Bond Graph - Example

Example

Consider a simple RC circuit:

R i1 --\/\/\-----+------ i2 → v1 | v2 C = ↓ic | ground ----+------

where v1 represents a voltage source that is implied as being connected to the resistor, and v2 represents an "output" measurement point.

If one follows the "flows" through the circuit diagram then the basic structure of 1 (common flow) and 0 (common effort) nodes can be identified. In general, one should be looking for common (shared) efforts and common (shared) flows, but commonality may not be immediately obvious to the new practitioner, so to get started one can place a 0 node wherever a distinct effort potential (voltage) can be identified, and 1 nodes wherever a flow is identified, and then bonds between the 0 and 1 nodes:

i1 v1 ir v2 i2 1 --- 0 --- 1 --- 0 ---- 1 | 1 ic

Note that i1 and ic both involve current flows to ground, so no power flows there, so no bonds are shown for those power flows.

Next, one can add the power dissipating elements next to and connected to the 1 junctions representing flows through components. The connecting bonds represent the power flows that are generated, stored or dissipated by those elements.

Se,in R | | | | v2 i2 1 --- 0 --- 1 --- 0 ---- 1 i1 v1 ir | | 1 --- C ic

Next, 0 or 1 junctions that only have one or two bonds can be optimized out of existence because the power flows on those bonds are identical.

R |ir v1 | v2 i2 Se,in ---- 1 --- 0 --- 1 i1 | | ic C

Note that because v2 is measured, we can equivalently assume that i1=0 or that the output bond has a full arrow, and we can re-arrange the bonds for a more regular graph, and we can assign direction of power flow:

R C |\ |\ |ir |ic v1 \ i2 Se,in ----- 1 ----- 0 ----- out i1 / / /

The half-arrows on the remaining 1 junction are assigned so that the power flows into passive elements (R and C), out of the source of effort, and arbitrarily for flows between the 0 and 1 junctions. If you can anticipate a convention that causes them to be positive then interpreting results will usually be easier. For example, the power flowing between the 1 and 0 junctions should flow away from the 1 junction like the power flowing to the resistor, so set the half-arrow to reflect that.

Causality is defined by first setting the causality for reactive elements and power sources according to their behavior. Sources of effort and capacitors should impose effort (causality stroke opposite the source), and sources of flow and inertial elements should impose flow (causality stroke near the source). Once this is done, all 0 junctions should have one causal stroke on the near end of its bonds, and all 1 junctions should have only one causal stroke on the opposite end of any of its bonds. Causality for bonds on resistive elements can go whichever way satisfies the junction at the other end of the bond.

R C --- |\ |\ | |ic --- v1 | | v2 \ Se,in ------| 1 |----- 0 ------- out i1 /| | / i2 /

Note that the output is assumed to draw no power from the circuit, so a full arrow is used instead of a half-arrow. For the purpose of modeling dynamics, this means the output full bond can be ignored and the diagram is simplified (though the variables have now been renumbered):

R --- |\ v3 | i3 | v1 | | v2 S_e,in ------| 1 |----- C i1 /| | i2 /

Although a systematic approach to formulating the bond graph was described above, in retrospect the central 1 junction in the final bond graph reflects the fact that the same current flows through the input voltage source, the resistor, and the capacitor. One attraction of bond graphs is that experienced bond graph users can bypass many steps on their way to modeling the dynamic system.

Having completed and simplified the bond graph, the diagram can now be used for its intended purpose: guiding the practitioner through the generation of differential equations that describe the dynamics of the system. This is accomplished by starting at each of the reactive elements in turn and working through the implications of each bond and junction. This process can, in more complicated diagrams, involve traversing the bonds in both directions at times, but having properly defined causality will prevent this apparent retracing of steps from leading to algebraic loops or integral equations.

is a "source of effort" (voltage source) that forces the dynamics. Note that the causality for a source of effort imposes effort on the junction. To avoid formulating integral equations, the causality stroke for the capacitor must also impose effort on the junction. Since every 1 junction should have exactly one flow causal stroke, bond 3 must show flow imposed by the R element (causal stroke away from 1 junction).

To derive the differential equation, start on bond 2 (attached to a reactive element) and write the differential equation for that reactive element:

One can follow an invisible path from through the C and back to the corresponding to writing this equation down. Because bond 2 is attached to a 1 junction (shared flow) where bond 3 determines the flow, we can extend our path through the diagram from the of bond 2 to the of bond 3 following the flow causal stroke, obtaining . Note that we ignore the half-arrows at this point because all flows on a 1 junction are equal, regardless of direction of power flow. Substituting, we expand the differential equation with more information about the system:

At this point, we can follow the causality path from through R and back to, writing the corresponding relation . We can substitute this relation into the differential equation:

Continuing to follow causality, the effort on bond 3 is related to all the other efforts on the 1 junction since they must all sum to zero. That is, using the half-arrows to define signs we can write and substitute this into the differential equation:

Since is an input, and is a state variable (effort on a C element), the equation is completely expanded. Had it not been completely expanded, it might be necessary to follow two causality paths beyond this point to eventually completely expand the differential equation.

For systems with multiple I and/or C elements, the process can be repeated once for each derivative of a state variable to form a system of (typically but not necessarily linear) differential equations. For example, suppose we put two of these RC circuits in series:

R → i2 R i1 --\/\/\-----+------------\/\/\-----+------ i3 → v1 | v2 | v3 C = ↓ic C = ↓ic | | ground ----+----------------------+---------

Although this circuit was constructed as two cascaded RC circuits, the behavior of the first RC circuit is now complicated by the fact that power flows out of its "output" where before no power flowed there. This changes the overall dynamics of this system, but bond graphs can guide the formation of correct equations anyway. The corresponding bond graph looks like

R C R C --- --- |\ |\ |\ |\ 6 | 4 | 2 | 3 | --- --- 1 | | 5 7 | | 8 9 \ SE ------| 1 |------ 0 ------| 1 |------ 0 ------ out /| | / /| | / /

where the ever-present effort/flow (voltage/current in this case) variables have been dropped and the bonds are simply numbered per typical bond graph convention (in this case the first four numbers were placed to avoid confusion with the signal numbering in the circuit diagram). Again, the output is assumed to draw no power so bonds 8 and 9 can effectively be removed in favor of a direct connection to bond 3:

R C R --- --- |\ |\ |\ 6 | 4 | 2 | --- 1 | | 5 7 | | 3 SE ------| 1 |------ 0 ------| 1 |------ C /| | / /| | /

As before, we can start with the derivative of a state variable and follow the bonds to form equations:

Continuing the expansion:

At this point, is defined in terms of inputs and state variables.

Similarly, can be obtained in terms of inputs and state variables:

It is conventional in state space representation to group terms by state variables and inputs:

and to express the equations in matrix form:

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