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
*=>

**
**

or

**
=> **

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 :

**
phase**
**magnitude**

**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 ** S ^{2} ** has a phase of 180 degrees, or
simply a sign change from "