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22/07/2021

Low Pass Butterworth Filter

 

First order Low Pass Butterworth Filter

  • It is a filter which has nearly ideal response, flat passband as well as flat stop band. 
  • As both passband and stop band are flat, it is also called as flat – flat filter.

Butterworth Filter Circuit

  • The first order low pass Butterworth filter employs RC circuit for filtering.  
  • The non – inverting amplifier configuration is used for it. The gain of the filter depends upon RF and R1.

first-order-low-pass-butterworth-filter.png


Input voltage at the non – inverting terminal

V1 = - jXC / ( R – jXC ) ( Vin ) 

     = [ - j / 2πfC / ( R – j / 2πfC ) ] ( Vin )

     = [ - j  / ( R2πfC – j  ) ] ( Vin )

     = [ 1  / ( 1 + R2πfC ) ] ( Vin )…… ( 1 )

Output voltage

Vo = ( 1 + RF / R1 ) V1……. ( 2 )

From equation ( 1 ) and ( 2 )

Vo = ( Vin )( 1 + RF / R1 ) / ( 1 + R2πfC ) 

Vo / Vin = ( 1 + RF / R1 ) / ( 1 + R2πfC )

 Vo / Vin = AF / [ 1 + j ( f / fH ) ]

Where


Gain of the filter = Vo / Vin

Passband gain of the filter AF = ( 1 + RF / R1 )

Frequency of input signal = f

High cut off frequency of filter = fH

Magnitude of gain of the filter = AF / √ ( 1 + ( f / fH )2

Phase Angle Φ = tan – 1 ( f / fH )

 

Frequency Response

  • The frequency response curve is shown in the figure. It shows relation between voltage gain and frequency.

Range of frequencies

Voltage gain

Low range < Higher cut off frequency

AF ( Constant )

Higher cut off frequency

0.707AF

Greater than higher cut off frequency

Gain decreases

 

Summary

  • The voltage gain is constant for the frequency less than the higher cut off frequency. 
  • The voltage gain is 0707 AF at higher cut off frequency. 
  • The voltage gain reduces when the frequency greater than the higher cut off frequency. 
  • The voltage gain decreases at constant rate after higher cut off frequency.



frequency-response-of-first-order-low-pass-butterworth-filter.png


Cut off frequency / - 3 dB frequency / Corner frequency / 3 dB frequency

  • When the gain of the filter is down by 70.7% ( 3 dB from 0 Hz ) of the maximum gain, it is called as cut off

Second order Low Pass Butterworth Filter

  • The second order low pass Butterworth filter is obtained by adding RC network to the first order low pass Butterworth filter. 
  • It is rolled off at – 40 dB /dec. 
  • The gain of the second order low pass Butterworth filter is set by R1 and RF whereas high cut off frequency is determined by R1, C1, R2 and C2.




second-order-low-pass-butterworth-filter.png


Gain of the amplifier =  AF / √ ( 1 + ( f / fH )2

High cut off frequency fH = 1 / 2π √ ( R1R2C1C2 )

 

Where

Passband of the filter AF = 1 + ( RF / R1 )

Frequency of the input signal = f


frequency-response-of-the-second-order-low-pass-filter.png



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