Transmission Lines
‰
Single stub impedance matching
Impedance matching can be achieved by inserting another
transmission line (stub) as shown in the diagram below
ZA = Z0
ZR
Z0
Z0S
dstub
Lstub
© Amanogawa, 2006 – Digital Maestro Series
192
Transmission Lines
There are two design parameters for single stub matching:
‰
The location of the stub with reference to the load dstub
‰
The length of the stub line Lstub
Any load impedance can be matched to the line by using single
stub technique. The drawback of this approach is that if the load is
changed, the location of insertion may have to be moved.
The transmission line realizing the stub is normally terminated by a
short or by an open circuit. In many cases it is also convenient to
select the same characteristic impedance used for the main line,
although this is not necessary. The choice of open or shorted stub
may depend in practice on a number of factors. A short circuited
stub is less prone to leakage of electromagnetic radiation and is
somewhat easier to realize. On the other hand, an open circuited
stub may be more practical for certain types of transmission lines,
for example microstrips where one would have to drill the insulating
substrate to short circuit the two conductors of the line.
© Amanogawa, 2006 – Digital Maestro Series
193
Transmission Lines
Since the circuit is based on insertion of a parallel stub, it is more
convenient to work with admittances, rather than impedances.
YA = Y0
YR = 1/ZR
Y0 = 1/Z0
dstub
Y0S
Lstub
© Amanogawa, 2006 – Digital Maestro Series
194
Transmission Lines
For proper impedance match:
1
YA = Ystub + Y ( d stub ) = Y0 =
Z0
Line admittance at location
dstub before the stub is applied
Input admittance
of the stub line
Ystub
+
Y (dstub)
YR = 1/ZR
Y0S
Lstub
© Amanogawa, 2006 – Digital Maestro Series
dstub
195
Transmission Lines
In order to complete the design, we have to find an appropriate
location for the stub. Note that the input admittance of a stub is
always imaginary (inductance if negative, or capacitance if positive)
Ystub = jBstub
A stub should be placed at a location where the line admittance has
real part equal to Y0
Y ( d stub ) = Y0 + jB ( d stub )
For matching, we need to have
Bstub = − B ( d stub )
Depending on the length of the transmission line, there may be a
number of possible locations where a stub can be inserted for
impedance matching. It is very convenient to analyze the possible
solutions on a Smith chart.
© Amanogawa, 2006 – Digital Maestro Series
196
Transmission Lines
First location
suitable for
stub insertion
1
0.5
y(dstub1)
2
3
θ2
θ1
0.2
0
0.2
0.5
zR
1
2
5
Constant
|Γ(d)| circle
-0.2
-3
yR
Load
location
Unitary
conductance
circle
-2
-0.5
-1
© Amanogawa, 2006 – Digital Maestro Series
y(dstub2)
Second location
suitable for stub
insertion
197
Transmission Lines
The red arrow on the example indicates the load admittance. This
provides on the “admittance chart” the physical reference for the
load location on the transmission line. Notice that in this case the
load admittance falls outside the unitary conductance circle. If one
moves from load to generator on the line, the corresponding chart
location moves from the reference point, in clockwise motion,
according to an angle θ (indicated by the light green arc)
4π
d
θ = 2β d =
λ
The value of the admittance rides on the red circle which
corresponds to constant magnitude of the line reflection coefficient,
|Γ(d)|=|ΓR |, imposed by the load.
Every circle of constant |Γ(d)| intersects the circle Re { y } = 1
(unitary normalized conductance), in correspondence of two points.
Within the first revolution, the two intersections provide the
locations closest to the load for possible stub insertion.
© Amanogawa, 2006 – Digital Maestro Series
198
Transmission Lines
The first solution corresponds to an admittance value with positive
imaginary part, in the upper portion of the chart
(
Y ( d stub1 ) = Y0 + j B d stub1
Line Admittance - Actual :
y ( d stub1 ) = 1 + j b ( d stub1 )
Normalized :
θ1
λ
Stub Location : d stub1 =
4π
(
)
− j b ( d stub1 )
− j B d stub1
Stub Admittance - Actual :
Normalized :

