Lumped, Inductorless Oscillators: How Far Can They Go?

The fundamental question of how much we can ultimately reduce the phase noise of a lumped, inductorless oscillator through careful design is addressed and it is shown that the fluctuation dissipation theorem of thermodynamics imposes a lower limit on the phase noise. An analytical formulation of this limit is presented and it is shown that the phase noise of ring oscillators with long-channel MOS devices is closer to this limit compared to that of the relaxation oscillators or ring oscillators with short channel MOS devices. Introduction Due to the increasing demand for high-level integration of electronic circuits, lumped, inductorless oscillators (LIOs) have become an extremely attractive choice for today’s IC designers. Relaxation and ring oscillators are used in several applications including clock recovery circuits for serial data communications (1) and on-chip clock distribution (2). However, applications in radio frequency (RF) circuits is quite limited mainly due to inferior phase noise behavior compared to inductor-based oscillators. Although several investigations have been performed to improve the phasenoise of LIOs (e.g. (3)), the fundamental question of whether or not there is a minimum achievable phase noise for this class of oscillators has not been addressed. This lingering question causes a great deal of uncertainty about the scalability of RFIC design without using tuned circuits. In this paper we address this question and show that one of the fundamental principles of thermodynamics sets a lower limit on the phase noise of LIOs. Using a simplified model, we then present a quantitative analysis of this minimum achievable phase noise. Finally, we present several examples from previously published results on relaxation and ring oscillators to compare the minimum achievable phase noise to the experimental results from real designs. The Physical Argument for Minimum Achievable Phase Noise The phase noise of an oscillator is an indication of the fact that the oscillator is not continuously oscillating with the same frequency. To build a stable (and hence low-phasenoise) oscillator, one should be able to enforce the period of oscillation in a reliable fashion. In other words, a constant with the dimension of time is required to dictate the oscillation period. In inductor-based oscillators (like the Colpitts) this constant is were L is the inductor and C is the capacitor. In transmission-line-based oscillators the ratio of establishes the time constant in which l is the length of the transmission line and v is the velocity of electromagnetic wave inside the transmission line. In LIOs the product of is usually the time constant. There is, however, a fundamental difference between this latter case and the first two ones. The fluctuation dissipation theorem of thermodynamics dictates that there exists a finite amount of thermal noise associated with any resistor. Thus, in contrast to inductor-based and transmission-line-based oscillators, the time constant of LIOs is inherently noisy because of the resistor noise. Consequently, even if the rest of the circuit is noise-free, the resistor noise imposes a lower limit on the phase noise of LIOs. To provide a quantitative prediction of this minimum achievable phase noise, we use a simple model for an LIO (Fig. 1). Only the equilibrium resistor noise (given by 4kT/R) is taken into account in this formulation of minimum achievable phase noise. Although ring oscillators do not completely resemble this model, the final result for the lower limit of phase noise is still applicable to them. In fact, by taking into account the transistor noise and noise bandwidth in critical circuit nodes of a ring oscillator, it can be shown that their minimum achievable phase noise is always slightly higher than that of the model given in Fig. 1. Fig. 1: (a) A typical RC relaxation oscillator. (b) The Schmitt comparator transfer function. (c) The waveform for the capacitor voltage R