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 L
j C

21
2
RP I R

21
4
mW I L
 
  
2 2 2
2 2 2
1 1 1 1
4 4 4
e C
C
W 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/sec
m e
R
W W
Q
P
At resonance   0

 

  
2
0
0 0
2
1
22 4
1
2
m
R
I L LW
Q
P RI R

 


  
2
2
0
0 0
2
0
1 1
2
2 4 1
1
2
e
R
I
W C
Q
P CRI R

X
Q
R
where 

 0
0
1
orX 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 L
Q
R R
L R
C R


1X
Q
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 L
j C
    0
Let   0  0
1
LC
 
 
   
 

 
    
           
    
 
2 2
0
22 2
2
0
1 1
1 1inZ R j L R j L R j L
LC
       

 
   
  0 0 0
2
0
2
R j L R jL
 

 
       
 0
2 2
QR
R j L R j

 0L
Q
R
I L R
C
 inZ
CV


where
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3-dB Bandwidth



0
2
BW


 
     
 
0
0
2
2
in
BW QR
Z R j R jBW QR
when 
1
BW
Q
   2inZ R jR R

   0
03-dB Bandwidth BW
Q
 
1
3-dB Bandwidth in % BW
Q
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 C
R j L

2
2
R
V
P
R

21
4
eW V C
 
  
2 2
2
2 2 2
1 1 1
4 4 4
m L
V V
W I L L
L L
At resonance, the angular resonant frequency  0 02 f
m eW W  0
1
LC
  0
1
inY
R
LI
LR C
 inY
V


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Parallel Resonant Circuit
• Quality Factor:
 


average energy stored
energy loss/sec
m e
R
W W
Q
P
At resonance   0

B
Q
G
where 

 0
0
1
orB C
L

1
G
R
Comparisons:


B R
Q
G L

B
Q CR
G
L
R
C
R
and
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3dB Bandwidth
• For parallel resonant circuits, as   0
   

    
1 1 1
2inY j C jC
R j L R



0
2
BWDefine
 




   0
0
1 1
2
2
in
BW Q Q
Y j jBW
R R R R
when 
1
BW
Q
  
1 1 2
inY j
R R R

   0
03-dB Bandwidth BW
Q
 
1
3-dB Bandwidth in % BW
Q
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
LR
Resonant
Circuit
Unloaded Q
• For series RLC circuit
• For parallel RLC circuit
 


 
 
0
0
1
L
L L
L
Q
R R C R R
 

  0
0
//
//L
L L
R R
Q C R R
L
• Define the external Q  eQ


 0
0
1
e
L L
L
Q
R R C
for series RLC


  0
0
L
e L
R
Q 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
LZo 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
sR
jX
jB LY  L L LY G jB
inZ
X : matching reactance
B : matching susceptance
We want to find X and B
 Case (a)   
1
ors s L
L
R R R
G
in sZ R

  

0in s
in s
Z R
Z R
Goal:
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L-Shape Matching – Case (a) Rs < RL
 
  
 
1
in s
L L
Z jX R
G jB jB
Real part:     1s L LR G X B B
Imaginary part:     0s L LR B B XG


sV
sR
jX
  Lj B B LG
lQsQ
Use the resonator concept:
Series RC or RL
Parallel RC or RL
s
s
X
Q
R l

 L
L
B B
Q
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 B
Q Q
R G
lReal part:     1s L LR G X B B
  2
1 1s LR G Q   
1
1
s L
Q
R G
Choose l    
1
1s
s L
Q Q Q
R G
  
1
1s s
s L
X R Q R
R G
and
    
1
1L L L L
s L
B G Q B G B
R 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
sR
jX
jB LZ  L L LZ R jX
inY
X : matching reactance
B : matching susceptance 
1
in
s
Y
R

  

1
0
1
in
s
in
s
Y
R
Y
R
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 1
in
L L s
Y jB
R 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 X
Q
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 X
Q BR Q Q
R
Real part:
 2
L s LQ R R R   1s
L
R
Q
R
Choose l    1s
s
L
R
Q Q Q
R
    1s
L L L L
L
R
X R Q X R X
R
(>0, inductance)
  
