Symmetrical transformer

The ideal symmetrical transformer, as shown in fig. 9.3, is determined by the following equations which introduce two more unknowns in the MNA matrix.

Figure 9.3: ideal three winding transformer
\includegraphics[width=4cm]{acstrafo}

$\displaystyle T_{1}\cdot\left(V_{2} - V_{3}\right) = V_{1} - V_{6} \quad \rightarrow \quad V_{1} - T_{1}\cdot V_{2} + T_{1}\cdot V_{3} - V_{6} = 0$ (9.30)

$\displaystyle T_{2}\cdot\left(V_{2} - V_{3}\right) = V_{5} - V_{4} \quad \rightarrow \quad - T_{2}\cdot V_{2} + T_{2}\cdot V_{3} - V_{4} + V_{5} = 0$ (9.31)

The new unknown variables $ I_{T1}$ and $ I_{T2}$ must be considered by the six remaining simple equations.

$\displaystyle I_{2} = T_{1}\cdot I_{T1} + T_{2}\cdot I_{T2} \quad I_{3} = -T_{1}\cdot I_{T1} - T_{2}\cdot I_{T2}$ (9.32)

$\displaystyle I_{1} = -I_{T1} \quad I_{4} = I_{T2} \quad I_{5} = -I_{T2} \quad I_{6} = I_{T1}$ (9.33)

The matrix representation needs to be augmented by two more new rows and their corresponding columns. For DC and AC simulation it is:

$\displaystyle \begin{bmatrix}.&.&.&.&.&.& -1 & 0\\ .&.&.&.&.&.& T_{1} & T_{2}\\...
...\\ 0 \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0 \end{bmatrix}$ (9.34)

Using the port numbers depicted in fig. 9.3, the scattering parameters of an ideal, symmetrical transformer with voltage transformation ratio (number of turns) $ T_1$ and $ T_2$, respectively, writes as follows.

$\displaystyle denom = 1+T_1^2+T_2^2$ (9.35)


$\displaystyle S_{11} = S_{66} = \frac{T_1^2}{denom}$   $\displaystyle \qquad S_{16} = S_{61} = 1-S_{11}$ (9.36)
$\displaystyle S_{44} = S_{55} = \frac{T_2^2}{denom}$   $\displaystyle \qquad S_{45} = S_{54} = 1-S_{44}$ (9.37)
$\displaystyle S_{22} = S_{33} = \frac{1}{denom}$   $\displaystyle \qquad S_{23} = S_{32} = 1-S_{22}$ (9.38)

$\displaystyle S_{12} = S_{21} = -S_{13} = -S_{31} = -S_{26} = -S_{62} = S_{36} = S_{63} = \frac{T_1}{denom}$ (9.39)

$\displaystyle -S_{24} = -S_{42} = S_{25} = S_{52} = S_{34} = S_{43} = -S_{35} = -S_{53} = \frac{T_2}{denom}$ (9.40)

$\displaystyle -S_{14} = -S_{41} = S_{15} = S_{51} = S_{46} = S_{64} = -S_{56} = -S_{65} = \frac{T_1\cdot T_2}{denom}$ (9.41)

An ideal symmetrical transformer is noise free.


This document was generated by Stefan Jahn on 2007-12-30 using latex2html.