Laplace Transform BIOE 4200 Why use Laplace Transforms

Laplace Transform BIOE 4200

Why use Laplace Transforms? l Find solution to differential equation using algebra l Relationship to Fourier Transform allows easy way to characterize systems l No need for convolution of input and differential equation solution l Useful with multiple processes in system

How to use Laplace l Find differential equations that describe system l Obtain Laplace transform l Perform algebra to solve for output or variable of interest l Apply inverse transform to find solution

What are Laplace transforms? l l l t is real, s is complex! Inverse requires complex analysis to solve Note “transform”: f(t) F(s), where t is integrated and s is variable Conversely F(s) f(t), t is variable and s is integrated Assumes f(t) = 0 for all t < 0

Evaluating F(s) = L{f(t)} l Hard Way – do the integral let let Integrate by parts

Evaluating F(s)=L{f(t)}- Hard Way remember let Substituting, we get: let It only gets worse…

Evaluating F(s) = L{f(t)} l This is the easy way. . . l Recognize a few different transforms l See table 2. 3 on page 42 in textbook l Or see handout. . l Learn a few different properties l Do a little math

Table of selected Laplace Transforms

More transforms

Note on step functions in Laplace l Unit step function definition: in conjunction with f(t)u(t) because of Laplace integral limits: l Used

Properties of Laplace Transforms l Linearity l Scaling in time l Time shift l “frequency” or s-plane shift l Multiplication by tn l Integration l Differentiation

Properties: Linearity Example : Proof :

Properties: Scaling in Time Example : Proof : let

Properties: Time Shift Example : Proof : let

Properties: S-plane (frequency) shift Example : Proof :

Properties: Multiplication by tn Example : Proof :

The “D” Operator 1. Differentiation shorthand 2. Integration shorthand if if then

Properties: Integrals Proof : Example : let If t=0, g(t)=0 for so slower than

Properties: Derivatives (this is the big one) Example : Proof : let

Difference in l The values are only different if f(t) is not continuous @ t=0 l Example of discontinuous function: u(t)

Properties: Nth order derivatives let NOTE: to take you need the value @ t=0 for called initial conditions! We will use this to solve differential equations!

Properties: Nth order derivatives Start with Now apply again let then remember Can repeat for

Relevant Book Sections l Modeling - 2. 2 l Linear Systems - 2. 3, page 38 only l Laplace - 2. 4 l Transfer functions – 2. 5 thru ex 2. 4