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Broadband Matching Bounds for Coupled Loads

The Bode-Fano bounds on the integral over all frequency of the logarithm of the reflection coefficient between a source and a load, connected to each other through a passive matching network, are well-known measures of the bandwidth of the matching network and load. When there are multiple coupled loads being driven by multiple independent sources, the notion of a single reflection coefficient is not obvious since energy delivered to one load port may reflect, in part, out another port. Establishing the maximum possible bandwidth of the matching network and loads requires the development of a broadband matching theory that applies to coupled systems. We define a bandwidth measure for coupled loads based on the integral logarithm of their collective power reflection, and develop upper bounds on this measure. These upper bounds can be used to examine the effects of coupling on bandwidth, and to evaluate the broadband performance of matching networks for an arbitrary number of coupled loads.