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微波電路講義8-1
Chapter 8 Microwave filters
8.3 Filter design by the insertion loss method
power loss ratio, maximally flat and equal-ripple LPF
prototypes
8.4 Filter transformations
impedance and frequency scaling, LPF HPF, BPF, BSF
8.5 Filter implementation
Richard’s transformation, Kuroda’s identities
8.6 Stepped-impedance low-pass filters
microstrip LPF
8.7 Coupled line filters
Z- and Y-inverters, microstrip BPF
8.8 Filters using coupled resonators
BSF, BPF
微波電路講義8-2
8.3 Filter design by the insertion loss method
• Power loss ratio (insertion loss)
filter
(lossless)
2
2
2 2
2
2
1
1 ( )
1
( )1
( ) ( )
( )1
( )
inc incLR
L inc R
P PP
P P P
M
M N
M
N
Discussion
1. ( ,available power from source)
for a lossless filter
insertion loss 10log 10log ( 10log ) (12.13)
inc avs in R
in L
incLR T
L
P P P P
P P
PIL P G
P
21( 20log , if 0)
return loss 10log 20log
S L
inc
R
S
PRL
P
Pinc
PLPinPR
PLR
LPF
1
微波電路講義8-3
2.2
2 2
2 2
*
* * *
2 2* *
)( ) is an even function of ( ) (p.173)
) )
( ) : real Re ( ) : , Im ( ) : ( ) ( )
( ) ( ), ( ) ( ), ( ) ( )
( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( )
M(ww w w
M(w N(w
v t V w even V w odd V w V w
I w I w Z w Z w w w
w w w w w w w w w w
2( ) 1 ( ) , :cutoff frequency, 1 ( ) 2 3
stopband attenuation 20 /
N
LR c LR c
c
wP w w P w dB
w
NdB decade
4. Equal ripple (Chebyshev, optimal) LPF
2 2
log(1 )
( ) 1 ( ) , ( ) 1
ripple 10 in passband, stopband attenuation 20 /
frequency responses (p.400, Fig. 8.21)
N
LR N LR c
c
wP w T P w
w
NdB decade
3. Maximally flat (Butterworth, binomial) LPF
微波電路講義8-4
5. Filter design procedure:
specification LPF prototype LPF, HPF, BPF, or BSF
transformation microstrip realization
6. Filter specification:
frequency range, BW, IL, stopband attenuation and frequencies,
input and output impedances, VSWR, group delay, phase
linearity, temperature range, and transient response.
• LPF prototype
go=1 g2
g1 g3 gN+1 wc=1
微波電路講義8-5
go=1 g1 g3
g2 g4 gN+1
Discussion
1. Maximally flat LPF prototype design equations
2
1
( ) /10
10log[1 ( ) ]
(2 1)1, 1, 2sin 1 2
2
log(10 1)
2log
N
c
c o N k
IL w
c
wIL
w
kw g g g k , ,...N
N
Nw
w
wc=1
微波電路講義8-6
2. Element values for maximally flat LPF prototype are given in
Table 8.3 of p.404, frequency response given in Fig. 8.26 of p.405.
