GENERAL IMPEDANCE TRANSFORMATION EQUATION
V
The reflection coefficient    is a function of position along the TL. Since we
V
1 
have Z  Z 0
.
1 
Z is also a function of position along the line.
1 
1   Le2 z '
e z '   Le z '
Z  Z0
 Z0
 Z0  z '
1 
1   Le2 z
e   L e  z
L 
Z L  Z0
Z L  Z0
Z L  Z 0  z
e
Z L  e z '  e   z '   Z 0  e z '  e   z ' 
Z L  Z0
Z  Z0
 Z0
Z L  Z 0  z
 z'
Z L  e z '  e   z '   Z 0  e  z '  e   z ' 
e 
e
Z L  Z0
e z ' 
Z  Z0
Z L  Z 0 tanh( z ')
Z 0  Z L tanh( z ')
Y  Y0
YL  Y0 tanh( z ')
Y0  YL tanh( z ')
The input impedance (admitance) of a line (terminated in
obtained by putting
Zin  Z 0
Yin  Y0
z'  .
Z L  Z 0 tanh( )
Z 0  Z L tanh( )
YL  Y0 tanh( )
Y0  YL tanh( )
Z L ) of length
is
For a Lossless Line:
  0,  z '  j z '
tanh( z ')  tanh( j z ')  j tan( z ') and
Z  Z0
Y  Y0
Z L  jZ 0 tan(  z ')
Z 0  jZ L tan(  z ')
YL  jY0 tan(  z ')
Y0  jYL tan(  z ')
Some Special Impedance Transformations on a lossless TL:
For a lossless TL:
Z  Z0
Z L  jZ 0 tan(  z ')
Z 0  jZ L tan(  z ')
1) Half-Wavelength Line
If
n

2
, n positive integer.
 
2

.n

2
 n tan   tan n  0 ,
Zin  Z L . This result can be generalized to any two points that are separated
by a multiple of a half- wavelength. Since tan( z ' n )  tan  z ' ( tan
function is periodic with  ) Z repears every half-wavelength along a
lossless TL.
2) The Quarter Wavelength Line
If  (2n  1)

4
, n=positive integer, then
tan    .
 
2

(2n  1).

4
 (2n  1)

Z 02
1
So, Zin 
or Zin 
ZL
ZL
Quarter wavelength lossless TL it is often refered as quarter wave
transformer.
2
and
i)
ii)
iii)
iv)
v)
A large value of load impedance  a small value of input impedance.
A small value of load impedance  a large value of input impedance.
Inductive load  capacitive input
Capacitive load  inductive input
Series resonant circuit Z L  circuit Z L parallel resonat circuit Z in
For Z L real, a quarter-wavelength long TL has all the properties of an
ideal transformer.
The quarter-wave transformer is useful in matching a resistive load
to a generator (necessary to deliver all the available generator power
to the load).
If Z G is real, then we must have Zin  ZG , or
Z 02
 ZG  Z 0  ZG RL
RL
Since the line can only be quarter-wavelength long at one freguency,
the transformer is frequency sensitive.
3) The Short-Circuited Line
Z L  0, Zin  jX in  jZ0 tan  or Zin  j tan   jX in




(

), Zin is inductive.
For
2
4
4) The open Circuited Line
Z L  , Zin   jZ0 cot 
Yin  j tan 




(

), Zin is capactive.
For
2
4
IMPEDANCE TRANSFORMATIONS ON A LOSSLESS LINE
For
 0,
Z  Z0
Z L  Z 0 tanh   z '
Z 0  Z L tanh   z '
Now consider,
i)
The Short Circuited Line
Z L  0 Z  Z0 tanh  z '  Z0 tanh  z ' j  z '
The input impedance of a line of length
Zin  tanh   j 
  Z0
:
tanh   j tan 
1  j tanh  tan 

[If   0,   n   n  , Zin  0 , short circuit repeats itself at every

2
 / 2 .]

1) If  n , Zin  Z0 tanh   Z0    0.5 Np 
2

2) If  (2n  1) , tan    Zin 
4
Z0
Z
 0 
tanh 

Z in is finite and real.
ii)
The Open Circuited Line
Z L  , YL  0 .
With z '  ,
Yin  Y0
YL  Y0 tanh( )
Y0  YL tanh( )
Yin  Y0 tanh(  j  )
1) If   n , Yin  Y0 tanh   Y0 

2) If   (2n  1) , Yin 
2

Z0
Y0
Y
1
 0 
tanh 
 Z0