By D. Atherton

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The latter is very straightforward when the relay has no dead zone as its output is a square wave of amplitude ! h , which has a fundamental component of amplitude 4h/r , either in phase with the input for D = 0 , or lagging when Δ is finite. 20 for no hysteresis, D = 0 ; for no dead zone, d = 0 ; and for an ideal relay, D = d = 0 . 1. Unfortunately this description is not correct for any input signal, x, and it has been used to obtain erroneous results for x a random signal. 11, one for single valued nonlinearities and the other for double valued nonlinearities.

5 enters the loop at the nonlinearity output. 5. This is due to the integrator in the loop, the input of which must have an average value of zero in the steady state.

Because the DF solution is approximate, the actual measured frequency of oscillation will differ from this value by an amount which will be smaller the closer the oscillation is to a sinusoid. 41%. A symmetrical square wave has a harmonic content which is 1/nth that of the fundamental for n odd. 21% of the fundamental. 9) gives the amplitude of the assumed sinusoidal limit cycle a as 5h/r . 20) gives 2 N (a) = 2 1/2 4h (a - D ) 2 a r - j 4h2 D from which a r 2 2 1/2 C (a) = - 1 = - r 6(a - D ) + jD @ .

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An Introduction to Nonlinearity in Control Systems by D. Atherton

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