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The Role of Damped Driven Oscillators in RLC Circuits

Published Date
Author First Page Sage

Key Takeaways

  • Learn what a damped driven oscillator is.

  • Find out how damped driven oscillation occurs in an RLC circuit.

  • Explore how resonance affects a damped driven oscillator.

Woman being pushed on a swing

The act of pushing a swing creates damped driven oscillation

An oscillator is a mechanism that produces repetitive movement centered on an equilibrium. The movement of an oscillator is depicted in a sinusoidal waveform. In an ideal setting, the oscillation would go on indefinitely, alternating between the positive and negative maximum amplitude. 

A swing is a real-life example of an oscillator. So is a pendulum, guitar string, and spring. These real-life oscillators do not perpetually oscillate. Bound by forces that resist its movement, an oscillator eventually stops. The term ‘damped oscillator’ describes an oscillator that’s subjected to gradually decreasing momentum. However, if necessary, an oscillator can be prevented from stopping by applying an external oscillating force to it.

Let’s take a closer look at the functions of damped driven oscillators. 

What is a Damped Driven Oscillator?

A damped driven oscillator is an oscillator that is bound by weakening movement and, yet, is supported by an external force for continuous movement. In physics, a damped driven oscillator is defined by the equation:

md2x dt2+ b dx dt+ kx = F0 cos ⍵t

Damped Driven Oscillation in an RLC Circuit

Damped driven oscillator with a series RLC circuit

The series RLC circuit as a damped driven oscillator

Damped driven oscillation is a phenomenon that is often observed in electronics. In fact, an electronic damped driven oscillator is a basic module that’s critical to various applications. To understand how damped driven oscillation works in electronics, let’s take a look at the series RLC circuit.

In a closed circuit, a capacitor will discharge and current will flow through the inductor, assuming that the capacitor is charged to the maximum and the voltage source is removed from the circuit. The inductor will build up stored magnetic energy in accordance with Lenz Law, which opposes the current flow. 

Once the capacitor is fully discharged, the inductor’s magnetic field forces a reverse current flow, which charges the capacitor in the opposite direction. This process will repeat, but is subject to the damping force of resistive elements in the circuit. 

To compensate for the loss of energy, a sinusoidal signal source is needed to ensure the RLC circuit continues to oscillate. The addition of the signal source turns the RLC circuit into a damped driven oscillator. Its response can be expressed in the following equation: 

d2q(t) dt2+ R Ldq(t) dt+ 1 LCq(t) = 1 LE0 cos ⍵t

Note that the damped driven oscillator of an RLC circuit is similar to the one defined for a mechanical oscillator. 

Damped Driven Oscillators and Resonance

Series RLC damped driven oscillator response at resonance

Maximum amplitude is achieved at resonance

Here’s a common misconception: a damped driven oscillator does not necessarily result in maximum amplitude, despite being able to ensure continuous oscillation. It’s quite similar to how pushing a swing may not always result in the swing swinging higher. 

In order to achieve maximum oscillation amplitude, external force needs to be driven at the resonant frequency of the oscillator. In an RLC circuit, the resonant frequency is given by the following formula: 

fr = 1 2π√LC

A series RLC circuit in resonance will have the capacitive and inductive reactance cancel each other out and appear as a short circuit. The impedance of the circuit is at its lowest and the current flows at the maximum amplitude when the AC signal is at the resonant frequency. 

The resonant damped driven oscillator is used in various applications, such as a clock signal source, voltage/current amplifiers, RF tuners, and in signal processing. 

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