This work presents an in-depth investigation of the nonlinear behavior of two mutually injection-locked oscillators through inductor coupling. An analytical formulation, solved through an innovative procedure, facilitates the understanding of the qualitative transformations in the system solutions when increasing the coupling factor. The analysis demonstrates that, in a manner similar to the unilaterally injection-locked oscillators, families of disconnected/connected curves are obtained when increasing this factor, although the patterns, associated with distinct operation modes, are more complex. Then, an accurate numerical method to predict the behavior of coupled transistor-based oscillators is presented, based on nonlinear admittance models of the individual oscillators. Mathematical conditions are derived to solve the coupled system through a two-level contour-intersection technique. In this way, all the solutions coexisting for a given set of element and parameter values are calculated simultaneously, in an exhaustive manner. The cases of two coupled oscillators at the fundamental frequency and at 1:3 frequency ratio are considered. Possible applications include the oscillator phase-noise reduction and the implementation of sensors using the phase shift between the two oscillator elements.

中文翻译：

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通过电感耦合进行振荡器互注入锁定的非线性分析


这项工作通过电感耦合对两个互注入锁定振荡器的非线性行为进行了深入研究。通过创新程序求解的分析公式有助于理解增加耦合因子时系统解决方案中的定性转换。分析表明，以类似于单侧注入锁定振荡器的方式，当增加该因子时，可以获得断开/连接曲线族，尽管与不同操作模式相关的模式更加复杂。然后，基于各个振荡器的非线性导纳模型，提出了一种准确的数值方法来预测基于耦合晶体管的振荡器的行为。通过两级轮廓相交技术导出数学条件来求解耦合系统。这样，对于给定的一组元素和参数值共存的所有解都以穷举的方式同时计算。考虑基频和频率比为 1:3 的两个耦合振荡器的情况。可能的应用包括振荡器相位噪声降低以及使用两个振荡器元件之间的相移来实现传感器。