Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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Nodal Analysis
In the analysis of any electric circuit, nodal analysis is the method of determining the potential difference or voltage between different nodes in an electric circuit. A node is a point where two or more branches connect each other. In the nodal analysi…
Question
**Figure 2: Two resistors from the circuit, each with a voltage potential difference across them, yielding current flows \( s_1 \) and \( f_2 \).** Diagram Explanation: - The left section shows a resistor labeled \( r_1 \) with a resistance of 50 Ω, which is connected between voltage points \( v_1 \) and \( v_2 \). The current flowing through this resistor is denoted as \( s_1 \). - The right section shows another resistor labeled \( r_2 \) with a resistance of 200 Ω, which is connected between voltage \( v_2 \) and the ground (earth), with the current through this resistor denoted as \( f_2 \). **(d) Use Ohm’s law for the two resistors (Figure 2) to write two linear equations that relate \( f_2, s_1, v_1, \) and \( v_2 \) in terms of \( r_1 \) and \( r_2 \).**
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Transcribed Image Text:**Figure 2: Two resistors from the circuit, each with a voltage potential difference across them, yielding current flows \( s_1 \) and \( f_2 \).** Diagram Explanation: - The left section shows a resistor labeled \( r_1 \) with a resistance of 50 Ω, which is connected between voltage points \( v_1 \) and \( v_2 \). The current flowing through this resistor is denoted as \( s_1 \). - The right section shows another resistor labeled \( r_2 \) with a resistance of 200 Ω, which is connected between voltage \( v_2 \) and the ground (earth), with the current through this resistor denoted as \( f_2 \). **(d) Use Ohm’s law for the two resistors (Figure 2) to write two linear equations that relate \( f_2, s_1, v_1, \) and \( v_2 \) in terms of \( r_1 \) and \( r_2 \).**
(e) Rewrite your system of equations as a single matrix equation of the form \[ B \begin{bmatrix} f \\ s \\ v \end{bmatrix} = 0. \] Keep in mind that \( f, s, v \in \mathbb{R}^2 \). Give an expression for the matrix \( B \). What are the dimensions of matrix \( B \)? (f) Are the columns of \( B \) linearly independent? Justify.
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Transcribed Image Text:(e) Rewrite your system of equations as a single matrix equation of the form \[ B \begin{bmatrix} f \\ s \\ v \end{bmatrix} = 0. \] Keep in mind that \( f, s, v \in \mathbb{R}^2 \). Give an expression for the matrix \( B \). What are the dimensions of matrix \( B \)? (f) Are the columns of \( B \) linearly independent? Justify.
Expert Solution
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Step 1: Summarize the details given:

Given Data:

Two DC resistive circuits with,

  • Voltages equals open parentheses v subscript 1 comma space v subscript 2 close parentheses
  • Currents equals open parentheses s subscript 1 comma space f subscript 2 close parentheses
  • Resistors equals open parentheses r subscript 1 comma space r subscript 2 close parentheses

To Find:

d. The linear equations relating open parentheses f subscript 2 comma v subscript 2 close parentheses in terms of open parentheses r subscript 1 comma space r subscript 2 close parentheses.

e. The expression and the dimension of B which in the form of B open square brackets table row f row s row v end table close square brackets equals 0.

f. Are the columns of the matrix B linearly independent or not?

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