Impedance Matching with Lumped Elements
jX1
jB2
Circuit used when G L  1
1
1
Z in  jX1 
 jX1 
 Z0
1
1
jB2 
jB2 
ZL
RL  jX L
YL
B2 ( X 1 RL  X L Z 0 )  RL  Z 0
X 1 (1  B2 X L )  B2 Z 0 RL  X L
X L  RL / Z 0 RL2  X L2  Z 0 RL
B2 
RL2  X L2
Z0
1 X L Z0
X1 


B2
RL
B2 RL
ENEE482-Dr. Zaki
1
jX1
jB
ZL
2
Circuit used when R L  1
1
 1/ Z0
Yin  jB2 
RL  j ( X 1  X L )
B2 Z 0 ( X 1  X L )  Z 0  RL
( X 2  X L )  B2 Z 0 RL
X 1   RL ( Z 0  RL )  X L
 ( Z 0  RL ) / RL
B2 
Z0
ENEE482-Dr. Zaki
2
Single-Stub Matching
Yin  1  jB
Load impedance
Input admittance=S
1 
Yin 
S
1 
If YL is real, then the reflection coefficien t is real
Let d 0 be the distance from the voltage - minimum point wher e
Yin  1  jB
d0 

S 1
cos 1
4
S 1

S
1
The stub length  0 
tan
2
S 1
ENEE482-Dr. Zaki
3
Series Stub
Voltage minimum
Z in  1 / S
S 1  j tan d 0
Z in  1  jX 
1  jS 1 j tan d 0
Input impedance=1/S

1 1  S
d0 
cos
4
1 S
1
X  (1  ) tan d 0
S
j tan  0   jX

1 1  S
0 
tan
2
S
ENEE482-Dr. Zaki
4
Double Stub Matching Network
a
b
jB2
b
jB1
YL
a

YL is transform ed into YL
YL  G L  jBL at the plane aa
The first stub adds a susceptanc e j B1 which moves the point
along constant conductanc e circle to P2
ENEE482-Dr. Zaki
5
x=1
YL
Pshort circuit
r=1
r=0.5
Smith Chart
0
Popen circuit
Real part of
Refl. Coeff.
x=-1
Move from P2 to P3 along a constant radius circle through an angle
  2 d
At the plane b - b the input admittance is Yb  Gb  jBb .
The P3 must lie on the G  1 circle. The stub will cancel jBb .
ENEE482-Dr. Zaki
6
x=1
Y
Pshort circuit
P2
G1=1
L
r=0.5
r=1
0
P3
Smith Chart
Popen circuit
Real part of
Refl. Coeff.
x=-1
Rotate the the G=1 circle through an angle -
The intersection of G=1 and the GL circle determine
The point P2
ENEE482-Dr. Zaki
7
x=1
YL
Pshort circuit
r=1
r=0.5
0
Popen circuit
Real part of
Refl. Coeff.
x=-1
Smith Chart
The shaded range is for the load impedance which
cannot be matched when d=1/8 wavelength
ENEE482-Dr. Zaki
8
Quarter-Wave Transformers
ZC=Z0
Z in
ZC=Z1
ZL
 /4
Z L  jZ1 tan(  / 4) Z12
 Z1

 Z0
Z1  jZ L tan(  / 4) Z L
Z1  Z 0 Z L  perfect match
Z L  jZ1t
Zin  Z1
Z1  jZ L t
, t  tan   tan  ( f )
Z in  Z 0
Z L  Z0


Z in  Z 0 Z L  Z 0  jt 2 Z 0 Z L
ENEE482-Dr. Zaki
9
1
 
 2 Z1 Z L


1
sec  
 Z L  Z1



2
If f is near f 0 then   /2 and sec 2  1 
 
Z L  Z1
2 Z1 Z L
cos 
If  m is the maximum value of reflection coefficien t
that can be tolerated
The correspond ing value of  is  m
 m  cos
1
2  m Z1 Z L
( Z L  Z 1 ) 1   m2
The band width f  2(f 0  f m ) is very small
ENEE482-Dr. Zaki
10


m
m
/2

3/2

Bandwidth characteristic for a single
Section quarter wave transformer


  Bandwidth  2   m 
2

ENEE482-Dr. Zaki
11
Theory of Small Reflection

1
Z1

T12
Z 2  Z1
1 
Z 2  Z1
T21  1  1 
1
Z2 3 Z
L
2
T21
, 2  1
2Z 2
Z 2  Z1
, T12 
2Z1
Z 2  Z1
ZL  Z2
3 
ZL  Z2
ENEE482-Dr. Zaki
12
  
