Lumped-element model

The lumped element model (also called lumped parameter model, or lumped component model) simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems (including electronics), mechanical multibody systems, heat transfer, acoustics, etc.

Mathematically speaking, 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.

Examples

Lumped element model in electrical systems

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 $L_{c}\ll \lambda$ , where $L_{c}$ denotes the circuit's characteristic length, and $\lambda$ 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.

Lumped element model in mechanical systems

The simplifying assumptions in this domain are:

all objects are rigid bodies;
all interactions between rigid bodies take place via kinematic pairs (joints), springs and dampers.

Lumped element model in acoustics

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.

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.

A reflex port 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.

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

A loudspeaker 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.

Lumped element model in heat transfer for buildings

The simplifying assumptions in this domain are:

all heat transfer mechanisms are linear, impying that radiation and convection are linearised for each problem;
all components (resitors and capacitors) have to maintain their value during the simulation for the RC-network to be considered a linear system.

Using Lumped Element Models (LEMs) to represent the building thermo-dynamics does not offer the same accuracy than more complex numerical methodologies such as EnergyPlus and IES <VE>; however, it allows to perform dynamic simulation of buildings with reduced computational cost. The simulation suite CitySim by Darren Robinson uses LEMs, what reduce greatly the computational times of performing the simulations. This allows to perform stochastic analysis and urban simulation.

Several publications can be found that describe how to generate LEMs 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. Ramallo-González's method (Dominant Layer Method) is the most accurate and simple so far.^[1] 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. In his thesis,^[2] Ramallo-González shows the whole process of obtaining the LEM of a complete building.

LEMs of buildings have also been used to evaluate the efficiency of domestic energy systems ^[3] In this case the LEMs allowed to run many simulations under different future weather scenarios.

External links

Advanced modelling and simulation techniques for magnetic components
IMTEK Mathematica Supplement (IMS), the Open Source IMTEK Mathematica Supplement (IMS) for lumped modelling - broken link

References

^ Ramallo-González, A.P., Eames, M.E. & Coley, D.A., 2013. Lumped Parameter Models for Building Thermal Modelling: An Analytic approach to simplifying complex multi-layered constructions. Energy and Buildings, 60, pp.174-184.
^ Ramallo-González, A.P. 2013. Modelling Simulation and Optimisation of Low-energy Buildings. PhD. University of Exeter.
^ Cooper, S.J.G., Hammond, G.P., McManus, M.C., Ramallo-Gonzlez, A. & Rogers, J.G., 2014. Effect of operating conditions on performance of domestic heating systems with heat pumps and fuel cell micro-cogeneration. Energy and Buildings, 70, pp.52-60.

[1] Ramallo-González, A.P., Eames, M.E. & Coley, D.A., 2013. Lumped Parameter Models for Building Thermal Modelling: An Analytic approach to simplifying complex multi-layered constructions. Energy and Buildings, 60, pp.174-184.

[2] Ramallo-González, A.P. 2013. Modelling Simulation and Optimisation of Low-energy Buildings. PhD. University of Exeter.

[3] Cooper, S.J.G., Hammond, G.P., McManus, M.C., Ramallo-Gonzlez, A. & Rogers, J.G., 2014. Effect of operating conditions on performance of domestic heating systems with heat pumps and fuel cell micro-cogeneration. Energy and Buildings, 70, pp.52-60.

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