21.5:
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.
21.5:
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.