Chapter 2-1
Resonators and Impedance Matching
with Lumped Elements
Chien-Jung Li
Department of Electronics Engineering
National Taipei University of Technology
Department of Electronic Engineering, NTUT
Resonators
• Lump Resonators
Series resonant circuit I R L
C
inZ
CV
1
inZ R j Lj C
21
2RP I R
21
4mW I L
2 2 2
2 2 2
1 1 1 1
4 4 4e C
CW V C I I
C C
At resonant frequency 0 02 f
m eW W 0
1
LC
0inZ R
Resonance occurs when the
average stored magnetic and
electric energies are equal, and
the input impedance is a purely
real impedance.
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Series Resonant Circuit
• Quality Factor:
average energy stored
energy loss/secm e
R
W WQ
P
At resonance 0
2
00 0
2
12
2 41
2
m
R
I LLW
QP R
I R
2
2
00 0
20
1 12
2 4 1
1
2
e
R
IW C
QP CR
I R
X
QR
where
0
0
1 or X L
C
Q is a measure of the loss of
a resonance circuit, and lower
loss implies higher Q.
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Comparisons
X LQ
R R
L R
C R
1XQ
R CR
Although the Q is defined, but the resonant frequency can’t be defined
(they are not resonators).
• For series RL and series RC Circuits
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Define the Bandwidth
• For series resonant circuits, as 0
1
inZ R j Lj C
0Let 0 0
1
LC
2 2
0
22 2
2
0
1 11 1inZ R j L R j L R j L
LC
0 0 0
2
0
2R j L R jL
0
2 2QR
R j L R j
0LQR
I L R
C
inZ
CV
where
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3-dB Bandwidth
0
2BW
0
0
22
in
BW QRZ R j R jBW QR
when 1
BWQ
2inZ R jR R
0
03-dB Bandwidth BWQ
1
3-dB Bandwidth in % BWQ
inZ
0
R
2R
2
• Define the bandwidth
When resonance occurs, the absolute
value of impedance is minimum in the
series resonant circuit.
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Parallel Resonant Circuit
Parallel resonant circuit
1 1
inY j CR j L
2
2R
VP
R
21
4eW V C
2 2
2
2 2 2
1 1 1
4 4 4m L
V VW I L L
L L
At resonance, the angular resonant frequency 0 02 f
m eW W 0
1
LC
0
1inY
R
LI
LR C
inY
V
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Parallel Resonant Circuit
• Quality Factor:
average energy stored
energy loss/secm e
R
W WQ
P
At resonance 0
B
QG
where
0
0
1 or B C
L
1G
R
Comparisons:
B RQ
G L
B
Q CRG
L
R
C
R
and
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3dB Bandwidth
• For parallel resonant circuits, as 0
1 1 1
2inY j C jCR j L R
0
2BWDefine
0
0
1 12
2in
BW Q QY j jBW
R R R R
when 1
BWQ
1 1 2
inY jR R R
0
03-dB Bandwidth BWQ
1
3-dB Bandwidth in % BWQ
inY
0
1
R
2
R
2
When resonance occurs, the absolute
value of admittance is minimum
(maximum impedance) in the parallel
resonant circuit.
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Loaded and Unloaded Q
LRResonant
Circuit
Unloaded Q
• For series RLC circuit
• For parallel RLC circuit
0
0
1L
L L
LQ
R R C R R
0
0
////L
L L
R RQ C R R
L
• Define the external Q eQ
0
0
1e
L L
LQ
R R Cfor series RLC
0
0
Le L
RQ R C
L
1 1 1
L eQ Q Q
for parallel RLC
for both cases
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Impedance Matching
Matching
Network
in sZ R
sV
sR
LZo sZ R
0in
Goal:
• The matching network is ideally lossless, to avoid unnecessary loss of
power, and is usually designed so that the impedance seen looking
into the matching network is Zo (matched with the transmission line) or Rs (matched with the source impedance when no line connected).
