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v,(t) t = 0 Find the current i, (t) 2 mA (4)2 k2S 2 k2 3 kΩ 6 V 20 mH and the node voltage v,(t)

v,(t) t = 0 Find the current i, (t) 2 mA (4)2 k2S 2 k2 3 kΩ 6 V 20 mH and the node voltage v,(t)

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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**Circuit Analysis Problem** In the given circuit diagram, you are tasked to find the current \( i_L(t) \) and the node voltage \( v_I(t) \). **Circuit Elements Description:** - **Current Source:** 2 mA - **Resistors:** - First resistor: 2 kΩ - Second resistor: 2 kΩ (in series with the first) - Third resistor: 3 kΩ (in series with the inductor) - **Inductor:** 20 mH - **Voltage Source (step function):** 6 V at \( t = 0 \) The circuit consists of a parallel connection between the current source and the first 2 kΩ resistor. This combination is in series with another 2 kΩ resistor. The node between these resistors and the next components is where \( v_I(t) \) is measured. Further along, there is a series connection of a 3 kΩ resistor and a 20 mH inductor, which are in series with a voltage source that steps from 0 V to 6 V at time \( t = 0 \). The current flowing through the inductor is denoted as \( i_L(t) \). **Objective:** Identify: - **\( i_L(t) \):** Current through the inductor. - **\( v_I(t) \):** Voltage at the specified node in the circuit. Use appropriate methodologies such as Kirchhoff’s laws, Thevenin’s theorem, or Laplace transforms to solve for the required quantities. **Equation Setup:** - Apply Kirchhoff’s Current Law (KCL) or Kirchhoff’s Voltage Law (KVL) where applicable. - Consider initial conditions due to the step function in the voltage source. **Steps for Solution:** 1. **Initial Analysis:** Calculate initial conditions at \( t = 0 \). 2. **Steady-State Analysis:** Determine the behavior as \( t \to \infty \). 3. **Transient Analysis:** Consider how the circuit transitions from the initial to steady state. This educational exercise involves steps that can help understand transient analysis in RL circuits.
Transcribed Image Text:**Circuit Analysis Problem** In the given circuit diagram, you are tasked to find the current \( i_L(t) \) and the node voltage \( v_I(t) \). **Circuit Elements Description:** - **Current Source:** 2 mA - **Resistors:** - First resistor: 2 kΩ - Second resistor: 2 kΩ (in series with the first) - Third resistor: 3 kΩ (in series with the inductor) - **Inductor:** 20 mH - **Voltage Source (step function):** 6 V at \( t = 0 \) The circuit consists of a parallel connection between the current source and the first 2 kΩ resistor. This combination is in series with another 2 kΩ resistor. The node between these resistors and the next components is where \( v_I(t) \) is measured. Further along, there is a series connection of a 3 kΩ resistor and a 20 mH inductor, which are in series with a voltage source that steps from 0 V to 6 V at time \( t = 0 \). The current flowing through the inductor is denoted as \( i_L(t) \). **Objective:** Identify: - **\( i_L(t) \):** Current through the inductor. - **\( v_I(t) \):** Voltage at the specified node in the circuit. Use appropriate methodologies such as Kirchhoff’s laws, Thevenin’s theorem, or Laplace transforms to solve for the required quantities. **Equation Setup:** - Apply Kirchhoff’s Current Law (KCL) or Kirchhoff’s Voltage Law (KVL) where applicable. - Consider initial conditions due to the step function in the voltage source. **Steps for Solution:** 1. **Initial Analysis:** Calculate initial conditions at \( t = 0 \). 2. **Steady-State Analysis:** Determine the behavior as \( t \to \infty \). 3. **Transient Analysis:** Consider how the circuit transitions from the initial to steady state. This educational exercise involves steps that can help understand transient analysis in RL circuits.
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