Relationships between Tsypkin, Hamel and approximate limit cycle analyses†

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Abstract
It is shown that the Hamel necessary conditions for the existence of simple limit cycles in relay control systems can be obtained from the corresponding Tsypkin conditions by a simple transformation using the Poisson sum formula of Fourier analysis. This relationship is an aid in understanding approximate analysis methods which are based on the exact conditions of Tsypkin and Hamel. A comparison of several different approximate analysis techniques, including the method of describing functions, is made using the notation and techniques introduced in Judd and Chirlian (1974). It is found that Tsypkin approximations, including the describing function, tend to be most accurate for high frequency designs, whereas Hamel-type approximations tend to be most accurate for low frequency designs. The complementary nature of these approximate analysis methods is a direct result of the Poisson sum formula relationship between the two exact methods.

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