Binomial multi-section matching network
with very low reflection coefficient.
ENRIQUE VEGA ARROYO1, EDGAR ALEJANDRO ANDRADE-GONZALEZ1, MARIO REYES-
AYALA1, HILARIO TERRES-PEÑA2, SANDRA CHÁVEZ SÁNCHEZ2
Electronics Department1, Energy Department2 Metropolitan Autonomous University
MEXICO
Abstract: - In this article, the methodology for the design of broadband impedance transformers is presented. The
design of a set of Binomial impedance transformers with different operating frequency and dielectric
characteristics is performed. These impedance transformers operate over frequency range from 1 to 2.8 GHz with
a reflection coefficient . The dielectric materials used were Duroid substrate and FR4 substrate,
Duroid substrate has dielectric constant εr = 2.2 and thickness of substrate 1.27 mm and for FR4 substrate
dielectric constant is εr = 4.4 and thickness of substrate 1.544 mm. The matching networks were simulated by
ADS. Proposed matching networks can be used for UWB communication applications.
Key-Words: - Binomial matching network, fractional bandwidth FBW, maximum reflection coefficient.
Received: July 22, 2021. Revised: March 8, 2022. Accepted: April 12, 2022. Published: May 6, 2022.
1 Introduction
Impedance matching is necessary to present
maximum power transfer, because if this does not
exist, a reflected signal is generated at the input of the
receiving block. The reflected signal represents
losses.
The matching networks present between the blocks
that are intended to be coupled, the conjugated
complex impedances to comply with a maximum
power transfer. There are narrowband and broadband
coupling networks.
Within narrowband coupling networks we can
mention L-type networks, stub-type networks and
quarter-wavelength transformer networks. Two types
are presented for broadband coupling networks:
Binomial type and Chebyshev type [1].
The Binomial-type network is based on the quarter-
wavelength matching network, so it is used for real
impedance matching.
The Binomial matching network is made up of
several sections of transmission lines of length
lambda quarters (figure 1), where each one has a
different impedance and those values are included
within the range of impedances that are to be coupled.
Fig. 3 Binomial Matching Network.
For the calculation of the Binomial matching
network, it is important to obtain the fractional
bandwidth (FBW) as a function of the number of
sections of the matching network.
The fractional bandwidth determines the percentage
of the central frequency that can be matched and it
depends on the maximum allowed reflection
coefficient , the sections number of matching
network and the values of the impedances to be
coupled. Let us remember that for a transfer of at least
90% of the power, it is necessary that it be coupled,
that is, its maximum reflection coefficient should be
less than -10 dB or 0.31 (magnitude) [3].
In this paper, the design of a set of Binomial-type
broadband matching networks is presented.
This work helps to the reader to understand the design
process of Binomial broadband matching networks
and also can be used for educational purposes.
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DOI: 10.37394/23204.2022.21.11
Enrique Vega Arroyo, Edgar Alejandro Andrade-Gonzalez,
Mario Reyes-Ayala, Hilario Terres-Peña, Sandra Chávez Sánchez
E-ISSN: 2224-2864
68
Volume 21, 2022
2 Matching network Design.
The calculation of broadband matching networks
begins by determining the value of some constants, a
number of coupling network transmission line
sections are proposed. The constants are A (equation
1) and the binomial coefficients are
(equation 2).
(Eq. 1)
(Eq. 2)
In equation 2, the Binomial coefficients are
calculated for the different values based on the
number of sections.
As the number of sections increases, the fractional
bandwidth of the matching network also increases
(equation 3). The maximum reflection coefficient
determines and limits the maximum bandwidth, since
if this value is exceeded, then it cannot be considered
coupled (only with values less than , depending on
the application).
(Eq. 3)
Now, the impedances of each section of the Binomial
matching network (equation 4) will be determined.
(Eq. 4)
Once the impedance of each section is obtained, the
dimensions W and d are determined by calculating
transmission lines at the operation frequency and
characteristics of the substrate using the effective
permittivity (equation 5).
(Eq. 5)
The width (W) of the microstrip is calculated
according to equation 6.
(Eq. 6)
Where:
(Eq. 7)
(Eq. 8)
Once the above has been calculated, the value W and
are obtained because the thickness of the substrate
is obtained. The wavelength is obtained by equation
9.
(Eq. 9)
And thus, the length of the transmission line section
() is determined. The above calculation is performed
for each transmission line section given the
previously calculated impedances to complete the
Binomial matching network.
3 Results.
The calculation and simulation of Binomial matching
networks were performed.
The design of a three-segment Binomial matching
network at a frequency of 2 GHz with transmission
line elements using the Duroid substrate is shown,
which has a dielectric constant εr = 2.2 and thickness
of substrate 1.27 mm. The maximum reflection
coefficient is proposed to be
Obtaining A and FBW from equations 1 and 3:
Determining the values of the impedances of the 3
sections.