1
1s
s s L
RQ
B
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
 s
s
s
X
Q
R
 s s sZ R jX
 p
p
p
R
Q
X



p p
p
p p
R jX
Z
R jX
    ps
s p
s p
RX
Q Q Q
R X

   

p p
s p s s
p p
R jX
Z Z R jX
R jX
  2
1s pR Q R
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Matching Bandwidth (I)
 Case (a)   
1
ors s L
L
R R R
G


sV
sR
jX
jB LY  L L LY G jB
inZ

 

in s
in s
Z R
Z R


sV
sR
jX
  Lj B B LG


sV
sR
jX eqjB
eqR

 2
1 1
1
eq
L
R
G Q
 2
11 L
eq
eq
G Q
B
QR Q

 
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Matching Bandwidth (II)


sV
sR
jX eqjB
eqR
Impedance matching at   0
      0
1
in eq s
eq
Z jX R R
jB
Imaginary part:  
1
0
eq
jX
jB
 1eqXB
Let  0X L  0eqB C  0
1
LC
Real part: eq sR R 
 2
1 1
1
s
L
R
G Q
  
1
1
L s
Q
G R
and ,where
and
Series resonant circuit
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Matching Bandwidth (III)


sV
sR
jX eqjB
eq sR R
inQ

1
2
L inQ Q
• Define QL and Qin for RLC resonator
      2
1in s L
eq
X
Q 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 s
s
eq
jX
jBZ 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 2
2 s sR R
L X
3-dB Bandwidth in % BW



    
0
22 2 1 2
1
1
s
L
s L
R
BW
X Q Q
R G
 Similarly, for case (b)
  

1 2 2
1
1L
s L
BW
Q Q
R G

0

1
2
1
2
 
1
1
s L
Q
R G
 Case (a)

1
s
L
R
G
 1s
L
R
Q
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 1
50s L
Q
R G
s
L
1
R < , 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 2
32%
| | 6.245L
BW
Q Q
 9dB 15dB16%, 8%BW BWand
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Three Elements Matching (High Q)
 Case (a) Pi-Shape Matching


sV
sR
2jX
3jB LY  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
3jB LY  L L LY G jB
VR
1jB
2bjX
VR
 2 2 2a bX X X
VR : virtual resistance (designed by yourself)

1
V
L
R
G
V sR R
Splitting into 2 “L-shape” matching networks


sV
sR
2ajX
3jB LY
 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
3jB LY
 L L LY G jB
1jB
2bjX
VR


sV
VR
in sZ R   0 in VZ R   0
  1 1s
V
R
Q
R
  2
1
1
V L
Q
R G
Since is small, you can get a high QVR
1
1
2
BW
Q
2
2
2
BW
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 RV 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
2ajB VR LZ
 L L LZ R jX
3jX
2bjB


sV
VR
in VZ R   0
  1 1V
s
R
Q
R
  2 1V
L
R
Q
R
Since is large, you can get a high QVR
1
1
2
BW
Q
2
2
2
BW
Q
 1 2min ,BW BW BW
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Department of Electronic Engineering, NTUT
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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Department of Electronic Engineering, NTUT
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
50
1 1 6.247
1.2492
s
V
R
Q
R
       2
1 2000
1 1 40
1.2492V L
Q
R G


1
2000
LG
 Pi-section matching network
     1 2max(| |,| |) max( 50/ 1, 2000/ 1) 2000/ 1v v vQ Q Q R R R
    

2
5% 1.249
(2000/ ) 1
v
v
BW R
R
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Department of Electronic Engineering, NTUT
Example – Pi Matching Network (II)


sV
 50sR
2ajX
1jB
 1.249VR
   100 50inZ f MHz
   100 0f MHz
VR
 1 6.247Q
 1
1 0
s
Q
B C
R
 198.85C pF
 2 1 0a VX R Q L  12.42L nH
 1 6.247Q


 1
1
0
1
s
Q
B
R L
 203.95C pF


 2 1
0
1
a VX R Q
C
 12.74L nH
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Department of Electronic Engineering, NTUT
Example – Pi Matching Network (III)
3jB
2bjX


sV
 1.249VR
   100 1.2492inZ f MHz
   100 0f MHz


1
2000
LG
 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
1
L LB G Q B
L
 79.58L nH


