Noise Model of Relaxation Oscillators Due to Feedback Regeneration Based on Physical Phase Change
Bosco Leung
备注：
The 2nd version Fixed the mistakes: major one is H(hamiltonian) in eq 50 of ver1 (now eq. 54 of ver 2) is replaced by L(Lagrangian). Footnotes are rearranged (e.g. footnote 1 in 2nd paragraph of Introduction section is removed)
提出了一种研究弛张振荡器再生噪声尖峰的新方法。在传统的相位噪声模型中，噪声尖峰没有得到令人满意的解释。本文试图将噪声尖峰/跳跃现象视为热力学系统中的相变（例如从气体到液体或铁磁体的磁化）来解释。两者都是由于再生（振荡器中的正反馈以及由于铁磁体中的正反馈导致的自旋对齐）。所用的数学工具是热力学中的配分函数，并将结果映射到热力学系统和弛张振子之间。理论的发展和公式推导出预测的跳跃的幅度，作为一个功能的设计参数，如再生参数或环路增益。公式表明，当再生参数/回路增益接近1时，噪声急剧增加，在相变中温度接近临界温度时，情况大致相同。对使用CMOS的电路（Eldo）进行了模拟，以及对铁磁体（伊辛模型）进行了蒙特卡罗模拟（Metropolis），两者都显示出与公式一致的跳跃行为。在0.13 μ m CMOS工艺制造的弛张振荡器上的测量验证了这种行为，其中当再生参数/环路增益接近1时噪声急剧增加，与理论公式密切匹配。使用该公式，设计人员可以通过其对再生参数/环路增益的影响来量化噪声尖峰的变化，该噪声尖峰依赖于设计参数，例如gm（器件增益）、R、I 0。
英文摘要：
A new approach to investigate noise spikes due to regeneration in a relaxation oscillator is proposed. Noise spikes have not been satisfactorily accounted for in traditional phase noise models. This paper attempts to explain noise spikes/jump phenomenon by viewing it as phase change in the thermodynamic system(for example, from gas to liquid or magnetization of ferromagnet). Both are due to regeneration (positive feedback in oscillator as well as alignment of spin due to positive feedback in ferromagnet). The mathematical tool used is the partition function in thermodynamics, and the results mapped between thermodynamic system and relaxation oscillator. Theory is developed and formula derived to predict the magnitude of the jump, as a function of design parameter such as regeneration parameter or loop gain. Formulas show that noise increases sharply as regeneration parameter/loop gain approaches one, in much the same way when temperature approaches critical temperature in phase change. Simulations on circuits (Eldo) using CMOS as well as Monte Carlo simulations (Metropolis) on ferromagnet (Ising model) were performed and both show jump behaviour consistent with formula. Measurements on relaxation oscillators fabricated in 0.13um CMOS technology verify such behaviour, where the sharp increase in noise when regeneration parameter/loop gain is close to one, matches closely with the theoretical formula. Using the formula the designer can quantify the variation of noise spikes dependency on design parameters such as gm (device transconductance), R, I0, via their influence on regeneration parameter/loop gain.
The narrow peak on the frequency dependance of the susceptibility of the harmonical oscillator in contact with thermostat with bosonic excitations. Exactly-solvable model
Volterra-series approach to stochastic nonlinear dynamics: linear response of the Van der Pol oscillator driven by white noise
Van der Pol方程是弛豫振荡的典型模型。这种显著的非线性自持振荡运动现象是自然界和电气工程中重要的节律过程的基础。真实系统中的弛豫振荡通常与环境噪声耦合，这进一步丰富了它们的动力学，但使得对此类系统的理论分析和方程参数值的确定成为一项困难的任务。在另一篇文章中，我们提出了一个类似的问题的另一个经典的非线性模型，D-Duffing振子的分析方法。在这里，我们扩展我们的技术的情况下，由白噪声驱动的范德波尔方程。我们分析了解决方案的统计，并提出了一种方法来估计参数值从振荡器的时间序列。我们使用生物系统中主动振荡的实验数据来演示我们的方法如何适用于实际观测，以及如何将其推广到更复杂的模型。
