Title : LC Oscillator and Damped Oscillation

Apparatus :

Voltage sensor  ( PASCO CI-6503 )                                       2

Capacitors )

Inductors ( 240-turn coil with two soft-iron C-cores, one soft iron C-core  )

Objective :

1.    Study the oscillation of voltage when a charged capacitor is connected

across an inductor.

2.    Determine the phase relationship between ,  the voltage

across capacitor and inductor.

Theory :

## Horizontal mass-spring system ( Simple Harmonic Oscillation )

If the displacement of the particle of mass m from the equilibrium position is x, the equation of motion is

Put

which is simple harmonic oscillation with the period T.

#### LC Oscillations

The LC circuit resembles a mass-spring system. Initially, the electrical energy from the capacitor is transferred into the magnetic energy of the inductor. When the electrical energy of the capacitor becomes zero, the process is reversed.  The magnetic energy from the inductor is transferred into the electrical energy of the capacitor. Electromagnetic oscillation occurs when energy is transferring between the capacitor and the inductor.

The following diagram shows eight stages in a cycle of oscillation of an LC circuit. The bar graph below the figure shows the stored electric and magnetic potential

energy.　click to play movie

The total energy U present at any instant in an oscillating LC circuit is given by

If we assume the circuit resistance to be zero, there is no energy transfer to Joule heat and U remains constant with time. This leads to

where U is a  constant and

Now. Q and I are not independent variables, being related by

Differentiating yields

Substituting these two expressions into Eq. (2) leads to

This equation is mathematically of exactly the same form as equation of motion of mass-spring system,

L C Circuit Analogy to Simple Harmonic Motion as shown in the following table

Mechanical Damped Harmonic Motion

The simple harmonic motion relates to the motion of a body acted on by a special kind of force and in friction-free conditions. The amplitude of SHM is a constant. If the amplitude of the oscillation gradually decreases to zero as a result of friction, the motion is said to be damped harmonic motion. The magnitude of the frictional force usually depends on the speed.

The equation of motion of the damped harmonic motion is given by

Electromagnetic Damped Oscillation

The resistance in the LC circuit will dissipate the energy. The variation of voltage across the capacitor is shown in the following figure.

Using a square wave

Apply a square wave to the LC-series combination, at each rising edge of the square wave, the capacitor would be charged. Since the charging current also passes through the inductor, oscillation would occur. At the falling edge, the charged capacitor would be discharged through the inductor. Again, oscillation would occur.

In the following figure, the first diagram represents the variation of voltage across the capacitor and the second diagram represents the variation of voltage across the inductor.

Procedure :

Hardware setup :

1.  Connect the Interface to the Computer, turn on  the interface, and turn on the computer.

2. Connect the Voltage Sensor to the interface

3. Connect the function generator ( OUTPUT ports on the interface ) directly across the LC

circuit and voltage sensor from analog channel A across  the inductor and voltage sensor

from analog channel B across the capacitor as shown in the following diagram.

Use the "Output" feature of the interface to supply a voltage to the inductor - capacitor circuit.

Use the Voltage Sensors to measure the voltage across the capacitor and inductor

Software setup :

4.  The signal generator is set to output a 5 volt,  "positive only" square wave at

10 Hz.

The Signal Generator is set to ' Auto' so it will start and stop automatically.

Data Recording :

For a given 240-turn air core coil, the variation of voltage across capacitor and inductor were measured.

The experiment was repeated with different combination of capacitances ( ) and inductances ( a 240 turns coil with one soft-iron core and a 240 turns coil with two soft-iron C-cores ).

Data Analysis :

Relationship between the period of LC Oscillation and the LC value

Variation of the period of LC Damped Oscillation

How does the amplitude of an LC  damped oscillation vary with time ?

Phase relationship between the voltage across capacitor and inductor

Discussion :

1.  What are the precautions of this experiment ?

2.  Design an experiment to investigate how the current fluctuates

after connecting the charged capacitor across the inductor.

3.  Discuss the application of LC circuit in our daily life.