## 3 The Synchronous Machine
### Electrical Equations  
In this section we present a \(dq0\) model of the machine. This model is more elaborated than the machine's quasi-static model, and may be used to describe fast transients, and high frequency effects. However, as all models, the model presented here is an approximation. Specifically, it is assumed that:  
* The machine is a magneto-quasi-static device. * Saturation of the magnetic materials and other sources of imbalance and harmonic distortion are ignored. * Self- and mutual inductances are composed of a constant term, in addition to a sinusoidal term varying with \(2\theta\). Although these assumptions may seem restrictive, they form the basis of the classic \(dq0\) model, and have been found to give excellent results in a wide variety of applications. The machine is described as a system of coupled inductors  
\[ \begin{bmatrix}\lambda_{a}\\ \lambda_{b}\\ \lambda_{c}\\ \lambda_{f}\end{bmatrix} = \begin{bmatrix}\ell_{aa}&\ell_{ab}&\ell_{ac}&\ell_{af} \\ \ell_{ba}&\ell_{bb}&\ell_{bc}&\ell_{bf}\\ \ell_{ca}&\ell_{cb}&\ell_{cc}&\ell_{cf}\\ \ell_{fa}&\ell_{fb}&\ell_{fc}&\ell_{ff}\end{bmatrix}\begin{bmatrix}-i_{a}\\ -i_{b}\\ -i_{c}\\ i_{f}\end{bmatrix} , \tag{79}\]  
where  
* \(\lambda_{a}\), \(\lambda_{b}\), \(\lambda_{c}\) are stator flux linkages. * \(i_{a}\), \(i_{b}\), \(i_{c}\) are the stator currents (generator output currents). The negative signs have been added since currents are positive when flowing out of the generator. * \(\lambda_{f}\) is field winding flux linkage. * \(i_{f}\) is field winding current. In addition, voltages are defined by  
\[v_{a}=-R_{a}i_{a}+\frac{\text{d}}{\text{d}t}\lambda_{a}, v_{b}=-R_{b}i_{b}+\frac{\text{d}}{\text{d}t}\lambda_{b}, \tag{80}\] \[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]  
where  
* \(v_{a}\), \(v_{b}\), \(v_{c}\) are the stator voltages (generator output voltages). * \(v_{f}\) is the field winding voltage. * \(R_{a}\) is the armature resistance. * \(R_{f}\) is the field winding resistance. The self- and mutual inductances in (79) depend on the rotor position. The stator self-inductances are given by  
\[\ell_{aa}=L_{aa}+L_{g2}\cos(2\theta), \tag{81}\] \[\ell_{bb}=L_{aa}+L_{g2}\cos(2\theta+120^{\circ}),\] \[\ell_{cc}=L_{aa}+L_{g2}\cos(2\theta-120^{\circ}),\]  
and the stator-to- stator mutual inductances are  
\[\ell_{ab}=\ell_{ba}=L_{ab}+L_{g2}\cos(2\theta-120^{\circ}), \tag{82}\] \[\ell_{bc}=\ell_{cb}=L_{ab}+L_{g2}\cos(2\theta),\] \[\ell_{ac}=\ell_{ca}=L_{ab}+L_{g2}\cos(2\theta+120^{\circ}),\]  
where \(L_{aa}\) is a positive constant, and \(L_{ab}\) is a negative constant. These inductances are composed of a constant term, in addition to a sinusoidal term varying with \(2\theta\). This additional term is required in case the rotor is not perfectly round (causing \"saliency effects\"). In addition, the stator-to-rotor mutual inductances are  
\[\ell_{af}=\ell_{fa}=L_{af}\cos(\theta), \tag{83}\] \[\ell_{bf}=\ell_{fb}=L_{af}\cos(\theta-120^{\circ}),\] \[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]0  
and the field winding self-inductance is  
\[v_{c}=-R_{c}i_{c}+\frac{\text{d}}{\text{d}t}\lambda_{c}, v_{f}=R_{f}i_{f}+\frac{\text{d}}{\text{d}t}\lambda_{f},\]1The model defined by the above equations does not have an equilibrium point, since the inductances depend on the rotor angle \(\theta\), which is time dependent. To this end, we convert the equations to the \(dq0\) domain.