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An exact graphical method is presented for the study of free and forced limit cycles in on-off control systems. This approach overcomes certain difficulties which arise when the Tsypkin-Hamel theory is applied to systems whose relay characteristics include dead zone or dissymmetry. The limit-cycle operation is determined from the intersection of characteristic curves, a construction which makes it possible to visualize separately the influence of the following: 1) the relay characteristic; 2) the dynamics of the linear system part; and 3) the wave shape of an external periodic forcing signal. Among other applications, the present method is very convenient for the analysis and design of switching-type power circuitry, and one such circuit is analyzed as an example problem.