signal models of single-phase COT control to design multiphase COT buck converters. These extensions rely on the assumption that multiphase and single-phase COT buck converters are equivalent from a small-signal perspective. This equivalence applies when there is no phase overlapping or when the duty cycle D < 1/N for an N-phase COT buck converter. However, this extension does not work when the duty cycle D>1/N (with phase overlap). In multiphase COT buck converters, total current ripple is used to modulate each phase duty cycle and achieve automatic phase interleaving. When the duty cycle D < 1/N, the total current ripple exhibits COT modulation after introducing a perturbation, so single-phase and multiphase COT control have identical small-signal models. However, if the duty cycle D > 1/N, then the total current ripple will exhibit variable on-time and off-time modulation after introducing a perturbation. Due to this difference in modulation principle, single-phase COT control models cannot be used for multiphase COT control design when the duty cycle D > 1/N. In this dissertation, we first developed a general describing function model for multiphase current-mode COT buck converters. As for single-phase current-mode control, the total current feedback in these converters makes the whole system dynamically non-linear. Hence, we treated the entire current loop and power stage as one entity. Unlike prior works, we derived a continuous-time frequency-domain model for multiphase current-mode COT control by performing a Fourier analysis directly on its perturbed waveform. In this way, we account for the change in total current modulation principle between the no-phase and phase-overlapping regions. The proposed model is quite general and applies to arbitrary phase numbers and phase overlapping numbers. We have also extended this model to multiphase current-mode COT buck converters with coupled inductors. Next, the proposed model for multiphase current-mode COT control was used to study the total current loop stability issue under phase overlapping conditions. Also, the minimum external ramp required for stability with different number of phase overlapping is derived. With two overlapping phases, the critical external ramp slope equals half of the sensed total current rising slope. However, the critical ramp slope with three overlapping phases equals the sensed total current rising slope and increases further with an increase in the number of overlapping phases. After that, we developed a general small-signal model for multiphase V² COT control, including the capacitance voltage ripple and phase overlapping effects. Using this model, it is demonstrated that direct output voltage feedback is inherently unstable in phase overlapping regions. In other words, unlike single-phase V² COT buck converters, even large ESR OSCON capacitors cannot ensure stability with phase overlapping. Hence, external ramp compensation is required to ensure stability when phase overlapping occurs. However, an external ramp can ensure stability only if the output voltage ripple has sufficient current strength. Hence, a physical current loop is required to ensure stability with low ESR ceramic capacitors. The critical limits derived were verified using SIMPLIS simulation and experimental results. In sum, this dissertation provided a mathematical framework to derive small-signal models for multiphase COT control methods based on phase manager under phase overlapping conditions. Small-signal models were derived for the popular multiphase current-mode and V² COT control under phase overlapping conditions. Using this model, we identified that total current loop of multiphase current-mode COT control becomes unstable without sufficient external ramp under phase overlapping conditions. Also, the stability boundaries with different number of phase overlapping are provided. The method proposed here can also be extended to other variations of multiphase COT control using phase manager.