Coupling Matrix-Based Co-Design of Filter-Oscillators

Abstract

This letter presents a methodology of designing integrated filter-oscillators using the coupling matrix technique. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N + 4$ </tex-math></inline-formula> coupling matrix is developed for the first time to describe the filter-oscillator topology. Accordingly, the corresponding group delay, loop gain, loop phase, and complex quality factor of the feedback filter can be calculated using the matrix. This helps rapidly predict oscillating frequencies at group delay or complex quality factor peak frequencies to achieve lower phase noise. Moreover, since the transistor and the transmission line (TL) entries are incorporated in the coupling matrix, the physical geometries can be directly determined, resulting in improved design efficiency. Two third-order filter-oscillators are implemented at 2.91 GHz (group delay peak) and 3.15 GHz (complex quality factor peak) frequencies, respectively. The developed filter-oscillators are measured with −121 and −145 dBc/Hz phase noise at 1-MHz offset frequency, validating this coupling matrix-based co-design approach.