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Design of RF and Microwave Filters
초고주파공학
(교재*의 5장)
서강대학교 전자공학과윤상원 교수
* “RF Circuit Design: Theory and Applications”, R. Ludwig & P. Bretchko
차 례
1. Introduction ; types of Filters --------------------------------------2. Characterization of Filters ------------------------------------------3. Approximate Design Methods --------------------------------------4. Lowpass Prototype Network ---------------------------------------5.1. Impedance Scaling ------------------------------------------------5.2. frequency Expansion ----------------------------------------------5.3. Lowpass to highpass transformation -----------------------------5.4. Lowpass to bandpass transformation ----------------------------5.5. Lowpass to bandstop transformation -----------------------------5.6. Immitance Inverters ----------------------------------------------5.7. Bandpass filters using J-, K- inverters ---------------------------6.1. LC filters -----------------------------------------------------------6.2. Distribute filters ---------------------------------------------------
34815181921222527283340
Microwave & Millimeter-wave Lab. 2
1. Introduction ; types of Filters --------------------------------------2. Characterization of Filters ------------------------------------------3. Approximate Design Methods --------------------------------------4. Lowpass Prototype Network ---------------------------------------5.1. Impedance Scaling ------------------------------------------------5.2. frequency Expansion ----------------------------------------------5.3. Lowpass to highpass transformation -----------------------------5.4. Lowpass to bandpass transformation ----------------------------5.5. Lowpass to bandstop transformation -----------------------------5.6. Immitance Inverters ----------------------------------------------5.7. Bandpass filters using J-, K- inverters ---------------------------6.1. LC filters -----------------------------------------------------------6.2. Distribute filters ---------------------------------------------------
34815181921222527283340
1. Introduction
Types of Filters
A. Lowpass Filters B. Highpass Filters
C. Bandpass Filters D. Bandstop Filters
attenuation
passband transitionband
stopband
freq
attenuation
passbandtransitionband
stopband
freqcutoffwc ; cutoff
Microwave & Millimeter-wave Lab. 3
Types of Filters
A. Lowpass Filters B. Highpass Filters
C. Bandpass Filters D. Bandstop Filters
attenuation
passband transitionband
stopband
freq
attenuation
passbandtransitionband
stopband
freqcutoffwc ; cutoff
atten
pass-band
transitionband
stop-band
freq
atten
pass-band
transitionband
stop-band
freqf1
stop-band
transitionband
f2
pass-band
transitionband
f1 f2
2. Filter Characterization
n Two-port Network ;
H(w)Input Output )()()( wqww jeHH =
Microwave & Millimeter-wave Lab. 4
H(w)Input Output
Fig. 1 Two-port Network
)()()( wqww jeHH =
1
Freq.
lH(w)l
q(w)
Characteristics of ideal bandpass filters ;
îíì
><££
=21
21
, 0 1
)(fffffor
fffforH w dand wtwq -=)(
Microwave & Millimeter-wave Lab. 5
Fig. 2 Characteristics of
ideal bandpass filter
1
Freq.
