GENERAL IMPEDANCE TRANSFORMATION EQUATION
The reflection coefficient
V
 
V
along the TL. Since we have
is a function of position
Z  Z0
1 
.
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 '
2 z
1 
1   Le
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'
 z '
 z'
 z '
Z

Z
 z'
Z
e

e

Z
e

e
L
0  z




L
0
e 
e
Z L  Z0
e z ' 
Z  Z0
Y  Y0
Z L  Z 0 tanh( z ')
Z 0  Z L tanh( z ')
YL  Y0 tanh( z ')
Y0  YL tanh( z ')
The input impedance (admitance) of a line (terminated in
Z L ) of length is obtained by putting z '  .
Zin  Z 0
Yin  Y0
Z L  Z 0 tanh( )
Z 0  Z L tanh( )
YL  Y0 tanh( )
Y0  YL tanh( )
For a Lossless Line:
  0,  z '  j z '
tanh( z ')  tanh( j z ')  j tan( z ') and
Z  Z0
Z L  jZ 0 tan(  z ')
Z 0  jZ L tan(  z ')
Yin  Y0
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

2

, n positive integer.    .n 2  n
2
tan   tan n  0 , Zin  Z L . This result can be generalized to
any two points that are separated by a multiple of a halfIf
n
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
3) The Short-Circuited Line
4) The open Circuited Line