Transmission Lines
Impedance Matching
A number of techniques can be used to eliminate reflections when
the characteristic impedance of the line and the load impedance are
mismatched.
Impedance matching techniques can be designed to be effective for
a specific frequency of operation (narrow band techniques) or for a
given frequency spectrum (broadband techniques).
A common method of impedance matching involves the insertion of
an impedance transformer between line and load
Z0
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Impedance
Transformer
ZR
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Transmission Lines
An impedance transformer may be realized by inserting a section of
a different transmission line with appropriate characteristic
impedance. A widely used approach realizes the transformer with a
line of length λ 4 .
The quarter-wavelength transformer provides narrow-band
impedance matching. The design goal is to obtain zero reflection
coefficient exactly at the frequency of operation.
λ/4
Z0
Zλ/4
|Γ|
Z0
ZR
f0
The length of the transformer is fixed at λ 4 for design
convenience, but is also possible to realize generalized transformer
lines for which the length of the transformer is a design outcome.
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Transmission Lines
A broadband design may be obtained by a cascade of λ 4 line
sections of gradually varying characteristic impedance.
Z0
λ/4
λ/4
λ/4
λ/4
Z1
Z2
Z3
Z4
|Γ|
Z0
ZR
|Γ|max
f0
It is not possible to obtain exactly zero reflection coefficient for all
frequencies in the desired band.
Therefore, available design approaches specify a maximum
reflection coefficient (or maximum VSWR) which can be tolerated in
the frequency band of operation.
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Transmission Lines
Another broadband matching approach may use a tapered line
transformer with continously varying characteristic impedance
along its length. In this case, the design obtains reflection
coefficients lower than a specified tolerance at frequencies
exceeding a minimum value.
Z0(x)
|Γ|
|Γ|max
x
fmin
Various taper designs are available, including linear, exponential,
and raised-cosine impedance profiles. An optimal design (due to
Klopfenstein) involves discontinuity of the impedance at the
transformer ends.
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Transmission Lines
Another narrow-band approach involves the insertion of a shunt
imaginary admittance on the line. Often, the admittance is realized
with a section (or stub) of transmission line and the technique is
commonly known as stub matching. The end of the stub line is
short-circuited or open-circuited, in order to realize an imaginary
admittance. Designs are also available for two or three shunt
admittances placed at specified locations on the line.
Other narrow-band examples involve the insertion of a series
impedance (stub) along the line, and the insertion of a series and a
shunt element in L-configuration.
ZR
ZR
ZR
The theory for several basic narrow-band matching techniques is
detailed in the following. Note that the effect of loss in the
transmission lines is always neglected.
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Transmission Lines
Matching I: Impedance Transformers
• Quarter Wavelength Transformer – A simple narrow band
impedance transformer consists of a transmission line section
of length λ 4
ZA
ZB
Z01
Z02
λ/4
Z01
ZR
dmax or dmin
The impedance transformer is positioned so that it is connected to
a real impedance ZA. This is always possible if a location of
maximum or minimum voltage standing wave pattern is selected.
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Transmission Lines
Consider a general load impedance with its corresponding load
reflection coefficient
ZR = RR + jX R ;
ZR − Z01
ΓR =
= Γ R exp ( jφ )
ZR + Z01
If the transformer is inserted at a location of voltage maximum dmax
1 + Γ (d)
1 + ΓR
= Z01
ZA = Z01
1 − Γ (d)
1 − ΓR
If it is inserted instead at a location of voltage minimum dmin
1 + Γ (d)
1 − ΓR
= Z01
ZA = Z01
1 − Γ (d)
1 + ΓR
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Transmission Lines
Consider now the input impedance of a line of length λ 4
Zin
Z0
ZA
L = λ/4
Since:
1 + Γ (d)
1 − ΓR
= Z01
ZA = Z01
1 − Γ (d)
1 + ΓR
we have
Zin =
lim
tan( β L )→∞
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ZA + jZ0 tan(β L)
Z0
→
jZA tan(β L) + Z0
Z02
ZA
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Transmission Lines
Note that if the load is real, the voltage standing wave pattern at the
load is maximum when ZR > Z01 or minimum when ZR < Z01 . The
transformer can be connected directly at the load location or at a
distance from the load corresponding to a multiple of λ 4 .
