This is the user sandbox of Pathgrow. A user sandbox is a subpage of the user's user page. It serves as a testing spot and page development space for the user and is not an encyclopedia article. Create or edit your own sandbox here. Other sandboxes: Main sandbox | Template sandbox Finished writing a draft article? Are you ready to request review of it by an experienced editor for possible inclusion in Wikipedia? Submit your draft for review!

This is a backup of my edits to the Bond Graph Page - my use only!!

A bond graph is a graphical representation of the energy flows though and between physical dynamical systems including those in the electrical, mechanical, hydraulic, thermal and chemical domains.  Because the concept of energy is common to all physical domains, the bond graph provides a unified description of all of these energy domains and can be thought of as a systematic use of the physical analogies introduced by the 19th century scientists James Clerk-Maxwell and Lord Kelvin.  The mechanical–electrical analogy is one example of a physical analogy.

Bond graphs use the concept of analogous power conjugate variables whose product is energy flow, or power; these variable pairs are called effort and flow and, for example, correspond to voltage and current in the electrical domain and force and velocity in the mechanical domain. These power conjugate variables are transmitted by bonds which connect bond graph components.

Bond graph  components are also based on analogies and, using the electrical and mechanical domains as examples,  include the  C component to represent  both mechanical spring and electrical capacitor, the  I component to represent both a mechanical inertia and  an electrical inductor and the R component to represent both mechanical damper and electrical resistor.

The electrical circuit notions of parallel and series connections are abstracted as 0-junctions and 1-junctions in bond graph terminology and again used as connection analogues for each physical domain.

The bond graph transformer  (TF) and gyrator (GY) components represent energy transformation within and between domains; thus an ideal gearbox in the rotational mechanical domain is represented by the  TF component  and an ideal DC motor transforming electrical into mechanical energy is represented by a GY component. Non-ideal transducers with flexibility, inertia and friction are modelled by including C, I and R components.

The concept of causality in the context of bond graphs is used not only to generate system equations in a number of forms including ordinary differential equation (ode), state-space and differential-algebraic equations (dae)  form suitable for simulation purposes but also to investigate dynamical system properties such as invertibility and zero dynamics. Causality can also be used to guide and correct modelling choices.

The bond graph use of energy flows leads to the systematic construction of hierarchical models of large multi-domain systems; thus the bond graph method provides a basis for constructing large computer models, or digital twins, of multi domain physical systems including both engineering systems and those found in the life sciences.

The bond graph approach is related to the  behavioral modelling approach of Jan C Willems, and the port-Hamiltonian approach of Arjan van der Schaft and B. M. Maschke.

The bond graph method was originally proposed by Henry Paynter.

The importance of analogies between physical domains was noted by Lord Kelvin and James Clerk-Maxwell. The bond graph can be thought of as a systematic approach to analogies. This section emphasises key features of bond graph analogies; more detail appears in a number of textbooks and tutorial papers.

As an introduction to the key features of bond graphs, the figure shows the bond graph of two analogous systems: one electrical and one mechanical. A brief description is given here and expanded in the following sections.

• The bond graph C component represents either the electrical capacitor C or the mechanical spring K.
• The bond graph I component represents either the electrical inductor L or the mechanical mass M.
• The bond graph R component represents either the electrical resistor R or the mechanical damper D.
• The harpoon symbol is a bond transferring energy; the harpoon direction corresponds to positive energy flow and is a sign convention. The conjugate variables are displayed on each bond for the purposes of illustration.
• The 1-junction represents the series connection of the electrical circuit where the same current i flows in each component and the connection of the three components in the mechanical system which share a common velocity. Thus the bond graph components share the same flow f, but the efforts are different, corresponding to the voltages and forces of the electrical and mechanical components respectively. Energy is conserved at the junction by requiring that the three efforts add to zero.

The bond graph uses three classes of analogy: analogies between variables, analogies between components and analogies between component connections; these are discussed in the following sections.

