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# Unified Equivalent Circuit Model and Optimal Design of V^{2}-Controlled Buck Converters

^{2}control has become widely applied in point-of-load buck converters. In this paper, a unified equivalent circuit model for V

^{2}control is proposed. The equivalent circuit model has clear physical meaning and is very helpful for design. Fig. 1 explicitly shows three feedback paths of V

^{2}control, and shows the frequency spectrum of each feedback path when the control signal is under modulation. Similar to current-mode control, the inductor current feedback does not have a low-pass filter, so all the sidebands are fed back to the modulator, and the sideband effect needs to be considered. The capacitor voltage loop is a direct feedback loop without any compensation. Therefore, the sidebands of the capacitor voltage also need to be taken into consideration. This complicates the scenario for finding an equivalent circuit model for V

^{2}control.

The analysis shows that the buck converter with V

^{2}control can be regarded as a non-ideal voltage source. The methodology used to derive an equivalent circuit model to represent this non-ideal voltage source is to establish the connection between V

^{2}control and current-mode control; as the equivalent circuit model of current-mode control is well established, the resulting equivalent circuit model is shown in Fig. 2. The non-ideal current source and non-ideal voltage source are virtually represented by the resonance between L

_{s}and C

_{e}and L

_{e}and C

_{o}, respectively. The damping factor of the double poles caused by the capacitor voltage sideband is determined by the damping resistance Rdamp, depending on the capacitor parameters.

Due to the low ESR of ceramic capacitors, when they are used V

^{2}has stability limitations to be applied. Several methods have been proposed to address this issue; one is using enhanced V

^{2}constant on-time control by adding current ramp information; the other method where adaptive voltage positioning is undesired is using the enhanced V

^{2}constant on-time control with a high-pass filter after current sensing; the third method is sensing the capacitor current. The complete equivalent circuit model for these three methods is also derived in this work and shown in Fig. 3. A unified equivalent circuit model for V

^{2}control is also proposed, as shown in Fig. 4; the model includes inductor current ramp R

_{i}for all modulation schemes. Furthermore, the model also includes external ramp compensation in constant frequency V

^{2}control. The proposed equivalent circuit models are experimentally verified, as shown in Fig. 5.

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