It is a mathematical statement (equation) that describes the transfer characteristics of a system. A transfer function defines the relationship between the input to a system and its output. It is typically written in the frequency domain (S-domain), rather than the time domain (t-domain). The Laplace transform is used to map the time domain representation to frequency domain representation.
A single-input x(t), single-output y(t) filter may be represented symbolically by figure A below, where x(t) is the I/P signal and y(t) is the O/P signal. The quantities Y(S) and X(S) shown on figure B, are respectively the Laplace transforms of y(t) and x(t), and H(S) is the transfer function which is the ratio of the transformed O/P and I/P variables.
Y(S) = H(S) * X(S) => H(S) = Y(S)/X(S)
figure A : time domain
figure B : frequency domain
The poles of a T.F are the values of the Laplace transform variable S, that cause the T.F to become infinite (roots of the denominator).
The zeros of a T.F are the values of the Laplace transform variable S that cause the T.F to become zero (roots of the nominator).
figure C : example circuit
Using Ohms Law for the circuit on figure C =>
then by applying Laplace transforms to the above equation =>
finally the transfer function obtained is
in order for the frequency response function to be found, the replacement S = jw takes place =>
S-plane is a two-dimensional space delivered by two orthogonal axes, the real number axis and the imaginary number axis. A point in the S-plane represents a complex number. When talking about control systems, complex numbers are typically represented by the letter S. Each complex number S has both a real component, typically represented by the letter sigma σ and an imaginary component, typically represented by the letter omega w.
S = σ + jw
Any point in the complex plane has an angle (or phase) and magnitude defined as :
φ is the angle between the real axis and the complex number S
Graphically, each complex number S is plotted in the S-plane as follows :
figure D : complex number plotting
If |H(jw)| is the amplitude or magnitude and φ(w) is the phase =>
note that: S is used for two reasons. First to make the notation simple and compact, and second, to get all the math into a standard form. S has both amplitude and phase associated with it. If S = jw , a single S has a phase of 90 degrees and S2 has a phase of 180 degrees, or simply a sign change from "+" to "-".