微波電路講義 6-1 Chapter 6 Microwave Resonators 6.1 Series and parallel resonant circuits series and parallel RLC resonators, quality factor Q 6.2 Transmission line resonators /2 and /4 resonators 6.5 Dielectric resonator concept 6.6 Excitation of resonators coefficient of coupling, gap-coupled microstrip resonator, determine Q u from 2-port measurement
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微波電路講義6-1
Chapter 6 Microwave Resonators
6.1 Series and parallel resonant circuits
series and parallel RLC resonators, quality factor Q
6.2 Transmission line resonators
/2 and /4 resonators
6.5 Dielectric resonator
concept
6.6 Excitation of resonators
coefficient of coupling, gap-coupled microstrip resonator,
determine Qu from 2-port measurement
微波電路講義6-2
6.1 Series and parallel resonant circuits
• series RLC resonator
2
2 2 2
2 2
2 2
2 2 2 2
2
1 1( ) (1 )
(1 )
2 ( ) 1,
/2 /2
1 1 1 1 1, ,
2 4 4 4
in
o o
in loss m eo
loss m e c
VZ R j L R j L
I j C LC
R j L R j L
P P j W W
LCI I
P I R W I L W V C IC
I R L
C+
V
-
inZ
( )inZ
01
3dB BW
R2
R
ω>ωo
ω<ωo
Q大
Q小
ω=ωo
( )inZ
微波電路講義6-3
2
2
average energy storedquality factor ( )
energy loss/second
2 / 42 1( ) , (loss )
/ 2
m e
loss
m oo o o
loss o
W WQ
P
I LW LQ R Q
P R RCI R
微波電路講義6-4
R L C
+
V
-
inZ
)(Z in
01
3dB BW
2R
R
• parallel RLC resonator
I
21 1
2
2 2
2
2 2 2
2
2
2
1 1 1 1 1( ) ( ) [ (1 )] ,
2 2 ( )
2
1 1 1 1 1, ,
2 4 4 4
2 / 42( ) , (loss )
/ 2
in o
o
in loss m e
loss m L e
eo o o o
loss o
Z j CR j L R j L LC
P P j W W
I I
VP W I L V W C V
R L
C VW RQ RC R Q
P LV R
ω<ωo
ω>ωo
Q大
Q小
=ωo
( )inZ
微波電路講義6-5
Discussion
1. At resonance, Wm=We, Pin=Ploss
2 2*2* *
, 2 2* *
2
3 , 3 ,2
3 , 3
Z1 1 1 1 1 1Re[ ]= Re[ ] Re[ ] Re[ ]=
2 2 2 Z 2 2 2
1 1 2series resonator:@ , , ( ) ( )
2 22
parallel resonator:@ , , ( )
inin av in in
in in in inin in
dB in av dB in av o
o
dB in av dB
V IVV IP VI V R I G
Z Z YZ Y
V RQR X P R P R X
R
G B P
2
,2
1 1 2( )
2 22
1( )
2 W
in av o
o
oo
I GQG P G B
G
QB
3. Half-power fractional bandwidth
2. Near resonant frequency2 1
series resonator: 2 , ( ) oin o
o o
LRQZ R j L R j Q
R RC
parallel resonator: , ( )1 2
1 2in o o
o
o
R R RZ Q RC
j RC Lj Q
微波電路講義6-6
(derivation of series resonator case)
2
2 2 2 2 2
1 1( ) ( ) ( )
( ) 2
2
o
o
oin
o o o o
Z R j L R jL R jLj C LC
R jL R jL R jL
R j L
(derivation of parallel resonator case)
1 1
2
1 1
2 2
1
1 1 1 1( ) [ ] [ ( ) ]
( )
1 1 1 1(1 )
(1 / )
1 1 1( ) [ ] [ ]
1[ 2 ]
1 2
o
in o
o
o o o o o o o
in o
o o o
Z j C j CR j L R j L
