Output of Op-Amp System (unit step input)

Discussion in 'Homework Help' started by tquiva, Mar 7, 2011.

  1. tquiva

    Thread Starter Member

    Oct 19, 2010
    176
    1
    Could someone please help me in solving these problems (attached below):

    From my understanding, I will need to:

    Determine the nodal voltages of each system, except for the input and output.
    For each capacitor, I will need to add a current source?

    But since the input of each system is a function of time, would I still need to convert the circuit into frequency domain, then solve for the output that way?

    I'm pretty confused how I should first start solving these problems.

    Could someone please assist me?
     
  2. Vahe

    Member

    Mar 3, 2011
    75
    9
    Write the differential equation in terms of the inductor current for the circuit using KVL going around the loop clockwise

    <br />
3 \frac{di(t)}{dt} + 5 i(t) = \delta(t)<br />

    with initial condition i(0)=1/3 \text{A}. Once you find a solution for i(t), the output voltage is simply 5 i(t) by Ohm's Law. So, how can we solve the differential equation above with the initial condition and there are a number of ways to do this. One method is to use Laplace transforms. Taking the Laplace transform (please review this in your text)of the differential equation above, we get

    <br />
3 (s I(s) - i(0)) + 5 I(s) = 1 \\<br />
3 (s I(s) - 1/3) + 5 I(s) = 1 \\<br />
3 s I(s) -1 + 5 I(s) = 1 \\<br />
(3 s + 5) I(s) = 2 \\<br />
I(s) = \frac{2}{3s+5} = \frac{2/3}{s+5/3}<br />

    Now solve for I(s) using inverse Laplace transform to find i(t) and from there you will get the output voltage. From above you should be able to get the following

    <br />
i(t) = \frac{2}{3} e^{-5t/3} u(t) \\<br />
out(t) = 5 i(t) = \frac{10}{3} e^{-5t/3} u(t)<br />

    where u(t) is the unit step function.

    Cheers,
    Vahe
     
  3. tquiva

    Thread Starter Member

    Oct 19, 2010
    176
    1
    Thank you so much!

    But how would I go about finding the zero input response and the total response?
     
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