1/26/2005
The Reflection Coefficient Transformation.doc
1/7
The Reflection Coefficient
Transformation
The load at the end of some length of a transmission line (with
characteristic impedance Z0 ) can be specified in terms of its
impedance ZL or its reflection coefficient ΓL .
Note both values are complex, and either one completely
specifies the load—if you know one, you know the other!
ΓL =
ZL − Z0
ZL + Z0
⎛ 1 + ΓL ⎞
⎟
⎝ 1 − ΓL ⎠
ZL = Z0 ⎜
and
Recall that we determined how a length of transmission line
transformed the load impedance into an input impedance of a
(generally) different value:
Z0 , β
Z
in
Z0 , β
Z
L
A
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/26/2005
The Reflection Coefficient Transformation.doc
2/7
where:
⎛ Z L cos β A + j Z 0 sin β A ⎞
Zin = Z 0 ⎜
⎟
A
A
+
β
β
Z
cos
j
Z
sin
L
⎝ 0
⎠
⎛ Z + j Z 0 tan β A ⎞
= Z0 ⎜ L
⎟
+
A
β
tan
Z
j
Z
0
L
⎝
⎠
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 )?
Z0 , β
Γ in = ?
Z0 , β
ΓL
A
A: Well, we could execute these three steps:
1. Convert ΓL to ZL:
⎛ 1 + ΓL ⎞
⎟
⎝ 1 − ΓL ⎠
ZL = Z0 ⎜
2. Transform ZL down the line to Zin :
⎛ Z L cos β A + j Z 0 sin β A ⎞
⎟
A
A
β
β
cos
sin
Z
j
Z
+
0
L
⎝
⎠
Zin = Z 0 ⎜
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/26/2005
The Reflection Coefficient Transformation.doc
3/7
3. Convert Zin to Γin :
Γin =
Zin − Z 0
Zin + Z 0
Q: Yikes! This is a ton of complex arithmetic—isn’t there an
easier way?
A: Actually, there is!
Recall in an earlier handout that the input impedance of a
transmission line length A , terminated with a load ΓL , is:
⎛ e + j β A + ΓL e − j β A ⎞
V ( z = −A )
Zin =
= Z0 ⎜ + j βA
⎟
I ( z = −A )
− ΓL e − j β A ⎠
⎝e
Note this directly relates ΓL to Zin (steps 1 and 2 combined!).
If we directly insert this equation into:
Γin =
Zin − Z 0
Zin + Z 0
we get an equation directly relating ΓL to Γin :
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/26/2005
The Reflection Coefficient Transformation.doc
4/7
+ j βA
+ Γ L e − j β A ) − (e + j β A − Γ L e − j β A )
Z 0 (e
Γin =
Z 0 (e + j β A + Γ L e − j β A ) + ( e + j β A − Γ L e − j β A )
2 ΓL e − j β A
=
2e + j β A
= ΓL e − j β A e − j β A
= ΓL e − j 2 β 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 2γ z
Now, for a lossless line, we know that γ = j β , so that:
Γ (z ) = Γ0 e j 2β z
Evaluating this function at the beginning of the line (i.e., at
z = z L − A ):
Γ (z = z L − A ) = Γ0 e
j 2 β (z L − A )
= Γ 0 e j 2 β zL e − j 2 β A
But, we recognize that:
Γ 0 e j 2 β zL = Γ ( z = z L ) = Γ L
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/26/2005
The Reflection Coefficient Transformation.doc
And so:
5/7
Γ ( z = z L − A ) = Γ 0 e j 2 β zL e − j 2 β A
= ΓL e − j 2 β A
Thus, we find that Γin is simply the value of function Γ ( z )
evaluated at the line input of z = z L − A !
Γin = Γ ( z = z L − A ) = Γ L e − j 2 β A
Makes sense! After all, the input impedance is likewise simply
the line impedance evaluated at the line input of z = z L − A :
Zin = Z ( z = z L − 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 2β A .
In other words, the magnitude of Γin is the same as the
magnitude of ΓL !
Γin = Γ L e
j (θ Γ −2 β A )
= Γ L (1)
= ΓL
If we think about this, it makes perfect sense!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/26/2005
The Reflection Coefficient Transformation.doc
6/7
Recall that the power absorbed by the load Γin would be:
V0+
in
Pabs
=
2
2 Z0
(
1 − Γin
2
)
while that absorbed by the load ΓL is:
L
Pabs
=
Z0, β
V0+
2
(1 − Γ )
2
2 Z0
L
Z0, β
Pa bi ns
Pa bL s
ΓL
A
Recall, however, that a lossless transmission line can absorb no
power! By adding a length of transmission line to load ΓL , we
have added only reactance. Therefore, the power absorbed by
load Γin is equal to the power absorbed by ΓL:
in
L
Pabs
= Pabs
V0+
2
2 Z0
2
(1 − Γ ) = 2 Z (1 − Γ )
2
in
1 − Γin
Jim Stiles
V0+
2
L
0
2
= 1 − ΓL
The Univ. of Kansas
2
Dept. of EECS
1/26/2005
The Reflection Coefficient Transformation.doc
7/7
Thus, we can conclude from conservation of energy that:
Γin = Γ L
Which of course is exactly the result we just found!
Finally, the phase shift associated with transforming the load
ΓL 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 ΓL , and then propagating a length A back up
the line:
Z0, β
Γ in = e
− jβ A
Γ Le
− jβ A
Γ
L
φ = βA
To emphasize this wave interpretation, we recall that by
definition, we can write Γin as:
V − (z = z L − A )
Γin = Γ ( z = z L − A ) = +
V (z = z L − A )
Therefore:
V − ( z = z L − A ) = Γin V + ( z = z L − A )
= e − j β A ΓL e − j β A V
Jim Stiles
The Univ. of Kansas
+
(z
= zL − A )
Dept. of EECS