• Maximum power is delivered when the load is matched to the line
(assuming the generator is matched), and power los in the feed line is
minimized.
• Impedance matching may be used to improve the signal-to-noise
ratio or amplitude/phase errors in some applications.
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Eight Ell Matching Sections (L-Shape)
LZ1C
2C
LZL
C
LZ1L
2L
LZC
L
LZC
L
LZ2C
1C
LZL
C
LZ2L
1L
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Lump Element Matching
• Two element (L-shape) matching
sV
sRjX
jBLY L L LY G jB
inZ
X : matching reactance
B : matching susceptance
We want to find X and B
Case (a) 1
or s s L
L
R R RG
in sZ R
0in s
in s
Z R
Z R
Goal:
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L-Shape Matching – Case (a) Rs < RL
1in s
L L
Z jX RG jB jB
Real part: 1s L LR G X B B
Imaginary part: 0s L LR B B XG
sV
sRjX
Lj B B LG
lQsQ
Use the resonator concept:
Series RC or RL
Parallel RC or RL
s
s
XQ
Rl
L
L
B BQ
G
When the impedances are matched,
the imaginary part of the input
impedance looking into the matching
network is equal to zero.
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L-Shape Matching – Case (a) Rs < RL
Imaginary part: 0s L LR B B XG
L
s
s L
X B BQ Q
R GlReal part: 1s L LR G X B B
2 1 1s LR G Q 1
1s L
QR G
Choose l 1
1s
s L
Q Q QR G
1
1s s
s L
X R Q RR G
and
1
1L L L L
s L
B G Q B G BR G
(>0, capacitance)
(<0, inductance)
(>0, inductance)
or 1L
s
R
R
When Q is chosen, X and B are also
decided.
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L-Shape Matching – Case (b) Rs > RL
sV
sRjX
jB LZ L L LZ R jX
inY
X : matching reactance
B : matching susceptance 1
in
s
YR
1
01
in
s
in
s
YR
YR
Again, we want to find X and B
Case (b) s LR R
Goal:
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L-Shape Matching – Case (b) Rs > RL
1 1in
L L s
Y jBR jX jX R
Real part: s L s LBR X X R R
Imaginary part: 0L s LX X BR R
Use resonator concept:
s sQ BR l
L
L
X XQ
R
sV
sR Lj X X
jB LR
lQsQ
Parallel
RC or RL
Series RC or RL
When the impedances are matched,
the imaginary part of the input
admittance looking into the matching
network is equal to zero.
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L-Shape Matching – Case (b) Rs > RL
Imaginary part: l
L
s s
L
X XQ BR Q Q
R
Real part:
2
L s LQ R R R 1s
L
RQ
R
Choose l 1ss
L
RQ Q Q
R
1sL L L L
L
RX R Q X R X
R(>0, inductance)
1
1s
s s L
RQB
R R R(>0, capacitance)
(<0, capacitance)
0L s LX X BR R
s L s LBR X X R R
When Q is chosen, X and B are also
decided.
