Alternative statements of the Rayleigh monotonicity law for linear time-invariant resistor networks driven by only voltages or only currents

摘要

Networks of linear time-invariant (LTI) resistors are a classical topic with many applications, used to model a variety of flow phenomena. In this short communication, we provide conceptually simple proofs of the following intuitively appealing bounds, which also imply the classical Rayleigh monotonicity law (RML): (i) For an LTI resistor network, driven by some node voltages, when a set of resistors is removed the total power dissipation is reduced by at least an amount that was consumed in the removed resistors. (ii) The complementary or dual result for such a network, but when current driven, is that shorting some resistors decreases the total power dissipation by at least as much as used to occur in these resistors before shorting. In fact, ideal diodes can be allowed as network branches and the same results hold. The short derivations assume that Dirichlet's and Thomson's minimum principles are known. With the expanding applications of a linear network theory, we hope that these variants of the monotonicity theorem will prove pedagogically useful.