f^{S}_{} |
= | + |
q^{ . }n_{0}^{ . }V_{0} |
0 | ||||

f^{S}_{} |
= | + | + |
q^{ . }n_{1}^{ . }V_{1} |
= | 0 | ||

f^{S}_{n0} |
= | I_{01} |
0 | |||||

f^{S}_{n1} |
= | I_{10} |
+ | I_{12} |
= | 0 | ||

= | (,) | = | - | |

I_{ij} |
= |
I(n_{i}, n_{j},,) |
= | - I_{ji} |

At the boundary, the constitutive relations are

f_{} |
= | - | = | 0 | ||

f_{n0} |
= | n_{0} |
- | N_{0} |
= | 0 |

f_{IC} |
= | I_{C} |
+ |
f^{S}_{n0} |
= | 0 |

f_{QC} |
= | Q_{C} |
+ |
f^{S}_{} |
= | 0 |

The boundary constitutive relations will be used to determine the quantity
values at the boundary while the segment constitutive relations will be used to build up an
expression for the boundary charge *Q*_{C} and for the boundary current
*I*_{C}. This is achieved by the boundary models which set the appropriate
entries in the transformation matrix
_{B} which reads

t_{x, y} |
n_{0} |
n_{1} |
I_{C} |
Q_{C} |
||

1 | ||||||

n_{0} |
||||||

n_{1} |
1 | |||||

I_{C} |
1 | |||||

Q_{C} |
1 |

The solution vector
**x** contains the following quantities

x |
= |
, , ... n_{0}, n_{1}, ... , I_{C}, Q_{C}, ... |

For voltage controlled contacts with *V*_{0} applied to the
contact one gets

f_{} |
= | - | V_{0} |
= | 0 |

When applying the current *I*_{0} to the contact
*f*_{} changes to

f_{} |
= | I_{C} |
- | I_{0} |
= | 0 |

The system matrix for iteration step *k* is

j_{x, y} |
n_{0} |
I_{C} |
Q_{C} |
r | ||

-1 | 1 |
f_{}^{k} |
||||

n_{0} |
-1 |
f_{n0}^{k} |
||||

-1 |
f_{}^{k} |
|||||

I_{C} |
- | - | -1 |
f_{IC}^{k} |
||

Q_{C} |
- |
-q^{ . }V_{0} |
-1 |
f_{QC}^{k} |

As the constitutive relations for the quantities , *n*_{0}, and *I*_{C}
are eliminated first, one ends up with the following matrix

j_{x, y} |
r | |

- |
f_{}^{k} - f_{IC}^{k} + f_{}^{k . }
+ f_{n0}^{k . } |

1999-05-31