Leeson's equation

Leeson's equation is an empirical expression that describes an oscillator's phase noise spectrum.

Leeson's expression^[1] for single-sideband (SSB) phase noise in dBc/Hz (decibels relative to output level per Hertz) is^[2]

$L(f_m) = 10 \log \bigg[ \frac{1}{2} \bigg( \bigg(\frac{f_0}{2 Q_l f_m}\bigg)^2 + 1\bigg)\bigg(\frac{f_c}{f_m} + 1\bigg)\bigg(\frac{FkT}{P_s}\bigg) \bigg]$

where f₀ is the output frequency, Q_l is the loaded Q, f_m is the offset from the output frequency (Hz), f_c is the 1/f corner frequency, F is the noise factor of the amplifier, k is Boltzmann's constant, in Joules/Kevin , T is absolute temperature in Kelvins, and P_s is the output power.^[3]

References

1. Leeson, D. B. (February 1966), "A Simple Model of Feedback Oscillator Noise Spectrum", Proceedings of the IEEE 54 (2): 329–330, doi:10.1109/PROC.1966.4682
2. Rhea, Randall W. (1997), Oscillator Design & Computer Simulation (Second ed.), McGraw-Hill, ISBN 0-07-052415-7, p. 115.
3. Engineeri, Analog Devices Inc (2011-08-30). Linear Circuit Design Handbook. Newnes. ISBN 9780080559155.

Further reading

• Rubiola, Enrico (2008), Phase noise and frequency stability in oscillators, Cambridge, ISBN 978-0-521-15328-7
• Rohde, Ulrich L. (20 October 2011), Noise in Oscillators with Active Inductors (PDF), p. 9

External links

• 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
• Enrico Rubiola, The Leeson effect, http://arxiv.org/pdf/physics/0502143.pdf. Superseded by Rubiola 2008.
• http://www.ieee.li/pdf/essay/phase_noise_basics.pdf
• http://www.researchgate.net/publication/48339964_An_experimental_investigation_into_the_validity_of_Leeson's_equation_for_low_phase_noise_oscillator_design