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.

(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) |

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).

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