Explanation Formula used: The expression for the equivalent resistor connected in series is as follows, R e q = R 1 + R 2 + R 3 ..... + R N (1) Here, R e q is the equivalent resistance between two points. R 1 , R 2 , R 3 …, R N are the resistance of the resistor connected in series. The expression for the time constant is as follows. τ = R e q C (2) Here, τ is the time constant. R e q is the equivalent resistance across the capacitor. C is the capacitance of 8 mF capacitor. The expression for the current flowing in the capacitor is as follows. i C = C d v d t (3) Here, v is the voltage across the capacitor and i C is the current flowing through the capacitor. The expression for the final response of the voltage is as follows. v ( t ) = v f ( t ) + v n ( t ) (4) Here, v ( t ) is the final response of the voltage, v f ( t ) is the forced response of the voltage and v n ( t ) is the natural response of the voltage. The expression for the natural response of the voltage is as follows, v n ( t ) = A e − t τ (5) Here, A is the arbitrary constant. The expression for the voltage is as follows v 0 = i 0 R 1 (6) Here, v 0 is the voltage across the independent source after source transformation, i 0 is the current through 5 cos 3 t u ( t ) A independent source and R 1 is the resistance of 4.7 Ω resistor. Calculation: The unit-step forcing function as a function of time which is zero for all values of its argument less than zero and which is unity for all positive values of its argument. u ( t − t 0 ) = { 0 t < t 0 1 t > t 0 Here, t 0 is the point at which argument of the unit step function become zero. So, the independent current source 5 cos 3 t u ( t ) A is 0 for t < 0 and 5 cos 3 t A for t > 0 . The redrawn circuit diagram is given in Figure 1 for t > 0 . Refer to the redrawn Figure 1, Source transformation is the process of simplifying a circuit solution, especially with mixed sources, by transforming voltage sources into current sources, and vice versa. The voltage source in series with resistor is converted into current source with a same resistor connected in parallel. Substitute 5 cos 3 t A for i 0 and 4.7 Ω for R 1 in equation (6), v 0 = ( 5 cos 3 t A ) ( 4.7 Ω ) = 23.5 cos 3 t V The redrawn circuit diagram is given in Figure 2. Refer to the redrawn Figure 2, Apply Kirchhoff’s voltage law in the circuit, v 0 = i C R 1 + i C R 2 + v (6) Substitute 4.7 Ω for R 1 and 10 Ω for R 2 and 23.5 cos 3 t V for v 0 in equation (6), 23.5 cos 3 t V = ( 4.7 Ω ) i C + ( 10 Ω ) i C + v 23.5 cos 3 t V = ( 14.7 Ω ) i C + v (7) Substitute C d v d t for i C in equation (7), 23.5 cos 3 t V = ( 14.7 Ω ) C d v d t + v (8) Substitute 22 μ F for C in equation (8), 23.5 cos 3 t V = ( 14.7 Ω ) ( 22 μ F ) d v d t + v 23.5 cos 3 t V = ( 14.7 Ω ) ( 22 × 10 − 6 F ) d v d t + v { ∵ 1 μ F = 10 − 6 F } Rearrange equation (7), 23.5 cos 3 t V ( 14.7 Ω ) ( 22 × 10 − 6 F ) = d v d t + v ( 14.7 Ω ) ( 22 × 10 − 6 F ) 72.67 × 10 3 cos 3 t V = d v d t + 3.092 × 10 3 v d v d t + 3.092 × 10 3 v = 72.67 × 10 3 cos 3 t V (9) The redrawn circuit diagram is given in Figure 3. Refer to the redrawn Figure 3, 10 Ω and 4.7 Ω resistor is connected in series. Substitute 10 Ω for R 1 and 4.7 Ω for R 2 in equation (1), R e q = 10 Ω + 4.7 Ω = 14.7 Ω So, the equivalent resistance across the capacitor is 14.7 Ω . The redrawn circuit diagram is given in Figure 4, Refer to the redrawn Figure 4, Substitute 14.7 Ω for R e q and 22 μ F for C in equation (2), τ = ( 14

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ISBN: 9781260540666

Author: Hayt

Publisher: MCG CUSTOM

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Chapter 8.8, Problem 15P

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Find the capacitor voltage v in the circuit for t>0.

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ENGINEERING CIRCUIT...(LL)>CUSTOM PKG.<

Ch. 8.1 - For the circuit in Fig. 8.2, what value of...Ch. 8.1 - Noting carefully how the circuit changes once the...Ch. 8.2 - In a source-free series RC circuit, find the...Ch. 8.3 - Prob. 4P Ch. 8.3 - Prob. 5P Ch. 8.4 - Prob. 6P Ch. 8.4 - Prob. 7P Ch. 8.4 - Prob. 8P Ch. 8.5 - Evaluate each of the following at t = 0.8: (a)...Ch. 8.6 - For the circuit of Fig. 8.37, find vc(t) at t...

