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21 Find v1,v2, and v3 using node analysis ia ww 5Ω 9v +1 22 ww 1Ω V3

21 Find v1,v2, and v3 using node analysis ia ww 5Ω 9v +1 22 ww 1Ω V3

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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**Transcription and Explanation for Educational Website** --- **Objective:** Determine the voltages \( v_1, v_2, \) and \( v_3 \) using node analysis in an electrical circuit. **Circuit Description:** The diagram represents an electrical circuit with the following components: 1. **Voltage Source:** - A 9V voltage source is present in series. 2. **Resistors:** - Resistor of 1Ω connected at node \( v_1 \). - Resistor of 5Ω between nodes \( v_1 \) and \( v_2 \). - Resistor of 1Ω between nodes \( v_2 \) and \( v_3 \). - Resistor of 5Ω connected at node \( v_3 \). 3. **Current Source:** - A controlled current source depicted as \( 3i_a \), which depends on the current \( i_a \) through the 1Ω resistor connected to \( v_1 \). 4. **Nodes:** - Node \( v_1 \) is connected to a 1Ω resistor and a 5Ω resistor. - Node \( v_2 \) is connected to the 9V source, the same 5Ω resistor as part of the path, another 1Ω resistor leading to \( v_3 \), and a dependent current source. - Node \( v_3 \) is connected to a 5Ω resistor and forms part of the loop with the other resistors and voltage source. **Instructions:** 1. **Node Analysis Method:** - Identify all nodes: \( v_1, v_2, v_3 \). - Apply Kirchhoff's Current Law (KCL) at each node. - Express the currents in terms of the node voltages. - Solve the resulting equations to find the unknown voltages \( v_1, v_2, \) and \( v_3 \). This circuit involves both independent and dependent sources, which requires calculating dependent variables based on currents through specified elements like \( i_a \). **Analysis:** - **Node \( v_1 \):** Analyze the sum of currents leaving or entering the node including \( i_a \). - **Node \( v_2 \):** Include contributions from the voltage source and account for the controlled current source. - **Node \( v_
Transcribed Image Text:**Transcription and Explanation for Educational Website** --- **Objective:** Determine the voltages \( v_1, v_2, \) and \( v_3 \) using node analysis in an electrical circuit. **Circuit Description:** The diagram represents an electrical circuit with the following components: 1. **Voltage Source:** - A 9V voltage source is present in series. 2. **Resistors:** - Resistor of 1Ω connected at node \( v_1 \). - Resistor of 5Ω between nodes \( v_1 \) and \( v_2 \). - Resistor of 1Ω between nodes \( v_2 \) and \( v_3 \). - Resistor of 5Ω connected at node \( v_3 \). 3. **Current Source:** - A controlled current source depicted as \( 3i_a \), which depends on the current \( i_a \) through the 1Ω resistor connected to \( v_1 \). 4. **Nodes:** - Node \( v_1 \) is connected to a 1Ω resistor and a 5Ω resistor. - Node \( v_2 \) is connected to the 9V source, the same 5Ω resistor as part of the path, another 1Ω resistor leading to \( v_3 \), and a dependent current source. - Node \( v_3 \) is connected to a 5Ω resistor and forms part of the loop with the other resistors and voltage source. **Instructions:** 1. **Node Analysis Method:** - Identify all nodes: \( v_1, v_2, v_3 \). - Apply Kirchhoff's Current Law (KCL) at each node. - Express the currents in terms of the node voltages. - Solve the resulting equations to find the unknown voltages \( v_1, v_2, \) and \( v_3 \). This circuit involves both independent and dependent sources, which requires calculating dependent variables based on currents through specified elements like \( i_a \). **Analysis:** - **Node \( v_1 \):** Analyze the sum of currents leaving or entering the node including \( i_a \). - **Node \( v_2 \):** Include contributions from the voltage source and account for the controlled current source. - **Node \( v_
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