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Problem 4: Derive an expression for the equivalent inductance for the circuit shown in Figure below: HINT: Write voltage equation and solve further to get Leg L1 Leq M 000

Problem 4: Derive an expression for the equivalent inductance for the circuit shown in Figure below: HINT: Write voltage equation and solve further to get Leg L1 Leq M 000

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 4: Equivalent Inductance Derivation **Objective:** Derive an expression for the equivalent inductance (\(L_{eq}\)) for the circuit shown in the figure. **Hint:** Write the voltage equation and solve further to obtain \(L_{eq}\). **Circuit Description:** The circuit diagram includes two inductors, \(L_1\) and \(L_2\), with mutual inductance \(M\) between them. The voltage across the circuit is given as \(v(t)\). The voltages across \(L_1\) and \(L_2\) are denoted as \(v_1(t)\) and \(v_2(t)\) respectively. - **Inductors:** - \(L_1\): First inductor - \(L_2\): Second inductor - **Mutual Inductance:** - \(M\): Mutual inductance between \(L_1\) and \(L_2\) - **Voltages:** - \(v_1(t)\): Voltage across \(L_1\) - \(v_2(t)\): Voltage across \(L_2\) - \(v(t)\): Total voltage - **Equivalent Inductance:** - \(L_{eq}\): Equivalent inductance of the entire circuit **Approach:** 1. **Voltage Equations:** - Write the voltage equations for each inductor considering both self-inductance and mutual inductance effects. 2. **Solve for \(L_{eq}\):** - Combine the individual voltage equations to express in terms of \(L_{eq}\). This derivation helps understand the impact of mutual inductance on the equivalent inductance in coupled inductors.
Transcribed Image Text:### Problem 4: Equivalent Inductance Derivation **Objective:** Derive an expression for the equivalent inductance (\(L_{eq}\)) for the circuit shown in the figure. **Hint:** Write the voltage equation and solve further to obtain \(L_{eq}\). **Circuit Description:** The circuit diagram includes two inductors, \(L_1\) and \(L_2\), with mutual inductance \(M\) between them. The voltage across the circuit is given as \(v(t)\). The voltages across \(L_1\) and \(L_2\) are denoted as \(v_1(t)\) and \(v_2(t)\) respectively. - **Inductors:** - \(L_1\): First inductor - \(L_2\): Second inductor - **Mutual Inductance:** - \(M\): Mutual inductance between \(L_1\) and \(L_2\) - **Voltages:** - \(v_1(t)\): Voltage across \(L_1\) - \(v_2(t)\): Voltage across \(L_2\) - \(v(t)\): Total voltage - **Equivalent Inductance:** - \(L_{eq}\): Equivalent inductance of the entire circuit **Approach:** 1. **Voltage Equations:** - Write the voltage equations for each inductor considering both self-inductance and mutual inductance effects. 2. **Solve for \(L_{eq}\):** - Combine the individual voltage equations to express in terms of \(L_{eq}\). This derivation helps understand the impact of mutual inductance on the equivalent inductance in coupled inductors.
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