Impedance Matching and Tuning
1
Impedance Matching and Tuningpeda ce atc g a d u g
• Impedance matching or tuning is important for the following reasons:reasons: Maximum power is delivered
Improve the SNR of the system Improve the SNR of the system
Reduce amplitude and phase errors
Lin
Figure 5.1 (p. 223)A lossless network matching an arbitrary load impedance to a transmission line.
2
Impedance Matching and Tuningpeda ce atc g a d u g
Other Discussion Matching network usually use lossless components: L C transmission Matching network usually use lossless components: L, C, transmission
line, transformer, …
There are many possible solutions available There are many possible solutions available
Use Smith chart to find the optimal design
F i h l i f i l hi k Factors in the selection of a particular matching network: Complexity
Bandwidth
Implementation
Adjustability
3
Matching with Lumped Elementsatc g w t u ped e e ts
Figure 5.2 (p. 223)i hi k ( ) k fL-section matching networks. (a) Network for zL
inside the 1 + jx circle. (b) Network for zL outside the 1 + jx circle.
4
Matching with Lumped Elementsatc g w t u ped e e ts
• Analytic Solutions (ZL = RL + j XL)LL ZRjxz 0 )( 1 theinside is :1 Case 0 )( 1 theoutside is :2 Case ZRjxz LL
jXZ01
:condition matching aFor
11
:condition matching aFor
jB
LL jXRjBjXZ0
:parts Re/Im into Separating1
0
:parts Re/Im into SeparatingXXjR
jBZ LL
LLL
LLL
XRBZBXXZRZXXRB
0
00
1
0
00
RBZXXRZXXBZ
LL
LL
LLLLL RZXRZRXB 0
220
:Solution
0
:Solution
XRZRX LLL
L
LL
ZZXX
XRB
00
22
1
0
0
ZRRZ
B LL
5LL BRRB
X
Matching with Lumped Elementsatc g w t u ped e e ts
jB- jB-jX jXLL L
jB jBjX- jX-C C
L CZ Y
L
LL
CC
C
6
L C
Matching with Lumped Elementsatc g w t u ped e e ts
• Smith chart solutionscircle1 theinsideis:1 Case jxzL jL
circle 1 jx
7
Matching with Lumped Elementsatc g w t u ped e e ts
circle 1 theoutside is :2 Case jxzL
i l1 jb circle 1 jb
8
Matching with Lumped Elementsatc g w t u ped e e ts
• Example 5.1 L-Section Impedance MatchingImpedance Matching
MHz 500, 100 , 100200 0
fZjZL
12:1Solution
jzL
jyjjbjyL
5.04.03.0 2.04.0
pF0 92bC
2.1 2.11 jxjz
nH838
pF0.92 2
0
0
ZxL
ZfC
9
nH 8.382 f
L
Matching with Lumped Elementsatc g w t u ped e e ts
• Example 5.1 L-Section Impedance MatchingImpedance Matching
MHz 500, 100 , 100200 0
fZjZL
12:1Solution
jzL
jyjjbjyL
5.04.07.0 2.04.0
pF61.21
C
2.1 2.11 jxjz
nH1.462
p 2
0
0
bf
ZL
Zxf
10
2 bf
Matching with Lumped Elementsatc g w t u ped e e ts
• Example 5.1 L-Section Impedance Matching
Figure 5.3b (p. 227) (b) The two possible L-section matching circuits. ( ) R fl i ffi i i d f f h hi i i f (b)
11
(c) Reflection coefficient magnitudes versus frequency for the matching circuits of (b).
Matching with Lumped Elementsatc g w t u ped e e ts
• Lumped elements (l < /10): parasitic C/L, spurious resonances, fringing fields, loss and perturbations caused by a ground plane.
nH10 nH10
pF 5.0pF 25
12
Matching with Lumped Elementsatc g w t u ped e e ts
frequency single:Bandwidth Estimating
may tuningeApproximatbandwidth
q yg
:ContoursFrequncy !! be Better
impedance of , as theoremreactance '
jXfsFoster
sadmittance and Impedances sadmittance of and
jB
increased. isfrequency as arcsclockwise echart tracSmith on the
13
Matching with Lumped Elementsatc g w t u ped e e ts
• Constant Q circles:BX
RFQGB
RXQ
21
LR
R
RRF
Q
0
L
14
Matching with Lumped Elementsatc g w t u ped e e ts
Q
matching LowBroadband
nRFQ 21
Q=1 074Q=1.074
15
Matching with Lumped Elementsatc g w t u ped e e ts
One-section High Q Matching v. s. 3-sections Low Q Matching
16
Single-Stub TuningS g e Stub u g
jBY i jBYin
(a) No Lumped Elements(b) Easy to fabricate in microstrip or stripline.
jBY 0
jXZ jXZ in
Figure 5.4 (p. 229)
jXZ 0
g (p )Single-stub tuning circuits.
(a) Shunt stub. (b) Series stub.
17
Single-Stub Tuning (Shunt)S g e Stub u g (S u t)
• Example 5.2
GHz 2 , 50 8060
0
fZjZL
S.C. yS.C.
i l1i t ti lSWR8.06.06.12.1
jbjyjz LL
4711for26047.11for 11.0
:circle 1intersectscircle SWR
11
jydjyd
jb
4050S C471095.0S.C.47.1
47.11for 26.0
1
22
ljlj
jyd
18
405.0S.C.47.1 2 lj
Single-Stub Tuning (Shunt)S g e Stub u g (S u t)
• Example 5.2
19
Single-Stub Tuning (Shunt)S g e Stub u g (S u t)
1 jXRYZ
0
0:
1
tjXRjZtjZjXRZZd
jXRYZ
LL
LLLL
1
0for ,tan21
1
1
ttd 0
1tan where
ZjBGYdt
tjXRjZ LL
0for ,tan21 1
BBBt
tt
ss
22
21 where
1
tZXRtRG
ZjBGY
L
11
stub, circuited-openan For
11
BBl so
ss
2200
20
tZXRZ
tZXtXZtRB
tZXR
LLL
LL
stub,circuited-shortaFor
tan2
1tan21
0
1
0
1
YB
Yl so
22
00
00
1 that sochosen is
ZXRZRX
ZYGdtZXRZ LL
tan21tan
21
,
0101
BY
BYl
s
s
0
0
022
0
f2
for
ZRZX
ZRZR
ZXRZRXt L
L
LLLL
2negative is resultant theIf
l
s
20
00 for 2 ZRZXt LL
Single-Stub Tuning (Series)S g e Stub u g (Se es)
• Example 5.3
50 80100
ZjZL
O.C. GHz 2,500
fZ
612 jzz
O.C.
3311for1200 :circle 1 intersects circle SWR
6.12
jzdjx
jzL
3970O C33133.11for 463.033.11for 120.0
22
11
ljjzdjzd
103.0O.C.33.1397.0O.C.33.1
2
1
ljlj
21
Single-Stub Tuning (Series)S g e Stub u g (Se es)
• Example 5.3
22
Single-Stub Tuning (Series)S g e Stub u g (Se es)
1 jBGZY
0
0 :
1
tjBGjYtjYjBGYYd
jBGZY
LL
LLLL
1
0for ,tan21
1
1
ttd 0
1tan where
YjXRZdt
tjBGjY LL
0for ,tan21 1
XXXt
tt
ss
22
21 where
1
tYBGtGR
YjXRZ
L
11
stub, circuited-openan For
0101
ZZlo
ss
2200
20
tYBGY
tYBtBYtGX
tYBG
LLL
LL
stub,circuited-shortaFor
tan21tan
21 0101
XXl
s
o
22
00
00
1 that sochosen is
YBGYGB
YZRdtYBGY LL
tan2
1tan21
,
0
1
0
1
ZX
ZXl ss
0
0
022
0
f2
for
YGYB
YGYG
YBGYGBt L
L
LLLL
2negative is resultant theIf
00
l
23
00 for 2 YGYBt LL
Double-Stub Tuningoub e Stub u g
24
Double-Stub Tuningoub e Stub u g
Figure 5.7 (p. 236)Double-stub tuning. (a) Original circuit with the load an(a) Original circuit with the load an arbitrary distance from the first stub. (b) Equivalent-circuit with load at the first stub.
25
first stub.
Double-Stub Tuningoub e Stub u g
pointon intersecti No:region Forbidden
circle 1 Rotatedwith jb
region forbidden reducingfor reduce d
frequency :2/or 0 d sensitive
8/3or 8/ aschosen generally are
d
26
Double-Stub Tuningoub e Stub u g
• Example 5.4060 80 , 50 LZ j Z
Stubs: open-circuited stubs,/ 8, 2 GHzd f
: a series resistor and capacitorSolution:
LZ
1 1
1.2 1.6 0.3 0.41.314 0.146
L Lz j y jb l
1 1
2
0.114 0.4821 3.381 1 38
b ly j
22
1 1.38y jb 23.38 0.204
1 38 0 350l
b l
27
2 21.38 0.350b l
Double-Stub Tuningoub e Stub u g0.995 pF
0.1460.204
0.995 pF
(c)0.4820.350
Figure 5.9b (p. 239)(b) The two double-stub tuning solutions. (c) Reflection coefficient magnitudes versus (b)
0.4820.350
28
frequency for the tuning circuits of (b).(b)
Double-Stub Tuningoub e Stub u g
• Analytic Solution
st :stub 1 theofleft theJust to
tBtBYt L
2
210
2
1410
:regionForbidden
11
linesion transmislength dBBjGY LL
YtYG
tY
02
2220
10
10
0102
nd :stub 2 theofleft thejust totYBBjGYY LL tGYGtY
BB
dtYG
LL
L
220
20
220
1
sin0
00
1002
fl/1 and tan where
YYZYdtBjBjGtjY LL
YGtGYGtYB
tBB
LLL
L
022
02
0
1
1
210
2
2
02
02
01
ofpart real
tBtBYtYGG
YY
LLL
tGB
L2
Blo 1tan1 :stub O.C.For
2
210
2
2
2
0
220
4111 tBtBYttYG
tt
LL
LL
Yl
Y
s 01
0
tan1 :stub S.C.For
2
29
2220
2012 tYtL
B2
The Quarter-Wave Transformere Qua te Wave a s o e
sec41
1
20 LZZ
Figure 5.10 (p. 241) A single-section quarter-t hi t f t th d i
2f
sec1
0
20
L
L
ZZ
ZZ
wave matching transformer. at the design frequency f0.
