Maxwell Matrices

The matrices generated by the software from the field simulators are in Maxwell matrix format. Since a standard SPICE component only has two terminals, the software derives a SPICE matrix format from the Maxwell matrix capacitance and conductance elements. You can then use the elements of the SPICE matrix directly in a circuit design (as an exported subcircuit), or to create a SPICE deck. This is only necessary for the capacitance and conductance matrices, because both formats yield the same results for the inductance and resistance matrices. Thus, using the definitions:

and

The Maxwell capacitance matrix equation, calculated from the energy (Ue) of the field solution for two conductors, is:

The self-capacitances (the diagonal matrix elements) are greater than zero, and the mutual capacitances (the off-diagonal matrix elements) are less than or equal to zero. This is because physically, placing a positive voltage on a conductor while leaving the other conductors grounded causes positive charge to accumulate on the excited conductor, while a corresponding negative charge develops on surrounding conductors.

In a SPICE matrix, the elements are the values of the ideal two-terminal components that must be connected between nodes to create the equivalent of the Maxwell matrix. The SPICE circuit equivalent for previous two-conductor matrix appears as:

where:

Thus, the SPICE matrix equation has the values:

This matrix is symmetric. Its diagonal entries are greater than or equal to zero (they may be exactly zero if the conductors are shielded from each other), and its off-diagonal entries are also greater than or equal to zero. This is because the SPICE matrix represents actual circuit values, rather than one where each conductor is included separately.

Modeling conductance to generate the conductance matrix is similar, except that charge is replaced by current.