Impedance Matching and Tuning
1
Impedance
peda ce Matching
atc g and
a d Tuning
u g
• Impedance matching or tuning is important for the following
reasons:

Maximum power is delivered

Improve the SNR of the system

Reduce amplitude and phase errors
in
L
Figure 5.1 (p. 223)
A lossless network matching an arbitrary load impedance to a transmission line.
2
Impedance
peda ce Matching
atc g and
a d Tuning
u g

Other Discussion

Matching network usually use lossless components: L
L, C
C, transmission
line, transformer, …

There are many possible solutions available

Use Smith chart to find the optimal design

F
Factors
iin the
h selection
l i off a particular
i l matching
hi network:
k

Complexity

Bandwidth

Implementation

Adjustability
3
Matching
atc g with
w t Lumped
u ped Elements
e e ts
Figure 5.2 (p. 223)
L-section
i matching
hi networks.
k (a)
( ) Network
k for
f zL
inside the 1 + jx circle.
(b) Network for zL outside the 1 + jx circle.
4
Matching
atc g with
w t Lumped
u ped Elements
e e ts
• Analytic Solutions (ZL = RL + j XL)
Case 1 : z L is inside the 1  jx ( RL  Z 0 )
Case 2 : z L is outside the 1  jx ( RL  Z 0 )
For a matching condition :
For a matching condition :
1
Z 0  jX 
jB  1 RL  jX L 
1
1
 jB 
Z0
RL  j  X  X L 
Separating into Re/Im parts :
Separating into Re/Im parts :
 B XRL  X L Z 0   RL  Z 0

 X 1  BX L   BZ 0 RL  X L
Solution :
 BZ 0  X  X L   Z 0  RL

 X  X L   BZ 0 RL
Solution :
X L  RL Z 0 RL2  X L2  Z 0 RL
B
RL2  X L2
X   RL Z 0  RL   X L
Z
1 X Z
X  L 0 0
B
RL
BRL
B
Z 0  RL 
RL
Z0
5
Matching
atc g with
w t Lumped
u ped Elements
e e ts
- jX
- jB
jX
jX
- jB
L
L
C
C
- jX
jB
jB
Y
Z
L
C
C
L
C
L
L
C
6
Matching
atc g with
w t Lumped
u ped Elements
e e ts
• Smith chart solutions
Case 1 : z L is inside the 1  jjx circle
1  jx circle
7
Matching
atc g with
w t Lumped
u ped Elements
e e ts
Case 2 : z L is outside the 1  jx circle
1  jb circle
i l
8
Matching
atc g with
w t Lumped
u ped Elements
e e ts
• Example 5.1 L-Section
Impedance Matching
Z L  200  j100 , Z 0  100 ,
f  500 MHz
Solution 1 :
z L  2  j1
y L  0.4  0.2 j   jb  j 0.3
y  0.4  0.5 j
z  1  j1.2  x  j1.2
b
C
 0.92
0 92 pF
2 f Z 0
L
x Z0
 38.8 nH
2 f
9
Matching
atc g with
w t Lumped
u ped Elements
e e ts
• Example 5.1 L-Section
Impedance Matching
Z L  200  j100 , Z 0  100 ,
f  500 MHz
Solution 1 :
z L  2  j1
y L  0.4  0.2 j   jb   j 0.7
y  0.4  0.5 j
z  1  j1.2  x   j1.2
1
C
 2.61 ppF
2 f x Z 0
 Z0
L
 46.1 nH
2 f b
10
Matching
atc g with
w t Lumped
u ped Elements
e e ts
• Example 5.1 L-Section Impedance Matching
Figure 5.3b (p. 227) (b) The two possible L-section matching circuits.
( )R
(c)
Reflection
fl i coefficient
ffi i magnitudes
i d versus frequency
f
for
f the
h matching
hi circuits
i i off (b).
(b)
11
Matching
atc g with
w t Lumped
u ped Elements
e e ts
• Lumped elements (l < /10): parasitic C/L, spurious resonances, fringing
fields, loss and perturbations caused by a ground plane.
 10 nH
 0.5 pF
 25 pF
12
Matching
atc g with
w t Lumped
u ped Elements
e e ts
Estimating Bandwidth :
single
g frequency
q
y
 bandwidth
Approximate tuning may
be Better!!
Frequncy Contours :
Foster' s reactance theorem
 as f , jX of impedance
and jB of admittances 
 Impedances and admittances
on the Smith chart trace clockwise
arcs as frequency is increased.
13
Matching
atc g with
w t Lumped
u ped Elements
e e ts
• Constant Q circles:
X B

