Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

When Kirchhoff's Laws are applied to the electrical network in the figure, the currents \(I_{1}, I_{2},\) and \(I_{3}\) (in amperes) are the solution of the system \(\left\\{\begin{aligned} I_{1}-I_{2}+I_{3} &=0 \\ 3 I_{1}+2 I_{2} &=7 \\\ 2 I_{2}+4 I_{3} &=8 \end{aligned}\right.\) Find the currents..

Step-by-Step Solution

Answer

The currents are: \(I_{1} = -1\) A, \(I_{2} = 1\) A, and \(I_{3} = 2\) A.

Step 1: Formulate the system of equations

The given system of equations is\[\begin{aligned}I_{1} - I_{2} + I_{3} &= 0 \3I_{1} + 2I_{2} &= 7 \2I_{2} + 4I_{3} &= 8\end{aligned}\]

Step 2: Simplify the system of equations

Let's simplify the third equation by dividing every term by 2. The system of equations now becomes:\[\begin{aligned}I_{1} - I_{2} + I_{3} &= 0, \3I_{1} + 2I_{2} &= 7, \I_{2} + 2I_{3} &= 4.\end{aligned}\]

Step 3: Solve for currents

One approach is substitution. From the first equation, we can express \(I_{1}\) as \(I_{1} = I_{2} - I_{3}\). Substituting this into the second equation yields \(3(I_{2} - I_{3}) + 2I_{2} = 7\), which simplifies to \(5I_{2} - 3I_{3} = 7\). Then, subtract the third equation from this resulting equation to obtain \(4I_{2} - I_{3} = 3\), which simplifies to \(I_{2} = \frac{1}{4}(3 + I_{3})\). Substituting this into the third equation yields \(I_{3} = 2\). We can substitute \(I_{3}\) into the equations to find that \(I_{2} = 1\) and \(I_{1} = -1\).

Key Concepts

Electrical Network Analysis

Electrical network analysis involves understanding and solving circuits composed of resistors, capacitors, inductors, and other electrical components. When we look at such a network, the goal is to determine unknown variables like current, voltage, and resistance.

Kirchoff's Laws are the cornerstone for analyzing these networks. There are two main laws:

• Kirchhoff's Current Law (KCL): This law states that the sum of currents entering a junction in a network equals the sum of currents leaving that junction. It ensures the conservation of charge.
• Kirchhoff's Voltage Law (KVL): This law states that the sum of electric potential differences around any closed loop in a network is equal to zero. It ensures the conservation of energy within the system.

Using these laws in tandem allows us to set up equations to solve for unknown quantities such as current values in different branches of the network.

Systems of Equations

In the exercise, Kirchhoff’s Laws help formulate a set of linear equations representing relationships in the electrical network. Solving these systems of equations requires finding values of variables that satisfy all the conditions simultaneously.

For the given network, the currents were represented by a set of three equations. Each equation comes from applying Kirchoff's Laws:

• The first equation utilizes KCL for junctions, ensuring current consistency.
• The second and third equations follow KVL for loops, setting the equal sum of voltages to zero.

This structured approach allows engineers and students alike to use mathematics to interpret physical real-world electrical systems.

Substitution Method

The substitution method is a popular algebraic technique for solving systems of equations. It involves isolating one variable and substituting its expression into other equations. This reduces the number of variables in equations, simplifying them into a form that is easier to solve.

In this problem, we expressed \(I_1\) in terms of \(I_2\) and \(I_3\) from the first equation:

\(I_1 = I_2 - I_3\).

Substituting \(I_1\) into other equations, the number of variables and equations can be reduced step by step until each unknown current finds a specific value. This method is straightforward because it systematically breaks down the problem into simpler parts, making the solution process more accessible.

Currents Calculation

Current calculation in the network is the final goal and result of analyzing a specific circuit using Kirchoff's laws. After simplifying and reducing our system of equations by substitution, we determined each current's magnitude.

From the strategic substitution and simplification process, we concluded:

• The value of \(I_3\) was directly calculated as 2 amperes.
• Using this, \(I_2\) became 1 ampere through further substitution into the simplified equations.
• \(I_1\) was then found to be -1 ampere after substituting known values back.

By following this logical chain of calculations, satisfying all initial equations and conditions, the specific values assigned to each current convey a complete understanding of the network’s behavior.