A bond graph is a graphical representation of a physical dynamic system. It allows the conversion of the system into a statespace representation. It is similar to a block diagram or signalflow graph, with the major difference that the arcs in bond graphs represent bidirectional exchange of physical energy, while those in block diagrams and signalflow graphs represent unidirectional flow of information. Bond graphs are multienergy domain (e.g. mechanical, electrical, hydraulic, etc.) and domain neutral. This means a bond graph can incorporate multiple domains seamlessly.
The bond graph is composed of the "bonds" which link together "single port", "doubleport" and "multiport" elements (see below for details). Each bond represents the instantaneous flow of energy (dE/dt) or power. The flow in each bond is denoted by a pair of variables called power variables, whose product is the instantaneous power of the bond. The power variables are broken into two parts: flow and effort. For example, for the bond of an electrical system, the flow is the current, while the effort is the voltage. By multiplying current and voltage in this example you can get the instantaneous power of the bond.
A bond has two other features described briefly here, and discussed in more detail below. One is the "halfarrow" sign convention. This defines the assumed direction of positive energy flow. As with electrical circuit diagrams and freebody diagrams, the choice of positive direction is arbitrary, with the caveat that the analyst must be consistent throughout with the chosen definition. The other feature is the "causality". This is a vertical bar placed on only one end of the bond. It is not arbitrary. As described below, there are rules for assigning the proper causality to a given port, and rules for the precedence among ports. Causality explains the mathematical relationship between effort and flow. The positions of the causalities show which of the power variables are dependent and which are independent.
If the dynamics of the physical system to be modeled operate on widely varying time scales, fast continuoustime behaviors can be modeled as instantaneous phenomena by using a hybrid bond graph. Bond graphs were invented by Henry Paynter.^{[1]}
The tetrahedron of state is a tetrahedron that graphically shows the conversion between effort and flow. The adjacent image shows the tetrahedron in its generalized form. The tetrahedron can be modified depending on the energy domain. The table below shows the variables and constants of the tetrahedron of state in common energy domains.
Energy Domain^{[2]}^{[Note 1]}  

