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 w0.

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.