System and Block Diagrams

Laplace Transform

F(s) = \mathscr L {F(t)} = \int^\infty_0 f(t)e^{-st} \mathrm{d}t

where s = \alpha + j\omega

The function F(s) is often much easier to manipulate than periodic function f(t).

As f(t) tends to infinity, sF(s) tends to 0.

\dot x_o = ax_o = ax_i

where x_o is the output and x_i is the input

Take the Laplace transform:

sX_o(s) + aX_o(s) = aX_i(s)

Rearrange to get equation for the transfer function:

G(s) = \frac{X_o}{X_i} = \frac{a}{s+a}

 X_o = GX_i

If X_i is a unit step, then:

X_i = \frac1s

and

X_o = \frac{a}{s(s+a)}

Taking the inverse gives:

X_0 = 1 - e^{-at}