On Various Extensions of the Shannon-Hagelbarger Concavity Theorem

The Shannon-Hagelbarger concavity theorem asserts that the one-port driving-point resistance of a network of linear two-terminal resistors is a concave function of its branch resistances. This theorem has been applied in many domains, including circuit optimization, graph algorithms, and network analysis. In this paper, we show that this concavity theorem can be extended to any linear networks consisting solely of two-terminal inductors or two-terminal capacitors. The main contribution of this paper is a unified proof methodology based on various forms of Tellegen's theorem. The methodology provides circuit insight into the concavity results and avoids the use of additional circuit elements as was the case in the original proof of Shannon and Hagelbarger.