Math Problem Statement

Solve it mathematically

Discuss, how you can solve the problem in the following figure, of finding the currents and voltage at various locations in a resistor circuit by modeling the problem as A*x = b.

Solution

To solve the problem of determining currents and voltages at various locations in the resistor circuit in Figure 12.8, we will use Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) to set up a system of linear equations. This system can be written in matrix form Ax=bA \vec{x} = \vec{b}, where AA represents the coefficients of the unknown currents and voltages, x\vec{x} represents the vector of unknowns, and b\vec{b} represents the constants in the equations (like voltages provided by the batteries).

Here are the steps:

1. Assign Currents to Each Branch

Define a current in each branch of the circuit. For simplicity, let:

  • I1I_1 be the current through the branch containing the R=10ΩR = 10 \, \Omega resistor between nodes 1 and 3.
  • I2I_2 be the current through the R=5ΩR = 5 \, \Omega resistor between nodes 2 and 3.
  • I3I_3 be the current through the R=15ΩR = 15 \, \Omega resistor between nodes 4 and 3.
  • I4I_4 be the current through the R=5ΩR = 5 \, \Omega resistor between nodes 1 and 4.
  • I5I_5 be the current through the R=10ΩR = 10 \, \Omega resistor between nodes 4 and 0 (ground).
  • I6I_6 be the current through the R=20ΩR = 20 \, \Omega resistor between nodes 2 and 0.

2. Apply Kirchhoff’s Current Law (KCL)

At each node, we use the rule that the sum of currents entering a node is equal to the sum of currents leaving the node.

  • Node 1: I1+I4=I0I_1 + I_4 = I_0 (from the battery).
  • Node 2: I2+I6=I0I_2 + I_6 = I_0.
  • Node 3: ( I_1 = I

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Kirchhoff's Current Law (KCL)
Kirchhoff's Voltage Law (KVL)
Ohm's Law

Formulas

Kirchhoff's Current Law: ∑i = 0
Kirchhoff's Voltage Law: ∑ξ - ∑iR = 0
Ohm's Law: V = IR

Theorems

Kirchhoff's Current Law
Kirchhoff's Voltage Law

Suitable Grade Level

Undergraduate Electrical Engineering