How to Simplify Backward Impedance Matching Equations?

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SUMMARY

The discussion focuses on simplifying backward impedance matching equations in transmission line theory. The user outlines their approach, which includes calculating the input impedance (Zin1), equivalent impedance (Zeq), and the corresponding impedance of Gamma_s (Zs). Despite following a systematic method, the user finds the resulting equations overly complex and seeks a more straightforward solution. The community suggests using LaTeX for clearer presentation of equations and calculations.

PREREQUISITES
  • Understanding of transmission line theory
  • Familiarity with complex impedance and conjugates
  • Knowledge of impedance matching techniques
  • Basic proficiency in LaTeX for mathematical notation
NEXT STEPS
  • Research methods for simplifying impedance matching equations
  • Learn about the Smith chart for visualizing impedance
  • Explore advanced transmission line theory concepts
  • Practice using LaTeX for formatting mathematical equations
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Electrical engineers, students studying transmission line theory, and anyone involved in RF design or impedance matching techniques will benefit from this discussion.

inter060708
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Homework Statement


https://scontent-lax3-1.xx.fbcdn.net/v/t34.0-12/14804959_10155526179417281_80533891_n.png?oh=649ee14155ac558b21c6553f003e418b&oe=5809EAA0

Homework Equations

The Attempt at a Solution


1) I determined the input impedance of the first transmission line (Zin1).
2) I calculated the equivalent impedance (Zeq) of 50 Ohms and Zin1
3) Found corresponding impedance of Gamma_s (Zs)
4) I switched the position of Zs with Zeq
5) Replaced Zs with its complex conjugate (Zs*)
6) Found input impedance of Zeq at length -l2
7) Set it equal to Zs*, solve for real and imaginary coefficients to get l1 and l2

The equation turns out to be very complicated and I am sure there is easier way, please help. Thank you
https://scontent-lax3-1.xx.fbcdn.net/v/t35.0-12/14647238_10155526274642281_613988889_o.jpg?oh=461d284cb7c90bcccfc46c1e3d960cb8&oe=5808D15D
 
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inter060708 said:

Homework Statement


https://scontent-lax3-1.xx.fbcdn.net/v/t34.0-12/14804959_10155526179417281_80533891_n.png?oh=649ee14155ac558b21c6553f003e418b&oe=5809EAA0

Homework Equations

The Attempt at a Solution


1) I determined the input impedance of the first transmission line (Zin1).
2) I calculated the equivalent impedance (Zeq) of 50 Ohms and Zin1
3) Found corresponding impedance of Gamma_s (Zs)
4) I switched the position of Zs with Zeq
5) Replaced Zs with its complex conjugate (Zs*)
6) Found input impedance of Zeq at length -l2
7) Set it equal to Zs*, solve for real and imaginary coefficients to get l1 and l2

The equation turns out to be very complicated and I am sure there is easier way, please help. Thank you
https://scontent-lax3-1.xx.fbcdn.net/v/t35.0-12/14647238_10155526274642281_613988889_o.jpg?oh=461d284cb7c90bcccfc46c1e3d960cb8&oe=5808D15D
Your picture is basically unreadable. Could you type your work into the forum instead? There is a LaTeX primer in the Help/How To section of the forum -- click on INFO at the top of the page.
 

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