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Series-to-Parallel Transformation
• Series-to-Parallel transformation
sR
sjX
pR
pjX
ss
s
XQ
R s s sZ R jX
p
p
p
RQ
X
p p
p
p p
R jXZ
R jX
ps
s p
s p
RXQ Q Q
R X
p p
s p s s
p p
R jXZ Z R jX
R jX
21s pR Q R
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Matching Bandwidth (I)
Case (a) 1
or s s L
L
R R RG
sV
sRjX
jBLY L L LY G jB
inZ
in s
in s
Z R
Z R
sV
sRjX
Lj B B LG
sV
sRjX eqjB
eqR
2
1 1
1eq
L
RG Q
211 L
eq
eq
G QB
QR Q
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Matching Bandwidth (II)
sV
sRjX eqjB
eqR
Impedance matching at 0
0
1in eq s
eq
Z jX R RjB
Imaginary part: 1
0eq
jXjB
1eqXB
Let 0X L 0eqB C 0
1
LC
Real part: eq sR R 2
1 1
1s
L
RG Q
1
1L s
QG R
and ,where
and
Series resonant circuit
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Matching Bandwidth (III)
sV
sRjX eqjB
eq sR R
inQ
1
2L inQ Q
• Define QL and Qin for RLC resonator
21in s L
eq
XQ R Q G Q Q
R
Find 3-dB bandwidth for
Let X L eqB C
as 0 ,let 0 0
1
LC 0
1
2
1 2 22
eqin s
in s ss
eq
jXjBZ R jL
Z R jL RjX R
jB
and
where and
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Matching Bandwidth (IV)
3-dB bandwidth for 2 occurs when
2 2 sL R
02 22 s sR R
L X
3-dB Bandwidth in % BW
0
22 2 1 2
11
s
L
s L
RBW
X Q Q
R G
Similarly, for case (b)
1 2 2
11L
s L
BWQ Q
R G
0
1
2
12
1
1s L
QR G
Case (a)
1
s
L
RG
1s
L
RQ
R
Case (b)
s LR R
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Example – L-Shape Matching (I)
• Find element values and 3-dB fractional bandwidth of the two-element
matching network.
1 2000
1 150s L
QR G
s
L
1R < , choose case (a)
G
6.245Q
6
6
50 6.245 2 100 10
6.245 / 2000 0 2 100 10
s
L L
X R Q L
B G Q B C
1st Solution: choose
Matching
Network
100in sZ f MHz R
sV
50sR 2000LR
100 0in f MHz
Goal:
LR
LR 4.9696C pF
496.96L nH
sV
50sR
-9
12
496.96 10 ( ) 496.96 ( )
4.9696 10 ( ) 4.9696 ( )
L H nH
C F pF
6.245
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Example – L-Shape Matching (II)
6
6
50 ( 6.245) 1/(2 100 10 )
( 6.245) / 2000 0 1/(2 100 10 )
s
L L
X R Q C
B G Q B L
6.245Q 2nd Solution: choose
12
9
5.097 10 ( ) 5.097 ( )
509.7 10 ( ) 509.7 ( )
C F pF
L H nH
LR
sV
50sR
509.7L nH
5.097C pF
3dB
1 2 232%
| | 6.245L
BWQ Q
9dB 15dB16%, 8%BW BWand
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Three Elements Matching (High Q)
Case (a) Pi-Shape Matching
sV
sR
2jX
3jBLY L L LY G jB
inZ in sZ R
0in s
in s
Z R
Z R
1jB
sV
sR
1jX
LZ L L LZ R jX
inZ in sZ R
0in s
in s
Z R
Z R
2jB
3jX
Case (b) T-Shape Matching
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Pi-Shape Matching
sV
sR
2ajX
3jBLY L L LY G jB
VR
1jB
2bjX
VR
2 2 2a bX X X
VR : virtual resistance (designed by yourself)
1
V
L
RG
V sR R
Splitting into 2 “L-shape” matching networks
sV
sR
2ajX
3jBLY
L L LY G jB
1jB
2bjX
VR
sV
sR
in sZ R 0 in VZ R 0
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Pi-Shape Matching
sV
sR
2ajX
3jBLY
L L LY G jB
1jB
2bjX
VR
sV
VR
in sZ R 0 in VZ R 0
1 1s
V
RQ
R 2
11
V L
QR G
Since is small, you can get a high Q VR
1
1
2BW
Q2
2
2BW
Q
1 2min ,BW BW BW
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T-Shape Matching
VR : virtual resistance (designed by yourself)
sV
sR
1jX
LZ L L LZ R jX
VR
2ajB
3jX
2bjB
VR V LR RV sR R
sV
sR
1jX
LZ
L L LZ R jX
in sZ R 0
2ajB
3jX
VR 2bjB
sV
VR
in VZ R 0
Splitting into 2 “L-shape” matching networks
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T-Shape Matching
sV
sR
1jX
in sZ R 0
2ajBVR LZ
L L LZ R jX
3jX
2bjB
sV
VR
in VZ R 0
1 1V
s
RQ
R 2 1V
L
RQ
R
Since is large, you can get a high Q VR
1
1
2BW
Q2
2
2BW
Q
1 2min ,BW BW BW
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Example – Pi and T Matching Networks
• Use Pi and T matching networks to achieve a matching BW<5%.
