Nonlinearities in a system

Homogeneity and superposition are the two properties of linear system. With superposition property we mean, output response of a system to the sum of inputs is the sum of the responses to the individual inputs. Therefore, if an input r1(t) gives an output for C1 (t) and output for C2(r) is provided by r2(t) then with r1(t)- r2(t) as input, we will get C1(t) + C2(t) as output. The superposition property can be expressed as the system response with respect to product of input and scalar. Saying it in specific terms, for a linear system the homogeneity property can be concluded as if for an input of r1(i) that yields an output of c(t), an input of Ar1(t) yields an output of Ac1(t); that is, multiplication of an input by a scalar yields a response that is multiplied by the same scalar.

Linearity can be visualized with the help of Fig. 1. Figure 1(a) represents a linear system in which input is twice of the output or you can say that f(x) = 0.5x; without considering value of X.  For instance, output will be ½ and 1 for input of 1 and 2 respectively. By using the superposition property, sum of the original inputs give inputs or 3 and it will give output as sum of original outputs means 1.5. For testing the property of homogeneity, we will take input as 2 and it will provide 1 as output value. Similarly multiplying the input by 2, also doubles the output.

Figure 1(a) the input of 4 is not producing output of 2. In this way, the readers can check that linearity property is not verified here and is not applicable to the relationship depicted in the Fig. 1(b). Some useful examples of physical nonlinearities are shown in the Fig. 2.  An Electronic Amplifier satisfies the property of linearity over specific range but at high value of input it exhibits non linearity known as saturation at high value of input voltages. A motor offers no response for low input value for voltage because of frictional forces which exhibits dead zone that is non linearity. Nonlinearity known as backlash is exhibited by gears which do not fit tightly. The input shifts only over a small range of values.

FIGURE 1 a. Linear system; b. nonlinear system

FIGURE 2 some physical nonlinearity
Without responding of output, the readers are required to verify that the curves represented in the Figure 2. does not justify the linearity definitions over complete range. Phase Detector is one more example of nonlinear subsystem. It is used in the FM radio receiver as part of phase locked loop. The output response for input is sine of the input. Linear approximations can be made by the designer for a nonlinear system. Design and analysis are simplified with linear approximations. It can be used till good approximation to reality is yielded.  For instance, if the range of input values about that point is small and translation of origin can be done to the point. Linear Amplification having little excursions about any point can be demonstrated by a physical device like electronic amplifiers.

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