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EMLAB
Chapter 8. Microwave Filters
1
EMLAB
Common filter responses
Low-pass filterHigh-pass filter
Band-pass filter Band-reject filter
Passive filter 2
EMLAB
Lumped element filter synthesis
1|||| 221
211 SS
2221 |)(||)(| HS
21111
*1111
211 |)(|1)()()()(|)(| HSSSSS
)(1
)(1)(
11
110
S
SZZin
Filter
H(s)
Filter
H(s)
+Vout
-
SR
LR+
Vin-
: conservation of energy
3
EMLAB
Example #1- 1st order low pass filter
2
1
2
1
22
1
1)(
p
V
VH
2
1
2
121111
1
|)(|1)()(
p
pHSS
1
111
1)(
pjpj
S
1p 1p
Pole 의 부호는 (–) 인 것을 선택한다(+) 인 것은 physically unrealizable
010
10
11
110
11
110
21
1,
21
1
1
)(1
)(1
Zpj
Z
p
jZ
pj
pj
pj
pj
ZS
SZZin
4
EMLAB
V_ACSRC1
LL1
2V
1V
1
02
p
ZL
pF63.7
nH159
MHz10021
C
L
p
m1freq=dB(S(2,1))=-3.000
99.85MHz
1E6
1E7
1E8
1E9
1E10
1E5
2E10
-20
-15
-10
-5
-25
0
freq, Hz
dB(S
(2,1
))
Readout
m1
01
2
ZpC
Circuit realization
CC1
V_ACSRC1
0Z
0Z
5
EMLAB
Example #2 – 3rd order Butterworth filter
6
1
6
1
2
1
22
1
1
1
1)(
ps
p
V
VsH
6
1
6
121111
1
|)(|1)()(
ps
ps
sHsSsS
122
)(
1
2
1
3
1
3
1
3/4
11
3/2
1
3
111
ps
ps
ps
ps
eps
eps
eps
ps
sSjjj
1p 1p
1222
122
,
122
1222
122
122
)(1
)(1
1
2
1
3
1
1
2
1
1
2
1
1
2
1
3
13
11
2
1
3
1
3
11
2
1
3
1
11
11
0
ps
ps
ps
ps
ps
ps
ps
ps
ps
ps
ps
ps
ps
ps
ps
ps
ps
ps
S
S
Z
Zin
1
12
1
1,
1
12
1
1
1
1
1
1
10
psp
sps
psp
sp
s
Z
Zin
6
EMLAB
1
0
p
Z
01
2
Zp
1
0
p
Z
0Z
0Z
01
1
Zp
1
02
p
Z
01
1
Zp 0Z
0Z
pF8.31 pF8.31
nH159
m1freq=dB(S(2,1))=-2.873
99.85MHz
1E6
1E7
1E8
1E9
1E10
1E5
2E10
-70
-60
-50
-40
-30
-20
-10
-80
0
freq, Hz
dB(S
(2,1
))
99.85M-2.873
m1
pF7.63
nH6.79
nH6.79
Example #2 – 3rd order Butterworth filter 7
EMLAB
Low pass filter proto-types : Ladder type
1,1 01 Zp 로 정한 필터
Value of n
g1 g2 g3 g4 g5 g6 g7
1 2.000 1.000
2 1.414 1.414 1.000
3 1.000 2.000 1.000 1.000
4 0.7654 1.848 1.848 0.7654 1.000
5 0.6180 1.618 2.000 1.618 0.618 1.000
6 0.5176 1.414 1.932 1.932 1.414 0.5176 1.000
NNsV
VsH
22
2
1
22
1
1
1
1)(
Butterworth filter
8
EMLAB
Filter transformation1. Impedance transformation
000 1 RZZ
00
Inductor.
LRLsLRZsLZ
a
0
01
Capacitor.
R
CC
sC
RZ
sCZ
b
2. Frequency transformation
]/[frequency offcut ]/[1frequency offcut sradsrad c
cc
LLL
sZsLZ
a
Inductor.
