# Modeling and Simulation in Multibond Graphs Applied to Three-Phase Electrical Systems
## 3 Modeling in Multibond Graphs  
In the connection of two elements, components or systems, there is always a power transfer \(P(t)\). This power in generalized terms described in bond graph modeling is defined as the product of effort \(e(t)\) and flow \(f(t)\). Likewise, the power link is represented by a power bond that is illustrated in Figure 2[32]. Bond graph modeling can be applied to systems of different energy domains, that is, to electrical, mechanical, hydraulic, and thermodynamic systems. The generalized variables for these different systems are indicated in Table 1.  
In bond graph modeling, two additional variables are required to characterize a system, in which variables are called energy variables which are momentum \(p(t)\) and displacement \(q(t)\). The time integral of an effort defines a momentum: \(p(t)=\int e(t)dt\) and the displacement is the time integral of a flow: \(q(t)=\int f(t)dt\). One of the fundamental properties in bond graph models is the causality that is applied to each of the elements that constitute part of a model. With the application of the causal stroke, each bond accurately indicates the input and output signals, as illustrated in Figure 3, which is a power bond with its causality [32]. \begin{table}
\begin{tabular}{c c c} \hline
**System** & **Effort [\(e(t)\)]** & **Flow [\(f(t)\)]** \\ \hline Electrical & Voltage [\(v(t)\)] & Current [\(i(t)\)] \\ \hline Mechanical & Force [\(F(t)\)] & Velocity [\(v(t)\)] \\  & Torque [\(\tau(t)\)] & Ang. velocity [\(\omega(t)\)] \\ \hline Hydraulic & Pressure [\(P(t)\)] & Volume flow rate [\(Q(t)\)] \\ \hline Thermodynamics & Temperature [\(T(t)\)] & Entropy flow [\(S(t)\)] \\ \hline \end{tabular}
\end{table} 
Table 1: Power variables. Figure 2: Power bond. According to the type of element, it has a given causality, as shown in Table 2.  
Many systems can generally be modeled as multiport power elements. For example, mechatronic systems that are constructed of mechanical, electrical and hydraulic subsystems but in turn have movements in three axes \((x,y,z)\); electrical power systems can be represented as systems with multiport phases \((a,b,c)\), likewise, hydraulic and thermal systems with a multiport can be modeled. Therefore, systems modeled with bond graphs can be represented with multibond graph models with the advantages of structural modeling, analysis and control that determine bond graphs which are generalized and extended with multibond graphs [18; 20]. Firstly, the notation of the variables generalized with multibond graphs are efforts \(\varepsilon(t)\) and flows \(f(t)\) with an underscore indicating that these variables determine vector bond notation, which is a composition of three bonds corresponding to the three perpendicular axes in a multibond graph. Figure 4 shows a multibond and its equivalent in bonds [18; 20]. Modeling with multibond graphs is common for the application to vectors with three elements for physical systems, but this does not limit the fact that it can be extended to higher-order vectors [18]. The power in a multibond is defined by  
\[\underline{P}(t)=\varepsilon^{T}(t)f(t) \tag{44}\]  
Figure 4: Bonds: (a) Single bond; and (b) Multibonds. Figure 3: Causal bond. where \(\underline{\epsilon}^{T}(t)\) is the transpose of the column vector  
\[\underline{\epsilon}(t)=\left[ \begin{array}{c}\epsilon^{x}(t)\\ \epsilon^{y}(t)\\ \epsilon^{z}(t)\end{array} \right]=\left[ \begin{array}{c}\epsilon^{x}(t)\\ \epsilon^{b}(t)\\ \epsilon^{c}(t)\end{array} \right] \tag{45}\]  
and  
\[\underline{f}(t)=\left[ \begin{array}{c}f^{x}(t)\\ f^{y}(t)\\ f^{z}(t)\end{array} \right]=\left[ \begin{array}{c}f^{a}(t)\\ f^{b}(t)\\ f^{c}(t)\end{array} \right] \tag{46}\]  
A general multibond graph indicating the generalized variables in each multibond and the junctions or multiport elements \((MP)\) is shown in Figure 5[18]. The essential elements in modeling with multibond graphs are described in Appendix A.