A unified model for injection-locked frequency dividers

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

“…This makes it possible to determine the following: (i) the amplitude and the phase of the output voltage in steady state, (ii) the locking range; (iii) the stability characteristics of locked oscillations, (iv) the amplitude and the phase of the output voltage during the transient preceding the steadystate. Note that previous approaches for analyzing ILFDs are limited only to the calculation of the locking range with the simplifying assumption that the amplitude of the output voltage is not dependent on the frequency [1,5,6]. The fixed points of (6) that correspond tȯ=̇= 0 can be found by solving the resulting nonlinear algebraic system for the amplitude and the phase .…”

“…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.…”

BPSK to ASK Signal Conversion Using Injection-Locked Oscillators—Part I: Theory

“…In the locked state, superharmonic ILOs act as frequency dividers, with the dividing factor being the nearest harmonic order of the ILO to the injected signal. For that reason, these components are also known as injection-locked frequency dividers (ILFDs) [12], [13]. The output of a superharmonic ILO or ILFD could be in any of possible phase states ( being the harmonic order).…”

“…1) Phase Locking: In this case, (i.e., ) and (8) reduces to (13) Its solution can be written as (14) where denotes the initial condition of the phase for The time is given by In this particular case, regardless of the injected power, the phase locking always takes place. According to (14), (15), and Fig.…”