1

Properties of Laplace transform

1. Linearity

Ex.

Proof.

Lecture 2

2

Properties of Laplace transform

2.Time delay

Ex.

Proof.f(t)

0 T

f(t-T)

t-domain s-domain

3

Properties of Laplace transform

3. Differentiation

Ex.

Proof.

t-domain s-domain

4

Properties of Laplace transform

4. Integration

Proof.

t-domain s-domain

5

Properties of Laplace transform

5. Final value theorem

Ex.

if all the poles of sF(s) are in

the left half plane (LHP)

Poles of sF(s) are in LHP, so final value thm applies.

Ex.

Some poles of sF(s) are not in LHP, so final value

thm does NOT apply.

6

Properties of Laplace transform

6. Initial value theorem

Ex.

Remark: In this theorem, it does not matter if

pole location is in LHS or not.

if the limits exist.

Ex.

7

Properties of Laplace transform

7. Convolution

IMPORTANT REMARK

Convolution

8

An advantage of Laplace transform

We can transform an ordinary differential

equation (ODE) into an algebraic equation (AE).

ODE AE

Partial fraction

expansionSolution to ODE

t-domain s-domain

1

2

3

9

Example 1

ODE with initial conditions (ICs)

1. Laplace transform

10

Properties of Laplace transform

Differentiation (review)

t-domain

s-domain

11

2. Partial fraction expansion

Multiply both sides by s & let s go to zero:

Similarly,

unknowns

Example 1 (cont’d)

12

3. Inverse Laplace transform

Example 1 (cont’d)

If we are interested in only the final value of y(t), apply

Final Value Theorem:

13

Example 2

S1

S2

S3

14

In this way, we can find a rather

complicated solution to ODEs easily by

using Laplace transform table!

15

Example: Newton’s law

We want to know the trajectory of x(t). By Laplace transform,

M

(Total response) = (Forced response) + (Initial condition response)

16

Summary & Exercises

Laplace transform (Important math tool!)

Definition

Laplace transform table

Properties of Laplace transform

Next

modeling of physical systems using Laplace

transform

Exercises

Exercises

Read Chapters 1 and 2.

Solve Quiz Problems.