Linear Approximation in Frequency Domain

Linear systems are defined by superposition and homogeneity. Superposition allows the response to multiple inputs to equal the sum of individual responses. Homogeneity ensures that scaling an input by a scalar scales the response by the same scalar. These properties enable predictable system behavior and simplify analysis.

A nonlinear system can be approximated as linear around an operating point for small deviations using Taylor series expansion. This method expresses a function in terms of its derivatives at a specific point. By neglecting higher-order terms for small deviations, a linear relationship is obtained, enabling standard linear analysis techniques.

Linearization is required because nonlinear components prevent direct application of standard control system analysis methods. A nonlinear RL circuit with a nonlinear resistor must be linearized to obtain a linear differential equation. Once linearized, the Laplace transform can be applied to derive the transfer function in the frequency domain.

Kirchhoff's voltage law is applied to the nonlinear RL circuit to derive the initial nonlinear differential equation. This equation describes the system behavior before linearization. The voltage law relates the applied voltage, inductance, resistance, and battery voltage, forming the foundation for subsequent linearization and transfer function derivation.

The steady-state operating point is found by setting the small-signal source to zero and solving for the equilibrium current. This equilibrium value represents the DC operating point around which the system is linearized. The nonlinear differential equation is then rewritten in terms of deviations from this equilibrium point for small-signal analysis.

After linearization, known values are substituted into the differential equation with zero initial conditions. The Laplace transform is then applied to convert the time-domain differential equation into an algebraic equation in the frequency domain. This transformation enables the derivation of the transfer function, which characterizes system behavior in the frequency domain.

Higher-order terms are neglected because they become negligibly small for small deviations around the operating point. Retaining only first-order terms from the Taylor series expansion produces a linear approximation valid within a small signal range. This simplification enables practical analysis while maintaining accuracy for small perturbations from equilibrium.