The transfer function is a fundamental concept in the analysis and design of linear time-invariant (LTI) systems. It offers a concise way to understand how a system responds to different inputs in the frequency domain. It serves as a bridge between the time-domain differential equations that describe system dynamics and the frequency-domain representation that facilitates easier manipulation and analysis.

To derive the transfer function, consider a general nth-order linear time-invariant differential equation of the form:

Here, c(t) is the output, r(t) is the input, and a_i and b_i are constant coefficients. Applying the Laplace transform to both sides, assuming all initial conditions are zero, the differential equation can be converted into an algebraic equation in terms of s, the complex frequency variable. Rearranging terms, we get:

The transfer function H(s) is defined as the ratio of the output C(s) to the input

R(s):

This expression shows that the transfer function is a rational function of s. The numerator is the polynomial formed by the input coefficients, and the denominator is the characteristic polynomial of the differential equation.

This transfer function indicates how the system's output c(t) responds to an input

r(t) in the frequency domain. The transfer function can be represented in a block diagram with the input R(s) on the left, the output C(s) on the right, and the transfer function H(s) inside the block. This visualization simplifies understanding and analyzing system behavior, especially when dealing with more complex systems.