ELG3106 Electromagnetic Engineering Fall 2022 Assignment 4
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ELG3106
Electromagnetic Engineering
Fall 2022
Assignment 4
λ/4 Transformer for Complex Loads
Introduction
A λ/4 transformer is to be used to match a line of lossless characteristic impedance Z01 , with
complex load ZL , to a feedline of lossless characteristic impedance Z01 . This requires the transformer lossless characteristic impedance Z02 and the load line length d to be found. Our purpose is to illustrate the design approach by doing a specific problem in an algorithmic fashion with a paper Smith Cart as assisted by numerical simulation using the Module 2.7 applets.
Theory
The input impedance at the source end of a line of length l is given by
Zin = Z0〈( ZL + jZ0 tan(bl))卜 (1)
Using this, a lossless line of length l = 入 4 , characteristic impedance Z0 , and real load ZL , has an input impedance of Zin = Z0(2)ZL , or Zin ZL = Z0(2) . Applying this understanding to the transmis- sion line of Figure 1 reveals that Z (d ) , the input impedance of the load line, must be real. Real wave impedances occur at voltage maxima and minima so we have two unique solutions.
Figure 1: An in-series λ/4 transformer inserted at either dmax or dmin .
Since in Fig. 1, the impedance to be matched is Zin = Z01 , the solutions for Z (d ) must satisfy
Z01Z (d ) = Z0(2)2 (2)
where d = dmax is the location of a voltage maximum and d = dmin is the location of a voltage minimum.
The computation and simulation activities
Let Z01 = 50 and ZL = (100 j200) . Use the Smith chart to find the two solutions Z (dmax ) and Z (dmin ) , along with the values dmax and dmin . Use a separate Smith chart for each solution. The Smith charts are to show all necessary annotations and information. You may wish to review Example 2- 15 in your textbook. Note that this example only shows half the problem on the Smith chart – that of determining dmax and dmin . While equation (2) must be used to find the transformer characteristic impedance value, the Smith chart can also be used to confirm that it is matched to the feedline. Use the “Module 2.7 Tutorial” app to guide you through this process. For each solution, confirm that your transformer values are indeed correct by using the “Module 2.7 Design” app to show that the reflection coefficient in the feedline is zero. Screenshots of these apps are shown in Figures 2 and 3. For convenience, you may assume air lines and a 1.0 GHz operating frequency.
The report
Your report should be given in standard technical report format. The task should be clearly stated. The theory must include at least that summarized above. Your description of the computation and simulation activities should contain a detailed algorithm for determining the λ/4 transformer parameters, and for confirming that it matches the feedline. Note that the “Module 2.7 Tutorial” app presents each step of this algorithm. Your results should include a summary of the solutions (tabulated) and screenshots of the “Module 2.7 Design” results, showing the panels: Set Line/Transformer, Output Data, and Phasor Plots (Reflection Coefficient). Your discussions should explain what is happening in each line segment. Your conclusion should explicitly state both the purpose of the λ/4 transformer and how it accomplishes it.
Figure 2: The “Module 2.7 Tutorial” applet.
Figure 3: The “Module 2.7 Design” applet.
2022-11-14