Commit 79ca0557 authored by Viviane's avatar Viviane
Browse files

added basic equations of synchronous machine

parent 3426a3e4
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% Packages and other configurations
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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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