Resource Lesson
LC Circuit
Printer Friendly Version
The LC circuits we will be investigating are those involving a DC power supply. Let's begin with a simple circuit containing a DC power supply (battery), two switches, a resistor, a capacitor, and an inductor.
When only switch A is closed (both B switches are open), only the left circuit containing the resistor, battery, and capacitor is connected and the capacitor becomes charged. Once the capacitor is fully charged and has attained the voltage the battery, switch A is opened and both switch B's are closed. At that time only the capacitor and the inductor are components in the active circuit.
Initially the charges on the positive capacitor plate (we are discussing conventional current) will begin circulating counter-clockwise. As the current from the capacitor dies out, the inductor reverses its emf to keep the charges flowing until the bottom plate of the capacitor becomes positively charged and the top plate holds all of the negative charge. Then the process reverses and the current flows clockwise until the capacitor's top plate is once again positively charged and its bottom plate negatively charged. This process of "filling and emptying" the capacitor's plates, and its subsequent electric field, continues at a frequency which we will define a little later in the lesson.
Note that in the animated gif provided by
Chetvorno (Own work) [CC0], via Wikimedia Commons
, the capacitor and inductor are presented on opposite sides compared to the introductory circuit presented at the top of the page.
Analogies to vibrating spring-mass systems
In a vibrating spring-mass system, the energy is shared between the elastic potential energy in the spring, U
_{s}
=
, and the kinetic energy, KE =
of the vibrating mass.
At any intermediate position during a vibration, some of the energy is kinetic and some is potential elastic, but the total amount of energy remains constant.
In an oscillating LC circuit, the energy is shared between the amount stored in the electric field of the capacitor and the amount storied in the magnetic field of the inductor.
Here are the analogies that equate the behavior of an oscillating spring-mass system and an resonating LC circuit.
mass
become
inductance, L
velocity
becomes
current, i
spring constant
becomes
C
^{-1}
displacement from equilibrium
becomes
charge, q
maximum displacement (amplitude)
becomes
Q
_{o}
(the maximum charge on the capacitor)
Substituting our new variables into our equation for the energy of a vibrating mass-spring system we get,
But what is Q
_{max}
or Q
_{o}
? This value comes from the functional equation for a capacitor: Q = CV where C is the capacitance and V is the voltage of the charging battery. When there is no charge on the capacitor (q = 0) we can calculate the maximum current.
But what about the frequency of the circuit mentioned earlier? What would be its expression? Once again, we will turn to our analogies.
where the units of "LC" are sec
^{2}
. The resonant frequency of the LC circuit is merely the reciprocal of its period,
In this presentation, the resistance in the circuit is considered minimal. That is, there are no energy losses to heat. In real circuits, the oscillations would eventually decay and die out.
Using the resonant frequency
So now that we have an expression for the frequency of an oscillating LC circuit, let's examine the position, velocity, and acceleration functions of our vibrating mass-spring system and the analogies for an LC circuit. In the top diagrams of a vibrating mass-spring, we started at a position of maximum compression. Let's "define" that amplitude to be negative and the amplitude at maximum extension to be positive. This would give us a negative cosine function for the instantaneous position of the vibrating mass.
spring-mass
LC circuit
position/charge
To obtain an equation for the instantaneous velocity of the vibrating mass, we take the derivative with respect to time of the mass' position function.
spring-mass
LC circuit
velocity/current
An one more time, to obtain an equation for the instantaneous acceleration of the vibrating mass, we would take the derivative with respect to time of the mass' velocity function.
spring-mass
LC circuit
acceleration/(di/dt)
Recall that the emf induced in an inductor is calculated according to the equation
Related Documents
Lab:
Labs -
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Labs -
Aluminum Foil Parallel Plate Capacitors
Labs -
Calculation of "g" Using Two Types of Pendulums
Labs -
Conical Pendulums
Labs -
Conical Pendulums
Labs -
Conservation of Energy and Vertical Circles
Labs -
Electric Field Mapping
Labs -
Electric Field Mapping 2
Labs -
Forces Between Ceramic Magnets
Labs -
Introductory Simple Pendulums
Labs -
Kepler's 1st and 2nd Laws
Labs -
Loop-the-Loop
Labs -
Magnetic Field in a Solenoid
Labs -
Mass of an Electron
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Oscillating Springs
