Math Problem Statement

Write a matrix equation that determines the loop currents for the given circuit using Kirchhoff's Voltage Law (KVL).

Solution

In this circuit, we are asked to write a matrix equation for the loop currents I1,I2,I3,I4I_1, I_2, I_3, I_4 based on the given resistances and voltage sources. We will use Kirchhoff's Voltage Law (KVL) for each loop to set up the system of equations. This law states that the sum of the potential differences (voltage) in any closed loop must be zero.

Loop Currents and Loops:

Let the loop currents I1,I2,I3,I4I_1, I_2, I_3, I_4 correspond to the clockwise currents in the four loops, and we assume:

  1. Loop 1 contains the current I1I_1,
  2. Loop 2 contains I2I_2,
  3. Loop 3 contains I3I_3,
  4. Loop 4 contains I4I_4.

Setting up the KVL Equations

We will sum the voltages in each loop, considering resistors shared between adjacent loops and the effect of voltage sources:

  1. For Loop 1: 20V6Ω(I1I2)5ΩI1=020V - 6\Omega(I_1 - I_2) - 5\Omega I_1 = 0 Simplified: (6+5)I16I2=20(6 + 5) I_1 - 6 I_2 = 20 11I16I2=2011 I_1 - 6 I_2 = 20

  2. For Loop 2: 34V6Ω(I2I1)2Ω(I2I3)=034V - 6\Omega(I_2 - I_1) - 2\Omega(I_2 - I_3) = 0 Simplified: 6I1+(6+2)I22I3=34-6 I_1 + (6 + 2) I_2 - 2 I_3 = 34 6I1+8I22I3=34-6 I_1 + 8 I_2 - 2 I_3 = 34

  3. For Loop 3: 19V2Ω(I3I2)3Ω(I3I4)=019V - 2\Omega(I_3 - I_2) - 3\Omega(I_3 - I_4) = 0 Simplified: 2I2+(2+3)I33I4=19-2 I_2 + (2 + 3) I_3 - 3 I_4 = 19 2I2+5I33I4=19-2 I_2 + 5 I_3 - 3 I_4 = 19

  4. For Loop 4: 23V3Ω(I4I3)2ΩI4=023V - 3\Omega(I_4 - I_3) - 2\Omega I_4 = 0 Simplified: 3I3+(3+2)I4=23-3 I_3 + (3 + 2) I_4 = 23 3I3+5I4=23-3 I_3 + 5 I_4 = 23

Matrix Equation Form

We can now represent these equations in matrix form AI=V\mathbf{A} \mathbf{I} = \mathbf{V}:

11 & -6 & 0 & 0 \\ -6 & 8 & -2 & 0 \\ 0 & -2 & 5 & -3 \\ 0 & 0 & -3 & 5 \end{pmatrix} \begin{pmatrix} I_1 \\ I_2 \\ I_3 \\ I_4 \end{pmatrix} = \begin{pmatrix} 20 \\ 34 \\ 19 \\ 23 \end{pmatrix}$$ ### Final Matrix Form: $$\begin{pmatrix} 11 & -6 & 0 & 0 \\ -6 & 8 & -2 & 0 \\ 0 & -2 & 5 & -3 \\ 0 & 0 & -3 & 5 \end{pmatrix} \begin{pmatrix} I_1 \\ I_2 \\ I_3 \\ I_4 \end{pmatrix} = \begin{pmatrix} 20 \\ 34 \\ 19 \\ 23 \end{pmatrix}$$ This matrix equation can now be solved using methods such as Gaussian elimination or using matrix inversion to find the loop currents $$I_1, I_2, I_3, I_4$$. --- Would you like details on solving this system or any other questions? Here are some related questions for deeper understanding: 1. How can we apply Gaussian elimination to solve this matrix equation? 2. What is the physical significance of each term in the KVL equations? 3. How do we handle dependent loops with shared resistances? 4. What happens if we reverse the polarity of one of the voltage sources? 5. How can we confirm that our loop current directions are correct? **Tip**: When solving electrical circuits using KVL, always ensure consistent direction for loop currents to avoid sign confusion.

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Math Problem Analysis

Mathematical Concepts

Electrical Engineering
Linear Algebra
Circuit Analysis

Formulas

Kirchhoff's Voltage Law (KVL): Σ Voltage = 0 in any closed loop
Matrix representation of linear equations

Theorems

Kirchhoff's Voltage Law (KVL)

Suitable Grade Level

Undergraduate (Electrical Engineering)