# Thread: Complicated Recursive formula or how to deal with non-commuting operators

1. ## Complicated Recursive formula or how to deal with non-commuting operators

The background is I'm trying to analyse a boost converter. There are two states, A and B. State A is trivial. State B is more complicated. The circuit switches between the states. It is in state A for time t' and state B for time t, going on forever. The last term is the derivative of V with respect to T evaluated at t.. I'm not sure how to write that in the formula. I'm not too hot on the proper way to write it but maybe everything should have t replaced by T and then evaluated at t... the time variable is set to 0 every time the state starts. The derivative is essentially the current of the circuit at the end of the previous state B, a change in t would require a recalculation from the first n, so this derivative term looks like it's going to be a real pain.

The formula I have is

$\displaystyle V_n=v(1-cos(wt)+t'wsin(wt))+V_{n-1}cos(wt) + sin(wt)\frac{dV_{n-1}}{dt}$

I initially solved it ignoring the last term but it turns out the last term may be quite important. We can take wt as small. I think one of the problems I've faced is that fact that d/dt and cos(wt) are non-commuting operators and I don't know how to deal with them really.

What I had done so far is
$
\displaystyle V_n=v(1-cos(wt)+t'wsin(wt))+V_{n-1}cos(wt)$

Writing out the first few terms I can see the pattern and then was able to generalise it as.
$\displaystyle V_n=\frac{v(1-cos(wt)+t'wsin(wt))(1-cos^n(wt))}{1-cos(wt)}
$

I hope the formulas display properly.

Thanks for any help.