Transformer

The two winding ideal transformer, as shown in fig. 9.2, is determined by the following equation which introduces one more unknown in the MNA matrix.

Figure 9.2: ideal two winding transformer

$$T\cdot\left(V_{2} - V_{3}\right) = V_{1} -V_{4} \quad \rightarrow \quad V_{1} - T\cdot V_{2} + T\cdot V_{3} - V_{4} = 0$$ (9.24)

The new unknown variable $I_{t}$ must be considered by the four remaining simple equations.

$$I_{1} = -I_{t} \quad I_{2} = T\cdot I_{t} \quad I_{3} = -T\cdot I_{t} \quad I_{4} = I_{t}$$ (9.25)

And in matrix representation this is for DC and for AC simulation:

$$\begin{bmatrix}.&.&.&.& -1\\ .&.&.&.& T\\ .&.&.&.& -T\\ .&.&.&.& ... ..._{3}\\ I_{4}\\ 0 \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0 \end{bmatrix}$$ (9.26)

It is noticeable that the additional row (part of the C matrix) and the corresponding column (part of the B matrix) are transposed to each other. When considering the turns ratio $T$ being complex introducing an additional phase the transformer can be used as phase-shifting transformer. Both the vectors must be conjugated complex transposed in this case.

Using the port numbers depicted in fig. 9.2, the scattering parameters of an ideal transformer with voltage transformation ratio $T$ (number of turns) writes as follows.

$$S_{14} = S_{22} = S_{33} = S_{41} = \frac{1}{T^2+1}$$ (9.27)

$$S_{12} = -S_{13} = S_{21} = -S_{24} = -S_{31} = S_{34} = -S_{42} = S_{43} = T\cdot S_{22}$$ (9.28)

$$S_{11} = S_{23} = S_{32} = S_{44} = T\cdot S_{12}$$ (9.29)

An ideal transformer is noise free.