Taking into account that r [approximately equal to] 1, (21) provides a very simple expression of the locking range as a function of the active and passive circuit parameters, which can be usefully employed for the design of ILFD with direct forcing signal.
Time waveforms of the input and output voltages, reported in Figure 9(a), demonstrated that the circuit under study can actually operate as a divide-by-three ILFD.
The comparison of the SPICE results with the results obtained by analytical expression (21), also reported in Figure 10, shows a good accuracy of (21), which can be effectively used to design the ILFD in Figure 5.
Moreover, the width of the locking range does not depend on the quality factor of the tank (Figure 10(c)) and, thus, it is not necessary to keep low the quality factor in order to widen the locking range as it happens in other ILFD topologies.
which, differently from the case of ILFD with saturation-like nonlinearity, do not depend on the circuit losses.
The phenomenon of injection locking , or frequency entrainment, of an oscillator through an external signal underlies the operation of injection-locked frequency dividers (ILFDs), which are nowadays realized on-chip, in a number of ways suited for RF integrated circuits.
Starting from the equivalent circuit usually employed for describing the LC frequency dividers and the classical result by Adler concerning injection-locked oscillators, we present the models of ILFDs with saturation-like nonlinearities and with polynomial-like nonlinearities.
In order to simplify the study of ILFDs aimed at obtaining simple formulas relating the LR to the circuit parameters, the actual ILFDs are usually reduced to a more simple equivalent circuit.
The possibility offered by the modern RF-CMOS technology to realize on-chip injection-locked frequency dividers (ILFDs) has led to a renewed interest for the study of the behavior of LC oscillators under the action of an external signal, firstly performed in a pragmatic, and effective, manner in a pioneering paper by Adler  and then in [2, 3].
Investigations were also devoted to the issue of developing a model of the ILFDs and a methodology for their analysis, at first addressed in [5-7] and subsequently in many papers (see  and references therein), which is motivated by the need for a better understanding of their operation in order to provide useful design insights.
However, the frequency dividers with direct injection have not been sufficiently treated in the literature from an analytical point of view, although they are widely used in applications for a very low input capacitance [11, 12] and a wide locking range [11,13], which is one of the most important figures of merit of ILFDs. In the present work, we wish to demonstrate the accuracy of the model of ILFDs developed in , here made more general, as well as the accuracy of the methodology for their analysis under steady-state operating conditions.
This simplifies the analysis, making it possible to capture the essential aspects of the synchronization phenomenon in ILFDs.