BASIC BOND GRAPH ELEMENTS
3.4 CAUSALITY CONSIDERATIONS FOR THE BASIC ELEMENTS The concept of causality was discussed in general terms in Section 2.4, and we
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
3.4.1 Causality for Basic 1-Ports
The effort and flow sources are the most easily discussed from a causal point of view, since, by definition, a source impresses either an effort or flow time history upon whatever system is connected to it. Thus, if we use the symbols Se— and Sf— for the abstract effort and flow sources, the only permissible causalities for these elements are
Se and Sf .
in which the causal stroke indicates the direction that the effort signal is oriented.
(Remember that a sign-convention half-arrow could be placed on either end of the bond without affecting the causality.) The causal forms for effort and flow sources are summarized in the first two rows of Table 3.6.
In contrast to the sources, the 1-port resistor is normally indifferent to the causality imposed upon it. The two possibilities may be represented in equation form as follows:
e=R(f ), f =−R1(e),
where we use the convention that the variable on the left of the equality sign represents the output of the resistor (the dependent variable), and that appearing in the function of the right side is the input (independent) variable for the element.
This convention is used commonly, but not universally, in writing equations and corresponds to the notation used in computer programming.
The correspondences between the causally interpreted equations and the causal strokes on the bond of theR— element are shown in the third row of Table 3.6.
As long as both the functions R and −1R exist and are known, either causal version of them could be used in a system model. It is possible, however, that the static relation betweeneandf shown in Figure 3.1 is multiple valued in one direction or the other; that is, either R or −1R might be multiple valued. In
TABLE 3.6. Causal Forms for Basic 1-Ports
Element Acausal Form Causal Form Causal Relation
Effort source Se Se e(t)=E(t)
Flow source Sf Sf f (t)=F (t)
Resistor R R e=R(f )
R f =−R1(e)
Capacitor C C e=−1C t
f dt
C f = d
dtC(e)
Inertia I I f =−1I t
e dt
I e= d
dtI(f )
such a case, the single-valued causality would be clearly preferable. In the linear case, with a finite slope of thee–f characteristic, the 1-port resistor is indifferent to the causality imposed upon it, although the resistance law would be written in two forms:
e=Rf or f =(1/R)e=Ge.
The constitutive laws of the C— and I— elements are expressed as static relations betweene andq=t
fdtandf andp=t
edt, respectively. In express- ing causal relations betweene’s andf’s, we will find that the choice of causality has an important effect. Taking the capacitor, we may rewrite the relations from Table 3.2 as follows:
e=−1C
t
f dt
, f = d
dtC(e), (3.22)
in which causality is implied by the form of the equation. Note that when f is the input to the C—, e is given by a static function of the time integral of f, but when e is the input, f is the time derivative of a static function of e. The correspondences between these causal equations and the causal stroke notation for the capacitance are shown in the fourth row of Table 3.6. The implications of the two types of causality, which are calledintegral causality andderivative causality, respectively, will be discussed in some detail in later chapters.
Since inertia is the dual∗ of the capacitor, similar effects occur with the two choices of causality. Rewriting the inertia element relations from Table 3.3, we have
f =−1I
t
e dt
, e= d
dtI(f ). (3.23) In this case, integral causality exists when e is the input to the inertia, and derivative causality exists whenf is the input. These observations are summarized in the fifth row of Table 3.6. Equations (3.22) and (3.23) are written in a form suitable for nonlinearC— andI— elements, but the distinction between integral and derivative causality remains for the special case of linear elements with their compliance and inertance parameters.
3.4.2 Causality for Basic 2-Ports
Proceeding now to the basic 2-ports, one might think initially that there would be a total of four possibilities for the assignment of causality of a transformer, namely, any combination of the two possible causalities for each of the two ports. However, there are only two possible causality assignments, as the defining
∗Dual elements have identical types of constitutive laws except that the roles of effort and flow are interchanged.
relations (3.9) and (3.11) show. As soon as one of the e’s or f’s has been assigned as an input to the —TF—, the other e or f is constrained to be an output by Eq. (3.9). Thus, in fact, the only two possible choices for causality for the transformer are T F and T F . The possible causalities are tabulated in the first row of Table 3.7. Again, causal equation equivalents to the causal stroke notation are given for all elements in Table 3.7.
