Locking Range of Frequency Dividers With Dual-Resonance Tank

Closed-form expressions are derived for predicting the locking ranges and the amplitudes of the locked oscillations of injection locking frequency dividers (ILFDs) with dual-resonance tank. The stability of the predicted locked-states is also assessed, showing that these are always stable in the locking ranges. These results are obtained by a proper transformation of the circuit equations that are reduced to a quasi-normal form by a suitable change of the circuit variables and then solved by applying the method of slowly varying amplitudes and phases. The derived formulas give results significantly more accurate than that resulting from the well-known Adler's equation, improperly applied in practice to ILFDs with dual resonance tank. Numerical results, obtained by using BSIM3 transistor models, validate the presented results.