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Show that a system of differential equations that describes the currents i(t) and i3(t) in the electrical network shown in the figure below is di 3 dt q= so diz L dt di 2 -R₁ dt E(t) = Li₁' + L Ai₁ E diz -R1 dt +L Then 13 = q' = CR₁12 - CR₂13' + R₂ 00000 L diz di3 dt dt Using Kirchhoff's first law we have i₁ = 12 + 13. By Kirchhoff's second law, on each loop we have E(t) = Li₁' + +L + R₂ so that in terms of C, R₁, R₂, 12 and 13 we have the following. di 3 dt + di3 dt + R₁₂ = E(t) iz R₂ 2/3 R₁ + R₁/2 = E(t) + = 0. = 0. IC

Show that a system of differential equations that describes the currents i(t) and i3(t) in the electrical network shown in the figure below is di 3 dt q= so diz L dt di 2 -R₁ dt E(t) = Li₁' + L Ai₁ E diz -R1 dt +L Then 13 = q' = CR₁12 - CR₂13' + R₂ 00000 L diz di3 dt dt Using Kirchhoff's first law we have i₁ = 12 + 13. By Kirchhoff's second law, on each loop we have E(t) = Li₁' + +L + R₂ so that in terms of C, R₁, R₂, 12 and 13 we have the following. di 3 dt + di3 dt + R₁₂ = E(t) iz R₂ 2/3 R₁ + R₁/2 = E(t) + = 0. = 0. IC

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
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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**Title: Differential Equations in Electrical Networks** **Introduction:** The goal is to show that a system of differential equations describes the currents \( i_2(t) \) and \( i_3(t) \) in the electrical network. **Differential Equations:** For the network shown below, the system of differential equations is given by: \[ L \frac{di_2}{dt} + L \frac{di_3}{dt} + R_1 i_2 = E(t) \] \[ -R_1 \frac{di_2}{dt} + R_2 \frac{di_3}{dt} + \frac{1}{C} i_3 = 0 \] **Network Diagram:** - The diagram includes an inductor \( L \), resistors \( R_1 \) and \( R_2 \), a capacitor \( C \), and an electromotive force (EMF) \( E \). - Currents \( i_1, i_2, \) and \( i_3 \) flow through different branches of the circuit. **Kirchhoff's Laws:** Using Kirchhoff’s first law: \[ i_1 = i_2 + i_3 \] By Kirchhoff’s second law, for each loop, we have: \[ E(t) = L \dot{i}_1 + \dots + \frac{q}{C} \] \[ E(t) = L \dot{i}_1 + \dots \] **Charge Equation:** In terms of \( C, R_1, R_2, i_2, \) and \( i_3 \), the charge \( q \) is: \[ q = \dots \] **Current Relationship:** \[ i_3 = \dot{q} = CR_1 i_2 - CR_2 i'_3 \] **Conclusion:** The derived equations connect the changes in current and charge in the circuit components, adhering to Kirchhoff's laws to solve for currents \( i_2(t) \) and \( i_3(t) \) effectively.
Transcribed Image Text:**Title: Differential Equations in Electrical Networks** **Introduction:** The goal is to show that a system of differential equations describes the currents \( i_2(t) \) and \( i_3(t) \) in the electrical network. **Differential Equations:** For the network shown below, the system of differential equations is given by: \[ L \frac{di_2}{dt} + L \frac{di_3}{dt} + R_1 i_2 = E(t) \] \[ -R_1 \frac{di_2}{dt} + R_2 \frac{di_3}{dt} + \frac{1}{C} i_3 = 0 \] **Network Diagram:** - The diagram includes an inductor \( L \), resistors \( R_1 \) and \( R_2 \), a capacitor \( C \), and an electromotive force (EMF) \( E \). - Currents \( i_1, i_2, \) and \( i_3 \) flow through different branches of the circuit. **Kirchhoff's Laws:** Using Kirchhoff’s first law: \[ i_1 = i_2 + i_3 \] By Kirchhoff’s second law, for each loop, we have: \[ E(t) = L \dot{i}_1 + \dots + \frac{q}{C} \] \[ E(t) = L \dot{i}_1 + \dots \] **Charge Equation:** In terms of \( C, R_1, R_2, i_2, \) and \( i_3 \), the charge \( q \) is: \[ q = \dots \] **Current Relationship:** \[ i_3 = \dot{q} = CR_1 i_2 - CR_2 i'_3 \] **Conclusion:** The derived equations connect the changes in current and charge in the circuit components, adhering to Kirchhoff's laws to solve for currents \( i_2(t) \) and \( i_3(t) \) effectively.
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