Generalized  Name  Generalized flow  Generalized effort  Generalized displacement  Generalized momentum  Resistance  Inertance  Compliance 
Symbol  
Linear
mechanical 
Name  Velocity  Force  Displacement  Linear momentum  Damping constant  Mass  Inverse of the spring constant 
Symbol  
Units  
Angular
mechanical 
Name  Angular velocity  Torque  Angular displacement  Angular momentum  Angular damping  Mass moment of inertia  Inverse of the angular spring constant 
Symbol  
Units  
Electromagnetic  Name  Current  Voltage  Charge  Flux linkage  Resistance  Inductance  Capacitance 
Symbol  
Units  
Hydraulic/
pneumatic 
Name  Volume flow rate  Pressure  Volume  Fluid momentum  Fluid resistance  Fluid inductance  Storage 
Symbol  
Units 
Using the tetrahedron of state, one can find a mathematical relationship between any variables on the tetrahedron. This is done by following the arrows around the diagram and multiplying any constants along the way. For example, if you wanted to find the relationship between generalized flow and generalized displacement, you would start at the f(t) and then integrate it to get q(t). More examples of equations can be seen below.
Relationship between generalized displacement and generalized flow.
Relationship between generalized flow and generalized effort.
Relationship between generalized flow and generalized momentum.
Relationship between generalized momentum and generalized effort.
Relationship between generalized flow and generalized effort, involving the constant C.
All of the mathematical relationships remain the same when switching energy domains, only the symbols change. This can be seen with the following examples.
Relationship between displacement and velocity.
Relationship between current and voltage, this is also known as Ohm's law.
Relationship between force and displacement, also known as Hooke's law. The negative sign is dropped in this equation because the sign is factored into the way the arrow is pointing in the bond graph.
If an engine is connected to a wheel through a shaft, the power is being transmitted in the rotational mechanical domain, meaning the effort and the flow are torque (τ) and angular velocity (ω) respectively. A word bond graph is a first step towards a bond graph, in which words define the components. As a word bond graph, this system would look like:
A halfarrow is used to provide a sign convention, so if the engine is doing work when τ and ω are positive, then the diagram would be drawn:
This system can also be represented in a more general method. This involves changing from using the words, to symbols representing the same items. These symbols are based on the generalized form, as explained above. As the engine is applying a torque to the wheel, it will be represented as a source of effort for the system. The wheel can be presented by an impedance on the system. Further, the torque and angular velocity symbols are dropped and replaced with the generalized symbols for effort and flow. While not necessary in the example, it is common to number the bonds, to keep track of in equations. The simplified diagram can be seen below.
Given that effort is always above the flow on the bond, it is also possible to drop the effort and flow symbols altogether, without losing any relevant information. However, the bond number should not be dropped. The example can be seen below.
The bond number will be important later when converting from the bond graph to statespace equations.
Single port elements are elements in a bond graph that can have only one port.
Sources are elements that represent the input for a system. They will either input effort or flow into a system. They are denoted by a capital "S" with either a lower case "e" or "f" for effort or flow respectively. Sources will always have the arrow pointing away from the element. An examples of sources include: motors (source of effort, torque), voltage sources (source of effort), and current sources (source of flow).
where J indicates a junction.
Sinks are elements that represent the output for a system. They are represented the same way as sources, but have the arrow pointing into the element instead of away from it.
Inertia elements are denoted by a capital "I", and always have power flowing into them. Inertia elements are elements that store energy. Most commonly these are a mass for mechanical systems, and inductors for electrical systems.
Resistance elements are denoted by a capital "R", and always have power flowing into them. Resistance elements are elements that dissipate energy. Most commonly these are a damper, for mechanical systems, and resistors for electrical systems.
Compliance elements are denoted by a capital "C", and always have power flowing into them. Compliance elements are elements that store potential energy. Most commonly these are springs for mechanical systems, and capacitors for electrical systems.
These elements have two ports. They are used to change the power between or within a system. When converting from one to the other, no power is lost during the transfer. The elements have a constant that will be given with it. The constant is called a transformer constant or gyrator constant depending on which element is being used. These constants will commonly be displayed as a ratio below the element.
A transformer applies a relationship between flow in flow out, and effort in effort out. Examples include an ideal electrical transformer or a lever.
Denoted
where the r denotes the modulus of the transformer. This means
and
A gyrator applies a relationship between flow in effort out, and effort in flow out. An example of a gyrator is a DC motor, which converts voltage (electrical effort) into Angular velocity (angular mechanical flow).
meaning that
and
Junctions, unlike the other elements can have any number of ports either in or out. Junctions split power across their ports. There are two distinct junctions, the 0junction and the 1junction which differ only in the how effort and flow are carried across. The same junction in series can be combined, but different junctions in series cannot.
0junctions behave such that all efforts values are equal across the bonds, but the sum of the flow values in equals the sum of the flow values out, or equivalently, all flows sum to zero. In an electrical circuit, the 0junction is a node and represents a voltage shared by all components at that node. In a mechanical circuit, the 0junction is a joint among components, and represents a force shared by all components connected to it.
An example is shown below.
Resulting equations:
