Integral Causality versus Differential Causality for Storage Elements Causality for Storage Elements

Examples 3.13–3.18 had storage elements with integral causality. Although this is the desirable outcome, this is not always achievable. Sometimes, the causal structure of the system is such that one or more of the energy stor-ing devices end up havstor-ing differential causality. Before we look at some of these examples, it is important to understand the different implications (in terms of the fi nal model) of an integral and differential causality.

For an energetic system, the state variables must at least uniquely defi ne the energy stored in the system. The minimum number of state variables required is determined by the number of independent energy storage elements in the system model. Energy storage devices that have been inte-grally causalled are independent.

An energy storage element that has derivative causality is not inde-pendent. Its stored energy is determined by the variables associated with the element from which the causal propagation began. Derivative causality on an energy storage element is not an error, but it can have undesirable consequences. It leads to implicit ordinary differential equations rather than explicit ordinary differential equations. Since explicit differential equations are more easily solved, the implicit equa-tions may lead to extensive algebra in deriving state equaequa-tions and may also lead to numerical diffi culties when simulating system behav-ior on a computer. If one or more energy storage elements in a model have derivative causality, the model developer may want to modify the model to eliminate the differential causalities. Although this is neither essential nor is it always possible, an attempt to change the causality is always recommended.

Examples 3.19–3.21 describe where derivative causality may occur.

EXAMPLE 3.19

The fi rst example is that of a circuit with a voltage source and a capacitor (Figure 3.67). The correctly causalled bond graph for this system is shown in Figure 3.68. The causal requirement for the 1 junction and the Se element are not violated, but the C element attains a derivative causality. If one tries to explore the origin/reason for this, it will eventually be clear that this circuit is not possible to construct. It is presumed that the circuit is being made to charge a capacitor. And charging a capacitor happens if a current starts fl owing through this circuit. However, without any resistance in the circuit, the current drawn could be infi nite. Thus, the circuit has to have some resistance in it, however small. If we add a resistance in the circuit, it looks like Figure 3.69 and its bond graph representation can be seen in Figure 3.70. Addition of the R element makes the actual circuit realistic and its model accurate. As a result, the causal description of the C element becomes integral as well.

V + C

−

FIGURE 3.67

A circuit with a voltage source and a capacitor.

C C1 1

1 Junction Se

Se1 FIGURE 3.68

The causalled bond graph for the circuit in Figure 3.67.

V +

−

R

C

FIGURE 3.69

Capacitor circuit modifi ed with resistance.

EXAMPLE 3.20

Let us look at another example. Here is a circuit with two inductors in series and a voltage source. The circuit and the causalled bond graph are shown in Figure 3.71.

Note that when we built the bond graph representation of the two Is, one will have a differential causality and another will have an integral causal-ity. This is because both the storage elements cannot have behavior that is

L1

L2 V

Se

Se1 l

I1

l I2

1 1 Junction

FIGURE 3.71

Circuit with two inductances and its corresponding bond graph.

R R1

1 1 Junction

C C 1 Se

Se1 FIGURE 3.70

Bond graph for Figure 3.69.

independent of each other. What happens in one will infl uence the other.

Hence, one is an integral causality and the other is a differential causality. If we modify this circuit by a single I equivalent (instead of using two I elements) as it is taught in EE circuits classes, the problem of differential causality will disappear and we will have a bond graph representation that will look as shown in Figure 3.72.

EXAMPLE 3.21

Consider the mechanical system and its initial bond graph shown in Figure 3.73.

After removing the ground velocity, the 1 junction labeled fi xed end, and 1

Modifi ed bond graph for system in Figure 3.71.

Se Rack velocity Rack and

pinion

FIGURE 3.73

A mechanical system and its corresponding initial bond graph.

simplifying, the bond graph representation looks like the fi rst one shown in Figure 3.74.

Note that after all the causal structure was determined one of the elements (the I element called pinion inertia) has a differential causality. Once again, this is a result of a modeling assumption that was made. Consider the transformer element, the rack and pinion. We are assuming that it is an ideal transformer, that is, there is no energy used to do anything other than to rotate the pinion.

This is possible if the teeth of the rack and pinion are ideally rigid. However, they are not and some of the energy is used to bend the teeth elastically. To modify the model, this elasticity needs to be accounted for using another C element after the transformer (or before), as shown in Figure 3.75. After adding this new C element, known as transformer compliance, all the storage elements now have integral causality.

R

Modifi ed bond graph and after the causal strokes are added.

Through Examples 3.19–3.21 we have demonstrated several ways to modify bond graph models to not only refl ect reality more closely but also to achieve integral causality for the storage elements in the model. In a very informative paper, Jose Granda (1984) offers a very insightful dis-cussion on this same topic. He summarized the different techniques that may be used to avoid differential causality. They are

1. Combine storage element into an equivalent storage element 2. Add an energy storage element

3. Add a resistive element 4. Remove some storage elements

Although these are general rules, these cannot be used without any relevance to the actual system. One must have some understanding of the system and the system model to determine which technique, if any, could be used in a given situation.