# Modeling and Simulation in Multibond Graphs Applied to Three-Phase Electrical Systems
## 4 Modeling of Three-Phase Circuits in the Physical Domain  
Considering the basic three-phase circuit of Figure 1 is defined by (39) to (43). A lemma to obtain a multibond graph of a three-phase circuit given in the coordinates \((d,q,0)\) is proposed. Lemma 2: _Consider a three-phase electrical system that is modeled by a multibond graph with a predefined integral causality assignment in coordinates \((a,b,c)\) according to the multiport junction structure of Figure 6 and then, a multibond graph with a predefined integral causality assignment in coordinates \((d,q,0)\) can be built if the different elements are connected according to the construction rules of multibond graphs and the multiport inertia in coordinates \((a,b,c)\) is replaced by a multiport inertia in coordinates \((d,q,0)\) connected through a \(1-\)junction to a multiport gyrator modulated by_  
\[x(-wL)=\left[ \begin{array}{ccc}0&wL_{d}&0\\ -wL_{q}&0&0\\ 0&0&0\end{array} \right] \tag{85}\]  
_and the multiport capacitor in coordinates \((a,b,c)\) is replaced by a multiport capacitor in coordinates \((d,q,0)\) connected through a \(0-\)junction to a multiport gyrator modulated by_  
\[x(-wC)=\left[ \begin{array}{ccc}0&wC_{d}&0\\ -wC_{q}&0&0\\ 0&0&0\end{array} \right] \tag{86}\]  
Proof: The basic elements of a three-phase electrical system are described by resistances, inductances and capacitances supplied by electrical currents and voltages. It is known that \(v^{dq0}=Pv^{abc}\) and \(i^{dq0}=Pi^{abc}\) where \(P\) is [PERSON]'s transformation matrix. The conversion of resistances from \((a,b,c)\) to \((d,q,0)\) is defined by  
\[v_{R}^{abc}=R^{abc}i_{R}^{abc} \tag{87}\]  
where \(i_{R}^{abc}\) are the electric currents through the resistors \(R^{abc}\) and the voltage across the resistors are \(v_{R}^{abc}\), premultiplying (87) by \(P\), and then  
\[Pv_{R}^{abc}=PR^{abc}i_{R}^{abc}=PR^{abc}PP^{-1}i_{R}^{abc}\]  
in coordinates \((d,q,0)\)  
\[v_{R}^{dq0}=R^{dq0}i_{R}^{dq0} \tag{88}\]  
where  
\[R^{dq0}=PR^{abc}P^{-1} \tag{89}\]  
in multibond graphs, the multiport resistor \(R^{abc}\) is changed by \(R^{dq0}\) as well as modulated by \(P\) and \(P^{-1}\) according to (89). For inductances in coordinates \((a,b,c)\), the voltage is given by  
\[v_{L}^{abc}=L^{abc}\frac{d\dot{i}_{L}^{abc}}{dt} \tag{90}\]and premultiplying by \(P\)  
\[Pv_{L}^{abc}=PL^{abc}\frac{d_{L}^{abc}}{dt}=PL^{abc}P^{-1}P\frac{d_{L}^{abc}}{dt} \tag{91}\]  
the derivative with respect to time \(i_{L}^{dq0}=Pi_{L}^{abc}\) determines  
\[\frac{d_{L}^{dq0}}{dt}=\frac{dP}{dt}i_{L}^{abc}+P\frac{d_{L}^{abc}}{dt} \tag{92}\]  
substituting (92) into (91)  
\[v_{L}^{dq0}=L^{dq0}\frac{d_{L}^{dq0}}{dt}-L^{dq0}\frac{dP}{dt}P^{-1}l^{dq0} \tag{93}\]  
however,  
\[\frac{dP}{dt}P^{-1}=\left[ \begin{array}{ccc}0&-w&0\\ w&0&0\\ 0&0&0\end{array} \right] \tag{94}\]  
then the term \(-L^{dq0}\frac{dP}{dt}P^{-1}l^{dq0}\) can be represented as a multiport gyrator which is shown in Figure 7.  
The constitutive relationship is given by  
\[v_{Lp}^{dq0}=X\Big{(}wL_{dq0}\Big{)}i^{dq0} \tag{95}\]  
where \(X\Big{(}wL_{dq0}\Big{)}=L^{dq0}\frac{dP}{dt}P^{-1}\) and then (93) is modeled according to Figure 8.  
The current through capacitances is defined by  
\[i_{C}^{abc}=C^{abc}\frac{dV_{C}^{abc}}{dt} \tag{96}\]  
premultiplying by \(P\)  
\[P_{C}^{abc}=PC^{abc}\frac{dV_{C}^{abc}}{dt}=PC^{abc}P^{-1}P\frac{dV_{C}^{abc}}{dt} \tag{97}\]  
Figure 8: Inductance in the coordinates \((d,q,0)\) modeled in multibond graph. Figure 7: Component \(v_{Lp}\) is represented by a multiport gyrator.