3/26/2007
5_5 The Theory of Small Reflections
1/1
5.5 – The Theory of Small Reflections
Reading Assignment: pp. 244-246
An important and useful approximation when considering
multi-section matching networks is the Theory of Small
Reflections.
HO: THE THEORY OF SMALL REFLECTIONS
We can use the Theory of Small Reflections to provide an
approximate analysis of a multi-section impedance
transformer (i.e., multi-section matching network).
HO: THE MULTISECTION TRANSFORMER
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Theory of Small Reflections
1/5
The Theory of
Small Reflections
Recall that we analyzed a quarter-wave transformer using the
multiple reflection view point.
Τ
Τ
V +(z )
V −( z )
Z0
Γ
Z1 =
−Γ
Z 0RL
RL
ΓL
A = λ
4
We found that the solution could be written as an infinite
summation of terms:
∞
V ( z ) = ∑Vn −( z )
−
n =1
where each term had a specific physical interpretation, in
terms of reflections, transmissions, and propagations.
For example, the first term was V1 − ( z = −A ) = Γ V + ( z = −A ) :
V +(z )
Z 0RL
Z0
RL
V1 −( z )
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Theory of Small Reflections
a5
b2
a1
−j
Τ
−Γ
Γ
ΓL
−j
Τ
b1
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a2
b5
While the second term was:
V +(z )
V2−( z )
Z 0RL
Z0
−j
Τ
V2−( z = −A ) =
−Γ
Γ
b1
a5
b2
a1
−Τ2 Γ L V + ( z = −A )
RL
ΓL
−j
Τ
a2
b5
Contrast these first two terms with the third term:
V +(z )
Z 0RL
Z0
RL
V3−( z )
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Theory of Small Reflections
V3 ( z = −A ) =
−Γ
Γ
b1
−j
Τ
−
2
a5
b2
a1
−Τ2 ( ΓL ) Γ V + ( z = −A )
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ΓL
−j
Τ
a2
b5
This third term is characterized by three reflections, and thus
there is a product of three reflection coefficient values Γ in the
term. In fact, all other terms in this infinite series will likewise
describe mechanisms with multiple reflections.
This is in contrast with the first two terms, which exhibit just a
single reflection coefficient in the product.
Q: How is this even remotely important?
A: This is an important observation when we consider the case
where Z0 and RL are very close in value. If this is true, then the
value Z 0RL must likewise be close in value to both Z0 and RL.
As a result, we find that the magnitudes of the marginal
reflection coefficients Γ and ΓL will be very small:
Γ 1
while the magnitudes of the marginal transmission coefficient
Τ will be approximately equal to one:
Τ 1 .0
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Theory of Small Reflections
4/5
This means that all higher-order terms (those with multiple
reflections and thus with products of multiple reflection
coefficients) will be very, very small:
n
Γ ≈0
for n ≥ 2
The result is that these higher order terms make little
difference in the final result—the first two (single reflection)
terms are the dominant terms in the infinite series.
Therefore IF Z0 and RL are very close in value, we find that we
can approximate the reflected wave using only the first two
terms of the infinite series:
V −( z = −A ) Γ V + ( z = −A ) − Τ2 ΓL V + ( z = −A )
= ( Γ − Τ2 ΓL )V + ( z = −A )
Now, if we likewise apply the approximation that Τ 1.0 , we
conclude for this quarter wave transformer (at the design
frequency):
V −( z = −A ) ( Γ − Τ2 ΓL )V + ( z = −A )
( Γ − Γ L )V + ( z = −A )
This approximation (using only single reflection terms) is known
as the Theory of Small Reflections, and allows us to use the
multiple reflection view point as an analysis tool (we don’t have
to consider an infinite number of terms!).
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Theory of Small Reflections
5/5
Note that this result is provided from the signal flow graph:
a1
b1
2
= Γ + ( − j ) ΓL
a1
−j
1 .0
ΓL
Γ
= Γ − ΓL
−j
1.0
b1
a2
−j
Γ
b1
b5
a5
a1
The approximate
SFG when
applying the
theory of small
reflections!
a5
b2
ΓL
−j
b5
However, we can only use this approximation when the marginal
reflection coefficient at each transition is very small (i.e., the
change in impedance is slight).
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Multisection Transformer
1/6
The Multi-section
Transformer
Consider a sequence of N transmission line sections; each
section has equal length A , but dissimilar characteristic
impedances:
Γ1
Γ0
Z0
Γin
Γ2
Z1
Z2
A
A
ΓN − 1
ΓN
"
ZN
RL
A
Where the marginal reflection coefficients are:
Γ0 Z1 − Z 0
Z1 + Z 0
Γn Z n +1 − Z n
Z n +1 + Z n
ΓN RL − Z N
RL + Z N
If the load resistance RL is less than Z 0 , then we should design
the transformer such that:
Z 0 > Z 1 > Z 2 > Z 3 " > Z N > RL
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Multisection Transformer
2/6
Conversely, if RL is greater than Z 0 , then we will design the
transformer such that:
Z 0 < Z 1 < Z 2 < Z 3 " < Z N < RL
In other words, we gradually transition from Z0 to RL !
