Editing Lumped-element model (section)

== Electrical systems ==
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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_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.

=== Lumped-element model ===
The lumped-element model of electronic [[Electrical network|circuits]] makes the simplifying assumption that the attributes of the circuit, [[Electrical resistance|resistance]], [[capacitance]], [[inductance]], and [[Gain (electronics)|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.

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-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.