lH(w)l
q(w)
→ not realizable→ approximation required
n Practical specifications ;
1) Passband
; lower cutoff frequency - upper cutoff frequency
2) Insertion loss
; must be as small as possible
3) Return Loss
; degree of impedance matching
4) Ripple
; variation of insertion loss within the passband
2f
)( )(log20 dBH w1f
Microwave & Millimeter-wave Lab. 6
1) Passband
; lower cutoff frequency - upper cutoff frequency
2) Insertion loss
; must be as small as possible
3) Return Loss
; degree of impedance matching
4) Ripple
; variation of insertion loss within the passband
)( 1log20 dBr
5) Group delay
; time to required to pass the filter
6) Skirt frequency characteristics
; depends on the system specifications
7) Power handling capability
wwqt
dd
d)(
-=
Microwave & Millimeter-wave Lab. 7
5) Group delay
; time to required to pass the filter
6) Skirt frequency characteristics
; depends on the system specifications
7) Power handling capability
3. Approximate Design Methods
1) based on Amplitude characteristics
A. Image parameter method B. Insertion loss method
a) J-K inverters
b) Unit element - Kuroda identity
2) based on Linear Phase characteristics
Microwave & Millimeter-wave Lab. 8
1) based on Amplitude characteristics
A. Image parameter method B. Insertion loss method
a) J-K inverters
b) Unit element - Kuroda identity
2) based on Linear Phase characteristics
3.1 Filter Design based on the insertion loss
Definition of Power Loss Ratio (PLR) ; impedance matching as well as frequency selectivity
[Sij]Pin
Prefl
Ptrans
Fig. 3 General filter network
Microwave & Millimeter-wave Lab. 9
[Sij]Pin
Prefl
Ptrans
Fig. 3 General filter network
← network synthesis procedures are required
inintrans
ininrefl
PSPTP
PSPP2
212
211
2
==
=G=
)()(1
11
2 ww
DN
PP
Ptran
inLR +=
G-=º
Approximation methods :
1) Maximally Flat (Butterworth) response
2) Chebyshev response
3) Elliptic Function response
Microwave & Millimeter-wave Lab. 10
Approximation methods :
1) Maximally Flat (Butterworth) response
2) Chebyshev response
3) Elliptic Function response
A. Maximally flat response
11.50.5 10 w/w c
PLR
Chebyshev
Maximally flat
Where, ;passband tolerance
; order of filter
Usually
→ degree of freedom=1 (order N)
N
cLR kP
221 ÷÷
ø
öççè
æ+=
ww
3.2 Approximation Methods
Microwave & Millimeter-wave Lab. 11
11.50.5 10 w/w c
PLR
Chebyshev
Maximally flat
Where, ;passband tolerance
; order of filter
Usually
→ degree of freedom=1 (order N)
Fig. 4 Comparison Between Maximally Flat and Chebyshev response
2kN
12 =k
B. Chebyshev response : equal ripple response in the passband
: Chebyshev Polynomial of order÷÷ø
öççè
æ+=
0
221ww
NLR TkPNT N
Microwave & Millimeter-wave Lab. 12
)()(2)(34)( ,12 ,)(
21
33
221
xTxxTxTxxxTxTxxT
nnn -- -=-=-==
; ripple (0.01 dB, 0.1 dB, etc.)
; order of filter
→ degree of freedom=2 (ripple and order)
2kN
10 w/wc
PLR
Chebyshev Response, N=4-1
1+k2
ws w
as
ar
wp
Elliptic function response N=5
attenuation
Microwave & Millimeter-wave Lab. 13
10 w/wc
PLR
Chebyshev Response, N=4-1
1+k2
ws w
as
ar
wp
Elliptic function response N=5
attenuation
Fig. 5 Chebyshev and Elliptic Function response
C. Elliptic Function responseequal ripple passband in both passband and stopband
: stopband minimum attenuation : transmission zero at stopband
degree of freedom=3 (order N, ripple, transmission zero at stopband )
sa
sw
Microwave & Millimeter-wave Lab. 14
C. Elliptic Function responseequal ripple passband in both passband and stopband
: stopband minimum attenuation : transmission zero at stopband
degree of freedom=3 (order N, ripple, transmission zero at stopband )sw
4. Lowpass Prototype Filter
; normalized to 1
...
...
R gN
g0=1g1
g2
g3g5
g4g6a
a'
sradgRL / 1 , 1 c0 =W== w
Microwave & Millimeter-wave Lab. 15
...
...
R gN
g0=1g1
g2
g3g5
g4g6a
a'
...
...
R
gNg0=1
g1
g2
g3g5
g4g6
a
a'
g7
Fig. 5 Lowpass prototype
n Maximally Flat response ;
n Equal Ripple response ;
11 02 ==®+= gRP LN
LR w
),( , ... 2, 1, ,2
12sin2 FHNiNigi =-
= p
Microwave & Millimeter-wave Lab. 16
n Maximally Flat response ;
n Equal Ripple response ;
ïî
ïíì
+-+==®+=
even 1212
odd 1,1)(1
22022
Nkkk
NRgTkP LNLR w
÷÷ø
öççè
æ
++
-+==
-==
--
-
1111ln ,
2sinh ,
212sin ,
42
22
11
1
kk
Nb
Nia
gbaa
g iiii
iii bbp
TypeElement No
Butterworth0.1 dB rippleChebyshev
0.5 dB rippleChebyshev
1 0.6180 1.1468 1.7058
Table1. Element values for Butterworth and chebyshev filters (n=5)
Microwave & Millimeter-wave Lab. 17
2 1.6180 1.3712 1.2296
3 2.0000 1.9750 2.5408
4 1.6180 1.3712 1.2296
5 0.6180 1.1468 1.7058
5. Impedance Scaling and Frequency Mapping
5.1 Impedance Scaling
Impedance level × 50; same reflection coefficient maintained
series branch (impedance) elements ;
shunt branch (admittance) elements ;
W=®W= 50 1 LL RR
Microwave & Millimeter-wave Lab. 18
5.1 Impedance Scaling
Impedance level × 50; same reflection coefficient maintained
series branch (impedance) elements ;
shunt branch (admittance) elements ;
( )iiii gggjgj 5050 ®® ww
( )50/50/ rrrr gggjgj ®® ww
5.2 Frequency Expansion
cutoff frequency 1 → lowpass cutoff frequency
mapping function ;series and shunt branch elements ;
cw
( )iciici gggjgj wwww ®®
www cf =)(
Microwave & Millimeter-wave Lab. 19
( )iciici gggjgj wwww ®®
PLR
w'1-1
PLR
wwc-wc
PLR
wc-wc
w
PLR
w-w1
-w0-w2w0w1 w2
(a) Lowpass Prototype response
(d) Lowpass to Bandpass Transformation
(b) Frequency expansion
(c) Lowpass to Highpass transformation
Microwave & Millimeter-wave Lab. 20
PLR
w'1-1
PLR
wwc-wc
PLR
wc-wc
w
PLR
w-w1
-w0-w2w0w1 w2
(a) Lowpass Prototype response
(d) Lowpass to Bandpass Transformation
(b) Frequency expansion
(c) Lowpass to Highpass transformation
Fig. 6 Various mapping relations derived from lowpass prototype network
5.3 Lowpass to Highpass transformation
(lowpass cutoff freq. 1 → highpass cutoff freq. ) mapping function ; series branch (impedance) elements ;
shunt branch (admittance) elements ;
cwwww /)( cf -=
( ))/(1 )/( iciici gggjgj wwww ®-®
Microwave & Millimeter-wave Lab. 21
...
...
R
gN' RL=1
g1'g3'g5'
g4' g2'
( ))/(1 )/( rcrrcr gggjgj wwww ®-®
Fig. 7 Highpass filter derived from lowpass prototype
5.4 Lowpass to bandpass transformation
(low cutoff freq. , high cutoff freq. )
mapping function ;
1w 2w
÷÷ø
öççè
æ-
-=
ww
ww
www
w 0
012
0)(f
12210
21
0
and
,1'0'
wwwwww
wwwwwww
-=D=®
±±=®±=±=®=
Microwave & Millimeter-wave Lab. 22
12210
21
0
and
,1'0'
wwwwww
wwwwwww
-=D=®
±±=®±=±=®=
n series branch element : impedance
n shunt branch element : admittance
ss
iiii Cj
Ljjggjgjggj
ww
www
ww
ww
ww
www 1 ;
200
0
01 +=
D+
D®÷÷
ø
öççè
æ-
D=
pp
rrrrr Lj
Cjjggjgjggj
ww
www
ww
ww
ww
ww
w 1 ; 200
0
0 +=D
+D
®÷÷ø
öççè
æ-
D=
Microwave & Millimeter-wave Lab. 23
n series branch element : impedance
n shunt branch element : admittance
pp
rrrrr Lj
Cjjggjgjggj
ww
www
ww
ww
ww
ww
w 1 ; 200
0
0 +=D
+D
®÷÷ø
öççè
æ-
D=
...
...
