BASIC BOND GRAPH ELEMENTS
3.3 THE 3-PORT JUNCTION ELEMENTS
FIGURE 3.10. A displacement-modulated transformer: (a) sketch of rigid, massless rotat- ing arm; (b) bond graph.*
We now introduce these two connection patterns as 3-port components them- selves, which, like the 2-ports of the previous section, are power conserving.
These 3-ports are called junctions, since they serve to interconnect other multi- ports into subsystem or system models. These 3-ports represent one of the most fundamental ideas behind the bond graph formalism. The idea is to represent in multiport form the two types of connections, which, in electrical terms, are called the series and parallel connections. As we shall see, such connections really occur in all types of systems, even though traditional treatments may not recognize the existence of the junctions as multiports.
First, consider the flow junction,0-junction, or common effort junction. The symbol for this junction is a zero with three bonds emanating from it (as will become evident, it is easy to extend the definition to a 4-, 5-, or more-port version of this 3-port element):
—— 0 —, —2
—1 0 —
3,
2
1 0
3.
This element is ideal in that power is neither dissipated nor stored. Using theinward power sign convention shown in the last version of the junction, this implies
e1f1+e2f2+e3f3=0. (3.16) The 0-junction is defined such that all efforts are the same, thus,
e1(t)=e2(t)=e3(t). (3.17) Combining (3.16) and (3.17) yields
f1(t)+f2(t)+f3(t)=0. (3.18) In words, the efforts on all bonds of a 0-junction are always identical, and the algebraic sum of the flows always vanishes. In other words, if power is flowing in on two ports of the three, then it must be flowing out of the third port.
The use of the 0-junction is suggested by Figure 3.11a. The most obvious examples of 0-junctions are the electrical conductors connected as shown to provide three terminal pairs and the pipe tee junction, which is an idealized version of the hardware store variety. The mechanical example may seem obscure, and it is contrived. Mechanical 0-junctions are just as necessary as electrical or hydraulic ones, but they do not appear so readily in gadget form. The two carts in the mechanical example of Figure 3.11a are supposed to be rigid and massless.
Note that V3= −V1−V2, which conforms with Eq. (3.18). If F is the force across the gap, X3, then F is the port effort for V1,V2, and V3, in accordance with Eq. (3.17). Such a force would, in fact, exist if F were due to a massless spring or damper connected between the two carts.
FIGURE 3.11. Basic 3-ports in various physical domains: (a) 0-junction; (b) 1-junction.
It is important to note that power conservation Eq. (3.16) and flow variable summation Eq. (3.18) change if not all the power half-arrows point inward as in the example above. Suppose that bond 3 had a half-arrow pointing outward instead of inward as is the case for bonds 1 and 2. Then a minus sign would appear in the power conservation equation and the flow summation, thus:
e1f1+e2f2−e3f3=0. (3.16a) f1(t)+f2(t)−f3(t)=0. (3.18a) This is why we stress that for a 0-junction, the flows add algebraically to zero. Depending on the orientation of the sign-convention half-arrows on the bonds, minus signs or plus signs will appear in the equations of the junction.
One simple way to decide on the proper signs in equations such as Eq. (3.18) or Eq. (3.18a) is to use aplus sign for all bonds withinward-pointing half-arrows and a minus sign for bonds with outward-pointing half-arrows. Note however, that the half-arrow directions have no effect on the strict equality of the efforts on all bonds of a 0-junction.
Before considering more examples in which the 0-junction is used, consider thedual of the 0-junction, that is, a multiport element in which the roles of effort and flow are interchanged. Such an element is aneffort junction, a 1-junction, or acommon flow junction. The symbol for this multiport is a 1 with three bonds:
—— 1 —, —2
—1 1 —
3,
e2 f2
e1
f1
1 e3
f3
This element is again power conserving according to Eq. (3.16) if all half- arrows point inward; however, this time the element is defined such that every bond has the identical flow, thus,
f1(t)=f2(t)=f3(t), (3.19) which, when combined with the power conserving idealization, requires
e1(t)+e2(t)+e3(t)=0. (3.20) As with the 0-junction, the constitutive equations for the 1-junction, Eqs.(3.19 and 3.20), combine to ensure power conservation in the form of Eq. (3.16) as long as all bonds have an inward sign convention.
