Problem 52 - Algebra 2 and Trigonometry

Question

Imaginary numbers are often used in electrical engineering. The impedance \(Z\) (measured in ohms) of a circuit measures the resistance of a circuit to alternating current \((\mathrm{AC})\) electricity. For two \(\mathrm{AC}\) circuits connected in series, the total impedance is the sum \(Z_{1}+Z_{2}\) of the individual impedances. Find the total impedance if \(Z_{1}=5 i\) ohms and \(Z_{2}=8 i\) ohms.

Step-by-Step Solution

Answer

The total impedance is \(13i\) ohms.

Step 1: Understand the Concept of Impedance

Impedance is a complex quantity that combines resistance and reactance to quantify opposition to AC electricity. In this problem, impedance is given as purely imaginary numbers because the circuits are purely inductive or capacitive at this stage.

Step 2: Identify the Impedance Values

We are given two impedances with imaginary numbers: \( Z_1 = 5i \) ohms and \( Z_2 = 8i \) ohms. These signify the circuit elements' opposition value in imaginary units.

Step 3: Add the Impedances

When circuits are connected in series, their total impedance is simply the sum of the individual impedances. Therefore, calculate \( Z_{total} = Z_1 + Z_2 \).

Step 4: Perform the Calculation

Substitute the given values into the equation: \( Z_{total} = 5i + 8i \). Add the imaginary components: \( 5i + 8i = 13i \).

Step 5: Present the Result

The total impedance for the series circuit is \( Z_{total} = 13i \) ohms, indicating it's inductive or capacitive depending on the context, but for the sake of this problem, it is purely imaginary.

Key Concepts

Imaginary Numbers

Imaginary numbers are a fascinating aspect of mathematics that play a crucial role in a variety of fields, especially in electrical engineering. They are defined as numbers that when squared give a negative result.
This means they "exist" on a different axis from the usual real number line, often referred to as the imaginary axis.
One fundamental component of imaginary numbers is the unit 'i', which is defined as the square root of -1: \( i = \sqrt{-1} \). This unit allows us to quantify the otherwise abstract idea of square roots of negative numbers.

• Imaginary numbers help us in solving equations that have no real solutions when considering real numbers alone.
• They can be represented as complex numbers in combination with real numbers, as \( a + bi \), where 'a' is the real part and 'b' the imaginary component.

In practical terms, electrical engineers embrace imaginary numbers because they enable the computation of quantities like impedance in AC circuits. This use extends far beyond theory, providing a basis for real-world applications.

Impedance in AC Circuits

In alternating current (AC) circuits, impedance is a critical concept that can be seen as the circuit's opposition to the flow of AC electricity. Unlike resistance in direct current (DC) circuits, impedance is more complex because it accounts for both resistance and reactance.
Reactance happens due to elements like capacitors and inductors, which store and release energy, causing shifts in the phase between voltage and current.

• Impedance is denoted by the symbol \( Z \) and is expressed as a complex number \( Z = R + jX \), where \( R \) is resistance and \( X \) is reactance.
• In the problem discussed, the impedance \( Z_1 = 5i \) and \( Z_2 = 8i \) are purely imaginary, implying either an entirely inductive or capacitive behavior.

Understanding and calculating impedance helps in designing and analyzing circuits to ensure they work as intended.
Engineers often add impedances in series or parallel to compute total impedance, influencing overall circuit functionality.

Series Circuits in Electrical Engineering

A series circuit is a fundamental type of electrical circuit that connects components along a single path, so the same current flows through each component.
In the context of AC circuits, understanding series circuits involves considering how impe dance affects the overall circuit.

• When impedances are connected in series, their total impedance is the arithmetic sum of the individual impedances: \( Z_{total} = Z_1 + Z_2 + ... + Z_n \). This is much like adding resistances in a DC circuit.
• For the specific problem we discussed, the total impedance \( Z_{total} \) was calculated by adding \( Z_1 = 5i \) and \( Z_2 = 8i \) to get \( Z_{total} = 13i \).

The concept of adding series impedances is vital for designing circuits that use alternating current, as it helps predict how circuit components will interact.
This ensures efficient circuit performance and can prevent potential problems in electrical systems.