Resistors relate efforts and flows to each other in a static fashion, i.e.:
e = R(f)
Similarly, capacitors relate efforts and generalized positions to each other, i.e.:
e = C(q)
Finally, inductors relate flows and generalized momentums to each other, i.e.:
f = I(p)
All of these relationships must be defined in the first and thrid quadrants only for the element to be physical. There cannot exist any other relationships in physics; thus, all bond graphs can be written in terms of the three variables R, C, and I.
In addition, energy can be transformed, either from one domain to another, or within a domain. The transformation of free energy can follow two different patterns:
e1 = m*e2
e1*f1 = e2*f2
or:
e1 = r*f2
e1*f1 = e2*f2
We call the former relationship a transformer, TF, whereas the latter relationship is called a gyrator, GY. The second equation in both cases ensures that no power is either lost or generated by the transformation.
In their causal forms, transformers and gyrators can be written as follows:
Thus, both the transformers and the gyrators have two possible causalities. Transformers have always one causality stroke at the transformer and the other away from it, whereas gyrators have either both causality strokes at the gyrator, or both away from it.
Finally, we need two types of sources, the effort source, Se, and the flow source, Sf:
The causality strokes of the two sources are fixed.
As we haven't discussed the causality strokes for the three basic modeling elements yet, let us cover that topic now:
Both capacitors and inductors have preferred causalities, so-called integral causalities, whereas the causality of the resistors is free.
That is all there is. These elements exhaust all of physics, and puritans among the bond graph practitioners would argue that no other elements should be offered.
If the preferred causality of a bond graph can be assigned in a unique way, the bond graph has neither algebraic loops nor structural singularities, i.e., can be described by an index-0 ordinary differential equation (ODE) system. If the modeler has free choice in assigning causality strokes, the model contains one or several algebraic loops. If a conflict exists on either capacitors or inductors, the model consists of a higher-index DAE system.