The bond graph use energy flow, or power, as the basis for abstracting analogies between different physical domains^[1]. Power conjugate variables are a pair of variables whose product is power and a list of these appears in the table for various physical domains. As indicated in the table, the bond graph uses the effort/flow analogy to categorise each of the two conjugate variables; the across/through analogy is also possible but not commonly used.^[2][3] The conventional bond graph symbol for effort is e, and that for flow is f.^[1]

As well as the two conjugate power variables e and f, the bond graph uses two integrated variables p and q where:

${\displaystyle p=\int ^{t}e(\tau )d\tau ~~~{\text{and}}~~~q=\int ^{t}f(\tau )d\tau }$

equivalently:

${\displaystyle {\dot {p}}={\frac {dp}{dt}}=e~~~{\text{and}}~~~{\dot {q}}={\frac {dq}{dt}}=f}$

The harpoon symbol shown in the figure represents a power bond, or bond,  transferring energy; the harpoon direction corresponds to positive energy flow and corresponds to a sign convention. The conjugate variables are displayed on each bond for the purposes of illustration.

The definitions relating p to e and q to f are indicated diagrammatically. Physical properties are encapsulated in constitutive equations relating the energy and power variables.  In the diagram, C represents the constitutive equation relating q and e,  I represents the constitutive equation relating p and f and  R represents the constitutive equation relating e and f. Instead of placing e, f, p and q on the four corners of a square as in the diagram, they can alternatively be placed at the four vertices of a tetrahedron; such a diagram is called the tetrahedron of state.^[4]

These three constitutive equations ${\displaystyle \Phi _{C},\Phi _{I}~{\text{and}}~\Phi _{R}}$ correspond to three components each relating e and f: two dynamic components C and I incorporating an integrator and the R component.^[4] The C and I components store, but do not dissipate energy, R dissipates, but does not store energy. It is also possible to define a memristor component linking p and q, but this component is not commonly used.^[5]

${\displaystyle \Phi _{C},\Phi _{I}~{\text{and}}~\Phi _{R}}$ may be linear or nonlinear functions relating the variables as:

${\displaystyle q=\Phi _{C}(e),~~p=\Phi _{I}(f)~~{\text{and}}~~e=\Phi _{R}(f)}$

in the linear case:

${\displaystyle q=Ce,~~p=If,~~e=Rf}$

where ${\displaystyle C,~I,~{\text{and}}~R}$ are scalar constants representing generalised capacitance, inertia and resistance respectively.

Because e and f are conjugate variables, they are carried on a single bond and therefore the three components C, I and R are connected to a single bond and thus have a single energy port though which energy flows. The components, and impinging bond are shown in the figure. By convention, bonds point into these three components.

Electrical circuit diagrams have two sorts of connection: parallel and series or common voltage and common current; they distribute, but do not store or dissipate energy. The bond graph analogy of the common voltage  and common current connections are the 0-junction (common effort) and 1-junction (common flow) respectively; they both distribute, but do not store or dissipate energy.^[4]

All bonds impinging on a 0-junctions have the same effort. As energy is distributed, not dissipated, it follows that the sum of energy inflows (indicated by bonds pointing in) must equal the sum of energy outflows (indicated by bonds pointing out). Hence, if the common effort is e, the power flow constraint implies that the sum of the   ${\displaystyle m}$ inflows ${\displaystyle f_{i}^{in}}$ must equal the sum of the ${\displaystyle n}$ outflows ${\displaystyle f_{j}^{out}}$:

${\displaystyle \sum _{i=1}^{m}f_{i}^{in}=\sum _{j=1}^{n}f_{j}^{out}}$

Using the same argument, the efforts impinging on a 1-junction are constrained by:

${\displaystyle \sum _{i=1}^{m}e_{i}^{in}=\sum _{j=1}^{n}e_{j}^{out}}$

Systems such as that the simple electrical and mechanical systems in the figure have no connection to the environment. Such connections may include external voltages and external forces (that is efforts) and external currents and external velocities (that is flows). Similarly, external measurements of efforts and flows are also important. For the purposes of building hierarchical system, it is convenient to define energy ports though which energy can flow between systems. These five possibilities correspond to five bond graph components:^[6][4]

• ${\displaystyle {\text{S}}_{e}}$, an effort source analogous to applying external voltages and external forces
• ${\displaystyle {\text{S}}_{f}}$, a flow source analogous to applying external currents and external velocities
• ${\displaystyle {\text{D}}_{e}}$, an effort sensor (detector) analogous to measuring voltage and forces
• ${\displaystyle {\text{D}}_{f}}$, a flow sensor (detector) analogous to measuring currents and velocities
• ${\displaystyle {\text{SS}}}$, a source/sensor component which acts both as ${\displaystyle {\text{S}}_{e}}$-${\displaystyle {\text{D}}_{f}}$ and as ${\displaystyle {\text{S}}_{f}}$-${\displaystyle {\text{D}}_{e}}$ pairs as well as an energy port for external connections.^[6]

