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) and , 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)