kb/data/en.wikipedia.org/wiki/Bond_graph-4.md

---
title: "Bond graph"
chunk: 5/11
source: "https://en.wikipedia.org/wiki/Bond_graph"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T14:13:40.582713+00:00"
instance: "kb-cron"
---

The concept of causality is visualised via the causal stroke notation. 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, 
  
    
      
        R
        f
      
    
    {\displaystyle Rf}
  
is replaced by  
  
    
      
        
          Φ
          
            R
          
        
        (
        f
        )
      
    
    {\displaystyle \Phi _{R}(f)}
  
and  
  
    
      
        e
        
          /
        
        R
      
    
    {\displaystyle e/R}
  
 by  
  
    
      
        
          Φ
          
            R
          
          
            −
            1
          
        
        (
        e
        )
      
    
    {\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  
  
    
      
        e
        =
        R
        f
      
    
    {\displaystyle e=Rf}
  
and 
  
    
      
        f
        =
        e
        
          /
        
        R
      
    
    {\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. 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.

==== R component ====

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.

==== C component ====
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 
  
    
      
        e
        :=
        q
        
          /
        
        C
         
         
        
          and
        
         
         
        q
        :=
        
          ∫
          
            t
          
        
        f
        (
        τ
        )
        d
        τ
      
    
    {\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  
  
    
      
        q
        :=
        C
        e
         
         
        
          and
        
         
         
        f
        :=
        
          
            
              d
              q
            
            
              d
              t
            
          
        
      
    
    {\displaystyle q:=Ce~~{\text{and}}~~f:={\frac {dq}{dt}}}
  
 and the corresponding block diagram. This is called derivative causality.

==== I component ====
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  
  
    
      
        f
        :=
        p
        
          /
        
        I
         
         
        
          and
        
         
         
        p
        :=
        
          ∫
          
            t
          
        
        e
        (
        τ
        )
        d
        τ
      
    
    {\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 
  
    
      
        p
        :=
        I
        f
         
         
        
          and
        
         
         
        e
        :=
        
          
            
              d
              p
            
            
              d
              t
            
          
        
      
    
    {\displaystyle p:=If~~{\text{and}}~~e:={\frac {dp}{dt}}}
  
 and the corresponding block diagram. This is called derivative causality.

=== Source-sensor components ===

The SS (source sensor) component acts as an effort source (
  
    
      
        
          S
          
            e
          
        
      
    
    {\displaystyle S_{e}}
  
), flow detector (
  
    
      
        
          D
          
            f
          
        
      
    
    {\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 (
  
    
      
        
          S
          
            e
          
        
      
    
    {\displaystyle S_{e}}
  
) flow detector (
  
    
      
        
          D
          
            f
          
        
      
    
    {\displaystyle D_{f}}
  
) combination. (b) The SS acts as a flow source (
  
    
      
        
          S
          
            f
          
        
      
    
    {\displaystyle S_{f}}
  
), effort detector (
  
    
      
        
          D
          
            e
          
        
      
    
    {\displaystyle D_{e}}
  
) combination.

=== Junctions ===
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.

=== Causal propagation ===

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 (SCAP):