Problem 74 - College Algebra and Calculus: An Applied Approach

Question

Circuit Analysis In Exercises 73 and 74, consider the circuit shown in the figure. The currents \(I_{1}, I_{2}\), and \(I_{3}\), in amperes, are the solution of the system of linear equations \(\left\\{\begin{aligned} 2 I_{1} \quad+4 I_{3}=& E_{1} \\ I_{2}+4 I_{3}=& E_{2} \\ I_{1}+I_{2}-I_{3}=& 0 \end{aligned}\right.\) where \(E_{1}\) and \(E_{2}\) are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. \(E_{1}=24\) volts, \(E_{2}=23\) volts

Step-by-Step Solution

Answer

The solutions for the currents are \(I_{1} = 4\) Amperes, \(I_{2} = 5\) Amperes, and \(I_{3} = 7\) Amperes

Step 1: Formulate the Coefficient matrix

The coefficient matrix for these equations (in order of \(I_{1}, I_{2}, I_{3}\)) is \[\[2, 0, 4],[0, 1, 4],[1, 1, -1]\]

Step 2: Compute the inverse of this Matrix

The inverse of the matrix is computed as \[\[-0.15, 0.20, 0.80],[0.20, 0.20, -0.20],[0.10, 0.20, -0.20]\]

Step 3: Multiply the inverse matrix by the voltage matrix

Multiplying the inverse matrix by the voltage matrix (\[24,23\]), we get \(I_{1} = 4, I_{2} = 5, I_{3} = 7\). This is the solution for the system of linear equations.

Key Concepts

System of Linear Equations

Understanding how to solve a system of linear equations is essential when analyzing electrical circuits, particularly when determining the currents flowing through the circuit. A system of linear equations is a set of equations where each equation is linear, which means that the variables are only to the first power and there are no products of variables. In circuit analysis, these equations represent the relationship between electrical currents, resistances, and voltages based on Kirchhoff's laws.

For instance, the given system of equations models a circuit with three unknown currents \(I_1, I_2, I_3\) influenced by two voltages \(E_1, E_2\). The aim is to find the values of the currents. This involves formulating the equations as per the given circuit, defining the relationship between currents and voltages, and finally solving for the unknowns, which is commonly done using matrix operations.

Coefficient Matrix

The coefficient matrix is a crucial concept in linear algebra and is highly important when solving a system of linear equations through matrix methods. It is a matrix composed solely of the coefficients of the variables in the system. Each row in the coefficient matrix corresponds to an individual equation from the system, and each column relates to a specific variable.

In our circuit analysis problem, the coefficient matrix reflects the relationship between the currents flowing in the circuit. For the given equations, the coefficient matrix is arranged by placing the coefficients of \(I_1, I_2, I_3\) from each equation into rows. Thus, the coefficient matrix effectively encapsulates the structure of the circuit's equations and becomes the foundation for solving the system using matrix algebra.

Matrix Inverse

The matrix inverse is a complex yet fundamental concept used in solving systems of linear equations. When a matrix is multiplied by its inverse, the result is the identity matrix, similar to how a number multiplied by its reciprocal equals 1. However, not all matrices have inverses; a matrix must be square (same number of rows and columns) and non-singular (having a determinant not equal to zero).

To solve for the currents in our electrical circuit problem, we need to multiply the inverse of the coefficient matrix by the voltage matrix. The process of finding the matrix inverse involves several steps that are computationally intensive and usually handled by calculators or computer software for larger matrices.

Electrical Currents and Voltages

In the realm of electrical engineering, understanding the relationship between currents and voltages is vital. Ohm's Law and Kirchhoff's laws are foundational principles that help us describe how currents flow through a network in response to applied voltages. Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) are used to write the equations that form the system of linear equations we solve in circuit analysis.

For a given application, like the textbook problem we have here, knowing the voltages (\(E_1, E_2\)) allows us to set up equations reflecting how the current flows through the circuit. Once we have those equations, we can apply methods such as matrix inversion to solve for the unknown currents, providing critical insights into the circuit's behavior.