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26-1
26.1 Introduction
Analog lters are essential in many dierent systems that electrical engineers are required to design
in their engineering career. Filters are widely used in communication technology as well as in other
applications. Although we discuss and talk a lot about digital systems nowadays, these systems alwayscontain one or more analog lters internally or as the interace with the analog world [SV01].
Tere are many dierent types o lters such as Butterworth lter, Chebyshev lter, inverse Chebyshev
lter, Cauer elliptic lter, etc. Te characteristic responses o these lters are dierent. Te Butterworth
lter is at in the stop-band but does not have a sharp transition rom the pass-band to the stop-band
while the Chebyshev lter has a sharp transition rom the pass-band to the stop-band but it has the
ripples in the pass-band. Oppositely, the inverse Chebyshev lter works almost the same way as the
Chebyshev lter but it does have the ripple in the stop-band instead o the pass-band. Te Cauer lter
has ripples in both pass-band and stop-band; however, it has lower order [W02, KAS89]. Te analog
lter is a broad topic and this chapter will ocus more on the methodology o synthesizing analog lters
only (Figures 26.1 and 26.2).
Section 26.2 will present methods to synthesize our dierent types o these low-pass lters. Tenwe will go through design example o a low-pass lter that has 3 dB attenuation in the pass-band, 30 dB
attenuation in the stop-band, the pass-band requency at 1 kHz, and the stop-band requency at 3 kHz
to see our dierent results corresponding to our dierent synthesizing methods.
26.2 Methods to Synthesize Low-Pass Filter
26.2.1 Butterworth Low-Pass Filter
ppass-band requency
sstop-band requency
pattenuation in pass-band
sattenuation in stop-band
26Analog Filter Synthesis26.1 Introduction ....................................................................................26-126.2 Methods to Synthesize Low-Pass Filter .......................................26-1
Butterworth Low-Pass Filter Chebyshev Low-Pass Filter Inverse
Chebyshev Low-Pass Filter Cauer Elliptic Low-Pass Filter
26.3 Frequency ransormations ........................................................26-10Frequency Transformat ions; Low-Pass to High-Pass Frequency
Transformat ions; Low-Pass to Band-Pass Frequency
Transformat ions; Low-Pass to Band-Stop Frequency
ransormation; Low-Pass to Multiple Band-Pass
26.4 Summary and Conclusion ...........................................................26-13Reerences ..................................................................................................26-13
Nam PhamAuburn University
Bogdan M.Wilamowski
Auburn University
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26-2 Fundamentals of Industrial Electronics
Butterworth response (Figure 26.3):
T j
n n( )
/
2
20
2
1
1=
+ ( )
Tere are three basic steps to synthesize any type o low-pass lters. Te rst step is calculating the
order o a low-pass lter. Te second step is calculating poles and zeros o a low-pass lter. Te third step
is design circuits to meet pole and zero locations; however, this part is another topic o analog lters, so
it will be not be covered in this work [W90, WG05, WLS92].
All steps to design Butterworth low-pass lter.
Step 1: Calculate order o lter:
n ns p
s p
= log[( )( )]
log( / )
/ / /10 1 10 11010 1 2
( needs to be rooundup to integer value)
[dB]
20
40
Magnitude
[dB]
20
40
Magnitude
FIGURE 26.1 Butterworth lter (lef), Chebyshev lter (right). AQ1
[dB]
20
40
Magnitude
[dB]
20
40
Magnitude
FIGURE 26.2 Inverse Chebyshev lter (lef), Cauer elliptic lter (right).
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Analog Filter Synthesis 26-3
Step 2: Calculate pole and zero locations:
Angle in is odd:
=
=
k
nk
n1800 1
1
2; , , ,
Angle in is even:
= +
=
0 5 180 0 12
2. ; , , ,
k
nk
n
Normalized pole locations:
a b
k k= = =
cos( ); sin( ); ( )
0 1
0
1 2
10 10 1 410 1 10 1
1
2=
=( )
[( ) / ( )];
/
/ / /( )
p s
n k
ks p
Qa
Step 3: Design circuits to meet pole and zero locations (not covered in this work) (Figure 26.4).
Example:
Step 1: Calculate order o flter:
n n=
= =log[(10 1)(10 1)]
log(3000 1000)3.1456 4
30 /10 3/10 1/ 2
/
Step 2: Calculate pole and zero locations
Normalized values o poles and 0 and Q:
0.38291 + 0.92443i 1.00059 1.30656
0.38291 0.92443i 1.00059 1.30656
0.92443 + 0.38291i 1.00059 0.54120
0.92443 0.38291i 1.00059 0.54120
Normalized values o zeros none.
0 dB
s
s
p
p
FIGURE 26.3 Butterworth lter characteristic.
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26-4 Fundamentals of Industrial Electronics
26.2.2 Chebyshev Low-Pass Filter
ppass-band requency
sstop-band requency
pattenuation in pass-band
sattenuation in stop-band
Chebyshev response (Figure 26.5):
T j Cn( ) / ( ( ))
2 2 21 1= +
Step 1: Calculate order o lter:
ns p
s p s p
=
+
ln[ * ( ) / ( )]
log[( (( )
/ / /
/ ) /
4 10 1 10 1
1
10 10 1 2
2 2
)) ]/1 2( needs to be roundup to integer value)n
[dB]
20
40
Magnitude
Phase
s -plane
90
180
270
FIGURE 26.4 Pole-zero locations, magnitude response, and phase o Butterworth lter.
Frequencies at whichCn= 0
Frequencies at which|Cn| = 1
|T6(j)|
Is here
Is here 1
00
1/1+2
FIGURE 26.5 Chebyshev lter characteristic.
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Analog Filter Synthesis 26-5
Step 2: Calculate pole and zero locations:
= +
+
90
90 1 180
n
k
n
( )
=
=
10 1110 1 2
1p
n
//
;sinh ( / )
a b a b Qa
k k k k k K k
k
= = = + =sinh( )cos( ); cosh( )sin( ); ;
2 2
2
Step 3: Design circuits to meet pole and zero locations (not covered in this work) (Figure 26.6).
Example:
Step 1: Calculate order o flter:
n =
+
ln[4 *(10 1) (10 1)]
log[(3000 1000) ((3000 1
30 /10 3/10 1/ 2
2
/
// 0000 ) 1) ]2.3535 3
2
= =1/2
n
Step 2: Calculate pole and zero locations
Normalized values o poles and 0 and Q:
0.14931 + 0.90381i 0.91606 3.06766
0.14931 0.90381i 0.91606 3.06766
0.29862
Normalized values o zeros none.
[dB]
Magnitude
x
x
s-plane
Phase
30
40
90
180
FIGURE 26.6 Pole-zero locations, magnitude response, and phase o Chebyshev lter.
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26-6 Fundamentals of Industrial Electronics
26.2.3 Inverse Chebyshev Low-Pass Filter
ppass-band requency
sstop-band requency
pattenuation in pass-band
sattenuation in stop-band
Inverse Chebyshev response (Figure 26.7):
T j
C
CIC
n
n
( )( / )
( / )
22 2
2 2
1
1 1=
+
Te method to design the inverse Chebyshev low-pass lter is almost the same as the Chebyshev low-
pass lter. It is just slightly dierent.
Step 1: Calculate order o lter
n = order o the Chebyshev lter
Step 2: Calculate pole and zero locations:
Pa b i n
i k npick k
i=+
= =
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