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LC Oscillations occur due to the continuous flow of energy that passes from the capacitor (C) to the inductor (L). It causes electric devices to glow. This is because all these circuits have capacitors and inductors wherein, electrical oscillations of constant amplitude and frequency are produced when a charged capacitor is allowed to discharge through a non-resistance. These oscillations are the LC oscillations.
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Key Terms: LC Oscillations, Energy, Inductors, Amplitude, Oscillations, Simple Harmonic Motion, Electrical Charge, Capacitor
Define LC Oscillations
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When a charged capacitor is connected to an inductor in the circuit, the electric current in the circuit and the charge on the capacitor undergo oscillations which are known as LC Oscillations. The electrical charge stored in the capacitor is the initial charge on it which may be termed as qm. It can be denoted by:
| UE=qm2 C/2 |
The inductor (L) in the circuit contains zero (0) charge, whereas the current varies in form of a sine wave varying with respect to time.
LC Oscillations
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Construction of LC Oscillations
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Let us assume the capacity of the capacitor as – C, and the inductance of the inductor as – L. The capacitor is charged to its full capacity and now we connect it end to end to the inductor in the circuit.
Working of LC Oscillations
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Working of LC Oscillations can be depicted as:
- When we turn the switch on, i.e., when we close the circuit, there is a current flow in the circuit that keeps on increasing while the charge on the capacitor keeps decreasing gradually. This induced current in the circuit produces a magnetic field inside the inductor. When the current reaches its maximum possible level (Im) in the circuit, the magnetic field at this point is specified as:
UB = 12LIm
LC Oscillator working and Circuit diagram
- As time proceeds, there is no further change observed in the inductor and thus the magnetic fields through the inductor start decreasing gradually. Due to this gradual fall of the magnetic field in the inductor in the circuit, there is an induction of current in the circuit.
- This induced current will be of the opposite polarization. This oppositely polarized current now flows through the capacitor in reversed direction thus charging the capacitor in turn. When the capacitor gets fully charged, the process will get reversed again thus forming magnetic fields in the circuit and again repeating the procedure.
- This to and fro motion continues in the circuit and the energy in the system keeps oscillating between the components with the change in the direction. This flow of energy (current) in the circuit from the capacitor to the inductor and again to the capacitor is known as what is called LC Oscillations.
- The charge in the circuit oscillates with a natural frequency which has been derived by applying the Kirchoff’s laws to the circuit and deriving the result to be: w=1√(LC)
LC Circuit
| Note: This oscillation of the energy between the LC circuit has been considered with various assumptions. The energy loss in the circuit is neglected. |
According to the theory, once the charged capacitor is placed in the inductor circuit, the energy will keep oscillating in the circuit forever, which in real life is not possible as there are various losses taking place while transferring energy from one component to another. There are various losses due to the radiations from the magnetic/electromagnetic waves generated in the circuit.
Equation of Simple Harmonic Motion
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Generally, charge as a function of time in simple harmonic motion will be given as:
| q(t) = Qo sin(ωt+Φ) |
As t = 0, q(0) = Q0, therefore, by putting this value in the above equation we get,
Qo = Qo sin(0+Φ)
⇒ sinΦ=1
⇒ Φ=π/2
So, charge as a function of time will be,
⇒ q(t)=Qo sin(ωt+π/2)
⇒ q(t)=Qo cos(ωt)
Now differentiating charge with respect to time will be,
⇒ i(t)=−dq(t)/dt
This is the current as a function of time.
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Things to Remember
- When a charged capacitor is connected to an inductor in the circuit, the electric current in the circuit and the charge on the capacitor undergo oscillations which are known as LC Oscillations.
- Generally, charge as a function of time in simple harmonic motion will be given as: q(t)=Qo sin(ωt+Φ)
- The electrical charge stored in the capacitor is the initial charge on it which may be termed as qm.
- The capacitor is charged to its full capacity and now we connect it end to end to the inductor in the circuit.
- This induced current in the circuit produces a magnetic field inside the inductor.
- The charge in the circuit oscillates with a natural frequency which has been derived by applying the Kirchoff’s laws to the circuit and deriving the result to be: w=1√(LC)
- This oscillation of the energy between the LC circuit has been considered with some assumptions: The energy loss in the circuit is neglected.
