A method for calculating phase noise based on a particle di#usion model in the phase plane is introduced. This approach allows for the derivation of closed-form solutions, which can be used to see the explicit dependencies of phase noise on circuit component values. The theory is applied to the Van der Pol and Colpitts oscillators, and comparisons to simulations show close matching.

Using a dynamical systems perspective, we develop a method to calculate phase noise in oscillators based on a particle diffusion model. Starting from the basic nonlinear equations of motion, a phase diffusion equation is formulated by viewing the system in state space. The resulting phase diffusion solution is then found approximately by using multiple scale perturbation theory to get closed-form analytical expressions. In this way, the effects of the nonlinearities on the phase diffusion become evident. Similar techniques are then also applied to finding the locking bandwidth of a 2:1 injection-locked frequency divider. Robert W. Brodersen, Chair 1 To my loving parents who have stood behind me every step of the way,

Abstract | In this tutorial paper, we review fundamentals of oscillator phase noise emphasizing its physical picture based on the concept of phase di®usion. This review is intended to provide an essential understanding of phase noise rather than a comprehensive phase noise modeling. We then summarize several phase noise reduction techniques for various types of oscillators. Index Terms | oscillators, phase noise, integrated circuits, analog integrated circuits, radio-frequency (RF) circuits. 1.

Parallel readout of a microcantilever array using single magnetic actuator and a single photo detector for concurrent detection is reported. The system includes MEMS cantilever array designed for different resonance frequencies, optical elements for laser beam shaping and focusing, one detector and feedback electronics, and single broadband actuator for parallel excitation. The cantilevers are made using a simple one-mask fabrication process with embedded amplitude gratings at the tips. A line shaped laser beam is used to illuminate the cantilevers. A single readout photodiode is placed at the first order diffraction beam location on the Fourier plane. The amplified photodiode signal is fed back into the magnetic actuation using a preamplifier and a broadband current amplifier. In this paper, we report for the first time parallel monitoring of the thermal resonance peaks of inherently frequency-multiplexed MEMS cantilevers. We demonstrated simultaneous self-sustained oscillations of seven cantilevers by using a single actuator and detector in air environment. The method is suitable for low-cost multiplexed portable biosensors.