λ
tan −1 
Stub Length : Lstub =
 Z0 s B
2π

λ
tan −1 Z0 s B
Lstub =
2π
(
© Amanogawa, 2006 – Digital Maestro Series
)
1
d stub1

 ( short )


(
)
( dstub1 ) )
(open)
199
Transmission Lines
The second solution corresponds to an admittance value with
negative imaginary part, in the lower portion of the chart
(
Y ( d stub2 ) = Y0 − j B d stub2
Line Admittance - Actual :
y ( d stub2 ) = 1 − j b ( d stub2 )
Normalized :
θ
Stub Location : d stub2 = 2 λ
4π
(
)
j b ( d stub2 )
Stub Admittance - Actual :
j B d stub2
Normalized :

1
λ
1
Stub Length : Lstub =
tan −  −
 Z0 s B d stub
2π
2

λ
Lstub =
tan −1 − Z0 s B d stub2
2π
(
© Amanogawa, 2006 – Digital Maestro Series
)
(
(

 ( short )


)
))
(open)
200
Transmission Lines
If the normalized load admittance falls inside the unitary
conductance circle (see next figure), the first possible stub location
corresponds to a line admittance with negative imaginary part. The
second possible location has line admittance with positive
imaginary part. In this case, the formulae given above for first and
second solution exchange place.
If one moves further away from the load, other suitable locations for
stub insertion are found by moving toward the generator, at
distances multiple of half a wavelength from the original solutions.
These locations correspond to the same points on the Smith chart.
First set of locations
Second set of locations
© Amanogawa, 2006 – Digital Maestro Series
λ
= d stub1 + n
2
λ
= d stub2 + n
2
201
Transmission Lines
Second location
suitable for stub
insertion
1
y(dstub2)
0.5
2
3
0
yR
θ2
0.2
0.2
0.5
Load
location
1
θ1
2
5
Unitary
conductance
circle
Constant
|Γ(d)| circle
-0.2
zR
-3
-2
-0.5
-1
© Amanogawa, 2006 – Digital Maestro Series
y(dstub1)
First location
suitable for stub
insertion
202
Transmission Lines
Single stub matching problems can be solved on the Smith chart
graphically, using a compass and a ruler. This is a step-by-step
summary of the procedure:
(a) Find the normalized load impedance and determine the
corresponding location on the chart.
(b) Draw the circle of constant magnitude of the reflection
coefficient |Γ| for the given load.
(c) Determine the normalized load admittance on the chart. This is
obtained by rotating 180° on the constant |Γ| circle, from the
load impedance point. From now on, all values read on the chart
are normalized admittances.
© Amanogawa, 2006 – Digital Maestro Series
203
Transmission Lines
(a) Obtain the normalized load
1
impedance zR=ZR /Z0 and find
its location on the Smith chart
0.5
2
3
zR
0.2
0
0.2
0.5
-0.2
1
2
5
180° = λ /4
yR
-3
(b) Draw the
constant |Γ(d)|
circle
-2
-0.5
(c) Find the normalized load
admittance knowing that
yR = z(d=λ /4 )
From now on the
represents admittances.
chart
© Amanogawa, 2006 – Digital Maestro Series
-1
204
Transmission Lines
(d) Move from load admittance toward generator by riding on the
constant |Γ| circle, until the intersections with the unitary
normalized conductance circle are found. These intersections
correspond to possible locations for stub insertion. Commercial
Smith charts provide graduations to determine the angles of
rotation as well as the distances from the load in units of
wavelength.
(e) Read the line normalized admittance in correspondence of the
stub insertion locations determined in (d). These values will
always be of the form
y ( d stub ) = 1 + jb
top half of chart
y ( d stub ) = 1 − jb
bottom half of chart
© Amanogawa, 2006 – Digital Maestro Series
205
Transmission Lines
First Solution
(d) Move from load toward
generator and stop at a
location where the real
part of the normalized line
admittance is 1.
05
First location suitable for