Matching
Network
100in sZ f MHz R
sV
50sR 2000LR
100 0in f MHz
Goal:
LR
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Example – Pi Matching Network (I)
sV
50sR
2ajX
3jB1jB
2bjX
1.249VR
sV
1.249VR
100 50inZ f MHz
100 0f MHz
100 1.2492inZ f MHz
100 0f MHz
VR LG
1
501 1 6.247
1.2492s
V
RQ
R 2
1 20001 1 40
1.2492V L
QR G
1
2000LG
Pi-section matching network
1 2max(| |,| |) max( 50 / 1, 2000 / 1) 2000 / 1v v vQ Q Q R R R
25% 1.249
(2000 / ) 1v
v
BW RR
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Example – Pi Matching Network (II)
sV
50sR
2ajX
1jB
1.249VR
100 50inZ f MHz
100 0f MHz
VR
1 6.247Q
11 0
s
QB C
R198.85C pF
2 1 0a VX R Q L 12.42L nH
1 6.247Q
1
1
0
1
s
QB
R L
203.95C pF
2 1
0
1a VX R Q
C
12.74L nH
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Example – Pi Matching Network (III)
3jB
2bjX
sV
1.249VR
100 1.2492inZ f MHz
100 0f MHz
1
2000LG
2 40Q
3 2 0L LB G Q B C 31.83C pF
2 2 0b VX R Q L 79.53L nH
2 40Q
3 2
0
1L LB G Q B
L 79.58L nH
2 2
0
1b VX R Q
C 31.85C pF
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Example – Pi Matching Network (IV)
2k198.85pF
12.42nH
sV
50 79.53nH
31.83pF 2k198.85pF
12.42nH
sV 79.58nH
31.85pF
2k12.74nH
203.95pF
sV
50 79.53nH
31.83pF 2k12.74nH
203.95pF
sV
5031.85pF
79.58nH
50
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Example – Pi Matching Network (V)
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Example – T Matching Network (I)
sV
50
1jX
100 50inZ f MHz
100 0f MHz
2ajBVR
3jX
2bjB
sV
T-section matching network
1 2max(| |,| |) max( ( / 50) 1, ( / 2000) 1) ( / 50) 1v v vQ Q Q R R R
25% 80050
( / 50) 1v
v
BW RR
80050VR
100 80050inZ f MHz
100 0f MHz
80050VR
LR
2LR k
1
800501 1 40
50V
s
RQ
R 2
800501 1 6.247
2000V
L
RQ
R
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Example – T Matching Network (II)
sV
50
1jX
100 50inZ f MHz
100 0f MHz
2ajB 80050VR
1 40Q
1 1 0sX R Q L 3.183L H
12 0a
V
QB C
R 0.795C pF
1 1
0
1sX R Q
C 0.796C pF
1
2
0
1a
V
QB
R L 3.185L H
1 40Q
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Example – T Matching Network (III)
3jX
2bjB
sV
100 80050inZ f MHz
100 0f MHz
80050VR
2LR k
2 6.247Q 3 2 0L LX R Q X L 19.884L H
22 0b
V
QB C
R 0.124C pF
3 2
0
1L LX R Q X
C 0.127C pF
2
2
0
1b
V
QB
R L 20.394L H
2 6.247Q
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Example – T Matching Network (IV)
2k0.795pF
3.183 H
sV
50 19.884 H
0.124pF
2k3.185 H
0.796pF
sV
50 19.884 H
0.124pF
2k0.795pF
3.183 H
sV
50 0.127pF
20.394 H
2k3.185 H
0.796pF
sV
500.127pF
20.394 H
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Example – T Matching Network (V)
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Cascaded L-Shape Matching (Low Q)
Case (a) 1
s V
L
R RG
sV
sR
1jX
1jBLY L L LY G jB
in sZ R
2jX
2jB
0VRVR
sV
VR
2jX
LY
L L LY G jB
2jB
sV
sR
1jX
1jB VR
in sZ R
0
in VZ R
0
Splitting into 2 “L-shape” matching networks
