# Output of Op-Amp System (unit step input)

#### tquiva

Joined Oct 19, 2010
176

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.

#### Vahe

Joined Mar 3, 2011
75
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