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# Unified Equivalent Circuit Model of V^{2} Control

^{2}control. The direct output volt-age feedback in V

^{2 }control contains inductor current, capacitor voltage and load current information, which can be separated into 3 feedback loops, as shown in Fig.1(a) (use constant on-time as an example). Fig.1(b) shows that for constant on time structure, the inductor current feedback turns the circuit into a controlled current source while the capacitor voltage feedback turns this controlled current source into a controlled voltage source, which will be realized by a current source in parallel with a resistor. Moreover, due to the sideband frequency in the capacitor voltage feedback, the voltage source is a non-ideal source, with the non-idealness shown by a R

_{e2}-L

_{e2}branch in Fig. 1(b), which forms a double pole at half of the switching frequency by resonating with the output capacitor. The damping of the double pole, which is shown as R

_{damp}in Fig.1(b), is related with the capacitor parameters and can be positive or negative. This explains why this structure is elegant when large ESR capacitors are employed (such as OSCON caps) while sub-harmonic oscillation occurs when low ESR capacitors are employed (such as ceramic caps). Fig. 2 shows a complete equivalent circuit model of V

^{2}control, which is applicable to all kinds of V

^{2}control. Generally speaking, the inductor current loop turns the circuit into a non-ideal current source, with the non-idealness shown by the R

_{e}-C

_{e}branch, which will form a double pole by resonating with the power state inductor. The capacitor voltage loop turns the non-ideal current source into a non-ideal voltage source, with the non-idealness shown by the R

_{e2}-L

_{e2}branch, which will form a double pole by resonating with power stage capacitor. The parameters of R

_{e}, C

_{e}, R

_{e2}and L

_{e2}are different between different modulation schemes. This equivalent circuit model can be used to examine all the transfer functions (control-to-output, output impedance, input impedance and audio susceptibility) and can be extended very easily to a multi-phase structure and V

^{2}control with a current ramp case.

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