Lumped element model
Encyclopedia
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems
Electrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...
(including electronics
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
), mechanical multibody system
Multibody system
A multibody system is used to model the dynamic behavior of interconnected rigid or flexible bodies, each of which may undergo large translational and rotational displacements.- Introduction :...
s, heat transfer
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...
, acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
, etc.
Mathematically speaking, the simplification reduces the state space
State space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...
of the system to a finite number, and the partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s (ODEs) with a finite number of parameters.
Lumped element model in electrical systems
The lumped element model of electronic circuitElectrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...
s makes the simplifying assumption that the attributes of the circuit, resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
, capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
, inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
, and gain
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...
, are concentrated into idealized electrical components; resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s, capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s, and inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s, etc. joined by a network of perfectly conducting wires.
The lumped element model is valid whenever
, where 
denotes the circuit's characteristic length, and 
denotes the circuit's operating wavelengthWavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
.
Otherwise, when the circuit length is on the order of a wavelength, we must consider more general models, such as the distributed element model
Distributed element model
In electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit are distributed continuously throughout the material of the circuit...
(including transmission line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
s), whose dynamic behaviour is described by the Maxwell 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 capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s for example, we can model the non-ideal capacitor as having a large lumped resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
connected in-parallel even though the leakage is, in reality distributed throughout the dielectric. Similarly a wire-wound resistor has significant inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
as well as resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
distributed along its length but we can model this as a lumped inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
in series with the ideal resistor.
Lumped element model in mechanical systems
The simplifying assumptions in this domain are:- all objects are rigid bodiesRigid bodyIn physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
; - all interactions between rigid bodies take place via kinematic pairKinematic pairA kinematic pair is a connection between two bodies that imposes constraints on their relative movement. Franz Reuleaux introduced the kinematic pair as a new approach to the study of machines that provided an advance over the notion of elements consisting of simple machines.Hartenberg & Denavit...
s (joints), springSpring (device)A spring is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication...
s and dampersDashpotA dashpot is a mechanical device, a damper which resists motion via viscous friction. The resulting force is proportional to the velocity, but acts in the opposite direction, slowing the motion and absorbing energy. It is commonly used in conjunction with a spring...
.
Lumped element model in acoustics
In this context, the lumped component model extends the distributed concepts of Acoustic theoryAcoustic theory
Acoustic theory is the field relating to mathematical description of sound waves. It is derived from fluid dynamics. See acoustics for the engineering approach....
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 capacitorCapacitorA capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
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 inductorInductorAn inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
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 correctionEnd correctionIn physics, end correction is the anomal difference between the frequency of a tuning fork and the corresponding sound waves inside of a tube. It is caused because generally there is space between the fork and the pipe end, causing the air column to vibrate a short distance beyond the edge of the...
. 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 resistorResistorA linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
. 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 loudspeakerLoudspeakerA loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...
drive unit (typically a wooferWooferWoofer is the term commonly used for a loudspeaker driver designed to produce low frequency sounds, typically from around 40 hertz up to about a kilohertz or higher. The name is from the onomatopoeic English word for a dog's bark, "woof"...
or subwooferSubwooferA subwoofer is a woofer, or a complete loudspeaker, which is dedicated to the reproduction of low-pitched audio frequencies known as the "bass". The typical frequency range for a subwoofer is about 20–200 Hz for consumer products, below 100 Hz for professional live sound, and below...
drive unit) may be approximated as a series connection of a zero-impedanceElectrical impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
voltageVoltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
source, a resistorResistorA linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
, a capacitorCapacitorA capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
and an inductorInductorAn inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
. The values depend on the specifications of the unit and the wavelength of interest.
External links
- Advanced modelling and simulation techniques for magnetic components
- IMTEK Mathematica Supplement (IMS), the Open Source IMTEK Mathematica Supplement (IMS) for lumped modelling