### added basic equations of synchronous machine

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 ... ... @@ -422,17 +422,32 @@ The flux linkage can also be expressed in terms of self and mutual inductance as where $L_{11}$ and $L_{22}$ are the self inductances of windings 1 and 2 respectively and defined as the flux linkage per unit current in the same winding. $L_{12}$ and $L_{21}$ are the mutual inductance between two windings, defined as the flux linkage with one winding per unit current in the other winding. \begin{equation} L_{11} = N_1(\Phi_{m1} + \Phi_{l1}) / i_1 L_{11} = \frac{N_1(\Phi_{m1} + \Phi_{l1})}{i_1} \end{equation} \begin{equation} L_{22} = N_2(\Phi_{m2} + \Phi_{l2}) / i_2 L_{22} = \frac{N_2(\Phi_{m2} + \Phi_{l2})}{i_2} \end{equation} \begin{equation} L_{12} = N_1 \Phi_{m2} / i_2 L_{12} = \frac{N_1 \Phi_{m2}}{i_2} \end{equation} \begin{equation} L_{21} = N_2 \Phi_{m1} / i_1 L_{21} = \frac{N_2 \Phi_{m1}}{i_1} \end{equation} The mutual fluxes $\Phi_{m1}$ and $\Phi_{m2}$ can be defined as in equations \ref{eq:MutualFlux1} and \ref{eq:MutualFlux2}, where $\Re_m$ is the reluctance of the path of the magnetizing fluxes. \begin{equation} \label{eq:MutualFlux1} \Phi_{m1} = \frac{N_1 i_1}{\Re_m} \end{equation} \begin{equation} \label{eq:MutualFlux2} \Phi_{m2} = \frac{N_2 i_2}{\Re_m} \end{equation} From these equations, it is possible to see that \begin{equation} L_{12}=L_{21}= \frac{N_1 N_2}{\Re_m} \end{equation} The terminal voltages $e_1$ and $e_2$ are defined as following: ... ... @@ -444,7 +459,103 @@ The terminal voltages $e_1$ and $e_2$ are defined as following: e_2 = \frac{d \Psi_2}{d t} + r_2 i_2 \end{equation} Equations \ref{eq:FluxLinkageInductances}, \ref{eq:VoltageEquation1} and \ref{eq:VoltageEquation2} give the performance of the circuit. Equations \ref{eq:FluxLinkageInductances}, \ref{eq:VoltageEquation1} and \ref{eq:VoltageEquation2} give the performance of the magnetically coupled circuit. \subsubsection{Synchronous Machine Basic equations} We can derive the equations of the synchronous machine similarly to the equations of the magnetic circuit. The flux linkage for stator phase windings a, b and c are given by \begin{equation} \begin{bmatrix} \Psi_a \\ \Psi_b \\ \Psi_c \end{bmatrix} = \begin{bmatrix} -l_{aa} & -l_{ab} & -l_{ac} & l_{afd} & l_{akd} & l_{akq} \\ -l_{ab} & -l_{bb} & -l_{bc} & l_{afd} & l_{akd} & l_{akq} \\ -l_{ac} & -l_{bc} & -l_{cc} & l_{afd} & l_{akd} & l_{akq} \end{bmatrix} \begin{bmatrix} i_a \\ i_b \\ i_c \\ i_{fd} \\ i_{kd} \\ i_{kq} \end{bmatrix} \end{equation} where $l_{aa}$, $l_{bb}$ and $l_{cc}$ are the self-inductances of the stator windings; $l_{ab}$, $l_{bc}$ and $l_{ac}$ are the mutual inductances between the stator windings; $l_{afd}$, $l_{akd}$ and $l_{akq}$ are the mutual inductances between stator and rotor windings; $i_a$, $i_b$ and $i_c$ are the instantaneous stator currents in phases a, b and c; $i_{fd}$, $i_{kd}$ and $i_{kd}$ are the field and amortisseur circuit currents. The equations of the stator phase to neutral voltages are \begin{equation} \begin{bmatrix} e_a \\ e_b \\ e_c \end{bmatrix} = \frac{d}{dt} \begin{bmatrix} \Psi_a \\ \Psi_b \\ \Psi_c \end{bmatrix} -R_a \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix} \end{equation} where $R_a$ is the armature resistance per phase. It is important to notice that the inductances are time-varying. They vary with the rotor position, due to the variation in the reluctance of the magnetic flux path (non-uniform air gap). The permeance of the magnetic flux path $P$ (inverse of the reluctance $\Re$) in function of the rotor position $\alpha$ can be writen as \begin{equation} P = P_0 + P_2 \cos 2 \alpha \end{equation} Inductance is directly proportional to permeance and the relative position between the phase a and the rotor is $\theta$. Therefore, we can write the self-inductance $l_{aa}$ as in equation \ref{eq:SelfInductanceA}. Knowing that the windings of phases b and c are identical to phase a and displaced from it by $120 \circ$ and $240 \circ$ respectively, we can also determine the self-inductances $l_{bb}$ and $l_{cc}$. A more detailed derivation of the self-inductances is in \cite{kundur1994power}. \begin{equation} \label{eq:SelfInductanceA} l_{aa} = L_{aa0} + L_{aa2} \cos 2 \theta \end{equation} \begin{equation} \label{eq:SelfInductanceA} l_{bb} = L_{aa0} + L_{aa2} \cos 2 (\theta - \frac{2 \pi}{3}) \end{equation} \begin{equation} \label{eq:SelfInductanceA} l_{cc} = L_{aa0} + L_{aa2} \cos 2 (\theta + \frac{2 \pi}{3}) \end{equation} The mutual inductance between two windings can be deduced evaluating the flux linking one phase due to the current in the other phase. The mutual inductance will also have a second harmonic variation due to the non-uniform air gap. \begin{equation} \label{eq:SelfInductanceA} l_{ab} = -L_{ab0} - L_{ab2} \cos (2 \theta + \frac{\pi}{3}) \end{equation} \begin{equation} \label{eq:SelfInductanceA} l_{bc} = -L_{ab0} - L_{ab2} \cos (2 \theta - \pi) \end{equation} \begin{equation} \label{eq:SelfInductanceA} l_{ac} = - L_{ab0} - L_{ab2} \cos (2 \theta - \frac{\pi}{3}) \end{equation} \subsubsection{Park's transformation} ... ...
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