c
c
CC
Cs
ZsC
Z
b
11
Capacitor. dB0
21S
dB3 10log
]/[1 sraddB0
21S
dB3 10log
]/[ sradC
FilterFilter
1
1
FilterFilter
0R
0R
9
EMLAB
Microstrip realization – Stepped impedance filter
Example 8.6
10
EMLAB
lYlZ
B
lZl
ZX
00
00
sin1
2
1
2tan
2
10 Z
10 Z
,0Z
lljY
ljZl
DC
BA
cossin
sincos
0
0
l
1Z
3Z
2Z
CZZ
C
DZZZ
C
AZZZ
1
1
1
123
12222
12111
00
03
02
01
1,
sin
12
tan
2tan
ZY
ljYZ
ljZZ
ljZZ
1l
Transmission line → T- network 11
EMLAB
Chevyshev filter
)(1
1)(
22
2
1
22
NTV
VsH
)()(2)(
34)(
12)(
)(
21
33
22
1
xTxxTxT
xxxT
xxT
xxT
nnn
12
EMLAB
ssH
1
1)(
LPF-to-HPF transformation
dB0
y
dB3
dB/dec20
111
1)'(
s
s
s
sH
dB0
y
dB/dec20
치환로 '
1
ss
10log
1c
LPF→HPF
. 1 지점임되는이s. 1 지점임되는이s
1c
dB3
13
EMLAB
RR1
V_ACSRC1
LL1
RR2
CC1
LPF→HPF
치환로 '
1
ss
Lss
LsL
1'
1
'1
1 LL1
11
1
LC
LsC
s
sC'
'1
CC1
LL1
CC1
RR2
V_ACSRC1
RR1
CC1
RR2
V_ACSRC1
RR1
11
1
CL
RR1
LL1R=
RR2
V_ACSRC1
14
EMLAB
ssH
1
1)(
dB0
y
dB3
dB/dec20
''
1
1)'(
0
0
0
ss
BW
sH
dB0
y
dB3
dB/dec20
치환로 '
' 0
0
0
s
s
BWs
10log
LPF→BPF
지점되는이 1'
' 0
0
0
s
s
BWs
jBW
j
BWj
BWj
j
j
BW
js
12
2
2122
20
20
220
2
0
0
20
2
1
'
dB/dec20
2' 1'
LPF-to-BPF transformation
1c
frequencyCenter;210
js js 1j2j
15
EMLAB
120
10
0
011
'
1'
'
'
LBW
s
sBW
L
s
s
BWLsL
LL1
'1
'
1
''
111200
0
0
sBW
Cs
BWC
ss
BW
CsC
치환로 '
' 0
0
0
s
s
BWs
LL1
CC1
CC1
RR1
V_ACSRC1
LL1
RR2 V_AC
SRC1
RR2R=50 Ohm
CC1
LL1
RR1
CC1
RR2
V_ACSRC1
RR1
CC1
LL1 R
R2V_ACSRC1
RR1
16
EMLAB
ssH
1
1)(
dB0
y
dB3
dB/dec20
''
11
1)'(
0
0
0
ss
BW
sH
dB0
y
dB3
dB/dec20
치환로
''
1
0
0
0
ss
BW
s
10log010log
LPF→BPF
dB/dec20
LPF-to-BRF transformation
1c
2' 1'
지점되는이 1'
' 0
0
0
s
s
BWs
jBW
j
BWj
BWj
j
j
BW
js
12
2
2122
20
20
220
2
0
0
20
2
1
'
frequencyCenter;210
js js 1j 2j
17
EMLAB
'1
'1
1
''
1
1
20
1
0
0
0
11
sBWLs
BWLss
BW
LsL
LL1
'
1'
1
'
'11 200
0
0
sCBWs
CBWs
s
BWCsC
치환로
''
1
0
0
0
ss
BW
s
LL1
CC1
CC1
RR1
V_ACSRC1
LL1
RR2
CC1
RR2
V_ACSRC1
RR1
CC1
LL1
RR2
V_ACSRC1
RR1
RR2
RR1
CC1
LL1
V_ACSRC1
18
EMLAB
Real capacitor characteristics
유효 영역
19
EMLAB
Real inductor characteristics
유효 영역
20
EMLAB
8.5 Filter implementation
LjjZ )( ljLjZ tan)(
p
ll
tantan
Richard’s transformation
8
1tan 1
l
l때일
CjjZ
1
)( ljY
jZtan
1)(
0
21
EMLAB
Kuroda’s identities
22
EMLAB
Figure 8.36a (p. 409)Filter design procedure for Example 8.5. (a) Lumped-element low-pass filter prototype. (b) Using Richard’s transformations to convert inductors and capacitors to series and shunt stubs. (c) Adding unit elements at ends of filter.
Example 8.523
EMLAB
Figure 8.36b (p. 410)(d) Applying the second Kuroda identity. (e) After impedance and frequency scaling. (f) Microstrip fabrication of final filter.
24
EMLAB
Figure 8.37 (p. 411)Amplitude responses of lumped-element and distributed-element low-pas filter of Example 8.5.