The conjugate effort and flow variables have different units in each energy domain thus models in one domain cannot be directly be connected by bonds to a different energy domain. However, because power has the same units (J/s or W) in each domain, the two-port power transducing components TF and GY can be used to provide such connections.^[4] As the two components transmit, but do not store or dissipate power, it follows that the power associated with the conjugate variables of the left-hand and right-hand bonds must be the same:

${\displaystyle e_{2}f_{2}=e_{1}f_{1}}$

Each component has a modulus ${\displaystyle m}$ associated with it. In the case of the TF component:

${\displaystyle e_{2}=me_{1}~~{\text{and}}~~f_{1}=mf_{2}}$

In the case of the GY component:

${\displaystyle e_{2}=mf_{1}~~{\text{and}}~~e_{1}=mf_{2}}$

For example, a frictionless, massless piston of area ${\displaystyle A}$ converts hydraulic power to mechanical power so that the hydraulic pressure ${\displaystyle P}$ is related to mechanical force ${\displaystyle F}$ by:

${\displaystyle F=AP}$

and hydraulic flow ${\displaystyle V}$ to piston velocity ${\displaystyle v}$ by:

${\displaystyle V=Av}$

Thus the bond graph analogy is the TF component with modulus ${\displaystyle m=A}$ where ${\displaystyle e_{1}=P,~~f_{1}=V}$ and ${\displaystyle e_{2}=F,~~f_{2}=v}$.

For example,  an ideal DC motor converts electrical power into rotational mechanical power so that the mechanical torque ${\displaystyle T}$is related to the electrical current ${\displaystyle i}$ by:

${\displaystyle T=ki}$

and back EMF (voltage) ${\displaystyle E}$ to angular velocity ${\displaystyle \Omega }$ by

${\displaystyle E=k\Omega }$

Thus the bond graph analogy is the GY component with modulus ${\displaystyle m=k}$ where ${\displaystyle e_{1}=E,~~f_{1}=i}$ and ${\displaystyle e_{2}=T,~~f_{2}=\Omega }$.

In both cases, non-ideal transduction behaviour can be modelled by including C ,I and R bond graph components in the model.

Bond graphs have been used to model systems relevant to the life sciences, including physiology and biology.^[7] In particular, the use of bond graphs to model biophysical systems was introduced by Aharon Katchalsky, George Oster, and Alan Perelson in the early 1970s.^[8][9] More recently, these ideas were used in the context of Systems Biology to provide an energy-based approach to modelling the biochemical reaction systems of cellular biology^[10][11][12] and to modelling the entire physiome.^[13]

The bond graph approach has a number of features which make it a good basis for building large computational models of the physiome.

• It is energy based, which implies that:
  • the models are physically-plausible^[14]
  • detailed balance (Wegscheider's conditions) for reaction kinetics are automatically satisfied^[15]
  • energy flow, usage and dissipation can be directly considered
• It is modular: bond graph components can themselves be bond graphs^[16][17]
• Energy transduction between physical domains is simply represented^[13] - see below
• Symbolic code, which may be used for simulation, can be automatically generated^[18]

The bond graph variables for biochemical systems are:

• Displacement: quantity of chemical species measured in moles, symbol ${\displaystyle x}$ (mol)
• Flow: rate of change of chemical species, symbol ${\displaystyle v}$ (mol/s)
• Effort: chemical potential, or Gibbs energy, per mole of a chemical species, symbol ${\displaystyle \mu }$ (J/mol)^[19]

Note that the product of effort and flow (μv) is, as always in the bond graph formulation, power (J/s).

As detailed below, the main features of the components used to model biochemical systems are: the R and C components are nonlinear, there is no I component required and the R component is replaced by a two-port Re component.^[12]

The bond graph zero (0) and one (1) components are no different in this context.

The C component integrates the flow ${\displaystyle v}$ to give the amount ${\displaystyle x}$ of species:

${\displaystyle x(t)=\int ^{t}v(\tau )d\tau }$

The effort, chemical potential ${\displaystyle \mu }$, is given by the formula:^[12][19]

${\displaystyle \mu =\mu ^{0}+RT\ln {\frac {x}{x^{0}}}}$

where ${\displaystyle \mu ^{0}}$ is the chemical potential corresponding to ${\displaystyle x=x^{0}}$, ${\displaystyle R}$ is the gas constant and ${\displaystyle T}$ is the absolute temperature in degrees Kelvin.