1junctions behave opposite of 0junctions. 1junctions behave such that all flow values are equal across the bonds, but the sum of the effort values in equals the sum the effort values out, or equivalently, all efforts sum to zero. In an electrical circuit, the 1 junction represents a series connection among components. In a mechanical circuit, the 1junction represents a velocity shared by all components connected to it.
An example is shown below.
Resulting equations:
Bond graphs have a notion of causality, indicating which side of a bond determines the instantaneous effort and which determines the instantaneous flow. In formulating the dynamic equations that describe the system, causality defines, for each modeling element, which variable is dependent and which is independent. By propagating the causation graphically from one modeling element to the other, analysis of largescale models becomes easier. Completing causal assignment in a bond graph model will allow the detection of modeling situation where an algebraic loop exists; that is the situation when a variable is defined recursively as a function of itself.
As an example of causality, consider a capacitor in series with a battery. It is not physically possible to charge a capacitor instantly, so anything connected in parallel with a capacitor will necessarily have the same voltage (effort variable) as that across the capacitor. Similarly, an inductor cannot change flux instantly and so any component in series with an inductor will necessarily have the same flow as the inductor. Because capacitors and inductors are passive devices, they cannot maintain their respective voltage and flow indefinitely—the components to which they are attached will affect their respective voltage and flow, but only indirectly by affecting their current and voltage respectively.
Note: Causality is a symmetric relationship. When one side "causes" effort, the other side "causes" flow.
In bond graph notation, a causal stroke may be added to one end of the power bond to indicate that the opposite end is defining the effort. Consider a constanttorque motor driving a wheel, i.e. a source of effort (SE). That would be drawn as follows:
Symmetrically, the side with the causal stroke (in this case the wheel) defines the flow for the bond.
Causality results in compatibility constraints. Clearly only one end of a power bond can define the effort and so only one end of a bond can have a causal stroke. In addition, the two passive components with timedependent behavior, I and C, can only have one sort of causation: an I component determines flow; a C component defines effort. So from a junction, J, the preferred causal orientation is as follows:
The reason that this is the preferred method for these elements can be further analyzed if you consider the equations they would give shown by the tetrahedron of state.
The resulting equations involve the integral of the independent power variable. This is preferred over the result of having the causality the other way, which results in derivative. The equations can be seen below.
It is possible for a bond graph to have a casual bar on one of these elements in the nonpreferred manner. In such a case a "casual conflict" is said to have occurred at that bond. The results of a causal conflict are only seen when writing the statespace equations for the graph. It is explained in more details in that section.
A resistor has no timedependent behavior: apply a voltage and get a flow instantly, or apply a flow and get a voltage instantly, thus a resistor can be at either end of a causal bond:
Sources of flow (SF) define flow, sources of effort (SE) define effort. Transformers are passive, neither dissipating nor storing energy, so causality passes through them:
A gyrator transforms flow to effort and effort to flow, so if flow is caused on one side, effort is caused on the other side and vice versa:
In a 0junction, efforts are equal; in a 1junction, flows are equal. Thus, with causal bonds, only one bond can cause the effort in a 0junction and only one can cause the flow in a 1junction. Thus, if the causality of one bond of a junction is known, the causality of the others is also known. That one bond is called the 'strong bond'
In order to determine the causality of a bond graph certain steps must be followed. Those steps are:
A walkthrough of the steps is shown below.
The first step is to draw causality for the sources, over which there is only one. This results in the graph below.
The next step is to draw the preferred causality for the C bonds.
Next apply the causality for the 0 and 1 junctions, transformers, and gyrators.
However, there is an issue with 0junction on the left. The 0junction has two causal bars at the junction, but the 0junction wants one and only one at the junction. This was caused by having be in the preferred causality. The only way to fix this is to flip that causal bar. This results in a causal conflict, the corrected version of the graph is below, with the representing the causal conflict.
One of the main advantages of using bond graphs is that once you have a bond graph it doesn't matter the original energy domain. Below are some of the steps to apply when converting from the energy domain to a bond graph.
The steps for solving an Electromagnetic problem as a bond graph are as follows:
These steps are shown more clearly in the examples below.
The steps for solving a Linear Mechanical problem as a bond graph are as follows:
These steps are shown more clearly in the examples below.
The simplifying step is the same regardless if the system was electromagnetic or linear mechanical. The steps are:
These steps are shown more clearly in the examples below.
Parallel power is when power runs in parallel in a bond graph. An example of parallel power is shown below.
Parallel power can be simplified, by recalling the relationship between effort and flow for 0 and 1junctions. To solve parallel power you will first want to write down all of the equations for the junctions. For the example provided, the equations can be seen below. (Please make note of the number bond the effort/flow variable represents).
By manipulating these equations you can arrange them such that you can find an equivalent set of 0 and 1junctions to describe the parallel power.
For example, because and you can replace the variables in the equation resulting in and since
, we now know that . This relationship of two effort variables equaling can be explained by an 0junction. Manipulating other equations you can find that which describes the relationship of a 1junction. Once you have determined the relationships that you need you can redraw the parallel power section with the new junctions. The result for the example show is seen below.
A simple electrical circuit consisting of a voltage source, resistor, and capacitor in series.
