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) As the current i ( t ) is same as the current through the capacitor, write the expression for current i ( t ) for t ≥ 0 as follows: i ( t ) = C d v ( t ) d t (10) 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 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 . α = 80 Ω 2 ( 5 mH ) = 80 ( 2 ) ( 5 × 10 − 3 ) rad / s = 8000 rad / s Substitute 8000 rad / s for α and 10 , 000 rad / s for ω 0 in Equation (1) as follows: ( 10 , 000 rad / s ) 2 > ( 8000 rad / s ) 2 The expression is satisfied

11th Edition

ISBN: 9780134747224

Author: Riedel

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 8.4, Problem 8AP

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

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Chapter 8 Solutions

EBK ELECTRIC CIRCUITS

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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