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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.]]
The '''lumped
Mathematically speaking, 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
The '''lumped
1. The change of the magnetic flux in time outside a conductor is zero.
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3. 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
=== Lumped
The lumped
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]]. Otherwise, when the circuit length is on the order of a wavelength, we must consider more general models, such as the [[distributed
▲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 [[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.
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.
== Thermal systems==
A '''lumped
The lumped
An early-discovered example of a lumped-capacitance system which exhibits mathematically simple behavior due to such physical simplifications, are systems which conform to ''Newton's law of cooling''. This law simply states that the temperature of a hot (or cold) object progresses toward the temperature of its environment in a simple exponential fashion. Objects follow this law strictly only if the rate of heat conduction within them is much larger than the heat flow into or out of them. In such cases it makes sense to talk of a single "object temperature" at any given time (since there is no spatial temperature variation within the object) and also the uniform temperatures within the object allow its total thermal energy excess or deficit to vary proportionally to its surface temperature, thus setting up the Newton's law of cooling requirement that the rate of temperature decrease is proportional to difference between the object and the environment. This in turn leads to simple exponential heating or cooling behavior (details below).
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To determine the number of lumps, the [[Biot number]] (Bi), a dimensionless parameter of the system, is used. Bi is defined as the ratio of the conductive heat resistance within the object to the [[convective heat transfer]] resistance across the object's boundary with a uniform bath of different temperature. When the [[thermal resistance]] to heat transferred into the object is larger than the resistance to heat being [[diffused]] completely within the object, the Biot number is less than 1. In this case, particularly for Biot numbers which are even smaller, the approximation of ''spatially uniform temperature within the object'' can begin to be used, since it can be presumed that heat transferred into the object has time to uniformly distribute itself, due to the lower resistance to doing so, as compared with the resistance to heat entering the object.
If the Biot number is less than 0.1 for a solid object, then the entire material will be nearly the same temperature with the dominant temperature difference will be at the surface. It may be regarded as being "thermally thin". The Biot number must generally be less than 0.1 for usefully accurate approximation and heat transfer analysis. The mathematical solution to the lumped
A Biot number greater than 0.1 (a "thermally thick" substance) indicates that one cannot make this assumption, and more complicated [[heat transfer]] equations for "transient heat conduction" will be required to describe the time-varying and non-spatially-uniform temperature field within the material body.
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==== Solution in terms of object heat capacity ====
If the entire body is treated as lumped
From the definition of heat capacity <math>C</math> comes the relation <math>C = dQ/dT</math>. Differentiating this equation with regard to time gives the identity (valid so long as temperatures in the object are uniform at any given time): <math>dQ/dt = C (dT/dt)</math>. This expression may be used to replace <math>dQ/dt</math> in the first equation which begins this section, above. Then, if <math>T(t)</math> is the temperature of such a body at time <math>t</math>, and <math>T_{env}</math> is the temperature of the environment around the body:
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== Acoustics ==
In this context, the lumped
*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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* all heat transfer mechanisms are linear, implying that radiation and convection are linearised for each problem;
Several publications can be found that describe how to generate
</ref>
==See also==
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