3.1.2 Electrostatic Energy

   Using the  capacitance matrix, charges and potentials are related by
 $$\begin{gather} \begin{bmatrix}\boldsymbol{q_{p,f}}\\ \boldsymbol{q_c} \end{bmatr... ...}\boldsymbol{\varphi_{p,f}}\\ \boldsymbol{\varphi_c} \end{bmatrix}. \end{gather}$$
The sub-matrices C_a, C_b, and C_c have dimensions $(N_p+N_f)\times (N_p+N_f)$, $N_c\times (N_p+N_f)$, and $N_c\times N_c$ 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 potential-nodes and floating-nodes and potentials of charge-nodes, $(\boldsymbol{q_{p,f}},\boldsymbol{\varphi_c})$, can be written in terms of the known quantities which are the potentials on potential-nodes and floating-nodes and the charges on charge-nodes.
 $$\begin{gather} \begin{bmatrix}\boldsymbol{q_{p,f}}\\ \boldsymbol{\varphi_c}\end{... ...bmatrix}\boldsymbol{\varphi_{p,f}}\\ \boldsymbol{q_c}\end{bmatrix}. \end{gather}$$
The potentials of the floating-nodes must be derived from the equations of the macro-nodes, which are the known sum of charge of each macro-node, and the individual potential differences given by the voltage sources comprising a macro-node. An example of a macro-node is shown in Fig. 3.2.

  

Figure 3.2: A macro-node 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 macro-node, which are for the example of Fig. 3.2
$$\begin{gather}\begin{bmatrix}V_{12}\\ V_{23}\\ V_{24}\end{bmatrix}= \begin{pmat... ...i_{f1}\\ \varphi_{f2}\\ \varphi_{f3}\\ \varphi_{f4}\end{bmatrix}. \end{gather}$$
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 macro-node
$$\begin{gather}q_{f\Sigma}=\sum_{i=1}^{N_{fm}} q_{fi} = q_{f1}+q_{f2}+q_{f3}+q_{f4}, \end{gather}$$
which is taken from (3.2) by adding all N_fm lines for the node charges comprising one macro-node, one has N_fm equations for the potentials of the N_fm floating-nodes.

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))
 $$\begin{align} E_C &= \frac{1}{2}\begin{bmatrix}\boldsymbol{q_{p,f}}^T,&\boldsymb... ...{p,f}}+\boldsymbol{q_c}^T\mathbf{C_c}^{-1}\boldsymbol{q_c} \right). \end{align}$$