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Title : LC Oscillator and Damped Oscillation Apparatus : Science Workshop Interface 750 1 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 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
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. 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
Now.
Q and I are not independent variables, being related by
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 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.
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 ( 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.
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, the
voltage 
Put
which is simple
harmonic oscillation with the period T.
)