The first step is to draw 0junctions at all of the nodes. The result is shown below.
The next step is to add all of the elements acting at their own 1junction. The result is below.
The next step is to pick a ground. The ground is simply an 0junction that is going to be assumed to have no voltage. For this case, the ground will be chosen to be the lower left 0junction, that is underlined above. The next step is to draw all of the arrows for the bond graph. The arrows on junctions should point towards ground (following a similar path to current). For resistance, inertance, and compliance elements, the arrows always point towards the elements. The result of drawing the arrows can be seen below, with the 0junction marked with a star as the ground.
Now that we have the Bond graph, we can start the process of simplifying it. The first step is to remove all the ground nodes. Both of the bottom 0junctions can be removed, because they are both grounded. The result is shown below.
Next, the junctions with less than three bonds can be removed. This is because flow and effort pass through these junctions without being modified, so they can be removed to allow us to draw less. The result can be seen below.
The final step is to apply causality to the bond graph. Applying causality was explained above. The final bond graph is shown below.
A more advanced electrical system with a current source, resistors, capacitors, and a transformer
Following the steps with this circuit will result in the bond graph below, before it is simplified. The nodes marked with the star denote the ground.
Simplifying the bond graph will result in the image below.
Lastly, applying causality will result in the bond graph below. The bond with star denotes a causal conflict.
A simple linear mechanical system, consisting of a mass on a spring that is attached to a wall. The mass has some force being applied to it. An image of the system is shown below.
For a mechanical system, the first step is to place a 1junction at each distinct velocity, in this case there are two distinct velocities, the mass and the wall. It is usually helpful to label the 1junctions for reference. The result is below.
The next step is to draw the R and C bonds at their own 0junctions between the 1junctions where they act. For this example there is only one of these bonds, the C bond for the spring. It acts between the 1junction representing the mass and the 1junction representing the wall. The result is below.
Next you want to add the sources and I bonds on the 1junction where they act. There is one source, the source of effort (force) and one I bond, the mass of the mass both of which act on the 1junction of the mass. The result is shown below.
Next you want to assign power flow. Like the electrical examples, power should flow towards ground, in this case the 1junction of the wall. Exceptions to this are R,C, or I bond, which always point towards the element. The resulting bond graph is below.
Now that the bond graph has been generated, it can be simplified. Because the wall is grounded (has zero velocity), you can remove that junction. As such the 0junction the C bond is on, can also be removed because it will then have less than three bonds. The simplified bond graph can be seen below.
The last step is to apply causality, the final bond graph can be seen below.
A more advanced linear mechanical system can be seen below.
Just like the above example, the first step is to make 1junctions at each of the distant velocities. In this example there are three distant velocity, Mass 1, Mass 2, and the wall. Then you connect all of the bonds and assign power flow. The bond can be seen below.
Next you start the process of simplifying the bond graph, by removing the 1junction of the wall, and removing junctions with less than three bonds. The bond graph can be seen below.
There is parallel power in the bond graph. Solving parallel power was explained above. The result of solving it can be seen below.
Lastly, apply causality, the final bond graph can be seen below.
Once a bond graph is complete, it can be utilized to generate the statespace representation equations of the system. Statespace representation is especially powerful as it allows complex multiorder differential system to be solved as a system of firstorder equations instead. The general form of the state equation can be seen below.
Where, is a column matrix of the state variables, or the unknowns of the system. is the time derivative of the state variables. is a column matrix of the inputs of the system. And and are matrices of constants based on the system. The state variables of a system are and values for each C and I bond without a causal conflict. Each I bond gets a while each C bond gets a .
For example, if you have the bond graph shown below.
Would have the following , , and matrices.
The matrices of and are solved by determining the relationship of the state variables and their respective elements, as was described in the tetrahedron of state. The first step to solve the state equations, is to list all of the governing equations for the bond graph. The table below, shows the relationship between bonds and their governing equations.
Bond Name  Bond with
Causality 
Governing Equation(s)  

"♦" denotes preferred causality  
One Port
Elements 
Source/ Sink, S  
Resistance, R:
Dissipated Energy 

Inertance, I:
Kinetic Energy 
♦  
Compliance, C:
Potential Energy 

♦  
Two Port
Elements 
Transformer, TR 
 
Gyrator, GY 
 
Multiport
Elements 
0 junction  One and only one
causal bar at the junction 

1 junction  one and only one causal
bar away from the junction 

For the example provided,
The governing equations are the following.
These equations can be manipulated to yield the state equations. For this example, you are trying to find equations that relate and in terms of , , and .
To start you should recall from the tetrahedron of state that starting with equation 2, you can rearrange it so that . can be substituted for equation 4, while in equation 4, can be replaced by due to equation 3, which can then be replaced by equation 5. can likewise be replaced using equation 7, in which can be replaced with which can then be replaced with equation 10. Following these substituted yields the first state equation which is shown below.
The second state equation can likewise be solved, by recalling that . The second state equation is shown below.
Both equations can further be rearranged into matrix form. The result of which is below.
At this point the equations can be treated as any other statespace representation problem.
A bibliography on bond graph modeling may be extracted from the following conferences :