1
T21
3
e  j
  1  T12T213 e  2 j  T12T2132 2 e  4 j  ...

 1  T12T213 e  2 j  2n 3n e  2 jn
n 0
Substitute for T12  1  2  1  1
T12T213 e  2 j
 1 
1  2 3 e  2 j
,
T21  1  1 
1  3 e  2 j
 2 j





e
1
3
1  13 e  2 j
If 1 and 3 are small compared to unity
ENEE482-Dr. Zaki
13
T12T213 e 2 j
1
1
T21
T213 e 2 j
T212 3 e 2 j
T12T2132 e 4 j
T12 2 32 e 4 j
T2122 32 e 4 j
T T   e
2
12 21 2
2
3
6 j
T212 32 e 6 j
2
Multiple reflection of waves for a circuit
with two reflection junctions
ENEE482-Dr. Zaki
14
Approximate Theory for Multi-Section
Quarter Wave Transformers

Z0
Z1

Z2

3 Z
L
ZN
A multi-section quarter-wave transformer
Assume Z L is real
Z1  Z 0
0 
 0
Z1  Z 0
Z n 1  Z n
, n 
 n
Z n 1  Z n
ZL  ZN
The last reflection coefficien t is N 
 N
ZL  ZN
ENEE482-Dr. Zaki
15
   0  1e 2 j   2 e 4 j  ....   N e 2 jN
Assume that the tranforme r is symmetrica l i.e.
 0   N , 1   N 1 ,  2   N  2 , ...etc
  e  jN [  0 (e jN  e  jN )  1 (e j ( N  2 )  e  j ( N  2 ) )  ...]
 2e  jN [  0 cos N  1 cos( N  2)  ...   n cos( N  2n)  ...
  ( N 1) / 2 cos  ]
for N odd
 2e  jN [  0 cos N  1 cos( N  2)  ...   n cos( N  n)  ...
  ( N ) / 2 cos  ]
ENEE482-Dr. Zaki
for N even
16
Binomial Transformer
A maximally flat passband characteri stics is obtained if   
and the first N - 1 derivative s w.r.t frequency ( or  ) vanish at the
matching frequency f 0 where   /2.
Choose ( )  A(1  e -2j ) N
( )  A e
 j N
e
j
e
 j N
    A2 N (cos  ) N
Note that ( )  0 for   /2 and
d
n

( ) / d n  0  0 at   /2 for n  1,2,..., N - 1
  /2 correspond s to the center frequency f 0 for which   /4
when   0 or  ,
ENEE482-Dr. Zaki
ZL  Z 0
 ( 0) 
ZL  Z0
17
 (0)  A2
C
A2
,
N
ZL  Z0
ZL  Z 0
ZL  Z 0
 2 j N
N ZL  Z 0
(1  e
) 2
ZL  Z 0
ZL  Z 0
( )  2  N
N
n
N
N
N  2 j
C
 ne
n 0
N ( N  1)( N  2)...( N  n  1)
N!


n!
( N  n)! n!
C nN  C NN n
, C 0N  1,
C1N  N  C NN1 , .....
Equate the desired passband response to the actual response
given by :
N
( )  A  C nN e  2 jn  0  1e  2 j  2 e  4 j  ....  N e  2 jN
n 0
n  2
N
ENEE482-Dr. Zaki
Z LZ 0 N
C n   N n
Z LZ 0
18
The characteri stic impedances Z n can be found from the  n
x 1
1 Z n 1 Z n 1 Z n
Use : lnx  2
ln

 n
x 1
2
Zn
Z n 1 Z n
Z n 1  Z n e 2  n
Z n 1
ZL
 N Z LZ 0
N
N
N
ln
 2  n  2[2
C n ]  2 C n ln
Zn
Z L Z 0
Z0
A2
 ( N 1)
ZL
ln(
)
Z0
Z2
ZL 
1  Z1
2 A  (0)  ln
 ln
 ...  ln

2  Z0
Z1
ZN 
1 Z1 Z 2 Z L
1 ZL
 ln(
...
)  ln
2 Z 0 Z1 Z N
2 Z0
N
ENEE482-Dr. Zaki
19
The maximum value of reflection coeficient that can be tolerated
over the passband is
 m  2 N A cos N  m
 m   / 2 is the lower edge of the passband
1/ N