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Cascaded L-Shape Matching (Low Q)
sV
VR
2jX
2Lj B B LG
sV
sR
1jX
1jB VR
in sZ R
0
in VZ R
0
1 1V
s
RQ
R 2
11
L V
QG R
We want to have the maximum bandwidth (find minimum Q)
Let 1 2Q Q Q sV
L
RR
G
2 2 2
2 11
s L
BWQQ
R G
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Cascaded L-Shape Matching (Low Q)
Case (b) s V LR R R
sV
sR
1jX
1jB LZ L L LZ R jX
in sZ R
2jX
2jB
VRVR 0
sV
sR
1jX
1jBLZ
L L LZ R jX
in sZ R
2jX
2jB
0
VR
sV
VR
in VZ R 0
Splitting into 2 “L-shape” matching networks
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Cascaded L-Shape Matching (Low Q)
sV
sR
1jX
1jB
in sZ R
2 Lj X X
2jB
0
VR
sV
VR
in VZ R 0
VR
1 1s
V
RQ
R 2 1V
L
RQ
R
Let 1 2Q Q Q V s LR R R
2 2 2
2 11
s L
BWQQ
R G
We want to have the maximum bandwidth (find minimum Q)
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Example – Cascaded L-Shape Matching (I)
• Use Cascaded L-shape matching networks to achieve BW>60%.
Matching
Network
100in sZ f MHz R
sV
50sR 2000LR
100 0in f MHz
Goal:
LR
Choose RV to let
50 2000 316.23V s LR R R
2 261.29%
1 2000 50 11s L
BW
R G
s V LR R R
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Example – Cascaded L-Shape Matching (II)
sV
50sR
1jX
1jB
100 50inZ f MHz
2jX
2jB
100 0f MHz
VR
sV
316.23VR
100 0f MHz
100 316.23inZ f MHz
LR
1
316.231 1 2.3075
50V
s
RQ
R
2
20001 1 2.3075
316.23L
V
RQ
R
1 2.3075Q
1 1 0sX R Q L 183.63L nH
11 0
V
QB C
R11.61C pF
1 2.3075Q
1 1
0
1sX R Q
C13.79C pF
1
1
0
1
V
QB
R L 218.11L nH
2 2.3075Q
2 2 0VX R Q L 1.161L H
22 0L
L
QB B C
R1.836C pF
2 2.3075Q
2 2
0
1VX R Q
C 2.181C pF
2
2
0
1L
L
QB B
R L1.379L H
316.23VR 2LR k
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Example – Cascaded L-Shape Matching (III)
2k11.61pF
183.63nH
sV
50 1.161 H
1.836pF
2k218.11nH
13.79pF
sV
501.161 H
1.836pF
2k11.61pF
183.63nH
sV
502.181pF
1.379 H
2k218.11nH
13.79pF
sV
502.181pF
1.379 H
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Example – Cascaded L-Shape Matching (IV)
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Matching For Larger Bandwidth
L-Shape
Matching
Network
sV
50sR
LR
L-Shape
Matching
Network
L-Shape
Matching
Network
• Use multi-section L-shape matching networks to extend BW.
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Summary
• Impedance matching is a procedure of transforming the complex load impedance ZL to match with the source impedance Zs or characteristics impedance Zo of the transmission line (Generally, Zs = Rs = Zo).
• The lossless lumped element matching networks that are commonly used include L-section, Pi-section, T-section, and cascaded L-section. L-section has medium but not adjustable matching bandwidth. Pi and T sections have rather small and adjustable matching bandwidth. Cascaded L-section can increase bandwidth as the number of stages increases.
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