# Analytically determine and draw the system response to a piecewise step input for an electrical system

Picture shows the electrical system. Analytically determine and draw system response $u_2(t)$ for the step input shown (use Laplace transform) (tp>>0).

So, I know to solve RC system transfer function and get G(s)=1/(RCs+1), but I don't understand how to do this problem, we have never done it in class and professor often asks this question. If anyone could help me out, thank you!

• Welcome to Engineering! This looks like a homework question. In order for such questions to be answered in this site, we need you to add details describing the precise problem you're having. What have you tried to solve this yourself? Please edit your question to include this information. – Wasabi Dec 10 '16 at 19:15

## 1 Answer

Because this is a linear and time-invariant system, you can use the principle of superposition.

If the input to $u_i(t)$ is $y_i(t)$, then

• The response to $\alpha u_i(t)$ is $\alpha y_i(t)$
• The response of $u_i(t)+u_j(t)$ is $y_i(t)+y_j(t)$
• The response of $u(t-t_p)$ is $y(t-t_p)$

The response of the system to a unit step input $\theta (t)$ is $1-e^{-\frac{t}{R C}}$. (I assume you have no problem with this.)

The input to the system can be split into two.

• $5 \theta (t)$ when $0\leq t\leq t_p$. The response of the system to this input is $5(1-e^{-\frac{t}{R C}})$.
• $5 \theta (t)-\theta(t-t_p)$ when $t_p < t$. The response is $5(1-e^{-\frac{t}{R C}})-(1-e^{-\frac{t-\text{tp}}{R C}})$.

The final response is a sum of these two piecewise functions.

I have plotted the result for $R=1$, $C=1$ and various values of $t_p$. The dashed line shows the response if the second step was not applied.

• Thanks a lot! How would I determine gain only from system repsonse ( Transient function), I know that T is when function reaches 63% ... – DomVl Dec 13 '16 at 22:05
• The gain comes from the steady-state response to a unit-step function. In this case it is 1. – Suba Thomas Dec 13 '16 at 22:13
• Is that input/(where function settles +1) ? – DomVl Dec 13 '16 at 22:23
• The input is a unit step. It is where the output settles. If you are using a unit step of magnitude m, it is ((steady-state value of output)/m). – Suba Thomas Dec 13 '16 at 22:31