Explanation Given data: Refer to Figure given in the textbook. The given values are, L = 25 mH C = 62.5 μ F R = 12.5 Ω The current source is 2 A. The initial current in the 25 mH inductor is 1 A and the initial voltage on the capacitor is 50 V. Formula used: Write the general expression for resonant radian frequency as follows, ω 0 = 1 L C (1) Write the general expression for neper frequency as follows, α = 1 2 R C (2) Write the general expression for damping frequency as follows, ω d = ω 0 2 − α 2 (3) Calculation: In the given circuit, the initial voltage across the capacitor is v ( 0 ) = 50 V and the initial current through the 25 mH inductor is i ( 0 ) = 1 A Substitute 12.5 for R and 62.5 μ for C in equation (2) as follows, α = 1 2 ( 12.5 ) ( 62.5 × 10 − 6 ) { ∵ 1 μ = 10 − 6 } = 640 rad s Substitute 62.5 μ for C and 20 m for L in equation (1) as follows, ω 0 = 1 ( 25 × 10 − 3 ) ( 62.5 × 10 − 6 ) { ∵ 1 μ = 10 − 6 , 1 m = 10 − 3 } = 800 rad s Substitute 800 for ω 0 and 640 for α in equation (3) as follows, ω d = ( 800 ) 2 − ( 640 ) 2 = 480 rad s As α 2 < ω 0 2 , the output response is underdamped. Hence, the output equation will be as follows, i L ( t ) = I f + B ′ 1 e − α t cos ω d t + B ′ 2 e − α t sin ω d t (4) In steady state condition, the inductor acts a short circuit. Hence, the current through the inductor is, I = I f = 2 A Substitute 2 for I f , 480 for ω d , and 640 for α in equation (4) as follows, i L ( t ) = 2 + B ′ 1 e − 640 t cos ( 480 t ) + B ′ 2 e − 640 t sin ( 480 t ) (5) Determine the values of the coefficient B ′ 1 and B ′ 2 by applying the initial condition to the above equation. Substitute 0 for t in equation (5) as follows, i L ( 0 ) = 2 + B ′ 1 e − 640 ( 0 ) cos ( 480 ) ( 0 ) + B ′ 2 e − 640 ( 0 ) sin ( 480 ) ( 0 ) 1 = 2 + B ′ 1 { ∵ e 0 = 1 , i L ( 0 ) = 1 A } B ′ 1 = − 1 As the given circuit is a parallel RLC circuit, the voltage across the inductor and the capacitor is the same

11th Edition

ISBN: 9780134746968

Author: James W. Nilsson, Susan Riedel

Publisher: PEARSON

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Chapter 8, Problem 32P

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Find the expression for iL(t) for t≥0.

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Chapter 8, Problem 31P

Chapter 8, Problem 33P

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Electric Circuits. (11th Edition)

Ch. 8.1 - The resistance and inductance of the circuit in...Ch. 8.2 - Use the integral relationship between iL and v to...Ch. 8.2 - Prob. 3AP Ch. 8.2 - Prob. 4AP Ch. 8.2 - Prob. 5AP Ch. 8.3 - Prob. 6AP Ch. 8.4 - Prob. 7AP Ch. 8.4 - Prob. 8AP Ch. 8.4 - Repeat Assessment Problems 8.7 and 8.8 if the 80 Ω...Ch. 8 - The resistance, inductance, and capacitance in a...

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