Problem 73 - Algebra and Trigonometry with Analytic Geometry

Question

The modulus of the impedance \(Z\) represents the total opposition to the flow of electricity in a circuit and is measured in ohms. If \(Z=14-13 i\), compute \(|Z|\).

Step-by-Step Solution

Answer

The modulus \(|Z| = \sqrt{365}\).

Step 1: Identify the components of the impedance

The given impedance is \(Z = 14 - 13i\). Here, the real part is \(a = 14\) and the imaginary part is \(b = -13\).

Step 2: Recall the formula for the modulus of a complex number

The modulus of a complex number \(a + bi\) is calculated using the formula \(|Z| = \sqrt{a^2 + b^2}\). This represents the distance of the point \((a, b)\) from the origin in the complex plane.

Step 3: Substitute the values into the formula

Now, substitute \(a = 14\) and \(b = -13\) into the modulus formula: \(|Z| = \sqrt{14^2 + (-13)^2}\).

Step 4: Calculate the squares

Calculate the squares of the real component and the imaginary component: \(14^2 = 196\) and \((-13)^2 = 169\).

Step 5: Compute the sum of the squares

Add the two squares: \(196 + 169 = 365\).

Step 6: Calculate the square root

Finally, calculate the square root of the sum from Step 5: \(|Z| = \sqrt{365}\).

Key Concepts

Impedance

Impedance is a critical concept in the study of electrical circuits. It refers to the total opposition that a circuit presents to the flow of alternating current (AC). Unlike resistance, which only applies to direct current (DC), impedance extends the idea to include the effects of capacitance and inductance within the circuit. This means that impedance is not just a real number but a complex one, expressed in the form of a complex number like 14 - 13i in circuits.

To understand impedance better:

• Resistance - The real part of impedance, affecting both DC and AC.
• Reactance - The imaginary part of impedance, which arises due to the presence of capacitance and inductance.
• Measured in Ohms (Ω) - Just like resistance, impedance is measured in ohms, but it encapsulates both resistive and reactive elements.

Understanding how to calculate and interpret impedance is vital for designing efficient electrical circuits and ensuring the correct flow of electricity.

Modulus of a Complex Number

The modulus of a complex number is a measure of its size or magnitude. When we have a complex number in the form of \(a + bi\), the modulus is the distance of this number from the origin in the complex plane. This distance is crucial as it gives us a sense of the 'size' of the impedance in electrical circuits, showing how much opposition there is to the current.

Here's how you find the modulus for a complex number \(Z = a + bi\):

• Formula - The formula is \(|Z| = \sqrt{a^2 + b^2}\).
• Application - For example, if \(a = 14\) and \(b = -13\), then \(|Z| = \sqrt{14^2 + (-13)^2}\).
• Interpretation - This result is particularly useful in assessing the effectiveness of components in a circuit.

The modulus gives a direct way to calculate the total impedance in practical electrical engineering problems.

Complex Plane

A complex number can be represented as a point in the complex plane, also known as the Argand plane. This plane is a way to visualize complex numbers graphically. The x-axis represents the real part of the complex number, and the y-axis represents the imaginary part.

Key points about the complex plane:

• Visual Representation - Each complex number \(a + bi\) corresponds to a point \((a, b)\) in the plane.
• Distances - The modulus of a complex number is the distance from the origin \((0, 0)\) to the point \((a, b)\).
• Utility in Circuits - Visualizing impedance in the complex plane helps engineers understand the behavior of circuits at different frequencies.

This visualization helps in analyzing the relationship between voltage, current, and impedance in AC circuits.

Electrical Circuits

Electrical circuits are networks that provide paths for electricity to flow. Understanding the role of impedance in these circuits is fundamental to ensuring they function correctly.

Consider the following aspects:

• Components - Circuits typically contain resistors, capacitors, and inductors, each contributing to the total impedance.
• AC vs. DC - Impedance plays an essential role in alternating current (AC) circuits, where both phase and magnitude of current and voltage vary.
• Impedance Matching - This is crucial for maximizing power transfer and ensuring that circuits operate efficiently and safely.

Knowledge of electrical circuits and how impedance affects them allows for designing and troubleshooting of complex electronic systems.