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    April Balce
    This document discusses modeling systems using transfer functions in the frequency domain. It introduces the Laplace transform, which allows representing inputs, outputs, and systems as separate algebraic entities. This simplifies modeling physical systems. The transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming all initial conditions are zero. Examples are provided of finding transfer functions from systems represented by differential equations and vice versa.

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    April Balce
    24 views11 pages
    This document discusses modeling systems using transfer functions in the frequency domain. It introduces the Laplace transform, which allows representing inputs, outputs, and systems as separate algebraic entities. This simplifies modeling physical systems. The transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming all initial conditions are zero. Examples are provided of finding transfer functions from systems represented by differential equations and vice versa.

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    This document discusses modeling systems using transfer functions in the frequency domain. It introduces the Laplace transform, which allows representing inputs, outputs, and systems as separate algebraic entities. This simplifies modeling physical systems. The transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming all initial conditions are zero. Examples are provided of finding transfer functions from systems represented by differential equations and vice versa.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    24 views11 pages

    Feedback

    Uploaded by

    April Balce
    This document discusses modeling systems using transfer functions in the frequency domain. It introduces the Laplace transform, which allows representing inputs, outputs, and systems as separate algebraic entities. This simplifies modeling physical systems. The transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming all initial conditions are zero. Examples are provided of finding transfer functions from systems represented by differential equations and vice versa.

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    © All Rights Reserved
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    of 11

    MODELING IN

    FREQUENCY DOMAIN
    In previous topic, we discussed the analysis and design
    sequence that included obtaining the system’s schematic and
    demonstrated this step for a position control system. To obtain a
    schematic, the control systems engineer must often make many
    simplifying assumptions in order to keep the ensuing model
    manageable and still approximate physical reality.

    The next step is to develop mathematical models from


    schematics of physical systems. We will discuss two methods: (1)
    transfer functions in the frequency domain and (2) state equations
    in the time domain. As we proceed, we will notice that in every
    case the first step in developing a mathematical model is to apply
    the fundamental physical laws of science and engineering. For
    example, when we model electrical networks,
    Ohm’s law and Kirchhoff’s laws, which are basic laws of electric
    networks, will be applied initially. We will sum voltages in a loop
    or sum currents at a node.
    LAPLACE TRANSFORM
    REVIEW
    A system represented by a differential equation is difficult to
    model as a block diagram. Thus, we now lay the groundwork for
    the Laplace transform, with which we can represent the input,
    output, and system as separate entities. Further, their
    interrelationship will be simply algebraic. Let us first define the
    Laplace transform and then show how it simplifies the
    representation of physical systems.

    The Laplace transform is defined as:

    where , a complex variable. Thus, knowing f(t) and that the


    integral in Eq. above exists, we can find a function, F(s), that is
    called the Laplace transform of f(t).1
    LAPLACE TRANSFORM
    REVIEW
    The inverse Laplace transform, which allows us to find f(t) given
    F(s), is
    LAPLACE TRANSFORM
    REVIEW
    Laplace Transform Table
    LAPLACE TRANSFORM
    REVIEW
    Laplace Transform Theorem
    LAPLACE TRANSFORM
    REVIEW
    Example 1. What is the Laplace Transform of 1?

    Example 1. What is the Laplace Transform of t?


    TRANSFER FUNCTION

    Transfer function is the ratio of Laplace Transform of output to


    the Laplace Transform of input, when all the initial conditions
    are assumed to be zero.

    input SYSTE output


    R(s) r(t) MG(s)g(t) c(t)
    C(s)
    Where: G(s) – Transfer Function of the System.
    C(s) – Output of the System.
    R(s) – Input of the System.
    TRANSFER FUNCTION

    Transfer function is the ratio of Laplace Transform of output to


    the Laplace Transform of input, when all the initial conditions
    are assumed to be zero.

    C(s)
    G(s) =
    R(s) initial condition is equal to zero.
    TRANSFER FUNCTION

    Example 3. Find the Transfer Function of the System given by:


    TRANSFER FUNCTION

    Example 4. Find the Transfer Function of the System given by:


    TRANSFER FUNCTION

    Example 4. Find the differential equation corresponding to the


    transfer function:

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