3.1.3 Work of Voltage Sources

  According to (2.9), the work done by the voltage sources is
 $$\begin{align} W =& \,\boldsymbol{\varphi_{p,f}}^T\boldsymbol{q_{p,f}} - \sum_{... ...a} -e(\varphi_{p,f_{\text{initial}}}-\varphi_{p,f_{\text{final}}}), \end{align}$$
where the sum over N_m macro-nodes considers the fact, that voltage sources in a macro-node can only shift charge relative from the node of lowest potential in the macro-node and not from ground level as grounded voltage sources do. The last term considers the work to replace a tunneled electron, since if it tunnels from or to a potential-node or floating-node its charge has to be replaced by a voltage source.