There are several methods to determine the control-to-output transfer function of a PFC. However, the common denominator to all of these approaches is the control law that is adopted by the PFC. For instance, if you think of a borderline or critical conduction mode PFC (BCM), then it is a constant on-time modulation, simply adjusting \$t_{on}\$ in relationship to the power that is processed. For a CCM type, the law might a bit more complex, like involving (or not) a multiplier to keep a constant gain across the input voltage range.
I have looked into this in my book on SPICE simulations (chapter 6) but also in my recent seminar on PFC that you can download from that list. It is number 17. The main idea to model a PFC is to determine its output power based on the control variable:
Once this expression is obtained, you run partial differentiation (or you perturb the nonlinear equations) and determine the small-signal coefficients:
Finally, a complete small-signal model is assembled and you can determine the transfer function quickly:
With the transfer function in hand, you can think of a compensation strategy for forcing the crossover frequency of your choice, with adequate margins.
This is the theoretical approach that I favor as a starting point in my design flow.
The second approach uses a SPICE or SIMPLIS simulations. In the below picture, you see the averaged model of the NCP1654, an old CCM PFC from onsemi which implements a so-called predictive control law (no input voltage sensing):
This circuit is part of my free 130+ ready-made templates which work, for most of them, with the free demo of SIMPLIS called Elements. With this kind of model, you have the power stage small-signal model and the compensation strategy (type 2) is automated from the left-side macro. The ac curves in the bottom of the picture show how you can quickly close the loop with a good margin. For those interested, I have described a similar process in LTspice with a PDF here.
If you now use a piece-wise linear (PWL) simulator like SIMPLIS, you can directly run the ac analysis of a PFC while its input is a sinusoidal voltage (you may have noticed that in my SPICE simulation, the boost stage is biased from a dc source).
This is the model of a CCM PFC based on the UC1854 introduced by Unitrode (now TI) years ago. After a few moments, you have the steady-state waveforms, the power stage and the compensated loop gain:
The compensator in this examples relies on an analog circuit (an OTA or an op-amp) but nothing prevents you from using a digital compensator. You have automated examples in my compensators list which implement a digital filtered PID or a simple PI.
And that is all for a Sunday morning!
