An analytical model for the power spectral density of a voltage-controlled oscillator and its analogy to the laser linewidth theory

We calculate the output power density spectrum for a simple voltage-controlled oscillator (VCO) circuit. The power spectral density of the oscillator is composed of a term related to the high-frequency fluctuations in the feedback loop and a term related to the low-frequency fluctuations of the frequency control voltage. The latter is treated stochastically in a similar fashion to the inhomogeneous line broadening of gas lasers due to the Doppler effect. This additional broadening causes a deviation of the power spectral density from the Lorentzian shape, that is, from the phase noise decay of -6 dB per octave. This is specially pronounced at not-too-large frequency offsets. The analogy between electrical oscillators and optical oscillators (lasers) allows the methods used in optical spectroscopy to be applied. The approach may be employed to synthesize oscillator spectra from the equivalent circuit parameters with small numerical effort. Furthermore, it allows experimental power density spectra to be decomposed into the contributions stemming from the high-frequency noise in the feedback loop and the low-frequency noise of the oscillation frequency. This should give better insight into the origin of the phase noise. Besides VCO's, this concept may be useful for oscillators subject to Gaussian supply and substrate noise.