3. Equal-ripple LPF prototype design equations
2
( )
10
1
1
( )
1 ( ) /10 10
1
10log[1 ( )]
10 1
cos( cos ) 0
( )
cosh( cosh )
cosh [10 1]/[10 1]
cos ( )
N
c
ripple dB
c
c
N
c
c
ripple dB
IL w
c
wIL T
w
wN w w
wT w
wN w w
w
Nw
w
微波電路講義8-7
NkN
kb
N
ka
Nkbb
aag
N
N even
N oddg
aggw
kk
kk
kkk
Noc
,...2,1 , sin ,2
)12(sin
,...3,2 , 4
]37.17
)1log(10ln[coth
2sinh
,4
coth
,1 ,
2 ,1 ,1
22
1
1
211
1
4. Element values for equal-ripple LPF prototype (p.406, Table 8.4) and
frequency response (p.407, Fig. 8.27)
微波電路講義8-8
8.4 Filter transformations
• Impedance scaling ooo RimpedanceRg 1
• Frequency scaling
:1 ' ( ) ( ) same impedance,
' '
scaled values:
' '
c LR LR
c c
L
c c
C
c c
w ww w P w P w
w w
w LjX j L jwL L
w w
w CjB j C jwC C
w w
scaled values: ' , ' , ' , 'S o L o L o
o
CR R R R R L R L C
R
scaled values: ' , ' , ' , 'oS o L o L
c o c
R L CR R R R R L C
w R w
• Impedance and frequency scaling
微波電路講義8-9
• Frequency scaling for HPF, BPF and BSF
C
L
LPF HPF BPF BSF
Cw
Lw
c
c
1
1
Lw
w
L
o
o
o o
C
w w C
1
o o
L
w L w
o
o
w
C
Cw
1
BWfractional: o
12
w
ww
微波電路講義8-10
Discussion
1. Ex. 8.3 design a maximally flat LPF, fc=2GHz, Zo=50, IL(3GHz)
=15dB
1 5 2 4 3
11 5
22 4
33
from Fig.8.26, 15 for 1 0.5@3 5
from Table 8.3, 0.618, 1.618, 2
' ' 0.98450
50' ' 6.438
' 3.18350
frequency response (p.412, Fig. 8.30) with the
c
c
c
c
wIL dB GHz N
w
C C L L C
CC C pF
w
LL L nH
w
CC pF
w
comparison of
equal-ripple LPF and linear phase LPF
微波電路講義8-11
2. LPF HPF frequency scaling
-1 1
w
PLR
-wc wc
w
PLR
0 ,1 , 1
1 11
11
cc c
cc
c
c
ww w w
w
jwL'L'w CjwC
w CjLPF HPFw
w L C'jwL j w L
w jwC'
微波電路講義8-12
. 3. LPF BPF frequency scaling
-1 1
w
PLR
-w2 -w1 w1 w2
w
PLR
-wo wo
2 12 1 1 2
10 ,1 , 1 ( ), ,
1 1 1
,( )
,( )
oo o
o o
o o
o oo o
o o
o oo o
w w www w w w w w w
w w w
CC' L'w CwC w CjwC
j jw w w Cw w w jwLPF BPF
Lw LwL w L L' C'
jwL j jw w w Lw w w jw
微波電路講義8-13
. 4. LPF BSF frequency scaling
-1 1
w
PLR
-w2 -w1 w1 w2
w
PLR
-wo wo
1 20 ,1 , 1
1 1
1( ),
11 ,
( )
o
o
o
o
o
o oo
o oo o
o o
w w ww w
w w
w jw
jwC jwC C w CjCC' L'w w
w w Cw wLPF BSF
LL' C'
jwL jL w w Lw ww jw
w w jwL Lw
微波電路講義8-14
5. Ex. 8.4 design a N=3, 0.5dB equal-ripple BPF, fo=1GHz, Zo=50,
BW=10%
1 3 2
11 3
1 3
1
2
2
22
from Table 8.4, 1.5963, 1.0967, 1
50' ' 127
' ' 0.19950
50' 0.726
' 34.9150
frequency response (p.415, Fig. 8.33)
L
o
o
o
o
L L C R
LL L nH
w
C C pFw L
L nHw C
CC pF
w
微波電路講義8-15
8.5 Filter implementation
• Richards’ transformation
, tanl = tan ( l/vp)
lumped elements commensurate lines with S.C. or O.C. stub
8
1 1 1
8
cin in
cin in
λZ j L L Z j L L,
λZ C Z ,
j C j C C
Discussion
1.
2. stub response repeats every 4c.8 48 8
c cc cf
c
21 tan tan tan 1
8 4
cc c c
c
λl
λ
微波電路講義8-16
• Kuroda’s identites
2Z
1
U.E.
1Z
U.E.
2
2
n
Z2
1
n
Z
U.E.
2Z1Z
2
2Zn
1
U.E.
1
2Zn
U.E. (unit element) : c/8 line1
22 1Z
Zn
微波電路講義8-17
Discussion
1. Use c/8 redundant lines to separate stubs.
2. series L (short stub) shunt C (open stub)
series C and shunt L change positions
1ZU.E.
