# Step response of a first order system

1. Feb 9, 2010

### VinnyCee

1. The problem statement, all variables and given/known data

Find the unit step response of the transfer function...

a) $$G(s)\,=\,\frac{4}{s\,+\,4}$$

b) $$G(s)\,=\,\frac{2}{0.2s\,+\,1}$$

2. Relevant equations

General first order step response equation...

$$C(s)\,=\,R(s)\,G(s)\,=\,\frac{a}{s(s\,+\,a)}$$, where $$R(s)\,=\,\frac{1}{s}$$

then do an inverse Laplace transform...

$$c(t)\,=\,1\,-\,e^{-at}$$

3. The attempt at a solution

Part a) is simple enough. I just plugged into formula above and got $$c(t)\,=\,1\,-\,e^{-4t}$$

However, part b) is where I am confused. To get the G(s) into the form needed (i.e. ~ $\frac{a}{s\,+\,a}$), I divided both the numerator and denominator by 0.2...

$$G(s)\,=\,\frac{2}{0.2s\,+\,1}\,=\,\frac{10}{s\,+\,5}\,=\,2\left(\frac{5}{s\,+\,5}\right)$$

But now the form is not exactly as needed in the first order system equations. What do I do?

I tried taking out a 2 from the numerator, and got an answer, just not sure if it's right though.

Is this right for part b)...

$$c(t)\,=\,2\,\left[1\,-\,e^{-5t}\right]$$

Last edited: Feb 9, 2010
2. Feb 9, 2010

### klouchis

That's right, remember that the Laplace transform is a linear operation. If f(t) has a Laplace transform of F(s), then a*f(t) has a Laplace transform of a*F(s). Assuming "a" is a scalar quantity. The same linearity is true for inverse Laplace transforms.