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Using the capacitance matrix, charges and
potentials are related by
The submatrices
C_{a},
C_{b}, and
C_{c} have dimensions
,
,
and
respectively. The capacitance matrix is a symmetric matrix, which allows
efficient storage and handling. Consequently
C_{a} and
C_{c} are
symmetric.
The unknown quantities of charges on potentialnodes and floatingnodes and potentials of
chargenodes,
,
can be written in terms of the known
quantities which are the potentials on potentialnodes and floatingnodes and the charges on
chargenodes.
The
potentials of the floatingnodes must be derived from the equations of the macronodes,
which are the known sum of charge of each macronode, and the individual potential
differences given by the voltage sources comprising a macronode. An example of a
macronode is shown in Fig. 3.2.
Figure 3.2:
A macronode is formed by a set of voltage sources which are not connected
to ground.

The voltage sources give a set of N_{fm}1
equations, where N_{fm} is the number
of nodes in a macronode, which are for the example
of Fig. 3.2
Voltage sources must not be short circuited. That is,
no loop consisting
solely of voltage sources may exist in the circuit.
With the additional equation for the charge of the macronode
which is taken from (3.2) by adding all N_{fm} lines for the node
charges comprising one macronode, one has
N_{fm} equations for the potentials of the N_{fm} floatingnodes.
The electrostatic energy of a circuit can be expressed using the relation
which links known charges and potentials with the unknown quantities,
(3.2), as, (see also (2.5))
Next: VoltageControlled Voltage Sources
Up: 3.1 Free Energy of
Previous: 3.1.1 Notation
Christoph Wasshuber