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

Venue: | IEEE Journal of Solid-State Circuits |

Citations: | 12 - 3 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.

### Citations

540 |
Semiconductor Devices
- Sze
- 2002
(Show Context)
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... |

165 | A General Theory of Phase Noise in Electrical Oscillators
- Hajimiri, Lee
- 1998
(Show Context)
Citation Context ...UCTION OSCILLATOR phase noise has been studied from several different angles, ranging from a mathematical physics treatment [1]–[4] to CAD-oriented methods [5], [6] and design-oriented approaches [7]–=-=[12]-=-, to name a few. The design-oriented approaches have evolved from a familiar linear time-invariant approach [7] to a more accurate time-varying theory [12], adding additional insight into the oscillat... |

149 | Phase Noise in Oscillators: A Unifying Theory and Numerical Methods for Characterization
- Demir, Mehrotra, et al.
- 2000
(Show Context)
Citation Context ...resonators, ring oscillators. I. INTRODUCTION OSCILLATOR phase noise has been studied from several different angles, ranging from a mathematical physics treatment [1]–[4] to CAD-oriented methods [5], =-=[6]-=- and design-oriented approaches [7]–[12], to name a few. The design-oriented approaches have evolved from a familiar linear time-invariant approach [7] to a more accurate time-varying theory [12], add... |

148 |
A Simplified Model of Feedback Oscillator Noise Spectrum
- Leeson
- 1965
(Show Context)
Citation Context ...TRODUCTION OSCILLATOR phase noise has been studied from several different angles, ranging from a mathematical physics treatment [1]–[4] to CAD-oriented methods [5], [6] and design-oriented approaches =-=[7]-=-–[12], to name a few. The design-oriented approaches have evolved from a familiar linear time-invariant approach [7] to a more accurate time-varying theory [12], adding additional insight into the osc... |

129 |
Topics in the Theory of Random Noise
- Stratonovich
(Show Context)
Citation Context ..., radio-frequency (RF) circuits, resonators, ring oscillators. I. INTRODUCTION OSCILLATOR phase noise has been studied from several different angles, ranging from a mathematical physics treatment [1]–=-=[4]-=- to CAD-oriented methods [5], [6] and design-oriented approaches [7]–[12], to name a few. The design-oriented approaches have evolved from a familiar linear time-invariant approach [7] to a more accur... |

119 |
Fundamentals of Statistical and Thermal Physics
- Reif
- 1965
(Show Context)
Citation Context ...to white noise, the variance of , which signifies the width of the probability distribution, is given by a key signature of any diffusion process subject to white noise (a.k.a., Wiener process) [18], =-=[19]-=-. The validity of (2) in the general time-varying case will be positively confirmed in Section IV. Extensive discussion of (2) can also be found in [20]–[22] in the special context of timing jitters i... |

87 | Jitter and Phase Noise in Ring Oscillators
- Hajimiri, Limotyrakis, et al.
- 1999
(Show Context)
Citation Context ...white noise (a.k.a., Wiener process) [18], [19]. The validity of (2) in the general time-varying case will be positively confirmed in Section IV. Extensive discussion of (2) can also be found in [20]–=-=[22]-=- in the special context of timing jitters in ring oscillators. The phase diffusion constant indicates how fast the phase diffusion occurs. 1 As will be seen shortly, this phase diffusion constant will... |

77 | A Study of Phase Noise in CMOS Oscillators - Razavi - 1996 |

73 |
Theory of oscillators
- Andronov, Vitt, et al.
- 1987
(Show Context)
Citation Context ...of the inductor onto a trajectory in the – state space, as shown in Fig. 2. For a stable oscillator, the limit cycle representing the steady-state oscillation is a closed curve due to the periodicity =-=[17]-=-. Regardless of its starting point, the state will be ultimately attracted to the limit cycle after the initial transient fades away, as shown in Fig. 2. This peculiar property of the self-sustained o... |

55 | Jitter in Ring Oscillators
- McNeill
- 1997
(Show Context)
Citation Context ...ant, different notations are often found in the literature. For example, in [1], ‡ is used as an indicator of the diffusion rate where h a ‡aP. The constant , used in the context of timing jitters in =-=[21]-=- and [22], is related to h through h a 3 aP. (2) (3) Fig. 5. Power spectral densities of the oscillator output for different diffusion constants for the same oscillation energy. the Lorentzian shape i... |

55 |
The Design of Low Noise Oscillators
- Hajimiri, Lee
- 1999
(Show Context)
Citation Context ...e to the periodic sensitivity of the state point to the perturbation. We use a unitless periodic function to denote this periodic sensitivity, the so-called impulse sensitivity function (ISF) 5 [12], =-=[34]-=-. Then the governing equation of the phase dynamics can be written as where (10) (11) signifies the charge amplitude. Additionally, the noise source is cyclostationary in general due to the periodic c... |

