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(c) X3(8): (d) X₁ (s): 8² +2 82-58-6' 23(t) is a two-sided signal 8-1 (8+2)(s² +8+1) • £₁(t) is a left-sided signal

(c) X3(8): (d) X₁ (s): 8² +2 82-58-6' 23(t) is a two-sided signal 8-1 (8+2)(s² +8+1) • £₁(t) is a left-sided signal

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Only 2c and 2d needed. Thanks

**Problem Statement:** Find the corresponding time function for each of the following rational Laplace transforms. For (a) and (b), use (at least) two different methods. Two suggested methods are provided below. For the remaining problems, using one method (of your choice) is adequate. --- **(a) \(X_1(s) = \frac{s + 2}{s^2 - 2s + 2}\), \(\text{Re}\{s\} > 1\)** - **Method 1**: Express \(X_1(s)\) as \(X_1(s) = \frac{A}{s - p_1} + \frac{B}{s - p_2}\) where \(p_1, p_2\) are the two poles and \(A, B\) are constants. - **Method 2**: Express \(X_1(s)\) as: \[ X_1(s) = \frac{C(s + \alpha)}{(s + \alpha)^2 + \omega_0^2} + \frac{D\omega_0^2}{(s + \alpha)^2 + \omega_0^2} \] for some properly chosen \(\alpha\) and \(\omega_0\), and constants \(C, D\). --- **(b) \(X_2(s) = \frac{4s}{(s + 4)^2}\), \(\text{Re}\{s\} > -4\) (repeated poles)** - **Method 1**: Express \(X_2(s)\) as \(X_2(s) = \frac{A}{s + 4} + \frac{B}{(s + 4)^2}\) for some constants \(A\) and \(B\). - **Method 2**: Express \(X_2(s)\) as \(X_2(s) = \frac{C}{(s + 4)} + \frac{d}{ds}(s + 4)\) for some constants \(C\) and \(D\). --- **(c) \(X_3(s) = \frac{s^2 + 2}{s^2 - 5s - 6}\); \(x_3(t)\) is a two-sided signal
Transcribed Image Text:**Problem Statement:** Find the corresponding time function for each of the following rational Laplace transforms. For (a) and (b), use (at least) two different methods. Two suggested methods are provided below. For the remaining problems, using one method (of your choice) is adequate. --- **(a) \(X_1(s) = \frac{s + 2}{s^2 - 2s + 2}\), \(\text{Re}\{s\} > 1\)** - **Method 1**: Express \(X_1(s)\) as \(X_1(s) = \frac{A}{s - p_1} + \frac{B}{s - p_2}\) where \(p_1, p_2\) are the two poles and \(A, B\) are constants. - **Method 2**: Express \(X_1(s)\) as: \[ X_1(s) = \frac{C(s + \alpha)}{(s + \alpha)^2 + \omega_0^2} + \frac{D\omega_0^2}{(s + \alpha)^2 + \omega_0^2} \] for some properly chosen \(\alpha\) and \(\omega_0\), and constants \(C, D\). --- **(b) \(X_2(s) = \frac{4s}{(s + 4)^2}\), \(\text{Re}\{s\} > -4\) (repeated poles)** - **Method 1**: Express \(X_2(s)\) as \(X_2(s) = \frac{A}{s + 4} + \frac{B}{(s + 4)^2}\) for some constants \(A\) and \(B\). - **Method 2**: Express \(X_2(s)\) as \(X_2(s) = \frac{C}{(s + 4)} + \frac{d}{ds}(s + 4)\) for some constants \(C\) and \(D\). --- **(c) \(X_3(s) = \frac{s^2 + 2}{s^2 - 5s - 6}\); \(x_3(t)\) is a two-sided signal
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