Explanation Formula used: Write the condition for over-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 under-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 for the given circuit as follows: ω 0 = 1 L C (4) Here, L is the inductance of the inductor, and C is the capacitance of the capacitor. Write the expression for neper frequency for the given circuit as follows: α = 1 2 R C (5) Here, R is the resistance of the resistor. Write the expression of required voltage response v ( t ) for over-damped response of a parallel RLC circuit as follows: v o ( t ) = A 1 e s 1 t + A 2 e s 2 t , t ≥ 0 (6) Here, A 1 , A 2 are arbitrary constants and s 1 , s 2 are roots of characteristic equation. Refer Table 8.1 in the textbook for Natural-Response parameters of parallel R L C circuit, and write the expression for characteristic root s 1 as follows: s 1 = − α + α 2 − ω 0 2 (7) Refer Table 8.1 in the textbook, and write the expression for characteristic root s 2 as follows: s 2 = − α − α 2 − ω 0 2 (8) Refer to Equation (8.10) in the textbook and write the expression to obtain the values of arbitrary constants. A 1 + A 2 = V 0 (9) Refer to Equation (8.11) in the textbook and write the expression to obtain the values of arbitrary constants. A 1 s 1 + A 2 s 2 = 1 C ( − V 0 R − I 0 ) (10) Calculation: From the given circuit, the switch 1 is closed for long time to the source and inductor. As the inductor gets short circuit at steady state, the entire source current flows through the inductor and it does not flow through 500 Ω resistor. Therefore, the current through the inductor for t < 0 is − 60 mA . The negative sign is due to the reverse direction of the current through the inductor. i L ( 0 − ) = − 60 mA I 0 = − 60 mA From the given circuit, the switch 2 is closed for long time to the source and capacitor through the resistor. 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 15 V. v C ( 0 − ) = 15 V V 0 = 15 V As it is required to find the required voltage response across the 100 Ω resistor, consider the value of R as 100 Ω . Substitute 100 Ω for R and 1 μ F for C in Equation (5) to obtain the value of α . α = 1 2 ( 100 Ω ) ( 1 μ F ) = 1 ( 2 ) ( 100 ) ( 1 × 10 − 6 ) rad / s = 5000 rad / s Substitute 62.5 mH for L and 1 μ F for C in Equation (4) to obtain the value of ω 0 . ω 0 = 1 ( 62.5 mH ) ( 1 μ F ) = 1 ( 62.5 × 10 − 3 ) ( 1 × 10 − 6 ) rad / s = 4000 rad / s Substitute 5000 rad / s for α and 4000 rad / s for ω 0 in Equation (1) as follows: ( 4000 rad / s ) 2 < ( 5000 rad / s ) 2 The expression is satisfied

Electric Circuits Plus Mastering Engineering with Pearson eText 2.0 - Access Card Package (11th Edition) (What's New in Engineering)

11th Edition

ISBN: 9780134814117

Author: NILSSON, James W., Riedel, Susan

Publisher: PEARSON

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Chapter 8, Problem 14P

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Chapter 8 Solutions

Electric Circuits Plus Mastering Engineering with Pearson eText 2.0 - Access Card Package (11th Edition) (What's New in Engineering)

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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Find v o ( t ) when t ≥ 0 for the given circuit using PSPICE.

Find vo(t) when t≥0 for the given circuit using PSPICE.

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Chapter 8 Solutions