TY - JOUR
T1 - Sum rules and constraints on passive systems
AU - Bernland, Anders
AU - Luger, Annemarie
AU - Gustafsson, Mats
PY - 2011
Y1 - 2011
N2 - 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.
AB - 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.
U2 - 10.1088/1751-8113/44/14/145205
DO - 10.1088/1751-8113/44/14/145205
M3 - Article
VL - 44
JO - Journal of Physics A: Mathematical and Theoretical
JF - Journal of Physics A: Mathematical and Theoretical
SN - 1751-8113
IS - 14
M1 - 145205
ER -