# Small signal modelling of buck converter, varying frequency

It seems that the most popular method of voltage mode control of buck converters is by varying the duty cycle. Is this because it is easier to do so because we can set the duty cycle using PWM ICs?

How do I start small-signal modelling of a buck converter (DCM mode) keeping input voltage Vin and duty cycle D constant, varying switching frequency? I want to get the transfer function $$\dfrac{\hat{v}(s)}{\hat{f}(s)}$$

EDIT 1:- For example, here's a question that asks to find this, [SMPC book, V Ramnarayan],

I'm not able to get the small signal model mentioned in part B, neither the answer to part C. I imagine a similar approach can then be taken for a buck converter.

• If the converter operates in continuous conduction mode (CCM), the switching frequency has no impact on the transmitted power. This is true for any of the switching cells operated in CCM. If the converter enters DCM, you express the output voltage with a large-signal equation featuring $F_{SW}$ and linearize it. This is a complicated exercise and I can show an example in an answer. I have not covered this control principle in my new book as it is rarely used beside frequency foldback for efficiency reasons. Apr 5, 2021 at 8:10
• @VerbalKint, yes it is in DCM, I forgot to mention that. My hardware circuit is built in such a way that it has provisions to be used as a synchronous buck, or a boost converter, because the inductor is connected externally. So I need to know this for either buck or boost, thank you.
– SM32
Apr 5, 2021 at 9:33
• @SM32 Can you tell me the name of this book? May 13, 2021 at 9:09

I have looked into a model like this long time ago. It was when the first PWM controllers having frequency foldback in light-load conditions were released. As the loop was going through different operating modes, it was important to check stability in all these loading conditions. One mode was when the peak current was frozen while the frequency was controlled through a voltage-controlled oscillator (VCO).

Most of the available small-signal models imply a fixed operating frequency where the error voltage controls either the duty ratio directly (voltage-mode control) or the inductor peak current (current-mode control). In continuous conduction mode (CCM), the transfer function and the dc transfer characteristic ignore the switching frequency and load values (ideal model). In discontinuous conduction mode (DCM), the switching frequency plays a role as well as the loading conditions for determining the output voltage. As thus, controlling the output via the switching frequency is a possibility if you freeze the peak current as in the previously-described example.

For many years now, I have adopted the PWM switch model to analyze power converters. Released in 1986 by Vatché Vorpérian, it cannot be beaten in terms of simplicity of analysis. The below figure shows you in the left side the PWM switch operated in peak-current-mode control with $$\V_c\$$ the control voltage. In all the equations, the frequency is fixed. In the right side, the model is tweaked to unveil the switching frequency contribution:

The difficulty now is to derive a small-signal approach with this large-signal model. This is not the place to show the complete linearization steps but I did it for the flyback converter exercise, look here. The model is invariant and you can reuse it in a buck converter. You first start with the large-signal model with equations reworked for future linearization:

When this is done, you start the linearization of the PWM switch operated in variable frequency. This is not a simple thing to do and the right-side window shows the many coefficients to determine:

When you have confirmed your model is correct, then you insert it in the buck configuration and you start the analysis using for instance the fast analytical circuits techniques or FACTs as described in my new book entirely dedicated to small-signal analysis of switching converters. I have covered many switching cells but did not touch variable frequency - except for the LLC converter - because it is rarely used as a control means.

• Thanks a lot for your answer, but I'm afraid I got a bit confused. I modified the question a little. Could your approach be applied to get the model written in Part B in my edit?
– SM32
Apr 7, 2021 at 6:29
• As explained in the post, this is a complicated matter. One way to simplify things rather than resorting to the complete model is to write an average equation of the output power. Have a look at slide 33 in the presentation I linked. Express the power delivered by the DCM-operated buck converter and apply partial differentiation to obtain a small-signal equation. It is a simplified approach but will perfectly work in your case. Apr 7, 2021 at 6:59