I was trying to calculate the Laplace of the Buck circuit, assuming the source signal was a square signal, when my professor told me that the output wouldn't be linear, and that the Laplace transform can't be used. Can anyone shed some light on this?

 

The Laplace and Fourier Transforms are linear.

That is if a is a constant and t is a function then a * L(t) v= L (a*t), where L(t) is the Laplace transform.

This is not the case with your output.

 

I was trying to calculate the Laplace of the Buck circuit, assuming the source signal was a square signal, when my professor told me that the output wouldn't be linear, and that the Laplace transform can't be used. Can anyone shed some light on this?

Hello there,

The buck circuit is non linear overall, but it is in fact piecewise linear, so there's no reason why we can not use Laplace Transforms to solve for the resulting output.

I am sure that the good professor meant that you can not use a *single application* of the Laplace Transform to solve this circuit, but using two (or perhaps more) applications it is possible to solve this kind of circuit for the output. Granted it will not be nearly as simple and easy, but it is nonetheless possible.

Looking at the basic operation of the circuit, there are two basic modes: one where the switch is on, and one where it is off. If there is another switch for synchronous operation then that would be turned on when the other is off and vice versa. If there is instead a diode (as there usually is) then that means a little more calculation is necessary.

The extra calculations are for the initial conditions of both the inductor and the capacitor. We need to solve for this as well as the output and if there is a diode we have to solve for the diode voltage as well so we know when it switches from 'on' to 'off' or any other states we care to include.

So the idea then is to solve the circuit for the first mode as if it had several inputs and several outputs, and for each output we would be looking for a possible change in mode. Once we see a change in mode, we then have to apply the transform again, and this gives us the transform for the next mode, and we must apply the initial conditions for the new mode from the final states we got from the previous mode. Since we know what causes a change in mode we can solve for the time when the mode will change, and in this way we know how long exactly we can use the current Transform before we have to switch back to the other one. Once we switch back, we do the same thing, and this means we will switch back to the original one again. This process repeats all throughout the solution period.

This sounds a little more complicated, and it is, but what we end up with is just two transforms that are solved for the output and perhaps a couple other things and most importantly it has to include the initial conditions. We can then solve for the output piece by piece.

This differs much from a numerical approach where we take tiny tiny steps and everything is calculated numerically, and is more like just solving two entirely different circuits and combining the results in a very particular way to get the total output.

If not anything else it is interesting to try this at least once

There is also a technique where we can transform the circuit into a linear circuit to study the overall operation, but that does not allow us to solve for the response during the individual switching periods. The Laplace Transform can be used there much easier because then the whole thing is just a regular linear system.