An original determination of the maximum phase shift range obtained for an array of N coupled oscillators

This paper presents an original approach, using a harmonic balance optimization method, allowing to predict the maximum phase shift range that can be practically obtained for an array of N coupled oscillators. Indeed, unlike what is predicted by the theory, the proposed analysis allows to show that the maximum value of the phase shift decreases by increasing the number N of coupled oscillators in the array. Hence, this method allows to find an empirical limit of the phase shift obtained between two adjacent oscillators which tends to ± 180°/N. The results obtained with the proposed method are compared with the measurement results of an array made of two and four coupled voltage controlled oscillators (VCOs) implemented in the QUBiC4X 0.25 µm BiCMOS SiGe:C process of NXP semiconductors. These measurement results confirmed that the maximum measured phase shift range is close to ± 180°/2 and ± 180°/4 for arrays made of two and four coupled VCOs respectively.