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SYSTEMS 1- Laplace Transforms and

Inverse Laplace Transform

Laplace Transform Review

A system represented by a differential equation is difficult to model as a block diagram.

Laplace transforms can be used to represent the input , output and system as separate entities.

Also, Laplace transforms makes the inter-relationship of the system’s input and output algebraic.

Laplace and the Inverse Laplace

Transform Laplace Transform

Inverse Laplace Transform

Laplace Transform Theorems

Table of Laplace Transforms

Sample Problems

1. Find the Laplace transform of

2. Derive the Laplace Transform of the

following time functions

u(t)

tu(t)

sin(wt) u(t)

cos(wt) u(t)

3. Using the Laplace transform pair and the Laplace transform theorems, derive the Laplace transform for the following time functions:

a.

b.

c.

4. Find the Inverse Laplace transform of

Euler’s Identity

INVERSE LAPLACE TRANSFORM

PARTIAL FRACTION EXPANSION

Case 1: Roots of the Denominator of F(s) are real and distinct

Case 2: Roots of the Denominator of F(s) are real and repeated

Case 3: Roots of the Denominator of F(s) are complex or Imaginary

More Examples

5. Find the inverse Laplace Transform of

the following (Fs):

a.

b.

c.

Laplace Transform solution of a

Differential Equation

Examples:

Find the solution of the differential

equation using Laplace Transforms:

assume zero initial conditions

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