Analysis and Designs of Magnetless Microstrip Circulators Based on Hybrid Static and Time-Modulated Resonators

In this article, we present a comprehensive analysis and designs of magnetless nonreciprocal circulators consisting of hybrid time-modulated and static resonators. The analysis is based on the derived three-port full-spectra coupling matrix formalism by introducing a new factor $\Upsilon $ , which maps the definitions of time-invariant and time-varying elements from conventional time-domain to the prescribed full-spectra-domain ( $\boldsymbol {w}$ -domain), and hence paves a way to extend the microwave circuit theory for a conventional linear time-invariant (LTI) network to a general time-varying network. Numerical examples show that multiple isolation and reflection poles can be generated within the circulator passband, indicating a wideband circulator of arbitrary-order nonreciprocity. Besides, the coupled static resonators work as a bandpass filter at the RF carrier frequency, leading to a high passband roll-off and bandstop filter at the intermodulation (IM) frequencies, resulting in a significant reduction of IM terms. To validate the proposed concept, the circulators of branch resonator order $N=1, 2$ , and 3 are designed, simulated, and measured using microstrip resonators loading with commercial varactors. The varactors are reversely-biased and their equivalent capacitances are modulated by sinusoidal waves of an incremental modulation phase of 120°. All the measurement results match well with the simulation and numerical predictions. The theory and concept presented in this article may find applications in wireless wideband in-band full-duplex (IBFD) communications.