If a 1-junction should have aninwardsign convention on bonds 2 and 3 but an outward sign convention on bond 3, the power conservation would be expressed as in Eq. (3.16a) and the effort summation of Eq. (3.20) would change to
e1(t)+e2(t)−e3(t)=0. (3.20a) The equality of the 1-junction flows, Eq. (3.19), is not affected by the power sign conventions on the bonds.
The 1-junction has a single flow, and thealgebraicsum of the effort variables on the bonds vanishes. Figure 3.11bshows some instances in which a 1-junction can be used to model physical situations. Both the electrical conductors and the hydraulic passages are arranged so that if 1-port components were attached to the ports, one could describe the resulting connection as a series connection. A single current or volume flow would circulate, and the voltages and pressures at the ports would sum algebraically to zero. In the mechanical example, the three forces are all associated with a common velocity, and the forces must sum to zero, since the cart is assumed to be massless.
An understanding of the meaning of the 0- and 1-junctions is important for anyone learning bond graph techniques, and it may be helpful to give some physical interpretation for these multiports in several physical domains.
Electrical
circuits: — 0 —,—
represents Kirchhoff’s current law for a node where three conductors join
—— 1 —,
represents Kirchhoff’s voltage law written along a loop in which a current flows and experiences three voltage drops
Mechanical
systems: — 0 —,—
represents geometric compatibility for a situation involving a single force and three velocities that algebraically sum to zero
—— 1 —,
represents dynamic equilibrium of forces asso- ciated with a single velocity—when an inertia element is involved, the junction enforces New- ton’s law for the mass element
Hydraulic
systems: — 0 —,—
represents the conservation of volume flow rate at a point where three pipes join
—— 1 —,
represents the requirement that the sum of pres- sure drops around a circuit involving a single flow must sum algebraically to zero
As might be expected, the existence of 0- and 1-junctions within complex systems is not always obvious, but in succeeding chapters formal techniques for modeling systems using these basic elements are presented.
To make clear the utility of the junctions, four elementary example systems are displayed in Figure 3.12. Note that only two bond graphs are involved. The series and parallel aspects of the junctions are more obvious in the electrical than in the mechanical cases. The reader should study these examples to understand how the sign conventions are transferred from the physical sketches to the bond graph.
Note that in Figure 3.12 the 1-ports have the sign-convention half-arrows as they were presented in the tables at the beginning of this chapter, but the junction signs are not all inward pointing. As we have noted, when the sign-convention arrows are changed from the inward-pointing convention used to introduce the 3-ports, Eqs. (3.18) and (3.20) must be modified with a minus sign for each port with an outward-pointing sign. Equations (3.17) and (3.19) remain invariant, however, under changes in a sign convention. Remember, a 0-junction has only a single effort and a 1-junction has only a single flow, independent of the sign convention. As an example, consider
e1 f1
e2f2
1 e3
f3, the equations of which are
f1=f2=f3, e1−e2−e3=0. (3.21) The reader should verify that the systems and bond graphs are consistent by writing equations such as (3.21).
FIGURE 3.12. Example systems involving basic 3-ports. (a) Systems using 0-junctions;
(b) systems using 1-junctions.
The slight generalization from 3-port junctions to 4- or n-port junctions is worth emphasizing. In bond graph symbolism, two similar 3-ports may be com- bined into a 4-port thus:
— — 0 — 0
—
—=— — 0
—
—
— — 1 — 1
—
—=— — 1
—
—.
An n-port 0- or 1-junction has a common effort or flow on all bonds, and the algebraic sum of the complementary power variables on the bonds vanishes.
Occasionally, 2-port junctions arise, and, in some cases, these are precisely equiv- alent to a single bond. The following bond graph identities are always valid:
0 = , 1 = .
However, with some sign patterns, the 2-port 0- and 1-junctions serve to reverse the sign definition of an effort or flow. For example,
e1 f1
0e2
f2
implies e1=e2, f1= −f2,
and e1
f1
1e2
f2
implies f1=f2, e1= −e2.
Such 2-ports are sometimes necessary when two multiports are to be joined by a bond, but the two multiports have been defined with signs that are not compatible with a single bond. For example, in connecting a spring, —C, to a mass, —I, one could define a common velocity but use a 2-port 1-junction to express the fact that the spring force is the negative of the force on the mass.
The resulting bond graph would then beC 1 I,in which the two passive 1-ports have the appropriate inward sign convention. In this case, both bonds have the same flow but one effort is the negative of the other.
The constitutive relations for 0- and 1-junctions are summarized in Table 3.5.
3.4 CAUSALITY CONSIDERATIONS FOR THE BASIC ELEMENTS