Explanation Given data: Refer to Figure given in the textbook. The value of the inductor is L = 0.5 mH . The value of the capacitor is C = 12.5 nF The value of the voltage source is 20 V and 28 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 = R 2 L (2) Write the general expression for damoing frequency as follows, ω d = ω 0 2 − α 2 (3) Calculation: Consider the circuit for t < 0 when the switch is at position ‘a’. In Figure 1, for t < 0 , the capacitor acts as an open circuit and inductor as a short circuit to find the initial conditions. The initial value of current through the inductor is calculated as follows, i L ( 0 − ) = 28 + 20 160 + 480 = 75 mA The initial value of capacitor voltage is calculated as follows, v o ( 0 − ) = 160 i L ( 0 − ) − 28 = ( 160 ) ( 75 × 10 − 3 ) − 28 = 12 − 28 = − 16 V Consider the circuit for t > 0 when the switch is at position ‘b’. In Figure 2, the 960 ohms and the 480 ohms resistor are connected in parallel. Therefore, the equivalent resistance for the parallel connection is, 960 Ω | | 480 Ω = 960 Ω × 480 Ω 960 Ω + 480 Ω = 320 Ω The modified circuit is shown in Figure 3. Substitute 320 for R and 0.5 m for L in equation (2) as follows, α = 320 2 × 0.5 × 10 − 3 { ∵ 1 m = 10 − 3 } = 320000 rad s Squaring the above expression as follows, α 2 = 320000 2 rad s Substitute 12.5 n for C and 0.5 m for L in equation (1) as follows, ω 0 = 1 ( 0.5 × 10 − 3 ) ( 12.5 × 10 − 9 ) { ∵ 1 m = 10 − 3 , 1 n = 10 − 9 } ω 0 = 400000 rad s Squaring the above expression as follows, ω 0 2 = 400000 2 rad s Substitute 400000 2 for ω 0 2 and 320000 2 for α 2 in equation (3) as follows, ω d = 400000 2 − 320000 2 = 240000 rad s In Figure 3, the value of the forced response V f is, V f = 20 V As α 2 < ω 0 2 , the output response is under damped. Hence, the output equation will be as follows, v o ( t ) = V f + B 1 e − α t cos ω d t + B 2 e − α t sin ω d t (4) Substitute 320000 for α , 20 for V f , and 240000 for ω d in equation (4) as follows, v o ( t ) = 20 + B 1 e − 320000 t cos 240000 t + B 2 e − 320000 t sin 240000 t (5) Substitute 0 for t in equation (5) as follows, v o ( 0 ) = 20 + B 1 e − 320000 ( 0 ) cos 240000 ( 0 ) + B 2 e − 320000 ( 0 ) sin 240000 ( 0 ) v o ( 0 ) = 20 + B 1 { ∵ e 0 = 1 } (6) Substitute − 16 for v o ( 0 ) in equation (6) as follows, − 16 = 20 + B 1 { v o ( 0 ) = v o ( 0 − ) = − 16 V } B 1 = − 36 V Differentiate equation (5) with respect to t

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Chapter 8, Problem 52P

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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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Capacitors Explained - The basics how capacitors work working principle; Author: The Engineering Mindset;https://www.youtube.com/watch?v=X4EUwTwZ110;License: Standard YouTube License, CC-BY

Find the expression for the voltage v o ( t ) for t ≥ 0 .

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Find the expression for the voltage vo(t) for t≥0.

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Chapter 8 Solutions