Low Pass Butterworth Filter

In this article, First order Low Pass Butterworth Filter circuit, Second order Low Pass Butterworth Filter circuit, gain of the filter and its frequency response is given.

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.

• 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.

Voltage Gain of First order Low Pass Butterworth Filter

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 / R₁ ) V₁……. ( 2 )

From equation ( 1 ) and ( 2 )

V_o = ( V_in )( 1 + R_F / R₁ ) / ( 1 + R2πfC ) 

V_o / V_in = ( 1 + R_F / R₁ ) / ( 1 + R2πfC )

 V_o / V_in = A_F / [ 1 + j ( f / f_H ) ]

Where

Gain of the filter = Vo / Vin

Passband gain of the filter AF = ( 1 + RF / R1 )

Frequency of input signal = f

High cut off frequency of filter = fH

Magnitude of gain of the filter = AF / √ ( 1 + ( f / fH )2

Phase Angle Φ = tan – 1 ( f / fH )

 

Frequency Response of First order Low Pass Butterworth Filter

• 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	AF ( Constant )
Higher cut off frequency	0.707AF
Greater than higher cut off frequency	Gain decreases

 

Summary of Frequency Response of First order Low Pass Butterworth Filter

• 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 R₁ and R_F whereas high cut off frequency is determined by R₁, C₁, R₂ and C₂.

Gain of the amplifier =  AF / √ ( 1 + ( f / fH )2

High cut off frequency fH = 1 / 2π √ ( R1R2C1C2 )

 

Where

Passband of the filter A_F = 1 + ( R_F / R₁ )

Frequency of the input signal = f

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