The formula for ${\displaystyle \mu }$ can be rewritten in a simplified form as:

${\displaystyle \mu =RT\ln Kx}$ where ${\displaystyle K={\frac {1}{x^{0}}}\exp {\frac {\mu ^{0}}{RT}}}$

Because of the special form of this particular C component it is sometimes given a special name Ce analogously to the special Re component.

The Re (reaction) component has two energy ports corresponding to the left (forward) and right (reverse) sides of a chemical reaction. The forward ${\displaystyle A^{f}}$and reverse ${\displaystyle A^{r}}$affinities are defined as the net chemical potential due to the species on the left and right sides of the reaction respectively. The Re component then gives the reaction flow ${\displaystyle v}$ as:^[12]

${\displaystyle v=\kappa \left(\exp {\frac {A^{f}}{RT}}-\exp {\frac {A^{r}}{RT}}\right)}$

where ${\displaystyle \kappa }$ (mol/s) is a rate constant.

Note that it is not possible to use the usual R component with 1-junction formulation as the flow depends on both the forward ${\displaystyle A^{f}}$and reverse ${\displaystyle A^{r}}$affinities rather than the difference ${\displaystyle A^{f}-A^{r}}$.^[10]

Sources:^[10][12]

The simple reaction ${\displaystyle {\ce {A <=> B}}}$ is represented by three components:

• C:A represents the species A with chemical potential ${\displaystyle \mu _{A}}$; the flow is ${\displaystyle -v}$.
• C:B represents the species B with chemical potential ${\displaystyle \mu _{B}}$; the flow is ${\displaystyle v}$.
• Re_r1 represents the reaction with flow ${\displaystyle v}$, forward affinity ${\displaystyle A^{f}=\mu _{A}}$and reverse affinity ${\displaystyle A^{r}=\mu _{B}}$.
• The bonds and junctions transfer chemical energy with effort and flow variables indicated.

Using the above equations, the flow ${\displaystyle v}$ is given by

${\displaystyle v=\kappa \left(\exp {\frac {\mu _{A}}{RT}}-\exp {\frac {\mu _{B}}{RT}}\right)=\kappa \left(K_{A}x_{A}-K_{B}x_{B}\right)}$

where the subscripts correspond to the species. This is the simple mass-action equation:

${\displaystyle v=k^{+}x_{A}-k^{-}x_{B}}$

where ${\displaystyle k^{+}=\kappa K_{A}{\text{ and }}k^{-}=\kappa K_{B}}$.

As discussed in section 1.4 of Keener & Sneyd,^[19] an enzyme-catalysed reaction reversibly transforming species ${\displaystyle {\ce {A}}}$ to species ${\displaystyle {\ce {B}}}$ via enzyme ${\displaystyle {\ce {E}}}$ and enzyme complex ${\displaystyle {\ce {C}}}$ can be written as the pair of reactions:

${\displaystyle {\ce {A + E <=> C <=> B + E}}}$

The enzyme complex ${\displaystyle {\ce {C}}}$ is formed from ${\displaystyle {\ce {A + E}}}$ and decomposes into the species ${\displaystyle {\ce {B}}}$ and releases enzyme ${\displaystyle {\ce {E}}}$. The bond graph shown in the figure shows how the enzyme is recycled.

The bond graph can be used to derive the properties of these reactions which are of generalised Michaelis-Menten form.^[10]

The bond graph TF (transformer) component represents energy transduction either within or between energy domains.^[20] (Note that the TF component has been called the TD (transduction) component^[8][9] - TF is more widely used.^[20]) This section focuses on transduction between the chemical domain with effort ${\displaystyle \mu }$ (J/mol) and flow ${\displaystyle v}$ (mol/s) and a generic domain with effort ${\displaystyle e}$ and flow ${\displaystyle f}$.

The key feature of the TF component is that it transmits energy without dissipation;^[20] hence, with reference to the figure:

${\displaystyle ef=\mu v}$

The transformer has a modulus m (with appropriate units) so that:

${\displaystyle f=mv}$

the energy formula then implies that:

${\displaystyle \mu =me}$

The stoichiometry of a chemical reaction determines how many of each chemical species occurs. Thus, for example, the reaction ${\displaystyle {\ce {A <=> m B}}}$ converts one mol of species ${\displaystyle {\ce {A}}}$ to m mol of species ${\displaystyle {\ce {B}}}$.

