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# Meeting w3 chapter 2 part 1

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## Meeting w3 chapter 2 part 1Presentation 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
Solution:
• Example 2
• Find the Laplace transform for
Solution:
• 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
cont.
• Unit Impulse Function
cont.
• 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: