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Under damped oscillations are the oscillations in which the amplitude of the oscillating body exponentially decays with time. For examples, Oscillations of Bob of a simple pendulum in air, Oscillations of prongs of tuning fork in air.

Over damped oscillation is when the oscillating body doesn't get to oscillate and transient response of displacement dies pretty slowly. For example, public transportation braking systems.

In LCR circuit, oscillating body is the charge flowing in the circuit responsible for current.

In any complete loop within a circuit, the sum of all voltages across components which supply electrical energy (such as cells or generators) must equal the sum of all voltages across the other components in the same loop.

On applying the law in LCR circuit, the following differential equation can be obtained,

$\frac{d^{2}q}{d{t}^2}+\frac{R}{L}\frac{dq}{dt}+\frac{1}{LC}q=0$

Form of solution of this differential equation is different for underdamped and over damped case.

When the resistance follows$R^2<\frac{4L}{C}$ , the solution so obtained is for underdamped.

When the resistance follows $R^2=\frac{4L}{C}$, the solution so obtained is for critically damped.

When the resistance follows$R^2>\frac{4L}{C}$ , the solution so obtained is for over damped.

Therefore,if the resistance of the resistor is relatively small, the circuit still oscillates, but with damped harmonic motion thus circuit is underdamped. If we increase the resistance, the oscillations die out more rapidly. When the resistance reaches a certain value, the circuit no longer oscillates; it is critically damped. For larger values of the resistance, the circuit is over damped.