The injection pulling of oscillators in phase-locked loop (PLL) circuits is analyzed by the linear approximation of nonlinear differential equations in terms of the oscillator phase. Applying this analysis to injection pulling in an injection-locked PLL (IL-PLL) and mutual injection between oscillators in a cascaded PLL, it is shown that the feedback loop becomes unstable due to the injection and the transfer function exhibits a large peak gain at the edge of the PLL bandwidth. These observations are confirmed by simulations based on linear model. It is also found that coupling from the second stage to the first stage can control the peak gain of transfer characteristics. Measurements on the test circuits of IL-PLL and cascaded PLL validate the proposed linear model. In particular, it reveals the dependence of jitter generation on the timing difference between the injection and oscillator. Additionally, in the cascaded PLL, it is shown that a properly controlled injection pulse mitigates the jitter generation caused by mutual injection pulling. This study presents important viewpoints on self and mutual interference in stand-alone and individual PLLs.

中文翻译：

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锁相环中的振荡器注入牵引的研究

通过根据振荡器相位对非线性微分方程进行线性逼近，分析了锁相环（PLL）电路中振荡器的注入拉动。将此分析应用于注入锁定PLL（IL-PLL）中的注入拉动和级联PLL中的振荡器之间的互注入，结果表明反馈环路由于注入而变得不稳定，并且传递函数在PLL带宽的边缘。这些观察结果通过基于线性模型的仿真得到了证实。还发现，从第二级到第一级的耦合可以控制传输特性的峰值增益。对IL-PLL和级联PLL的测试电路进行的测量验证了所提出的线性模型。特别是，它揭示了抖动产生与注入和振荡器之间的时间差的相关性。此外，在级联PLL中，还显示出适当控制的注入脉冲可减轻由相互注入拉动引起的抖动产生。这项研究提出了关于独立和独立PLL中自干扰和互干扰的重要观点。