2Z
U.E.
2
2
n
Z
U.E.
1Z
2
1
Z
2
2Zn
1
U.E.
1
2Zn
2
1
n
Z
2n:1
1:n2
1
22 1Z
Zn
微波電路講義8-18
3. derivation of (a)
U.E.
1Z
U.E.
2
2
n
Z
2Z
2
1
n
Zl l
l
l
l,Zo
1
21
1cos sin
1sin cos
1 0
open-circuited shunt stub1
1 0
1 1 0
short-circuited shunt stub11
o
o
o
o
o
o
j Zl jZ lA B
jjY l lC D
Z
j
Z
Y
j Z
Y
微波電路講義8-19
Z
1 open-circuited series stub
10 1
0 11
short-circuited series stub0 1
o
o
Z
jZ
j Z
1
2
2 1
1
2 12
1 2 2
2
2 1
22
2
2
1 22
22
2 1
2 2
1 0 1 1
1 11
1 1
1 1 ( 11
111
1 1 0 1
1 ( )
1
1 1
L
R
j ZA B
j jC D
Z Z
j Z
Zj )
Z Z Z
j Zj Z
A B nn
j nC D
Z
jZ Z
n
Zj n
Z Z
2 2
1
1Z
nZ
微波電路講義8-20
4. Microstrp LPF design procedure
series L, shunt C c/8 series short stub, c/8 shunt open stub
add redundant c/8 Zo lines c/8 shunt open stubs
consider discontinuity effects
5. Ex.8.5 design a 3dB equal-ripple LPF with fc=4GHz, N=3,
Zin, Zout=50
L1=3.3487 L3=3.3487
C2=0.7117
Zo=3.3487 Zo=3.3487
Zo=1.405
l = c /8
微波電路講義8-21
Zo=4.35 Zo=4.35
Zo=1 Zo=1Zo=1.299
Zo=1.299Zo=1.405
50 217.5 217.5 50
64.9 70.3 64.9
frequency response (p.421, Fig.8.37)
frequency repeats every 16GHz
S21 @8GHz, 24GHz (l=’/4, 3’/4,…)
l = c/8
(a) (b)
Kuroda’s identity
Zo=1.405
Zo=3.3487 Zo=3.3487
微波電路講義8-22
6. Similar procedures can be used for bandstop filters, but Kuroda
identities are not useful for highpass or bandpass filters. (p.421)
/4 /4
Series capacitor transformation is not
available in Kuroda identities.
/4
微波電路講義8-23
8.6 Stepped-impedance LPF
• Short transmission line section
l,,Zo
,4
h hZ l
2
ltanjZo
lsinj
Zo
lsinjZo 2
ltanjZo
2
ltanj
Zo
2
ltanj
Zo
,4
l lZ l
h hjwL jZ l
1 l
l
Z
jwC j l
low Zo high Zo
Discussion
1. (derivation from notes 4-12, 13)
微波電路講義8-24
l,,Zo cot csc cot csc
,csc cot csc cot
o o
l
Z jZ Y jY
1211 ZZ 1222 ZZ
12Z
12
11 12
2
csc
cos 1(cot csc ) ( )
sin sin
2sin2 tan
22sin cos
2 2
o
o o
o o
Z jZ
Z Z jZ jZ
jZ jZ
1211 YY 1222 YY
12Y
12
11 12
csc
tan1 cos 1 2[ ]
sin sin
o
o o
Y jY
jY Y
jZ Z
微波電路講義8-25
2. Microstrp LPF design procedure
L, C select proper Zh, Zl at cutoff frequency RoL= Zh lh,
Ro/C= Zl / ll lh , ll consider parasitics of L,C and discontinuity
effects
3. Considerations of Zl
4
1
1
1 1 2sin
1sin 2sin open stubs?
