## Virtual damping and Einstein relation in oscillators (2003)

Venue: | IEEE Journal of Solid-State Circuits |

Citations: | 8 - 2 self |

### BibTeX

@ARTICLE{Ham03virtualdamping,

author = {Donhee Ham and Ali Hajimiri},

title = {Virtual damping and Einstein relation in oscillators},

journal = {IEEE Journal of Solid-State Circuits},

year = {2003},

volume = {38},

pages = {407--418}

}

### OpenURL

### Abstract

Abstract—This paper presents a new physical theory of oscillator phase noise. Built around the concept of phase diffusion, this work bridges the fundamental physics of noise and existing oscillator phase-noise theories. The virtual damping of an ensemble of oscillators is introduced as a measure of phase noise. The explanation of linewidth compression through virtual damping provides a unified view of resonators and oscillators. The direct correspondence between phase noise and the Einstein relation is demonstrated, which reveals the underlying physics of phase noise. The validity of the new approach is confirmed by consistent experimental agreement. Index Terms—Analog integrated circuits, LC oscillators, oscillators, phase noise, radio-frequency (RF) circuits, resonators, ring oscillators. I.

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Citation Context ...sing the Einstein relation. To this end, we recast the expression for the phase-diffusion con6 In semiconductor physics, the Einstein relation is usually expressed as ha" a � „a�, where " is mobility =-=[38]-=-. It can be easily shown that this expression is equivalent to (18) [19].sHAM AND HAJIMIRI: VIRTUAL DAMPING AND EINSTEIN RELATION IN OSCILLATORS 415 stant in (17) into a different form by expressing t... |

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Citation Context ...or design. The second step of the NCR minimization is typically exercised through oscillation amplitude maximization, but there have recently been emphases on the more proper tank energy maximization =-=[26]-=-, [30], [32]. However, these two quantities, and NCR, have been understood rather separately but not in the context of the two-step procedure, and understanding of how phase noise behaves as a whole d... |

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Citation Context ...inimizing in (7) in placing the resonator in the positive feedback loop. As will be seen in Section VI, the linewidth compression ratio strongly depends on the noise-to-carrier ratio (NCR) defined as =-=[30]-=- (7) (8) NCR (9) where and are the thermal and tank energy, respectively. Therefore, the maximum linewidth compression in the second step may be achieved through the minimization of the NCR. 4 Each of... |

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Citation Context ...d the tank energy of the oscillator are given by and , respectively, as mentioned earlier. Although there have recently been emphases on the importance of the NCR in design of oscillators [30], [32], =-=[33]-=-, the link between the phase noise and the NCR has not been clearly explained. Now, the above equation in association with the Einstein relation elucidates the link, spelling out that the NCR originat... |

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Citation Context ...ent system. As can be seen, currently available silicon-based on-chip oscillators whose phase-noise values range typically between 100 dBc/Hz and 140 dBc/Hz at the offset frequency (e.g., [13], [14], =-=[25]-=-–[31]) have virtual damping time constants roughly between milliseconds and seconds. B. Experimental Verification We can observe the virtual damping phenomenon experimentally as well. Typical LC oscil... |

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Citation Context ...he second step of the NCR minimization is typically exercised through oscillation amplitude maximization, but there have recently been emphases on the more proper tank energy maximization [26], [30], =-=[32]-=-. However, these two quantities, and NCR, have been understood rather separately but not in the context of the two-step procedure, and understanding of how phase noise behaves as a whole depending on ... |

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Citation Context ... time, if there were no such perturbation, the state point would travel along the limit cycle to end up at point . This shift from to is responsible for the phase change due to the noise perturbation =-=[5]-=-, [12]. For small enough perturbations, is proportional to and is inversely proportional to the oscillation amplitude . For the same , the resultant phase change also depends on where the state point ... |

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Citation Context ... formula for thermal noise. As another example, for the MOS transistor whose channel thermal noise intensity is given by , where is the MOS thermal noise factor and is the channel transconductance at =-=[39]-=-, is equal to . On the other hand, if the noise is of nonthermal origin (e.g., shot noise), does not necessarily represent the loss since the fluctuation-dissipation relation only holds true for the t... |

3 | Virtual Damping in Oscillators
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Citation Context ...o fundamental that the phase-noise formula can be directly obtained through a cogent physical argument utilizing the Einstein relation, without going through the mathematical derivation in Section IV =-=[40]-=-. C. NCR Versus Phase Noise Using the definition of NCR in (9), we can rewrite (25) as NCR (26) where the thermal energy and the tank energy of the oscillator are given by and , respectively, as menti... |

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Citation Context ...e diffusion constant , as we guessed earlier. From the mathematical perspective, (6) per se is not surprisingly new as it results from the standard probability calculation similar to the one shown in =-=[24]-=-, which alone does not necessarily provide a physical meaning in conjunction with phase diffusion or its far-reaching implications. However, the virtual damping concept indeed has important impact, le... |

2 |
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Citation Context ...ystem. As can be seen, currently available silicon-based on-chip oscillators whose phase-noise values range typically between 100 dBc/Hz and 140 dBc/Hz at the offset frequency (e.g., [13], [14], [25]–=-=[31]-=-) have virtual damping time constants roughly between milliseconds and seconds. B. Experimental Verification We can observe the virtual damping phenomenon experimentally as well. Typical LC oscillator... |

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Citation Context ...evolution of the phase probability distribution , introduced in Section II. [See Fig. 4(c).] This type of conversion is a standard exercise in stochastic theory [18] and in this special case leads to =-=[35]-=- where (13) (14) Equation (13) resembles the typical diffusion equation [18], [19] except that the diffusion rate is not constant but periodic, due to the time-varying effect represented by . Seemingl... |

1 |
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Citation Context ... oscillator design. These studies have helped circuit designers understand the evolution of noise in oscillators, leading to more accurate phase-noise predictions and lower noise designs (e.g., [13], =-=[14]-=-). However, the currently available works on the phase noise assume rather phenomenological or pure mathematical standpoints and a more fundamental yet intuitive understanding of the phase-noise pheno... |