Leeson's equation is an empirical expression that describes an

oscillator Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...

phase noise In signal processing, phase noise is the frequency-domain representation of random fluctuations in the phase of a waveform, corresponding to time-domain deviations from perfect periodicity (jitter). Generally speaking, radio-frequency engineers ...

spectrum. Leeson's expression for single-sideband (SSB) phase noise in dBc/Hz (decibels relative to output level per hertz) and augmented for

flicker noise Flicker noise is a type of electronic noise with a 1/''f'' power spectral density. It is therefore often referred to as 1/''f'' noise or pink noise, though these terms have wider definitions. It occurs in almost all electronic devices and can show ...

: :

L(f_m) = 10 \log \bigg \frac \bigg( \bigg(\frac\bigg)^2 + 1\bigg)\bigg(\frac + 1\bigg)\bigg(\frac\bigg) \bigg /math>
where  is the output frequency,  is the loaded

quality factor In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...

, is the offset from the output frequency (Hz), is the corner frequency, is the

noise factor Noise figure (NF) and noise factor (''F'') are figures of merit that indicate degradation of the signal-to-noise ratio (SNR) that is caused by components in a signal chain. These figures of merit are used to evaluate the performance of an amplifie ...

of the amplifier, is

Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...

in joules/kelvin, is absolute temperature in kelvins, and is the available power at the sustaining amplifier input.https://www.ieee.li/pdf/essay/phase_noise_basics.pdf There is often misunderstanding around Leeson's equation, even in text books. In the 1966 paper, Leeson stated correctly that " is the signal level at the oscillator active element input" (often referred to as the power through the resonator now, strictly speaking it is the available power at the amplifier input). F is the device noise factor, however this does need to be measured at the operating power level. The common misunderstanding, that is the oscillator output level, may result from derivations that are not completely general. In 1982, W. P. Robins (IEEE Publication "Phase noise in signal sources") correctly showed that the Leeson equation (in the -20 dB/decade region) is not just an empirical rule, but a result that follows from a linear analysis of an oscillator circuit. However, a used constraint in his circuit was that the oscillator output power was approximately equal to the active device input power. The Leeson equation is presented in various forms. In the above equation, if is set to zero the equation represents a linear analysis of a feedback oscillator in the general case (and flicker noise is not included), it is for this that Leeson is most recognised, showing a -20 dB/ decade of offset frequency slope. If used correctly, the Leeson equation gives a useful prediction of oscillator performance in this range. If a value for is included, the equation also shows a curve fit for the flicker noise. The for an amplifier depends on the actual configuration used, because radio-frequency and low-frequency

negative feedback Negative feedback (or balancing feedback) occurs when some function (Mathematics), function of the output of a system, process, or mechanism is feedback, fed back in a manner that tends to reduce the fluctuations in the output, whether caused by ...

can have an effect on . So for accurate results, must be determined from added noise measurements on the amplifier using R.F., with the actual circuit configuration to be used in the oscillator. Evidence that is the amplifier input power (often contradicted or very unclear in text books) can be found in the derivation in further reading which also shows experimental results, Enrico Rubiola, The Leeson Effect also shows this in a different form.

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*Ali M. Niknejad, Oscillator Phase Noise, University of California, Berkeley, 2009 http://rfic.eecs.berkeley.edu/~niknejad/ee242/pdf/eecs242_lect22_phasenoise.pdf, stating "Leeson modified the above noise model to account for several experimentally observed phenomena". Also, "In Leeson’s model, the factor ''F'' is a fitting parameter rather than arising from any physical concepts. It’s tempting to call this the oscillator "noise figure", but this is misleading." *John van der Merwe, An Experimental Investigation into the Validity of Leeson's Equation for Low Phase Noise Oscillator Design, December 2010, https://scholar.sun.ac.za/bitstream/handle/10019.1/5424/vandermerwe_experimental_2010.pdf and http://www.researchgate.net/publication/48339964_An_experimental_investigation_into_the_validity_of_Leeson's_equation_for_low_phase_noise_oscillator_design *Enrico Rubiola, The Leeson effect, . Superseded by . {{Electronic oscillators Electronic oscillators

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