l l
ll l
ll l
ZZ C
l CZj l j C
l CZ
4. Considerations of Zh
4
1
2 fabrication ?1
sin sin 1sin capacitor coupling?2
l h
h h h
h h
h
Z LL
jZ l j L l LZ l
Z
微波電路講義8-26
5. Ex.8.6 design a maximally flat LPF with fc=2.5GHz, IL(4GHz)=20
dB, Zin, Zout =50, Zh=120, Zl =20→N=6
L2=1.414 L4=1.932 L6=0.517
C1=0.517 C3=1.932 C5=1.414
lh =RoL/ Zh,
ll =CZl /Ro
frequency response (p.425, Fig.8.41)
no repetition of frequency response
微波電路講義8-27
8.7 Coupled line filters
• Coupled line BPF element
2
2 2( 1 )( 1 )
C
j C j C
1 2
3 4/4
1
C
C 2
2
C1jC
C1jC
2
2
C1j
C1j
4
2 2 2 4 2[ (1 )] 4 4 1C C C C 1
2 2 2 2 4 2[ 2 1 ] 4 (1 ) 4 4j C C C C C C
1 4+ =1
1if ,
2C 1 4=0, =1
微波電路講義8-28
1 2
3 4/4
2
1 1
2 22 1
23 1
24
2
1
2 22 1
23
24
0 1 0 0
1 0 0 0 1
00 0 1
0 00 1 0
0 1 0 0
1 0 0 1
0 0 1
0 1 0
j C Cb a
b j C C j C a
b CaC j C
bC j C
j C Cb
b j C C j C a
b C j C
bC j C
1
2 2
11
2
2 211
0
[ (1 )+ ] 0
0 0=
0 0
[ 1 1 ]
C
Ca
C C a
jaj C C j C C a
微波電路講義8-29
•Impedance and admittance inverters (p.422, Fig. 8.38)
L
2
inZ
KZ
impedance inverter admittance inverter
L
2
inY
JY
,4
K
1
,4J
2 2/ /
o oZ jX Zoo YjBY
2
1
tan ,2
1 ( )
2tan
o
o
o
θ KK Z X
K
Z
X
Z
2
1
tan ,2
1 ( )
2tan
o
o
o
θ JJ Y B
J
Y
B
Y
ZLJ , 90 YLK, 90
2 2/ /
-C
C
-C
1K
C
C
-C-CJ C
微波電路講義8-30
(derivation of impedance inverter)
,4
K
20
cos sin,
sin cos 04
loo
o
Z K jKl jZ lA B
jjY l lC D l
K
2
2
1 0cos sin cos sin cos sin (sin sin )2 2 2 2 2 2
11 11
sin cos sin cos ( sin cos ) cos sin2 2 2 2 2 2
cos sin 02
1 1( sin cos
o oo o o
o
o o o
o
o
Z ZjZ jZ jZ
X X
j j ZjjX
Z Z Z X X
Z
X
jZ X
1
2
2 2
22
2 2
2 (1) (2)tan
1 ( )
) sin cos ...(1)2 tan2 2
2 /2 2tan , tan
sin sin ...(2) 2(sin sin ) 1 ( ) 1 tan222
o
o
o o
oo oo
ooo
o
KX X
KZ
ZZ Zj
K X KK ZX
Z K K ZZ KZjZ jKX ZX Z
2 2/ /
o oZ jX Z Y1 0
1
A B
C D Y
微波電路講義8-31
(derivation of impedance inverter)
,4
K
20
cos sin,
sin cos 04 1
1 1 11 11 0 0
100 1 0 1
loo
o
Z K jKl jZ lA B
jjY l lC D l
KK
CA B j
j C j C CC D j C
j C
Y1 0
1
A B
C D Y
-C
C
-CZ
1
0 1
A B Z
C D
微波電路講義8-32
(derivation of admittance inverter)
1,
4J
2cos sin 0,
sin cos04
loo
o
jY Jl jZ lA B
JjY l lC D l
jJ
2
2
1 11cos sin cos sin cos sin ( sin cos )
12 2 2 2 2 2
0 1sin cos sin cos (sin sin ) cos sin2 2 2 2 2 2
cos sin 02
1 1( sin cos
o
o o o
o oo o o
o
o
Yj jj
Y Y B Y BjB
Y YjY jY jY
B B
Y
B
jY B
1
2
2 2
222 2
2 (1) (2)tan
1 ( )
) sin cos ...(1)2 tan2 2
2 /2 2tan , tan
sin sin ...(2) 2(sin sin ) 1 ( ) 1 tan222
o
o
o o
oo oo
ooo
o
JB B
JY
YY Yj
J B JJ YB
Y J J YY JYjY jJB YB Y
oo YjBY
2 2/ /
Z
1
0 1
A B Z
C D
微波電路講義8-33
(derivation of admittance inverter)
1,
4J
21
0cos sin,
sin cos04
1 11 01 0 1 0
1 10 1 0
loo
o
Y Jl jZ lA B
jJjY l lC D l
jJJ C
A Bj C j C
C D j C j Cj C
Y1 0
1
A B
C D Y
Z
1
0 1
A B Z
C D
C
-C-C
微波電路講義8-34
Discussion
1.
