| 1. |
- Bernland, Anders, et al.
(författare)
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Sum rules and constraints on passive systems
- 2011
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Ingår i: Journal of Physics A: Mathematical and Theoretical. - : IOP Publishing. - 1751-8113 .- 1751-8121. ; 44:14
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Bernland, Anders, et al.
(författare)
-
Sum rules and constraints on passive systems
- 2010
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A passive system is one that cannot produce energy, a property that naturally poses constraints on the system. A system on 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.
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| 3. |
- Bernland, Anders, et al.
(författare)
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Sum rules and constraints on passive systems with applications in electromagnetics
- 2010
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Ingår i: [Host publication title missing]. ; , s. 33-36
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Konferensbidrag (refereegranskat)abstract
- A passive system is one that cannot produce energy, a property that naturally poses constraints on the system. In this paper there is a review of some results on linear, time translational invariant, continuous, causal and passive systems, where it turns out that Herglotz functions are related to the Fourier transform of the impulse response of such systems. Some well known facts of this function class is considered, and a set of integral identities and an outline of the proof of these are presented. The identities may be used to derive sum rules and constraints on various physical systems. The theory is illuminated with two examples from electromagnetics: the first revisits Fano’s matching equations, while the latter makes a link to the Kramers-Kronig relations and discusses physical limitations on metamaterials.
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| 4. |
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| 5. |
- Ivanenko, Yevhen, et al.
(författare)
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Passive Approximation and Optimization Using B-Splines
- 2019
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Ingår i: SIAM Journal on Applied Mathematics. - : SIAM PUBLICATIONS. - 0036-1399 .- 1095-712X. ; 79:1, s. 436-458
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Tidskriftsartikel (refereegranskat)abstract
- A passive approximation problem is formulated where the target function is an arbitrary complex-valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The norm used is a weighted L-p-norm where 1 <= p <= infinity. The approximating functions are Herglotz functions generated by a measure with Holder continuous density in an arbitrary neighborhood of the approximation domain. Hence, the imaginary and the real parts of the approximating functions are Holder continuous functions given by the density of the measure and its Hilbert transform, respectively. In practice, it is useful to employ finite B-spline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finite-dimensional convex optimization problem which is amenable for numerical solution. A constructive proof is given here showing that the convex cone of approximating functions generated by finite uniform B-spline expansions of fixed arbitrary order (linear, quadratic, cubic, etc.) is dense in the convex cone of Herglotz functions which are locally Holder continuous in a neighborhood of the approximation domain, as mentioned above. As an illustration, typical physical application examples are included regarding the passive approximation and optimization of a linear system having metamaterial characteristics, as well as passive realization of optimal absorption of a dielectric small sphere over a finite bandwidth.
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| 6. |
- Nedic, Mitja, et al.
(författare)
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Herglotz functions and applications in electromagnetics
- 2020
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Ingår i: Advances in Mathematical Methods for Electromagnetics. - : Institution of Engineering and Technology. - 9781785613845 - 9781785613852 ; , s. 491-514
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- Herglotz functions inevitably appear in pure mathematics, mathematical physics, and engineering with a wide range of applications. In particular, they are the pertinent functions to model passive systems, and thus appear in modeling of electromagnetic phenomena in circuits, antennas, materials, and scattering. In this chapter, we review the basic theory of Herglotz functions and its applications to determine sum rules and physical bounds for passive systems.
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