1 1 
m 
 m  cos  
 
 2  A  
The fractional bandwidth is :
1/ N


4 m
f 2( f 0  f m )
4
1 1   m 

 2
 2  cos  
 
f0
f0


 2  A  
ENEE482-Dr. Zaki
20
Example
Z0
Z1
Z2
Z3
ZL
Design a three section binomial transformer to match
a 50 Ohms load to a 100 Ohms line. Calculate the bandwidth
For max reflection =.05 over the passband.
For 3 sections N  3, Z L  50, Z 0  100 
A2
3
ZL  Z0
1 ZL
 4 ln
ZL  Z0
Z0
2
3!
3!
3
C 
 1 , C1 
3 ,
(3  0)!0!
(3  1)!1!
3!
3!
C 23 
 3, C 33 
1
(3  2)!2!
(3  3)!3!
3
0
ENEE482-Dr. Zaki
21
n  0 : lnZ 1  ln Z 0  2 3 C 03 ln
ZL
Z0
50
ln Z 1  ln 100  2 ln
 4.518  Z 1  91.7
100
Z
n  1 : lnZ 2  ln Z 1  2 3 C13 ln L
Z0
3
50
ln Z 2  ln 91.7  2 (3) ln
 4.26  Z 2  70.7
100
Z
n  2 : lnZ 3  ln Z 2  2 3 C 23 ln L
Z0
3
50
ln Z 3  ln 70.7  2 (3) ln
 4.00  Z 3  54.5
100
3
ENEE482-Dr. Zaki
22
Chebyshev Transformer
Z L  Z0
Z L  Z0
m
m
/2 m
Tn ( x) : Chebyshev polynomial of degree n

T1 ( x)  x
T2 ( x)  2 x 2  1
T3 ( x)  4 x 3  3x
Tn ( x)  2 xTn 1 ( x)  Tn  2 ( x)
Tn (cos  )  cos n
ENEE482-Dr. Zaki
23
 cos  

1 cos  




Consider Tn 
 cos n cos

cos  m 
 cos  m 

  2e  jN [  0 cos N  1 cos( N  2)  ....
  n cos( N  2n)  .......]
 A e  jN TN (sec m cos )
When   0 we have :
ZL  Z0
1 ZL

 A TN (sec m )  ln
ZL  Z0
2 Z0
ln( Z L / Z 0 )
A
2TN (sec m )
1  jN Z L TN (sec m cos )
 e
ln
2
Z 0 TN (sec m )
ENEE482-Dr. Zaki
24
In the passband the maximum value of TN (sec m cos ) is unity
ln( Z L / Z 0 )
m 
2TN (sec m )
1 1
TN (sec m )   m ln( Z L / Z 0 )
2
,
1
1 ln( Z L / Z 0 ) 

sec m  cos cos
2m
N

n
(cos  ) n  2  n e  jn (1  e 2 j ) n  2  n e  jn  C mn e 2 jm
m 0
 2  n 1 [C 0n cos n  C1n cos( n  2)  .......  C mn cos( n  2m)  ....]
T1 (sec m cos )  sec m cos
T2 (sec m cos )  2(sec m cos ) 2  1  sec 2  m (1  cos 2 )  1
T3 (sec m cos )  sec 3  m (cos3 m  3cos )  3 sec  m cos 
T4 (sec m cos )  sec 4  m (cos4 m  4cos2  3)  4 sec 2  m (cos 2  1)
ENEE482-Dr. Zaki
25
Example
Design a two section Chebyshev transformer (two sections) to
Match a line of load impedance =2. The maximum tolerance
Value of  is 0.05.
1
2
T2 (sec m )  2 sec  m  1 
 6.67
3(0.05)
sec m  1.96, and  m  1.04
2  0 cos 2  1   m T2 (sec m cos )
  msec 2 m cos2   m (sec 2 m  1)
1
 0   m sec 2 m   2  0.099
2
1   m (sec 2 m  1)  0.148
ENEE482-Dr. Zaki
Z1  e 2  0 Z 0  1.219, Z 2  e 2 1 Z1  1.639
26
ENEE482-Dr. Zaki
27
Design of Complex Impedance
Termination
Amplifier
Zc
Input
Matching
network
Zs
Zc
ZL Output
Matching
network
Microwave amplifier circuit
ENEE482-Dr. Zaki
28
2
1 V0 c
The available power from the network is :
2 4 RT
Z T  RT  jX T
2
4 RT R L
1 V0 c
The power delivered to R L 
2 4 RT Z T  Z L
2
2
1 V0 c

