Observations Concerning the Locking Range in a Complementary Differential $LC$ Injection-Locked Frequency Divider—Part I: Qualitative Analysis

Modeling, Analysis, and Experimental Validation of Frequency Dividers with Direct Injection

“…Finally, it should be highlighted that even if our experiments were performed in MHz range as in [8,16], similar results are expected at GHz range because it was shown in [5] that approximate models based on algebraic characteristics for the active parts of the divider can be effectively used at GHz frequencies.…”

“…Thus, we conclude that (8) provides the frequency interval wherein the injectionlocked frequency divider is able to lock the input signal, that is, the so-called locking range. Expression (8) clearly shows that the locking range of direct-injection frequency dividers, differently from the locking range of frequency dividers with injection via tail device [6,9], does not depend on the circuit losses, and thus, it can be widened without lowering the quality factor and the phase noise of the divider.…”

“…It should be observed that the values assumed by the parameters of the algebraic characteristics (3) and (4) and used to model the transconductor and the injection circuit, obviously depend on the technology used as well as on the transistor sizes, but this modeling approach can be advantageously applied independently on the device technology. The parasitics and the delay of devices, due to the high frequency, are included, as a rule [5,6,8], into the capacitance of the tank. This simplifies the analysis, making it possible to capture the essential aspects of the synchronization phenomenon in ILFDs.…”

“…Investigations were also devoted to the issue of developing a model of the ILFDs and a methodology for their analysis, at first addressed in [5][6][7] and subsequently in many papers (see [8] and references therein), which is motivated by the need for a better understanding of their operation in order to provide useful design insights. Among these, mention should be made of papers [9,10] where a comprehensive methodology is provided for the analysis of most commonly used topologies of ILFDs, namely, with injection through a tail device and with direct injection.…”

Experimental characterization of Arnold tongues in injection-locked CMOS LC frequency dividers with tail and direct injection

“…II, we recall the Arnold tongue locking scenario from nonlinear dynamics [13]. Existing experimental methods for characterizing the Arnold tongue bifucation diagram [6], [10] are described in Sec. III.…”