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This paper aims to find a closed-form expression for the frequency and amplitude of single-ended ring oscillators when transistors experience all regions.

In this paper, the analytical relationships presented for ring oscillator amplitude and frequency are approximately derived due to the nonlinear nature of this oscillator, taking into account the differential equation that governs the ring oscillator and its output waveform.

In the case where the transistors experience the cut-off region, the relationships presented so far have no connection between the frequency and the dimensions of the transistor, which is not valid in practice. The relationship is presented for the frequency, including the dimensions of the transistor. Also, a simple and approximately accurate relationship for the oscillator amplitude is provided in this case.

The validity of these relationships has been investigated by analyzing and simulating a single-ended oscillator in 0.18 µm technology.

The purpose of this paper is to find a closed relation for the phase noise of LC oscillators.

The governing equation of oscillators is generally a stochastic nonlinear differential equation. In this paper, a closed relation for the phase noise of LC oscillators was obtained by approximating the I–V characteristic of the oscillator with third-degree polynomials and analyzing its differential equation.

This relation expresses phase noise directly in terms of circuit parameters, including the sizes of the transistors and the bias. Next, for evaluation, the phase noise of the cross-coupled oscillator without tail current was calculated with the proposed model. In this approach, the obtained equations are expressed independently of technology by combining the obtained phase noise relation and g_m/I_D method.

A technology-independent method using the g_m/I_D method and the closed relationship is provided to calculate phase noise.