1. ## Help with a Heaviside function please

I'm trying to solve an RC circuit using laplace transforms, where we have 2f' + f = V(t) where f is the voltage across the capacitor and V(t) is the supply voltage. the circuit has initial conditions of f(0) = 0 so I need to try and find an expression for the voltage across the capacitor with time. thats all fine the problems with the supply voltage function which is H(t-b)sin(at-ab)
I've tried to laplace transform it like in the picture below but I dont seem to be able to find any straight forward way to do this. If anyone could give me a nod in the right sort of direction on transforming the input voltage function it'd be greatly appreciated

2. Originally Posted by def77
I'm trying to solve an RC circuit using laplace transforms, where we have 2f' + f = V(t) where f is the voltage across the capacitor and V(t) is the supply voltage. the circuit has initial conditions of f(0) = 0 so I need to try and find an expression for the voltage across the capacitor with time. thats all fine the problems with the supply voltage function which is H(t-b)sin(at-ab)
I've tried to laplace transform it like in the picture below but I dont seem to be able to find any straight forward way to do this. If anyone could give me a nod in the right sort of direction on transforming the input voltage function it'd be greatly appreciated
You have:

$2f'(t)+f(t)=V(t)$

Take Laplace transforms:

$2sF(s)-f(0)+F(s)=\mathcal{L}V(s)$

so:

$F(s)=\frac{\mathcal{L}V(s)+f(0)}{2s+1}$

which leaves you with the problem of finding the LT of $V$:

$\mathcal{L}V(s)=\int_{t=0}^{\infty} H(t-b)\sin(at-ab)e^{-st}\;dt=\int_{t=b}^{\infty}\sin(at-bt)e^{-st}\;dt$

This can probably be done with a shift theorem, but put $t'=t-b$, then:

$\mathcal{L}V(s)=
e^{-sb}\int_{t'=0}^{\infty}\sin(at')e^{-st'}\;dt'=\mathcal{L}[sin(at)](s)$

CB