Explanation Given data: The natural response of an R L C circuit is described by the differential equation, d 2 v d t 2 + 2 d v d t + v = 0 (1) The value of initial voltage, v ( 0 ) = 10 V . The value of d v ( 0 ) d t = 0 . Formula used: Write an expression to find the voltage response of the circuit when the roots of characteristic equations are equal. v ( t ) = ( A 1 t + A 2 ) e − α t (2) Here, A 1 and A 2 are constants, and α is the neper frequency. When the roots of characteristic equation are equal, s 1 = s 2 = − α (3) Write an expression to solve quadratic equation. s 1 , 2 = − b ± b 2 − 4 a c 2 (4) Here, a is the coefficient of second order term, b is the coefficient of first order term, and c is the coefficient of constant term. Calculation: The characteristic equation of the equation (1) is written as follows, s 2 + 2 s + 1 = 0 (5) From the equation (5), a = 1 b = 2 c = 1 Substitute 1 for a , 2 for b , and 1 for c in equation (4) to find s 1 , 2 . s 1 , 2 = − 2 ± ( 2 ) 2 − 4 ( 1 ) ( 1 ) 2 = − 2 ± 4 − 4 2 = − 2 ± 0 2 = − 1 , − 1 Therefore, the roots of characteristic equation are equal. s 1 = s 2 = − 1 Substitute − 1 for s 1 in equation (3) to find α . − 1 = − α α = 1 Substitute 1 for α in equation (2) to find v ( t ) . v ( t ) = ( A 1 t + A 2 ) e − t (6) Substitute 0 for t in equation (6) to find v ( 0 ) . v ( 0 ) = ( A 1 ( 0 ) + A 2 ) e − ( 0 ) { ∵ e 0 = 1 } v ( 0 ) = A 2 (7) Substitute 10 V for v ( 0 ) in equation (7) to find A 2

6th Edition

ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Series and Parallel Circuit

Circuits can be classified as parallel and series depending on the manner in which components are connected to one another and the electrical source. The resultant output of the circuit depends on the electrical component.

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Chapter 8, Problem 11P

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