State-space model of a buck converter in DCM

Question

I have a problem getting the state-space representation of a buck converter in DCM.

There are two switch cycles between one period.

If the switch is closed the circuit looks like this:

After applying KVL you get for the state variables:

\begin{align} \dfrac{\mathrm di_L}{\mathrm dt} &= \dfrac{1}{L}\cdot(U_{in}-u_C)\\ \dfrac{\mathrm du_C}{\mathrm dt} &= \dfrac{1}{C}\cdot(i_{L}-i_{R}) \end{align}

The same procedure can be followed when the switch is open:

So the state variables are:

\begin{align} \dfrac{\mathrm di_L}{\mathrm dt} &= -\dfrac{u_C}{L}\\ \dfrac{\mathrm du_C}{\mathrm dt} &= \dfrac{1}{C}\cdot(i_{L}-i_{R}) \end{align}

So, there is a procedure for extracting the state variables.

I would like to know how to get:

\begin{align} \dfrac{di_L}{dt}=~...\\ \dfrac{du_C}{dt}=~... \end{align}

when the buck converter is working in DCM.

In State-space average Modeling of DC-DC Converters with parasitic in Discontinuous Conduction Mode (DCM) on page 20-21 the authors describe how they got to a solution, but I don't understand their method (or the way they describe it).

Answer 1

There are three states each cycle:

1. Switch closed, your first equation applies (less the transistor drop).
2. Switch open, diode conducting; your second equation applies (less a diode drop, but I assume from your use of a IGBT that you consider a diode drop to be negligible).
3. Switch open, diode not conducting; in theory the coil has neither current nor voltage (in practice it'll be ringing like a bell), and the cap voltage is only affected by the output current

Your challenge is that the time that the converter stays in the switch open, diode conducting state depends on the coil current when the switch opens -- the switch-over happens when the circuit wants it to, not from any external control.