To determine Find the expression of i o ( t ) for t ≥ 0 in the given circuit. Explanation Formula used: Write the condition for critically damped response for the given circuit as follows: α 2 = ω 0 2 (1) Here, α is the neper frequency and ω 0 is the resonant radian frequency. Write the condition for under-damped response for the given circuit as follows: ω 0 2 > α 2 (2) Write the condition for over-damped response for the given circuit as follows: ω 0 2 < α 2 (3) Write the expression of i o ( t ) for over-damped response as follows: i o ( t ) = I f + D ′ 1 t e − α t + D ′ 2 e − α t , t ≥ 0 (4) Here, I f is the final value of the current, D ′ 1 , D ′ 2 are arbitrary constants, and Write the expression for resonant radian frequency as follows: ω 0 = 1 L C (5) Here, L is the inductance of the inductor and C is the capacitance of the capacitor. Write the expression for neper frequency as follows: α = 1 2 R C (6) Write the expression to obtain the arbitrary constants as follows: I f + D ′ 2 = I 0 (7) Write the expression to obtain the arbitrary constants as follows: D ′ 1 − α D ′ 2 = V 0 L (8) Calculation: From the given circuit, the switch is open for long. As the inductor gets open circuit at initial state, the current through the inductor for t < 0 is 0 A. i L ( 0 − ) = 0 A I 0 = 0 A From the given circuit, the capacitor is connected to the source irrespective of the position of switch. As the capacitor gets open circuit at steady state, the entire source voltage reflects across the capacitor. Therefore, the voltage across the capacitor for t < 0 is 60 V. v C ( 0 − ) = 60 V V 0 = 60 V From the given circuit, at steady state ( t = ∞ ) , the capacitor gets open circuit and the inductor gets short circuit. Therefore, the final current through the inductor is calculated as follows: I f = 60 V 20 Ω = 3 A Substitute 20 Ω for R and 31.25 μ F for C in equation (6) to obtain the value of α . α = 1 2 ( 20 Ω ) ( 31.25 μ F ) = 1 ( 2 ) ( 20 ) ( 31.25 × 10 − 6 ) rad / s = 800 rad / s Substitute 50 mH for L and 31.25 μ F for C in equation (5) to obtain the value of ω 0 . ω 0 = 1 ( 50 mH ) ( 31.25 μ F ) = 1 ( 50 × 10 − 3 ) ( 31.25 × 10 − 6 ) rad / s = 800 rad / s Substitute 800 rad / s for α and 800 rad / s for ω 0 in equation (1) as follows: ( 800 rad / s ) 2 = ( 800 rad / s ) 2 The expression is satisfied

Electrical Circuits and Modified MasteringEngineering - With Access

10th Edition

ISBN: 9780133992793

Author: NILSSON

Publisher: PEARSON

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Chapter 8, Problem 31P

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Chapter 8, Problem 30P

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Chapter 8 Solutions

Electrical Circuits and Modified MasteringEngineering - With Access

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 - The switch in the circuit shown has been in...Ch. 8.4 - Prob. 8AP Ch. 8 - Prob. 1P Ch. 8 - Prob. 2P

Ch. 8 - The resistance in Problem 8.1 is decreased to80 Ω...Ch. 8 - Prob. 4P Ch. 8 - Prob. 5P Ch. 8 - The natural voltage response of the circuit in...Ch. 8 - Prob. 7P Ch. 8 - Prob. 8P Ch. 8 - The natural response for the circuit shown in Fig....Ch. 8 - Prob. 10P Ch. 8 - The two switches in the circuit seen in Fig.P8.11...Ch. 8 - The resistor in the circuit of Fig. P8.11 is...Ch. 8 - The resistor in the circuit of Fig.P8.11 is...Ch. 8 - The switch in the circuit of Fig. P8.17 has been...Ch. 8 - The inductor in the circuit of Fig. P8.17 is...Ch. 8 - The inductor in the circuit of Fig. P8.17 is...Ch. 8 - Design a parallel RLC circuit (see Fig. 8.1) using...Ch. 8 - Prob. 18P Ch. 8 - Prob. 19P Ch. 8 - Find υ(t) for t ≥ 0 in the circuit in Problem 8.19...Ch. 8 - Prob. 21P Ch. 8 - Prob. 22P Ch. 8 - The initial value of the voltage υ in the circuit...Ch. 8 - Prob. 24P Ch. 8 - Prob. 25P Ch. 8 - Prob. 26P Ch. 8 - Prob. 27P Ch. 8 - Prob. 28P Ch. 8 - Prob. 29P Ch. 8 - Prob. 30P Ch. 8 - The switch in the circuit in Fig. P8.31 has been...Ch. 8 - Prob. 32P Ch. 8 - There is no energy stored in the circuit in Fig....Ch. 8 - For the circuit in Fig. P8.30, find υo for t ≥...Ch. 8 - The switch in the circuit in Fig. P8.36 has been...Ch. 8 - Prob. 36P Ch. 8 - Prob. 37P Ch. 8 - Prob. 38P Ch. 8 - Prob. 39P Ch. 8 - Find the voltage across the 80 nF capacitor for...Ch. 8 - The initial energy stored in the 31.25 nF...Ch. 8 - In the circuit in Fig. P8.42, the resistor is...Ch. 8 - Design a series RLC circuit (see Fig. 8.3) using...Ch. 8 - Change the resistance for the circuit you designed...Ch. 8 - Prob. 45P Ch. 8 - Prob. 46P Ch. 8 - The switch in the circuit shown in Fig. P8.48 has...Ch. 8 - The switch in the circuit in Fig. P8.48 has been...Ch. 8 - The initial energy stored in the circuit in Fig....Ch. 8 - The resistor in the circuit shown in Fig. P8.50 is...Ch. 8 - The resistor in the circuit shown in Fig. P8.50 is...Ch. 8 - Prob. 52P Ch. 8 - The two switches in the circuit seen in Fig. P8.53...Ch. 8 - Prob. 55P Ch. 8 - Prob. 57P Ch. 8 - Prob. 58P Ch. 8 - Prob. 59P Ch. 8 - Prob. 60P Ch. 8 - Prob. 61P Ch. 8 - Derive the differential equation that relates the...Ch. 8 - The voltage signal of Fig. P8.63(a) is applied to...Ch. 8 - The circuit in Fig. P8.63 (b) is modified by...Ch. 8 - Prob. 65P Ch. 8 - Prob. 66P Ch. 8 - Prob. 67P Ch. 8 - Prob. 68P

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