6.3.1 Capacitance Extraction

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6.3.1 Capacitance Extraction

**Figure 6.15:** In circuit with conductors there are $\frac{1}{2}n(n-1)$ capacitors.
$\begin{figure} \vspace{0.2cm} \centerline{\epsfig{file=INTnetCap.eps,width=0.82\linewidth}} \vspace{0.2cm} \end{figure}$

In a circuit with 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 in a system with 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 partial capacitance pairs is obtained by solving Laplace's equation:

$\begin{displaymath} div(\underline{\varepsilon}~grad~\varphi)=0 \end{displaymath}$

(6.1)

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

Next: 6.3.2 Resistance and Thermal Up: 6.3 Capacitance, Resistance and Previous: 6.3 Capacitance, Resistance and

Rui Martins
1999-02-24