To determine Find v o ( t ) when t ≥ 0 for the given circuit using PSPICE. Explanation 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 critically damped response for a parallel RLC circuit as follows: α 2 = ω 0 2 (2) Write the condition for over-damped response for a parallel RLC circuit as follows: ω 0 2 < α 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 ) in the given circuit for underdamped response as follows: v o ( t ) = B 1 e − α t cos ω d t + B 2 e − α t sin ω d t , t ≥ 0 (6) Here, B 1 , B 2 are arbitrary constants and ω d is the damped frequency. Write the expression for damped frequency in the circuit as follows: ω d = ( ω 0 ) 2 − α 2 (7) Write the expression to obtain the value of B 1 . B 1 = V 0 (8) Write the expression to obtain the value of B 2 . − α B 1 + ω d B 2 = 1 C ( − V 0 R − I 0 ) (9) 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 50 V source is connected to the capacitor for t < 0 . As the capacitor gets open circuit at steady state, the initial value of voltage across capacitor is equal to the voltage across the 4 k Ω resistor. Therefore, write the expression for initial value of voltage across the capacitor as follows: V 0 = v C ( 0 − ) = v C ( 0 + ) = v 4 k Ω Use voltage division rule and find the initial value of voltage across the capacitor 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 (10) 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 ) (11) 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 (11) 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 (10) 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

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Chapter 8, Problem 17P

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