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Oscillations In An LC Circuit01:30

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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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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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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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An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
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Fabrication and Characterization of High-Q Silicon Nitride Membrane Resonators
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Modeling nonlinearities in MEMS oscillators.

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    Summary
    This summary is machine-generated.

    This study introduces a unified mathematical model for microelectromechanical system (MEMS) oscillators, integrating resonator and circuit nonlinearities. The model accurately predicts MEMS oscillator performance, enhancing device design and analysis.

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    Area of Science:

    • Electrical Engineering
    • Mechanical Engineering
    • Materials Science

    Background:

    • Microelectromechanical systems (MEMS) resonators exhibit nonlinear behavior crucial for oscillator design.
    • Accurate modeling of MEMS oscillator nonlinearities is essential for performance prediction and optimization.
    • Existing models often treat resonator and circuit nonlinearities separately, limiting comprehensive analysis.

    Purpose of the Study:

    • To develop a unified mathematical model for MEMS oscillators that incorporates both resonator and circuit nonlinearities.
    • To validate the proposed model against experimental data for MEMS resonators and oscillators.
    • To derive analytical expressions for steady-state output power and frequency.

    Main Methods:

    • Transforming the nonlinear mechanical model of MEMS resonators into the electrical domain.
    • Developing a nonlinear electrical model for MEMS resonators, considering prominent nonlinearities.
    • Investigating essential nonlinearities within oscillator circuits, specifically transimpedance amplifier-based designs.
    • Deriving closed-form expressions for steady-state output power and frequency.

    Main Results:

    • The nonlinear electrical model of the MEMS resonator was validated against experimental amplitude-frequency response data.
    • A comprehensive mathematical model for MEMS oscillators integrating resonator nonlinearities was proposed.
    • Closed-form expressions for output power and frequency were derived for square-wave and sine-wave MEMS oscillators.
    • The derived expressions showed good agreement with experimental and simulation results.

    Conclusions:

    • The unified model effectively captures the essential nonlinearities in MEMS oscillators.
    • The proposed modeling approach facilitates accurate prediction of MEMS oscillator performance.
    • This work provides a valuable tool for the design and analysis of advanced MEMS-based oscillators.