Explanation Given data: Refer to Figure corresponding to the Assessment Problem 8.7 in the textbook for the given circuit. The switch in the given circuit has been in position ‘a’ for a long time and it moves to position ‘b’ at t = 0 . Formula used: Write the condition for under-damped response for a parallel RLC circuit as follows: ω 0 2 > α 2 (1) Here, α is the neper frequency and ω 0 is the resonant radian frequency. Write the condition for over-damped response for a parallel RLC circuit as follows: ω 0 2 < α 2 (2) Write the condition for critically damped response for a parallel RLC circuit as follows: α 2 = ω 0 2 (3) Write the expression for resonant radian frequency when the switch moves to position b in the given circuit as follows: ω 0 = 1 L C (4) Here, L is the inductance of the inductor, which is 5 mH and C is the capacitance of the capacitor, which is 2 μ F . Write the expression for neper frequency when the switch moves to position b in the given circuit as follows: α = R 2 L (5) Here, R is the resistance of the resistor, which is 80 Ω . Writ the expression for damped frequency when the switch moves to position b in the given circuit as follows: ω d = ( ω 0 ) 2 − α 2 (6) Write the expression of v C ( t ) for t ≥ 0 in the given circuit as follows: v C ( t ) = V f + B ′ 1 e − α t cos ( ω d t ) + B ′ 2 e − α t sin ( ω d t ) , t ≥ 0 (7) Here, V f is the final value of the voltage, B ′ 1 , B ′ 2 are arbitrary constants, and ω d is the damped frequency of the response. Write the expression to obtain the value of B ′ 1 . V f + B ′ 1 = V 0 (8) Write the expression to obtain the value of B ′ 2 . − α B ′ 1 + ω d B ′ 2 = I 0 C (9) Calculation: From the given circuit, the inductor is not connected to the source when t < 0 . Therefore, the initial value of current through the inductor as follows: i L ( 0 + ) = 0 A I 0 = i L ( 0 + ) = 0 A From the given circuit, use voltage division rule and write the expression to find the initial value of voltage across the capacitor as follows: v C ( 0 + ) = ( 15 k Ω 15 k Ω + 9 k Ω ) ( 80 V ) = ( 15 24 ) ( 80 V ) = 50 V Therefore, V 0 = v C ( 0 + ) = 50 V From the given circuit, the final value of the voltage is 100 V. V f = 100 V Substitute 2 μ F for C and 5 mH for L in Equation (4) to obtain the value of ω 0 . ω 0 = 1 ( 5 mH ) ( 2 μ F ) = 1 ( 5 × 10 − 3 ) ( 2 × 10 − 6 ) rad / s = 10 , 000 rad / s Substitute 5 mH for L and 80 Ω for R in Equation (5) to obtain the value of α

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ISBN: 9781292060545

Author: James W. Nilsson, Susan Riedel

Publisher: Pearson Education Limited

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Chapter 8.4, Problem 8AP

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Find the expression of vC(t) for t≥0 in the given circuit.

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Chapter 8.4, Problem 7AP

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Chapter 8 Solutions

Electric Circuits, Global 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 - 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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