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Meeting w3   chapter 2 part 1

Meeting w3 chapter 2 part 1






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    Meeting w3   chapter 2 part 1 Meeting w3 chapter 2 part 1 Presentation Transcript

    • Chapter 2 Analog Control System Eddy Irwan Shah Bin Saadon Dept. of Electrical Engineering PPD, UTHM [email_address] 019-7017679
      • Outline:
      • Introduction
      • Laplace Transform – Table/ Theorem/ Eg.
      • Common Time Domain Input Function
      • Transfer Function – Open/ Closed Loop & Eg.
      • Electrical Elements Modelling – Table & Eg.
      • Mechanical Elements Modelling - Table & Eg.
      • Block Diagram Reduction - Table & Eg.
      • System Response – Poles/ Zeros, Second Order, Steady State Error, Stability Analysis
      • Intro - Objective of this chapter
      • After completing this chapter you will be able to:
      • Describe the fundamental of Laplace transforms.
      • Apply the Laplace transform to solve linear ordinary differential equations.
      • Apply Mathematical model, called a transfer function for linear time-invariant electrical, mechanical and electromechanical systems.
    • 2. What is Laplace Transform?
      • Laplace transform is a method or techniques used to transform the time ( t ) domain to the Laplace/frequency ( s ) domain
      • What is algebra & calculus?
      Time Domain Frequency Domain Differential equations Input q(t) Output h(t) Algebraic equations Input Q(s) Output H(s) Calculus Algebra Laplace Transformation Inverse Laplace Transformation
    • Laplace Transform (cont.)
      • The Laplace transform solution consists of the following three steps:
      • the Laplace transformation of q1(t) and (r dhldt + h = Gq) to frequency domain
      • the algebraic solution for H(s)
      • the inverse Laplace transformation of H(s) to time domain h(t).
      • The calculus solution is shown as step 4.
    • Definition of the Laplace Transform
      • Laplace transform is defined as
      • Inverse Laplace transform is defined as
      L L -1
    • Laplace Theorem
    • Laplace Table
    • Example 1
      • Find the Laplace transform for
    • Example 2
      • Find the Laplace transform for
    • Example 3
      • Find the inverse Laplace transform of
      Solution: Expanding F(s) by partial fraction: Where, Then, taking the inverse Laplace transform
    • Example 4
      • Given the ,solve for y ( t ) if all initial conditions are
      • zero. Use the Laplace transform method .
      Solution: Substitute the corresponding F ( s ) for each term: Solving for the response: Where, K 1 = 1 when s=0 K 2 =-2 when s=-4 K 3 = 1 when s=-8 Hence
    • 3. Common Time Domain Input Functions
      • Unit Step Function
      • Unit Ramp Function
      • Unit Impulse Function
    • 4. Transfer Function
      • Definition:
      • Ratio of the output to the input; with all initial conditions are zero
      • If the transformed input signal is X(s) and the transformed output signal is Y(s) , then the transfer function M(s) is define as;
      • From this,
      • Therefore the output is
    • TF of Linear Time Invariant Systems
      • In practice, the input-output relation of lines time-invariant system with continuous-data input is often described by a differential equation
      • The linear time-invariant system is described by the following n th-order differential equation with constant real coefficients;
      c(t) is output r(t) is input
    • cont.
      • Taking the Laplace transform of both sides,
      • If we assume that all initial conditions are zero, hence
      • Now, form the ratio of output transform, C(s) divided by input transform. The ratio, G(s) is called transfer function.
    • cont.
      • The transfer function can be represented as a block diagram
      • General block diagram
    • Block Diagram of Open Loop System
    • Block Diagram of Closed Loop System
    • Example 1
      • Problem: Find the transfer function represented by
      • Solution:
      • Taking the Laplace transform of both sides, assuming zero initial conditions, we have
      • The transfer function, G(s) is
    • Example 2
      • Problem: Use the result of Example 1 to find the response, c(t), to an input, r(t)=u(t), a unit step and assuming zero initial conditions.
      • Solution:
      • Since r(t)=u(t), R(s)=1/s, hence
      • Expanding by partial fractions, we get
      • Finally, taking the inverse Laplace transform of each term yields
    • Example 3
      • Problem: Find the transfer function, G(s)=C(s)/R(s), corresponding to the differential equation
      • Solution:
    • Example 4
      • Problem: Find the differential equation corresponding to the transfer function,
      • Solution:
    • Example 5
      • Problem: Find the ramp response for a system whose transfer function is,
      • Solution: