Analog Tutorial Lesson 8: Designing Active Higher-Order Cascaded Op-Amp Filters

What You Will Learn

In this tutorial you will cascade two second-order lowpass filters to design fourth-order Butterworth and Chebyshev lowpass filters.

Designing a Cascaded Fourth-Order Butterworth Lowpass Filter

The following is a list of parts needed for this part of the tutorial lesson:

Part Name	Part Type	Part Value
VCC	DC Voltage Source	15V
VEE	DC Voltage Source	-15V
V1	AC Voltage Source	1V
R1 - R4	Resistor	10k
C1	Capacitor	17n
C2	Capacitor	15n
C3	Capacitor	42n
C4	Capacitor	6n
X1 - X2	LM741 Op-Amp	Defaults

You can cascade two second-order lowpass filters to design a fourth-order lowpass filter as shown in the figure below. The transfer function of the cascaded systems is given by:

[math]H(s) = H_1(s) . H_2(s) = \frac {\omega_{01} ^2 \omega_{02} ^2 } { \left( s - s_1 \right) \left( s - s_2 \right) \left( s - s_3 \right) \left( s - s_4 \right) } [/math]

The four poles of the fourth-order filter are nothing but the two pairs of poles of the constituent second-order lowpass stages. These poles can be determined according to the type of the filter response specification. For example, for the Butterworth filter, the poles are given by:

N = 4	Butterworth Poles
Pole 1	-0.924 + j 0.383
Pole 2	-0.924 - j 0.383
Pole 3	-0.383 + j 0.924
Pole 4	-0.383 - j 0.924

Based on the Butterworth poles, and assuming R1 = R2 = R3 = R4 = R = 10k, we can calculate the values of the four capacitors:

C1 = 17nF

C2 = 15nF

C3 = 42nF

C4 = 6nF

The Fourth-Order Cascaded Active Bandpass Filter with Butterworth poles.

Place and connect the parts with the specified values as shown in the above figure. Run an AC frequency sweep of your active filter with the stop frequency of the sweep set to 50kHz. You should get a frequency response like the figure shown below with a cutoff frequency of 1kHz. You can see from the graph that frequency response of the fourth-order filter drops after the cutoff much faster and the value of gain at f = 5kHz is below -55dB.

Start Frequency	1Hz
Stop Frequency	50kHz
Steps/Interval	50
Interval Type	Decade
Preset Graph Plots	Custom: Gain = VDB(OUT) - VDB(IN)

The frequency response of the fourth-order active loswpass filter based on the Butterworth design.

Designing a Cascaded Fourth-Order Chebyshev Lowpass Filter

The following is a list of parts needed for this part of the tutorial lesson:

Part Name	Part Type	Part Value
VCC	DC Voltage Source	15V
VEE	DC Voltage Source	-15V
V1	AC Voltage Source	1V
R1 - R4	Resistor	10k
C1	Capacitor	187n
C2	Capacitor	1.5n
C3	Capacitor	77.2n
C4	Capacitor	16.7n
X1 - X2	LM741 Op-Amp	Defaults

Next, we consider the Chebyshev filter design. With a ripple factor ε = 1, the maximum peak-to-peak ripples would be 3dB. For a fourth-order design, the poles are given by:

N = 4	Chebyshev Poles
Pole 1	-0.085 + j 0.947
Pole 2	-0.085 - j 0.947
Pole 3	-0.206 + j 0.392
Pole 4	-0.206 - j 0.392

Based on the 3-dB Chebyshev poles, and assuming R1 = R2 = R3 = R4 = R = 10k, we can calculate the values of the four capacitors:

C1 = 187nF

C2 = 1.5nF

C3 = 77.2nF

C4 = 16.7nF

The Fourth-Order Cascaded Active Bandpass Filter with Chebyshev poles.

Place and connect the parts with the specified values as shown in the above figure. Run an AC frequency sweep of your active filter with the stop frequency of the sweep set to 50kHz. You will get a frequency response like the figure shown below with a cutoff frequency of 1kHz. You can see from the graph that frequency response of this fourth-order filter drops even faster than the previous case and the value of gain at f = 5kHz is -70dB. However, now you have introduced a ripple in the frequency response with a maximum peak-to-peak amplitude of 3dB!

Start Frequency	1Hz
Stop Frequency	50kHz
Steps/Interval	50
Interval Type	Decade
Preset Graph Plots	Custom: Gain = VDB(OUT) - VDB(IN)

The frequency response of the fourth-order active loswpass filter based on the 3dB Chebyshev design.

 

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