# 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

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.

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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

$
3 \frac{di(t)}{dt} + 5 i(t) = \delta(t)
$

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

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

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

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

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?