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
    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


    Mar 3, 2011
    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.

  3. tquiva

    Thread Starter Member

    Oct 19, 2010
    Thank you so much!

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