Help with a Heaviside function please

def77 · #1 $Old$ October 31st, 2009, 06:08 AM

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

CaptainBlack · #2 $Old$ October 31st, 2009, 02:43 PM

Quote:

Originally Posted by def77 $View Post$

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