4/1/2009
The Theory of Small Reflections
1/9
The Theory of
Small Reflections
Recall that we analyzed a quarter-wave transformer using the
multiple reflection view point.
Τ
Τ
V +(z ) = a Z 0 e − j β
(z + A )
Z0
V −( z ) = b Z 0 e + j β
Γ −Γ
Z1 =
Z 0RL
ΓL
RL
(z + A )
A = λ
4
We found that the solution could thus be written as an infinite
summation of terms (the propagation series):
∞
b = a ∑ pn
n =1
where each term had a specific physical interpretation, in
terms of reflections, transmissions, and propagations.
For example, the third term was path:
a
Z0
Jim Stiles
Z 0RL
RL
p3 a
The Univ. of Kansas
Dept. of EECS
4/1/2009
The Theory of Small Reflections
2/9
a5
b2
a1
Τ
p3 = Τ2 ( ΓL ) Γ e − j 2 β A
2
e −j βA
−Γ
Γ
ΓL
Τ
b1
e −j βA
a2
b5
Now let’s consider the magnitude of this path:
2
2
2
2
p3 = Τ ΓL Γ e − j 2 β A
= Τ ΓL
Γ
Recall that Γ = Γ L for a properly designed quarter-wave
transformer :
Γ=
and so:
2
RL − Z 1
= ΓL
RL + Z 1
2
2
p3 = Τ ΓL Γ = Τ ΓL
3
For the case where values RL and Z 1 are numerically “close” in
—i.e., when:
RL − Z 1 RL + Z 1
we find that the magnitude of the reflection coefficient will be
very small:
R − Z1
ΓL = L
1. 0
RL + Z 1
As a result, the value Γ L
3
Jim Stiles
The Univ. of Kansas
will be very, very, very small.
Dept. of EECS
4/1/2009
The Theory of Small Reflections
3/9
Moreover, we know (since the connector is lossless) that:
2
2
2
1 = Γ + Τ = ΓL + Τ
and so:
2
Τ = 1 − ΓL
2
2
≈1
We can thus conclude that the magnitude of path p3 is likewise
very, very, very small:
2
3
3
p3 = Τ ΓL ≈ ΓL 1
This is a classic case where we can approximate the propagation
series using only the forward paths!!
Recall there are two forward paths:
a
Z0
p2
Z 0RL
p1
a
b ( p1 + p2 ) a
(
e −j βA
Τ
)
= Γ+Τ2 Γ L e j 2 β A a
−Γ
Γ
p1 = Γ
RL
Τ
ΓL
e −j βA
p2 = Τ2 Γ e − j 2 β A
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/1/2009
The Theory of Small Reflections
4/9
Therefore IF Z0 and RL are very close in value, we find that we
can approximate the reflected wave using only the direct paths
of the infinite series:
b ( p1 + p2 ) a
(
)
= Γ+Τ2 Γ L e j 2 β A a
Therefore:
V −( z ) = b Z 0 e + j β (z + A )
(
)
≅ Γ+Τ2 ΓL e j 2 β A a Z 0 e + j β (z + A )
Now, if we likewise apply the approximation that Τ 1.0 , we
conclude for this quarter wave transformer (at the design
frequency):
b ( p1 + p2 ) a
(
)
= Γ+ ΓL e j 2 β A a
Therefore:
V −( z ) = b Z 0 e + j β (z + A )
(
)
≅ Γ+ ΓL e j 2 β A a Z 0 e + j β (z + A )
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/1/2009
The Theory of Small Reflections
5/9
This approximation, where we:
1. use only the direct paths to calculate the
propagation series,
2. approximate the transmission coefficients as
one (i.e., Τ = 1 ).
is known as the Theory of Small Reflections, and allows
us to use the propagation series as an analysis tool (we
don’t have to consider an infinite number of terms!).
