Analytical Approach to the Study of Injection-Locked Frequency Dividers

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

“…This is possible resorting to the methods of asymptotic analysis for weakly nonlinear systems. Applying the well-known solution method of slowly varying amplitude and phase (SVA) to (5), as shown in detail in [9], we obtain a couple of equations for the amplitude and the phase, that is, the so-called averaging equationṡ…”

“…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 can be shown that this system can be solved in a closed form [9], leading to the following physically meaningful solution:…”

“…A divide-by-2 injection-locked frequency divider with complementary topology and direct injection (Figure 1) can be modeled by the equivalent circuit in Figure 2 [9]. The model is formed by the parallel connection of an LC tank, whose losses are represented by an equivalent resistance , a nonlinear two-terminal transconductor described by the current-voltage relationship nl (V), and a controlled current source described by the current-voltage relationship in (V, V in ), where V denotes the differential voltage between the drain nodes of the ILFD.…”

“…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.…”

“…In the present work, we wish to demonstrate the accuracy of the model of ILFDs developed in [9], here made more general, as well as the accuracy of the methodology for their analysis under steady-state operating conditions. This allows us to determine the amplitude and the phase shift of the locked oscillation, as well as the locking range by analytical expressions.…”