EECS 412 Filter Design Project

What is Filter in electrical engineering field ? An electric filter can be used to eliminate unwanted

constituents, such as electric noise, from an electrical signal.. And, lowpass filter lets all the signals

whose frequencies  below the cutoff frequency pass through the  filter without obstruction.  In this

project, I design the Butterworth lowpass filter with the cutoff frequency 1000 radius per second.

Butter worth lowpass transfer function:

                                             H_L  = ( + or - ) 1/ ( D(s) )

( Please note: The filter transfer function is the ration of the coutput voltage to the input voltage )

The D(s) with a cutoff frequency = 1 rad/s in fifth order is:

                                              (s² + 1.618s + 1 ) ( s² + 1.618s + 1 ) ( s + 1 )

Then our transfer function is :

                                              -1/ (  (s² + 0.618s + 1 ) ( s² + 1.618s + 1 ) ( s + 1 ) )

which is equal to :

                                             ( -1/ ( s + 1 ) ) ( 1/ (s² + 1.618s + 1 ) ) ( 1/ (s² + 0.618s + 1 ) )

And, we want the cutoff frequency at 1000 rad/s, so apply the frequency scaling to each s term:

                                            { -1/ [ ( s/1000) + 1 ] } *  

                                            { 1/ [(s/1000 ) ² + (1.618s/1000) + 1 )  *

                                            { 1 / [ ( s/1000 ) ² +  ( 0.618s/1000 ) + 1 ] }

It finally becomes like this:

                                            { -303.7737/  ( s + 1000) } *  

                                            { 1382/ [(s² + 1618 s + 1000000 )  *

                                            { 2382 / [  s² +  618s + 1000000 ] }

Use these three equations to apply the Sallen and Key Filters and the first order Filter, we will have

Our butterworth lowpass filter at cutoff frequency at 1000 rad/s.

(  Refrences: Continuous Time Active Filter Design ( EECS 412 Text Book ),

                       Introduction to Electric Circuits ( EECS 210 Text Book ),

                       Microelectronic Circuits ( EECS 340.342 Text Book ).

                      The main design method of this project is to refer to the Chapter 16 of Introduction to 

Electric Circuits.                                                                                                                                            )