25. Driven RLC Circuits

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• Syllabus
• Electric Fields

  • 1. Introduction to Electric Fields
  • 2. Math Review, Fields
  • 3. Electric Fields and Discrete Charge Distributions
  • 4. Electric Fields and Continuous Charge Distributions
  • 5. Gauss's Law
• Electric Potential

  • 6. Discrete and Continuous Distributions of Charge
  • 7. Equipotential Lines and Electric Fields
  • 8. Gauss's Law and Configuration Energy
• Capacitors

  • 9. Conductors and Insulators, Conductors as Shields
  • 10. Capacitance and Capacitors, Energy Stored in Capacitors
  • 11. Capacitors and Dielectrics
• Circuits

  • 12. Current, Current Density, Resistance, and Ohm's Law
  • 13. Batteries and Circuit Elements
  • 14. DC Circuits
  • 15. DC Circuits with Capacitors
• Magnetic Fields and Forces

  • 16. Magnetic Field
  • 17. Magnetic Forces
  • 18. Magnetic Dipoles
• Creating Magnetic Fields

  • 19. Biot-Savart Law
  • 20. Ampere's Law
• Faraday's Law

  • 21. Faraday's Law
  • 22. Inductance and Magnetic Energy
  • 23. RL Circuits
• Oscillating Circuits

  • 24. Undriven RLC Circuits
  • 25. Driven RLC Circuits
• Maxwell's Equations

  • 26. The Displacement Current and Maxwell's Equations
  • 27. Poynting Vector and Energy Flow in a Capacitor
• Electromagnetic Waves

  • 28. Maxwell's Equations and Electromagnetic Waves
  • 29. Energy and Momentum in Electromagnetic Waves
  • 30. Generating EM Waves, Dipole Radiation and Polarization
• Nature of Light

  • 31. Interference
  • 32. Diffraction
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Learning Objectives

• To become familiar with the concept of a circuit driven by an external power supply and how it differs from a circuit which is not driven.
• To comprehend the analogy between a driven mass spring system and a driven RLC circuit.
• To understand the meaning of the phasor diagram for a capacitor and an emf, for an inductor and an emf, and for a resistor and an emf.
• To understand the meaning of the phasor diagram when all three circuits are present in series with a driving emf.
• To understand the meaning of the phrase "the current leads the emf" as opposed to the phrase "the current lags the emf."

Preparation

Course Notes

Read through the course notes before watching the video. The course note files may also contain links to associated animations or interactive simulations.

Alternating-Current Circuits (PDF - 1.2MB)

Lecture Video

Video Excerpts

Learning Activities

Guided Activities

Read through the class slides. They explain all of the concepts from the module.

Slides (PDF)

Self-Assessment

Do the Concept Questions first to make sure you understand the main concepts from this module. Then, when you are ready, try the Challenge Problems.

Concept Questions

Concept Questions (PDF)

Solutions (PDF)

Challenge Problems

Challenge Problems (PDF)

Solutions (PDF)

Problem Solving Help

Watch the Problem Solving Help videos for insights on how to approach and solve problems related to the concepts in this module.

Problem 1: Driven RLC Circuits

A circuit consists of a generator with frequency ω, a resistor with resistance R, a capacitor with capacitance C, and an inductor with an inductance L. Why does the circuit end up oscillating at the frequency of the driving electromotive force? Give an expression for the amplitude of the driven current in the circuit. Plot this current as a function of our driving frequency ω. At what frequency does it maximize, and what is the maximum current? How does the shape of this curve depend on the resistance in the circuit? What is the phase difference between the current and the driving electromotive force? In what frequency regime does the capacitor play the primary role, and in what frequency range does the inductor play the primary role?

Problem 2: A Driven RLC Circuit

A driven RLC circuit has a resistance of 300 Ohms, an inductance of 0.25 Henries, and a capacitance of 8 x 10^-6 Farads. It is driven by a 120 V power supply at an angular frequency ω=400 radians/sec. Find the current in the circuit, the voltage you would read when you put voltmeters across the resistor, the capacitor, and the inductor, and the lead or lag of the current with respect to the driving voltage. Make sure you specify whether the current leads or lags the voltage. Is this circuit dominated by the capacitor or the inductor, or neither one?

Problem 3: Another Driven RLC Circuit

A driven RLC circuit has a total impedance Z of 150 Ohms. The root mean square voltage of the generator is 160 Volts, and the resistance in the circuit is 110 Ohms. What is the mean power delivered to the circuit in Watts?

Problem 4: An RLC Circuit at Resonance

A driven RLC circuit is driven at its resonant frequency by a generator with an amplitude of 90 Volts. The capacitance is 2.5 x 10^-6 Farads, the inductance is 0.90 Henries, and the resistance is 400 Ohms. At resonance, we have ωL = 1/ωC = 600 Ohms. What is the root mean square voltage measured individually across the capacitor and the inductor? If we put a voltmeter across both the capacitor and the inductor together, what root mean square voltage will we read?

Problem 5: A Driven RLC Circuit

In a driven RLC circuit, suppose you put a voltmeter across the combination of the resistance and the inductor. What is the ratio of the voltage read across that the RL combination to the generator source voltage?

Problem 6: An RLC Circuit

The tuning RLC circuit in an FM radio has an inductance of 1 x 10^-6 Henries, but we do not know the resistance or the capacitance. We vary the capacitance to tune into one FM station which broadcasts at an angular frequency ω = 6 x 10⁸ radians per second. What is the value of the capacitance when we are receiving this station? There is an annoying nearby station which radiates at a frequency ω = 5.99 x 10⁸ radians per second, but our tuning is so sharp that the power across the resistor due to this second station is only 1% of the power of the station we want to hear at ω = 6 x 10⁸ radians per second. What is the resistance of the circuit?

Problem 7: An Unusual RLC Circuit

Suppose instead of the usual arrangement of a series RLC circuit, we have all of the elements (power source, capacitor, inductor, resistor) in parallel instead of in series. How does this differ from our usual series RLC circuit? What is the expression for the total current flowing in the circuit in terms of R, L, C, and the driving amplitude of the power source V_s? If R is fixed, when is the current a maximum? Suppose R, L, and C are all fixed, is there a value of ω where the current is a maximum?

 

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