R G
1  Q 2  FR
Q

R 
 FR  0 
RL 

14
Matching
atc g with
w t Lumped
u ped Elements
e e ts
Broadband 
Low Q matching
1  Q 2  n FR
Q=1 074
Q=1.074
15
Matching
atc g with
w t Lumped
u ped Elements
e e ts
One-section High Q Matching v. s. 3-sections Low Q Matching
16
Single-Stub
S
g e Stub Tuning
u g
Yini   jB
(a) No Lumped Elements
(b) Easy to fabricate in
microstrip or stripline.
 Y0  jB
Z in   jX
 Z 0  jX
Figure
g
5.4 (p
(p. 229))
Single-stub tuning circuits.
(a) Shunt stub. (b) Series stub.
17
Single-Stub
S
g e Stub Tuning
u g (S
(Shunt)
u t)
• Example 5.2
S.C.
Z L  60  j80 
Z 0  50 , f  2 GHz
S.C.
y
z L  1.2  j1.6  y L  0.6  j 0.8
SWR circle
i l intersects
i t
t 1  jb circle
i l :
d1  0.11 for y1  1  j1.47
d 2  0.26 for y2  1  j1.47
 j1.47  S.C.  l1  0.095 
 j1.47  S.C.
S C  l2  0.405 
18
Single-Stub
S
g e Stub Tuning
u g (S
(Shunt)
u t)
• Example 5.2
19
Single-Stub
S
g e Stub Tuning
u g (S
(Shunt)
u t)
Z L  1 YL  RL  jX L
1
1
tan
t,
for t  0
d  2

  1   tan 1 t , for t  0
 2
t  Bs Bs   B 

RL  jX L   jZ 0t
 d : Z  Z0
Z 0  j RL  jX L t

where t  tan d
Y  G  jB  1 Z


For an open - circuited stub,
RL 1  t 2
where G  2
2
RL   X L  Z 0t 


d is chosen so that G  Y0  1 Z 0

X L  RL Z 0  RL   X L2 Z 0
t
for RL  Z 0
RL  Z 0
2
 B   1 1  B 
1
tan  
tan 1  s  
 2
 Y0  2
 Y0 
For a short - circuited stub,,
lo
RL2t  Z 0  X Lt  X L  Z 0t 
B
2
Z 0 RL2   X L  Z 0t 



 1 1  Y0  1
1  Y0 


tan  

tan   
 2
B
 Bs  2
If the resultant l is negative    2
ls
t   X L 2Z 0 for
f RL  Z 0
20
Single-Stub
S
g e Stub Tuning
u g (Se
(Series)
es)
• Example 5.3
Z L  100  j80 
Z 0  50 ,
O.C.
O.C. f  2 GHz
z
z L  2  j1.6
SWR circle intersects 1  jx circle :
d1  0.120 for z1  1  j1.33
d 2  0.463 for z 2  1  j1.33
 j1.33  O.C.
O C  l1  0.397 
 j1.33  O.C.  l2  0.103 
21
Single-Stub
S
g e Stub Tuning
u g (Se
(Series)
es)
• Example 5.3
22
Single-Stub
S
g e Stub Tuning
u g (Se
(Series)
es)
YL  1 Z L  GL  jBL
d:
1
1
tan
t,
for t  0
d  2

  1   tan 1 t , for t  0
 2
t  X s X s   X 

GL  jBL   jY0t
Y  Y0
Y0  j GL  jBL t

where t  tan d
Z  R  jX  1 Y


For an open - circuited stub,
GL 1  t 2
where R  2
2
GL  BL  Y0t 


d is chosen so that R  Z 0  1 Y0

BL  GL Y0  GL   BL2 Y0
t
for GL  Y0
GL  Y0
2
 1 1  Z 0  1
Z 
 
tan 1  0 
tan 
 2
X 
 X s  2
For a short - circuited stub,,
lo
GL2t  Y0  BLt BL  Y0t 
X
2
Y0 GL2  BL  Y0t 



 1 1  X 
1
1  X s 



tan  

tan 

 2
 Z 0  2
 Z0 
If the resultant l is negative    2
ls
t   BL 2Y0 for
f GL  Y0
23
Double-Stub
oub e Stub Tuning
u g
24
Double-Stub
oub e Stub Tuning
u g
Figure 5.7 (p. 236)
Double-stub tuning.
(a) Original circuit with the load an
arbitrary distance from the first stub.
(b) Equivalent-circuit with load at the
first stub.
25
Double-Stub
oub e Stub Tuning
u g
Forbidden region :
No intersection point
with Rotated 1  jb circle
 reduce d for reducing
forbidden region
d  0 or  / 2 : frequency
sensitive
d are generally chosen
as  / 8 or 3 / 8
26
Double-Stub
oub e Stub Tuning
u g
• Example 5.4
Z L  60  j80 , Z 0  50 
Stubs: open-circuited stubs,
d   / 8, f  2 GHz
Z L : a series resistor and capacitor
Solution:
zL  1.2  j1.6  yL  0.3  j 0.4
b1  1.314  l1  0.146
b1  0.114  l1  0.482
y2  1  j 3.38
y2  1  j1.38
1 38
b2  3.38  l2  0.204
b2  1.38
1 38  l2  00.350
350
27
Double-Stub
oub e Stub Tuning
u g
0.995 pF
0.204
0.146
0.995 pF
(c)
Figure 5.9b (p. 239)
(b) The two double-stub tuning solutions.
(c) Reflection coefficient magnitudes versus
frequency for the tuning circuits of (b).
0.350
0.482
(b)
28
Double-Stub
oub e Stub Tuning
u g
Forbidden region :
• Analytic Solution st
Just to the left of the 1 stub :
Y1  GL  j BL  B1 
length d transmission line
2
0  1

