Linear response formula for open systems

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.

A second-order differential equation characterizes a source-free series RLC circuit, marking its distinct mathematical representation. The complete solution of this equation is a blend of two unique solutions, each linked to the circuit's roots expressed in terms of the damping factor and resonant frequency.

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:

When an RL (Resistor-Inductor) circuit is connected to a DC source, the complete response of the circuit can be divided into two parts: the transient response and the steady-state response.
The transient response of the circuit is its temporary reaction to the sudden application of the DC source. This response is characterized by a current that exponentially decays to zero as time approaches infinity. During this transitional period, the inductor behaves like a short circuit, causing the source...

In electrical engineering, the analysis of networks composed of passive linear components — resistors (R), capacitors (C), and inductors (L) — is fundamental. These components are organized into circuits where the relationship between input and output can be analyzed using transfer functions. The transfer function of an RLC circuit, which relates the voltage across a capacitor to the input voltage, can be derived using Kirchhoff's laws.
To derive the transfer function, consider an RLC circuit...

Consider the operation of an automobile ignition system, a crucial component responsible for generating a spark by producing high voltage from the battery. This system can be described as a simple series RLC circuit, allowing for an in-depth analysis of its complete response.
In this context, the input DC voltage serves as a forcing step function, resulting in a forced step response that mirrors the characteristics of the input. Applying Kirchhoff's voltage law to the circuit yields a...