Dr. Vahid Nayyeri
Microwave Ciruits Design
The Theory of Small Reflections
Γ?
Γ1 =
Z 2 − Z1
Z 2 + Z1
Γ3 =
Γ2 = −Γ1
T21 = 1 + Γ1 =
ZL − Z2
ZL + Z2
2Z 2
Z1 + Z 2
2 Z1
T12 = 1 + Γ2 =
Z1 + Z 2
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Massachusetts Institute of Technology
RF Cavities and Components for Accelerators
USPAS 2010
22
Dr. Vahid Nayyeri
Microwave Ciruits Design
Γ = Γ1 + T12T21Γ3e
−2 jθ
2
−4 jθ
12 21 3 2
+T T Γ Γ e
∞
1
x
=
∑
1− x
n =0
n
for
x <1
+  = Γ1 + T12T21Γ3e
T12T21Γ3e −2 jθ
Γ = Γ1 +
1 − Γ2 Γ3e − 2 jθ
− 2 jθ
∞
n n − 2 jnθ
Γ
∑ 2 Γ3 e
n =0
Γ2 = −Γ1
2Z 2
T21 = 1 + Γ1 =
Z1 + Z 2
2 Z1
T12 = 1 + Γ2 =
Z1 + Z 2
(1 + Γ1 )(1 − Γ1 )Γ3e −2 jθ
(1 − Γ12 )Γ3e −2 jθ Γ1 + Γ12 Γ3e −2 jθ + (1 − Γ12 )Γ3e −2 jθ
Γ = Γ1 +
= Γ1 +
=
− 2 jθ
− 2 jθ
1 + Γ1Γ3e
1 + Γ1Γ3e
1 + Γ1Γ3e − 2 jθ
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Massachusetts Institute of Technology
RF Cavities and Components for Accelerators
USPAS 2010
23
Microwave Ciruits Design
Dr. Vahid Nayyeri
Γ1 + Γ3e −2 jθ
Γ=
1 + Γ1Γ3e − 2 jθ
If the discontinuities between the impedances Z1, Z2, and Z2, ZL are
small, then |Γ1 Γ3| <<1.
Γ ≅ Γ1 + Γ3e −2 jθ
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Dr. Vahid Nayyeri
Microwave Ciruits Design
Z1 − Z o
Γ0 =
Z1 + Z o
ZL − ZN
ΓN =
ZL + ZN
Z n +1 − Z n
Γn =
Z n +1 + Z o
Γ ≅ Γ1 + Γ3e −2 jθ
Γ = Γ(θ ) ≅ Γ0 + Γ1e −2 jθ + Γ2 e −4 jθ +  + ΓN − 2 e −2 j ( N − 2 )θ + ΓN −1e −2 j ( N −1)θ + ΓN e −2 jNθ
Assume that the transformer is symmetrical
Γ0 = ΓN
Γ1 = ΓN −1
Γ2 = ΓN − 2
etc.
Γ(θ ) = Γ0 + Γ1e −2 jθ + Γ2 e −4 jθ +  + Γ2 e −2 j ( N − 2 )θ + Γ1e −2 j ( N −1)θ + Γ0 e −2 jNθ
(
)
(
)
(
)
Γ(θ ) = Γ0 1 + e − 2 jNθ + Γ1 e − 2 jθ + e − 2 j ( N −1)θ + Γ2 e − 4 jθ + e − 2 j ( N − 2 )θ + 
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Dr. Vahid Nayyeri
Microwave Ciruits Design
[ (
)
(
)
(
)
]
Γ(θ ) = e − jNθ Γ0 e jNθ + e − jNθ + Γ1 e j ( N − 2 )θ + e − j ( N − 2 )θ + Γ2 e j ( N − 4 )θ + e − j ( N − 4 )θ + 
1


Γ(θ ) = 2e − jNθ Γ0 cos Nθ + Γ1 cos( N − 2 )θ +  + Γn cos( N − 2n )θ +  + ΓN / 2 
2


For N even
1


Γ(θ ) = 2e − jNθ Γ0 cos Nθ + Γ1 cos( N − 2 )θ +  + Γn cos( N − 2n )θ +  + Γ( N −1) / 2 cos θ 
2


For N odd
Finite Fourier Cosine Series
By choosing the Γns and enough sections (N) we can achieve the
required response.
Binomial Multisection
Matching Transformers
Chebyshev Multisection
Matching Transformers
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Microwave Ciruits Design
Dr. Vahid Nayyeri
The Binomial MultiSection Transformer
We first consider the Binomial Function:
This function has the desirable properties that:
and that:
In other words, this Binomial Function is maximally flat at the Point
θ=π/2 , where it has a value of Г(θ=π/2 ) = 0
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Microwave Ciruits Design
Dr. Vahid Nayyeri
By letting f=0 (or θ=0) he electrical length (βl) of each section will
likewise approach zero. Thus, the input impedance will simply be
equal to RL, thus
So
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Microwave Ciruits Design
Dr. Vahid Nayyeri
Let’s expand of the Binomial Function:
Where
Compare this to an N-section transformer function:
Of course, we also know that these marginal reflection coefficients are
physically related to the characteristic impedances of each section as:
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Microwave Ciruits Design
Dr. Vahid Nayyeri
You undoubtedly have previously used the approximation:
Now, we know that the values of Zn+1 and Zn in a multi-section
matching network are typically very close, such that
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Microwave Ciruits Design
Then we have
Where
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Dr. Vahid Nayyeri
Microwave Ciruits Design
Bandwidth
In the figure above Гm is maximum allowed reflection
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Dr. Vahid Nayyeri
Microwave Ciruits Design
So we have
Doing some math we will have
On the other word
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Dr. Vahid Nayyeri
Microwave Ciruits Design
Chenushev MultiSection Transformer
Reading Assignment: section 5.7
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Dr. Vahid Nayyeri
Microwave Ciruits Design
Tapered Lines
As we know
Remember that
we will have:
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Dr. Vahid Nayyeri
Microwave Ciruits Design
Dr. Vahid Nayyeri
Finally the reflection coefficient will be
If Z(z) is known, Г(θ) can be found as a function of frequency. Alternatively, if
Г(θ) is specified, then in principle Z(z) can be found by inversion. This latter
procedure is difficult, and is generally avoided in practice
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Microwave Ciruits Design
Exponential Taper
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Dr. Vahid Nayyeri
Microwave Ciruits Design
Triangular Taper
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Dr. Vahid Nayyeri
Microwave Ciruits Design
Klopfenstein Taper
where
And I1 is modified Bessel function
It can be shown that
where
Clearly
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Dr. Vahid Nayyeri
Microwave Ciruits Design
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Dr. Vahid Nayyeri