I've been learning power electronics and am trying to design a dual-switch forward converter. One of the things I've been trying to do is derive the small-signal equivalent circuit of the non-ideal forward converter. The part I'm having difficulty with is how to write the state equations when the ESR of the capacitor is included.
In order to better understand the model of the forward converter I'm following along with a state-space-averaging example of a non-ideal buck converter. This Paper describes on page 8 how perform a state space averaging on a non-ideal buck converter, which is similar to the forward converter since the forward converter is a buck-derived topology and the equations are nearly identical. The circuit is:
Where Vg(t) is the input voltage, Ig(t) is the input current and v(t) is the output voltage.
Now, we need to write the state equations for the circuit during the first sub-interval when the transistor is on. During this interval the transistor is replaced by an equivalent resistance Rt and the free-wheeling diode is an open-circuit. The state equation of this linear circuit are of the form:
Which are the equations for the first interval. Where R1 is the inductor parasitic resistance, R2 is the capacitor ESR, Rt is the on resistance of the transistor and R is the load resistance. I'm confused about the how the author derived these state equations. I'm sure there's something simple I'm missing in the circuit analysis, but I'm not able to get the KVL and KCL equations for my inductor voltage and capacitor current to match in order to get the SSA model of the converter in equillibirum. If someone could walk me through there derivation or provide hints I would greatly appreciate it. I've done SSA on various other converters and understand the process well, but the added complexity that the ESR's introduce is giving me difficulty. Thank you.