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| [[File:Ohm's Law with Voltage source TeX.svg|right|thumb|Representation of a lumped model made up of a voltage source and a resistor.]]
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| The '''lumped element model''' (also called '''lumped parameter model''', or '''lumped component model''') simplifies the description of the behaviour of spatially distributed physical systems into a [[topology]] consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in [[electrical network|electrical systems]] (including [[electronics]]), mechanical [[multibody system]]s, [[heat transfer]], [[acoustics]], etc.
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| Mathematically speaking, the simplification reduces the [[State space (controls)|state space]] of the system to a [[finite set|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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| == Examples ==
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| === Lumped element model in electrical systems ===
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| The lumped element model of electronic [[Electrical network|circuit]]s makes the simplifying assumption that the attributes of the circuit, [[Electrical resistance|resistance]], [[capacitance]], [[inductance]], and [[gain]], are concentrated into idealized [[electrical component]]s; [[resistor]]s, [[capacitor]]s, and [[inductor]]s, etc. joined by a network of perfectly [[Electrical conduction|conducting]] wires.
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| The lumped element model is valid whenever <math>L_c \ll \lambda</math>, where <math>L_c</math> denotes the circuit's characteristic length, and <math>\lambda</math> denotes the circuit's operating [[wavelength]].
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| Otherwise, when the circuit length is on the order of a wavelength, we must consider more general models, such as the [[distributed element model]] (including [[transmission line]]s), whose dynamic behaviour is described by the [[Maxwell's equations]]. Another way of viewing the validity of the lumped element model is to note that this model ignores the finite time it takes signals to propagate around a circuit. Whenever this propagation time is not significant to the application the lumped element model can be used. This is the case when the propagation time is much less than the [[period (physics)|period]] of the signal involved. However, with increasing propagation time there will be an increasing error between the assumed and actual phase of the signal which in turn results in an error in the assumed amplitude of the signal. The exact point at which the lumped element model can no longer be used depends to a certain extent on how accurately the signal needs to be known in a given application.
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| Real-world components exhibit non-ideal characteristics which are, in reality, distributed elements but are often represented to a [[first-order approximation]] by lumped elements. To account for leakage in [[capacitor]]s for example, we can model the non-ideal capacitor as having a large lumped [[resistor]] connected in-parallel even though the leakage is, in reality distributed throughout the dielectric. Similarly a [[wire-wound resistor]] has significant [[inductance]] as well as [[Electrical resistance|resistance]] distributed along its length but we can model this as a lumped [[inductor]] in series with the ideal resistor.
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| === Lumped element model in mechanical systems ===
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| The simplifying assumptions in this domain are:
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| * all objects are [[rigid body|rigid bodies]];
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| * all interactions between rigid bodies take place via [[kinematic pair]]s (''joints''), [[spring (device)|spring]]s and [[dashpot|dampers]].
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| === Lumped element model in acoustics === | |
| In this context, the lumped component model extends the distributed concepts of [[Acoustic theory]] subject to approximation. In the acoustical lumped component model, certain physical components with acoustical properties may be approximated as behaving similarly to standard electronic components or simple combinations of components.
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| *A rigid-walled cavity containing air (or similar compressible fluid) may be approximated as a [[capacitor]] whose value is proportional to the volume of the cavity. The validity of this approximation relies on the shortest wavelength of interest being significantly (much) larger than the longest dimension of the cavity.
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| *A [[reflex port]] may be approximated as an [[inductor]] whose value is proportional to the effective length of the port divided by its cross-sectional area. The effective length is the actual length plus an [[end correction]]. This approximation relies on the shortest wavelength of interest being significantly larger than the longest dimension of the port.
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| *Certain types of damping material can be approximated as a [[resistor]]. The value depends on the properties and dimensions of the material. The approximation relies in the wavelengths being long enough and on the properties of the material itself.
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| *A [[loudspeaker]] drive unit (typically a [[woofer]] or [[subwoofer]] drive unit) may be approximated as a series connection of a zero-[[Electrical impedance|impedance]] [[voltage]] source, a [[resistor]], a [[capacitor]] and an [[inductor]]. The values depend on the specifications of the unit and the wavelength of interest.
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| ==See also== | |
| * [[Lumped matter discipline]]
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| ==External links==
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| * [http://www.jat.co.kr/eda/saber/mpp.pdf Advanced modelling and simulation techniques for magnetic components]
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| *[http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/ IMTEK Mathematica Supplement (IMS)], the Open Source IMTEK Mathematica Supplement (IMS) for lumped modelling - '''broken link'''
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| [[Category:Mechanics]]
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| [[Category:Acoustics]]
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| [[Category:Components]]
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| [[Category:Electronic circuits]]
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