Laplace Transforms

Laplace Transform in Circuit Analysis

The Laplace transform* is a technique for analyzing linear time-invariant systems such as electrical circuits It provides an alternative functional description that often simplifies:

The process of analyzing the behaviour of the systemThe synthesis of a new system based on a set of specifications

* After Pierre-Simon Laplace (1749 – 1827)

Laplace Transforms

Mechatronic System

Mechanical System

Sensors

Actuators

Output Signal Conditioning& Interfacing

Input Signal Conditioning& Interfacing

Operator

Display System

Control Architecture

Laplace Transforms

Introduction to Transformations

A mathematical transformation employs rules to change the form of data without altering its meaningPopular 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)

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Laplace Transforms

Laplace Transform

A powerful tool for circuit analysisThe steps involved are

A set of differential equations describing a circuit converted to the complex frequency domainThe variables of interest are solvedConvert from frequency domain back to time domain

Laplace Transforms

Implant Defibrillator ProblemImplant defibrillator manufacturer Guidant found that the close spacing between a wire and device component could potentially arc between them and cause a short circuitIn March 2005, a 21-year-old college student who had a Guidant defibrillator implanted in his chest died suddenlyThe type of defibrillator in his death was short-circuiting at a rate of about once a month from 2003 to 2004; but this finding was not reported until February 2005

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, it is reusable, and thus can be used for temporary prototypes and experimenting with circuit design more easily.

A breadboard with a completed circuit

Laplace Transforms

Printed Circuit BoardsPrinted circuit boards (PCBs) are used to mechanically support and electrically connect electronic components using conductive pathways, or traces, etched from copper sheets laminated onto a non-conductive substrate.

PCB for mobile phones

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 able to perform auto-routing.

Screenshot of PCB design software

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Laplace Transforms

Soldering

Electrical components need to be physically attached to the right locations on the printed circuit board. This is accomplished using soldering.

Laplace Transforms

Surface Mount TechnologySurface 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).

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Laplace Transforms

Definition of Laplace Transform

∫∞ −==0

)()()( dtetfsFtfL st

ω+σ= js

[ ] ∫∞+α

∞+α

−

π==

j

j

st dsesFj

tfsFL )(21)()(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

Laplace Transforms

Laplace Transform Properties (1)Linearity:

Scaling:

Laplace Transforms

Laplace Transform Properties (2)Time Shift:

u(t-a) = 0 for t<a and u(t-a) = 1 for t>a

Frequency Shift:

Laplace Transforms

Laplace Transform Properties (3)

Time Differentiation:

Integrating by parts

With one more differentiation

In the general case

Laplace Transforms

Laplace Transform Properties (4)

Time Integration:

Integrating by parts

The first term is zero

Laplace Transforms

Laplace Transform Properties (5)

Frequency Differentiation:

Taking derivative wrt x

Laplace Transforms

Laplace Transform Properties (6)

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

Laplace Transforms

Laplace Transform PropertiesSummary

The properties of the Laplace Transform allow us to obtain transform properties without performing the integral.

Laplace Transforms

Laplace Transform of Circuit Elements

ssFtfL

t )()(0

=⎥⎦⎤

⎢⎣⎡∫∫

∞

=0

)(1)( dttiC

tv

)(11)( sIsC

sV =

Voltage Source

CssIsVsZC

1)()()( ==

)()( tidtdLtV = )0()()( FssFtf

dtdL −=⎥⎦

⎤⎢⎣⎡

LssIsVsZL ==)()()(

ssV 1)( = RZR =Resistor

Capacitor

Inductor

)()( ssLIsV =

Laplace Transforms

Laplace Transforms

Transfer Function

)()()(

sXsYsH =

For excitation X(s) and response Y(s) in the complex frequency domain. The transfer 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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