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The Reflection Coefficient
Transformation
The load at the end of some length of a transmission line (with
characteristic impedanceZ 0 ) can be specified in terms of its
impedanceZ L or its reflection coefficient Γ L.
Note both values are complex, and either one completely specifies the load—if you know one , you know the other!
0 0 0
and 1
L L L L L L
Z Z
Z Z
Z Z
− ⎛^ + Γ ⎞
Recall that we determined how a length of transmission line transformed the load impedance into an input impedanc e of a (generally) different value:
Z 0 , β
A
Z 0 , β Z i n Z L
where:
0 0 0 0 0 0 cos sin cos sin tan tan
L in L L L
Z j Z
Z Z
Z j Z
Z j Z
Z
Z j Z
A A
A A
A
A
Q: Say we know the load in terms of its reflection coefficient.
How can we express the input impedance in terms its reflection
coefficient (call this Γin )?
A: Well, we could execute these three steps:
1. Convert Γ L toZ L:
0
L L L
Z Z
2. TransformZL down the line toZ in :
0 0 0
cos sin cos sin
L in L
Z j Z
Z Z
Z j Z
A A
A A
Z 0 , β Γ L
A
Z 0 , β Γ^ i n =^?
( ) ( ) ( ) ( )
0 0
2
j j j j L L in j j j j L L j L j j j L j L
Z e e e e
Z e e e e
e
e
e e
e
β β β β β β β β β β β β β
− − −
A A A A A A A A A A A A A
Q: Hey! This result looks familiar. Haven’t we seen something
like this before?
A: Absolutely! Recall that we found that the reflection coefficient function Γ ( z) can be expressed as:
Γ (^) ( z (^) ) = Γ 0 e 2 γz
Now, for a lossless line, we know that γ = jβ, so that:
Γ (^) ( z (^) ) = Γ 0 e j^2 βz
Evaluating this function at the beginning of the line (i.e., at
z = zL − A ):
( ) 0 2 (^ ) 2 2 0
L L
j z L j z j
z z e
e e
β β β
− −
A A
A
But, we recognize that:
Γ 0 ej^2 βzL = Γ (^) ( z = zL)= ΓL
And so: ( ) 0 2 2 2
L j^ zL^ j j L
z z e e
e
β β β
− −
A A
A
Thus, we find that Γin is simply the value of function Γ( z)
evaluated at the line input of z = zL − A!
Γin = Γ (^) ( z = zL− A) = ΓL e −j^2 βA
Makes sense! After all, the input impedance is likewise simply
the line impedance evaluated at the line input of z = zL − A :
Z in = Z (^) ( z = zL − A)
It is apparent that from the above expression that the reflection coefficient at the input is simply related to ΓL by a
phase shift of 2 β A.
In other words, the magnitude of Γin is the same as the
magnitude of Γ (^) L!
( 2 )
(1)
j in L L L
Γ = Γ e θ^ Γ^ − β
A
If we think about this, it makes perfect sense!
Thus, we can conclude from conservation of energy that:
Γin = ΓL
Which of course is exactly the result we just found!
Finally, the phase shift associated with transforming the load Γ L down a transmission line can be attributed to the phase shift associated with the wave propagating a length A down the line, reflecting from load Γ L , and then propagating a length A back up the line:
To emphasize this wave interpretation, we recall that by definition, we can write Γin as:
( ) ( ) ( )
L in L L
V z z
z z
V z z
−
A
A
A
Therefore:
( ) ( ) ( )
L in L j j L L
V z z V z z
e β e β V z z
− + − − +
= A^ Γ A = −
A A
A
Z 0 , β Γ L
φ = βA
j j
i n e Le
Γ = −^ βA^ Γ −^ βA
