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Journal ArticleDOI

Phase Noise in Oscillators: A Mathematical Analysis of Leeson's Model

01 Dec 1977-IEEE Transactions on Instrumentation and Measurement (IEEE)-Vol. 26, Iss: 4, pp 408-410
TL;DR: In this paper, the authors studied how an oscillator reacts to internal noise that occurs in the active element of the oscillator and provided theoretical analysis to express the Leeson's model in a more general form.
Abstract: This paper is devoted to one aspect of the study of phase noise in oscillators: how an oscillator reacts to internal noise that occurs in the active element. The following theoretical analysis will lead us to express the Leeson's model in a more general form.
Citations
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Journal ArticleDOI
TL;DR: The Langevin equations describing the deterministic and stochastic behaviour of an oscillator by perturbation methods are derived and applied to a Van der Pol oscillator exhibiting parametric sideband amplification and to a realized oscillator demonstrating the applicability of the theory to technically relevant circuits.
Abstract: In this paper the general correlation spectrum of an oscillator with white and f -α noise sources is derived from the Langevin equations describing the deterministic and stochastic behaviour of an oscillator by perturbation methods. The treatment of f -α noise and the influence of the finite measuring time on the noise spectra are included in a time domain calculation. The theory is applied to a Van der Pol oscillator exhibiting parametric sideband amplification and to a realized oscillator demonstrating the applicability of the theory to technically relevant circuits

297 citations

Journal ArticleDOI
05 Feb 1998
TL;DR: In this article, an integrated voltage-controlled oscillator (VCO) at a frequency of 2 GHz is implemented in a f/sub T/= 25 GHz standard bipolar process, where the phase noise of the VCO is -136 dBc/Hz at 4.684 MHz, when the integration bandwidth and the transmit output power of 25 dBm are taken into account.
Abstract: An integrated voltage-controlled oscillator (VCO) at a frequency of 2 GHz is implemented in a f/sub T/= 25 GHz standard bipolar process. The phase noise of the VCO is -136 dBc/Hz at 4.7 MHz frequency offset. The LC-resonator uses vertically coupled on-chip inductors and integrated tuning diodes. Due to the poor performance of integrated resonators on silicon ICs, oscillators with phase noise meeting requirements of wireless applications are difficult to integrate. With fully integrated designs only the standards for cordless phones, for instance DECT, can be achieved. The critical point in the DECT-specification is the emission of the transmitter due to intermodulation in the third adjacent channel, that must be <-47 dBm. This value is measured with an integration bandwidth of 1 MHz centered at the nominal center frequency. With a channel-spacing of 1.728 MHz the third adjacent channel is located 5.184 MHz from the actual transmit channel frequency. The beginning of the integration bandwidth is at an offset frequency of 4.684 MHz related to the nominal frequency of the transmit channel. This is the offset frequency, at which the specification must be met. The resulting noise requirement is -132 dBc/Hz at a offset frequency of 4.684 MHz, when the integration bandwidth and the transmit output power of 25 dBm are taken into account.

118 citations

Journal ArticleDOI
TL;DR: Measured phase noise for a microresonator-based oscillator is found to agree with the developed analytical and simulated noise models, and the capacitive transduction is shown to be the dominant mechanism for low-frequency 1/f-noise mixing into the carrier sidebands.
Abstract: Phase noise in capacitively coupled micro-resonator-based oscillators is investigated. A detailed analysis of noise mixing mechanisms in the resonator is presented, and the capacitive transduction is shown to be the dominant mechanism for low-frequency 1/f-noise mixing into the carrier sidebands. Thus, the capacitively coupled micromechanical resonators are expected to be more prone to the 1/f-noise aliasing than piezoelectrically coupled resonators. The analytical work is complemented with simulations, and a highly efficient and accurate simulation method for a quantitative noise analysis in closed-loop oscillator applications is presented. Measured phase noise for a microresonator-based oscillator is found to agree with the developed analytical and simulated noise models.

103 citations


Cites background or methods from "Phase Noise in Oscillators: A Mathe..."

  • ...The model is based on well-known Leeson’s model for the phase noise [7] [8] [10] and is expanded to incorporate the 1/f -noise aliasing in microresonators....

