Analog Filter Synthesis - Auburn Universitywilambm/pap/2011/K10147_C026.pdf · Butterworth Low-Pass...
Transcript of Analog Filter Synthesis - Auburn Universitywilambm/pap/2011/K10147_C026.pdf · Butterworth Low-Pass...
26-1
26.1 Introduction
Analog filters are essential in many different 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 always contain one or more analog filters internally or as the interface with the analog world [SV01].
There are many different types of filters such as Butterworth filter, Chebyshev filter, inverse Chebyshev filter, Cauer elliptic filter, etc. The characteristic responses of these filters are different. The Butterworth filter is flat in the stop-band but does not have a sharp transition from the pass-band to the stop-band while the Chebyshev filter has a sharp transition from the pass-band to the stop-band but it has the ripples in the pass-band. Oppositely, the inverse Chebyshev filter works almost the same way as the Chebyshev filter but it does have the ripple in the stop-band instead of the pass-band. The Cauer filter has ripples in both pass-band and stop-band; however, it has lower order [W02, KAS89]. The analog filter is a broad topic and this chapter will focus more on the methodology of synthesizing analog filters only (Figures 26.1 and 26.2).
Section 26.2 will present methods to synthesize four different types of these low-pass filters. Then we will go through design example of a low-pass filter that has 3 dB attenuation in the pass-band, 30 dB attenuation in the stop-band, the pass-band frequency at 1 kHz, and the stop-band frequency at 3 kHz to see four different results corresponding to four different synthesizing methods.
26.2 Methods to Synthesize Low-Pass Filter
26.2.1 Butterworth Low-Pass Filter
ωp—pass-band frequencyωs—stop-band frequencyαp—attenuation in pass-bandαs—attenuation in stop-band
26Analog Filter Synthesis
26.1 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 Transformations ........................................................26-10Frequency Transformations; Low-Pass to High-Pass • Frequency Transformations; Low-Pass to Band-Pass • Frequency Transformations; Low-Pass to Band-Stop • Frequency Transformation; Low-Pass to Multiple Band-Pass
26.4 Summary and Conclusion ...........................................................26-13References ..................................................................................................26-13
Nam PhamAuburn University
Bogdan M. WilamowskiAuburn University
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Butterworth response (Figure 26.3):
T j
n n( )
/ω
ω ω2
202
11
=+ ( )
There are three basic steps to synthesize any type of low-pass filters. The first step is calculating the order of a low-pass filter. The second step is calculating poles and zeros of a low-pass filter. The third step is design circuits to meet pole and zero locations; however, this part is another topic of analog filters, so it will be not be covered in this work [W90, WG05, WLS92].
All steps to design Butterworth low-pass filter.
Step 1: Calculate order of filter:
n n
s p
s p= − −log[( )( )]
log( / )
/ / /10 1 10 110 10 1 2α α
ω ω ( needs to be rooundup to integer value)
[dB]
–20
–40
Magnitude
[dB]
–20
–40
Magnitude
FIGURE 26.1 Butterworth filter (left), Chebyshev filter (right). AQ1
[dB]
–20
–40
Magnitude
[dB]
–20
–40
Magnitude
FIGURE 26.2 Inverse Chebyshev filter (left), Cauer elliptic filter (right).
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Analog Filter Synthesis 26-3
Step 2: Calculate pole and zero locations:Angle if n is odd:
Ω = ± ° = −k
nk n180 0 1 1
2; , , ,…
Angle if n is even:
Ω = ± +
° = −0 5 180 0 1 22
. ; , , ,kn
k n…
Normalized pole locations:
a bk k= − = ± =cos( ); sin( ); ( )Ω Ω ω0 1
ω
ω ωα α0
1 2
10 10 1 410 1 10 112
=− −
=( )
[( ) / ( )];
/
/ / /( )p s
n kks 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 of filter:
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 locationsNormalized values of poles and ω0 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 of zeros ⇒ none.
0 dB
αs
ωs
αp
ωp
FIGURE 26.3 Butterworth filter characteristic.
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26.2.2 Chebyshev Low-Pass Filter
ωp—pass-band frequencyωs—stop-band frequencyαp—attenuation in pass-bandαs—attenuation in stop-band
Chebyshev response (Figure 26.5):
T j Cn( ) / ( ( ))ω ε ω2 2 21 1= +
Step 1: Calculate order of filter:
ns p
s p s p= − −
+ −ln[ * ( ) / ( )]
log[( (( )
/ / /
/ ) /
4 10 1 10 11
10 10 1 2
2 2
α α
ω ω ω ω )) ]/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 of Butterworth filter.
Frequencies at whichCn= 0
Frequencies at which|Cn| = 1
|T6( jω)|
Is here
Is here1
00
1/√1 + ε2
FIGURE 26.5 Chebyshev filter characteristic.
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Analog Filter Synthesis 26-5
Step 2: Calculate pole and zero locations:
Ω = ° + ° + − °90 90 1 180
nk
n( )
ε γ εα= −
=
−
10 1 110 1 2 1p
n/ /
; sinh ( / )
a b a b Q
ak k k k k Kk
k= = = + =sinh( )cos( ); cosh( )sin( ); ;γ γ ω ωΩ Ω 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 of filter:
n = − −
+ln[4 * (10 1) (10 1)]
log[(3000 1000) ((3000 1
30/10 3/10 1/2
2
/// 0000 ) 1) ]
2.3535 32 −= ⇒ =1/2 n
Step 2: Calculate pole and zero locationsNormalized values of poles and ω0 and Q:
−0.14931 + 0.90381i 0.91606 3.06766
−0.14931 − 0.90381i 0.91606 3.06766−0.29862
Normalized values of zeros ⇒ none.
[dB]
Magnitude
x
x
s-plane
Phase
–30
–40
–90
–180
α
FIGURE 26.6 Pole-zero locations, magnitude response, and phase of Chebyshev filter.
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26.2.3 Inverse Chebyshev Low-Pass Filter
ωp—pass-band frequencyωs—stop-band frequencyαp—attenuation in pass-bandαs—attenuation in stop-band
Inverse Chebyshev response (Figure 26.7):
T j C
CICn
n( ) ( / )
( / )ω ε ω
ε ω2
2 2
2 21
1 1=
+
The method to design the inverse Chebyshev low-pass filter is almost the same as the Chebyshev low-pass filter. It is just slightly different.
Step 1: Calculate order of filtern = order of the Chebyshev filter
Step 2: Calculate pole and zero locations:
P
a b i ni k npic
k ki=
+= = − <1 1
22 1 1 3 5,
cos[ * / ( )]; : , ,find zeros ω
Π…
Notes: two conjugate poles on the imaginary axis.
Step 3: Design circuits to meet pole and zero locations (not covered in this work) (Figure 26.8).
Example:
Step 1: Calculate order of filter:
n = − −
+ln[4 * (10 1) (10 1)]
log[(3000 1000) ((3000 1
30/10 3/10 1/2
2
// / 0000 ) 1) ]
2.3535 32 1/2−= ⇒ =n
Passband Stopband
Gain= c
√1+ c2
FIGURE 26.7 Inverse Chebyshev filter characteristic.
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Step 2: Calculate pole and zero locationsNormalized values of poles and ω0 and Q:
−0.6613 + 1.29944i 1.45803 1.10240
−0.6613 − 1.29944i 1.45803 1.10240−1.60734
Normalized values of zeros:3.4641i 3.4641i 3.4641
−3.4641i −3.4641i 3.4641
26.2.4 Cauer Elliptic Low-Pass Filter
Cauer elliptic response (Figure 26.9):
T jw
R w Ln( )
( , )2
2 21
1=
+ ε
To design the Cauer elliptic filter is more complicated than designing three previous filters. In order to calculate the transfer function of this filter, a mathematic process is summarized as below. Although the low-pass Cauer elliptic filter has ripples in both stop-band and pass-band, it has lower order than the three previous filters (Figure 26.10). That is the advantage of the Cauer elliptic filter:
k p
s=
ωω
(26.1)
′ = −k k1 2 (26.2)
q k
k0
0 5 11
= − ′+ ′
. ( )( )
(26.3)
[dB]
–20
–40
Phase
s-plane
x
x
Magnitude
–180
–90
FIGURE 26.8 Pole-zero locations, magnitude response, and phase of inverse Chebyshev filter.
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26-8 Fundamentals of Industrial Electronics
q q q q q= + + +0 05
09
0132 15 150 (26.4)
D
s
p= −
−10 110 1
0 1
0 1
.
.
α
α
(26.5)
n Dq
≥ log( )log( / )
161
(26.6)
Λ = +−
12
10 110 1
0 05
0 05n
p
pln
.
.
α
α (26.7)
Magnitude
Phase
[dB]
–20
–40
–90
0
x
x
s-planea
FIGURE 26.10 Pole-zero locations, magnitude response, and phase of Cauer elliptic filter.
1G=√1+ ε2
1G=√1+ ε2L i
2
FIGURE 26.9 Cauer elliptic filter characteristic.
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Analog Filter Synthesis 26-9
σ0
1 4 1
02 1 2 1
1 2 1 22
=− +
+ −
+
=
∞∑q q m
q m
m m m
m
m m
/ ( )( ) sinh[( ) ]
( ) cosh( )
Λ
Λmm=
∞∑ 1
(26.8)
ω σ σ= +( ) +
1 10
2 02
kk
(26.9)
Ωi
m m m
m
m m
q q m n
q m=
− +
+ −
+
=
∞∑2 1 2 1
1 2 1 2
1 4 1
02
/ ( )( ) sinh(( ) / )
( ) cosh
πµ
ππµnm
=
∞∑ 1
(26.10)
µ =−
=
i n
i ni r
for odd
for even 12
1 2, , ..., (26.11)
V kki ii= − −
( )1 12
2
Ω Ω (26.12)
Ai
01 21=
Ω (26.13)
B Vi
i i
i0
02 2
02 2 21
= ++
( ) ( )( )
σ ωσ
ΩΩ
(26.14)
B Vi
i
i1
0
02 2
21
=+
σσ Ω
(26.15)
H
BA
BA
i
ii
r
i
ii
rp
0
00
01
0 05 0
01
10=
=
−
=
∏
∏
σ
α
for odd n
for even n.
(26.16)
Example:
n = 1.9713 ⇒ n = 2. This filter is the second low pass filter.Normalized values of poles and ω0 and Q:
−0.31554 + 0.97313i 0.85360 1.35259
−0.31554 + 0.97313i 0.85360 1.35259
Normalized values of zeros:
4.18154i 4.18154i 4.18154−4.18154i −4.18154i 4.18154
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26-10 Fundamentals of Industrial Electronics
26.3 Frequency Transformations
Four typical methods of deriving a low-pass transfer function that satisfies a set of given specifications are presented. However, there are a lot of applications in the real world of designing, which require not only the low-pass filters but also the band-pass filters, high-pass filters, and band-rejection filters. A designer can design any type of filters by designing a low-pass filter first. When a low-pass filter is achieved, the desired filter can be derived by “frequency transformation.” In other words, the under-standing of methods to design a low-pass filter is the basic but not the trivial task.
26.3.1 Frequency Transformations Low-Pass to High-Pass
Z s Ss
jj
( ) ; ;= = = ⇒− ≤ ≤−
1 1 1 1 1Ω Ω
Ωω ω
frequency of low-pass passband11 1≤ ≤ω frequency of high-pass passband
Frequency transformation transforms the pass-band of the low-pass, centered around Ω = 0, into that of the high-pass, centered around ω = ∞ (Figure 26.11). Similarly, it transforms the low-pass stop-band that is centered around Ω = ∞ into that of the high pass, centered around ω = 0. Consequently, the fre-quency transformation function Z(s) has a zero in the center of the pass-band of the high-pass (at ω = ∞) and a pole in the center of the high-pass, stop-band (at ω = 0) [SV01]:
T SS S Q
T ss Qs
s( )( / )
( )( / ) ( / ) ( /
=+ +
⇒ =+ +
=ωω ω
ωω ω ω
02
20 0
202
20 0
2
2
021 1 )) ( / )+ +s Q sω0
2 2
T(S): low-pass transfer function; T(s): high-pass transfer function.
26.3.2 Frequency Transformations Low-Pass to Band-Pass
Z s S sBs
sB s B
Bc c c
c
cc( ) ( ) ; ;= = + = + ⇒ = − = = −
2 2 2 22
1 2 2 1ω ω ω
ωω ω
ωω ω ω ω ωΩ
Frequency transformation transforms the pass-band of the low-pass, centered around Ω = 0, into that of the band-pass, centered around ω = ωc. Similarly, it transforms the low-pass stop-band that is centered around Ω = ∞ into that of the band-pass, centered around ω = 0 (Figure 26.12). Consequently,
1 ω
1
Ω
–1
–11sS=
FIGURE 26.11 Frequency transformations low-pass to high-pass.
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Analog Filter Synthesis 26-11
the frequency transformation function Z(s) has zeros in the center of the pass-band of the band-pass (at ω = ± ωc) and poles in the center of the band-pass, stop-band (at ω = 0 and ω = ∞) [SV01]:
T SS S Q
T s s Bs Bs Q Bc
( )( / )
( )( / ) (
=+ +
⇒ =+ + +
ωω ω
ωω ω ω
02
20 0
2
2 202
40
3 2 202 22 2
02 4) ( / )s B s Qc c+ +ω ω ω
T(S): low-pass transfer function; T(s): band-pass transfer function.
26.3.3 Frequency Transformations Low-Pass to Band-Stop
Z s S Bss
B Bc c
c( ) ; ;= =+
⇒ = −−
= = −2 2 2 22
1 2 2 1ωω
ω ωω ω ω ω ωΩ
Frequency transformation transforms the pass-band of the low-pass, centered around Ω = 0, into that of the band-stop, centered around ω = 0 and ω = ∞ (Figure 26.13). Similarly, it transforms the low-pass, stop-band that is centered around Ω = ∞ into that of the band-stop, centered around ω = ωc. Consequently, the frequency transformation function Z(s) has zeros in the center of the pass-band of the band-stop (at ω = 0 and ω = ∞) and poles in the center of the band-stop, stop-band (at ω = ± ωc) [SV01]:
BΩ
ω
ω1 ωc ω2
Ω0
–Ω0
FIGURE 26.13 Frequency transformations low-pass to band-stop.
BΩ
ω
ω1 ωc ω2
Ω0
–Ω0
FIGURE 26.12 Frequency transformations low-pass to band-pass.
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T SS S Q
T s s ss
c c( )( / )
( )(
=+ +
⇒ = + ++
ωω ω
ω ω ω ω ωω ω
02
20 0
202 4
02 2 2
02 4
02 4
200
3 202 2 2
02 4
022Bs Q B s B s Qc c c/ ) ( ) ( / )+ + + +ω ω ω ω ω ω
26.3.4 Frequency Transformation Low-Pass to Multiple Band-Pass
Frequency transformation transforms the pass-band of the low-pass, centered around Ω = 0, into that of the multiple band-pass, centered around ω = 0 and ω = ωz1. Similarly, it transforms the low-pass, stop-band that is centered around Ω = ∞ into that of the multiple band-pass, centered around ω = ωp1 and ω = ∞. Consequently, the frequency transformation function Z(s) has zeros in the center of the pass-band of multiple band-pass and at ωz1 (at ω = 0 and ω = ±ωz1) and poles in the center of the band-stop of multiple pass-band (at ω = ±ωc and ω = ∞) [SV01] (Figure 26.14):
Z s S s sB s B
z
P
z
P( ) ( )
( )( )( )
= = ++
⇒ = −−
212
212
212
212
ωω
ω ω ωω ω
Ω
Transfer functions from the low-pass frequency S to the frequency s of other types of filters are recog-nized and can be written under the following form:
Z s H s s ss s s
z z zn
p p( ) ( )( ) ( )
( )( ) (= + + +
+ + +
212 2
22 2 2
21
2 22
2 2ω ω ω
ω ω ω…… ppn
2 )
Or Ω( ) ( )( ) ( )( )( ) (
ω ω ω ω ω ω ωω ω ω ω ω
= − − −− −
H z z zn
p p
212 2
22 2 2
21
2 22
2 2…… −− ω pn
2 )
Z(s) has zeros where the desired filter has pass-bands and poles where it has stop-bands. The function Z(s) is called Foster Reactance function. For example, we can write the transfer function of the filter (Figure 26.15) as
Z s Hs ss
Hz
p
z
p( ) ( )
( )( ) ( )
( )= +
+= −
−
2 2
2 2
2 2
2 2ω
ωω ω ω ω
ω ωor Ω
Ω
ω
ω1
ωz0 ωz1ωp1
ω2 ω3
Ω0
–Ω0
FIGURE 26.14 Frequency transformation low-pass to multiple band-pass.
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Analog Filter Synthesis 26-13
The transfer function has zeros at ω = 0, ω = ωz and poles at ω = ωp and ω = ∞.At corner frequencies ω1 = 1 kHz, ω2 = 4 kHz, ω3 = 6 kHz, the values of Ω (ω) are equal to 1, −1, and 1,
respectively. Therefore, the transformation Ω (ω) can be rewritten into multi-equations corresponding to ω = ω1, ω2, ω3. Three equations with three unknowns always have solutions:
1 1 12 2
12 2= −
−H Z
P
ω ω ωω ω( )
− = −−
1 2 22 2
22 2
H Z
P
ω ω ωω ω( )
1 3 32 2
32 2= −
−H Z
P
ω ω ωω ω( )
ω
ω
z
p
H
2
2
22
8
13
=
=
=
; so the Foster Transfer Function iss S s ss
= ++
( / ) ( / )1 3 22 38
3
2
26.4 Summary and Conclusion
Analog filters have been used broadly in communication. Understanding the methods to synthesiz-ing analog filters is extremely important and is the basic step to design analog filters. Four different synthesizing methods were presented, each method will result in different characteristics of filters. Besides that, this chapter also presented steps to design other types of filters from the low-pass filter by writing the frequency transfer function.
References
[SV01] R. Schaumann and M.E. Van Valkenburg, Analog Filter Design, Oxford University Press, Oxford, U.K., 2001.
[W02] S. Winder, Analog and Digital Filter Design, Newnes, Woburn, MA, 2002.
1 kHz
30 dB
2 dB
4 kHzωp ωz 6 kHz
FIGURE 26.15 Frequency transformation by foster reactance function.
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[KAS89] M.R. Kobe, J. Ramirez-Angulo, and E. Sanchez-Sinencio, FIESTA-A filter educational synthesis teaching aid, IEEE Trans. Educ., 32(3), 280–286, August 1989.
[W90] B.M. Wilamowski, A filter synthesis teaching-aid, in: Proceedings of the Rocky Mountain ASEE Section Meeting, Golden, CO, April 6, 1990.
[WG05] B.M. Wilamowski and R. Gottiparthy, Active and passive filter design with MATLAB, Int. J. Eng. Educ., 21(4), 561–571, 2005.
[WLS92] B.M. Wilamowski, S.F. Legowski, and J.W. Steadman, Personal computer support for teaching analog filter analysis and design courses, IEEE Trans. Educ., E-35(4), 351–361, 1992.
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