Chap 3 Laplace Transforms and Applications LS 1 Basic Concepts Bilateral Laplace Transform: where is a complex variable Region of Convergence (ROC): The region of s for which the integral converges Transform Pairs 1 Impluse Function:, for all s 2 Real Exponential Function Causal: for 3 Step Function:, for 4 Real Exponential Function Anticausal: for
LS 2 Problem 31 Find the Laplace transform of the following (a) (b) for
LS 3 for Properties and Relations for Causal Signals Causal Signal: Unilateral Laplace Transform: Proofs: 2 Derivative: for 4 Integral:
LS 4 for 5 Time Shift ( ): 6 Frequency Shift ( ): 7 Convolution: 8 Final Value Theorem (FVT): 9 Initial Value Theorem (IVT): Problem 32 Use FVT and IVT to find and of the following equations a b c
LS 5 Transfer Function Let then where is called the transfer function of input and output Alternatively, for ordinary differential equation, Take the Laplace transform of both sides, we have Note: require, and That is zero initial condition for both input and output Problem 33: Find the transfer function of the following equation Since, let the output be (impulse response), then the transfer function That is a transfer function is the Laplace transform of the impulse response
LS 6 Problem 34: Find the transfer function of the following impulse response: Zeros and Poles Suppose a transfer function form: can be factored into the following Zeros: Poles: Problem 35: Find the zeros and poles of the following transfer function Region of Convergence (a) Causal Systems: the unit impulse response of an Nth-order causal LTI system is and the corresponding transfer function transform of, is, the Laplace ROC: (b) Anticausal Systems:
LS 7 ROC: (c) Noncausal Systems ROC: Problem 36: Find the ROCs for the following a b c d
Stability (a) Causal Systems: All poles must be on the left side of y-axis ROC include y-axis (b) Antiausal Systems: All poles must be on the right side of y-axis ROC include y-axis (c) Noncausal Systems: and Problem 37: Given the following transfer function LS 8 determine the stability for the following regions of convergence a b c The Evaluation of Inverse Transforms where the line to lie in the ROC Partial Fraction Expansion (PFE), Simple Poles Let, in general where has degree smaller than N Then, where Thus can be expanded in partial fraction forms as
LS 9 (a) Causal Signal (b) Anticausal Signal Problem 38: Given the transform Find the inverse transform for the following ROCs a b c Partial Fraction Expansion, Multiple Poles: Example: Expansion: From Table 31, we have Example: Problem 39: Find the expansion and the corresponding time signal of the following
LS 10 Solution of Linear Differential Equations by Laplace Transform Starting with a system s differential equations: Take the Laplace transform of both sides: Then, Complex solution of 1 Transfer Function: include the contributions from two parts:
LS 11 2 Initial Conditions: Problem 310: Let, a Use Laplace transform to find for b Check with IVT and FVT Problem 311: Let Find the solution for Convolution Since Therefore, Problem 312: An LRC circuit with the unit impulse response is subjected to a unit step input a Use Laplace transforms to find the output b Use convolution integral to find the output c Use MATLAB function step to find the output Sinusoidal Static-State Solution Let, then
LS 12 Therefore, If the system is stable, we have Therefore, Problem 313: An analog notch filter that is realized with op-amps, resistors, and capacitors can be modeled by the differential equation Let, find static-state solution
LS 13 System Diagrams or Structures Let, then We have System Diagram Symbols 1 A branch represents the flow of a signal and its direction 2 A value on a branch representing the multiplication factor to the signal flowing through 3 Signals flowing into a node represent a sum operation 4 Each signal flowing out a node is the sum of all input signals Problem 314: Draw a system diagram of the following
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LS 15 Transform Domain Diagram Problem 315 Draw the system diagram of the following
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LS 17 Mason s Gain Rule and Applications Purpose: for computing transfer functions where : the graph determinant, Example Fig 317 : loop gain for kth loop Not touching only for,, : the gain of the kth path from input to output, : cofactor of the kth path, formed by striking out from all terms associated with loops that are touched by kth path Therefore,, Problem 316 Calculate the transform function of Fig 317a from to,
LS 18 Simplification Rules a Series rule b Parallel rule c Self-loop rule d Splitting rule Example 36 a The unstable system of Fig E36a1 can be stabilized by adding a feedback path with a gain of f, as in Fig E36a2 Find f b Find the resulting transfer function of the parallel-connected three first-order systems in Fig E36b c Find the resulting transfer function of the series-connected three first-order systems in Fig E36b3
P313 Find the transform function of the following LS 19
LS 20 Example 31 Consider the mechanical system in Fig E31a By Newton s law, we have where is the position of the mass a Apply Laplace transform to the equation b Find c Given the following data:,,,,, and Find the poles and expand in partial fractions d Use Table 31 to find Find e Use MATLAB to find and by state-space equations f What if? Example 32 Suppose we have an airplane whose ailerons are oscillated sinusoidally and we want to solve the roll rate The inertia torque is, and aileron torque is Therefore, a Find for,, and b Find by partial fraction expansion c Use MATAB to plot with the result of b d Use MATLAB residue to verify b Note: [R,P,K] = residue(b,a); where Example 33 An unstable spacecraft is stabilized by the addition of an analog filter, with the result that the relationship between the output position and input satisfy a Find the steady-state output when Assume
zero initial condition b Estimate the time at which the steady-state solution is reached c USE MATLAB to plot d What if? Example 34 A bandpass filter is subjected to a step input and the response (plotted in Fig E34a) is a What is the transfer function b Find the analytical solution of the unit impulse response Compared with MATLAB solution c Find for LS 21