Lumped-element model: Difference between revisions

Revision as of 20:02, 19 November 2022

Revision as of 20:53, 30 January 2023

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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, such as electrical circuits, into a [[Topology (electrical circuits)|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. This may be contrasted to [[distributed parameter system]]s or models in which the behaviour is distributed spatially and cannot be considered as localized into discrete entities.

Mathematically speaking, the simplification reduces the [[State space (controls)|state space]] of the system toas follows:

* The system has a [[counting number|finite]] [[dimension]],.
* and the [[partial differential equation]]s (PDEs) of theThe continuous (infinite-dimensional) time and space model of the physical system intois modeled by [[ordinary differential equation]]s (ODEs), withinstead aof finitepartial numberdifferential ofequations parameters(PDEs).

* The differential equations have a finite number of parameters.

== Electrical systems ==

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==== Thermal purely resistive circuits ====

A useful concept used in heat transfer applications once the condition of steady state heat conduction has been reached, is the representation of thermal transfer by what is known as thermal circuits. A thermal circuit is the representation of the resistance to heat flow in each element of a circuit, as though it were an [[electrical resistor]]. The heat transferred is analogous to the [[electric current]] and the thermal resistance is analogous to the electrical resistor. The values of the thermal resistance for the different modes of heat transfer are then calculated as the denominators of the developed equations. The thermal resistances of the different modes of heat transfer are used in analyzing combined modes of heat transfer. The lack of "capacitative" elements in the following purely resistive example, means that no section of the circuit is absorbing energy or changing in distribution of temperature. This is equivalent to demanding that a state of steady state heat conduction (or transfer, as in radiation) has already been established.