Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007
340
Impedance Matching by Using a Multi-stub System
CAROLINA REGOLI
Universidad Central de Venezuela
Electrical Engineering School
Los Chaguaramos, Ciudad Universitaria, Caracas
VENEZUELA
cregoli@elecrisc.ing.ucv.ve
Abstract: Nowadays lots of applications in telecommunications require a wide bandwidth to transmit more information and improve the quality on the transmission of this information. In order to guarantee this condition low
losses systems are designed; these losses can be classified in dissipation losses and mismatch losses. There are
different classic methods for impedance matching, but they reach the impedance matching in a very narrow bandwidth, whereas the bandwidth used in different applications is generally greater in accordance with the quantity
of information to be transmitted. A solution proposed in this paper to reach an impedance matching into a wider
bandwidth is through the connection of various stub lines in parallel all along the transmission line, so we can have
a multiple variables system, with more grades of liberty. A software that automatizes the calculations to reach
the impedance matching was programmed, by using the minimum squares and Nelder-Mead methods. Once the
user introduces the data of Standing Wave Ratio (SWR) and bandwidth desired, the load impedance and the work
frequency of the system, the program determines the minimum number of stubs necessary to match the system,
and the longitudes and distances along the line where they should be located.
Key–Words: SWR, stub, impedance matching, optimization
1
Introduction
The impedance matching is a fundamental stage in the
design process of any telecommunication system. The
essential reasons why it is required to match the system correctly are two: To protect the generator from
the effects caused by the reflected waves at the load,
and to avoid the transmission losses when the load is
mismatched from the line.
Generally the matching network is constructed
with non dissipative elements, to avoid unnecessary
power losses, and it is designed in order the network
input impedance to the line to be Zo. When this is
achieved, the reflections in the transmission line disappear. This means that the reflection coefficient is
null and therefore the SWR is equal to one. There
are several classic techniques for impedance matching, like for example λ/4 line and one stub matching, among others. In general, in a communication
system the impedance matching is needed in a bandwidth, which depends on the application the system is
designed for. The classic methods mentioned before
only reach the impedance matching at one frequency.
This paper proposes to use an optimization algorithm to achieve the impedance matching into the
desired frequency range, from the knowledge of the
system requirements. This algorithm is based on the
connection of several stub lines in parallel all along
the transmission line to reach an impedance matching
into a wider bandwidth. The software automatizes the
calculus to reach the impedance matching, and give
to the user the minimum number of stubs necessary
to match the system, and the longitudes and distances
along the line where they should be located.
2
Impedance matching methods
There are a lot of classic methods to achieve
impedance matching in a telecommunication system.
Some of them are mentioned below:
◦ λ/4 transformer
◦ series-parallel matching
◦ parallel-series matching
◦ one stub matching
The problem with these methods is that they reach
the impedance matching at only one frequency, instead of reaching this into the bandwidth desired by
the user. However, we will explain here the one
stub matching method, because from this one an other
method that improves the impedance matching is originated: two stub matching, and also it is the foundation of this work.
Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007
2.1
341
One stub matching
The objective is to achieve the impedance matching
for the system at one specific frequency, by connecting one stub line in parallel to the line at a distance
ds from the load, with longitude ls , like it is shown in
figure 1. In general ds and ls have to be calculated.
Figure 2: SWR for one stub
2.2
Figure 1: Connection of one stub
From figure 1 the following equations can be obtained:
Ya = Ys + Y 0
(1)
where
Y 0 = Y0 ·
1−Γ
1+Γ
Two stub matching
This is an other method for impedance matching,
where instead of connecting one stub to the line, the
user adds an other stub. The circuit obtained is as
shown in figure 3.
(2)
being Γ the reflection coefficient at a distance ds from
the load, defined by:
Γ = Γl · e−2γds
Y0 − Yc
Γl =
Y0 − Yc
(3)
(4)
The stub line reflection coefficient, if it is ended
in a short circuit, is:
Γs = −e
−2jβls
(5)
and its input admittance is:
Ys = Y0s ·
1 − Γs
1 + Γs
(6)
In order to reach the impedance matching Ya must
be equal to Y0 . Now, ls and ds values can be obtained
from the previous equations. This way, the system is
matched at the frequency given by:
β=
2π
λ
(7)
where λ = vp /f .