stub insertion
1
dstub1=(θ1/4π)λ
2
3
zR
0.2
0
(e) Read here the
value of the
normalized line
admittance
y(dstub1) = 1+jb
θ1
0.2
0.5
1
2
5
-0.2
-3
yR
Load
location
-2
-0.5
Unitary
conductance
circle
-1
© Amanogawa, 2006 – Digital Maestro Series
206
Transmission Lines
Second Solution
(d) Move from load
toward generator and
stop at a location
where the real part of
the normalized line
05
admittance is 1.
1
2
3
θ2
0.2
0
0.2
0.5
zR
1
2
5
-0.2
-3
yR
Load
location
-2
-0.5
(e) Read here the
value of the
normalized line
admittance
y(dstub2) = 1 - jb
-1
© Amanogawa, 2006 – Digital Maestro Series
Unitary
conductance
circle
Second location suitable
for stub insertion
dstub2=(θ2/4π)λ
207
Transmission Lines
(f) Select the input normalized admittance of the stubs, by taking
the opposite of the corresponding imaginary part of the line
admittance
line: y ( d stub ) = 1 + jb
→
stub: ystub = − jb
line: y ( d stub ) = 1 − jb
→
stub: ystub = + jb
(g) Use the chart to determine the length of the stub. The
imaginary normalized admittance values are found on the circle
of zero conductance on the chart. On a commercial Smith chart
one can use a printed scale to read the stub length in terms of
wavelength. We assume here that the stub line has
characteristic impedance Z0 as the main line. If the stub has
characteristic impedance Z0S ≠ Z0 the values on the Smith chart
must be renormalized as
Y0
Z0 s
± jb ' = ± jb
= ± jb
Y0 s
Z0
© Amanogawa, 2006 – Digital Maestro Series
208
Transmission Lines
1
0.5
2
3
Short circuit
0.2
0
0.2
-0.2
y=∞
0.5
1
2
5
(g) Arc to determine the length of a
short circuited stub with normalized
input admittance - jb
-3
(f) Normalized input
admittance of stub
-0.5
ystub = 0 - jb
-2
-1
© Amanogawa, 2006 – Digital Maestro Series
209
Transmission Lines
1
0.5
(g) Arc to determine the length of an
open circuited stub with normalized
input admittance - jb
0.2
0
2
0.2
0.5
1
2
3
5
-0.2
y=0
-3
(f) Normalized input
admittance of stub
Open circuit
-0.5
ystub = 0 - jb
-2
-1
© Amanogawa, 2006 – Digital Maestro Series
210
Transmission Lines
1
(f) Normalized input
admittance of stub
ystub = 0 + jb
0.5
2
3
(g) Arc to determine the length of a
short circuited stub with normalized
input admittance + jb
0.2
0
0.2
0.5
1
2
y=∞
Short circuit
5
-0.2
-3
-2
-0.5
-1
© Amanogawa, 2006 – Digital Maestro Series
211
Transmission Lines
1
(f) Normalized input
admittance of stub
ystub = 0 + jb
0.5
0.2
0
2
(g) Arc to determine the length of an
open circuited stub with normalized
input admittance + jb
0.2
0.5
1
2
3
5
-0.2
y=0
-3
Open circuit
-2
-0.5
-1
© Amanogawa, 2006 – Digital Maestro Series
212
Transmission Lines
First Solution
1
0.5
2
3
0.2
0
matching
condition
0.2
0.5
1
2
5
-0.2
-3
yR
-2
-0.5
After the stub is inserted,
the admittance at the stub
location is moved to the
center of the Smith chart,
which
corresponds
to
normalized admittance 1
and reflection coefficient 0
(exact matching condition).
If you imagine to add
gradually
the
negative
imaginary admittance of
the inserted stub, the total
admittance would follow
the yellow arrow, reaching
the match point when the
complete stub admittance
is added.
-1
© Amanogawa, 2006 – Digital Maestro Series
213
Transmission Lines
First Solution
If the stub does not have
the proper normalized input
admittance, the matching
condition is not reached
1
0.5
2
Effect of a stub with
negative susceptance of
insufficient magnitude
3
Effect of a stub with
positive susceptance
0.2
0
0.2
0.5
1
2
5
-0.2
-3
yR
-2
-0.5
Effect of a stub with
negative susceptance of
excessive magnitude
-1
© Amanogawa, 2006 – Digital Maestro Series
214
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