Buck converter - CCM DC model

A buck converter is a simple DCDC converter. To extract its steady state model and, successively, the AC model, a DC model must be found. Let's do some calculations with components substituted with ideal models (no losses, stray components, saturation effects, voltage drops).

To find a DC model, it's convenient to analyse separately the circuit during the on-time and the off-time.

Since there are two reactive elements, let's write their constitutive equations related to voltage and current that must equal 0 over the whole period; it is simply required to calculate the average values between D and D' instants $$\overline{v_L} = D(v_i-v_o) + D'(-v_o) = 0$$ $$\overline{i_C} = D(i_L-\frac{v_o}{R}) + D'(i_L-\frac{v_o}{R}) = 0$$ From the first equation, we can find the input-output relation at steady-state $$D(v_i-v_o) = D'v_o \; \rightarrow \; D = \frac{v_o}{v_i}$$ From the second equation, instead, it is possible to find the inductor average current $$D(i_L-\frac{v_o}{R}) = -D'(i_L-\frac{v_o}{R}) \; \rightarrow \; i_L=\frac{v_o}{R}$$

All converters have many non-idealities, so, to refine the model, we have to consider them in our analysis. Let's take a look at the following scheme.

The average values over v_L and i_C are $$\overline{v_L} = D(v_i-v_o-R_{DSon}i_L-R_{DCR}i_L) + D'(-v_o-v_f-R_{DCR}i_L) = 0$$ $$\overline{i_C} = D(i_L-\frac{v_o}{R}) + D'(i_L-\frac{v_o}{R}) = 0$$ From the first equation, we can find the input-output relation in steady-state, with additional parasitic terms $$D(v_i-v_o-R_{DSon}i_L-R_{DCR}i_L) = D'(v_o+v_f+R_{DCR}i_L) \; \rightarrow \; D=\frac{v_o+v_f+R_{DCR}i_L}{v_i+v_f-R_{DSon}i_L}$$ From the second equation, we can see that the inductor current is slightly higher than the ideal one when the duty cycle is lower than 0.5 or lower when the duty cycle is higher than 0.5. However, it is difficult to estimate Q - which is the amount of charge taken from the output capacitor causing v_o to decrease during turn off (D') - so it is often considered equal to 0C. $$D(i_L-\frac{v_o}{R}+\frac{Qf_{sw}}{D}) = -D'(i_L-\frac{v_o}{R}-\frac{Qf_{sw}}{D}) \; \rightarrow \; i_L = \frac{v_o}{R} + f_{sw}Q\frac{1-2D}{D} $$