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.
Input voltage at
the non – inverting terminal
V1 = - jX_{C} / ( R – jX_{C} ) ( V_{in} )
= [ - j / 2πfC / (
R – j / 2πfC ) ] ( V_{in} )
= [ - j / ( R2πfC – j
) ] ( V_{in} )
= [ 1 / ( 1 + R2πfC ) ] ( V_{in} )…… ( 1 )
Output voltage
V_{o} = (
1 + R_{F} / R_{1} ) V_{1}……. ( 2 )
From equation ( 1
) and ( 2 )
V_{o} = (
V_{in} )( 1 + R_{F} / R_{1} ) / ( 1 + R2πfC )
V_{o} / V_{in}
= ( 1 + R_{F} / R_{1} ) / ( 1 + R2πfC )
V_{o} / V_{in} = A_{F}
/ [ 1 + j ( f / f_{H} ) ]
Where
Gain
of the filter = V_{o} / V_{in} |
Passband gain of the filter A_{F}
= ( 1 + R_{F} / R_{1} ) |
Frequency
of input signal = f |
High cut off frequency of filter = f_{H} |
Magnitude
of gain of the filter = A_{F} / √ ( 1 + ( f / f_{H} )2 |
Phase Angle Φ = tan ^{– 1} ( f /
f_{H} ) |
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 |
A_{F} (
Constant ) |
Higher cut off frequency |
0.707A_{F} |
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 A_{F }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.
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 R_{1} and R_{F} whereas high cut off frequency is determined by R_{1}, C_{1}, R_{2} and C_{2}.
Gain of the amplifier = A_{F} / √ ( 1 + ( f / f_{H}
)2 |
High cut off frequency f_{H} = 1 / 2π √
( R_{1}R_{2}C_{1}C_{2} ) |
Where
Passband of the
filter A_{F} = 1 + ( R_{F} / R_{1} )
Frequency of the input signal = f
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