# Control of interleaved boost converter

In a school project we will try to implement a control system for a boost power factor correction (PFC) converter. The converter circuit can be seen below (http://www.ti.com/lit/an/slyt517/slyt517.pdf).

The control system works as follows: An outer loop controls the output DC voltage of the boost, and an inner loop controls the current into the converter such that it has the same shape as the input voltage Vs.

Our problem right now is that we don't really know how to create a transfer function for the boost converter. And we need a transfer function in order to tune our PI and PID regulators.

Does anybody know of a simple way to get a transfer function for an interleaved boost? All of the papers that we have read have very advanced derivations that are hard to follow, and the end result is usually very complex and difficult to use. Any guidance would be appreciated.

## 1 Answer

Note, actual answer is at the bottom, see TLDR bold text.

Does anybody know of a simple way to get a transfer function for an interleaved boost?

No. That is not a simple thing to do.

All of the papers that we have read have very advanced derivations that are hard to follow

That's because that is how you derive a transfer function. And they're hard for you to follow, but there is nothing inherently wrong or problematic with those papers. If you can't follow the derivations, there is no way to make them more clear, there is not some magic way to show the derivation such that it also teaches LTI system theory, Laplace transforms, linear differential calculus, and how to derive transfer functions with just one example derivation.

The cold, hard truth is that this stuff is a pain, it is not simple, and when they say it is only complex in the frequency domain, they don't mean literally. Sorry, bad math joke. While it is very possible to work through these derivations, I am not sure anyone would describe doing so using the word 'easy', or even without using expletives.

the end result is usually very complex and difficult to use.

And? There is no ease-of-use requirement in the definition of a transfer function. Transfer functions are discovered (derived), not made. They are what they are. And there is exactly one of them for a given topology. One transfer function. So if the equations you found were difficult to use, that just means the transfer function for what you want is difficult to use. That's a bummer to be sure, but there is nothing you can do about it.

This is, unfortunately, a case of the answer you need and the answer you want being two different things. It isn't what you want, and that's unfortunate, but there is one answer, you already found it, and what you want does not exist. If it makes you feel any better, I keep asking if kittens stay kittens forever, and the answer is never the one I want either.

Now, I am not about to tell you anything you probably haven't already found. But this is actually not that bad, because this is really just two boost converters that are 180° out of phase and divide up the duty cycle between them. Beyond that, the topology is the same as two boost converters in parallel.

And you don't need to find the transfer function for your converter because there is nothing unique about it, someone else has derived the transfer function and it will hold true for any thing that is of this topology. Only the actual numbers will differ, and that is just 'plug and chug' - putting in values for variables.

And you don't need to find the transfer function at all. PID controllers are not limited to things we have perfect analytical solutions for. The only requirement is that a system behaves in a way that can be approximated to something that is linear. And that is really kind of arbitrary anyway, but I digress.

Indeed, even the 'formal' transfer functions you use for an analytical solution, at least in the context of describing circuit topologies, are approximations themselves, and are not actually true linear time-invariant systems. If you ignore the existence of parasitics and things like quantized circuit effects (shot noise etc), then you can pretend they're LTI systems.

It is perfectly feasible to achieve high PID performance using an approximation (in this case). You can also use one of several heuristic means of tuning a PID control loop - some variation of the Åström–Hägglund (relay) method would probably be most appropriate. I'd avoid the Ziegler–Nichols method, it is more of a 'eliminate all disturbances, overshoot be damned' tuning method. This needs a softer touch.

But you wanted the transfer function for what is called an n x m interleaving topology. n being the number of boost converters (2 in your case), m being the number of switches per converter (which is simply 1 in your case and can be ignored). Note that this means that the transfer function is the same even with an arbitrarily huge number of individual converters, and likewise any number of switches per converter. At least, if they are all 2π/m radians out of phase from each other, which is what makes them interleaved rather than simply 'a bunch of boost converters in parallel and fighting each other for regulation supremacy'.

TLDR, as good as you're going to get in terms of an actual, analytical transfer function:

You're not going to get any easier than this (Warning: PDF). On the second page it has all the small signal transfer functions you could want, duty to output voltage, duty to inductor current, inductor current to output voltage. Conveniently, it also tells you exactly what each and every variable in said transfer functions is, and exactly how to calculate it for your specific circuit in a table directly to the left. I do not think there is any way it can possibly be made easier. Remember, m = number of switches per phase/converter, n = number of phases/converters. 1, and 2 in your case. These are totally generalized and work for any n x m interleaved boost topology, including 1 x 1 (simply normal boost converter) or an 2 x 1 (one switch per phase, two phases).