# Confusion on State Space Equation of Buck Converter

I'm trying to represent a buck converter in terms of state space equations.

I'm assuming CCM operation and I had derived the state space equations for two modes of operations:

• Mode 1: Switch is ON (0 < t < dT)
• Mode 2: Switch is OFF (dT < t < T)

My workings are displayed below: The confusion arose when I'm checking my state space equations with my lecture notes. My state space equation for mode 1 is identical with the referred state space model, so I had no problem for that. However, for mode 2, I found that my state space equation is the negative version of the referred state space equations.

After investigation, I found that the assumption on polarity of voltages and direction of current are the key differences of my work and the reference. Based on what I have learnt, I know that after disconnected from the power supply, the polarity of inductor voltage should change to maintain the current in the same direction, and the direction of capacitor current should change to maintain the same voltage polarity.

However, the reference seemed to neglect this phenomena. May I know why is this?

Just in case anyone wonder what is in my lecture note: The two state variables $x_1$ (the inductor current) and $x_2$ (the capacitor voltage) do not change directions between the on- and off-states. You can see that during the off-state, as correctly drawn in your lecture notes, the inductor current still splits between $C$ and $R$. If I may further comment, I would recommend to use the PWM switch model rather than the state-space averaging (SSA) technique for analyzing switching converters. See here for more details: slides 46/47 for the buck converter SSA derivation and the rest of the pages for the PWM switch model analysis.