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

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

The change of the magnetic flux in time outside a conductor is zero. ${\frac {\partial \phi _{B}}{\partial t}}=0$
The change of the charge in time inside conducting elements is zero. ${\frac {\partial q}{\partial t}}=0$
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 $L_{c}\ll \lambda$ , 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.

Thermal systems

A lumped-capacitance model, also called lumped system analysis,^[2] reduces a thermal system to a number of discrete “lumps” and assumes that the temperature difference inside each lump is negligible. This approximation is useful to simplify otherwise complex differential heat equations. It was developed as a mathematical analog of electrical capacitance, although it also includes thermal analogs of electrical resistance as well.

The lumped-capacitance model is a common approximation in transient conduction, which may be used whenever heat conduction within an object is much faster than heat transfer across the boundary of the object. The method of approximation then suitably reduces one aspect of the transient conduction system (spatial temperature variation within the object) to a more mathematically tractable form (that is, it is assumed that the temperature within the object is completely uniform in space, although this spatially uniform temperature value changes over time). The rising uniform temperature within the object or part of a system, can then be treated like a capacitative reservoir which absorbs heat until it reaches a steady thermal state in time (after which temperature does not change within it).

An early-discovered example of a lumped-capacitance system which exhibits mathematically simple behavior due to such physical simplifications, are systems which conform to Newton's law of cooling. This law simply states that the temperature of a hot (or cold) object progresses toward the temperature of its environment in a simple exponential fashion. Objects follow this law strictly only if the rate of heat conduction within them is much larger than the heat flow into or out of them. In such cases it makes sense to talk of a single "object temperature" at any given time (since there is no spatial temperature variation within the object) and also the uniform temperatures within the object allow its total thermal energy excess or deficit to vary proportionally to its surface temperature, thus setting up the Newton's law of cooling requirement that the rate of temperature decrease is proportional to difference between the object and the environment. This in turn leads to simple exponential heating or cooling behavior (details below).

Method

To determine the number of lumps, the Biot number (Bi), a dimensionless parameter of the system, is used. Bi is defined as the ratio of the conductive heat resistance within the object to the convective heat transfer resistance across the object's boundary with a uniform bath of different temperature. When the thermal resistance to heat transferred into the object is larger than the resistance to heat being diffused completely within the object, the Biot number is less than 1. In this case, particularly for Biot numbers which are even smaller, the approximation of spatially uniform temperature within the object can begin to be used, since it can be presumed that heat transferred into the object has time to uniformly distribute itself, due to the lower resistance to doing so, as compared with the resistance to heat entering the object.

If the Biot number is less than 0.1 for a solid object, then the entire material will be nearly the same temperature, with the dominant temperature difference being at the surface. It may be regarded as being "thermally thin". The Biot number must generally be less than 0.1 for usefully accurate approximation and heat transfer analysis. The mathematical solution to the lumped-system approximation gives Newton's law of cooling.

A Biot number greater than 0.1 (a "thermally thick" substance) indicates that one cannot make this assumption, and more complicated heat transfer equations for "transient heat conduction" will be required to describe the time-varying and non-spatially-uniform temperature field within the material body.

The single capacitance approach can be expanded to involve many resistive and capacitive elements, with Bi < 0.1 for each lump. As the Biot number is calculated based upon a characteristic length of the system, the system can often be broken into a sufficient number of sections, or lumps, so that the Biot number is acceptably small.

Some characteristic lengths of thermal systems are:

Plate: thickness
Fin: thickness/2
Long cylinder: diameter/4
Sphere: diameter/6

For arbitrary shapes, it may be useful to consider the characteristic length to be volume / surface area.

Thermal purely resistive circuits

A useful concept used in heat transfer applications once the condition of steady state heat conduction has been reached, is the representation of thermal transfer by what is known as thermal circuits. A thermal circuit is the representation of the resistance to heat flow in each element of a circuit, as though it were an electrical resistor. The heat transferred is analogous to the electric current and the thermal resistance is analogous to the electrical resistor. The values of the thermal resistance for the different modes of heat transfer are then calculated as the denominators of the developed equations. The thermal resistances of the different modes of heat transfer are used in analyzing combined modes of heat transfer. The lack of "capacitative" elements in the following purely resistive example, means that no section of the circuit is absorbing energy or changing in distribution of temperature. This is equivalent to demanding that a state of steady state heat conduction (or transfer, as in radiation) has already been established.

The equations describing the three heat transfer modes and their thermal resistances in steady state conditions, as discussed previously, are summarized in the table below:

Equations for different heat transfer modes and their thermal resistances.
Transfer Mode	Rate of Heat Transfer	Thermal Resistance
Conduction	${\dot {Q}}={\frac {T_{1}-T_{2}}{\left({\frac {L}{kA}}\right)}}$	${\frac {L}{kA}}$
Convection	${\dot {Q}}={\frac {T_{\rm {surf}}-T_{\rm {envr}}}{\left({\frac {1}{h_{\rm {conv}}A_{\rm {surf}}}}\right)}}$	${\frac {1}{h_{\rm {conv}}A_{\rm {surf}}}}$
Radiation	${\dot {Q}}={\frac {T_{\rm {surf}}-T_{\rm {surr}}}{\left({\frac {1}{h_{r}A_{\rm {surf}}}}\right)}}$	${\frac {1}{h_{r}A}}$ , where $h_{r}=\epsilon \sigma (T_{\rm {surf}}^{2}+T_{\rm {surr}}^{2})(T_{\rm {surf}}+T_{\rm {surr}})$

In cases where there is heat transfer through different media (for example, through a composite material), the equivalent resistance is the sum of the resistances of the components that make up the composite. Likely, in cases where there are different heat transfer modes, the total resistance is the sum of the resistances of the different modes. Using the thermal circuit concept, the amount of heat transferred through any medium is the quotient of the temperature change and the total thermal resistance of the medium.

As an example, consider a composite wall of cross-sectional area $A$ . The composite is made of an $L_{1}$ long cement plaster with a thermal coefficient $k_{1}$ and $L_{2}$ long paper faced fiber glass, with thermal coefficient $k_{2}$ . The left surface of the wall is at $T_{i}$ and exposed to air with a convective coefficient of $h_{i}$ . The right surface of the wall is at $T_{o}$ and exposed to air with convective coefficient $h_{o}$ .

Using the thermal resistance concept, heat flow through the composite is as follows:

{\dot {Q}}={\frac {T_{i}-T_{o}}{R_{i}+R_{1}+R_{2}+R_{o}}}={\frac {T_{i}-T_{1}}{R_{i}}}={\frac {T_{i}-T_{2}}{R_{i}+R_{1}}}={\frac {T_{i}-T_{3}}{R_{i}+R_{1}+R_{2}}}={\frac {T_{1}-T_{2}}{R_{1}}}={\frac {T_{3}-T_{o}}{R_{0}}}

[1]

[2]