24/07/2021

High Pass Butterworth Filter

 First order High Pass Butterworth Filter

  • The high pass filter is obtained by changing frequency determining component – resistors and capacitors in the low pass filter. 
  • The first order high pass filter is obtained by interchanging R and C in the first order low pass filter. 
  • Figure shows first order high pass Butterworth filter.


first-orde-rhigh-pass-butterworth-filter.png


Lower cut off frequency

  • It is frequency at which gain of the amplifier reduces from maximum to 70.7% of its maximum value. 
  • It is from frequency response graph, one can say that lower cut off frequency is determining parameter for stop band frequencies and pass band frequencies.

Impedance of the capacitor = – jXC

                                            = – j ( 1 / ωC )

                                            = – j / 2πfC

                                            = 1 / j2πfC

Input voltage at point P according to voltage divider rule

VP = Vin { R / ( R – jXC ) }

     = Vin { R / ( R + 1 / j2πfC  ) }

     = Vin { R / (( j2πfRC + 1) / j2πfC  ) }

     = Vin { j2πfRC / ( 1 + j2πfRC ) }

Let us consider that

Lower cut off frequency fL = 1 / 2πRC

Divide numerator and denominator by 1 /2πRC

        = Vin { ( jf ) / ( 1 / 2πRC ) +( j2πfRC / 2πRC )

        = Vin { jf / ( 1 / fL + jf ) }

        = Vin { jf / ( 1 +jfL f ) / fL }

        = Vin { jf fL / ( 1 + jfL f ) }

Divide numerator and denominator by fL2

         = Vin { j( f / fL ) / ( 1 + j( f / fL )  }

Output voltage Vo = AF VP

Where AF = gain of the OP – Amp 

                 = ( 1 + RF / R1 )

Output voltage Vo = AF VP

                 = ( 1 + RF / R1 ) { j( f / fL ) / ( 1 + j( f / fL ) } Vin

 

Voltage gain

Vo / Vin = ( 1 + RF / R1 ) { j( f / fL ) / ( 1 + j( f / fL ) }

Magnitude of voltage gain

= ( 1 + RF / R1 ) { ( f / fL ) / ( 1 + √ ( f / fL )2 }

= AF{ ( f / fL ) / ( 1 + √ ( f / fL )2 }

 

Frequency response of high pass Butterworth filter

 

  • When the frequency decrease that of lower cut off frequency, the high pass Butterworth filter rolls at – 20 dB / dec. 
  • The voltage gain reduced to 0.707AF at lower cut off frequency. 
  • The voltage gain becomes constant when frequency increases beyond lower cut off frequency.


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



Frequency

Voltage gain

Low frequency

< AF

Lower cut off frequency

= 0.707AF

Greater than lower cut off frequency

Constant

 

Second order High Pass Butterworth Filter

  • The second order high pass Butterworth filter can be obtained by interchanging frequency determining components R & C in the second order low pass filter. 
  • The second order high pass Butterworth filter is shown in the Figure.


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

Magnitude of voltage gain

= AF / √ { 1 + ( fL / f )2 }


Where

    f = Input frequency

   fL = Low cut off frequency

        =  1 / 2π √ ( R) ( R) ( C) ( C)

  AF = Pass band gain



frequencyresponseofsecondorderhighpassbutterworthfilter.png



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