])(1[
])(1[ ,
2 2
2
ooooo
ooooe
JZJZZZ
JZJZZZ
J
-90oZo, Zo,
1 2
3 4
1 1 1 11 1 1 1 1 1
1 1 1 12 1 1 1 1 2
1 13 1 1 1 1 3
For the left circuit: ,
,
,
e e o oe e o o
oe oe oo oo
j j j j j j j je e o oe e o o
oe oe oo oo
ee e o o
oe
V V V VV V V V V I
Z Z Z Z
V V V VV V e V e V e V e I e e e e
Z Z Z Z
V VV V V V V I
Z
1 1
1 1 1 14 1 1 1 1 4,
e o o
oe oo oo
j j j j j j j je e o oe e o o
oe oe oo oo
V V
Z Z Z
V V V VV V e V e V e V e I e e e e
Z Z Z Z
design equation #1
1/2 -1/2
1/2 1/2
微波電路講義8-35
4 4
1 11 14 1 1 411 14
4 14 11 4 1 10 0
2
1 1
112 3 1 1
414
1 1
11 11
41
, ,
( ) cotports 2, 3 open, I I 0 2
I 0
( )csc2
I I
j
o o
oe ooe o
oe oooe oo
e o
oe oo
V Z Z I V VZ Z
V Z Z I I I
V V e jZ Z Z
V V
jZ ZZ Z Z
V V
Z Z
Z Z
ZA B
C D
2 244
14
41
44
41 41
( ) ( ) coscos 2
( )sin
1 sin2 cos
oe oo oe oo oe oo
oe oo oe oo
oe oo
oe oo oe oo
Z Z Z Z Z Z ZZ j
Z Z Z Z Z
Z Z Zj
Z Z Z Z Z Z
微波電路講義8-36
)1(
)1(
22,2
sin2sincos,coscossin)( ""
cossin)(sincos
cossincossin)(
cossin
sincos
0
0
cossin
sincos
22
22
2
2
22
2
22
2
222
ooooo
ooooe
o
o
oooeo
o
oooe
oooe
oooe
o
oooe
oooeoo
oo
o
oo
o
o
o
o
o
ZJJZZZ
ZJJZZZ
JZY
JZZ
J
YJZ
ZZ
ZZ
ZZ
j
J
YjjJ
ZZ
ZZ
J
YJZ
J
YJZ
J
YjjJ
J
jJjZ
J
YJZ
jY
jZ
jJJ
j
jY
jZ
J
-90oZo, Zo,
微波電路講義8-37
2.
Zo, 2C L
2design equation #2 ,
2
o
o o o
ZL C
Z
Z1 Z1 1 : -1
Z2
cot2sin
2cos)1(
2sin
11
21
10
01
11
21
2cos2sin
2sin2cos
1
2
2
1
2
2
1
2
2
2
11
2
1
2
1
2
2
2
11
2
1
o
o
o
o
o
jZZ
jZZ
Z
Z
jZZ
Z
Z
Z
Z
ZZ
Z
Z
Z
Z
Z
Z
ZZ
Z
Z
jY
jZ
微波電路講義8-38
1 : -1
2sin
ojZ
cotojZ cotojZ 1 : -1
w
wjZ oo
oooo
o
w
w
w
w
w
w
w
wwl
c
wl
sin)sin(2sin2
0cot ,2
for
C L
Zin
oooo
oooo
ooo
o
o
o
o
oin
ZwLwC
w
ZL
w
ZjwL
w
jw
Lw
jw
w
Lwwjw
wwww
Lwwj
LCww
Lwwj
LCw
jwL
jwLjwC
Z
2
1,
2
2""
22
)(
1)2(
)(
)(1
)(
11
1
2
2
2
2
2
22
微波電路講義8-39
1
1
11
12
1 ,
2 ,
2
1
NNo
N
iio
i
o ggZJ
ggZJ
gZJ
3. derivation of design equation #3 (8.121)
“N=2” (p.434-435, Fig. 8.45 (e), (f))
J3J2J1Zo
L’1C’1
L’2 C’2
Zo
Yin
微波電路講義8-40
1000 00
lef : right: ,100 0
0 0
o
o o
o
o o
o o
JZjjZ jZ
JZt JjY jY
jJ JZ JZ
J
-90°
1 : ZoJ
Zo, o/4 Zo, o/4
ZoJ:1
J2
-90°
1 : ZoJ3
Zo
ZoJ1:1
C1 L1 C2 L2
Yin
微波電路講義8-41
g1 g3 gNg2
4. Coupled line BPF design
2 1
shunt element shunt element
, ,
series element series element
o i
o i o o
i i
i o o
o i oo
Z g
g ZL C
g Z
g Z
微波電路講義8-42
JN+1J2J1
1 1
1 11
1 1design equation #3 , ,
2 22i N
o o N No i i
J J JZ g Z g gZ g g
JN+1
-90oJ2
-90o
J1
-90o
Zo, 2 Zo, 2Zo Zo
1 2 3 N+1
2
2
[1 ( ) ]πdesign eq. #1 θ ,
2 [1 ( ) ]
oe o o o
oo o o o
Z Z JZ JZ
Z Z JZ JZ
微波電路講義8-43
5. Ex.8.7 design a 0.5dB ripple equal-ripple coupled line BPF, N=3,
fo=2GHz, BW=10%, Zo=50
i gi ZoJi Zoe Zoo
1 1.5963 0.3137 70.61 39.24
2 1.0967 0.1187 56.64 44.77
3 1.5963 0.1187 56.64 44.77
4 1.0000 0.3137 70.61 39.24
frequency response
(p.436, Fig.8.46)
IL(1.8GHz) 20dB
1
1
11
12
1 ,
2 ,
2
1
NNo
N
iio
i
o ggZJ
ggZJ
gZJ
2 2[1 ( ) ], [1 ( ) ]oe o o o oo o o oZ Z JZ JZ Z Z JZ JZ
4. Microstrp coupled line BPF design procedure
LPF prototype gi Ji Zoei, Zooi microstrip line Wi, Si
consider discontunuity effects
微波電路講義8-44
8.8 Filters using coupled resonators
• /4 stub
ooo
o
ZC
ZL
4 ,
4
Zo, o/4 O.C.
L
C
design equation
O.C. line: ,tan 4 4 2 2
( )tan
2 2tan( )
2 2
1 1series LC: ( ) ( )
( )( ) 2( )2 ( )
o o oin
o o o
o oin o o
o o
o
oin
o
o o oo o
o o
Z w ww w c wZ l
j c c f w w
Z w wwZ jZ jZ
w w wj
w
wL L wZ jwL j w LC j
jwC C C w ww LC
w w w w w wL Lj j j L w w Z
C w w C w
4 ow L
4,
4
o
o o o
ZC L
Z
Zo, o/4 S.C. C L
design equation
微波電路講義8-45
1. BSF using /4 open-circuit stub
g1 g3 gNg2
Discussion
J=1/Zo
-90o
J=1/Zo
-90o
J=1/Zo
-90o
微波電路講義8-46
4BSF: open-circuit shunt stub
4
(BPF: short-circuit shunt stub )4 4
ooi
i
ooi
i
ZZ
g
ZZ
g
design equation
Zo, o/4 Zo, o/4 Zo, o/4
Zoi, o/4
Derivation for N=2 is given in p.439-440.
