6.3.1 Capacitance Extraction

Figure 6.15: In circuit with $n$ conductors there are $\frac{1}{2}n(n-1)$ capacitors.

In a circuit with $n$ nets (made physically with conductive materials) there are always $\frac{1}{2}n(n-1)$ capacitors $C_{i,j}$ across them. These may be desired capacitors by the integrated circuit designer, or parasitic ones which have unwanted influence on the circuit performance. Whatever the case, our simulator returns a netlist with all capacitors in a SPICE like circuit description format:

• C$_{1\_0}$  1   0   value [unit]
• C$_{1\_2}$  1   2   value [unit]
• C$_{1\_3}$  1   3   value [unit]
• C$_{2\_0}$  2   0   value [unit]
• ...
  

The energy method is used to calculate the capacitance values, because high numerical accuracy is achieved. The energy $W$ in a system with $n$ conductors is

$W=\frac{1}{2}\sum\limits_{i=1}^{n}~\sum\limits_{j=i+1}^{n}C_{i,j}(\psi_i-\psi_j)^2$.

Therefore, we must apply $\frac{1}{2}n(n-1)$ different potential conditions and solve the resulting linear system to obtain all capacitor values. The electric potential $\varphi$ for $n-1$ partial capacitance pairs is obtained by solving Laplace's equation:

$$div(\underline{\varepsilon}~grad~\varphi)=0$$ (6.1)

where $\underline{\varepsilon}$ is the permittivity tensor, and with this potential contributions the unknown ones are calculated by superposition.