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A root-locus method for the analysis of nonlinear servomechanisms is developed, using the complex convolution theorem of the Laplace transformation. The nonlinear servomechanisms that can be treated by this method are systems that contain nonlinear elements whose input-output relationships can be expressed in the form of power series. With some approximation this includes systems containing ideal relays and linear gains with dead zone. Transfer functions are developed for odd-power and odd-root nonlinear elements. These transfer functions are valid only for sinusoidal inputs and are functions of the frequency as well as the Laplace Transform variable s. The method is illustrated by a root-locus analysis on a system that includes an ideal relay. Comparison with existing methods using the Kochenburger describing function and with transient solutions shows that the new method is reasonably accurate. The usual root-locus synthesis techniques may be applied to improve nonlinear system performance.