SOLVE STEP BY STEP IN DIGITAL FORMATGiven the transfer functionH(s):Find y(t) when x(t)=8(t)=s+2(s + 1)² + 4

Sol:- First find the h(t) using Inverse Laplace Transform:- L-1s+2s+12+4=L-1s+1s+12+4+1·1s+12+4Use…

Answered: SOLVE STEP BY STEP IN DIGITAL FORMAT…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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SOLVE STEP BY STEP IN DIGITAL FORMAT
Given the transfer function
H(s):
Find y(t) when x(t)=8(t)
=
s+2
(s + 1)² + 4

Expert Solution

Step 1

Sol:-

First find the h(t) using Inverse Laplace Transform:-

$L^{- 1} \{\frac{s + 2}{{(s + 1)}^{2} + 4}\} = L^{- 1} \{\frac{s + 1}{{(s + 1)}^{2} + 4} + 1 \cdot \frac{1}{{(s + 1)}^{2} + 4}\} Use the linearity property of Inverse Laplace Transform : For functions f (s), g (s) and constants a, b : L^{- 1} \{a \cdot f (s) + b \cdot g (s)\} = a \cdot L^{- 1} \{f (s)\} + b \cdot L^{- 1} \{g (s)\} = L^{- 1} \{\frac{s + 1}{{(s + 1)}^{2} + 4}\} + 1 \cdot L^{- 1} \{\frac{1}{{(s + 1)}^{2} + 4}\} L^{- 1} \{\frac{s + 1}{{(s + 1)}^{2} + 4}\} Apply inverse transform rule : if L^{- 1} \{F (s)\} = f (t) then L^{- 1} \{F (s - a)\} = e^{a t} f (t) For \frac{s + 1}{{(s + 1)}^{2} + 4} : a = - 1, F (s) = \frac{s}{s^{2} + 4} = e^{- t} L^{- 1} \{\frac{s}{s^{2} + 4}\} L^{- 1} \{\frac{s}{s^{2} + 4}\} : \cos (2 t) = e^{- t} \cos (2 t) S i m i l a r l y : - L^{- 1} \{\frac{1}{{(s + 1)}^{2} + 4}\} : e^{- t} \frac{1}{2} \sin (2 t) Laplace Inverse Transform of \frac{s + 2}{{(s + 1)}^{2} + 4} : e^{- t} \cos (2 t) + \frac{1}{2} e^{- t} \sin (2 t)$