Brian N. Limketkai and Robert W. Brodersen

A method for calculating phase noise based on a particle diffusion model in the phase plane is introduced. This approach allows for the derivation of closed-form solutions, which can be used to see the explicit dependencies of phase noise on circuit component values. The theory is applied to the Van der Pol and Colpitts oscillators, and comparisons to simulations show close matching. Introduction The need for a regular time base has made the oscillator an important component of many systems. Unfortunately, because of the presence of noise, no real oscillator can be truly periodic. Instead, the actual period of oscillation fluctuates about some mean period. This is manifested in frequency space as a peak about the mean frequency with spectral content in nearby frequencies, instead of a delta function as in a truly periodic oscillator. This spread is referred to as phase noise. Many have looked at calculating this spread [5] [6] [7] [8] [9]. Most of these methods rely on linear techniques, which fail to capture the inherently nonlinear effects of oscillators. One fully nonlinear model [9] is effective in calculating phase noise but focuses on exact solutions, making it complicated and thus more suitable for simulators. Recently, a similar view to that presented in this paper was published [5] explaining the phase diffusion with fluctuation-dissipation methods. While the concepts are similar, an equationbased method to handling time-varying noise was not made explicit. This paper presents a perturbative method for calculating this spread while accounting for nonlinearities in the oscillator, which is based solely on hand analysis and equations, and results in approximate closed-form expressions. Phase Plane Perspective Fig. 1. Path of a noisy oscillator in phase space. A point in phase space characterizes the complete state of the system, allowing for predictions of the past and Fig. 2. Ensemble of oscillators flowing through phase space around the limit cycle. future of the deterministic system. As time passes, the state of an oscillator traces out a trajectory which tends towards an isolated closed orbit called a limit cycle. A single noisy oscillator would consist of a “jagged” (see Fig. 1) orbit as noise disturbs the otherwise smooth path of the oscillator’s state. Solving for the actual path by considering the fluctuations of the noise and its effects on the trajectory can get very involved. One can also solve the same problem by considering the flow of the ensemble of oscillators (see Fig. 2). Solving for the density of realizable oscillators instead of the one realized path transforms the problem into a more tractable problem. The resulting equation describing the density flow is none other than the advection-diffusion equation [4] with nonconstant coefficients, which are due to the non-constant flow field (restoring forces and damping) in the phase plane. This is due to the nonlinear behavior of the oscillator, which presents damping that is a function of the actual voltage in the tank. This PDE is sometimes referred to as the Fokker-Planck equation. Perturbation Analysis The main idea behind perturbation theory [1] [2] [3] [4] is solving a problem by expanding about nearby known solutions. For our problem, two limiting cases become obvious for further inspection: oscillations with strong and weak nonlinearities. This paper focuses on the latter case, which exhibits near sinusoidal solutions. In order to analyze a variety of oscillator circuit topologies, we solve for a general nonlinearity, f , which is also assumed to be time-independent. We find the autocorrelation function of the output voltage which can then be transformed into the power spectral density for the phase noise. Starting with the non-dimensionalized differential equation, q̈ + q = f (q, q̇) + w(t) (1) where is small and w(t) is a white noise source, which we assume to be small compared to the amplitude of oscillation. The natural frequency of this oscillator is 1 because of the non-dimensionalization. Now turning this into a system of first-order equations, we have q̇ = v v̇ = −q + f (q, v) + w(t) (2) The two phase space variables are q and v (which are proportional to the charge and voltage in parallel-tank electrical oscillators). Their dynamics characterize the oscillator. The associated advection-diffusion equation [4] describing the flow of the phase space density is then ηt +∇ · ((v1− q2) η + f (q, v) η2− Dηv2) = 0 (3) where η is the phase space probability density, D is the diffusion constant which depends on the magnitude of the noise, and 1 and 2 are the unit vectors in the qand vdirections, respectively. We pulled out a factor of from the diffusion term because the noise is assumed to be O( ). To solve this, we consider solutions of the form η(t, q, v) = η(t, q, v) + η(t, q, v) + · · · (4) where the superscripts are not to be confused with exponents. Equating coefficients, we get η t +∇ · (v1− q2) η = 0 (5) which is just the equation for a linear harmonic oscillator with sinusoidal solutions. Using the method of multiple scales [1] [2] [3] [4], we introduce a slow time variable T ≡ t. We also convert to polar coordinates to get the first-order equation η t − η θ + η T = −∇ · {( f (q, v) η −Dη v )