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Oscillations In An LC Circuit01:30

Oscillations In An LC Circuit

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An idealized LC circuit of zero resistance can oscillate without any source of emf by shifting the energy stored in the circuit between the electric and magnetic fields. In such an LC circuit, if the capacitor contains a charge q before the switch is closed, then all the energy of the circuit is initially stored in the electric field of the capacitor. This energy is given by
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RLC Circuit as a Damped Oscillator01:30

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An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
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In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
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When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.
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Design Example: Underdamped Parallel RLC Circuit01:17

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Consider designing an oscillator circuit, a crucial component in various electronic devices and systems. The objective is to create an oscillator circuit with specific characteristics: a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. To accomplish this, a parallel RLC circuit is employed, known for its ability to sustain oscillations at a resonant frequency. In this case, the damping factor is pivotal in achieving the desired performance.
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Types of Damping01:20

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If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...
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Amplitude death in oscillator networks with variable-delay coupling.

Aleksandar Gjurchinovski1, Anna Zakharova2, Eckehard Schöll2

  • 1Institute of Physics, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, P. O. Box 162, 1000 Skopje, Macedonia.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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Summary

We investigated amplitude death in coupled oscillators with time-varying delays. Our method, generalizing master stability functions, shows improved control over constant delays for stabilizing oscillator networks.

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Area of Science:

  • Nonlinear dynamics
  • Complex networks
  • Systems theory

Background:

  • Amplitude death is a phenomenon in coupled oscillators where oscillations cease.
  • Time-varying delays in coupling can significantly alter network dynamics.
  • Understanding stability conditions is crucial for designing and controlling complex systems.

Purpose of the Study:

  • To investigate amplitude death in networks of delay-coupled limit cycle oscillators.
  • To analyze the impact of time-varying delays and self-feedback on stability.
  • To generalize the master stability function formalism for variable-delay systems.

Main Methods:

  • Generalizing the master stability function (MSF) formalism to incorporate distributed delays.
  • Analyzing amplitude death regimes in a ring network of Stuart-Landau oscillators.
  • Comparing the efficacy of time-varying delays against constant delays.

Main Results:

  • The proposed method, incorporating high-frequency delay modulations, effectively analyzes amplitude death.
  • Time-varying delays offer superior control over amplitude death compared to constant delays.
  • The odd-number property of local node dynamics restricts steady-state stabilization, irrespective of network topology.

Conclusions:

  • Time-varying delays provide a powerful tool for controlling amplitude death in oscillator networks.
  • The generalized MSF formalism is effective for analyzing complex delay-coupled systems.
  • Local node properties play a fundamental role in network stability, independent of global parameters.