Previous Year Questions
- RMS value of AC is _______ of the peak value… [VITEEE 2006]
- The instantaneous values of alternating current and voltages in a circuit... [JIPMER 2003]
- The average power dissipated in the A.C. circuit is 2 watts. If a current... [KCET 2014]
- An alternating voltage of 220 V, 50 Hz frequency is applied across... [VITEEE 2018]
- An inductive circuit contains a resistance of 10 Ω and... [VITEEE 2011]
- An AC supply gives 30 Vrms which is fed on... [JIPMER 2003]
Sample Questions
Ques. In an ideal parallel LC circuit, the capacitor is charged by connecting it to a DC source that is then disconnected. The current in the circuit
i) Becomes zero instantaneously
ii) Grows monotonically
iii) Oscillates instantaneously
iv) Decays monotonically (2 marks)
Ans.
This is the equation of harmonic motion.
Therefore, the current starts oscillating instantaneously.
Ques. In modern days, the incoming frequency of a radio receiver is superposed with a locally produced frequency that produces an intermediate frequency and this is always constant. This makes the tuning of the receiver very simple. This is used in superheterodyne oscillators. (2 marks)
Ans. A superheterodyne receiver, which is often shortened to superhet, is a type of radio receiver which uses frequency mixing to convert a received signal to a fixed intermediate frequency (IF) which can be more conveniently processed than the original carrier frequency.
Ques. Are LC oscillations possible in real life? (2 marks)
Ans. NO, LC oscillations are practically possible in real life because there are always certain resistance losses present which gradually decrease the amount of energy flow thus damping the energy to zero after a certain point.
Ques. What are the conditions necessary for the LC oscillations to take place? (2 marks)
Ans. To start with the oscillations, the phase shift for the circuit must be equal to 360 degrees, and also the magnitude gain for the loop to be permissible for oscillations must be greater than 1.
Ques. Do LC oscillations get damped? If yes, why so? (2 marks)
Ans. YES, LC oscillations do get damped with the rise in time due to the resistance and the circuit factors which gradually lower the energy transfer from one component and along the line of circuit.
Ques. Why is the term LC Oscillator keyed to the system? What is the factor that oscillates? (2 marks)
Ans. The LC oscillator has been keyed to the system because the circuit contains the components Capacitor (C) and an inductor (L) which are the main components of the circuits within which the energy oscillates.
The energy (current ) is the factor that oscillates within the closed circuit.
Ques. The values of inductance and capacitance in an LC circuit are 0.5 H and 8 μF respectively and current in the circuit is maximum, then the angular frequency of alternating emf applied in the circuit will be
i) 5 x 103 Radian/sec
ii) 50 Radian/sec
iii) 5 x 102 Radian/sec
iv) 5 Radian/sec (2 marks)
Ans. As we know that the maximum current flows when the circuit is at resonance. Thus, resonance frequency is given by,
Hence, this gives
ω = 5 x 102 Radian/sec
Ques. In an oscillating LC circuit, the maximum charge on the capacitor is Q. The charge on capacitor when the energy is stored equally between electric and magnetic field is:
i)Q/2
ii) Q/√3
iii) Q/√2
iv) Q (4 marks)
Ans. In the case of maximum charge on the capacitor, the whole energy is stored in the capacitor in the form of electric field that is
At the time when energy is distributed equally between electric and magnetic field, meaning, in the capacitor is,
Now, if the charge on the capacitor is Q1, then
Ques. Show that in the free oscillations of an LC circuit, the sum of energies stored in the capacitor and inductor is constant in time. (3 marks)
Ans. At a constant t, charge q on the capacitor and the current ‘i’ are given by,
This sum is constant in time as q0 and C both are time-independent.
Ques. If an LC circuit is considered to be analogous to a harmonically oscillating spring block system, which energy of the LC circuit would be analogous to potential energy and which one would be analogous to kinetic energy? (3 marks)
Ans. When a charged capacitor C with an initial charge q0 is discharged through an inductance L, the charge and current in the circuit start to oscillate simply harmonically. If the resistance of the circuit is zero, no energy is lost as heat. We can also consider an idealized situation where energy is not radiated away from the circuit. The total energy associated with the circuit is constant.
The oscillation of the LC circuit is an electromagnetic analog to the mechanical oscillation of a block-spring system. Hence, the total energy of the system remains conserved.
If an LC circuit is considered to be analogous to a harmonically oscillating spring block system, the energy associated with moving charges i.e., magnetic energy analogous to kinetic energy and electrostatic energy analogous to potential energy.
NOTE: An oscillator is an Amplifier with either ‘Positive feedback’ or ‘ regenerative feedback.’
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