Best Windband Impedance Matching Bounds for Lossless 2-Ports

摘要

The selection of a lossless 2-port to maximize the wideband power transfer from a generator to a load is a ubiquitous problem in electrical engineering. The mathematical problem is to maximize the wideband transducer power gain over a class of lossless 2-ports. As a numerical optimization problem, wideband impedance matching is difficult because the wideband transducer power gain is a nonlinear, nondifferentiable badly scaled multivariable function. Therefore, any information on the global solution is valuable to the engineer for assessing the quality of suboptimal solutions computed by numerical optimizers. In his classic 1950 paper, Fano determined a theoretical upper bound on the transducer power gain 16. Specifically, the transducer power gain of any lossless 2-port cannot exceed Fano's bound. Development of Fano's approach continued through the 1960s, However, computing these bounds required solving a highly nonlinear system of multivariate inequalities amenable only for simple cases. In the early 1970s, Helton made the amazing connection between operator theory and electrical engineering. Powerful Hardy space techniques were coupled to the electrical engineer's Smith chart computations. In this framework, Nehari's Theorem gave an upper bound on the transducer power gain computable as an (easy) eigenvalue problem. This report shows that continuity conditions make this Nehari bound tight.