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_{L}^{2}
= 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} / R_{1} )
Output voltage V_{o}
= A_{F} V_{P}
= ( 1 + R_{F} / R_{1} ) { j( f / f_{L} ) / ( 1 + j( f / f_{L} ) } V_{in}
_{ }
Voltage
gain |
V_{o} / V_{in} = ( 1 + R_{F}
/ R_{1 }) { j( f / f_{L} ) / ( 1 + j( f / f_{L} ) } |
Magnitude of voltage gain |
= ( 1 + R_{F} / R_{1} ) {
( f / f_{L} ) / ( 1 + √ ( f / f_{L} )^{2} } = A_{F}{ ( f / f_{L} ) /
( 1 + √ ( f / f_{L} )^{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 |
< A_{F} |
Lower cut off frequency |
= 0.707A_{F} |
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 |
= A_{F}
/ √ { 1 + ( f_{L} / f )^{2} } |
Where
f = Input frequency
f_{L} = Low cut off frequency
= 1 / 2π √ ( R_{2 }) ( R_{3 }) ( C_{2 }) ( C_{3 })
A_{F} = Pass band gain
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