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ENGEE334-20A, Transmission Lines and Antennae – Final Assignment
SPECIAL INSTRUCTIONS:
1. This final assignment consists of three (3) questions.
2. Attempt all the questions and submit your answers (including your approach, MATLAB scripts and
plots) via the Moodle link.
3. This assignment has a strict deadline (19-06-20, 12 pm). 10% will be deducted for late submission.
4. If you have any difficulties in understanding the questions, please email me any time.
Question 1: Transmission Lines [25 marks]
Hint for Q1.1-Q1.3:
Power dividers, or power splitters, are used to split an input signal into two output signals, or
combine two signals into one in the opposite direction. One popular power divider that has
been widely implemented in various industrial applications is T-junction divider. Figure 1
illustrates the transmission-line model of a lossless T-junction divider.
As a power divider, microwave signals fed into port 1 will be divided into two parts. Ports 2
and 3 are the two outputs. The Section A of this power divider is terminated by Section B that
consists of two parallel-connected transmission lines. The physical length of Section A before
the junction is l1, whereas l2 and l3 are the lengths of the two transmission lines after the
junction. The characteristic impedance of the system is Z0 = 50 Ω.
Figure 1. Transmission-line model of a lossless T-junction divider.
Q1.1: Suppose that the voltage at the junction is V0, as shown in Figure 1. Find the characteristic
impedances Z1 and Z2 so that the powers delivered to Z1 and Z2 are in a 2:1 ratio. Based on the
calculated Z1 and Z2, prove that the Section A is perfectly matched. Here assume the input impedance
of transmission lines Z1 and Z2 equal to their characteristic impedance.
[7 marks]
Q1.2: Compute the reflection coefficients at Port 2 and Port 3.
[7 marks]
Q1.3: For the power divider configured as in Figure 2, compute the average power that can be
delivered to each load (ZL1 and ZL2).
[6 marks]
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Figure 2. Power divider for Q1.3.
Hint for Q1.4-Q1.5:
| | In Figure 1, if l2 ≠ l3, there will be a phase difference between the two outputs. Then, the structure in Figure 1 also serves as a phase shifter. One typical application of phase shifters is the Balun – a balance to unbalance device – that has been widely used to match between a balanced microwave component (e.g., dipole antennas) and an unbalanced component (e.g., coaxial cables). Figure 3 illustrates how to excite a dipole antenna with a coaxial cable via a Balun. One |
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essential requirement for Baluns is that the output signals of Port 2 and Port 3 should be
totally out of phase (i.e., a phase difference of 180°).
Figure 3. Excite a dipole antenna with a coaxial cable.
Q1.4: Suppose that in Figure 1, both transmission lines after the junction are terminated by an openended load. If the power divider needs to be implemented as a Balun, what is the relation between the
length l2 and l3?
[2 marks]
Q1.5: Briefly explain, with your knowledge of dipole antennas, why the two outputs should be out of
phase.
[3 marks]
(This is the end of Question 1. Please continue on attempting Question 2)
Balun
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Question 2: Matching circuits and Two-port networks [45 marks]
As shown in Figure 4, this question provides a guidance to design matching networks for an antenna
(the load) that intends to be connected to a 50-Ω feedline. The antenna is designed for Bluetooth
transmissions, whose central operating frequency is 2.45 GHz. Follow the below design steps and
attempt all the questions.
Figure 4. Matching network.
| Step a. | The input impedance of an antenna (Zin, equivalent to the ZL in Figure 4) has been provided in an Excel file in this assignment folder, entitled “Input Impedance”. The frequency range |
of the provided input impedances is from 2 to 3 GHz, stored in three columns. The first
column is the discrete frequency points measured in GHz, the second and the third columns
are the real and the imaginary parts of Zin of the antenna, respectively.
| Step b. | Use the command xlsread to read this Excel file into MATLAB. Use three sequences to store the variables of the frequency, the real and the imaginary parts of the Zin. The reflection coefficient of the antenna can therefore be computed, which is the S11 of the antenna. |
| Step c. |
Below questions are for Steps a – c.
Q2.1: With MATLAB, compute the S11 of the antenna for each discrete frequency point from 2 to 3
GHz. Convert your computed results to decibels (dB) and plot your results versus the frequency. From
your plot, specify the resonant frequency (f0, i.e., the frequency point that has the minimal S11) of this
antenna.
[5 marks]
Q2.2: Suppose that the power delivered to MM’ is 1 mW, compute the standing wave ratio on the
feedline and the power that can be delivered to the antenna without the matching network at f0.
[4 marks]
Step d. As you can observe from the plot, the S11 of the antenna over the whole frequency band is
larger than –10 dB. This means that the majority of the microwave energy fed to the antenna
is reflected due to the mismatch between the antenna and the feedline.
Below questions are for the Step d.
Q2.3: Use the Smith chart to design two possible lumped-element matching circuits to match the
antenna to the 50-Ω feedline, so the antenna can work at the desired frequency of 2.45 GHz. You may
use the “Interactive Smith Chart” module that was provided with Lab 2 to assist your design.