For the gyrator, Eqs. (3.11) show that as soon as the causality for one bond has been determined, the causality for the other is also. Thus, the only permissible causal choices for the —GY— are GY and GY . The choices for the causality for the gyrator are summarized in the second row of Table 3.7.
3.4.3 Causality for Basic 3-Ports
The causal properties of 3-port 0- and 1-junctions are somewhat similar to those of the basic 2-ports. Although each bond of the 3-ports, considered alone, could have either of the two possible causalities assigned, not all combinations of bond causalities are permitted by the constitutive relations of the element. For example, the constitutive relations for the 0-junction given in Table 3.5 indicate that all efforts on all the bonds are equal and the flows must sum to zero. Thus, if on any bond the effort is an input to a 0-junction, then all other efforts are determined, and on all other bonds they must be outputs of the 0-junction. Conversely, if all the flows on all bonds except one are inputs to the 0-junction, the flow on the remaining bond is determined and must be an output of the junction. A typical TABLE 3.7. Causal Forms for Basic 2-Ports and 3-Ports
Element Acausal Graph Causal Graph Causal Relations
Transformer e1=me2
f2=mf1
1 T F 2 1 T F 2
1 T F 2 f1=f2/m
e2=e1/m
Gyrator 1 GY 2 1 GY 2 e1=rf2
e2=rf1
1 GY 2 f1=e2/r
f2=e1/r
0-Junction 1 0 e2=e1
3
2 1
0
3 2
e3=e1
f1= −(f2+f3)
1-Junction 1 1 f2=f1
3
2 1
1
3 2
f3=f1 e1= −(e2+e3)
permissible causality for a 0-junction is shown in the third row of Table 3.7.
Here the causal stroke on the end of bond 1 nearest the 0 indicates that e1 is an input to the junction and that all other bonds must have causal strokes at the end away from the 0. To interpret the diagram another way, the flows on bonds 2 and 3 are inputs to the 0-junction. These considerations are also expressed by the causal equations shown in Table 3.7. For a 3-port 0-junction, then, there are only three different permissible causalities in which each of the three bonds in succession plays the role assigned to bond 1 in the example shown in the table.
For an n-port 0-junction this description of the constraints on causality is still valid, and there are exactlyn different permissible causal assignments.
For a 1-junction the same considerations apply as for a 0-junction except that the roles of the efforts and flows are interchanged. Table 3.5 indicates that flows on all the bonds are equal and the efforts sum to zero. Thus, if the flow on any single bond is an input to the 1-junction, the flows on all other bonds are determined and must be considered outputs of the junction. Alternatively, when the efforts on all bonds except one are inputs to the 1-junction, the effort on the remaining bond is determined and must be an output of the junction. A typical permissible causality is shown in the fourth row of Table 3.7. In this example, bond 1 plays the special role of determining the common flow at the junction, and the remaining bonds supply effort inputs that suffice to determine the effort on bond 1. Clearly, there are three permissible causalities for a 3-port 1-junction, and there arenpermissible different causal assignments for ann-port 1-junction.
Although the causal considerations have been stated for all the basic multiports defined so far (summarized in Tables 3.6 and 3.7), it can hardly be clear as to what all the implications of causality are. The study of causality is very important, and bond graphs are uniquely suited to this study. However, only when some real system models have been assembled is it clear why causal information is so important. In the next chapter, system models are built up using the basic multiports just discussed. Using the rules of causality, it is then possible to predict many important features of these systems even before the constitutive laws for all the elements have been decided upon. For instance, it will be possible to predict the mathematical order of the system model before any equations are written and before a decision has been made about whether the model should be linear or nonlinear. In addition, causal considerations will prove invaluable in writing a set of state equations or in setting up computational program for system simulation.