 2 2
0
1
b VX R Q
C
 31.85C pF
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Department of Electronic Engineering, NTUT
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
50
31.85pF
79.58nH
50
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Department of Electronic Engineering, NTUT
Example – Pi Matching Network (V)
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Department of Electronic Engineering, NTUT
Example – T Matching Network (I)


sV
50
1jX
   100 50inZ f MHz
   100 0f MHz
2ajB VR
3jX
2bjB


sV
 T-section matching network
     1 2max(| |,| |) max( ( /50) 1, ( / 2000) 1) ( /50) 1v v vQ Q Q R R R
    

2
5% 80050
( /50) 1
v
v
BW R
R
 80050VR
   100 80050inZ f MHz
   100 0f MHz
 80050VR
LR
 2LR k
       1
80050
1 1 40
50
V
s
R
Q
R
       2
80050
1 1 6.247
2000
V
L
R
Q
R
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Department of Electronic Engineering, NTUT
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
 1
2 0a
V
Q
B C
R
 0.795C pF


 1 1
0
1
sX R Q
C
 0.796C pF


 1
2
0
1
a
V
Q
B
R L
 3.185L H
 1 40Q
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Department of Electronic Engineering, NTUT
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
 2
2 0b
V
Q
B C
R
 0.124C pF


  3 2
0
1
L LX R Q X
C
 0.127C pF


 2
2
0
1
b
V
Q
B
R L
 20.394L H
 2 6.247Q
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Department of Electronic Engineering, NTUT
Example – T Matching Network (IV)
2k0.795pF
3.183 H


sV
50 19.884 H
0.124pF
2k3.185 H
0.796pF


sV
50 19.884 H
0.124pF
2k0.795pF
3.183 H


sV
50 0.127pF
20.394 H
2k3.185 H
0.796pF


sV
50
0.127pF
20.394 H
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Department of Electronic Engineering, NTUT
Example – T Matching Network (V)
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Department of Electronic Engineering, NTUT
Cascaded L-Shape Matching (Low Q)
 Case (a)  
1
s V
L
R R
G


sV
sR
1jX
1jB LY  L L LY G jB
in sZ R
2jX
2jB
  0
VRVR


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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Department of Electronic Engineering, NTUT
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
R
Q
R
  2
1
1
L V
Q
G R
We want to have the maximum bandwidth (find minimum Q)
Let  1 2Q Q Q  s
V
L
R
R
G
 

2 2 2
2 1
1
s L
BW
QQ
R G
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Department of Electronic Engineering, NTUT
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
1jB LZ
 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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Department of Electronic Engineering, NTUT
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
R
Q
R
  2 1V
L
R
Q
R
Let  1 2Q Q Q V s LR R R
2 2 2
2 1
1
s L
BW
QQ
R G
 

We want to have the maximum bandwidth (find minimum Q)
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Department of Electronic Engineering, NTUT
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 2
61.29%
1 2000 50 11
s L
BW
R G
 s V LR R R
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Department of Electronic Engineering, NTUT
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
316.23
1 1 2.3075
50
V
s
R
Q
R
       2
2000
1 1 2.3075
316.23
L
V
R
Q
R
 1 2.3075Q
 1 1 0sX R Q L  183.63L nH
 1
1 0
V
Q
B C
R
 11.61C pF
 1 2.3075Q


 1 1
0
1
sX R Q
C
 13.79C pF


 1
1
0
1
V
Q
B
R L
 218.11L nH
 2 2.3075Q
 2 2 0VX R Q L  1.161L H
  2
2 0L
L
Q
B B C
R
 1.836C pF
 2 2.3075Q


 2 2
0
1
VX R Q
C
 2.181C pF


  2
2
0
1
L
L
Q
B B
R L
 1.379L H
 316.23VR  2LR k
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Department of Electronic Engineering, NTUT
Example – Cascaded L-Shape Matching (III)
2k11.61pF
183.63nH


sV
50 1.161 H
1.836pF
2k218.11nH
13.79pF


sV
50 1.161 H
1.836pF
2k11.61pF
183.63nH


sV
50
2.181pF
1.379 H
2k218.11nH
13.79pF


sV
50
2.181pF
1.379 H
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Department of Electronic Engineering, NTUT
Example – Cascaded L-Shape Matching (IV)
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Department of Electronic Engineering, NTUT
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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Department of Electronic Engineering, NTUT
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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