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 49b
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→∞?
Verified step by step guidance1
Step 1: Understand the RL circuit and the role of phasor diagrams. In an RL circuit, the resistor (R) and inductor (L) are connected in series. The phasor diagram represents the voltage and current relationships in the circuit using vectors in the complex plane. The angular frequency ω determines the behavior of the circuit.
Step 2: Analyze the behavior of the circuit as ω → 0. At very low frequencies, the inductive reactance (X_L = ωL) approaches zero because it is proportional to ω. This means the inductor behaves like a short circuit, and the voltage across the resistor (V_R) will equal the total voltage applied to the circuit.
Step 3: Analyze the behavior of the circuit as ω → ∞. At very high frequencies, the inductive reactance (X_L = ωL) becomes very large, effectively acting as an open circuit. In this case, the current in the circuit approaches zero, and the voltage across the resistor (V_R) also approaches zero.
Step 4: Use the phasor diagram to visualize these limits. At ω → 0, the phasor for V_R aligns with the total voltage phasor because the inductor has negligible reactance. At ω → ∞, the phasor for V_R shrinks to zero as the inductor dominates the circuit and blocks current flow.
Step 5: Summarize the results. For ω → 0, V_R equals the total voltage. For ω → ∞, V_R approaches zero. These results are consistent with the behavior of the RL circuit in the frequency domain.
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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 by converting differential equations into algebraic equations. This allows for easier calculations of voltages and currents in circuits with resistors, inductors, and capacitors.
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Guided course
08:32
Phasors
RL Circuit Behavior
An RL circuit consists of a resistor (R) and an inductor (L) connected in series or parallel. The behavior of the circuit changes with frequency; at low frequencies (ω→0), the inductor behaves like a short circuit, while at high frequencies (ω→∞), it acts like an open circuit. Understanding these limits is crucial for determining the voltage across the resistor (VR) in different frequency scenarios.
Recommended video:
Guided course
07:50LR Circuits
Impedance
Impedance is the total opposition a circuit offers to the flow of alternating current (AC) and is represented as a complex number. In an RL circuit, the impedance combines resistance and inductive reactance, which varies with frequency. At ω→0, the impedance is purely resistive, while at ω→∞, the inductive reactance dominates, affecting the voltage across the resistor and the overall circuit behavior.
Recommended video:
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08:40Impedance in AC Circuits
Related Practice
Textbook Question
You have a resistor and a capacitor of unknown values. First, you charge the capacitor and discharge it through the resistor. By monitoring the capacitor voltage on an oscilloscope, you see that the voltage decays to half its initial value in 2.5 ms. You then use the resistor and capacitor to make a low-pass filter. What is the crossover frequency fc?
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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 an expression for the crossover frequency ωc.
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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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