Chapter 31: Problem 25
A series ac circuit contains a 250-\(\Omega\) resistor, a 15-mH inductor, a 3.5-\(\mu\)F capacitor, and an ac power source of voltage amplitude 45 V operating at an angular frequency of 360 rad/s. (a) What is the power factor of this circuit? (b) Find the average power delivered to the entire circuit. (c) What is the average power delivered to the resistor, to the capacitor, and to the inductor?
Short Answer
Step by step solution
Calculate the Impedance of the Inductor
Calculate the Impedance of the Capacitor
Calculate the Total Impedance of the Circuit
Calculate the Magnitude of the Total Impedance
Determine the Power Factor of the Circuit
Find the Average Power Delivered to the Entire Circuit
Calculate the Average Power Delivered to the Resistor
Calculate the Average Power Delivered to the Capacitor and Inductor
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Impedance Calculation
To calculate the impedance of an inductor, use \( Z_L = j \omega L \), where \( j \) denotes the imaginary unit, \( \omega \) is the angular frequency, and \( L \) is the inductance. For a capacitor, the formula is \( Z_C = \frac{-j}{\omega C} \), where \( C \) is the capacitance.
Once you find these individual impedances, total impedance in a series circuit combines resistance and reactance: \( Z = R + Z_L + Z_C \). Here, resistance \( R \) is a purely real number, while inductive and capacitive reactance are imaginary. This mix requires converting into polar form for magnitude: \(|Z| = \sqrt{R^2 + (X_L - X_C)^2}\).
This magnitude represents the "total opposition" to alternating current, factoring in both resistive and reactive components, essential for complete analysis.
Power Factor
A power factor of 1 (or 100%) means that all electrical power is effectively used for work. In practical circuits, values less than 1 suggest inefficiencies due to reactive components like inductors and capacitors. These components cause phase shifts: inductors delay current by 90°, while capacitors lead voltage by 90°.
An efficient circuit aims to have a power factor as close to 1 as possible. A lower power factor indicates more energy is stored rather than used, leading to higher energy costs and less efficient power use. Addressing these inefficiencies often involves reactive power compensation or power factor correction techniques.
Average Power
In resistive components, average power reflects real, consumable power, calculated as \( P_R = I_{rms}^2 R \). Inductors and capacitors ideally do not consume energy over time. Instead, they store energy temporarily and return it to the system, leading to zero average power over a complete cycle: \( P_C = P_L = 0 \).
Understanding average power helps in efficient energy management, allowing for the design of circuits that minimize wasted energy, ensuring consumer loads receive necessary power for optimal functionality.