The case where ${\displaystyle m=1}$ corresponds to the simple reaction of the first example above. Using the same approach for general ${\displaystyle m}$, the reaction flow is:^[10][12]

${\displaystyle v=\kappa \left(\exp {\frac {\mu _{A}}{RT}}-\exp {\frac {m\mu _{B}}{RT}}\right)=\kappa \left(K_{A}x_{A}-(K_{B}x_{B})^{m}\right)}$

This section looks at the case where the generic domain is the electrical domain so that effort is (electrical) voltage ${\displaystyle {\mathcal {V}}}$ ( ${\displaystyle e={\mathcal {V}}}$) and the flow is current (${\displaystyle f=i}$). Consider the flow ${\displaystyle v}$ of charged ions where the charge on the molecule is ${\displaystyle z\epsilon }$ (Coulomb) where ${\displaystyle \epsilon }$ is the charge on the electron measured in Coulomb; the charge associated with a mole of ions is thus ${\displaystyle z\epsilon N_{A}}$where ${\displaystyle N_{A}}$is the Avogadro constant. The equivalent current is then

${\displaystyle i=z\epsilon N_{A}v=z{\mathcal {F}}v}$ where ${\displaystyle {\mathcal {F}}=\epsilon N_{A}}$is the Faraday constant; thus the corresponding TF modulus is:

${\displaystyle m=z{\mathcal {F}}}$ (C/mol)

Again, it follows that

${\displaystyle \mu =me=z{\mathcal {F}}{\mathcal {V}}}$

In this context, the bond graph TF component can be used to model energy flows associated with action potential,^[21] membrane transporters,^[22] cardiac action potential,^[23] and the mitochondrial electron transport chain.^[21]

Consider a long rigid molecule such as actin where a sub unit of length ${\displaystyle \delta }$ (m) is added at a rate of ${\displaystyle v}$ (mol/sec). Then the tip velocity ${\displaystyle V}$ is given by:^[24]

${\displaystyle V=\delta N_{A}v}$ where ${\displaystyle N_{A}}$is the Avogadro constant.

Thus the modulus ${\displaystyle m=\delta N_{A}}$ (m/mol) and

${\displaystyle \mu =mF=\delta N_{A}F}$ where ${\displaystyle F}$ is the corresponding force at the tip.

These formulae have been used^[24] to generate force/velocity curves for actin filaments. The approach provides a useful alternative to the Brownian Ratchet approach^[25] as the bond graph TF component can be potentially used with modular bond graph models of cellular systems.^[11]

Causality is a word with many uses and connotations. In the context of bond graphs, however, it has a limited, precise but important meaning and allows the bond graph model of a system to be converted to various other forms including a (nonlinear) state-space representation.^[26] The causality concept can also be used to examine structural properties, including inversion, of the system represented by a bond graph as well as to expose modelling errors.^[27]

The concept of causality  is visualised via the causal stroke notation.^[26] This notation is introduced in the three figures where the R, C and I components are connected to a bond augmented by the causal stoke: a short line perpendicular to the bond and located at either (but not both) end of the bond. (For clarity, the figures correspond to linear components; in the nonlinear case, ${\displaystyle Rf}$is replaced by ${\displaystyle \Phi _{R}(f)}$and ${\displaystyle e/R}$ by ${\displaystyle \Phi _{R}^{-1}(e)}$ and similarly for the C and I components. Whereas an acausal (without strokes) bond graph represents a set of equations (where the left and right sides of an equation can be swapped without change of meaning), a causal (with strokes on each bond) represents a set of assignment statements whereby the value of the left-hand side of the assignment statement (represented here by :=) becomes the value of the expression on the right-hand side of the assignment statement. Thus, for example, the constitutive equation of a linear resistor can be written as ${\displaystyle e=Rf}$and ${\displaystyle f=e/R}$ without changing the meaning; but in contrast, the two assignment statements e:=Rf and f := e/R are different. In particular, in the first case, f must be known to compute e and in the second case, e must be known to compute f.

The assignment statement representation can be graphically visualised as a block diagram where each assignment statement is represented as a block with input representing the right-hand of the assignment statement and output representing the left-hand side of the assignment statement.^{[28] [26]} The block diagrams for each causality are shown in the figures for each component. Note that each causality of a component leads to a different block diagram.

The figure shows the causality of the R component with linear constitutive equation. (a) Flow is imposed on R and R imposes effort; this corresponds to the assignment statement e := Rf and the block diagram. (b) Effort is imposed on R and R imposes flow; this corresponds to the assignment statement f := e/R and the corresponding block diagram.

