To determine Find the value of current flowing through the inductor i o ( t ) for t ≥ 0 . Explanation Given data: Refer to Figure given in the textbook. The given values are, L = 250 mH C = 6.25 μ F R = 125 Ω The voltage source is 25 V. Formula used: Write the general expression for resonant radian frequency as follows, ω 0 = 1 L C (1) Write the general expression for neper frequency as follows, α = 1 2 R C (2) Write the general expression for damping frequency as follows, ω d = ω 0 2 − α 2 (3) Calculation: The given circuit is modified as shown in Figure 1 for t < 0 . In Figure 1, the voltage across the capacitor is v o ( 0 − ) = 0 V and the current through the inductor is i o ( 0 − ) = 0 A The given circuit is modified as shown in Figure 2 for t > 0 . In Figure 2, the voltage source with series resistance is converted into current source with parallel resistance using source transformation. The value of current source is calculated by using ohms law as follows, I = 25 125 = 0.2 A The modified circuit is shown in Figure 3. Substitute 125 for R and 6.25 μ for C in equation (2) as follows, α = 1 2 ( 125 ) ( 6.25 × 10 − 6 ) { ∵ 1 μ = 10 − 6 } = 640 rad s Substitute 6.25 μ for C and 250 m for L in equation (1) as follows, ω 0 = 1 ( 250 × 10 − 3 ) ( 6.25 × 10 − 6 ) { ∵ 1 μ = 10 − 6 , 1 m = 10 − 3 } = 800 rad s Substitute 800 for ω 0 and 640 for α in equation (3) as follows, ω d = ( 800 ) 2 − ( 640 ) 2 = 480 rad s As α 2 < ω 0 2 , the output response is underdamped. Hence, the output equation will be as follows, i o ( t ) = I f + B ′ 1 e − α t cos ω d t + B ′ 2 e − α t sin ω d t (4) Substitute 0.2 for I , 480 for ω d , and 640 for α in equation (4) as follows, i o ( t ) = 0.2 + B ′ 1 e − 640 t cos ( 480 t ) + B ′ 2 e − 640 t sin ( 480 t ) { ∵ I = I f } (5) Determine the values of the coefficient B ′ 1 and B ′ 2 by applying the initial condition to the above equation. That is, i o ( 0 ) = 0 . Substitute 0 for t in equation (5) as follows, i o ( 0 ) = 0.2 + B ′ 1 e − 640 ( 0 ) cos ( 480 ) ( 0 ) + B ′ 2 e − 640 ( 0 ) sin ( 480 ) ( 0 ) 0 = 0.2 + B ′ 1 { ∵ e 0 = 1 } B ′ 1 = − 0

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Chapter 8, Problem 33P

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Chapter 8 Solutions

Electric Circuits (10th Edition)

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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