Let's Treat Leeson Right

Your June issue featured an interesting article by Eoin Carey on the effects of VCO tuning bandwidth on phase noise. The article proposed that the phase noise of two otherwise similar VCOs that were based on the same design approaches and technologies would differ according to their tuning bandwidths. The article based its analysis on the well-known equation for phase noise presented by D.B. Leeson in 1966 ("A simple model of feedback oscillator noise spectrum," Proceedings of the IEEE, Vol. 54, February 1966, pp. 329-330).

Unfortunately, Mr. Carey's rendering of Mr. Leeson's incomplete empirical expression for phase noise :

L(ΔΩ) = 10log{(2FkT/Psig)0/2QΔΩ)2>1/f3)/|ΔΩ|>}

somewhat simplified the formula in terms of several noise contributions. For comparison, consider Leeson's phase-noise equation as presented (on p. 302) in my text Communications Receivers, Principles & Design:

L(fm) = 10log{0 2/(2fmQload)2> (1 + fc/fm)(FkT/2Ps av) + (2kTRKO2)/fm2}

Although different notation has been used throughout, such as ΔΩ and fm for the frequency offset in which the phase noise is predicted, the former treatment of Leeson's equation overly minimizes the contributions of several key factors, including the average power at the input of the oscillator (Ps av) and the equivalent noise resistance of the tuning diode (R).

Ironically, Mr. Carey's phase-noise analysis also neglects my previous work on the noise contributions of variable-capacitance devices, such as tuning diodes, on the phase noise of a tunable oscillator, such as a VCO. In my text Digital PLL Frequency Synthesizers, Theory and Design, these devices are referred to as "voltage-sensitive" devices, and an expression is presented for determining the capacitance (and thus tuning bandwidth) of such devices based on their tuning voltages.


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