The figure shows the causality of the C component with linear constitutive equation. (a) Flow is imposed on C and C imposes effort; this corresponds to the assignment statements ${\displaystyle e:=q/C~~{\text{and}}~~q:=\int ^{t}f(\tau )d\tau }$ and the corresponding block diagram; this is called integral causality. (b) Effort is imposed on C and C imposes flow; this corresponds to the assignment statements ${\displaystyle q:=Ce~~{\text{and}}~~f:={\frac {dq}{dt}}}$ and the corresponding block diagram. This is called derivative causality.

The figure shows the causality of the I component with linear constitutive equation. (a) Effort is imposed on I and I imposes flow ; this corresponds to the assignment statements ${\displaystyle f:=p/I~~{\text{and}}~~p:=\int ^{t}e(\tau )d\tau }$ and the corresponding block diagram. This is called integral causality. (b) Flow is imposed on I and I imposes effort; this corresponds to the assignment statements ${\displaystyle p:=If~~{\text{and}}~~e:={\frac {dp}{dt}}}$ and the corresponding block diagram. This is called derivative causality.

The SS (source sensor) component acts  as an effort source (${\displaystyle S_{e}}$), flow detector (${\displaystyle D_{f}}$) combination when the causal stroke is distant from the SS component and vice versa. The figure shows the causality of the SS (source/sensor) component. (a) The SS acts as an effort source (${\displaystyle S_{e}}$) flow detector (${\displaystyle D_{f}}$) combination. (b) The SS acts as a flow source (${\displaystyle S_{f}}$), effort detector (${\displaystyle D_{e}}$) combination.

As, by definition, all efforts associated with bonds impinging on a 0-junction are the same, it follows that exactly one bond can impose effort causality. Similarly, all flows associated with bonds impinging on a 1-junction are the same, it follows that exactly one bond can impose flow causality. Thus if a bond imposes effort causality on a 0-junction, the junction imposes effort on the other bonds and if  a bond imposes flow causality on a 1-junction, the junction imposes flow on the other bonds.

When one-port components (sources, C, I and R) are connected by a junction structure consisting of 0-junctions, 1-junctions, TF and GY, the causality assigned to each one port component propagates though the junction structure because of the causal constraints imposed by the junction structure components. This propagation can be applied systematically using the sequential causal assignment procedure:^[26]

1. Choose any source (SS, Se or Sf)  and assign the required causality. Immediately extend the causal implications though the bond graph as far as possible using the constraints on the junction elements (0 and 1) and the TF and GY elements.
2. Repeat step 1 for all of the source components.
3. Choose any storage element  (C or I)  and assign the integral causality. Immediately extend the causal implications though the bond graph as far as possible using the constraints on the junction elements (0 and 1) and the TF and GY elements.
4. Repeat step 3 until all C and I elements have been assigned causality.

The figure shows the result of this procedure on the simple example. In particular, the I component imposes flow causality onto the 1-junction thus the one junction imposes flow causality on to the other two components, C and R. This means that both the C and R components imposes effort causality onto the one junction; thus the C component is in integral causality and the R component corresponds to the assignment ${\displaystyle e_{R}:=Rf}$.

Although this procedure can be accomplished manually for small systems,  a computer-based approach is more generally useful.

There are three possible results of this procedure.

1. It completes with all C and I components in integral causality and all bond causalities assigned; the resultant system is a causal bond graph and can be converted into a state-space system.
2. All bonds have causality assigned but one or more C or I component has derivative causality.
3. All C and I components have integral causality, but some bond do not have causality assigned.

A bond graph system representation contains the constitutive equations of each component, embedded in the structure of bond and junctions.

The question of how to manipulate a set of equations into a form suitable for analogue computation was posed, and partially answered by Lord Kelvin.^[29] This approach underlies the conversion of a bond graph model to a state-space form suitable for digital computation.^[30]

The bond graph uses the notion of (bond graph) causality to provide a systematic and constructive way to investigate whether a state-space representation exists, and, if so, what is that representation; this causality approach is well suited to computational implementation and has an intuitive representation on the bond graph itself using the causal stroke notation.

A causal bond graph can be put into state-space form if^[26]:

• every bond has a causal stroke and
• every component has allowed causality
• all C and I components are in integral causality.

Thus  the sequential causal assignment procedure (SCAP) provide a graphical approach to determining whether  a system represented by a bond graph  has a state-space representation.