R
CN RL=1
C1L1L3L5
C4L4
C5
C2
C3
L2LN
Fig. 8 Bandpass filter derived from the lowpass prototype
Example : Design a bandpass filter having a 0.5dB equal-ripple response, with N=3. The f0 is 1GHz, bandwidth is 10%, and the input and output impedance 50Ω.
step 1 : from the element values of lowpass prototype
(0.5dB ripple Chebyshev)
step 2 : apply impedance scaling
step 3 : apply bandpass transformation
0000.1 ,5963.1 ,0967.1 ,5963.1 4321 ==== gggg
Microwave & Millimeter-wave Lab. 24
Example : Design a bandpass filter having a 0.5dB equal-ripple response, with N=3. The f0 is 1GHz, bandwidth is 10%, and the input and output impedance 50Ω.
step 1 : from the element values of lowpass prototype
(0.5dB ripple Chebyshev)
step 2 : apply impedance scaling
step 3 : apply bandpass transformation
HZgLFZgCHZgL 815.79 , 022.0/ , 815.79505963.1 031022011 =====´==
012
022
202
3101
3011
/)( 91.34/' 726.0/'
' 199.0/'' 127/'
wwww
www
-=D=D==D=
==D===D=
pFCCnHCL
CpFLCLnHLL R=50 W
RL=50 W
L3'=127nH
C3'=0.199pF
L2'=0.726nH C2'=34.91pF
L1'=127nH
C1'=0.199pF
5.5 Lowpass to bandstop transformation
(low cutoff freq. , high cutoff freq. )
mapping function ;
inverse of bandpass mapping function
1w 2w1
0
00
12)(-
÷÷ø
öççè
æ-
-=
ww
ww
www
wf
Microwave & Millimeter-wave Lab. 25
12210
21
0
and
,1'0'
wwwwww
wwwwwww
-=D=®
±±=®±=±=®=
n series branch element : admittance
n shunt branch element : impedance
ss
iiii Lj
Cjjgg
jgjggjw
www
ww
www
ww
www 1 ;
20
-1
0
001 +=
D+
D®÷÷
ø
öççè
æ-
D=
pp
rrrrr Cj
Ljjggjgjggj
ww
www
ww
ww
ww
www 1 ;
20
-1
0
00
+=D
+D
®÷÷ø
öççè
æ-
D=
Microwave & Millimeter-wave Lab. 26
pp
rrrrr Cj
Ljjggjgjggj
ww
www
ww
ww
ww
www 1 ;
20
-1
0
00
+=D
+D
®÷÷ø
öççè
æ-
D=
Fig. 9 Bandstop network derived from the lowpass prototype
...
...
R
CN
RL=1C1
L1L3L5 C4
L4
C5
C2
C3
L2LN
5.6 Immitance Inverters
n K ; impedance inverter →
n J ; admittance inverter →
K(or J)
immittanceinverter
ZL(or YL)Zin(or Yin)
Fig. 10 Immitance inverter
Lin ZKZ /2=
Microwave & Millimeter-wave Lab. 27
n K ; impedance inverter →
n J ; admittance inverter →
ex. simplest form of inverter : λ/4 transformer
series LC → J-inverter + shunt LC shunt LC → K-inverter + series LC
Lin ZKZ /2=
Lin YJY /2=
5.7 Bandpass filters using J-, K-inverters
g0 Rn+1
LosslessLowpassNetwork
Zin(w) or Glow
gn+1 R0
Lossless Bandpass Network
Zin'(w) orGband
Fig. 11 Equivalent Network for lowpass prototype and bandpass network
Reflection coefficient ;
lowpass :
bandpass :
If (mapping relation)
G
Microwave & Millimeter-wave Lab. 28
Reflection coefficient ;
lowpass :
bandpass :
If (mapping relation)
G
1/)'(1/)'(
0
0
+-
=GgZgZ
in
inLow w
w
1/)(1/)(
0
0
+-
=GRZRZ
in
inBand w
w
)()'(/)(/)'( 00 wwww BandLowinin RZgZ G=G®=
...
...
R0
g1 gn+1
gn
gn-
1
g5
g4
g3
a
a'
g2
Zin(w')
Microwave & Millimeter-wave Lab. 29
...
...
R0
Rn+1
CnLnL2L1
L4
C1 C2
K01 K12 Kn,n+1
Zin(w)
Fig. 12 Lowpass network and bandpass network
n From the partial fraction expansion including bandpass mapping relation
: fractional bandwidth, : center frequency
n In the same manner, J-inverter values are derived as
1
011,
1
12
01,
10
10001 , ,
+
++
+
++ ===
nn
nnnn
ii
iiii gg
LWRK
ggLL
WKggLWR
Kwww
W 0w
Microwave & Millimeter-wave Lab. 30
n From the partial fraction expansion including bandpass mapping relation
: fractional bandwidth, : center frequency
n In the same manner, J-inverter values are derived as
1
011,
1
12
01,
10
10001 , ,
+
++
+
++ ===
nn
nnnn
ii
iiii gg
CWGJ
ggCC
WJggCWG
Jwww
n Typical immittance inverters ;
CK w/1= LK w=
-L
L
-L-C
C
-C
Microwave & Millimeter-wave Lab. 31
Fig. 13 Impedance(K-) inverters
CK w/1= LK w=
X=negative
F
F=positive
Z0 X=positive
F
F=negative
Z0
CJ w= LJ w/1=
-C
C
-C -L
L
-L
Microwave & Millimeter-wave Lab. 32
B=positiveY0
F/2 F/2
F=negative
Y0
B=negative
F/2 F/2
F=positive
Fig. 14 Admittance(J-) inverters
6.1. LC filters
A. C-coupled bandpass filters
...
...
Y0
Yb
CnLnL2L1 L4
C1C2J01 J12 Jn,n+1
Fig. 14 Bandpass filter network using ideal J-inverters
Microwave & Millimeter-wave Lab. 33
Fig. 14 Bandpass filter network using ideal J-inverters
...
...
Y0
Yb
CnLnL2L1C1
C2J01 Jn,n+1
J-inverter
-C12
C12
Fig. 15 Bandpass filter network containing practical inverters
YaL1
C1-Ca'J01 Ya L1C1
Yin1Yin2
C01
Ca'
Fig. 16 Inverter of first and last stages
( ) ( )201
012
01
201
2
012
201
1
/1/1/
/1/11
'
aa
a
ain
aa
in
YCCj
YCYC
CjYY
CjYJ
Y
ww
ww
w
w
++
+=
+=
+=
Microwave & Millimeter-wave Lab. 34
( ) ( )201
012
01
201
2
012
201
1
/1/1/
/1/11
'
aa
a
ain
aa
in
YCCj
YCYC
CjYY
CjYJ
Y
ww
ww
w
w
++
+=
+=
+=
By equating the real and imaginary part of and1inY 2inY
aa YCifCJCC <<== 01010101 ,' ww
B. L-coupled bandpass filter
......
Zb
Cn+1C1 C3C2
Cp1CpnCp2
Lp1 Lp2 Lpn
Za
Fig.17 C-coupled Bandpass filter
Microwave & Millimeter-wave Lab. 35
B. L-coupled bandpass filter
......
Zb
Ln+1L1 L3L2
Cp1CpnCp2
Lp1 Lp2 Lpn
Za
Fig.18 L-coupled Bandpass filter
Example : Design a LC bandpass filter. The f0 is 2.8 GHz, bandwidth is 500 MHz, and the input and output impedance 50Ω.
