In this work we propose an optimal back plane biasing (OBB) scheme to be used in a UTBB FD SOI technology that minimizes the energy per operation consumption of sub threshold digital CMOS circuits. By using this OBB scheme, simulations show that more than 30% energy savings can be obtained with low threshold voltage (LVT) devices in comparison with classic symmetric back plane biasing (SBB) schemes. Additionally, this OBB scheme allows to adjust the performance of the circuit with very small energy penalties. A very simple and intuitive model, for sub threshold digital CMOS circuits, was developed to justify the benefits obtained by OBB. The results predicted by the model are confirmed with extensive simulation results. We show that the OBB approach can be applied easily to a given circuit just based on the information provided by a logic simulation of the circuit (or even an analysis of its structure) and simple electrical simulations of the pMOS and nMOS transistors. Finally, we show that the variability in the energy consumption is improved by using OBB and suggests that new sizing methodologies must be studied to fully benefit from the wide back plane voltage range available in UTBB FD SOI technology for the design of robust energy efficient digital circuits.