52 |
Design issues in CMOS differential LC oscillators
- Hajimiri, Lee
- 1999
(Show Context)
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... |

51 |
A Study of Oscillator Jitter Due to Supply and Substrate Noise
- Herzel, Razavi
- 1999
(Show Context)
Citation Context ... section are found (perhaps apart from certain notational differences). Essentially the same equations have reappeared in numerous subsequent developments to analyze phase noise (e.g., [6], [15], and =-=[16]-=-, to name a few). We will refer only to original source(s) in this section. The phase diffusion concept reviewed in this section will be crucial in understanding the material presented in later sectio... |

49 |
Annalen der Physik
- Einstein
- 1918
(Show Context)
Citation Context ... the friction (energy loss) of the environment in which the diffusion process occurs. This important notion is perfectly captured in the Einstein relation, which he derived to explain Brownian motion =-=[36]-=-, [37]. We will begin with a brief introduction to the Einstein relation in the context of Brownian motion. This mechanical example facilitates intuitive understanding, appealing to our everyday exper... |

48 |
Concepts and methods in optimization of integrated LC VCOs
- Ham, Hajimiri
- 2001
(Show Context)
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... |

42 | Hajimiri A. Oscillator Phase Noise: a Tutorial
- Lee
- 2000
(Show Context)
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... |

30 |
A 2.6 GHz/5.2 GHz CMOS voltage-controlled oscillator
- Lam, Razavi
- 1999
(Show Context)
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... |

28 | Analysis of timing jitter in CMOS ring oscillators - Weigandt, Kim, et al. - 1994 |

25 | A Filtering Technique to Lower LC Oscillator Phase Noise - Hegazi, Sjöland, et al. - 2001 |

17 |
Classical noise V. Noise in self-sustained oscillators”, Phys
- Lax
- 1967
(Show Context)
Citation Context ...oise, radio-frequency (RF) circuits, resonators, ring oscillators. I. INTRODUCTION OSCILLATOR phase noise has been studied from several different angles, ranging from a mathematical physics treatment =-=[1]-=-–[4] to CAD-oriented methods [5], [6] and design-oriented approaches [7]–[12], to name a few. The design-oriented approaches have evolved from a familiar linear time-invariant approach [7] to a more a... |

11 |
An Analytical Model for the Power Spectral Density of a Voltage-Controlled Oscillator and Its Analogy to the Laser Linewidth Theory
- Herzel
- 1998
(Show Context)
Citation Context ...ns of this section are found (perhaps apart from certain notational differences). Essentially the same equations have reappeared in numerous subsequent developments to analyze phase noise (e.g., [6], =-=[15]-=-, and [16], to name a few). We will refer only to original source(s) in this section. The phase diffusion concept reviewed in this section will be crucial in understanding the material presented in la... |

10 | Noise in oscillators - Edson - 1960 |

9 | A Stochastic Theory of Line-Shape and Relaxation - Kubo - 1961 |

9 | A Packaged 1.1GHz CMOS VCO with Phase Noise of -126dBc/Hz at a 600kHz Offset - Hung, Kenneth |

8 | Abidi, “physical processes of phase noise - A |

7 | A low-phase-noise CMOS LC oscillator with a ring structure - Kim, Kim - 2000 |

6 | Phase noise in LC oscillators - Kouznetsov, Meyer - 2000 |

6 | A 1.3 GHz low-phase noise fully tunable CMOS - Svelto, Deantoni, et al. - 2000 |

4 |
Phase Noise to Carrier Ratio in LC Oscillators
- Huang
- 2000
(Show Context)
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 ... |

3 |
Analysis of white and f noise in oscillators,” Int
- Kärtner
- 1990
(Show Context)
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 ... |

3 |
Nonlinear effects of noise in electronic clocks
- Grivet, Blaquiere
- 1963
(Show Context)
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... |

3 |
Noise in Solid-State DeVices and Circuits
- Ziel
- 1986
(Show Context)
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
- Ham, Hajimiri
- 2002
(Show Context)
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... |

2 |
A 10-GHz CMOS distributed voltage-controlled oscillator
- Wu, Hajimiri
- 2000
(Show Context)
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... |

2 |
Statistical Electronics: Noise Processes in Integrated Communication Systems
- Ham
- 2002
(Show Context)
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 |
A low-noise transformerbased 1.7-GHz CMOS
- Straayer, Cabanillas, et al.
- 2002
(Show Context)
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... |