Electromagnetic decoupling and complexity

In radio-frequency systems that drive coupled dissipative loads, the matching network between the amplifiers and their loads needs to account for the coupling. With N amplifiers driving N loads, a favorite choice is a “decoupling” network, which is a lossless reciprocal network that has N input ports connected to the sources and N output ports connected to the loads. The decoupling network transforms the coupled impedance of the loads into the uncoupled characteristic impedance of the sources. Any incident signal at the input ports of the network is transferred, without reflection, to the loads. Decoupling networks can be realized by generalized II-networks of lumped and distributed impedances, depending on the design frequency. Although the impedance requirements of the network are unique, its realization is not, and networks that involve many impedances can be complex to lay out on circuit boards. In this paper, we establish that a decoupling network requires a minimum of N² + N impedances for N arbitrarily coupled loads, and provide a systematic method for realizing this lower bound.