The following figure shows a typical curve of the
Standing Wave Ratio obtained by matching with one
stub. The impedance matching is reached only at one
frequency:
Figure 3: Circuit with two stubs
In this case, the equations to obtain the values of
l1 , d1 , l2 y d2 are the followings:
Ya = Y1 + Y10
1 − Γl · e−2γd1
Y10 = Y0 ·
1 + Γl · e−2γd1
Y0 − Yc
Γl =
Y0 − Yc
1 + e−2jβl1
Y1 = Y0s1 ·
1 − e−2jβl1
Yb = Y2 + Y20
1 − Γa · e−2γd2
Y20 = Y0 ·
1 + Γa · e−2γd2
Y0 − Ya
Γa =
Y0 − Ya
1 + e−2jβl2
Y2 = Y0s2 ·
1 − e−2jβl2
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
The figure 4 shows a typical curve of the Standing
Wave Ratio obtained by matching with two stubs. It is
possible to see that the impedance matching into the
Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007
bandwidth is better that the one obtained in figure 2,
by using the one stub matching method.
going to be maximized or minimized, depending on
the case.
In this paper the function to be minimized is
SWR. This relation depends on the reflection coefficient , through the following equation:
SW R =
Figure 4: SWR for two stubs
2.3
342
1 + |Γ|
1 − |Γ|
(16)
It is easier to work with |Γ|, because its range of variation is 0 < |Γ| < 1, instead of SWR that varies between zero and infinity.
Finally, the objective function chose was |Γ|,
knowing that if this function is minimized, automatically SWR is minimized too.
It was decided to minimize the objective function
by minimum squares, because in other works developed before the results obtained were very good [2].
Matching at several frequencies
Based on the previous theory, if impedance matching
is wanted in a specific bandwidth, various stubs can be
connected along the line. This way a SWR near to one
can be obtained along all the bandwidth, as shown in
figure 5. In this figure there are the system ideal and
real responses, which should be into a minimum error
margin. As it is very difficult to achieve a SWR equal
to one (null reflection coefficient) for all the frequencies, it is necessary to impose a maximum value of it,
which can not be overcome on the desired bandwidth.
3.1
Minimum squares
The function considered G(f, a1 , a2 , . . . , an ) is con∆f
tinuous in f at the interval I = [f0 − ∆f
2 ; f0 + 2 ],
and depends on the N parameters a1 , a2 , . . . , an . For
each Ai set there is a function G and the problem is
to find the function G which better approximates into
the interval I to the previous selected reference function (R).
Having the reference function evaluated in several
points into the interval I, the quadratic errors can be
determinate as follows:
F (a1 , ··, an ) =
M
X
[G(fi , a1 , ··, an ) − R(fi )]2 (17)
i=1
The expression (17) is the objective function to
be minimized, and that occurs when ∇F = 0. The
solution to this problem is reached when the optimum
values of a1 , a2 , . . . , an are obtained.
3.2
Figure 5: SWR obtained with various stubs
Due to the difficulty of the calculus for several
stubs along the line, it is better to use a numerical solution than an analytical one, which is a very good
reason to develop a software to solve the problem.
3 Numerical solution
The first thing that has to be done in an optimization
problem is to define the objective function, which is
Numerical method used
The function to be minimized is a nonlinear function,
and it is also very complex. For these reason it was important to choose an adequate optimization method, so
the time used by the program to make the calculations
could be minimum. In order to reach this, the NelderMead method was used. It belongs to a class of methods that use comparisons between different values of
the objective function and do not require the use of
any derivatives, called direct search methods.
Nelder-Mead method is generally used for unconstrained optimization. In this case there are some constraints, such as the values of distances and longitudes,
that have to be positive; these constraints were programmed apart.
Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007
In this method, once the function is evaluated at
several points, the one that gives the worst value of
the objective function is replaced for an other one, by
using one of the four basic operations, shown in Figure 6. Then the function is evaluated again at this new
point, if there is an improvement respect to one of the
others points, then the algorithm starts again with the
new worst point. If not, it is necessary to apply an
other different operation that the one was applied first.
Figure 6: Basics operations in Nelder-Mead
4
Programming
The objective of the software is to find the number of
stubs required to reach the impedance matching into
the bandwidth selected by the user, and their longitudes and location along the transmission line. For
this paper the stubs and transmission line have the
same characteristic impedance, and they do not have
attenuation losses.
343
impedance of the system, the interval of frequencies
and the characteristic impedance of the system. Also
the user has to put the maximum value of SWR allowed into the bandwidth as a tolerance margin, so
the method convergence can be assured. It can also
be chose the maximum number of stubs desired. The
outputs are the longitudes and distances of the stub
lines, and the quantity of stubs necessaries to find the
best solution. The user interface is shown in Figure
7.The function to be minimized is the quadratic error
between ρ = |Γ| and the reference function, given in
(18):
2
R = ∆ · ( )2m · (f − f0 )2m
(18)
B
where ∆ is adjusted in accordance with the maximum
value of SWR allowed in the bandwidth [2]. The value
of m adjusts the form of R to the desired form of SWR
into the interval.
The equations used for ρ = |Γ| are the same explained in Section 2. The only difference is that the
Γc , for each stub is the result of the Γ obtained at the
connection point of the previous stub and so on.
The Nelder-Mead method can converge a local
optimum, depending on the initial conditions. In order to assure the convergence to the global optimum,
is very important an adequate selection of the initial
conditions. For this reason a subprogram was made
to calculate the best initial conditions. After that, the
main program calculates the number of stubs, their
longitudes and locations along the transmission line.
The program has a limit of eight possible stubs to
be connected. Of course, if it is necessary this limit
could be change . Once the user introduces the data,
the program starts calculating the number of stubs that
minimizes the quadratic error between ρ and the reference function. If the condition of maximum value of
SWR introduced by the user is not satisfied, the program shows a message suggesting the change of this
condition or the bandwidth.
Once the program finds the solution, the user can
also see the SWR curve at the point of the last stub
connection.
5
Results obtained
The program was validated for different examples,
and the solutions obtained were always satisfactory.
An example is presented here, for the following data:
◦ Interval of frequencies: 1.1GHz to 1.3GHz
◦ Z0 = 50Ω
Figure 7: User Interface
The input data for the software are:
◦ Zc = 150 − j60Ω
load
◦ maximum SWR= 1.15
Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007
stub
1
2
3
4
5
6
7
l (mm)
24.5371
63.3895
65.3817
61.4128
60.2661
60.369
64.2648
d (mm)
39.9823
38.8459
5.83867
4.07737
65.0554
95.5695
40.3593
Table 1: Results obtained with the program
The results obtained are shown in Table 1.
Each stub distance is referred to the previous stub
connection point, except for the first stub, which reference point is the load, as shown in Figure 8.
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to achieve the system impedance matching. The problem with the classic impedance matching methods is
that they only solve the problem at one frequency or
in a very narrow band.
The software developed in this work is very useful, because it can solve the problem for an impedance
matching into the bandwidth required for a specific
application. This way is easier to guarantee the quality of the system, because the software finds a very
good solution for a specific problem.
It is very important to emphasize that if the bandwidth is very broad, or the maximum value of SWR
desired is very small, the software might not found a
solution. In this case, the program shows a message
and suggests the user to change one of those attributes.
Finally it would be very interesting to design and
simulate a system, to check the results obtained. It
could also help making a hardware to verify the results
by metering the SWR.
References:
Figure 8: SWR curve for 7 stubs
Finally, the SWR obtained at the seventh stub
connection point is shown in Figure 9. It is obvious
that the condition of maximum SWR desired is satisfied in this case.
Figure 9: SWR curve for 7 stubs
6
Conclusion
Telecommunications systems, day by day, need a
greater bandwidth, so more information can be transmitted, but it is necessary to guarantee a good quality
of the system. In order to reach this, is very important
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