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Forced Harmonic Oscillators Explained

Key Takeaways

  • Understand what a forced harmonic oscillator is.

  • Review examples of forced harmonic oscillators in electronics.

  • Clarify how a forced harmonic oscillator behaves at resonance.

Playground time was always fun for me and my son, except when he chose to play on the swings. As a daredevil, my son was never content with slow and gentle movement. Instead, he would coax the swing to fly higher and higher with his legs, eventually forcing me to intervene.

His ability to use his legs to increase the amplitude of his swinging motion always reminded me of a forced harmonic oscillator. In this piece, we are going to look at what forced harmonic oscillation is and how it is used in electronic design. 

What Is a Forced Harmonic Oscillator

Forced harmonic oscillation using the example of a swing

Pushing a swing is forced oscillation. 

First, let’s define the phrase harmonic oscillator as it relates to basic mechanics. A harmonic oscillator can be visualized as a pendulum that oscillates naturally in ideal conditions. Being in ideal conditions means the pendulum is not subjected to any gravitational or frictional forces. In such a scenario, the pendulum will continue to oscillate indefinitely at a constant amplitude.

In real life, an ideal harmonic oscillator does not exist because other conflicting forces do exist; what you’ve got instead is a damped oscillator. Just like a child playing on a swing, a damped oscillator is subjected to a frictional force that gradually reduces the oscillation. Eventually, the oscillator will stop vibrating and rest in an idle state.

A forced harmonic oscillator refers to a damped oscillator being subjected to an external force. When an external force is applied to an oscillator, it will undergo a transient state where the oscillator will shift from its natural frequency to the applied force’s frequency. 

Forced harmonic oscillation can be represented by the following equation:

m(d2x/dt2) + b(dx/dt) + kx = F0 cos ωdt

Forced Harmonic Oscillators in Electronics Design

Forced harmonic oscillator with an RLC circuit.

An RLC circuit is an example of a forced harmonic oscillator.

Incidences of forced harmonic oscillation are not only limited to a physical mechanism; the same principle is also observed in electronics design. RLC circuits are a common example of a forced harmonic oscillator in electronics and consist of a resistor, inductor, and capacitor. Harmonic oscillation occurs when either the inductor or capacitor has stored energy. When a charged capacitor is discharged, the current flows towards the inductor. This causes the inductor to build up a magnetic field, which will eventually collapse sending current back to the capacitor.

The process of current flowing back and forth between the inductor and the capacitor is harmonic oscillation. However, the resistance in the circuit causes damped oscillation. To enable continued oscillation, the RLC circuit is connected to an AC signal source which turns it into a forced harmonic oscillator. 

Forced Harmonic Oscillators and Resonance

Graph of forced harmonic oscillation at resonant frequency

Resonance happens when an oscillator is forced to oscillate at its natural frequency. 

Being in forced oscillation means that an external force is applied to an oscillating system. It doesn’t guarantee that the system will oscillate at a higher amplitude. As with the example of riding on a swing, only pumping your legs at the appropriate time will drive the swing higher. 

The frequency of the applied force is crucial in determining the system’s oscillation response. At a specific frequency, the system will oscillate at its peak amplitude. The frequency for the oscillator to respond with maximum amplitude is called the resonant frequency. When an exerted force matches the oscillator’s resonant frequency, it is said to be in resonance. 

In an RLC circuit, the resonant frequency can be obtained via the following equation.

 fr= 1/2π√LC

An RLC circuit in resonance will result in peak frequency response. Its impedance will be purely resistive as both capacitive and inductive reactance will be at the same but opposite magnitude. For a series RLC circuit, the impedance will be at a minimum at resonance and the opposite is true for a parallel RLC circuit. 

When tasked with designing forced harmonic oscillator circuits, you’ll need a complete, integrated, and simulation-driven PCB design and analysis software. Allegro has what it takes to deliver well-performing, accurately designed circuits with its robust range of innovative features.

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts