USC Electrical Engineering
ELCT 301
Project
44#1
BUTTERWORTH FILTER DESIGN
Objective
The main purpose of this laboratory exercise is to learn the procedures for designing
Butterworth filters. In addition, the theory behind these procedures will be discussed in
recitation and the hands-on methods practiced in the laboratory. This exercise will focus
on the fourth-order filter but the methods are valid regardless of the order of the filter. In
this lab only even order filters will be considered.
Background
Read pgg.728-738 Hambley or pgg.1254-1267 Sedra Smith and the handout. The
Butterworth and Chebyshev filters are high order filter designs which have significantly
different characteristics, but both can be realized by using simple first order or biquad
stages cascaded together to achieve the desired order, passband response, and cut-off
frequency. The Butterworth filter has a maximally flat response, i.e., no passband ripple
and a roll-off of -20dB per pole. In the Butterworth scheme the designer is usually
optimizing the flatness of the passband response at the expense of roll-off. The
Chebyshev filter displays a much steeper roll-off, but the gain in the passband is not
constant. The Chebyshev filter is characterized by a significant passband ripple that often
can be ignored. The designer in this case is optimizing roll-off at the expense of passband
ripple.
Both filter types can be implemented using the simple Sallen-Key configuration.
The Sallen-Key design (shown in Figure A-1) is a biquadratic or biquad type filter,
meaning there are two poles defined by the circuit transfer function
H ( s) 
H0
,
( s /  0 )  (1 / Q )( s /  0 )  1
2
(1)
where H0 is the DC gain of the biquad circuit and is a function only of the ratio of the two
resistors connected to the negative input of the opamp. The quantity Q is called the
quality factor and is a direct measure of the flatness of the passband (in particular, a large
value of Q indicates peaking at the edge of the passband.) When C1 = C2 = C and R1 = R2
=R for the Sallen-Key design, the value of Q depends exclusively on the gain of the opamp stage and the cut-off frequency depends on R and C, or
© 2007 Santi
Q
1
,
(3  H 0 )
fC 
1
2 RC
and
(2)
.
(3)
1
USC Electrical Engineering
ELCT 301
Project
44#1
Notice that the quality factor Q and the DC gain H0 cannot be independently set in the
Sallen-Key circuit.
Choice of the value of Q depends on the desired characteristics of the filter. In particular
the designer must consider both the quality and the stability of the circuit. Calculations
with different values of Q demonstrate that for the second order case the flattest response
occurs when Q = 0.707, hence the maximally flat response of the Butterworth filter will
also be realized when Q = 0.707. In addition the circuit is only stable when H0 < 3.
Higher values will result in poles in the right half-plane and an unstable circuit.
The Butterworth filter is an optimal filter with maximally flat response in the
passband. The magnitude of the frequency response of this family of filters can be
written as
H( f ) 
H0
1   f / fC 
2n
(4)
,
where n is the order of the filter and fc is the cutoff frequency . For a single biquad
section, i.e., n = 2, the gain for which Q will equal 0.707, and hence yield a Butterworth
type response, is found to be 1.586 from equation (2).
In the case of a fourth order Butterworth filter (n = 4), the correct response can be
achieved by cascading two biquad stages. Each stage must be characterized by a specific
value of gain (or Q) that achieves both a flat response and stability. For a fourth order
Butterworth filter, the Q factors of the two stages must be Q = 0.541 and Q = 1.307.
From (2) it follows that one stage must have a gain of 1.152 and the other stage a gain of
2.235. As a result the overall DC gain of the fourth-order Butterworth realized with two
biquad stages will be equal to the product of the DC gain of the two stages, i.e., 2.575.
The gain values required to cascade biquad sections to achieve even-order filters are
given in Table A-I. Note that the number of biquad stages needed to realize a filter of
order n is n/2.
Other types of filters such as high pass, and bandpass filters can be designed the
same way, i.e., by cascading the appropriate filter sections to achieve the desired
bandpass and roll-off. The high-pass dual to the Sallen-Key low pass filter can be
realized by simply exchanging the positions of R and C in the circuit of Figure A-1.
Prelab
1. Do the following four paper designs and write all component values in your lab
notebook using the corner frequency assigned to you on the class webpage:
a. Second order lowpass filter
b. Fourth order lowpass filter
c. Second order highpass filter
d. Fourth order highpass filter
2. What is the passband gain of the four filters? Is it 0dB (gain of 1) or not?
© 2007 Santi
2
USC Electrical Engineering
ELCT 301
Project
44#1
3. Simulate the two fourth order filters using Spice. Produce Bode plots for
magnitude and phase for the two filters.
