Extensive research has proposed the use of multiple on-die power supplies (VDD) for reducing power consumption in CMOS circuits. We present a detailed study and design techniques for power delivery systems in dual VDD CMOS circuits. We first show that the total current to be delivered by the voltage supplies is significantly reduced (by 27%-46%) in dual VDD circuits. This current reduction prompts various design strategies that can be employed to design the power delivery system. We describe issues that arise at the system, board and package levels and propose a high-level model for the same. We then provide a new placement driven approach for designing on-die dual VDD power grids. Compared to already existing methods, the dual VDD grids generated by our approach reduce the worst case and average voltage drop by up to 12.3% and 6.8% respectively with no area overhead and sometimes improving wire congestion. We also show that dual VDD circuits can afford lower on-die decoupling capacitance budgets.