Question

A.C. Fundamentals - Laplace Transforms of Some Useful Functions

Answer 1

Laplace transform is a mathematical technique used to transform a function of time into a function of a complex variable s, which simplifies solving linear time-invariant systems in the frequency domain. Here are the Laplace transforms of some commonly encountered functions:

Unit Step Function (u(t)):
The unit step function is defined as follows:

u(t) = 1 for t >= 0
u(t) = 0 for t < 0

Laplace Transform:
L{u(t)} = 1/s

Impulse Function (δ(t)):
The impulse function (also known as Dirac delta function) is used to model an instantaneous event.

Laplace Transform:
L{δ(t)} = 1

Exponential Function (e^(at)):
Laplace Transform:
L{e^(at)} = 1 / (s - a)

Cosine and Sine Functions (cos(ωt) and sin(ωt)):
Laplace Transforms:
L{cos(ωt)} = s / (s^2 + ω^2)
L{sin(ωt)} = ω / (s^2 + ω^2)

Ramp Function (t):
Laplace Transform:
L{t} = 1 / s^2

Exponential Decay Function (e^(-at)u(t)):
Laplace Transform:
L{e^(-at)u(t)} = 1 / (s + a)

Periodic Rectangular Function (f(t) = 1 for 0 <= t <= T, 0 otherwise):
Laplace Transform:
L{f(t)} = (1 - e^(-sT)) / (sT)

Periodic Impulse Train (sum of δ(t - nT)):
Laplace Transform:
L{∑ δ(t - nT)} = ∑ e^(-nTs)

These are just a few examples of Laplace transforms for common functions. The Laplace transform has properties such as linearity, shifting, differentiation, integration, and convolution that make it a powerful tool in analyzing linear time-invariant systems. When dealing with more complex functions or combinations of functions, these properties can be used to find their Laplace transforms.