Linear Approximation in Frequency Domain

Linear systems exhibit superposition, summing responses to individual inputs, and homogeneity, scaling responses consistently to an input multiplied by a scalar.

A nonlinear system can be considered linear about an operating point for small changes.

The Taylor series expansion links a function's value to its derivatives at a specific point and deviations from that point. Neglecting higher-order terms for small deviations gives a linear relationship.

Consider an RL circuit with a non-linear resistor. The presence of the nonlinear component necessitates system linearization before deriving the transfer function.

Kirchhoff's voltage law is utilized to derive a nonlinear differential equation.

The steady-state current is found by setting the small-signal source to zero.

The expression is rewritten in terms of the current's equilibrium value. The characteristics of the non-linear resistor are used to derive the linearized differential equation.

The known values are substituted, and the Laplace transform is applied with zero initial conditions.

The expression for the voltage across the inductor around the equilibrium point is subjected to the Laplace transform and simplified to obtain the transfer function.