## Oscillator Phase Noise: A Tutorial (2000)

Venue: | IEEE J. Solid-State Circuits |

Citations: | 42 - 3 self |

### BibTeX

@ARTICLE{Lee00oscillatorphase,

author = {Thomas H. Lee and Ali Hajimiri},

title = {Oscillator Phase Noise: A Tutorial},

journal = {IEEE J. Solid-State Circuits},

year = {2000},

volume = {35},

pages = {326--336}

}

### OpenURL

### Abstract

Linear time-invariant (LTI) phase noise theories provide important qualitative design insights but are limited in their quantitative predictive power. Part of the difficulty is that device noise undergoes multiple frequency translations to become oscillator phase noise. A quantitative understanding of this process requires abandoning the principle of time invariance assumed in most older theories of phase noise. Fortunately, the noise-to-phase transfer function of oscillators is still linear, despite the existence of the nonlinearities necessary for amplitude stabilization. In addition to providing a quantitative reconciliation between theory and measurement, the time-varying phase-noise model presented in this tutorial identifies the importance of symmetry in suppressing the upconversion of 1 noise into close-in phase noise, and provides an explicit appreciation of cyclostationary effects and AM--PM conversion. These insights allow a reinterpretation of why the Colpitts oscillator exh...

### Citations

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(Show Context)
Citation Context ...Hz” actually applies to the argument of the log, not to the log itself; doubling the measurement bandwidth does not double the decibel quantity. As lacking in rigor as “dBc/Hz” is, it is common =-=usage [1]-=-. Equation (12) tells us that phase noise (at a given offset) improves as both the carrier power and increase, as predicted earlier. These dependencies make sense. Increasing the signal power improves... |

164 | A general theory of phase noise in electrical oscillators
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Citation Context .... In fact, it is surprisingly simple to demonstrate that oscillators are fundamentally time-varying systems. Recognizing this truth is the main key to developing a more accurate theory of phase noise =-=[3]-=-. To show that time invariance fails to hold, consider explicitly how an impulse of current affects the waveform of the simplest resonant system, a lossless LC tank (Fig. 3). Assume that the system is... |

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Citation Context ...r of unity (inside the braces) to account for the noise floor, and a multiplicative factor (the term in the second set of parentheses) to provide a 1 behavior at sufficiently small offset frequencies =-=[2]-=-. With these modifications, the phase-noise spectrum appears as in Fig. 2. It is important to note that the factor is an empirical fitting parameter and therefore must be determined from measurements,... |

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Citation Context ...r are the same, the ISF for different nodes will differ only in phase by multiples of 2 , as shown in Fig. 15. Therefore, the total phase due to all the sources is given by (15) through superposition =-=[13]-=- (29) Expanding the term in brackets in a Fourier series, it can be observed that it is zero except at dc and multiples of , i.e., (30) which means that for fully correlated sources, only noise in the... |

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Citation Context ...nonlinear behavior. One observation is that a single-frequency sinusoidal disturbance injected into an oscillator gives rise to two equal-amplitude sidebands, symmetrically disposed about the carrier =-=[7]-=-. Since LTI systems cannot perform frequency translation and nonlinear systems can, nonlinear mixing has occasionally been proposed to explain phase noise. Unfortunately, the amplitude of the sideband... |

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Citation Context ...fact that we are dealing with a power quantity here). This relationship can be, and has been, verified experimentally. The foregoing result may be extended to the general case of a white noise source =-=[6]-=- (21) Equation (20) implies both upward and downward frequency translations of noise into the noise near the carrier, as illustrated in Fig. 8. This figure summarizes what the foregoing equations tell... |

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Citation Context ...an be approximated as 1 Note that the amplitude change is only equal to the normalized voltage change 1†a† if the impulse is injected at the peak of the sinusoidal tank voltage. first or second or=-=der [12]-=-. The function typically will thus be either a dying exponential or a damped sinusoid. For a first-order system (33) Therefore the excess amplitude response to an arbitrary input current is given by t... |

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Citation Context ...form. The LTV model is able to accommodate a cyclostationary white noise source with ease, since such a source may be treated as the product of a stationary white noise source and a periodic function =-=[14]-=- (27)s332 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 35, NO. 3, MARCH 2000 Here, is a stationary white noise source whose peak value is equal to that of the cyclostationary source, and is a periodic u... |

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Citation Context ...ration that does so is Clapp’s modification to the Colpitts oscillator (Fig. 11). Differential implementations of oscillators with tapped resonators have recently made an appearance in the literatur=-=e [5]-=-, [9], [10]. These types of oscillators become increasingly attractive as supply voltages scale downward, where conventional resonator connections lead to -constrained signal swings. Use of tapping al... |

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Citation Context ...n that does so is Clapp’s modification to the Colpitts oscillator (Fig. 11). Differential implementations of oscillators with tapped resonators have recently made an appearance in the literature [5]=-=, [9]-=-, [10]. These types of oscillators become increasingly attractive as supply voltages scale downward, where conventional resonator connections lead to -constrained signal swings. Use of tapping allows ... |

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Citation Context ...that substrate noise appears as a common-mode source. This latter consideration is particularly important in the case of a lightly doped substrate, since such a substrate may not act as a single node =-=[11]-=-. It is also important that the orientation of all the stages be kept identical. The interconnecting wires between the stages must be identical in length and shape, and a common supply line should fee... |

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Citation Context ...t does so is Clapp’s modification to the Colpitts oscillator (Fig. 11). Differential implementations of oscillators with tapped resonators have recently made an appearance in the literature [5], [9]=-=, [10]-=-. These types of oscillators become increasingly attractive as supply voltages scale downward, where conventional resonator connections lead to -constrained signal swings. Use of tapping allows signal... |

1 |
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Citation Context ...e oscillation waveform. In general, it is most practical (and most accurate) to determine through simulation, but there are also analytical methods (some approximate) that apply in special cases [4], =-=[8]. -=-In any event, to develop a feel for typical shapes of ISF’s, consider two representative examples, first for an LC and a ring oscillator in Fig. 5(a) and (b). Once the ISF has been determined (by wh... |

1 |
S’87–M’91) received the
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(Show Context)
Citation Context ...ffsets and thus ultimately produce a noise floor. Last, there is almost always a 1 region at small offsets (we will ignore here the eventual flattening of the spectrum at extremely small offsets [4], =-=[15]-=-). A modification to (12) provides a means to account for these discrepancies (13) These modifications, due to Leeson, consist of a factor to account for the increased noise in the 1 region, an additi... |