[10 marks]
Q2.4: Based on the two solutions obtained in Q2.3, design two shunt shorted stub tuning circuits to
match the antenna to the feedline at 2.45 GHz (note: the distance d at which the lumped element
inserted does not have to be changed).
[5 marks]
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| Step e. | Analyse the performance of matching networks The matching network between MM’ and AA’ can be regarded as a two-port network. After inserting the matching networks, the Zin in Figure 4 becomes the new input impedance of the antenna. For the matching with shunt shorted stub tuning circuit, it resembles the lossless T junction divider in Figure 1, where port 1 connects the feedline, port 2 connects the antenna and port 3 connects a short-ended load, where Z0 = Z1 = Z2 = 50 Ω. |
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Below questions are for the Step e.
Q2.5: Compute the scattering parameters [S] for both matching circuits obtained in Q2.3 at 2.45 GHz.
[5 marks]
Q2.6: For both matching circuits obtained in Q2.4, write a short script to plot: 1) the input impedance
seen looking into the matching circuit (i.e., Zin in Figure 4) versus the frequency from 2 to 3 GHz, and
2) the reflection coefficient seen looking into the matching circuit versus the frequency from 2 to 3
GHz. Indicate the operation frequency of the combined matching circuit plus the antenna.
[6 marks]
Q2.8: At 2.45 GHz, suppose that the power delivered to MM’ is 1 mW. Pickup one matching circuit
obtained in Q2.4, compute the power delivered to the short-circuited stub and the part terminated by
the antenna. Further compute the standing-wave ratio on the antenna-terminated line and the energy
that can be delivered to the antenna. Compare with the results obtained in Q2.2, state your conclusion.
[10 marks]
(This is the end of Question 2. Please continue on attempting Question 3)
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Question 3: Antennae Analysis [30 marks]
This question will guide you to design a quarter-wave monopole antenna with MATLAB. Some basic
knowledge of a monopole antenna can be found below.
| | A quarter-wave monopole antenna resembles a half-wave dipole antenna. Unlike a dipole antenna which has two arms, the monopole antenna only has one arm (refer to Figure 5). The image theory implies that the monopole antenna has another imaginary part on the other side of the conductive ground. Both the radiation resistance and the total radiated power of a monopole antenna are the half of the values for the dipole antenna operating at the same frequency. |
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Figure 5. Monopole antenna structure.
The monopole antenna can be designed and simulated with the Antenna Toolbox of the MATLAB.
The help document on this topic is available at:
https://www.mathworks.com/help/antenna/ref/monopole.html. The units used in the Antenna Toolbox
are metre (length) and Hertz (frequency).
Follow the below design steps and attempt all the questions.
Step a. Antenna design:
| | Following the steps and codes outlined in the Lab 5 for designing a dipole antenna, construct a monopole antenna that can operate at 1 GHz. Use 0.015 (15 mm) as the width of the antenna. Visualise the antenna in the MATLAB. The length and width of the conductive ground need to be specified when constructing the antenna. In this question, the ground is a square with a side length of l, use 0.3 (300 mm) for l. You may want to use the following commands. |
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a =
monopole(‘Height’,height,’Width’,width,’GroundPlaneLength’,plength,’GroundPlaneWidth’,pwidth);
%a is the name of the antenna, height and width are the length and the width of the antenna,
respectively; plength and pwidth are the length and width of the ground, respectively
Step b. Antenna analysis: after designing the dimensions of the antenna, its performance, including
the S11 and radiation patterns, can be examined in MATLAB.
Below questions are for Steps a–b.
Q3.1: Compute the length of this monopole antenna. Indicate its input impedance and plot the S11 of
the antenna from 0.5 to 1.5 GHz with a frequency resolution of 1 MHz.
[10 marks]
Q3.2: Compute and plot the 3D radiation pattern of the antenna, and its 2D radiation pattern on the
azimuthal plane. Indicate the directions with the maximal radiation and the half-power beamwidth.
[5 marks]
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Step c. Influence from the conducting ground
| | The monopole antenna is proposed based on the image theory, where the conductive ground serves as the “mirror”. Therefore, the performance of the antenna is significantly affected by the size of this |
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conducting ground. Note: this influence from the ground to the monopole antenna also
applies for other types of microwave devices where a ground is required.
Below question is for the Step c.
Q3.3: Change the side length l of the conducting ground to 1) 0.1 (100 mm) and 2) 0.4 (400 mm). For
each l, redo Q3.1 and Q3.2. State the differences of the computed S11, gain and radiation pattern with
these obtained for l = 0.3.
[10 marks]
Step d. Compare with the dipole antenna
Below question is for the Step d.
Q3.4. Compare the radiation pattern you obtained for the monopole when l = 0.3, and the one you
obtained for the dipole antenna designed in Lab 5, which also operates at 1 GHz. State your
observations in terms of the antenna gain, beamwidth and the radiation directions.
[5 marks]
(This is the end of this final assignment)
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