Consider again the quarter-wave matching network SFG. Note
there is one branch ( −Γ = S22 of the connector), that is not
included in either direct path.
a
e −j βA
Τ
−Γ
Γ
p1 = Γ
ΓL
e −j βA
Τ
p2 = Τ2 Γ e − j 2 β A
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/1/2009
The Theory of Small Reflections
6/9
With respect to the theory of small reflections (where only
direct paths are considered), this branch can be removed from
the SFG without affect.
a
e −j βA
Τ
ΓL
Γ
e −j βA
Τ
p1 = Γ
p2 = Τ2 Γ e − j 2 β A
Moreover, the theory of small reflections implements the
approximation Τ = 1 , so that the SFG becomes:
a
1 .0
e −j βA
ΓL
Γ
e −j βA
1 .0
p1 = Γ
p2 = Γ e − j 2 β A
Reducing this SFG by combining the 1.0 branch and the e − j β A
branch via the series rule, we get the following approximate
SFG:
a
Γin =
b
a
= Γ+ ΓL e j 2 β A
Jim Stiles
e −j βA
Γ
b
ΓL
e −j βA
The Univ. of Kansas
The approximate
SFG when applying
the theory of
small reflections!
Dept. of EECS
4/1/2009
The Theory of Small Reflections
7/9
Note this approximate SFG provides precisely the results of
the theory of small reflections!
Q: Why is that?
A: The approximate “theory of small reflections SFG”
Contains all of the significant physical propagation mechanisms
of the two forward paths, and only the two significant
propagation mechanisms of the two forward paths.
Namely:
1. The reflection at the connector (i.e., Γ ).
2. The propagation down the quarter-wave transmission
line (e − j β A ), the reflection off the load ( ΓL ), and the
propagation back up the quarter-wave transmission line
( e − j β A ).
a
Z0
p2
Z 0RL
p1
p2
a
e −j βA
The approximate
SFG when applying
the theory of
small reflections!
p1
ΓL
Γ
e −j βA
b
Jim Stiles
RL
The Univ. of Kansas
Dept. of EECS
4/1/2009
The Theory of Small Reflections
a
From
series
rule
Γ
8/9
a
From
parallel
rule
ΓL e − j 2 β A
b
Γ + ΓL e − j 2β A
b
Q: But wait! The quarter-wave transformer is a matching
network, therefore Γin = 0 . The theory of small reflections,
however, provides the approximate result:
Γin ≈ Γ + Γ L e − j 2 β A
Is this approximation very accurate? How close is this
approximate value to the correct answer of Γin = 0 ?
A: Let’s find out!
Recall that Γ = Γ L for a properly designed quarter-wave
matching network, and so:
Γin ≈ Γ + Γ L e − j 2 β A
(
= ΓL 1 + e − j 2 β A
)
Likewise, A = λ 4 (but only at the design frequency!) so that:
⎛ 2π
2β A = 2 ⎜
⎝ λ
⎞λ
⎟ =π
⎠4
where you of course recall that β = 2π λ !
Jim Stiles
The Univ. of Kansas
Dept. of EECS
4/1/2009
The Theory of Small Reflections
Thus:
9/9
(
)
(1 + e )
Γin ≈ Γ L 1 + e − j 2 β A
= ΓL
−jπ
= Γ L (1 − 1 )
=0
!!!
Q: Wow! The theory of small reflections appears to be a
perfect approximation—no error at all!?!
A: Not so fast.
The theory of small reflections most definitely provides an
approximate solution (e.g., it ignores most of the terms of the
propagation series, and it approximates connector transmission
as Τ = 1 , when in fact Τ ≠ 1 ).
As a result, the solutions derived using the theory of small
reflections will—generally speaking—exhibit some (hopefully
small) error.
We just got a bit “lucky” for the quarter-wave
matching network; the “approximate” result Γin = 0
was exact for this one case!
Æ The theory of small reflections is an approximate
analysis tool!
Jim Stiles
The Univ. of Kansas
Dept. of EECS