The lumped-element model (also called lumped-parameter model, or lumped-component model) is a simplified 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. It is useful in electrical systems (including electronics), mechanical multibody systems, heat transfer, acoustics, etc. This is in contrast to distributed parameter systems or models in which the behaviour is distributed spatially and cannot be considered as localized into discrete entities. The simplification reduces the state space of the system to a finite dimension, and the partial differential equations (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into ordinary differential equations (ODEs) with a finite number of parameters.
Electrical systems
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. The self-imposed constraints are:
The change of the magnetic flux in time outside a conductor is zero. ∂ Φ B ∂ t = 0 {\displaystyle {\frac {\partial \Phi _{B}}{\partial t}}=0}
The change of the charge in time inside conducting elements is zero. ∂ q ∂ t = 0 {\displaystyle {\frac {\partial q}{\partial t}}=0}
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. The third assumption is the basis of the lumped-element model used in 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 circuits makes the simplifying assumption that the attributes of the circuit, resistance, capacitance, inductance, and gain, are concentrated into idealized electrical components; resistors, capacitors, and inductors, etc. joined by a network of perfectly conducting wires. The lumped-element model is valid whenever L c ≪ λ {\displaystyle L_{c}\ll \lambda } , where L c {\displaystyle L_{c}} denotes the circuit's characteristic length, and λ {\displaystyle \lambda } 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 lines), 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 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 capacitors 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 resistance distributed along its length but we can model this as a lumped inductor in series with the ideal resistor.
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