Some approaches to transistor sizing have used convex posynomial formulations to guarantee a solution that is globally optimal [1,2]. The convex posynomials are fit to experimental data characterizing a gate. More recent circuit optimization methods have generally avoided posynomial models, due to perceived inaccuracies in the models. We wish to find an optimization formulation that includes sizing with threshold voltage and gate voltage assignment.
Previous heuristic approaches [3-5] to transistor threshold voltage assignment to minimize power have not considered simultaneous sizing of transistors and threshold changes. We extend previous work by including supply voltage and performing simultaneous optimization of all gate parameters.
We chose posynomials to formulate the problem of simultaneous transistor sizing with supply voltage and threshold voltage assignment to combinational circuits. Using convex posynomials guarantees a global minimum, and avoids heuristic approaches where circuit variables are optimized independently. The convex posynomials are used for static timing analysis and power analysis. Power analysis includes all components of circuit power: switching capacitance, short-circuit power through a gate when it switches, and leakage current for the dominant leakage state (primarily subthreshold leakage). Posynomials are fit to a continuous range of supply voltage, threshold voltage, drive strength ratio, and drive strength of a gate for a range of input slew and output load capacitance conditions. The fidelity of the posynomial models for delay and power analysis is verified versus SPICE circuit simulations.
Given the globally optimal solution from the continuous sizing, continuous supply voltage, and continuous threshold voltage formulation, we discretize supply voltage and threshold voltage. Upper and lower bounds on the minimum power for a circuit can be easily determined from the posynomial formulation. A limited branch and bound during the iterative discretization ensures a discretized solution near the global optimum of the continuous problem.