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In the circuit in Fig. P7.8, the voltage and current expressions are v = 400e-51 V, t2 0; i = 10e A, t2 0. Find a) R. b) 7 (in milliseconds). c) L. d) the initial energy stored in the inductor. e) the time (in milliseconds) it takes to dissipate 80% of the initial stored energy. Figure P7.8 L. R.

In the circuit in Fig. P7.8, the voltage and current expressions are v = 400e-51 V, t2 0; i = 10e A, t2 0. Find a) R. b) 7 (in milliseconds). c) L. d) the initial energy stored in the inductor. e) the time (in milliseconds) it takes to dissipate 80% of the initial stored energy. Figure P7.8 L. R.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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### Problem 7.8: Analysis of an RL Circuit **Given:** In the circuit shown in Figure P7.8, the switch has been open for a long time. The voltage and current expressions are: \[ v = 400e^{-5t} \, \text{V,} \quad t \geq 0^{+} \] \[ i = 10e^{-5t} \, \text{A,} \quad t \geq 0^{+} \] **Figure P7.8:** - A simple RL circuit with a resistor \( R \) and an inductor \( L \) in series with a voltage source. The current \( i \) flows clockwise. **Tasks:** 1. **Find \( R \):** Determine the resistance in the circuit. 2. **Find \( \tau \) (in milliseconds):** Calculate the time constant \( \tau \) of the circuit, where \( \tau = \frac{L}{R} \). 3. **Find \( L \):** Determine the inductance of the inductor. 4. **Calculate the initial energy stored in the inductor:** Use the initial current to find the energy, given by the formula: \[ W = \frac{1}{2} L i^2 \] 5. **Determine the time (in milliseconds) it takes to dissipate 80% of the initial stored energy:** Calculate the time required for the energy in the inductor to decrease to 20% of its initial value. ### Explanation of the Diagram - **Diagram Setup:** - The circuit layout is a simple series connection. - The inductor \( L \) and resistor \( R \) are connected, with the direction of current marked. - The circuit is exposed to a decaying exponential voltage function over time. ### Concepts Involved - **RL Circuit Dynamics:** Understanding how voltage and current decrease over time in an RL circuit. - **Time Constant \( \tau \):** Governs the rate at which current and voltage change. - **Energy Storage in Inductors:** Calculating how much energy is stored and how it dissipates over time. This exercise helps in understanding exponential responses in circuits and the energy dynamics associated with inductors.
Transcribed Image Text:### Problem 7.8: Analysis of an RL Circuit **Given:** In the circuit shown in Figure P7.8, the switch has been open for a long time. The voltage and current expressions are: \[ v = 400e^{-5t} \, \text{V,} \quad t \geq 0^{+} \] \[ i = 10e^{-5t} \, \text{A,} \quad t \geq 0^{+} \] **Figure P7.8:** - A simple RL circuit with a resistor \( R \) and an inductor \( L \) in series with a voltage source. The current \( i \) flows clockwise. **Tasks:** 1. **Find \( R \):** Determine the resistance in the circuit. 2. **Find \( \tau \) (in milliseconds):** Calculate the time constant \( \tau \) of the circuit, where \( \tau = \frac{L}{R} \). 3. **Find \( L \):** Determine the inductance of the inductor. 4. **Calculate the initial energy stored in the inductor:** Use the initial current to find the energy, given by the formula: \[ W = \frac{1}{2} L i^2 \] 5. **Determine the time (in milliseconds) it takes to dissipate 80% of the initial stored energy:** Calculate the time required for the energy in the inductor to decrease to 20% of its initial value. ### Explanation of the Diagram - **Diagram Setup:** - The circuit layout is a simple series connection. - The inductor \( L \) and resistor \( R \) are connected, with the direction of current marked. - The circuit is exposed to a decaying exponential voltage function over time. ### Concepts Involved - **RL Circuit Dynamics:** Understanding how voltage and current decrease over time in an RL circuit. - **Time Constant \( \tau \):** Governs the rate at which current and voltage change. - **Energy Storage in Inductors:** Calculating how much energy is stored and how it dissipates over time. This exercise helps in understanding exponential responses in circuits and the energy dynamics associated with inductors.
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