# Small signal model of buck converter

I want to find the transient response of an open loop buck converter for a step change in load current.

To obtain the output impedance Zo = vo/io, which would help me to find the transient response of the output voltage for a sudden change in load currrent, I referred to a paper where the author calculates the output impedance through circuit averaging followed by small signal modelling as shown below.

I have a requirement where I have some R L C network present at the input side and then it should be connected to a buck converter.

To find the transient response of Vo for a change in load current io, I need to obtain Zo for this modified network.

I am using circuit averaging and then small signal model for this circuit to calculate Zo.

Could you please suggest if my approach below is correct for the desired transient response or any modifications are needed?

• You should have a look at my APEC 2017 seminar in which you will find all the materials on the buck interaction with the input filter. It should be enough to keep you busy on this Sunday afternoon : ) Feb 26 at 13:53
• Now, for your average model, the front-end $RLC$ elements shall not be averaged for the determination of $Z_{out}$. Use the PWM switch small-signal model supplied by $V_{in}$ via the $RLC$ filter and compute $Z_{out}$ from here. Feb 26 at 14:08

The front-end EMI filter will degrade the output impedance of the switching converter and can potentially bring instability when the loop is closed in certain conditions. To determine the open-loop output impedance with this filter, I recommend to go with the PWM switch model and gradually simplify the circuit with intermediate sanity check: make sure the plots you obtain from the simplified versions are always identical in phase in magnitude with the first one:

In this first picture, there is no EMI filter and the output impedance classically starts with the inductive ohmic loss $$\r_L\$$ paralleled with the load resistance. Then it is inductive, resonate and goes down again in the capacitive slope before ending up as the ESR of the capacitor $$\r_C\$$. From the large-signal model, use the small-signal version to start looking a potential analysis.

Now, you can add a front-end EMI filter and see the effects on both models:

Immediately you see the glitch in the phase and magnitude responses. This glitch is brought by the $$\RLC\$$ filter resonance which affects the source: without the filter, the node $$\V_{in}\$$ is at 0 V for the analysis and the current delivered by the buck as no effect on node a. With the $$\RLC\$$ front-end filter, this is no longer the case and now node a is modulated, affecting the whole transfer function. Damping is necessary and that is what I detail in my APEC 2017 seminar.

For the final schematic, rearrange and simplify the circuitry to put in a friendly shape where all is arranged in a meaningful. Make sure the final plot is rigorously identical to the first one and there you go:

You can install a test generator $$\I_T\$$ (the stimulus) producing a voltage response $$\V_T\$$ across the generator's terminals. The output impedance is $$\Z_{out}(s)=\frac{V_T(s)}{I_T(s)}\$$ You have to solve a 4th-order transfer function and I recommend you use the fast analytical circuits techniques or FACTs. You have an example you can download from my page.

• Thanks a lot for all the information. Mar 6 at 15:08
• No problem, good luck with the determination of $Z_{out}$! Mar 6 at 16:55