Capacitance Matrices
At the simplest level, capacitance is proportional to the amount of energy stored in the electric field in and around a structure. In a circuit, this is the energy stored in the electric field resulting from the voltage differential across a dielectric:
where:
- Ue is the energy stored in the electric field
- C is the capacitance
- v is the voltage across the dielectric
Q3D Extractor uses AC conduction to compute the capacitance between two lines by simulating the electric field that arises when a voltage differential is applied and then computing the energy stored in the field. The solver uses the following equation to solve for capacitance in terms of Ue:
To compute the elements in the capacitance matrix for an n conductor transmission line, the Q3D Extractor performs a sequence of field simulations. In each field simulation, 1 volt is applied to a single conductor and 0 volts are applied to all other conductors. The energy stored in the electric field is given by the following relationship:
where:
- Uij is the energy stored in the electric field linking conductor i with conductor j.
- Di is the electric flux density produced when 1 volt is applied to conductor i and 0 volts are applied to all other conductors.
- Ej is the electric field produced when 1 volt is applied to conductor j and 0 volts are applied to all other conductors.
The capacitance coupling i and j is therefore:
Since a standard SPICE component only has two terminals, the software derives a SPICE matrix format from the Maxwell matrix capacitance elements.