Lumped-element model

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[edit]

Lumped-matter discipline[edit]

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.^[1] The self-imposed constraints are:

Plate: thickness

: thickness/2

Fin

Long : diameter/4

cylinder

: diameter/6

Sphere

all objects are ;

rigid bodies

all interactions between rigid bodies take place via (joints), springs and dampers.

kinematic pairs

The simplifying assumptions in this domain are:

A rigid-walled cavity containing air (or similar compressible fluid) may be approximated as a 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.

capacitor

A 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.

reflex port

Certain types of damping material can be approximated as a . 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.

resistor

A drive unit (typically a woofer or subwoofer drive unit) may be approximated as a series connection of a zero-impedance voltage source, a resistor, a capacitor and an inductor. The values depend on the specifications of the unit and the wavelength of interest.

loudspeaker

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.

Heat transfer for buildings[edit]

A simplifying assumption in this domain is that 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 lumped-element models of buildings. In most cases, the building is considered a single thermal zone and in this case, turning multi-layered walls into lumped elements can be one of the most complicated tasks in the creation of the model. The dominant-layer method is one simple and reasonably accurate method.^[4] In this method, one of the layers is selected as the dominant layer in the whole construction, this layer is chosen considering the most relevant frequencies of the problem.^[5]

Lumped-element models of buildings have also been used to evaluate the efficiency of domestic energy systems, by running many simulations under different future weather scenarios.^[6]

Fluid systems[edit]

Fluid systems can be described by means of lumped-element cardiovascular models by using voltage to represent pressure and current to represent flow; identical equations from the electrical circuit representation are valid after substituting these two variables. Such applications can, for example, study the response of the human cardiovascular system to ventricular assist device implantation.^[7]