Problem 53 - Algebra 2 and Trigonometry

Question

In certain circuits, the total impedance \(Z_{T}\) is given by the formula \(Z_{T}=\frac{Z_{1} Z_{2}}{Z_{1}+Z_{2}} .\) Find \(Z_{T}\) when \(Z_{1}=-3 i\) and \(Z_{2}=4 i\)

Step-by-Step Solution

Answer

The total impedance \(Z_T\) is \(12i\).

Step 1: Identify the Given Variables

We are given the impedances \(Z_1 = -3i\) and \(Z_2 = 4i\). The goal is to find the total impedance \(Z_T\) using these values.

Step 2: Substitute the Values into the Formula

Substitute the given values into the formula for total impedance. \[ Z_T = \frac{Z_1 \cdot Z_2}{Z_1 + Z_2} = \frac{(-3i) \cdot (4i)}{-3i + 4i} \]

Step 3: Simplify the Numerator

Calculate the product in the numerator: \((-3i) \cdot (4i) = -12i^2\). Since \(i^2 = -1\), we have \(-12i^2 = 12\).

Step 4: Simplify the Denominator

Calculate the sum in the denominator: \(-3i + 4i = i\).

Step 5: Calculate the Total Impedance

Substitute the simplified terms back into the equation: \[ Z_T = \frac{12}{i} \] To simplify \(\frac{12}{i}\), multiply numerator and denominator by \(-i\): \[ Z_T = \frac{12 \cdot (-i)}{i \cdot (-i)} = \frac{-12i}{i^2} = \frac{-12i}{-1} = 12i \]

Step 6: Final Result

The total impedance \(Z_T\) is \(12i\).

Key Concepts

Understanding Complex Numbers

Complex numbers are a fascinating part of mathematics, often denoted as numbers that have both a real and an imaginary part. This is typically written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. In the context of electrical engineering, especially in circuit analysis, complex numbers play a crucial role in representing impedances, voltages, and currents.

• The real part \( a \) represents the actual components such as resistance in a circuit.
• The imaginary part \( bi \) often represents components like inductance and capacitance.

By using complex numbers, engineers can effectively analyze and design circuits because they can easily handle phase differences and magnitude in alternating current (AC) circuits. Since a complex number combines these two parts, it becomes a powerful tool to solve complex circuit problems with ease.

Deciphering the Imaginary Unit

The imaginary unit, represented by \( i \), is the cornerstone of complex numbers. It allows us to extend the realm of numbers beyond the real line. The defining property of \( i \) is that \( i^2 = -1 \). This property is key when working with complex impedances in electrical engineering.

• When we multiply two imaginary numbers, the result incorporates \( i^2 \), converting part of the product back into a real value.
• For example, in the step-by-step solution, we calculated \((-3i) \cdot (4i) = -12i^2\). Given \(i^2 = -1\), this becomes \(12\).

Understanding how to manipulate the imaginary unit is critical in circuit analysis, as it simplifies the process of working with phasors and phase angles. This process allows engineers to efficiently solve complex problems involving frequency-dependent behaviors in a system.

Practical Approach to Circuit Analysis

Circuit analysis using complex numbers is a foundational skill in electrical engineering. It involves using mathematical techniques to determine the various electrical parameters within a circuit. By employing complex impedance, engineers can design and analyze circuits that operate efficiently under alternating current conditions.

• Impedance is a measure of how much a circuit resists the flow of electricity, represented as a complex number \(Z = R + jX\).
• \(R\) represents resistance, while \(X\) represents reactance (inductive or capacitive).
• The formula for total impedance in a parallel circuit, \(Z_{T} = \frac{Z_{1}Z_{2}}{Z_{1}+Z_{2}}\), is crucial in simplifying complex circuit calculations.

In our example, we used this formula to find the total impedance \(Z_T\) by taking into account both the imaginary contributions from \(Z_1\) and \(Z_2\). This streamlined approach highlights how complex numbers, through the use of impedance, facilitate the analysis of more intricate circuit systems.