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__FREQUENCY RESPONSE / NORMALISATION AND FILTER
CIRCUITS__

**Second Order System
**

It
is a circuit or a system which contains *2
or more energy storage elements* and it can be described by second order
differential equations. They provide a simple approximation to ideal filters and
they are used as building blocks of more expensive filters.

The frequency response that
characterises these second order filters includes three filter parameters, the
gain** k**, the quality factor **Q** and the corner frequency **w**_{0}.

The transfer function of a
second order filter shown on *figure I* below, is of the form :

*
figure I : second order system circuit*

=>
,

,
=>

the denominator is set equal
to zero in order for its roots to be found. **D** is the discriminant of the
denominator :

For the case that **2
different real roots** to exist then :

,
,
,

When **Q is equal to 1/2**
then the **two real roots are equal**.

When** Q is less than 1/2**
then the** two roots are complex and conjugate**.

For plotting the frequency
response of a second order system :

**Each
pole **derived from the transfer function will result, as in first order, a **
+6db/octave** declination on the asymptotic response line, at a frequency
calculated for each pole.

**Each
zero** derived from the transfer function will result, as in first order, a **
-6db/octave** inclination on the asymptotic response line, at a frequency
calculated for each zero.

For the
case where we have **two identical poles or zeroes** then the declination or
inclination on the response line would be** (****+/-)
12db/octave**.