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APPLICATIONS OF LAPLACE TRANSFORM

INTRODUCTION:

  • Laplace transform is an integral transform method using which we convert signals which are function of time (i.e. in time domain) into signals which are functions of complex angular frequency (i.e. in frequency domain). It is named after Pierre Simon Laplace, a French mathematician and astronomer who used Laplace transform in probability.

  • Any signal x(t) can be transformed using Laplace by:

  • Laplace Transform


    APPLICATIONS:

    Laplace transform is mainly employed to transform complicated differential equations into relatively simple polynomial equations. After solving the equation using polynomial equation the original differential equation can be obtained using inverse of Laplace. So this transform has its applications in each and every field which is concerned with solving complex differential equations. It also has applications where a signal is converted from time domain to frequency domain for analysis.

  • A few of the applications are mentioned below:


  • IN ELECTRICAL NETWORKS:

    To analyze electrical networks containing resistors, inductors and capacitors, the equations obtained are differential equations (often even of higher order) whose solution is quite complex and lengthy. However, the same equation when solved in Laplacian domain, the solution is far simpler. This is done by replacing each component in the circuit by its Laplace equivalent. We can also restore the Laplace solution in differential form using inverse Laplace transform.


    ELECTRICAL COMPONENTLAPLACE EQUIVALENT
    Resistor - R ohm R(remains same)
    Inductor - L Henry Ls along with voltage source of L*i(0) in series of opposite polarity or current source (i(0))/s in parallel of same polarity.
    Capacitor - C Farad 1/Cs along with voltage source (v(0))/s of same polarity in series or current source C*v(0) in parallel of opposite polarity.
    Voltage Source - V Volts V/s
    Current Source - I Amps I/s



    IN SIMPLE HARMONICS:

    The equation simple harmonic motion is an second order differential equation. Some standard conditions can be directly solved using concepts of calculus, however in some situations, transforming the same in Laplace domain makes the same calculation easy and more understandable.


    IN DIGITAL SIGNAL PROCESSING:

    Apart from using Laplace transform to solve differential equations, this mathematical tool is also applicable to transform continuous time domain signal to frequency domain. The imaginary part of ā€œsā€ in Laplacian contains the frequency component. Since processing of signals is convenient in frequency domain, it is important that we transform the signal.


    IN COMMUNICATIONS:

    The signals (of radio, cellular phones, etc.) are in frequency domain. However, for transmission they are converted in time domain. Again, at the receiving end they are decoded by converting back in time domain. In such transformations, Laplace is the mathematical tool which is used.


    IN CONTROL SYSTEMS:

    The various transfer functions are often calculated in Laplace domain. In control systems, it not only helps in analysis of system but also in modelling of the system as large differential equations are involved which get simplified.


    IN PROCESS CONTROL:

    Laplace transform is important to analyze the variables, which when altered, bring about the change in output. This is a very important aspect of engineering as a system is only efficient when it produces desired output, with minimum error. Such error causing variables are guided by differential equations and hence Laplace transform is used for analysis.










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