Lumped-element model: Difference between revisions

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[[File:Ohm's Law with Voltage source TeX.svg|thumb|Representation of a lumped model consisting of a voltage source and a resistor.]]

The '''lumped-element model''' (also called '''lumped-parameter model''', or '''lumped-component model''') is ana [[idealization (philosophy of science)|idealizedsimplified]] representation of a [[physical system]] or circuit that assumes all components are concentrated at a single point and their behavior can be described by idealized mathematical models. The lumped-element model simplifies the system or circuit behavior description into a [[Topology (electrical circuits)|topology]]. It is useful in [[electrical network|electrical systems]] (including [[electronics]]), mechanical [[multibody system]]s, [[heat transfer]], [[acoustics]], etc. This is in contrast to [[distributed parameter system]]s or models in which the behaviour is distributed spatially and cannot be considered as localized into discrete entities.

The simplification reduces the [[State space (controls)|state space]] of the system to a [[counting number|finite]] [[dimension]], and the [[partial differential equation]]s (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into [[ordinary differential equation]]s (ODEs) with a finite number of parameters.

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=== Lumped-matter discipline ===

The '''lumped-matter discipline''' is a set of imposed assumptions in [[electrical engineering]] that provides the foundation for '''lumped-circuit abstraction''' used in [[Network analysis (electrical circuits)|network analysis]].<ref>Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare ([http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-002-circuits-and-electronics-spring-2007/video-lectures/6002_l1.pdf PDF]), [[Massachusetts Institute of Technology]].</ref> The self-imposed constraints are:

# The change of the magnetic flux in time outside a conductor is zero. <math display="block">\frac{\partial \phi_BPhi_B} {\partial t} = 0</math>

# The change of the charge in time inside conducting elements is zero. <math display="block">\frac{\partial q} {\partial t} = 0</math>

# Signal timescales of interest are much larger than [[propagation delay]] of [[electromagnetic waves]] across the lumped element.

The first two assumptions result in [[Kirchhoff's circuit laws]] when applied to [[Maxwell's equations]] and are only applicable when the circuit is in [[steady state (electronics)|steady state]]. The third assumption is the basis of the lumped-element model used in [[Network analysis (electrical circuits)|network analysis]]. Less severe assumptions result in the [[distributed-element model]], while still not requiring the direct application of the full Maxwell equations.

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{{Main|Newton's law of cooling}}

'''Newton's law of cooling''' is an [[empirical relationship]] attributed to English physicist [[Isaac Newton|Sir Isaac Newton]] (1642–1727). This law stated in non-mathematical form is the following:

{{Quotation|The rate of heat loss of a body is proportional to the temperature difference between the body and its surroundings.}}