step 1 : from the element values of lowpass prototype
step 2 : apply impedance scaling
step 3 : apply bandpass transformation using J-inverters
Step 4 : simulation
Microwave & Millimeter-wave Lab. 36
Step 5 : Realization
0.5 pF 0.5 pF
6.8
nH
air-coil
2.7
nH
chip 1 p
F1.5
pF
1 pF 0.5 pF
6.8
nH
air-coil
2.7
nH
chip 1 p
F1.5
pF
1 pF
6.8
nH
air-coil
2.7
nH
chip
0.5
pF
5 p
F
0.5 pF 1 pF
6.8
nH
air-coil
2.7
nH
chip
0.5
pF
5 p
F
1 pF
Microwave & Millimeter-wave Lab. 37
Insertion loss < 3.1 dB
Return loss > 15.5 dB
Attenuation @ 3.3 GHz : 15 dB
Step 6. improvement
20 pF9.5 nH air-coil 9.5 nH air-coil 6.8 nH air-coil
6.8
nH
air-coil
6.8
nH
air-coil
6.8
nH
air-coil
0.5
pF
1 p
F
1 p
F
1 p
F
1 pF 1.5 pF 0.5 pF 0.5 pF 0.5 pF 0.5 pF
6.8
nH
air-coil
6.8
nH
air-coil
6.8
nH
air-coil
2.7
nH
chip
2.7
nH
chip
2.7
nH
chip
0.5
pF 1
pF
1.5
pF
Microwave & Millimeter-wave Lab. 38
20 pF9.5 nH air-coil 9.5 nH air-coil 6.8 nH air-coil
6.8
nH
air-coil
6.8
nH
air-coil
6.8
nH
air-coil
0.5
pF
1 p
F
1 p
F
1 p
F
1 pF 1.5 pF 0.5 pF 0.5 pF 0.5 pF 0.5 pF
6.8
nH
air-coil
6.8
nH
air-coil
6.8
nH
air-coil
2.7
nH
chip
2.7
nH
chip
2.7
nH
chip
0.5
pF 1
pF
1.5
pF
C-couplingLC filter
L-couplingLC filter
+ =
Microwave & Millimeter-wave Lab. 39
27 dB
6.2 Distributed filters
At microwave frequencies :
Resonators made of Lumped elements are lossy(low Q) or bulky → Distributed Resonators
Distributed resonators ; quarter-wavelength or half-wavelength transmission lines such as microstrip lines, coaxial lines and waveguides
Microwave & Millimeter-wave Lab. 40
At microwave frequencies :
Resonators made of Lumped elements are lossy(low Q) or bulky → Distributed Resonators
Distributed resonators ; quarter-wavelength or half-wavelength transmission lines such as microstrip lines, coaxial lines and waveguides
A. Combline filters: cellular base stations as well as handy phone
Fig. 17(a) Top View of Combline Filter
conductor
air or ceramic
a
Microwave & Millimeter-wave Lab. 41
Fig. 17(a) Top View of Combline Filter
conductor
air or ceramic
a
Fig. 17(b) Side View of Combline Filter
conductor tuning screw
L
Fig. 19 (a) Top view of Combline Filter
Fig. 19 (b) Side view of Combline Filter
n Instead of lumped element inductors distributed inductors (L < λ/4) are used.
n Overall equivalent circuit :
In
out
Fig. 18 Coupled line
Yoe
Yoe
YooIn OutYoe Yoe
(Yoe-Yoo)/2
Fig. 20 Coupled line
Microwave & Millimeter-wave Lab. 42
n Instead of lumped element inductors distributed inductors (L < λ/4) are used.
n Overall equivalent circuit :
In OutYoe Yoe
(Yoe-Yoo)/2
L1 L2 L4L3
Lc1 Lc5Lc4Lc3Lc2
C1 C2 C3 C4
Cc1 Cc2 Cc3
Fig. 20 Coupled lineFig. 21 Equivalent circuit of Fig.20
Fig. 22 Equivalent circuit of Fig. 19
B. Microstrip filters: Compact, light weight and low cost
Fig. 21 Side-couple microstrip filter
Microwave & Millimeter-wave Lab. 43
Fig. 23 Side-coupled Microstrip filterFig. 21 Side-couple microstrip filter
n Practical specifications ;
1) Passband
; lower cutoff frequency - upper cutoff frequency
2) Insertion loss
; must be as small as possible
3) Return Loss
; degree of impedance matching
4) Ripple
; variation of insertion loss within the passband
2f
)( )(log20 dBH w1f
Microwave & Millimeter-wave Lab. 44
1) Passband
; lower cutoff frequency - upper cutoff frequency
2) Insertion loss
; must be as small as possible
3) Return Loss
; degree of impedance matching
4) Ripple
; variation of insertion loss within the passband
)( 1log20 dBr
5) Group delay
; time to required to pass the filter
6) Skirt frequency characteristics
; depends on the system specifications
7) Power handling capability
wwqt
dd
d)(
-=
Microwave & Millimeter-wave Lab. 45
5) Group delay
; time to required to pass the filter
6) Skirt frequency characteristics
; depends on the system specifications
7) Power handling capability