Textbook Question
A series RLC circuit consists of a 75 Ω resistor, a 0.12 H inductor, and a 30 μF capacitor. It is attached to a 120 V/60 Hz power line. What is the peak current I?
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Ch 32: AC Circuits
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 32, Problem 49d
Use a phasor diagram to analyze the RL circuit of FIGURE P32.49. In particular, Find an expression for the crossover frequency ωc.
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Identify the components of the RL circuit: The circuit consists of a resistor (R) and an inductor (L) connected in series. The total impedance of the circuit is the combination of the resistive and inductive reactance components.
Write the expression for the impedance of the RL circuit: The impedance \( Z \) is given by \( Z = \sqrt{R^2 + (\omega L)^2} \), where \( \omega \) is the angular frequency, \( R \) is the resistance, and \( L \) is the inductance.
Define the crossover frequency \( \omega_c \): The crossover frequency is the frequency at which the inductive reactance \( \omega L \) equals the resistance \( R \). This is the point where the magnitudes of the resistive and inductive components are equal.
Set up the equation for the crossover frequency: At \( \omega_c \), \( \omega_c L = R \). Solve for \( \omega_c \) to find \( \omega_c = \frac{R}{L} \).
Conclude with the expression for the crossover frequency: The crossover frequency \( \omega_c \) is determined by the ratio of the resistance \( R \) to the inductance \( L \), and is given by \( \omega_c = \frac{R}{L} \). This is the desired expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Phasor Diagrams
Phasor diagrams are graphical representations of sinusoidal functions, where each phasor represents the amplitude and phase of a sinusoidal voltage or current. In the context of AC circuits, phasors simplify the analysis of circuit components like resistors, inductors, and capacitors by converting differential equations into algebraic equations. This allows for easier manipulation and understanding of the relationships between voltages and currents in the circuit.
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Phasors
RL Circuit
An RL circuit consists of a resistor (R) and an inductor (L) connected in series or parallel. The behavior of an RL circuit is characterized by its impedance, which combines resistance and inductive reactance. The inductor's reactance increases with frequency, affecting the circuit's overall response to alternating current (AC) and leading to phenomena such as phase shifts between voltage and current.
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Crossover Frequency (ωc)
The crossover frequency (ωc) in an RL circuit is the frequency at which the inductive reactance equals the resistance, resulting in a specific phase relationship between voltage and current. At this frequency, the circuit transitions from being dominated by resistive behavior to inductive behavior. Understanding ωc is crucial for analyzing the frequency response of the circuit and determining how it will behave under different AC conditions.
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Related Practice
Textbook Question
A series RLC circuit consists of a 75 Ω resistor, a 0.12 H inductor, and a 30 μF capacitor. It is attached to a 120 V/60 Hz power line. What is the phase angle ϕ?
Textbook Question
Use a phasor diagram to analyze the RL circuit of FIGURE P32.49. In particular, What is VR in the limits ω→0 and ω→∞?
Textbook Question
The small transformers that power many consumer products produce a 12.0 V rms, 60 Hz emf. Design a circuit using resistors and capacitors that uses the transformer voltage as an input and produces a 6.0 V rms output that leads the input voltage by 45°.
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Textbook Question
Use a phasor diagram to analyze the RL circuit of FIGURE P32.49. In particular, Find expressions for I, VR, and VL.
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Textbook Question
A series RL circuit is built with a 110 Ω resistor and a 5.0-cm-long, 1.0-cm-diameter solenoid with 800 turns of wire. What is the peak magnetic flux through the solenoid if the circuit is driven by a 12 V, 5.0 kHz source?
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