Ch. 8.7 - Prob. 11P Ch. 8.7 - The voltage source 60 40u(t) V is in series with...Ch. 8.7 - Prob. 13P Ch. 8.8 - Prob. 14P Ch. 8.8 - Prob. 15P Ch. 8 - A source-free RC circuit has R = 4 k and C = 22 F,...Ch. 8 - A source-free RC circuit has v(0) = 12 V and R =...Ch. 8 - The resistor in the circuit of Fig. 8.51 has been...Ch. 8 - Prob. 4E Ch. 8 - Prob. 5E Ch. 8 - Prob. 6E Ch. 8 - Prob. 7E Ch. 8 - Prob. 8E Ch. 8 - Prob. 9E Ch. 8 - The switch in Fig. 8.56 has been closed for a long...Ch. 8 - For the circuit in Fig. 8.56, find (a) the total...Ch. 8 - Design a capacitor-based circuit that can achieve...Ch. 8 - (a) Graph the function f (t) = 10e2t over the...Ch. 8 - The current i(t) flowing through a 1 k resistor is...Ch. 8 - Radiocarbon dating has a similar exponential time...Ch. 8 - For the circuit of Fig. 8.4, compute the time...Ch. 8 - Design a circuit which will produce a current of 1...Ch. 8 - Prob. 18E Ch. 8 - Prob. 19E Ch. 8 - Referring to the circuit shown in Fig. 8.11,...Ch. 8 - Prob. 21E Ch. 8 - With the assumption that the switch in the circuit...Ch. 8 - The switch in Fig. 8.57 has been closed since...Ch. 8 - The switch in the circuit of Fig. 8.58 has been...Ch. 8 - Assuming the switch initially has been open for a...Ch. 8 - (a) Obtain an expression for v(t), the voltage...Ch. 8 - For the circuit of Fig. 8.61, determine ix, iL,...Ch. 8 - Prob. 28E Ch. 8 - Prob. 29E Ch. 8 - Prob. 30E Ch. 8 - Prob. 31E Ch. 8 - (a) Obtain an expression for vx as labeled in the...Ch. 8 - Prob. 33E Ch. 8 - Prob. 34E Ch. 8 - Prob. 35E Ch. 8 - Prob. 36E Ch. 8 - Prob. 37E Ch. 8 - The switch in Fig. 8.70 is moved from A to B at t...Ch. 8 - Prob. 39E Ch. 8 - Prob. 40E Ch. 8 - Evaluate the following functions at t = 1, 0, and...Ch. 8 - Prob. 42E Ch. 8 - Prob. 43E Ch. 8 - Prob. 44E Ch. 8 - You can use MATLAB to represent the unit-step...Ch. 8 - With reference to the circuit depicted in Fig....Ch. 8 - For the circuit given in Fig. 8.75, (a) determine...Ch. 8 - Prob. 48E Ch. 8 - Prob. 49E Ch. 8 - You build a portable solar charging circuit...Ch. 8 - The switch in the circuit of Fig. 8.78 has been...Ch. 8 - The switch in the circuit of Fig. 8.78 has been...Ch. 8 - Prob. 53E Ch. 8 - Prob. 54E Ch. 8 - Prob. 55E Ch. 8 - For the circuit represented in Fig. 8.82, (a)...Ch. 8 - Prob. 58E Ch. 8 - Prob. 59E Ch. 8 - For the circuit given in Fig. 8.85, (a) determine...Ch. 8 - The circuit depicted in Fig. 8.86 contains two...Ch. 8 - Prob. 62E Ch. 8 - Prob. 63E Ch. 8 - A series RL circuit has a voltage that steps from...Ch. 8 - For the two-source circuit of Fig. 8.89, note that...Ch. 8 - (a) Obtain an expression for iL as labeled in Fig....Ch. 8 - Obtain an expression for i(t) as labeled in the...Ch. 8 - Obtain an expression for i1 as indicated in Fig....Ch. 8 - Plot the current i(t) in Fig. 8.93 if (a) R = 10 ;...Ch. 8 - A dc motor can be modeled as a series RL circuit...Ch. 8 - Prob. 71E Ch. 8 - Prob. 72E Ch. 8 - A series RC sequentially switched circuit has R =...Ch. 8 - Refer to the circuit of Fig. 8.95, which contains...Ch. 8 - In the circuit of Fig. 8.95, a 3 mF capacitor is...Ch. 8 - Prob. 78E

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