Sum rules and constraints on passive systems

Anders Bernland, Annemarie Luger, Mats Gustafsson

Research output: Contribution to journalArticlepeer-review

42 Citations (SciVal)


A passive system is one that cannot produce energy, a property that
naturally poses constraints on the system. A system in convolution form is fully described by its transfer function, and the class of Herglotz functions, holomorphic
functions mapping the open upper half plane to the closed upper half plane, is
closely related to the transfer functions of passive systems. Following a well-known
representation theorem, Herglotz functions can be represented by means of positive
measures on the real line. This fact is exploited in this paper in order to rigorously
prove a set of integral identities for Herglotz functions that relate weighted integrals
of the function to its asymptotic expansions at the origin and infinity.
The integral identities are the core of a general approach introduced here to derive
sum rules and physical limitations on various passive physical systems. Although
similar approaches have previously been applied to a wide range of specific applications,
this paper is the first to deliver a general procedure together with the necessary
proofs. This procedure is described thoroughly, and exemplified with examples from
electromagnetic theory.
Original languageEnglish
Article number145205
JournalJournal of Physics A: Mathematical and Theoretical
Issue number14
Publication statusPublished - 2011

Subject classification (UKÄ)

  • Electrical Engineering, Electronic Engineering, Information Engineering


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