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 = – 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 |
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
fL = Low cut off frequency
= 1 / 2π √ ( R2 ) ( R3 ) ( C2 ) ( C3 )
AF = Pass band gain
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