ABE425 Engineering Measurement Systems ABE425 Engineering Measurement Systems Laplace Transform Dr. Tony E. Grift Dept. of Agricultural & Biological Engineering University of Illinois

Pierre-Simon Laplace “The French Newton” ( ) Why do we need a Laplace Transform? Definition Laplace Transform Laplace Transform of functions Unit step function Ramp function Exponential function Cosine/Sine Impulse function (dirac delta) Laplace Transform of operations Convolution

The Laplace transform converts one function into another function Lowercase in t domain Uppercase in s domain Always write t=… t is a dummy variable that gets integrated out s is a constant under integration wrt time. How do you know? Time (t) domain Laplace (s) domain

The Laplace transform is a linear operation since integration itself is linear. Red frame: Important result

Laplace transform of unit step function Definition Laplace Transform The variable s is a constant under integration with respect to t

Laplace transform of a ramp function Blue frame: You should know this already Integration by parts

Laplace transform of an exponential function

Laplace transform of a cosine function Euler’s formula

Laplace transform of a sine function Euler’s formula

Laplace transform of an impulse (Dirac  “function”) I: Use McLaurin’s expansion Writing as a McLaurin series

Laplace transform of an impulse (Dirac  “function”) I: Use McLaurin’s expansion

Laplace transform of an impulse (Dirac  “function”) II: Use l’Hopital’s rule (easier) L’Hopital’s rule

Laplace Transform of a function shifted in time

Inverse Laplace Transform of a function shifted in s- domain (Frequency shift). Since we do not have an inverse Laplace transform we show the proof in reverse Examples

Laplace transform of operations: differentiation and integration

Laplace transform of a differentiation operation Integration by parts

Laplace transform of impulse (Dirac  function) III: Use Laplace Transform of differentiation (much easier)

Check Laplace Transform of a differentiation operation Example Check:

Laplace transform of the definite integral of a function Integration by parts

Laplace Transforms of common functions

Time domainLaplace domain Laplace table Partial fraction expansion Differential equation Algebraic equation 1.Solving a differential equation using Laplace Transform Input Function (here unit step) Solve for Solution 1.Take d.e. including input function, transform equation to Laplace domain. Solve for Y 2.Convert factors into terms using Partial fraction expansion (PFE) 3.Use Laplace table to return to time domain and obtain solution y(t)

Classical example of an electrical first order system. Kirchhoff Voltage Law: i In input-output notation <- differential equation Capacitor voltage:

The unit of RC is second (which is why it is called a time constant). Ohm’s Law Capacitor equation Current is charge per second Unit of RC = second

Classical example of an electrical first order system cont. Now we’ll use the Laplace Transform determine the response of this system to a step input. For simplicity let’s change to y for output and x for input, Remember lowercase variables live in the time domain.

Transform the dynamic system to the Laplace domain using a unit step function as input. Uppercase variables live in the Laplace domain. Initial charge on capacitor assumed 0

Partial fraction expansion allows for splitting factors into terms that can be recognized from the Laplace transform table. Partial Fraction Expansion

The inverse transform consists of recognizing known Laplace transforms. Output for = 20 s

The final value theorem is very handy to determine what the steady state value of the output of a system (or an error in PID control) will be without having to go through determining the response in the time domain. The initial value is the logical counterpart, but hardly ever used.

Initial value theorem proof Interchange the limits

Final value theorem proof Interchange the limits

Example initial and final value theorem. We have a first order system and a unit time step excitation. Unit step Transfer function Output in s domain Initial value theorem Final value theorem

ABE425 Engineering Measurement Systems ABE425 Engineering Measurement Systems Laplace Transform Dr. Tony E. Grift The End Dept. of Agricultural & Biological Engineering University of Illinois