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  • ...As prior work on microresonators has focused on the power-handling capacity, only little attention has been given to near-carrier noise in micro-oscillators, but considerable theoretical and experimental work has been done on phase noise in conventional oscillators [7]–[11]....

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Journal ArticleDOI
01 Apr 1991
TL;DR: In this paper, it was shown that the fre- quency noise of the probing signal, at Fourier frequencies equal to even multiples of the modulation frequency, is translated into the frequency band of the selectively amplified resonator response, which sets a limit to its achievable frequency stability.
Abstract: It is shown that in passive frequency standards, the fre- quency noise of the probing signal, at Fourier frequencies equal to even multiples of the modulation frequency, is translated into the frequency band of the selectively amplified resonator response. Then an addi- tional perturbation of the slaved frequency source arises, which sets a limit to its achievable frequency stability. A quantitative estimate of this limit is given for the first and the third harmonic locking tech- niques. As an example, numerical values are given assuming a good crystal oscillator' as a frequency source. It is concluded that the limi- tation considered is very serious and may impinge on the expected fre- quency stability of newly developed frequency standards in which the resonance line can be observed with an enhanced signal to noise ratio. Such is the case, for instance, in devices using laser optically pumped cesium beam, rubidium cell, or stored ion(+

102 citations

Book ChapterDOI
Qiuting Huang1
TL;DR: In this paper, the steady-state amplitude of a CMOS LC Colpitts oscillator is analyzed and its response to small interferences is discussed. And the problem of signal dependency of noise sources is also addressed.
Abstract: An analysis is presented in this contribution to describe the steady-state amplitude of a CMOS LC Colpitts oscillator, as well as its response to small interferences. The problem of signal dependency of noise sources is also addressed. The general conclusions of the analysis are applicable to most LC oscillators. The procedure to perform a general analysis for an arbitrary LC oscillators is outlined. Controlled experiments are used to verify each important conclusion for the Colpitts analysis and implications on design are discussed.

100 citations

References
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Journal ArticleDOI
01 Feb 1966

2,440 citations


"Phase Noise in Oscillators: A Mathe..." refers background in this paper

  • ...Thiskindofspectral density hasoften beenobserved withquartz crystal oscillators [ 1 ], [2]....

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Book
28 Oct 2017
TL;DR: In this article, the spectral density S y (f) of the function y(t) where the spectrum is considered to be one-sided on a per hertz basis is defined.
Abstract: Consider a signal generator whose instantaneous output voltage V(t) may be written as V(t) = [V 0 + ??(t)] sin [2??v 0 t + s(t)] where V 0 and v 0 are the nominal amplitude and frequency, respectively, of the output. Provided that ??(t) and ??(t) = (d??/(dt) are sufficiently small for all time t, one may define the fractional instantaneous frequency deviation from nominal by the relation y(t) - ??(t)/2??v o A proposed definition for the measure of frequency stability is the spectral density S y (f) of the function y(t) where the spectrum is considered to be one sided on a per hertz basis. An alternative definition for the measure of stability is the infinite time average of the sample variance of two adjacent averages of y(t); that is, if y k = 1/t ??? tk+r = y(t k ) y(t) dt where ?? is the averaging period, t k+1 = t k + T, k = 0, 1, 2 ..., t 0 is arbitrary, and T is the time interval between the beginnings of two successive measurements of average frequency; then the second measure of stability is ?? y 2(??) ??? (y k+1 - y k )2/2 where denotes infinite time average and where T = ??. In practice, data records are of finite length and the infinite time averages implied in the definitions are normally not available; thus estimates for the two measures must be used. Estimates of S y (f) would be obtained from suitable averages either in the time domain or the frequency domain.

725 citations

Journal ArticleDOI
TL;DR: In this article, a test set based on high-pass filtering of phase noise was used to measure the short-term frequency instability of the best quartzcrystal oscillators in both time and frequency domains.
Abstract: A recently developed theoretical analysis has shown that it is possible to measure the Allan variance (a time-domain measure of frequency instability) without any statistical treatment of data from an electronic counter. The measurement is made via high-pass filtering of phase noise with a test set similar to the one used for frequency-domain measurements. The unique test set described in this paper relies on this principle and is capable of measuring the short-term frequency instability of the best quartzcrystal oscillators in both time and frequency domains.

18 citations