40
tZjZ
2near for cos2
0
0 L
L
ZZ
L
L
flttZjZtZjZZZ
1
11in
at2tantanwhere
LL
L
ZZtjZZZZ
ZZZZ
flt
00
0
0in
0in
0
2
at ,2tan tan where
L
LL
ZZZ
ZZtjZZZZ
02
1
000in 2
Figure 5.11 (p. 242) Approximate behavior of the reflection coefficient magnitude for a single-section quarter-wave transformer operating near
30
its design frequency.
The Quarter-Wave Transformere Qua te Wave a s o e
22 :Bandwidth m
2
02 sec
211
ZZZZ L
0
2
0
2cosor
ZZZZ
ZZ
Lmm
Lm
0
2
thenlines, TEM assume weIf1 ZZLm
m
00 24
2 ff
fv
vfl p
p
Fi 5 12 ( 243) R fl ti ffi i t
0
0
2
1
0
2
1cos4242
ZZZZ
ff
L
L
m
mm
Figure 5.12 (p. 243) Reflection coefficient magnitude versus frequency for a single-section quarter-wave matching transformer with various load mismatches
31
m with various load mismatches.
The Theory of Small Reflectionse eo y o S a e ect o s
• Single-Section Transformer jj eTTeTT 42
232112
2321121
n
jnnnj eeTT0
232
2321121
1
jn
n
eTT
xx
x
232112
01for ,
11
j
j
e
eeTT
231
232
321121 1
j
j
e
ee
231
231
31
1
Figure 5.13 (p. 244) Partial reflections and transmissions on a single section matching transformer
31
32
on a single-section matching transformer.
The Theory of Small Reflectionse eo y o S a e ect o s
• Multisection Transformer
Figure 5 14 (p 245) PartialFigure 5.14 (p. 245) Partial reflection coefficients for a multisection matching transformer.
010 ZZ
ZZ
jN
Njj eee
:lsymmetricamadebecanrTransforme
242
210
1
1
01
nnn ZZ
ZZZZ
jN
NjNjjNjNjN
nNNNe
eeeee 2cos2coscos2
:lsymmetrica made becan r Transforme22
10
1
NL
NLN
nn
ZZZZZZ
jNN
nj
NNN
NnNNNe
2cos2coscos2
even for ,21 2cos2coscos2
2/
10
llymonotonica vary :n
NL
Z
N
njN
ZZZ
NnNNNe
d iGi
odd for ,cos 2cos2coscos2
2/)1(
10
33
NZZZ , ,design ,Given 21
The Bode-Fano Criterione ode a o C te oCircuit limit Fano-Bode
Figure 5.22 (p. 262) The Bode-Fano limits for RC and RL loads matched with passive and lossless networks (ω0 is the center frequency of the matching bandwidth). (a) Parallel RC. (b) S i RC
34
(b) Series RC.
The Bode-Fano Criterione ode a o C te oCircuit limit Fano-Bode
Figure 5.22 (p. 262) The Bode-Fano limits for RC and RL loads matched with passive and lossless networks (ω0 is the center frequency of the matching bandwidth). (c) Parallel RL. (d) S i RL
35
(d) Series RL.
The Bode-Fano Criterione ode a o C te o
Figure 5.23 (p. 263)Illustrating the Bode-Fano criterion. (a) A possible reflection coefficient(a) A possible reflection coefficient response. (b) Nonrealizable and realizable reflection coefficient responsesreflection coefficient responses.
:Given 1. RC
fi il00 unless ,0 .2 m
m
or3sfrequencie ofnumber
finite aat only 0
CR
m
matchharder toisloadhighor
or .3
Q
CR
m
36
match harder to is load high Q
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