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DOI: 10.37394/23204.2022.21.11
Enrique Vega Arroyo, Edgar Alejandro Andrade-Gonzalez,
Mario Reyes-Ayala, Hilario Terres-Peña, Sandra Chávez Sánchez
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Subsequently, the physical dimensions of the
transmission lines are calculated to present the same
electrical length according to their values of
impedances.
Table 1. Physical dimensions of the Binomial
matching network.
Impedance
W (mm)
(mm)
Z1
3.62
27.45
Z2
2.85
27.63
Z3
2.21
27.82
Once the calculations have been made and using the
Advanced Design System (ADS), the Source and
load impedances are configured, as well as the
parameters of the transmission line sections of the
Binomial impedance transformer, as shown in figure
2.
Fig. 2 Binomial matching network in ADS.
Figure 3 shows the frequency response of the 3-
section Binomial matching network.
Fig. 3 Scattering parameter S11using Duroid
substrate.
The scattering parameter S11 allows observing the
required reflection coefficient (0.05).
From here, it can be seen that since FBW=86.31%,
therefore at a frequency of 2 GHz, there will be a
theoretical bandwidth of 1.7262 GHz and from the
simulation, a bandwidth of 1.741 GHz is obtained
with the coefficient maximum reflection of 0.05.
For the FR4 substrate the return losses S11 is shown
in figure 4.
Fig. 4 Scattering parameter S11using FR4 substrate.
Hence, the bandwidth is 1.616 GHz with a maximum
reflection coefficient of 0.05.
The calculation was performed for 16 different types
of Binomial coupling networks at different
frequencies as shown in table 2.
Table 2. Binomial matching networks for different
operation conditions.
F (GHz)
Z0(Ω)
ZL(Ω)
m
εr
H(mm)
1
1
50
100
0.05
2.2
1.27
2
1.5
50
100
0.05
2.2
1.27
3
2
50
100
0.05
2.2
1.27
4
2.5
50
100
0.05
2.2
1.27
5
1
50
100
0.05
4.4
1.544
6
1.5
50
100
0.05
4.4
1.544
7
2
50
100
0.05
4.4
1.544
8
2.5
50
100
0.05
4.4
1.544
9
1
50
75
0.05
2.2
1.27
10
1.5
50
75
0.05
2.2
1.27
11
2
50
75
0.05
2.2
1.27
12
2.5
50
75
0.05
2.2
1.27
13
1
50
75
0.05
4.4
1.544
14
1.5
50
75
0.05
4.4
1.544
15
2
50
75
0.05
4.4
1.544
16
2.5
50
75
0.05
4.4
1.544
4 Conclusion
Despite having only made matching networks with
three sections, as the number of sections of the
Binomial transformer increases, the bandwidth
increases. The frequency response of the Binomial
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DOI: 10.37394/23204.2022.21.11
Enrique Vega Arroyo, Edgar Alejandro Andrade-Gonzalez,
Mario Reyes-Ayala, Hilario Terres-Peña, Sandra Chávez Sánchez
E-ISSN: 2224-2864
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matching network turns out to be flat in the
bandwidth shown.
Bandwidth is increased by employing substrates with
low dielectric constant. A very small reflection
coefficient was chosen, but for an acceptable value
(less than 0.3), the bandwidth of the coupling
network is several gigahertz.
Acknowledgment
This work was supported by the research project
“Diseño, Desarrollo y evaluación de antenas de
banda ultra ancha (UWB) para aplicaciones en la
banda de 3.1 GHz a 10.6 GHz.” (EL002-20). From
Metropolitan Autonomous University –
Azcapotzalco.
References:
[1]. Pozar, D. M. (2012). Microwave engineering,
4to ed. Hoboken, NJ: J. WileyR. S.
Kshetrimayun, “An Introduction to UWB
Communication Systems,” IEEE Potentials,
Vol. 28, No. 2, pp. 9-13, April, 2009.
[2]. J.M. Drozd; W.T. Joines, "Using the binomial
transformer to approximate the Q distribution
for maximally flat quarter-wavelength-coupled
filters," IEEE Transactions on Microwave
Theory and Techniques, Vol. 46, No. 10,
october, 1998.
[3]. I. De Coster, E. Van Lil & A. Van de Capelle,
"Comparison of Design Methods for Binomial
Matching Transformers," Journal of
Electromagnetic Waves and Applications, 14:9,
1229-1239, DOI: 10.1163/156939300X01139
(2000).
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
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DOI: 10.37394/23204.2022.21.11
Enrique Vega Arroyo, Edgar Alejandro Andrade-Gonzalez,
Mario Reyes-Ayala, Hilario Terres-Peña, Sandra Chávez Sánchez
E-ISSN: 2224-2864
71
Volume 21, 2022
