Question

Team Project: The following equations result when Kirchhoff's law is applied to a certain four-loop network: $$\begin{array}{l}14.7 I_{1}-25.7 I_{2}+14.7 I_{3}+19.3 I_{4}=26.2 \\\17.2 I_{1}+19.6 I_{2}-19.2 I_{3}+24.4 I_{4}=-28.3 \\\18.6 I_{1}-22.7 I_{2}+24.7 I_{3}+17.3 I_{4}=23.3 \\\27.7 I_{1}+19.6 I_{2}-33.2 I_{3}-42.3 I_{4}=25.4\end{array}$$ Find the four currents by a calculator.

Short answer

Use matrix operations on a calculator with matrices A and B to find the solution. The currents \(I_1, I_2, I_3,\) and \(I_4\) are given by the resulting solution vector.

Step-by-step solution

Set Up the Matrix Equation

Organize the system of equations into an augmented matrix that can be used for calculation. For the system of equations given, the matrix A will consist of the coefficients of the currents, and B will be a column matrix consisting of the constants on the right side of the equations. Matrix A:\[\begin{bmatrix}14.7 & -25.7 & 14.7 & 19.3 \17.2 & 19.6 & -19.2 & 24.4 \18.6 & -22.7 & 24.7 & 17.3 \27.7 & 19.6 & -33.2 & -42.3 \end{bmatrix}\]Matrix B:\[\begin{bmatrix}26.2 \-28.3 \23.3 \25.4\end{bmatrix}\]

Use a Calculator to Solve the System

Input the matrices A and B into a calculator that has the ability to do matrix operations. Select the appropriate function for solving linear systems (often 'solve', 'rref', or similar). The calculator will perform the matrix operation \( A^{-1}B \) (where \( A^{-1} \) is the inverse of Matrix A) to find the matrix of unknowns X, representing the currents \( I_1, I_2, I_3, \) and \( I_4 \).

Interpret the Solution

The calculator will display a column matrix with the values for \( I_1, I_2, I_3, \) and \( I_4 \), which are the currents in the network. Make sure to interpret the signs correctly: a positive value indicates current flow in the assumed direction, while a negative value indicates the opposite flow direction.

Key concepts

Linear Algebra

Linear algebra is a fundamental field of mathematics that is essential for understanding systems of linear equations, which are prevalent in various scientific and engineering disciplines, including electrical engineering. At its core, linear algebra deals with vectors, matrices, and linear transformations. It provides the language and framework to describe systems of equations, where each equation can be seen as a linear combination of variables.

In the context of electrical networks, linear algebra allows us to represent Kirchhoff's laws in a compact matrix form. This approach can significantly simplify the analysis and calculation of complex networks by using matrix operations to solve for unknown variables, such as currents in different branches of the circuit.

System of Equations

A system of equations is a set of simultaneous equations, each containing multiple variables, that are all true at the same time. When analyzing electrical networks, Kirchhoff's laws generate a system of equations where the unknowns are the currents or voltages within the network.

For instance, when multiple loops exist in a network, as in our example, Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) can be applied to each loop and junction to create a system that describes the behavior of the entire network. Solving this system gives us the values of the unknown currents. Kirchhoff's laws thus help predict how electric currents will distribute themselves throughout a network.

Matrix Operations

Matrix operations are computations involving matrices, including addition, multiplication, and finding the inverse of a matrix. These operations are critical for solving systems of linear equations, especially when dealing with a large number of equations and unknowns, which is often the case in complex electrical circuits.

In our example, matrix inversion and multiplication are used to solve the system. By setting up the coefficients of the currents into matrix A, and the constants into matrix B, we perform the operation A^{-1}B to find matrix X, which contains the values of the currents. The ability of modern calculators and computer software to perform these operations quickly and accurately is invaluable for electrical engineers and others working with such systems.

Electrical Networks

Electrical networks, or circuits, are formed by connecting electrical components like resistors, inductors, capacitors, and sources of electromotive force in a loop or mesh. Kirchhoff's laws are the cornerstone for analyzing electrical networks, where the sum of voltages around any closed loop is zero (KVL), and the sum of currents at any junction is zero (KCL).

By applying these laws to different parts of a network, we can create a system of linear equations that reveals the currents and voltages within the network. Understanding the interplay of these elements and applying linear algebra and matrix operations is crucial for anyone looking to dissect complex networks and troubleshoot or optimize their performance.