Simulation of power delivery networks with Joule heating effects for 3D integration

In this paper, we present the first work on simulation of power delivery network (PDN) with both Joule heating and convection effects. The finite volume formulations of DC voltage drop equation and thermal equation with convection boundary conditions are explained. In thermal simulation, both Joule heating effect from the PDN and convection effect are considered in solving the steady-state heat equation. In electrical DC voltage drop simulation, by updating the temperature-dependent electrical resistivity of the PDN, voltage distribution is obtained with temperature effects. By iterating between the electrical DC voltage drop and thermal simulations until results converge, the simulation not only enables accurate estimation of system level voltage drop with convection effects, but also provides accurate temperature distribution with convection and Joule heating effects. The simulation results show that even with natural convection effects, the temperature effect on system level IR drop is about 5-10%.