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Math Help - Help with a Heaviside function please

  1. #1
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    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
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  2. #2
    Grand Panjandrum
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    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)=<br />
e^{-sb}\int_{t'=0}^{\infty}\sin(at')e^{-st'}\;dt'=\mathcal{L}[sin(at)](s)

    CB
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