Problem 72 - Algebra 2

Question

ELECTRICITY. The impedance in one part of a series circuit is \(3+4 j\) ohms, and the impedance in another part of the circuit is \(2-6 j .\) Add these complex numbers to find the total impedance in the circuit.

Step-by-Step Solution

Answer

The total impedance is \(5 - 2j\) ohms.

Step 1: Understanding the Components

We are given two complex numbers representing impedances in a series circuit. The first impedance is given as \(3 + 4j\) ohms, and the second impedance is \(2 - 6j\) ohms. To find the total impedance in a series circuit, we need to add these two complex numbers.

Step 2: Separating Real and Imaginary Parts

The complex numbers have both real and imaginary components. For \(3 + 4j\), the real part is 3 and the imaginary part is 4. For \(2 - 6j\), the real part is 2 and the imaginary part is -6.

Step 3: Adding Real Parts

Add the real parts of the complex numbers: \(3 + 2 = 5\). This will be the real part of the total impedance.

Step 4: Adding Imaginary Parts

Add the imaginary parts of the complex numbers: \(4 + (-6) = -2\). This will be the imaginary part of the total impedance.

Step 5: Determining Total Impedance

Combine the sum of the real parts and the sum of the imaginary parts to express the total impedance as a complex number. The total impedance is \(5 - 2j\) ohms.

Key Concepts

Impedance

Impedance is a fundamental concept in electrical engineering, especially when dealing with alternating current (AC) circuits. It describes how much a circuit resists or impedes the flow of an AC current. Impedance is analogous to resistance in DC circuits, but it is more complex because it contains both a real part and an imaginary part.
- **Real Part**: This represents the resistive aspect which dissipates energy (like heat) from the circuit.- **Imaginary Part**: This corresponds to the reactive component (inductive or capacitive), which stores and returns energy.
Impedance in AC circuits is a complex number, typically expressed in the form of \( Z = R + jX \), where \( R \) is the resistance (real part) and \( jX \) is the reactance (imaginary part). The presence of both parts means that impedance affects both the magnitude and phase of the current and voltage in a circuit.

Series Circuit

A series circuit is an electrical circuit in which components are connected end-to-end such that current flows through each component sequentially without branching off into other paths.
- **Current Consistency**: In a series circuit, the same current flows through all components because there's only one pathway for flow.- **Voltage Addition**: The total voltage drop across a series circuit is the sum of the voltage drops across individual components.
For series circuits dealing with impedances, the total impedance is simply the sum of the individual impedances. If you have two impedances, say \( Z_1 = 3 + 4j \) ohms and \( Z_2 = 2 - 6j \) ohms, the total impedance in the series circuit is calculated by adding these impedances directly, resulting in an impedance of \( 5 - 2j \) ohms.

Real and Imaginary Parts

Complex numbers such as the ones used to represent impedance have two components:
- **Real Part**: This is the component of the complex number not associated with the imaginary unit \( j \). It stands on its own and is purely 'real,' as in \( a \) from \( a + bj \). For impedance, this represents the resistance.- **Imaginary Part**: This involves the imaginary unit \( j \) and is denoted as \( bj \) in a complex number. In electrical systems, this part accounts for the reactance, which is dependent on frequency.
Both real and imaginary parts are important in calculations. To add or subtract complex numbers, we handle real and imaginary parts separately:- Add the real parts: From \( 3 + 4j \) and \( 2 - 6j \), real parts are \( 3 \) and \( 2 \), yielding \( 3 + 2 = 5 \).- Add the imaginary parts: Imaginary parts \( 4j \) and \(-6j\) add to \( 4 + (-6) = -2 \).
Hence, the result of combining these components is a new complex number, \( 5 - 2j \), describing the total impedance.