The goal of this paper is to transform classical absolute stability criteria into nonlinear design procedures which employ efficient numerical tools, such as LMI’s. We first analyze an earlier circle criterion design, and show that its feasibility is limited by conditions on the unstable part of the zero dynamics and the relative degree. Then, we develop an extended circle criterion design which eliminates the relative degree obstacle. The restrictions on the zero dynamics are relaxed by using the Popov multiplier, which also reduces controller complexity. The results are illustrated on several physically motivated design examples.