25
EMLAB
Figure 8.38a/b (p. 412)Impedance and admittance inverters. (a) Operation of impedance and admittance inverters. (b) Implementation as quarter-wave transformers.
Impedance and admittance inverter 26
EMLAB
1
1
aC
g1aC
1g
2g
3g
1
10 g
1
1
aC
g2g1
3
3
aC
g3aC
3
3
aC
g1
LZ
1g
K
21
K
g KLZ
inZ inZ
L
L
in Zsg
KZ
Ksg
KZ
1
221
2
1
121
aCK
g
Low pass filter using admittance inverter
LZLin Z
KZ
2
K
C 값이 여러 가지이므로 제작하기 어려움 .하나의 C 값으로 통일 .
L, C 가 모두 필요하므로 제작하기 어려움 . L 을 제거 .
27
EMLAB
LZ
2g2K
2aC2K
LZinZ inZ
La
La
La
in
ZK
sCK
K
ZKK
sCK
K
ZKK
sC
KK
Z21
222
21
22
21
222
21
21
21
2
22
21
1
aCK
L
2
1K 1K
L
in
Zsg
Z1
1
2
L
in
Zsg
Z1
1
2
2
221 ,1
aC
gKK
2aCLZ1
2
2
aC
g
2
2
aC
g1
28
EMLAB
1aC
1
1
aC
g
1aC 1
1
aC
g1
3
3
aC
g
3aC 3
3
aC
g1
2aC1
2
2
aC
g
2
2
aC
g1
1
1
aC
g
1aC 21
21
aa CC
gg1
3aC 3
3
aC
g1
2aC 32
32
aa CC
gg
1
1
aC
g
1aC 21
21
aa CC
gg1
3
3
aC
g1
32
32
aa CC
gg
1aC1aL 2aL 3aL
LPF→BPF
120
11
1
10
0
01
,
'
'
aa
aa
aa
C
BWL
BW
CC
Cs
s
BWsC
치환로
29
EMLAB
Band pass filter implementation using resonators30
EMLAB
Admittance inverter equivalents 31
EMLAB
2
21S
Resonator coupling & resonant freq. split 32
EMLAB
)(
1,
)(
121
m
r
m
rCCLCCL
C
C
CCCCL
CC
LC
CCCCL
CCCCC
CCLCCL
m
mm
m
mm
mm
mm
rr
02
21
))((
))((
/1/1
)(
1
)(
1
2102,10
11
gg
BW
CKC
Cm
Inter-resonant coupling - 등가 회로 33
EMLAB
2aC1
1
aC
g
1aC 21
21
aa CC
gg1
3
3
aC
g1
32
32
aa CC
gg
3aC1aL 2aL 3aL
eKC
2,1K1
1L eKL
C
Inter-resonator coupling
2102,10
11
gg
BW
CK
120
11
1 ,a
aa
a C
BWL
BW
CC
약하게 coupling 시킨다 .
34
EMLAB
C
1eK 1L
eKC
L2eK 2
eK
102
0
3 221
g
BW
C
L
KQ
BW
e
dB
100
3 2
g
BWBW dB
External Q
동일한 간격의 gap
35
EMLAB
2100
1
gg
BW
약하게 coupling 시킨다 .
Gap 을 변화시키면서 resonant freq. 의 차이가Proto-type 에 의해 정해진 값이 나오게 한다 .
m1freq=dB(S(1,2))=-13.401
140.0MHz
m2freq=dB(S(1,2))=-5.794
143.6MHz
139 140 141 142 143 144 145 146 147 148138 149
-60
-40
-20
0
-80
10
freq, MHz
dB(S
(1,2
))
Readout
m1
Readout
m2dB
(S(1
,1))
Example – Inter-resonator coupling 36
EMLAB
m1freq=dB(S(2,1))=-2.963
142.4MHz
m2freq=dB(S(2,1))=-0.001
141.9MHzm1freq=dB(S(2,1))=-2.963
142.4MHz
m2freq=dB(S(2,1))=-0.001
141.9MHz
139 140 141 142 143 144 145 146 147 148138 149
-30
-20
-10
-40
0
freq, MHz
dB(S
(2,1
))
142.4M-2.963
m1
141.9M-513.7u
m2dB
(S(1
,1))
100
3 2
g
BWBW dB
동일한 간격의 gap
Gap 을 변화시키면서 3dB bandwidth 가 Proto-type 에 의해 정해진 값이 나오게한다 .
dBBW3
Example – External coupling 37
EMLAB
f0 맞추기
Coupling
계수 구하기
Qex 구하기
Filter design by simulator38
EMLAB
12-pole hair pin filter39