Laplace Transforms Laplace Transform in Circuit .Laplace Transforms Laplace Transform in Circuit

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  • 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

    02-02-DNA_Circuits.wmv

  • 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

    += js

    [ ] +

    +

    ==

    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 ta

    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()()( FssFtfdt

    dL =

    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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