## 3 The Synchronous Machine
### Electrical Equations
Omitting the algebraic details, this yields  
\[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]2 \[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]3  
and  
\[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]4 \[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]5  
where  
* \(v_{d}\), \(v_{q}\), \(v_{0}\) are the \(dq0\) transformation of \(v_{a}\), \(v_{b}\), \(v_{c}\) (stator voltages). * \(\lambda_{d}\), \(\lambda_{q}\), \(\lambda_{0}\) are the \(dq0\) transformation of \(\lambda_{a}\), \(\lambda_{b}\), \(\lambda_{c}\) (stator flux linkages). * \(L_{d}\) is called the direct-axis synchronous inductance. * \(L_{q}\) is called the quadrature-axis synchronous inductance. * \(L_{0}\) is called the zero-sequence inductance. Notes:  
1. The reference angle for the \(dq0\) transformation is the rotor electrical angle \(\theta\). 2. The \(dq0\) variables do not depend directly on \(\theta\), and define a time-invariant model. The inductances in (86) are given by  
\[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]6 \[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]7 \[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]8  
Throughout this section we use the inductances \(L_{d}\), \(L_{q}\), \(L_{0}\) directly, and do not use the original machine inductances \(L_{aa}\), \(L_{ab}\), \(L_{g2}\). We also define the synchronous inductance as  
\[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]9  
For a perfectly round rotor with \(L_{g2}=0\), we have \(L_{s}=L_{d}=L_{q}\). To complete the basic model, the machine output power (three-phase power flowing out of the stator and into the grid) is  
\[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]0  
and the electric power decelerating the rotor is  
\[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]1  
Following (90), the electric torque is given by  
\[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]2  
A complete state-space model of the synchronous machine is obtained by merging the equations above. Using Equations (78), (85), (86) and (91), and again omitting the algebraic details, yields\[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]3 \[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]4 \[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]5 \[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]6 \[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]7 \[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]8  
where the inputs are \(v_{d}\), \(v_{q}\), \(v_{0}\), \(v_{f}\), \(P_{ref}\), and the constants are defined as \(K:=\left(\frac{\mathrm{poles}}{2}\right)^{2}\frac{1}{l\omega_{s}}\), \(\beta:=2L_{d}L_{ff}-3L_{af}^{2}\). Several optional outputs (in addition to the state variables) are  
\[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\]9 \[\ell_{bb}=L_{aa}+L_{g2}\cos(2\theta+120^{\circ}),\]0 \[\ell_{bb}=L_{aa}+L_{g2}\cos(2\theta+120^{\circ}),\]1 \[\ell_{bb}=L_{aa}+L_{g2}\cos(2\theta+120^{\circ}),\]2 \[\ell_{bb}=L_{aa}+L_{g2}\cos(2\theta+120^{\circ}),\]3  
Following are two examples demonstrating how to use the model above in simulation. Consider first a synchronous generator feeding a symmetrically configured resistive load \(R_{L}\). We wish to simulate the transient that follows a sudden 3-phase short circuit at the armature terminals. A signal flow diagram of the simulation is shown in \[\ell_{bb}=L_{aa}+L_{g2}\cos(2\theta+120^{\circ}),\]4 Using the state-space description in (92), the currents \(i_{d}\), \(i_{q}\), \(i_{0}\) are inputs to the machine's model, and the voltages \(v_{d}\), \(v_{q}\), \(v_{0}\) are its outputs.