Question

The laplace transform for ramp function with a time delay can be expressed as follows

Answer 1

Step 1
Let the time delay is T,
So the ramp function will be
\(f(t)=(t-T)u(t-T)\)
Where
\(u(t-T)=\begin{cases}1 & t>T\\0 & t
Step 2
Let the Laplace transform of f(t) is F(s)
So the Laplace transform is
\(F(s)=\int_0^\infty f(t)e^{-st}dt\)
\(\Rightarrow F(s)=\int_T^\infty T(t-T)e^{-st}dt\)
let
x=t-T
\(\Rightarrow dx=dt\)
So we have
\(F(s)=\int_0^\infty xe^{-s(x+T)}dx\)
\(\Rightarrow F(s)=e^{-sT}\int_0^\infty xe^{-sx}dx\)
\(\Rightarrow F(s)=\frac{e^{-sT}}{s^2}\)