微波電路講義8-47
2. Ex.8.8 design a 0.5dB ripple equal-ripple /4 stub BSF,
N=3, fo=2GHz, BW=15%, Zo=50
i gi Zoi
1 1.5963 265.9
2 1.0967 387
3 1.5963 265.9
frequency response (p.440, Fig.8.49)
3. Design procedure for microstrip BPF or BSF using /4 stub
LPF prototype gi /4 open-circuit or short-circui stubs Zoi
microstrip line consider discontunuity effect
JN+1
90o
J2
90o
J1
90o
Zo, Zo, Zo Zo
4. BPF using capacitively coupled series resonator
微波電路講義8-48
jB2Zo, Zo, Zo Zo
2/2/1 1 2/2/2 2
jB1 jBN+1
2/2/1N 1N
Zo, 1 Zo, 2Zo ZojB1 jB2 jBN+1
1 11 1
2
2 2 ( ), (tan tan )
2 2 2
1 ( )
o i i i io i
o o
ii
o i
B Bl
Y Y
JB
Z J
design
equation
微波電路講義8-49
5. Design procedure for microstrip BPF using capacitively coupled
resonator
LPF prototype gi Ji Bi, i Ci, i microstrip gap width
and line length consider discontunuity effect
6. Ex.8.9 design a 0.5dB ripple equal-ripple BPF using capacitively
coupled resonators, fo=2GHz, BW=10%, Zo=50, IL(2.2GHz) >
20dB
i gi ZoJi Bi(x10-3) Ci(pF) i(deg)
1 1.5963 0.3137 6.96 0.554 155.8
2 1.0967 0.1187 2.41 0.192 166.5
3 1.5963 0.1187 2.41 0.192 166.5
4 1.0000 0.3137 6.96 0.554
frequency response (p.443, Fig.8.51)
微波電路講義8-50
7. BPF using capacitively coupled shunt stub resonator
JN,N +1
-90o
J12
-90o
J01
-90o C1 L1 C2 L2 CN LN
C1 L1 C2 L2 CN LN
C01
-C01
C12
-C12 -C12
CN,N+1
-CN,N+1
01 , 1 , 1
1 11
, ,4 44
o o n n o N N
N Nn n
Z J Z J Z Jg g gg g
design equation
, 1 , 10101 , 1 , 1
2 200 0 01 0 0 , 1
, ,1 ( ) 1 ( )
n n N N
n n N N
N N
J JJC C C
ww Z J w Z J
微波電路講義8-51
C12
Zo, l1C01
Zo, l2CN,N+1
Zo, l2
C’1 L1 C’2 L2 C’N LN
C01 C12 CN,N+1
design equation
'
1, , 1 1, , 1,
4 2
n n n n n n n n n n n n n
o o nn
C C C C C C C C C
Z w Cl
微波電路講義8-52
Zo, /4Cn
YL
Zo, /4
YL
l
L o nY Y jw C
1
1 1 1 tan
tan tan
tan
1tan
1
1tan
o L
L o o L oo
o L
L lo
L
o oL
o
Z jZ lY
Z jZ lZ Z Z jZ lZZ jZ l
Y j llZ
Y jZ ZjY l
Z
2
2
o o o no n
o o
l l Z w Cjw C j j l
Z Z
Y Y
(derivation of the change of stub length due to a shunt capacitor)
微波電路講義8-53
8. Design procedure for BPF using capacitively coupled shunt stub
resonator
LPF prototype gi Ji-1,i Ci Ci li resonator length
consider discontunuity effect
9. Ex.8.10 design a 0.5dB ripple 3rd order equal-ripple BPF using
capacitively coupled shunt resonators, fo=2GHz, BW=10%, Zo=50,
i gi ZoJi-1,i Ci-1,i(pF) Ci(pF) li() l
1 1.5963 0.2218 0.2896 -0.3652 -0.04565 73.6º
2 1.0967 0.0594 0.0756 -0.1512 -0.0189 83.2º
3 1.5963 0.0594 0.0756 -0.3652 -0.04565 73.6º
4 1.0000 0.2218 0.2896
frequency response (p.447, Fig.8.54)
ADS examples: Ch8_prj
