To determine Find the expression of v o ( t ) for t ≥ 0 . Explanation Given data: Refer to Figure 8.17 in the textbook for the given circuit. The inductor in the circuit is increased to 125 mH . L = 125 mH 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 o ( 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 data, the voltage response is required to be determined across the inductor in the circuit. As the source is not connected to the inductor for t < 0 , the initial value of the current through the inductor is 0 A. I 0 = 0 A From the given circuit, the initial value of voltage across the inductor is the final value of voltage across the capacitor. The final value of voltage across the capacitor is the voltage across 4 k Ω resistor. Therefore, write the expression for initial value of voltage across the inductor as follows: V 0 = v o ( 0 − ) = v o ( 0 + ) = ( v C ) f = v 4 k Ω Use voltage division rule and find the initial value of voltage across the inductor as follows: V 0 = ( 50 V ) ( 4 k Ω 6 k Ω + 4 k Ω ) = ( 50 V ) ( 4 10 ) = 20 V Find the equivalent resistance (Thevenin’s resistance) of the circuit for t ≥ 0 by redrawing the given circuit as shown in Figure 1. From the circuit in Figure 1, write the expression for equivalent resistance (Thevenin’s resistance) as follows: R Th = v Th i T (11) From the circuit in Figure 1, write the expression for v Th as follows: v Th = − 10 i ϕ + i T ( 150 Ω ‖ 60 Ω ) (12) Use current division rule and write the expression for i ϕ in the circuit as follows: i ϕ = − i T ( 150 Ω 150 Ω + 60 Ω ) = ( − 150 210 ) i T Substitute ( − 150 210 ) i T for i ϕ in Equation (12) as follows: v Th = − 10 ( − 150 210 ) i T + i T ( 150 Ω ‖ 60 Ω ) = ( 1500 210 ) i T + i T [ ( 150 Ω ) ( 60 Ω ) 150 Ω + 60 Ω ] = ( 1500 210 ) i T + ( 9000 210 ) i T = ( 10 , 500 210 ) i T Substitute ( 10 , 500 210 ) i T for v Th in Equation (11) to obtain the Thevenin’s resistance in the circuit for t ≥ 0 . R Th = ( 10 , 500 210 ) i T i T = 50 Ω Modify the expression in Equation (5) for t ≥ 0 as follows: α = 1 2 R Th C Substitute 50 Ω for R Th and 8 μ F for C to obtain the value of α

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 19P

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

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