DAMPED AND UNDAMPED OSCILLATIONS
The oscillatory circuit, also called the L-C circuit or tank circuit, consists of an inductive coil of inductance L connected in parallel with a capacitor of capacitance C. The values of L and C determines the frequency of oscillations produced by the circuit. The most important point is that both the capacitor and inductor are capable of storing energy—the capacitor stores energy in its dielectric field whenever a pd exists across its plates while the inductor stores energy in its magnetic field whenever current flows through it.
For understanding the operation of an oscillatory circuit, let the capacitor be charged from a dc source with the polarity as shown in figure (a). A potential difference will be across the plates of the capacitor because of the accumulation of electrons in the lower plate of the capacitor. The electrons get accumulated in the lower plate due to the supply from the negative terminal of the battery. Thus, a potential energy will be formed in the capacitor. Now when the capacitor is fully charged and the switch S is opened, as shown in figure (b), the capacitor cannot discharge through L.
Suppose the switch S is kept in position ‘b’. The current starts flowing in the circuit but the self induced emf in the coil opposes the current flow. Thus the rate of rise of current is slow. Maximum current flows in the circuit when the capacitor is fully discharged. Due to flow of current, magnetic field is set up which stores the energy given by the electric field,as shown in figure (c). Thus, at the instant the capacitor gets completely discharged, the electrostatic energy stored in the capacitor gets converted into the magnetic field energy associated with the inductor L.
When the capacitor is completely discharged, the magnetic field begins to collapse and a counter or back emf is developed which, according to Lenz’s law, keeps the current flowing in the same direction. The capacitor now starts getting charge but with opposite polarity, as shown in fig.(d). In this case, the energy associated with the magnetic field is again converted into electrostatic energy. In an ideal case (that is, both the L and C are loss-free), the capacitor is charged to the value it had initially while the magnetic field energy reduces to zero.
After the collapsing field has recharged the capacitor, the capacitor now begins to discharge with a current flow in the opposite direction. The electric field starts collapsing whereas magnetic field starts building up again but in opposite direction. Fig. (e) shows the condition when the capacitor gets fully discharged. The sequence of charging and discharging continues, that is, the process of transformation of dielectric energy into magnetic energy and vice-versa is repeated again and again. This situation is similar to an oscillating pendulum, in which the energy keeps on interchanging between potential and kinetic energy. Thus the charge and discharge of a capacitor through inductor results in oscillating current and hence electrical oscillations are set up in the L-C or tank circuit.
The frequency of oscillation is the same as the resonant frequency of the tank circuit and it is given as fo = 1/2∏√LC
The interchange of energy between L and C would continue indefinitely if there were no losses in the tank circuit. But since there are losses, the indefinite interchange of energy cannot be proved practically. The losses include the energy that is lost in the form of heat generated in the coil resistance, capacitor leakage resistance and connecting wires. Some energy is even lost in the form of electro-magnetic waves that are radiated out from the circuit through which an oscillatory current is flowing. As a result, while the total energy is consumed in overcoming the losses, the oscillating current goes on decreasing with time and eventually becomes zero.
The rate at which the energy from the L-C circuit dissipates must be same as the energy that is supplied to the L-C circuit. In case of electronic oscillators, the transistor and power supply source are provided to feed energy to the for meeting the losses at right time. Thus sustained or undamped oscillations are produced by electronic oscillator circuits.
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