2 2
1


Y0  1  t 2 GLY0  GL2t 2
B1   BL 
t

where t  tan d and Y0  1 / Z 0
part off Y2   Y0
1  t 2 Y0  BLt  B1t 
G  GLY0 2 
0
2
t
t
2
4t 2 Y0  BLt  B1t  
1 t 2 
1  1 

 GL  Y0
2
2
2
2
2t 

Y0 1  t

2
L


Y 1 t
2
0
Y0
1 t 2
 0  GL  Y0 2 
t
sin 2 d
 just to the left of the 2 nd stub :
GL  j BL  B1  Y0t 
Y2  Y0
Y0  j t GL  j BL  j B1 
reall
4t 2 Y0  BLt  B1t 


 Y0 1  t 2 GLY0  GL2t 2  GLY0
B2 
GL t
lo
1
1  B 
For O.C. stub : 
tan  
 2
 Y0 
For S.C. stub :
ls


 1 1  Y0 
tan  
2
B
29
Thee Quarter-Wave
Qua te Wave Transformer
a so e

Figure 5.10 (p. 241) A single-section quarterwave matching
t hi transformer.
t
f
th design
d i
  0  4 att the
frequency f0.
Z in  Z1

1
 4Z 0 Z L  2
1 
sec 
2
 Z L  Z 0  
Z L  Z0
2 Z0Z L
cos for
f  near  2
Z L  j Z1t
Z1  j Z L t
where t  tan  l  tan     2 , at f 0 

Z
2
1
Z in  Z 0
Z L  Z0

Z in  Z 0 Z L  Z 0  j 2t Z 0 Z L
 Z0Z L

Figure 5.11 (p. 242) Approximate behavior of
the reflection coefficient magnitude for a singlesection quarter-wave transformer operating near
its design frequency.
30
Thee Quarter-Wave
Qua te Wave Transformer
a so e


Bandwidth :   2   m 
2

 2 Z0Z L

1

 2  1
sec 
 Z L  Z0

m


2
m
2 Z0Z L
or cos m 
1  m2 Z L  Z 0
If we assume TEM lines, then
2 f v p  f
 l

v p 4 f0 2 f0
 
2 Z0Z L
4 m
4
f
1
m

 2
 2  cos
f0


 1  m2 Z L  Z 0
Fi
Figure
5.12
5 12 (p.
( 243) Reflection
R fl ti coefficient
ffi i t
 magnitude versus frequency for a single section quarter-wave matching transformer
 with various load mismatches.
mismatches
31
Thee Theory
eo y of
o Small
S a Reflections
e ect o s
• Single-Section Transformer
  1  T12T213e 2 j  T12T2132 2 e 4 j  
 1  T12T213e
 2 j

  e
n 0
n n  2 jn
2 3

1
x 
, for x  1

1 x
n 0
n
T12T213e 2 j
  1 
1  2 3e 2 j
1  3e  2 j

1  13e 2 j
 1  3e  2 j
Figure 5.13 (p. 244) Partial reflections and transmissions
on a single
single-section
section matching transformer.
transformer
32
Thee Theory
eo y of
o Small
S a Reflections
e ect o s
• Multisection Transformer
Figure 5.14
5 14 (p.
(p 245) Partial
reflection coefficients for a
multisection matching transformer.
Z  Z0
0  1
Z1  Z 0
Z n 1  Z n
n 
Z n 1  Z n
N 
ZL  ZN
ZL  ZN
Z n : vary monotonically
   0  1e 2 j  2e 4 j    N e 2 jN
Transformer can be made symmetrical :
 
 
   1 2 N / 2 ,
for N even
 
   e  jN 0 e jN  e  jN  1 e j  N 2   e  j  N 2   
   2e  jjN 0 cos N  1 cosN  2    n cos N  2n 
   2e  jN 0 cos N  1 cosN  2    n cos N  2n 

   ( N 1) / 2 cos ,
for N odd
 Given
Gi
 , design
d i Z1 , Z 2 ,  Z N
33
Thee Bode-Fano
ode a o Criterion
C te o
Circuit
Bode - Fano limit
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
Series
i RC.
RC
34
Thee Bode-Fano
ode a o Criterion
C te o
Circuit
Bode - Fano limit
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
Series
i RL.
RL
35
Thee Bode-Fano
ode a o Criterion
C te o
Figure 5.23 (p. 263)
Illustrating the Bode-Fano criterion.
(a) A possible reflection coefficient
response.
(b) Nonrealizable and realizable
reflection coefficient responses
responses.
1. Given RC :
 
 m 
2. m  0, unless   0
 m  0 only
l at a finite
fi i
number of frequencies
3. R or C 
   or m 
high Q load is harder to match
36