Problem 45 - College Algebra with Modeling and Visualization

Question

Electricity \(\quad\) Complex numbers are used in the study of electrical circuits. Impedance \(Z\) (or the opposition to the flow of electricity. voltage \(V\) and current \(I\) can all be represented by complex numbers. They are related by the equation \(Z=\frac{V}{I} .\) Find the value of the missing variable. $$ I=1+2 i \quad Z=3-4 i $$

Step-by-Step Solution

Answer

The voltage \( V \) is \(-5 + 2i\).

Step 1: Recall the relationship between impedance, voltage, and current

Given the formula for impedance, voltage, and current in a circuit as \[ Z = \frac{V}{I}. \]Our goal is to find the missing variable, which is the voltage \( V \). We are given that \( I = 1 + 2i \) and \( Z = 3 - 4i \).

Step 2: Solve for voltage V using the given Z and I

To find \( V \), we rearrange the formula:\[ V = Z \cdot I. \]Substitute the values of \( Z \) and \( I \) from the problem:\[ V = (3 - 4i) \cdot (1 + 2i). \]

Step 3: Multiply the complex numbers

Perform the multiplication of the complex numbers:\[ V = (3 - 4i)(1 + 2i) = 3(1) + 3(2i) - 4i(1) - 4i(2i). \]

Step 4: Simplify the expression

Simplify the expression using the definition of imaginary numbers, where \( i^2 = -1 \):- \( = 3 + 6i - 4i - 8i^2. \)- Since \(i^2 = -1\), replace \(-8i^2\) with \(8\).- Combine like terms:\[ = 3 - 8 + 6i - 4i = -5 + 2i. \]

Step 5: Write down the final expression for V

Combining all simplified results:\[ V = -5 + 2i. \]This is the value of the voltage represented as a complex number.

Key Concepts

Impedance

Impedance is a fundamental concept in electrical circuits, defining the opposition a circuit offers to the flow of alternating current (AC). Unlike pure resistance found in direct current (DC) circuits, impedance encompasses both resistance and reactance. Reactance occurs due to capacitors and inductors present in a circuit, and it can both store and release energy over time.
One important trait of impedance is its expression as a complex number. This is where the use of complex numbers becomes necessary in electrical engineering. Impedance is denoted by the symbol \( Z \), represented as a combination of real and imaginary numbers. The magnitude of impedance indicates the absolute opposition, while the angle represents the phase difference between voltage and current.

• Real Part: Corresponds to the resistive portion.
• Imaginary Part: Associated with the reactive components like inductors and capacitors.

Understanding impedance allows engineers to design circuits that manage the flow of AC effectively, ensuring stability and efficiency.

Electrical Circuits

In the world of electrical engineering, circuits are pathways that facilitate the flow of electric current. They comprise various elements like resistors, capacitors, inductors, and sources of voltage. The study of circuits involves analyzing how electricity is distributed across these components.
Electrical circuits can be categorized into two primary types:

• DC Circuits: These circuits have currents flowing in one direction, often powered by batteries.
• AC Circuits: Currents reverse direction periodically, suited for domestic and industrial power supply.

An essential part of analyzing electrical circuits involves using complex numbers to describe the relationship between voltage, current, and impedance. With AC circuits, impedance plays a crucial role in determining how voltage and current interact over time.
Each component within a circuit influences its total impedance, contributing either resistive or reactive properties. By calculating total impedance, engineers can predict circuit behavior, determining factors like power dissipation, voltage drops, and current distribution.

Voltage and Current

Voltage and current are key elements in the functioning of electrical circuits. Voltage, often described as electrical pressure, is the potential difference between two points, prompting current flow.
Voltage is measured in volts (V), and in the context of AC circuits, it can be represented as a complex number to account for phase differences. It results from sources like batteries or generators, providing the necessary push to move electrons through a circuit.
Current, measured in amperes (A), represents the flow rate of electric charge. It is directly affected by both voltage and impedance. The relationship between these quantities is succinctly described by Ohm's Law for AC, expressed as \( V = IZ \).

• Direct Proportionality: Higher voltage leads to higher current if impedance remains constant.
• Complex Relationship: Current phase can lead or lag voltage depending on the impedance's reactive components.

Understanding voltage and current interplay is crucial for effective circuit design and analysis. It ensures that electrical systems function as desired, safely and efficiently.