Review on Laplace Transforms

Definition: The Laplace transform is defined by the linear transformation

where is an arbitrary complex number and f(t) is of exponential order

(i.e. is bounded for large t).

Some Laplace Transform Pairs

Unit Impulse

where

Unit Step

where

Exponential

where

Ramp

where

 

Table 4.1: Laplace Transform Table

Properties/ theorems of Laplace Transform

Superposition / Linearity

Let's use this relationship to find , where

where

From eqns. (4.7) and (4.9) we have

or

Note that this relationship can be derived from the basic definition in eqn. (4.3).

Time Delay / Time Shift

where

Letting and , gives

Note that the time delay is given formally by the use of the delayed unit step

function, . If we define a causal time function as one that is identically zero

for a negative argument [i.e. for t < t0], then the use of the unit step

notation is unnecessary. Therefore, we denote such a function as and its

Laplace transform is where .

Differentiation

where

Integrating this expression by parts gives

and

For the 2nd derivative, we have

In general, the nth derivative is

Integration

where

Integrating by parts gives

and

Note that Matlab has built-in capability within the Symbolic Toolbox to generate

Laplace transforms and inverse Laplace transforms (see commands LAPLACE

and ILAPLACE, respectively).

 

Inverse Transforms

Definition: The inverse Laplace transform is given by

This is a formal definition. In practice, one usually does not need to perform a

contour integration in the complex plane. Instead, a "dictionary" of Laplace

transform pairs is generated and some simple rules allow one convert

between the time domain solution, f(t), and the frequency or s-plane solution, F(s).

Laplace transform solutions to nth-order linear time-invariant systems are typically

of the form:

where the are referred to as the zeros of F(s) and the

are the poles of F(s).

There are two principal methods for finding when F(s) is a ratio of

polynomials; Partial Fraction Expansion and the Method of Residues. These will

be illustrated with examples.

Partial Fraction Expansion

Non-Repeated Linear Factors

Repeated Linear Factors

Complex Roots and Quadratic Factors

Solving for A gives

Now clearing fractions gives

Equating like terms

Bs2 = s2, 2C = 4, (2B+C)s = 4s

Or

B = 1, C = 2, 2+2 = 4

Finally, rewriting the quadratic term as follows

or

gives

and from Laplace table & theorem

Method of Residues

Statement of Method: If F(s) is a ratio of polynomials in s, then

where the residue of an nth-order pole at s = s1 is given by

First Order Pole:

Second Order Pole:

Non-Repeated Linear Factors

Repeated Linear Factors

Complex Roots and Quadratic Factors

Note that given , the roots are given by

and for this case the roots of the quadratic term are

Therefore, F(s) can be written in terms of linear factors, or

Expanding the middle term, for example, gives

Evaluating the first and last terms in a similar manner gives

The Partial Fraction Expansion technique and Method of Residues are very

powerful and they can be utilized in most cases of interest. Both methods can be

complicated algebraically, but conceptually it is straightforward to find f(t) given

F(s) and vice versa.

Note: residue function in MATLAB can be used for partial fraction.

e.g.

[r,p,k] = residue([3 5 3 6],[1 6 11 6])

Solving LTIV Differential EquationsSolving LTIV differential equations by Laplace involve 2 steps:

1. Take Laplace transform of each term in differential equation and rearrange

to obtain the Laplace expression

2. Take the inverse Laplace

Example 1:

Example 2: