High-Frequency Modeling and Analyses for Buck and Multiphase Buck Converters
Future microprocessor poses many challenges to its dedicated power supplies, the voltage regulators (VRs), such as the low voltage, high current, fast load transient, etc. For the VR designs using multiphase buck converters, one of the results from these stringent challenges is a large amount of output capacitors, which is undesired from both a cost and a motherboard real estate perspective. In order to save the output capacitors, the control-loop bandwidth must be increased. However, the bandwidth is limited in the practical design. The influence from the switching frequency on the control-loop bandwidth has not been identified, and the influence from multiphase is not clear, either. Since the widely-used average model eliminates the inherent switching functions, it is not able to predict the converter's high-frequency performance. In this dissertation, the primary objectives are to develop the methodology of high-frequency modeling for the buck and multiphase buck converters, and to analyze their high-frequency characteristics.
First, the nonlinearity of the pulse-width modulator (PWM) scheme is identified. Because of the sampling characteristic, the sideband components are generated at the output of the PWM comparator. Using the assumption that the sideband components are well attenuated by the low-pass filters in the converter, the conventional average model only includes the perturbation-frequency components. When studying the high-frequency performance, the sideband frequency is not sufficiently high as compared with the perturbation one; therefore, the assumption for the average model is not good any more. Under this condition, the converter response cannot be reflected by the average model. Furthermore, with a closed loop, the generated sideband components at the output voltage appear at the input of the PWM comparator, and then generate the perturbation-frequency components at the output. This causes the sideband effect to happen. The perturbation-frequency components and the sideband components are then coupled through the comparator. To be able to predict the converter's high-frequency performance, it is necessary to have a model that reflects the sampling characteristic of the PWM comparator. As the basis of further research, the existing high-frequency modeling approaches are reviewed. Among them, the harmonic balance approach predicts the high-frequency performance but it is too complicated to utilize. However, it is promising when simplified in the applications with buck and multiphase buck converters. Once the nonlinearity of the PWM comparator is identified, a simple model can be obtained because the rest of the converter system is a linear function.
With the Fourier analysis, the relationship between the perturbation-frequency components and the sideband components are derived for the trailing-edge PWM comparator. The concept of multi-frequency modeling is developed based on a single-phase voltage-mode-controlled buck converter. The system stability and transient performance depend on the loop gain that is affected by the sideband component. Based on the multi-frequency model, it is mathematically indicated that the result from the sideband effect is the reduction of magnitude and phase characteristics of the loop gain. With a higher bandwidth, there are more magnitude and phase reductions, which, therefore, cause the sideband effect to pose limitations when pushing the bandwidth.
The proposed model is then applied to the multiphase buck converter. For voltage-mode control, the multiphase technique has the potential to cancel the sideband effect around the switching frequency. Therefore, theoretically the control-loop bandwidth can be pushed higher than the single-phase design. However, in practical designs, there is still magnitude and phase reductions around the switching frequency in the measured loop gain. Using the multi-frequency model, it is clearly pointed out that the sideband effect cannot be fully cancelled with unsymmetrical phases, which results in additional reduction of the phase margin, especially for the high-bandwidth design. Therefore, one should be extremely careful to push the bandwidth when depending on the interleaving to cancel the sideband effect.
The multiphase buck converter with peak-current control is also investigated. Because of the current loop in each individual phase, there is the sideband effect that cannot be canceled with the interleaving technique. For higher bandwidths and better transient performances, two schemes are presented to reduce the influence from the current loop: the external ramps are inserted in the modulators, and the inductor currents are coupled, either through feedback control or by the coupled-inductor structure. A bandwidth around one-third of the switching frequency is achieved with the coupled-inductor buck converter, which makes it a promising circuit for the VR applications.
As a conclusion, the feedback loop results in the sideband effect, which limits the bandwidth and is not included in the average model. With the proposed multi-frequency model, the high-frequency performance for the buck and multiphase buck converters can be accurately predicted.