The describing function technique

The describing function method is applicable to any non-linearity which has the characteristic that if the input is a sinusoidal signal then the output is a periodic function 38 401. Because of its simplicity and wide range of applicability, the describing function technique is one of the most versatile procedures for analysing non-linear effects. 

Many non-linearities are such that, for a sinusoidal input (e.g. = Afsinto/ = Ms n(2nft)), the output will exhibit the same period (where the period =1//= 2/r/o)) as the input signal. Moreover, these outputs satisfy the condition pertaining to odd periodic functions, viz. that the signal over the second half of the period is identical to that which would be obtained if the signal over the first half of the period were rotated n radians about the mid-point of the whole period. An example of such a function of period 2 is shown in Fig. 7.79. Clearly, if this function over the period 0 to it is rotated it radians about the point at = it, it will then coincide with itself over the period it to 2it. Odd functions of period 2/rcan be represented by a Fourier series of the form(l7)  

The describing function approach will now be applied to two common non-linear elements which fulfil these conditions 

The Describing Function of an On-Off Element The operating characteristics of an on-off element (Fig. 7.80b) can be written as  

The effect of applying a sinusoidal perturbation to this element is also shown in Fig. 7.80. Hence, from equation 7.193  

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Although there is no universal procedure for analysing non-linear systems, there are several methods which can be applied to particular types or classes of system. One of these (viz. the describing function technique) is discussed in this section. First, however, the linearisation procedure employed in Section 7.5.2 is expanded to include relationships containing more than one independent variable. 

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