Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.

In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear. This approximation is achieved through the Taylor series expansion, which 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.

Consider an RL circuit containing a nonlinear resistor. To analyze this system, linearization is necessary before deriving the transfer function.

The first step involves applying Kirchhoff's voltage law to the circuit, resulting in a nonlinear differential equation that describes the system. For instance, the voltage law equation might take the form:

Where V(t) is the applied voltage, L is the inductance, R is the resistance, and E represents the battery voltage.

To find the steady-state current, we set the small-signal source to zero and solve for the equilibrium current i₀. The nonlinear differential equation is then rewritten in terms of deviations from this equilibrium:

The characteristics of the nonlinear resistor are used to derive the linearized differential equation. For small deviations in current, the voltage equation can be written as:

Substituting this approximation into the voltage law equation, we obtain a linear differential equation. With known values substituted and assuming zero initial conditions, the Laplace transform is applied to convert the differential equation into an algebraic equation in the Laplace domain.