Damped oscillation

An oscillator is anything that has a rhythmic periodic response. A damped oscillation means an oscillation that fades away with time. Examples include a swinging pendulum, a weight on a spring, and also a resistor - inductor - capacitor (RLC) circuit.

Suppose we have an RLC circuit, which has a resistor + inductor + capacitor in series.When the switch closes at time t=0 the capacitor will discharge into a series resistor and inductor.

Now, the voltages and current in this circuit can be given by

${\begin{aligned}I(t)&={\frac {V_{0}}{\beta L}}e^{-at}\sin(\beta t)\\V_{c}(t)&=V_{0}e^{-at}\cos(\beta t)\end{aligned}}$

where

${\begin{aligned}\beta &={\sqrt {{\frac {1}{LC}}-{\frac {R^{2}}{4L^{2}}}}}\\a&={\frac {R}{2L}}\end{aligned}}$

and V= initial voltage C = capacitance (farads) R = resistance (ohms) L = inductance (henrys) e = base of natural log (2.71828...)

The above equation is the current for a damped sine wave. It represents a sine wave of maximum amplitude (V/BL) multiplied by a damping factor of an exponential decay. The resulting time variation is an oscillation bounded by a decaying envelope.

Critical Damping

We can use these equations to discover when the energy fades out smoothly (over-damped) or rings (under-damped).

Look at the term under the square root sign, which can be simplified to: R²C²-4LC

• When R²C²-4LC is positive, then α and β are real numbers and the oscillator is over-damped. The circuit does not show oscillation
• When R²C²-4LC is negative, then α and β are imaginary numbers and the oscillations are under-damped. The circuit responds with a sine wave in an exponential decay envelope.
• When R²C²-4LC is zero, then α and β are zero and oscillations are critically damped.

The circuit response shows a narrow peak followed by an exponential decay.