The design of a damped free-yaw system for a novel downwind two-bladed wind turbine is presented. The major advantage of a damped free-yaw system is the reduced load transfer from the blades to the support structure. The amount of damping provided by the yaw system and the bandwidth of the individual pitch controller used for yaw control significantly impact the loads on the wind turbine. In order to find the optimal yaw damping and bandwidth of the individual pitch controller, a novel grey-box model matching methodology, exploiting the generalized Nyquist stability criterion, is presented to obtain a control-oriented model of the nonlinear yaw dynamics. By obtaining control-oriented models for different yaw damping values, an integrated design procedure of the yaw damper and yaw controller is carried out. The results of the design approach are verified through high-fidelity simulations of the two-bladed wind turbine and indicate that the design is a tradeoff between blade loads and tower loads.