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Small-Signal Model of Multiphase Constant-On Time Control with Phase Overlapping

Year: 2023 | Author: Sundaramoorthy Sridhar | Paper: H5.3
Waveforms
Fig.1. Two-phase COT controlled buck with two-phase overlapping: (a) perturbed waveforms, and (b) control-to-output transfer function comparison.
  Multiphase constant-on time (COT) controlled voltage regulators (VRs) that power modern processors must have a fast transient response, be able to handle large current, and offer better efficiency. To achieve these goals, the phase number of these VRs continues to increase. With a large phase count, these VRs operate with phase overlapping; i.e., duty cycle D > 1/N even within the practical duty cycle range. When D < 1/N, the total current waveform exhibits COT modulation. Multiphase COT control is equivalent to single-phase COT control from a small-signal perspective. However, with D > 1/N, the total current waveform exhibits variable on-time and off-time modulation, as shown in Fig. 1(a). Hence, the single-phase model is not applicable for compensator design in the phase-overlapping region, as shown in Fig. 1(b). This work presents a general describing function model for multiphase COT control that is applicable to the whole duty cycle range. To develop this model, a general modulation law for total current waveform was first derived from the perturbed time-domain waveforms. This law revealed that with M overlapping phases, the total current has M rising intervals and M-1 falling intervals within each fixed on-time pulse Ton. Also, this behavior remains the same irrespective of phase number. Then, using the modulation law, and performing Fourier analysis on the perturbed waveforms, the control, line, and output describing functions were derived. These describing functions are found to be accurate beyond switching frequency, as shown in Fig. 2. To obtain better design insights, a rational transfer function approximation for control-to-output transfer function was also derived. With M ? 1, i.e., phase overlapping, this transfer function has a moving zero, static zero, and a double pole due to the external ramp. However, With M=1, i.e., no-phase overlapping, a pole-zero cancellation occurs and the proposed model reduces to the single-phase model. Thus, proving its generality.
Transfer function verification
Fig.2. Control-to-output transfer function verification for two-phase overlapping.

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