jZ j C j C j C
R j L j L R L
Rj C
R j RC
微波電路講義6-7
4. Locus of r= ± x on the Smith chart
2 2
2 2 2 2
2 2 2 2
2 2 2
1 1 21
1 1 (1 ) (1 )
1 2 2 1 2
( 1) ( 2)
r i r i i
r i r i r i
r i i r i i
r i
jz r jx j
j
r x
(derivation)
Locus of g = ± b on the Smith chart
(derivation)2 2
2 2 2 2
2 2 2 2
2 2 2
1 1 21
1 1 (1 ) (1 )
1 2 2 1 2
( 1) ( 2)
r i r i i
r i r i r i
r i i r i i
r i
jy g jb j
j
g b
微波電路講義6-8
5. Lossless resonator lossy resonator: (1 )2
o o
j
Q
6. Unloaded Q, Qu, loaded Q, QL, external Q, Qe
1 1 1,
for series RLC circuit
for parallel RLC circuit
L e U
o
L
e
L
o
Q Q Q
L
RQ
R
L
resonator
Qu (Qo)RL
(derivation)
(1 )2
1 1
(1 )2
1
lossless series resonator 2 2( )
2[ (1 )] 2 22
lossless parallel resonator [ 2 ] [ 2( ) ]
2[ (1 )] [ 22
oo o
o o
in o
j LQ
Q Ro
in o
in o
j
Q
in o
Z j L j L
LjZ j L j L R j L
Q Q
Z j C j C
jZ j C j C
Q
1 11] [ 2 ]
oQ RCoC j CQ R
微波電路講義6-9
6.2 Transmission line resonators
• Short-circuited /2 line
Zo, ,
n/2
( )in o
o
Z Z l j
R L C
2
2
1, ,
2o
in
oo
o
Z R jL
ZR Z l L C
L
0
<<1
(derivation)
tanh tanh tan tanh( )
tanh 1 tan tanh
, tan tan( ) , tanh2 2 2 / 2
(
1
L
L oin o o o
o L Z
lpo o o
p p o o o o o
oin o o
o
Z Z l l j lZ Z Z j l Z
Z Z l j l l
vl l l l
v v f
l j
Z Z Z
j l
2) 2 , ( )
2 2 2
oo
o o o oU o
o o
Z
L ll j R j L Q
R Z l l l
微波電路講義6-10
• Open-circuited /2 line
Zo, ,
n/2
o
oin
jl
ZZ
R L C
2
1 2
1, ,
2o
in
o
o o
RZ
j RC
ZR C L
l Z C
(derivation)
1tanh 1 tan tanh
tanh tanh( ) tanh tan
/, ( )
1 2 2 2 21
L
L o o oin o o o
o L Z
o
o o o oU o o o
o o
o o
j lZ Z l Z j l l
Z Z Z ZZ Z l j l l j l
l j
Z Z l ZRQ RC
j RC l Z ll j j
l
微波電路講義6-11
• Short-circuited /4 line
Zo, ,
(2n+1)/4
2
oin
o
ZZ
l j
R L C
2
1 2
1, ,
4o
in
o
o o
RZ
j RC
ZR C L
l Z C
0
(derivation)
tanh tanh tan cot cot tanh 1 tanh( )
tanh 1 tan tanh cot cot tanh
,cot tan( )4 4 4 / 2 2 2 2
L
L oin o o o o
o L Z
po o o
p p o o o o
Z Z l l j l j l j l lZ Z Z j l Z Z
Z Z l j l l j l j l l
vl l
v v f
2
12
, ( )1 2 4 4 2
2 2
o
o o o oin o U o o o
o o
o o
j lZ ZR
Z Z Q RCj RC l Z l
j l l j
微波電路講義6-12
( )2
oU oQ w
Discussion
1. Transmission line resonator
2
7
104.7 air2, ,
2 104.7 2.08 Teflon
0.022 / air1 1 1.84 10
2 ln ( ) , 2
5.813 10 / copper0.022 2.08 0.032 / Teflon
Te
o roU o c d
s
os
c
fQ
c
RNp m
b R
a a bS m
Np m
flon tan 0.03 /2
104 7 2380 air
2 0.022(5 )
104.7 2.081218 Teflon
2 (0.022 0.03)
o r
d
U
kNp m
.
Q GHz
2. Ex. 6.1 /2 coaxial line resonator, b=4mm, a=1mm, f=5GHz,
Teflon r=2.08, tan=0.0004, calculate QUair and QUTeflon