Capacitance computation is one of the primary goals of an electrostatic analysis. For the definition of ground (partial) and lumped capacitance matrices see Vago and Gyimesi([240]). The knowledge of capacitance is essential in the design of electrostatic devices, Micro Electro Mechanical Systems (MEMS), transmission lines, printed circuit boards (PCB), electromagnetic interference and compatibility (EMI/EMC) etc. The computed capacitance can be the input of a subsequent MEMS analysis using the electrostructural transducer element TRANS126; for the theory, see TRANS126 - Electromechanical Transducer.

The capacitance matrix of an electrostatic system can be computed (by the CMATRIX command macro). The capacitance calculation is based on the energy principle. For details, see Gyimesi and Ostergaard([250]) and for its successful application, see Hieke([251]). The energy principle constitutes the basis for inductance matrix computation, as shown in Inductance, Flux and Energy Computation.

The electrostatic energy of a linear three electrode (the third is ground) system is:

(5–179)

where:

W = electrostatic energy

V1 = potential of first electrode with respect to ground

V2 = potential of second electrode with respect to ground

By applying appropriate voltages on electrodes, the coefficients of the ground capacitance matrix can be calculated from the stored static energy.

The charges on the conductors are:

(5–180)

(5–181)

where:

The charge can be expressed by potential differences, too:

(5–182)

(5–183)

where:

The lumped capacitances can be obtained by lumped capacitors as shown in Figure 5.4: Lumped Capacitor Model of Two Conductors and Ground. Lumped capacitances are suitable for use in circuit simulators.

Figure 5.4: Lumped Capacitor Model of Two Conductors and Ground

In some cases, one of the electrodes may be located very far from the other electrodes. This can be modeled as an open electrode problem with one electrode at infinity. The open boundary region can be modeled by infinite elements or simply closing the FEM region far enough away by an artificial Dirichlet boundary condition. In this case, the ground key parameter (GRNDKEY on the CMATRIX command macro) should be activated. This key assumes that there is a ground electrode at infinity.

The previous case should be distinguished from an open boundary problem without an electrode at infinity. In this case the ground electrode is one of the modeled electrodes. The FEM model size can be minimized in this case, too, by infinite elements. When performing the capacitance calculation, however, the ground key (GRNDKEY on the CMATRIX command macro) should not be activated since there is no electrode at infinity.