ZA=Real
ZB
Z01
Z02
Z01
ZR=Real
d1
λ/4
n λ/4 ; n=0,1,2…
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Transmission Lines
If the load impedance is real and the transformer is inserted at a
distance from the load equal to an even multiple of λ 4 , then
ZA = ZR ;
λ
λ
d1 = 2 n = n
4
2
but if the distance from the load is an odd multiple of λ 4
2
Z01
ZA =
ZR
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;
λ
λ λ
d1 = (2 n + 1) = n +
4
2 4
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Transmission Lines
The input impedance of the impedance transformer after inclusion
in the circuit is given by
2
Z02
ZB =
ZA
For impedance matching we need
2
Z02
Z01 =
ZA
⇒
Z02 = Z01 ZA
The characteristic impedance of the transformer is simply the
geometric average between the characteristic impedance of the
original line and the load seen by the transformer.
Let’s now review some simple examples.
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Transmission Lines
‰
Real Load Impedance
ZA
ZB
Z01 = 50 Ω
Z02 = ?
RR = 100 Ω
λ/4
2
Z02
ZB =
= Z01 ⇒ Z02 = Z01 RR = 50 ⋅ 100 ≈ 70.71 Ω
RR
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Transmission Lines
Note that an identical result is obtained by switching Z01 and RR
ZA
ZB
Z01 = 100 Ω
Z02 = ?
RR = 50 Ω
λ/4
2
Z02
ZB =
= Z01 ⇒ Z02 = Z01 RR = 100 ⋅ 50 ≈ 70.71 Ω
RR
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Transmission Lines
Another real load case
ZA
ZB
Z01 = 75 Ω
Z02 = ?
RR = 300 Ω
λ/4
2
Z02
ZB =
= Z01 ⇒ Z02 = Z01 RR = 75 ⋅ 300 = 150 Ω
RR
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Transmission Lines
Same impedances as before, but now the transformer is inserted at
a distance λ 4 from the load (voltage minimum in this case)
ZB
Z01 = 75 Ω
2
Z01
752
ZA =
=
= 18.75 Ω
RR 300
ZA
Z02
λ/4
Z01
RR = 300 Ω
λ/4
2
Z02
ZB =
= Z01 ⇒ Z02 = Z01 ZA = 75 ⋅ 18.75 = 37.5 Ω
ZA
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Transmission Lines
‰
Complex Load Impedance – Transformer at voltage maximum
ZB
Z01 = 50 Ω
ZA
Z02
Z01
λ/4
dmax
ZR = 100 + j 100Ω
100 + j100 − 50
ΓR =
≈ 0.62
100 + j100 + 50
1 + ΓR
≈ 213.28Ω
ZA = Z0
1 − ΓR
Z02 = Z01 ZA = 50 ⋅ 213.28 = 103.27 Ω
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Transmission Lines
‰
Complex Load Impedance – Transformer at voltage minimum
ZB
Z01 = 50 Ω
ZA
Z02
Z01
λ/4
dmin
ZR = 100 + j 100Ω
100 + j100 − 50
ΓR =
≈ 0.62
100 + j100 + 50
1 − ΓR
ZA = Z0
≈ 11.72 Ω
1 + ΓR
Z02 = Z01 ZA = 50 ⋅ 11.72 = 24.21 Ω
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Transmission Lines
• Generalized Transformer
If it is not important to realize the impedance transformer with a
quarter wavelength line, one may try to select a transmission line
with appropriate length and characteristic impedance, such that the
input impedance is the required real value
ZA
Z01
Z02
ZR = RR + jXR
L
RR + jX R + jZ02 tan(β L)
Z01 = ZA = Z02
Z02 + j ( RR + jX R ) tan(β L)
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Transmission Lines
After separation of real and imaginary parts we obtain the equations
Z 02 ( Z 01 − RR ) = Z 01 X R tan(β L)
tan(β L) =
Z 02 X R
2
Z 01RR − Z 02
with final solution
Z 02 =
Z 01RR − RR2 − X R2
tan ( β L ) =
1 − RR / Z 01
(
2
2
−
−
−
R
Z
Z
R
R
X
1
/
(
01 )
01 R
R
R
R
XR
The transformer can be realized as long as the result for
Note that this is also a narrow band approach.
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)
Z02 is real.
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Transmission Lines
Matching II – Shunt Admittance
We wish to insert a parallel (shunt) reactance on the transmission
line to obtain impedance matching. Since the design involves a
parallel circuit, it is more convenient to consider admittances:
Y1
Y1’
Z0
jX
ZR
Y0 = 1/Z0
ds
−jB
YR = 1/ZR
ds
The shunt may be inserted at locations ds where the real part of the
line admittance is equal to the characteristic admittance Y0
Y1' = Y0 + jB
Matching is obtained by using a shunt susceptance −jB so that
Y1 = [ Z (d s )]−1 − jB = Y1' − jB = Y0
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Transmission Lines
To solve this design problem, we need to find the suitable locations
ds (where the real part of the line admittance is equal to Y0) and the
corresponding values of the shunt susceptance −B.
The shunt element may be also realized by inserting a segment of
transmission line of appropriate length, called a stub.
In order to obtain a pure susceptance, the stub element may consist
of a short-circuited or an open-circuited transmission line with
input admittance –jB.
Y1
Y1’
Y0
−jB
Y1
Y1’
YR
ds
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Y0
−jB
YR
ds
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Transmission Lines
The line admittance at location ds can be expressed as a function of
reflection coefficient

Y1' = Y0 + jB =  Z 0

1 + Γ (d s ) 
1 − Γ (d s ) 
−1
1 − Γ (d s )
= Y0
1 + Γ (d s )
For more general results, we introduce normalization:
1 − Γ (d s )
Y1'
'
= y1 = 1 + jb =
= normalized admittance
Y0
1 + Γ (d s )
b = normalized susceptance
Then, the line reflection coefficient can be expressed in terms of b
1 − Γ (d s )
1−
'
y1 =
⇒ Γ (d s ) =
1 + Γ (d s )
1+
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y1' 1 − (1 + jb)
− jb
=
=
'
y1 1 + 1 − jb 2 + jb
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Transmission Lines
Γ (d s ) = Γ R exp(− j 2β d s )
− jb
Γ R = Γ R exp ( jθ ) =
exp( j 2β d s )
2 + jb
Since we know that
b
=
4 + b2
exp ( ∓ j π 2 − tan −1 ( b 2 ) ) exp( j 2β d s + j 2nπ )
− for b > 0;
+ for b < 0
Added to account for
periodic behavior
The absolute value of the load reflection coefficient provides b
ΓR =
b
4 + b2
⇒
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b2 =
4 ΓR
2
1− ΓR
2
⇒ b=±
2 ΓR
2
1− ΓR
B = b ⋅ Y0 = b / Z 0
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Transmission Lines
Finally, the phase of the load reflection coefficient yields ds
∠ Γ R = θ = ∓ π 2 + 2 β d s − tan −1 ( b 2 ) + 2 n π
4π
⇒ 2 β ds =
d s = θ ∓ π 2 − tan −1 ( b 2 ) − 2 n π
λ
λ
ds =
(θ ± π 2 + tan −1 ( b 2 ) − 2 n π )
4π
+ for b > 0;
− for b < 0
The last term accounting for periodic behavior of the solution gives
λ
λ
2nπ = n
4π
2
indicating that the solutions repeat every λ 2 along the line.
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