Transmission Lines and Antennae - Transmission Line Model - Engineering Assignment Help

Assignment Task :

Question 1: Transmission Lines

• 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.

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. 

Q1.2: Compute the reflection coefficients at Port 2 and Port 3. 

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).

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 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°). 

Q1.4: Suppose that in Figure 1, both transmission lines after the junction are terminated by an open- ended load. If the power divider needs to be implemented as a Balun, what is the relation between the length l2 and l3? 

Q1.5: Briefly explain, with your knowledge of dipole antennas, why the two outputs should be out of phase. 

Question 2: Matching circuits and Two-port networks 

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. 

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. Step c. The reflection coefficient of the antenna can therefore be computed, which is the S11 of the 

antenna. 

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. 

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. 

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. 

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). 

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