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ELG3106

Electromagnetic Engineering

Fall 2022

Assignment 4

λ/4 Transformer for Complex Loads

Introduction

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 = 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.