Many simulators support non-ideal transformers (e.g. mutual inductor in SPICE). An often used model consists of finite inductances and an imperfect coupling (straw inductance). This model has three parameters: Inductance of the primary coil , inductance of the secondary coil and the coupling factor .
This model can be replaced by the equivalent circuit depicted in figure 9.4. The values are calculated as follows.
The Y-parameters of this component are:
Furthermore, its S-parameters are:
Also including an ohmic resistance and for each coil, leads to the following Y-parameters:
Building the S-parameters leads to too large equations. Numerically converting the Y-parameters into S-parameters is therefore recommended.
The MNA matrix entries during DC analysis and the noise correlation matrices of this transformer are:
A transformer with three coupled inductors has three coupling factors , and . Its Y-parameters write as follows (port numbers are according to figure 9.3).
A more general approach for coupled inductors can be obtained by using the induction law:
Realizing this approach with the MNA matrix is straight forward: Every inductance needs an additional matrix row. The corresponding element in the matrix is . If two inductors are coupled the cross element in the matrix is . For two coupled inductors this yields:
Obviously, this approach has an advantage: It also works for zero inductances and for unity coupling factors and is extendible for any number of inductors. It has the disadvantage that it enlarges the MNA matrix.
The S-parameter matrix of this component is obtained by converting the Z-parameter matrix of the component. The Z-parameter matrix can be constructed using the following scheme: The self-inductances on the main diagonal and the mutual inductances in the off-diagonal entries.
This matrix representation does not contain the second terminals of the inductances. That's why the Z-parameter matrix must be converted into the Y-parameter matrix representation which is then extended to contain the additional terminals.