Electrical Engineering ⇒ Topic : Linear and Non-linear Systems

Linear and Non-linear Systems

If a system satisfies the property of homogeneity and the property of superposition, the system is said to be linear.

Homogeneity

Let the system produces output y(t) for an input x(t). If the input is scaled 'a' times, the output will also be scaled by 'a' times for all values of 'a' and x(t). Therefore, we can write mathematically as follows:

                        x(t) → y(t)

                      ax(t) → ay(t)

Superposition

If two inputs x₁(t) and x₂(t) are applied to a system simultaneously, the overall output will be the combined effects of x₁(t) and x₂(t).

                          x₁(t) + x₂(t) → y₁(t) + y₂(t)

Combining these properties, we can write the following for a linear system:

If x₁(t) → y₁(t) and x₂(t) → y₂(t), then

                                              a₁x₁(t) + a₂x₂(t) → a₁y₁(t) + a₂y₂(t)

If any system does not satisfy the above two conditions, the system is called non-linear. Usually,non-linearity arises in a physical system due to non-linear input and output relation of its components.