Note that since RL is real, and since we assume lossless
transmission lines, all Γn will be real (this is important!).
Likewise, since we gradually transition from one section to
another, each value:
Z n +1 − Z n
will be small.
As a result, each marginal reflection coefficient Γn will be real
and have a small magnitude.
This is also important, as it means that we can apply the “theory
of small reflections” to analyze this multi-section transformer!
The theory of small reflections allows us to approximate the
input reflection coefficient of the transformer as:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Multisection Transformer
Γ1
Γ0
Z0
Γin
The approximate
SFG when
applying the
theory of small
reflections!
Γ2
Z1
Z2
A
A
a1
a0
e −j βA
"
......
Γ2
e −j βA
b1
ΓN
ZN
RL
A
a2
Γ1
e −j βA
b0
ΓN − 1
e −j βA
Γ0
3/6
b2
......
aN −1
aN
e −j βA
ΓN −1
ΓL = ΓN
e −j βA
bN −1
bN
b0
= Γin ( β )
a0
Γ 0 + Γ1 e − j 2 β A + Γ2 e − j 4 β A + " + ΓN e − j 2N β A
N
= ∑ Γn e − j 2n β A
n =0
Q: So why is the input reflection coefficient a function of β ?
Isn’t β a constant?
A:
Nope. Recall that for a lossless line:
β = ω LC
In other words, beta (and thus the electrical length of each
transmission line section) is a function of frequency! Recall:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Multisection Transformer
βA = ω
A
vp
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= ωT
where:
T A
vp
= propagation time through 1 section
Therefore, we can alternatively express the input reflection
coefficient as a function of frequency:
Γin (ω ) = Γ 0 + Γ1e − j 2ωT + Γ2e − j 4ωT + " + ΓN e − j 2N ωT
N
= ∑ Γn e − j (2nT )ω
n =0
We see that the function Γin (ω ) is expressed as a weighted set
of N basis functions! I.E.,
N
Γin (ω ) = ∑ cn Ψ (ω )
n =0
where:
cn = Γn
and
Ψ (ω ) = e
− j (2nT )ω
We find, therefore, that by selecting the proper values of basis
weights cn (i.e., the proper values of reflection coefficients Γn ),
we can synthesize any function Γin (ω ) of frequency ω , provided
that:
1. Γin (ω ) is periodic in ω = 1 2T
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Multisection Transformer
5/6
2. we have sufficient number of sections N.
Q: What function should we synthesize?
A: Ideally, we would want to make Γin (ω ) = 0 (i.e., the
reflection coefficient is zero for all frequencies).
Bad news: this ideal function Γin (ω ) = 0 would require an
infinite number of sections (i.e., N = ∞ )!
Instead, we seek to find an “optimal” function for Γin (ω ) , given
a finite number of N elements.
Once we determine these optimal functions, we can find the
values of coefficients Γn (or equivalently, Zn ) that will result in
a matching transformer that exhibits this optimal frequency
response.
To simplify this process, we can make the transformer
symmetrical, such that:
Γ 0 = ΓN , Γ1 = ΓN −1 , Γ2 = ΓN −2 , ""
Note that this does NOT mean that:
Z 0 = Z N , Z1 = Z N −1 , Z2 = Z N −2 , ""
We find then that:
Jim Stiles
The Univ. of Kansas
Dept. of EECS
3/26/2007
The Multisection Transformer
(
)
6/6
(
Γ (ω ) = e − jN ωT ⎡Γ 0 e jN ωT + e − jN ωT + Γ1 e j (N −2)ωT + e − j (N −2)ωT
⎣
(
+ Γ2 e
j (N − 4 )ωT
and since:
+e
− j (N − 4 )ωT
)
) + "⎤⎦
e j x + e − j x = 2 cos ( x )
we can write for N even:
Γ (ω ) = 2 e − j N ωT ⎡⎣Γ 0 cosN ωT + Γ1 cos (N − 2 ) ωT
1
⎤
+ " + Γn cos (N − 2n ) ωT + " + ΓN 2 ⎥
2
⎦
whereas for N odd:
Γ (ω ) = 2 e − j N ωT ⎡⎣Γ 0 cosN ωT + Γ1 cos (N − 2 ) ωT
+ " + Γn cos (N − 2n ) ωT + " + Γ (N −1) 2 cosωT ⎤⎦
The remaining question then is this: given an optimal and
realizable function Γin (ω ) , how do we determine the necessary
number of sections N, and how do we determine the values of all
reflection coefficients Γn ??
Jim Stiles
The Univ. of Kansas
Dept. of EECS