Theory of Machines

A-4

LAPLACE TRANSFORMS

The Laplace transform F(s) of a function f(t) is defined as follows:

where ‘s’ is a complex variable called the Laplace operator and is equal to jw(s = σ + jω) · e^−st is called the kernel of the Laplace transformation.

1. Step input

f(t) = 0 for t < 0

= u₀ for t ≥ 0

For a unit step,

2. Pulse input

f(t) = 0 for t < 0 and t > t₀

= u₀ for 0 < t < t₀

3. Exponentially decaying function

f(t) = e^−at

4. Ramp function