Dipole Assemblies

The fifth level is the assembly of dipoles, made by association of two or three dipoles belonging to different types. Thus, we can have four dipole assemblies capacitive -i- inductive, capacitive + conductive, inductive + conductive, and capacitive + inductive -i- conductive. Examples of dipole assemblies are an electric LC oscillator, a mass attached to a spring, outflow from a fluid reservoir, a chemical reaction with multiple reactants, etc. Note that the association of two dipoles of the same type does not form an assembly, which requires an association of two different types of dipoles. Two capacitive dipoles, for instance, when energy exchange is possible between them, merely form another capacitive dipole, that combines the properties of both dipoles. 

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The maximum perturbation of this kind of dipole assembly converges to approximately 0.24 V, as seen from the plateau in Fig. 17.5, and from the plot of maximum potential against number of dipoles in Fig. 17.6. 

Dipole assembly Association of dipoles of the same energy variety but different natures. 

An argued introduction and discussion of time will be made in Chapter 9, when dealing with the conversion of energy permitted by associating dipoles of different energy subvarieties, called dipole assemblies. In this chapter, the dynamics of dipoles, which are considered without any time involvement, will not be developed. The dynamic regimes, or kinetics out of stationary regime, will be treated in Chapter 8. 

A dipole can be connected to other dipoles, whatever the energy snbvariety or variety. A dipole, or multipole, association belongs to the following higher organization level of dipole assemblies (above multipoles). 

This choice is based on the rule used for a dipole assembly. In a mounting in series, the efforts are added to form the total effort (and the flow is common to both dipoles), whereas in parallel mounting it is the converse, flows are added as the effort is shared by both dipoles. In any case, the basic quantities or the impulses are involved in such an operation, so it is not very easy to justify physically an addition of induction quantities for modeling influence. The energies-per-entity are entitled in the Formal Graph theory to be formed from several contributions contrary to entity numbers. This rule applies equally to the poles. 

A multipole is merely an extension of the concept of dipole by increasing the number of poles. As for the dipole, by definition, poles must be of the same energetic natnre, that is, all belonging to the same energy subvariety or all being purely conductive in the same energy variety. The case of mixed natures (for instance, capacitive-inductive) is relevant to the upper Formal Object called dipole assembly. 

The correspondence between efforts in the storing dipole and in the conductive dipole (see Chapter 11 devoted to dissipation in dipole assemblies) makes these coefficients play the role of a proportionality factor between the conductive dipole effort and one of the pole efforts (see Graph 8.33). 

Table 9.1 lists the case studies of dipole assemblies given in this chapter. (See also Figure 9.1 for the position of the dipole assemblies.)... 

Multi pole Dipole assembly Coupled assemblies... 

FIGURE 9.1 Position of the dipole assembly along the complexity scale of Formal Objects. 

Once this mathmatical form for the evolution operator is established, one has to show that the same operator works for aU energy subvarieties. Let us take a system (a dipole assembly) containing the two subvarieties of the same energy variety. The total energy is the sum of energies of the subvarieties, according to the additive property of energy defined in Chapter 2 ... 

Note that when the system is isolated, the received power is naturally zero, as it has been considered in the previous dipole assembly. In such a situation. Equation 9.49 set to zero is known as Tellegen s theoran (TeUegen 1952). 

Naturally, when several units are used, they need to be connected together and this is ensured by supplementary square units, on the same template as for the dipoles constituting a multipole. This case of more complex structures will be tackled in subsequent chapters. In this chapter, only elementary dipole assemblies will be treated, so elementary Formal Graphs (square unit) will be sufficient. 

To discuss the way to represent a dipole assembly having in common only one eneigy-per-entity (i.e., elementary), we take again the case of three dipoles, one capacitive, one inductive, and one conductive. 

The two Formal Graphs in Graph 9.4 model the most general dipole assemblies using only one common node which are composed of the three natures of dipoles simultaneously. Simpler Formal Graphs using fewer properties are drawn when a lower number of dipole natures form the assembly, as will be shown in this chapter. 

To illustrate how elementary dipole assemblies are modeled by a Formal Graph, two cases are worth presenting before discussing several case studies. The first case deals with the subject of the isolation or the connection of a dipole assembly with an external source of power the second case deals with the representation of the most important system treated in this chapter, which is the oscillator. 

An effort or a flow, which is a gate for communicating with the exterior, can be imposed (or supplied) by an external system (another dipole or dipole assembly, for instance). An example of external effort is when a force is imposed on a mass placed in a gravitational field. An external flow may correspond to the convection phenomenon, when a fluid transporting an object imposes its own velocity. 

GRAPH 9.4 Serial RLC (common flow and summed effort) dipole assembly (left) and parallel RLC (common effort and summed flow) dipole assembly (right). No indication is made about the isolation or the connection with a power supply in these models. 

When modeling a dipole assembly, the question needs to be taken into account because such a system may work whether in an autonomous manner or in relationship with another system, considered as external in this case. When power is supplied to the dipole assembly, it influences the way energy is exchanged or converted between dipoles. The consequence is that the energetic behavior is different according to the isolation or the connection of the assembly to exterior. 

This indication is necessary when using a Formal Graph for simulation owing to its similarity with neural networks used in computation. For theoretical studies—for just modeling the physical law governing a process—the fact that a dipole assembly is powered or not may be less important. 

Graphs 9.6 and 9.7 give in the three languages—equivalent circuit, algebraic equation, and Formal Graph—the models of a powered dipole, or dipole assembly, featured by an admittance Yq. 

GRAPH 9.5 Equivalence between the two Formal Graphs of a dipole assembly with an external generator imposing an effort. The Formal Graph on the right is the compact version in which the circle around the effort indicates that it is imposed. 

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