Theory of Broadband Transmission of Frequency Modulated Signals through High-Q Resonators with Continuous Time-Varying Matching Networks

Impedance match to a high-Q load with a passive network is fundamentally narrow band as its linear, time-invariant (LTI) property is subject to the Bode-Fano limit. However, such a limit may be partially lifted for frequency modulated signals when a time-varying matching network is used whose reactance varies continuously following the instantaneous frequency of the signal. The mismatch, however, may increase for high-speed modulations as the result of transient changes of energy balance among the different reactive components. As the information bandwidth of a modulated signal is primarily determined by the modulation speed, a general theory quantifying the modulation speed limit for such dynamic impedance matching is derived based on circuit laws. The analyses are applied to parallel LC resonators with one type of time-varying reactance, two types of time-varying reactance, and gyrator coupled time-varying reactance. The analyses are validated with nonlinear circuit simulations. It is concluded that while a single type of time-varying reactance can help to broaden the modulation rates of impedance matching, two paired types of time-varying reactance (inductance and capacitance) can lift the modulation speed limit completely. Furthermore, when a pair of single-type time-varying reactance coupled with a gyrator is used, the modulation speed limit can be similarly removed. In general, the gyrator adds another dimension in dynamic impedance matching in either lowering the resonant frequency or suppressing the mismatch. The proposed theory can be applied to the dynamic matching of electrically small antennas, which may help to achieve a radiation efficiency bandwidth product that goes beyond the limit imposed by its radiation quality factor, whose lower bound is defined by the famous Harrington-Chu’s limit.