Purpose This paper aims to find a closed-form expression for the frequency and amplitude of single-ended ring oscillators when transistors experience all regions. Design/methodology/approach 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. Findings 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. Originality/value The validity of these relationships has been investigated by analyzing and simulating a single-ended oscillator in 0.18 µm technology.

The analytical relationships presented for amplitude and frequency of the ring oscillator are derived approximately due to the nonlinear nature of this oscillator. 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. In this paper, considering the circuit’s governing equation and the ring oscillator’s output waveform, a relation for the frequency is presented, 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 0.18μm technology.

Purpose The purpose of this paper is to find a closed relation for the phase noise of LC oscillators. Design/methodology/approach 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. Findings 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 gm/ID method. Originality/value A technology-independent method using the gm/ID method and the closed relationship is provided to calculate phase noise.

The analytical relationships presented for amplitude and frequency of the ring oscillator are derived approximately due to the nonlinear nature of this oscillator. 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. In this paper, considering the circuit's governing equation and the ring oscillator's output waveform, a relation for the frequency is presented, 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 0.18µm technology.