M L  Pin
2 4 RT
ML  M
ENEE482-Dr. Zaki
29
l
l
ZL
=
Zc
jB1
ZL
G=1
Stub
Transmission Line Matching Network
jX1
jX2
jB2
ZL
G=1
ZL
jB1
G=1
Alternative Matching Networks
ENEE482-Dr. Zaki
30
j1
Y’in
1
YL
G=2
ZL= 0.4-j0.2
2
Y”in
G=1
-j1
Design Procedure for the Matching Network with Shunt Stub
ENEE482-Dr. Zaki
31
VS Impedance
I
Z S  Z in
Mismatch Factor
2
1 VS
Pin 
Rin
2 Z S  Z in
If Z in  Z S* then Rin  RS ,
X in   X S
Maximum available power is obtained
2
Pava
1 VS

2 4 RS
1 VS  4 Rin RS 

  Pava M
Pin 
2
2 4 RS  Z in  Z S 


 4R R 
in S
 is called the impedance mismatch factor
M 
2
ENEE482-Dr.
 Z in Zaki
 Z S 
2
32
Z22-Z12
Z11-Z12
ZT
ZS
Z12
VS
Zin
ZL
ZL
Voc
ML
Thevenin equivalent
network
A T matching network
Z 12VS
Voc 
Z 11  Z S
  Z 11 Z 22  Z 12
ENEE482-Dr. Zaki
,
  Z 22 Z S
ZT 
Z 11  Z S
2
33
Impedance Transformation and Matching
Review of Transmission Lines and Smith Chart
V ( z )  V  ( z )  V  ( z )  V0 e z  V0 ez
I ( z )  I  ( z )  I  ( z )  I 0 e z  I 0 ez ,
V0
V0
    Z 0 : Characters tic Impedance of the T.L.

I0
I0
V0 z V0 z
I(z) 
e 
e
Z0
Z0
V0
  REFLECTION COEFFICEINT  
Zg
V0
Vg
Z=0
Finite Transmission
ENEE482-Dr.
Zaki
Z0
Z=L
ZL
Line terminated with load impedance
L
34
V L  V0 e L


V
V
 V0 e L , I L  0 e L  0 e L ,
Z0
Z0
1
1
(VL  I L Z 0 )e L V0  (VL  I L Z 0 )e L
2
2
IL
V ( z )  [( Z L  Z 0 )e  ( L  z )  ( Z L  Z 0 )e  ( L  z ) ]
2
I
I ( z )  L [( Z L  Z 0 )e  ( L  z )  ( Z L  Z 0 )e  ( L  z ) ]
2Z 0
Let z   L  z
V ( z )  I L ( Z L cosh z   Z 0 sinh z )
V0 
IL
( Z L sinh z   Z 0 cosh z )
Z0
Z L  Z 0 tanh z 
V ( z )
Z ( z ) 
 Z0
I ( z )
Z 0  Z L tanh z 
I ( z ) 
Z in ( z   L)  Z 0
ENEE482-Dr. Zaki
Z L  Z 0 tanh L
Z 0  Z L tanh L
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Standing wave ration (SWR) S:
Vmax
S
Vmin
1 
S -1

; 
1 
S 1
Smith Chart:
Z L  Z0

  e L
Z L  Z0
The normalized impedance z L
ZL
1  L
zL 
 r  jx 
Z0
1  L
j L
1


e
z 1
  r  i  L
; zL 
zL  1
1   e j L
j  2 j
Z in 1  () 1  L e 2 j 1  L e L
Z in 



 2 j
Z 0 1  ( ) 1  L e
1  L e j L 2 j
ENEE482-Dr. Zaki
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(1  r )  ji
r  jx 

(1  r )  ji
1  r2  i2
2i
r
; x
2
2
(1  r )  i
(1  r ) 2  i2
2
r 2
 1 
2
(r 
)  i  
 ; Equation of a circle of a radius
1 r
1 r 
r
and centered at r 
and i  0
1 r
2
 1 


1 r 
2
1
1

1
(r  1)   i      ; Equation of a circle of radius
x
x

 x
1
and centered at r  1 and i 
x
2
ENEE482-Dr. Zaki
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Imaginary part of
Refl. Coeff.
x=1
Pshort circuit
r=1
0
r=0.5
Popen circuit
Real part of
Refl. Coeff.
x=-1
ENEE482-Dr. Zaki
Smith Chart
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Review of Transmission Lines and Smith Chart
V ( z )  V  ( z )  V  ( z )  V0 e z  V0 ez
I ( z )  I  ( z )  I  ( z )  I 0 e z  I 0 ez ,
V0
V0
    Z 0 : Characters tic Impedance of the T.L.

I0
I0
V0 z V0 z
I(z) 
e 
e
Z0
Z0
V0
  REFLECTION COEFFICEINT  
V0
Z=L
Z Z=0
g
Vg
Z0
ZL
L
Finite Transmission Line terminated with load impedance
ENEE482-Dr. Zaki
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