# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 5 Case Study
### Synchronous Generator  
Synchronous generators represent one of the main sources of electrical energy in three-phase power systems. These synchronous generators are powered by hydroelectric or thermoelectric plants [42; 43]. The schematic diagram of the cross-section of a synchronous generator with a pair of poles is illustrated in Figure 16. The field winding uses direct current and produces a magnetic field that induces voltages in the three phases in the armature windings. The circuits used for the analysis of a synchronous generator are shown in Figure 17. The three-phase armature winding represents the stator circuits carrying current in all three phases, and the field winding describes the rotor circuit. Figure 18 shows an MBGI of a synchronous generator on the \(d\)-\(q\) axis. Figure 18 contains two sections:  
* Stator circuits formed by three phase windings on the \(d\)-\(q\) axis where \(R_{d}\) and \(L_{d}\) denote the resistance and self-inductance on the \(d\)-axis circuit; \(R_{q}\) and \(L_{q}\) denote the resistance and self-inductance on the \(q\)-axis circuit; \(M\) is the mutual inductance between the stator and rotor; \(V_{d}\) and \(V_{q}\) are the supply voltages on _theed-\(q\)_ axis; and the mechanical part whose elements are \(T_{L}\) the mechanical torque, \(J\) the moment of inertia, and \(D\) the mechanical friction. This section is modeled by a multibond graph where \(V_{d}\), \(V_{q}\), and \(T_{L}\) are effort sources that determine a multiport effort source \(\underline{MSE}:V_{dqTL}\); the resistances on the \(d\)-\(q\) axis and the friction give a multiport resistance \(\overline{\mathbb{R}}:\mathbb{R}_{dqD}\); the inductances \(L_{d}\), \(L_{q}\), and \(J\) represent a multiport field \(\mathbb{I}:M_{dqFL}\) with the mutual inductance \(M\) linking the inductance rotor circuit \(L_{f}\); the relationships between the electrical circuits and the mechanical part are represented by a multiport gyrator \(\mathbb{M}\mathbb{G}\mathbb{Y}:X(d_{o},q_{0},0)\). * The rotor circuit formed by the resistance and inductance on the field winding \(R:R_{f}\) and \(L_{f}\) and the supply voltage for this winding \(MSE:V_{f}\). Figure 16: Schematic diagram of a three-phase synchronous machine. Figure 17: Stator and rotor circuits of a synchronous generator. In order to demonstrate that the multibond graph of Figure 18 represents a synchronous generator, the mathematical model in state variables is obtained. The multibond graph has the following key vectors:  
\[\underline{x}(t) = \left[ \begin{array}{c}\underline{p_{4}(t)}\\ \underline{p_{5}(t)}\end{array} \right];\,\,\underline{\bullet}(t)=\left[ \begin{array} []{c}\underline{e_{4}(t)}\\ \underline{e_{5}(t)}\end{array} \right];\,\,\underline{z}(t)=\left[ \begin{array} []{c}\underline{f_{4}(t)}\\ \underline{f_{5}(t)}\end{array} \right]\] \[\underline{D_{in}}(t) = \left[ \begin{array}{c}\underline{f_{2}(t)}\\ \underline{f_{6}(t)}\end{array} \right];\,\underline{D_{out}}(t)=\left[ \begin{array} []{c}\underline{e_{2}(t)}\\ \underline{e_{6}(t)}\end{array} \right];\,\underline{u}(t)=\left[ \begin{array} []{c}\underline{e_{1}(t)}\\ \underline{e_{7}(t)}\end{array} \right]\] \[\underline{I_{GY}}(t) = \underline{f_{3}(t)};\,\underline{O_{GY}}(t)=\underline{e_{3}}(t)\]  
The constitutive relations are:  
\[\mathbf{F}^{-1} = \left[ \begin{array}{cc}\mathbf{L}_{dqJ}&\mathbf{M}_{sR}^{T}\\ \mathbf{M}_{sR}&L_{f}\end{array} \right] \tag{77}\] \[\mathbf{L} = diag\Big{\{}\mathbf{R}_{d\sharp D},R_{f}\Big{\}}\] (78) \[\mathbf{X}_{GY} = \mathbf{X}(-d_{0},-q_{0},0)=\left[ \begin{array}{ccc}0&0&P_{q0} \\ 0&0&-P_{d0}\\ -P_{q0}&P_{d0}&0\end{array} \right] \tag{79}\]  
where:  
\[\mathbf{R}_{d\sharp D} = diag\big{\{}R_{d},R_{q},D\big{\}}\] \[\mathbf{L}_{d\sharp J} = diag\big{\{}L_{d},L_{q},I\big{\}}\] \[\mathbf{M}_{sR} = \left[ \begin{array}{cc}M&0&0\end{array} \right]\]  
and the multiport junction structure is defined by:  
\[\left[ \begin{array}{c}\underline{e_{4}(t)}\\ \underline{e_{5}(t)}\\ \underline{f_{3}(t)}\\ \underline{f_{2}(t)}\\ \underline{f_{6}(t)}\end{array}