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

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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 - 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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