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*********** ************* Determine a system of first-order differential equations that describes the currents i₂(t) and i3(t) in the electrical network shown in the figure below. E diz dt dl3 i₁ + m R₁ + R₁₂ + R₂ 42₂ R3 12+ R₁3 = E ₁ = E

*********** ************* Determine a system of first-order differential equations that describes the currents i₂(t) and i3(t) in the electrical network shown in the figure below. E diz dt dl3 i₁ + m R₁ + R₁₂ + R₂ 42₂ R3 12+ R₁3 = E ₁ = E

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Analyzing First-Order Differential Equations in Electrical Networks** **Objective:** Determine a system of first-order differential equations describing the currents \(i_2(t)\) and \(i_3(t)\) in the given electrical network. **Overview:** The circuit diagram includes: - Voltage source \(E\). - Three resistors \(R_1\), \(R_2\), and \(R_3\). - Two inductors \(L_1\) and \(L_2\). - Three currents \(i_1\), \(i_2\), and \(i_3\). **Circuit Description:** 1. **Components:** - **Voltage Source \(E\):** Supplies voltage to the circuit. - **Resistors:** - \(R_1\) is connected in series with the inductor \(L_1\). - \(R_2\) is in parallel with \(L_1\) and \(R_1\). - \(R_3\) is in series with inductor \(L_2\). - **Inductors:** - \(L_1\) is in the branch with \(R_1\) and also in the loop influenced by \(R_2\). - \(L_2\) is coupled with \(R_3\). - **Currents:** - \(i_1\) flows through \(R_1\) and \(L_1\). - \(i_2\) flows through \(R_2\). - \(i_3\) flows through \(R_3\) and \(L_2\). 2. **Connections and Branches:** - Current \(i_1\) splits into \(i_2\) and \(i_3\). - \(i_2\) flows through \(R_2\) and impacts the behavior of the inductor \(L_1\). - \(i_3\) flows through \(R_3\) and the inductor \(L_2\) completes the loop back to the source \(E\). **Equations to Determine:** Using Kirchhoff’s voltage and current laws, we need to establish differential equations to describe the behavior of currents \(i_2(t)\) and \(i_3(t)\).
Transcribed Image Text:**Title: Analyzing First-Order Differential Equations in Electrical Networks** **Objective:** Determine a system of first-order differential equations describing the currents \(i_2(t)\) and \(i_3(t)\) in the given electrical network. **Overview:** The circuit diagram includes: - Voltage source \(E\). - Three resistors \(R_1\), \(R_2\), and \(R_3\). - Two inductors \(L_1\) and \(L_2\). - Three currents \(i_1\), \(i_2\), and \(i_3\). **Circuit Description:** 1. **Components:** - **Voltage Source \(E\):** Supplies voltage to the circuit. - **Resistors:** - \(R_1\) is connected in series with the inductor \(L_1\). - \(R_2\) is in parallel with \(L_1\) and \(R_1\). - \(R_3\) is in series with inductor \(L_2\). - **Inductors:** - \(L_1\) is in the branch with \(R_1\) and also in the loop influenced by \(R_2\). - \(L_2\) is coupled with \(R_3\). - **Currents:** - \(i_1\) flows through \(R_1\) and \(L_1\). - \(i_2\) flows through \(R_2\). - \(i_3\) flows through \(R_3\) and \(L_2\). 2. **Connections and Branches:** - Current \(i_1\) splits into \(i_2\) and \(i_3\). - \(i_2\) flows through \(R_2\) and impacts the behavior of the inductor \(L_1\). - \(i_3\) flows through \(R_3\) and the inductor \(L_2\) completes the loop back to the source \(E\). **Equations to Determine:** Using Kirchhoff’s voltage and current laws, we need to establish differential equations to describe the behavior of currents \(i_2(t)\) and \(i_3(t)\).
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