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

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ISBN: 8220102801448

Author: Alexander

Publisher: YUZU

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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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Find the expression for voltage v(t).

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EBK FUNDAMENTALS OF ELECTRIC CIRCUITS

Ch. 8.2 - The switch in Fig. 8.4 was open for a long time...Ch. 8.2 - For the circuit in Fig. 8.7, find: (a) iL(0+),...Ch. 8.3 - If R = 10 , L = 5 H, and C = 2 mF in Fig. 8.8,...Ch. 8.3 - The circuit in Fig. 8.12 has reached steady state...Ch. 8.4 - In Fig. 8.13, let R = 2 , L = 0.4 H, C = 25 mF,...Ch. 8.4 - Refer to the circuit in Fig. 8.17. Find v(t) for t...Ch. 8.5 - Having been in position a for a long time, the...Ch. 8.6 - Find i(t) and v(t) for t 0 in the circuit of Fig....Ch. 8.7 - Determine v and i for t 0 in the circuit of Fig....Ch. 8.7 - For t 0, obtain v0(t) in the circuit of Fig....

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Find the expression for voltage v ( t ) .

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