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.

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 = – jX_C

                                            = – j ( 1 / ωC )

                                            = – j / 2πfC

                                            = 1 / j2πfC

Input voltage at point P according to voltage divider rule

V_P = V_in { R / ( R – jX_C ) }

     = V_in { R / ( R + 1 / j2πfC  ) }

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

     = V_in { j2πfRC / ( 1 + j2πfRC ) }

Let us consider that

Lower cut off frequency f_L = 1 / 2πRC

Divide numerator and denominator by 1 /2πRC

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

        = V_in { jf / ( 1 / f_L + jf ) }

        = V_in { jf / ( 1 +jf_L f ) / f_L }

        = V_in { jf f_L / ( 1 + jf_L f ) }

Divide numerator and denominator by f_L²

         = V_in { j( f / f_L ) / ( 1 + j( f / f_L )  }

Output voltage V_o = A_F V_P

Where A_F = gain of the OP – Amp 

                 = ( 1 + R_F / R₁ )

Output voltage V_o = A_F V_P

                 = ( 1 + R_F / R₁ ) { j( f / f_L ) / ( 1 + j( f / f_L ) } V_in

 

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 0.707A_F at lower cut off frequency. 
• The voltage gain becomes constant when frequency increases beyond lower cut off frequency.

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.

Magnitude of voltage gain

= AF / √ { 1 + ( fL / f )2 }

Where

    f = Input frequency

   f_L = Low cut off frequency

        =  1 / 2π √ ( R₂ ) ( R₃ ) ( C₂ ) ( C₃ )

  A_F = Pass band gain

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