Subsections

# Non-ideal transformer

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 .

## Mutual inductors with two or three of inductors

This model can be replaced by the equivalent circuit depicted in figure 9.4. The values are calculated as follows.

 (9.42) (9.43) (9.44) (9.45)

The Y-parameters of this component are:

 (9.46) (9.47) (9.48)

Furthermore, its S-parameters are:

 (9.49)

 (9.50)

 (9.51)

 (9.52)

 (9.53)

 (9.54)

Also including an ohmic resistance and for each coil, leads to the following Y-parameters:

 (9.55) (9.56) (9.57)

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:

 (9.58)

 (9.59)

 (9.60)

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).

 (9.61) (9.62) (9.63) (9.64) (9.65) (9.66) (9.67)

## Mutual inductors with any number of inductors

A more general approach for coupled inductors can be obtained by using the induction law:

 (9.68)

where and is the voltage across and the current through the inductor, respectively. is its inductance. The inductor is coupled with other inductances . The corresponding coupling factors are and are the currents through the inductors.

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:

 (9.69)

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.

 (9.70)

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.

 (9.71)

The resulting Y-parameter matrix can be converted into the appropriate S-parameters numerically by eqn. (15.7).

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