4. In Spice apply to the lowpass filter a 1V square wave and simulate the response of
the lowpass filter. Do this for three cases: square wave frequency equal to 0.1 fc,
0.5 fc, and fc, where fc is the cutoff frequency of the lowpass filter. Comment on
the results.
5. In the lab you will have to measure the frequency response of the filter. You will
measure a total of 20 points. Select 20 frequencies equally spaced on a log scale
centered around the cutoff frequency, starting one decade below and ending one
decade above the cutoff frequency. For example, if you were choosing 5 points
only, you would select the following points: 0.1fc - 0.316 fc - fc - 3.16 fc - 10 fc .
Lab Project
1. Build the fourth-order lowpass Butterworth filter you designed for the Prelab.
Notice that there are quite a few components. Try to be orderly and test the
Sallen Key sections one at a time.
2. Measure the cutoff frequency of the lowpass filter using the following
procedure: measure the low-frequency gain for a sinusoidal input and increase
the frequency until the gain drops by 3 dB. That is the definition of cutoff
frequency.
3. Make the measurements needed to create a magnitude and phase Bode plot of
the filter frequency response. Apply a 1-Volt sinusoidal signal at the
frequencies you calculated in the prelab, observe input and output and
measure gain and phase shift.
4. Perform a preliminary check comparing the response to the model predictions
from the prelab.
5. For the lowpass filter perform experimentally step 4 of the prelab (square
wave response). Compare with simulation.
6. Exchange resistors R1, R2 and capacitors C1, C2 and make a few
measurements to verify that the circuit becomes a highpass filter. Determine
the cutoff frequency.
Lab Report
In the lab report discuss the cutoff frequency measurement that you made. Based
on component tolerances, predict the expected range of values that you would expect to
see and comment on your actual measurement.
Create a magnitude and phase Bode plot based on your experimental
measurements. Make sure you use the correct scales for the axes (for example the
magnitude Bode plot is a log-log plot with the y-axis expressed in dB). Compare the
frequency response results with Spice results and discuss. Report also on the lowpass
filter square wave response, both the simulation and experimental results. Explain the
result.
© 2007 Santi
3
USC Electrical Engineering
ELCT 301
Project
44#1
Appendix
C2
R1
R2
+
VIN
VOUT
(H0-1)Rf
C1
Rf
Figure A-1
The Sallen-Key lowpass filter
For R  R1  R2 and C  C1  C 2 the Sallen Key filter has a gain
H ( s) 
H0
1  RCs (3  H 0 )  ( RC ) 2 s 2
This transfer function can be written in normalized pole-zero form as
H ( s) 
1
H0
1
with Q 
and  0  2f c 
2
s
s
RC
(3  H 0 )
1
 2
 0Q  0
Notice that the DC gain H0 and the Q of the filter cannot be independently set.
Poles
2
4
6
8
© 2007 Santi
Butterworth
Chebyshev (0.5dB)
Chebyshev (2.0dB)
(DC Gain H0)
n
Gain
n
Gain
1.586
1.231
1.842
0.907
2.114
1.152
0.597
1.582
0.471
1.924
2.235
1.031
2.660
0.964
2.782
1.068
0.396
1.537
0.316
1.891
1.586
0.768
2.448
0.730
2.648
2.483
1.011
2.846
0.983
2.904
1.038
0.297
1.522
0.238
1.879
1.337
0.599
2.379
0.572
2.605
1.889
0.861
2.711
0.842
2.821
2.610
1.006
2.913
0.990
2.946
Table I Design Table for Butterworth and Chebyshev filters
4
USC Electrical Engineering
ELCT 301
Project
44#1
Butterworth Design Example
Design Example. You want to design a fourth-order Butterworth filter having a corner frequency fc =
3kHz. Specify all component values needed for the design.
You need to use two cascaded Sallen-Key second-order sections. We will call them section A and section
B. From the Table above the DC gains of the two sections are:
H 0 A  1.152
H 0 B  2.235
For each section we need to decide values for capacitor C and resistors R, Rf and ( H 0  1) R f . We can
arbitrarily select values for C and Rf. Let us choose C = 10nF and Rf = 10k. The corner frequency of the
R-C filter must be 3kHz and therefore
R
1
 5.3k  5.1k
2Cf c
where the last value is a standard 5% resistor value. So resistors R1 and R2 will have this value. Regarding
the feedback resistors the values are:
( H 0 A  1) R f  1.52k  1.5k
( H 0 B  1) R f  12.35k  12k
Comment:
how do you choose the arbitrary values for C and Rf? Typically you want all resistors to
be in the range 1k - 1M (this is related to the output current capability of opamps) and capacitors to be
in the range 100pF – 100nF, so that you can use standard plastic capacitors.
© 2007 Santi
5
USC Electrical Engineering
ELCT 301
44#1
© 2007 Santi
6
Project