Labs -
RC Time Constants
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Sand Springs
Labs -
Simple Pendulums: Class Data
Labs -
Simple Pendulums: LabPro Data
Labs -
Telegraph Project
Labs -
Video LAB: A Gravitron
Labs -
Video LAB: Circular Motion
Labs -
Video LAB: Looping Rollercoaster
Labs -
Water Springs
Resource Lesson:
RL -
A Comparison of RC and RL Circuits
RL -
A Derivation of the Formulas for Centripetal Acceleration
RL -
A Guide to Biot-Savart Law
RL -
A Special Case of Induction
RL -
Ampere's Law
RL -
Capacitors and Dielectrics
RL -
Centripetal Acceleration and Angular Motion
RL -
Conservation of Energy and Springs
RL -
Continuous Charge Distributions: Charged Rods and Rings
RL -
Continuous Charge Distributions: Electric Potential
RL -
Coulomb's Law: Beyond the Fundamentals
RL -
Coulomb's Law: Suspended Spheres
RL -
Derivation of Bohr's Model for the Hydrogen Spectrum
RL -
Derivation: Period of a Simple Pendulum
RL -
Dielectrics: Beyond the Fundamentals
RL -
Eddy Currents plus a Lab Simulation
RL -
Electric Field Strength vs Electric Potential
RL -
Electric Fields: Parallel Plates
RL -
Electric Fields: Point Charges
RL -
Electric Potential Energy: Point Charges
RL -
Electric Potential: Point Charges
RL -
Electricity and Magnetism Background
RL -
Electrostatics Fundamentals
RL -
Energy Conservation in Simple Pendulums
RL -
Famous Experiments: Cathode Rays
RL -
Famous Experiments: Millikan's Oil Drop
RL -
Gauss' Law
RL -
Generators, Motors, Transformers
RL -
Gravitational Energy Wells
RL -
Induced Electric Fields
RL -
Induced EMF
RL -
Inductors
RL -
Introduction to Magnetism
RL -
Kepler's Laws
RL -
Magnetic Field Along the Axis of a Current Loop
RL -
Magnetic Forces on Particles (Part II)
RL -
Magnetism: Current-Carrying Wires
RL -
Maxwell's Equations
RL -
Meters: Current-Carrying Coils
RL -
Motional EMF
RL -
Parallel Plate Capacitors
RL -
Period of a Pendulum
RL -
RL Circuits
RL -
Rotational Kinematics
RL -
Shells and Conductors
RL -
SHM Equations
RL -
Simple Harmonic Motion
RL -
Spherical, Parallel Plate, and Cylindrical Capacitors
RL -
Springs and Blocks
RL -
Symmetries in Physics
RL -
Tension Cases: Four Special Situations
RL -
The Law of Universal Gravitation
RL -
Thin Rods: Moment of Inertia
RL -
Torque on a Current-Carrying Loop
RL -
Uniform Circular Motion: Centripetal Forces
RL -
Universal Gravitation and Satellites
RL -
Vertical Circles and Non-Uniform Circular Motion
Review:
REV -
Drill: Electrostatics
REV -
Drill: Induction
REV -
Electrostatics Point Charges Review
REV -
Review: Circular Motion and Universal Gravitation
Worksheet:
APP -
Big Al
APP -
Maggie
APP -
Ring Around the Collar
APP -
The Birthday Cake
APP -
The Electrostatic Induction
APP -
The Satellite
APP -
The Spring Phling
APP -
The Tree House
APP -
Timex
CP -
Centripetal Acceleration
CP -
Centripetal Force
CP -
Coulomb's Law
CP -
Electric Potential
CP -
Electrostatics: Induction and Conduction
CP -
Induction
CP -
Magnetism
CP -
Power Transmission
CP -
Satellites: Circular and Elliptical
CP -
Transformers
NT -
Bar Magnets
NT -
Circular Orbits
NT -
Electric Potential vs Electric Potential Energy
NT -
Electrostatic Attraction
NT -
Induction Coils
NT -
Lightning
NT -
Magnetic Forces
NT -
Meters and Motors
NT -
Pendulum
NT -
Photoelectric Effect
NT -
Potential
NT -
Rotating Disk
NT -
Spiral Tube
NT -
Van de Graaff
NT -
Water Stream
WS -
Basic Practice with Springs
WS -
Capacitors - Connected/Disconnected Batteries
WS -
Charged Projectiles in Uniform Electric Fields
WS -
Combinations of Capacitors
WS -
Coulomb Force Extra Practice
WS -
Coulomb's Law: Some Practice with Proportions
WS -
Electric Field Drill: Point Charges
WS -
Electric Fields: Parallel Plates
WS -
Electric Potential Drill: Point Charges
WS -
Electrostatic Forces and Fields: Point Charges
WS -
Electrostatic Vocabulary
WS -
Induced emf
WS -
Inertial Mass Lab Review Questions
WS -
Introduction to Springs
WS -
Kepler's Laws: Worksheet #1
WS -
Kepler's Laws: Worksheet #2
WS -
Magnetic Forces on Current-Carrying Wires
WS -
Magnetic Forces on Moving Charges
WS -
More Practice with SHM Equations
WS -
Parallel Reading - The Atom
WS -
Pendulum Lab Review
WS -
Pendulum Lab Review
WS -
Practice with Ampere's Law
WS -
Practice with Induced Currents (Changing Areas)
WS -
Practice with Induced Currents (Constant Area)
WS -
Practice: SHM Equations
WS -
Practice: Uniform Circular Motion
WS -
Practice: Vertical Circular Motion
WS -
SHM Properties
WS -
Standard Model: Particles and Forces
WS -
Static Springs: The Basics
WS -
Universal Gravitation and Satellites
WS -
Vertical Circular Motion #1
TB -
36A: Magnets, Magnetic Fields, Particles
TB -
36B: Current Carrying Wires
TB -
Advanced Capacitors
TB -
Basic Capacitors
TB -
Centripetal Acceleration
TB -
Centripetal Force
TB -
Electric Field Strength vs Electric Potential
TB -
Exercises on Current Carrying Wires
PhysicsLAB
Copyright © 1997-2022
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton