An analytical model and a methodology are presented for the analysis of CMOS injection-locked frequency dividers with direct injection, which are used in modern wireless communication systems. The amplitude and phase of the oscillation in the synchronous operation mode, as well as the locking range, are found in explicit form. The width of the locking range is determined by the condition of the existence and stability of synchronous oscillations. The accuracy of the model and of the presented formulas is validated through a comparison with experimental results, which are in good agreement with the analytical ones.

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 [

Investigations were also devoted to the issue of developing a model of the ILFDs and a methodology for their analysis, at first addressed in [

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 [

A divide-by-2 injection-locked frequency divider with complementary topology and direct injection (Figure

Circuit diagram of an injection-locked frequency divider with complementary topology and direct injection.

Equivalent model of the ILFD in Figure

In the circuit in Figure

(a) Locally active two-terminal sub-circuit of the ILFD in Figure

The two complementary MOS switches (

From the previous observations, it follows that the behavior of the ILFD in Figure

First, we observe that usually the dividers operate with small amplitudes of the injection signal, that is, with

In order to measure the nonlinear characteristics of the circuit under test, that is,

Auxiliary circuit used to apply voltages

Nonlinear characteristic of the locally active two-terminal sub-circuit, shown in Figure

(a) Nonlinear characteristics of the injection circuit shown in Figure

On the other hand, if the circuit under test is the injection circuit shown in Figure

The schematization of the divider as in Figure

It should be observed that the values assumed by the parameters of the algebraic characteristics (

If a sinusoidal synchronization signal

It should be observed that the exact periodic solution of (

The formulation (

Numerical integration of system (

Phase portrait of system (

(a) Steady-state amplitude and (b) phase for the ILFD in Figure

Finally, note that the presented formulas provide several interesting and useful design insights to circuit designers, which are usually interested in design variables such as oscillation amplitude, lock range, and stability of lock, which are here given in a simple explicit form in terms of circuit parameters.

In order to experimentally verify the circuit model and the presented formulas, a prototype of the circuit in Figure

The presented methodology for predicting the behavior of the injection-locked frequency divider provides the analytical expressions for the output amplitude

Then, we compared the results obtained by formulas (

Finally, it should be highlighted that even if our experiments were performed in MHz range as in [

We have experimentally verified the possibility of predicting accurately the behavior of injection-locked frequency dividers with direct injection through simple formulas that provide useful design insights to circuit designers. This is made possible by the dynamical model (