Electro Mechanics 2

8/12/2019 Electro Mechanics 2

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Laplace Transforms

Laplace Transform in CircuitAnalysis

The Laplace transform* is a technique foranalyzing linear time-invariant systems suchas electrical circuits

It provides an alternative functionaldescription that often simplifies:

The process of analyzing the behaviour of the

system The synthesis of a new system based on a set of

specifications

* After Pierre-Simon Laplace (1749 1827)


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Laplace Transforms

Mechatronic System

Mechanical System

Sensors

Actuators

Output Signal Conditioning& Interfacing

Input Signal Conditioning

& Interfacing

Operator

Display System

Control Architecture


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Laplace Transforms

Introduction toTransformations

A mathematical transformation employs rules tochange the form of data without altering itsmeaning

Popular transformations used in signals Fourier (suited to solving problems where input

domain is either repetitive or if the input is on a loop)

Z (suited for problems where the input is discrete

instead of continuous) Laplace (suited to solving problems with known initial

values)

02-01-ImplantDefribillator.wmv


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Laplace Transforms

Laplace Transform

A powerful tool for circuit analysis The steps involved are

A set of differential equations describing a circuit

converted to the complex frequency domain The variables of interest are solved

Convert from frequency domain back to time domain


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Laplace Transforms

Implant Defibrillator Problem

Implant defibrillatormanufacturer Guidant foundthat the close spacingbetween a wire and devicecomponent could potentiallyarc between them and cause

a short circuit In March 2005, a 21-year-old

college student who had aGuidant defibrillatorimplanted in his chest died

suddenly The type of defibrillator in his

death was short-circuiting ata rate of about once a monthfrom 2003 to 2004; but this

finding was not reported untilFebruary 2005


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Laplace Transforms

Breadboard (protoboard)A breadboard (protoboard) is a construction base for a one-of-a-kind electronic

circuit, a prototype. Because the solderless breadboard does not require soldering, itis reusable, and thus can be used for temporary prototypes and experimenting with

circuit design more easily.

A breadboard with a

completed circuit


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Laplace Transforms

Printed Circuit Boards

Printed circuit boards (PCBs) are used to mechanically support and electricallyconnect electronic components using conductive pathways, or traces, etched from

copper sheets laminated onto a non-conductive substrate.

PCB for mobile phones


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Laplace Transforms

Printed Circuit Board DesignPrinted circuit board designs are normally very complex. Hence, this is normally

done on computer software developed for this purpose. Most such software are ableto perform auto-routing.

Screenshot of PCB design

software

02-02-DNA_Circuits.wmv


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Laplace Transforms

Surface Mount Technology

Surface mount technology (SMT) is a method for constructing electronic circuits in

which the components are mounted directly onto the surface of printed circuit boards

(PCBs). Electronic devices so made are called surface-mount devices or SMDs. In

the industry it has largely replaced the previous construction method of fitting

components with wire leads into holes in the circuit board (also called through-hole

technology).

02-03-WedgeBonding.wmv

02-04-BallBonding.wmv


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Laplace Transforms

Definition of LaplaceTransform

==0

)()()( dtetfsFtfL st

+= js

[ ]

+

+

==

j

j

st

dsesFjtfsFL )(2

1

)()(

1

Laplace Transform is defined as

s is a complex variable given by

The inverse Laplace transform is defined as

A list of Laplace transform pairs

Uniqueness of Laplace Transform enables us

to avoid the complex integration


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Laplace Transforms

Laplace Transform Properties (1)Linearity:

Scaling:


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Laplace Transforms

Laplace Transform Properties (2)Time Shift:

u(t-a) = 0 for ta

Frequency Shift:


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Laplace Transforms

Laplace Transform Properties (3)

Time Differentiation:

Integrating by parts

With one more differentiation

In the general case


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Laplace Transforms


Time Integration:

Integrating by parts

The first term is zero


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Laplace Transforms


Time Periodicity:

With the time-shift property

Using the identity

Periodic function

Decomposition of periodic function

The transform of a periodic function is the transform of the

first period of the function divided by 1 e-Ts


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Laplace Transforms

LaplaceTransformProperties

Summary

The properties of the

Laplace Transform allow

us to obtain transform

properties without

performing the integral.


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Laplace Transforms

Laplace Transform of Circuit Elements

s

sFtfL

t )()(

0=

=0

)(1

)( dttiC

tv

)(11

)( sIsC

sV =

Voltage Source

CssI

sVsZC

1

)(

)()( ==

)()( tidt

d

LtV = )0()()( FssFtfdt

d

L =

Ls

sI

sVsZL ==

)(

)()(

ssV

1

)( = RZR =Resistor

Capacitor

Inductor

)()( ssLIsV =


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Laplace Transforms


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Laplace Transforms

Transfer Function

)(

)()(

sX

sYsH =

For excitation X(s) and response Y(s) in the complex frequency domain. Thetransfer function is given by

The transfer function of a circuit describes how the output behaves with

respect to the input. It also indicates how a signal is processed as it passes

through a network.

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Electro Mechanics 2