## 3 The Synchronous Machine
### Simplified Machine Model  
This section presents a simplified dynamic model of the machine. We assume that  
* Round rotor: \(L_{g2}=0\), or equivalently \(L_{d}=L_{q}=L_{s}\). * Constant field current: \(i_{f}=I_{f}=\text{const}\). * Balanced voltages and currents: \(v_{0}=0\), \(i_{0}=0\). We will now show that:  
* The machine may be described as an EMF source behind a series impedance. * The EMF is proportional to the frequency \(\omega\). * The electric torque is proportional to the current \(i_{q}\). Using (85), (86) with \(L_{d}=L_{q}=L_{s}\), \(i_{f}=I_{f}\), \(v_{0}=0\), and \(i_{0}=0\) results in  
\[ \begin{split}\lambda_{d}&=-L_{s}i_{d}+L_{af}I_{f}, \\ \lambda_{q}&=-L_{s}i_{q},\\ \lambda_{f}&=-\frac{3}{2}L_{af}i_{d}+L_{ff}I_{f}, \end{split}  \tag{97}\]  
and  
\[ \begin{split} v_{d}&=-R_{a}i_{d}+\frac{\text{d}}{ \text{d}t}\lambda_{d}-\omega\lambda_{q},\\ v_{q}&=-R_{a}i_{q}+\frac{\text{d}}{\text{d}t}\lambda _{q}+\omega\lambda_{d},\\ v_{f}&=R_{f}I_{f}+\frac{\text{d}}{\text{d}t}\lambda _{f}.\end{split}  \tag{98}\]  
Substitution of (97) in (98) yields  
\[ \begin{split} v_{d}&=-R_{a}i_{d}-L_{s}\frac{\text{d }}{\text{d}t}i_{d}+\omega L_{s}i_{q},\\ v_{q}&=-R_{a}i_{q}-L_{s}\frac{\text{d}}{\text{d}t }i_{q}-\omega L_{s}i_{d}+\omega L_{af}I_{f},\\ v_{f}&=R_{f}I_{f}-\frac{3}{2}L_{af}\frac{\text{d }}{\text{d}t}i_{d}.\end{split}  \tag{99}\]  
Note that \(v_{f}\) does not affect the other quantities, and may be considered an output of the model. We now define the constant voltage  
\[V_{E}=\omega_{s}L_{af}I_{f}=\text{const}. \tag{100}\]  
Using this constant, the resulting equations are  
\[ \begin{split} v_{d}&=-R_{a}i_{d}-L_{s}\frac{\text{d }}{\text{d}t}i_{d}+\omega L_{s}i_{q},\\ v_{q}&=-R_{a}i_{q}-L_{s}\frac{\text{d}}{\text{d }t}i_{q}-\omega L_{s}i_{d}+\frac{\omega}{\omega_{s}}V_{E},\end{split}  \tag{101}\]  
which may be rearranged as  
\[ \begin{split} v_{d}&=\underbrace{\omega_{s}V_{E}}_{ \text{stator}}\\ \underbrace{v_{q}}_{\text{stators}}&=\underbrace{ \frac{\omega}{\omega_{s}}V_{E}}_{\text{induced EMF}}\\ \underbrace{\begin{subarray}{c}\text{$\omega$}\\ \text{voltage drop}\\ \text{on series }\text{is }\text{inductance}\end{subarray}}\\ \underbrace{-R_{a}i_{q}}_{\text{voltage drop}}\\ \underbrace{-\left(L_{s}\frac{\text{d}}{\text{d}t}i_{q}+\omega L _{s}i_{d}\right)}_{\text{voltage drop}}.\end{split}  \tag{102}\]Using these expressions, the machine can be modeled as a source of induced EMF behind a series impedance, as shown in Figure 13. The induced EMF is given by \(e_{d}=0\), \(e_{q}=\frac{\omega}{\omega_{s}}V_{E}\), \(e_{d}=0\), where \([e_{d},e_{q},e_{0}]^{\mathsf{T}}\) is the \(dq0\) transformation of \([e_{a},e_{b},e_{c}]^{\mathsf{T}}\). Notes:  
1. The induced EMF is proportional to the frequency, where at steady-state and at nominal frequency the induced EMF (peak value) is \(V_{E}=\omega_{s}L_{af}I_{f}\). 2. The induced EMF is equal to the machine's open-circuit voltage. This can be seen in Figure 13, assuming that the stator currents are zero. In addition, using (91), (97), and (100), the electric torque acting on the rotor is  
\[T_{e}=\frac{3}{2}\frac{\text{poles}}{2}(\lambda_{d}i_{q}-\lambda_{q}i_{d})= \frac{3}{2}\frac{\text{poles}}{2}L_{af}I_{f}i_{q}=\frac{3}{2}\frac{\text{poles}}{2 \omega_{s}}V_{E}i_{q}. \tag{103}\]  
We see that the electric torque is proportional to the stator current \(i_{q}\). Based on these results, a simplified model of the machine may be obtained by separating the dynamics of the induced EMF from the dynamics of the series impedance (\(R_{a}\) and \(L_{s}\)). This series impedance, although physically a part of the machine, can be modeled as if it is a part of the transmission network. The machine is then described by the simple dynamics of the induced EMF. Using (78) and (103),  
\[ \begin{split}\frac{\text{d}}{\text{d}t}\omega&=K \left(3P_{ref}-\frac{2\omega_{s}}{\text{poles}}T_{e}-\frac{1}{D}(\omega- \omega_{s})\right),\\ T_{e}&=\frac{3}{2}\frac{\text{poles}}{2\omega_{s} }V_{E}i_{q}.\end{split}