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Laplace Transforms
Post: #1

Dr. Dillon

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Laplace Transforms
What Are Laplace Transforms?
A Laplace transform is an example of an improper integral : one of its limits is infinite.
A Calculation
To What End Does One Use Laplace Transforms?
Then What?
How Do You Transform an Differential Equation?
How Do You Find Inverse Laplace Transforms?
Inverse Laplace Transforms
Why Use Such Dangerous Machines?
When Might You Have To?
The Dirac Delta Function
The Laplace Transform of the
Dirac Delta Function
Typical Scenario
What Do We Expect You to Be Able to Do?
Know the definition of the Laplace transform
Know the properties of the Laplace transform
Know that the inverse Laplace transform is an improper integral
Know when you should use a Laplace transform on a differential equation
Know when you should not use a Laplace transform on a differential equation
Be able to solve IVPs using Laplace transforms…
Post: #2
The Laplace Transform

.ppt  The Laplace Transform.ppt (Size: 1.33 MB / Downloads: 73)

Laplace transform is another method to transform a signal from time domain to frequency domain (s-domain)
The basic idea of Laplace transform comes from the Fourier transform
As we have seen in the previous chapter, not many functions have their Fourier transform such as t, t2, et etc.

Circuit element models

Apart from the transformations
we must model the s-domain equivalents of the circuit elements when there is involving initial condition (i.c.)
Unlike resistor, both inductor and capacitor are able to store energy
Post: #3

.pdf  1LAPLACE TRANSFORMS.pdf (Size: 179.72 KB / Downloads: 97)


Laplace transforms provide a method for representing and analyzing linear systems
using algebraic methods. In systems that begin undeflected and at rest the Laplace ’s’
can directly replace the d/dt operator in differential equations. It is a superset of the phasor
representation in that it has both a complex part, for the steady state response, but also a
real part, representing the transient part. As with the other representations the Laplace s is
related to the rate of change in the system.

A Few Transform Tables

Laplace transform tables are shown in Figure 17.5, Figure 17.7 and Figure 17.8.
These are commonly used when analyzing systems with Laplace transforms. The transforms
shown in Figure 17.5 are general properties normally used for manipulating equations,
and for converting them to/from the s-domain.


It is not necessary to develop a transfer functions for a system. The equation for the
voltage divider is shown in Figure 17.28. Impedance values and the input voltage are converted
to the s-domain and written in the equation. The resulting output function is manipulated
into partial fraction form and the residues calculated. An inverse Laplace transform
is used to convert the equation into a function of time using the tables.


• Transfer and input functions can be converted to the s-domain
• Output functions can be calculated using input and transfer functions
• Output functions can be converted back to the time domain using partial fractions.
Post: #4

.ppt  LAPLACE TRANSFORMS1.ppt (Size: 639.5 KB / Downloads: 41)


Transforms -- a mathematical conversion from one way of thinking to another to make a problem easier to solve
Definition -- Partial fractions are several fractions whose sum equals a given fraction
Purpose -- Working with transforms requires breaking complex fractions into simpler fractions to allow use of tables of transforms

Different terms of 1st degree

To separate a fraction into partial fractions when its denominator can be divided into different terms of first degree, assume an unknown numerator for each fraction

Repeated terms of 1st degree (1 of 2)

When the factors of the denominator are of the first degree but some are repeated, assume unknown numerators for each factor
If a term is present twice, make the fractions the corresponding term and its second power
If a term is present three times, make the fractions the term and its second and third powers

Effect of Control Actions

Proportional Action
Adjustable gain (amplifier)
Integral Action
Eliminates bias (steady-state error)
Can cause oscillations
Derivative Action (“rate control”)
Effective in transient periods
Provides faster response (higher sensitivity)
Never used alone

Basic Controllers

Proportional control is often used by itself
Integral and differential control are typically used in combination with at least proportional control
eg, Proportional Integral (PI) controller:

Summary of Basic Control

Proportional control
Multiply e(t) by a constant
PI control
Multiply e(t) and its integral by separate constants
Avoids bias for step
PD control
Multiply e(t) and its derivative by separate constants
Adjust more rapidly to changes
PID control
Multiply e(t), its derivative and its integral by separate constants
Reduce bias and react quickly

Root-locus Analysis

Based on characteristic eqn of closed-loop transfer function
Plot location of roots of this eqn
Same as poles of closed-loop transfer function
Parameter (gain) varied from 0 to 
Multiple parameters are ok
Vary one-by-one
Plot a root “contour” (usually for 2-3 params)
Quickly get approximate results
Range of parameters that gives desired response

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