Wavelengths along a Smith Chart

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Discussion Overview

The discussion revolves around understanding the mathematical principles behind using a Smith chart to find input impedance and return loss at a specific distance from a load on a transmission line. Participants explore the periodic nature of input impedance as a function of line length and the reasoning behind deducting multiples of λ/2.

Discussion Character

  • Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant questions the reasoning behind deducting multiples of λ/2 when analyzing input impedance on a Smith chart, specifically asking how the remainder of λ/8 is derived from an original length of 5λ/8.
  • Another participant clarifies that deducting λ/2 prevents unnecessary rotations on the Smith chart, providing a numerical example to illustrate the calculation.
  • A further inquiry is made regarding the specific number of λ/2 deductions, with a participant seeking clarification on why 4 is used instead of another number.
  • Responses indicate that 1/2 λ is equivalent to 4/8 λ, which helps to resolve the initial confusion about the deduction process.

Areas of Agreement / Disagreement

Participants engage in a constructive dialogue, with some reaching an understanding of the mathematical reasoning involved, while questions about the specific deductions remain open for further clarification.

Contextual Notes

The discussion includes assumptions about the periodicity of input impedance and the specific calculations involved in using the Smith chart, which may depend on the definitions and context of the problem presented.

OnceMore
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Hello,

This may be something that is more mathematical than electrical, but I figured here was the best place.

I am going through a worked example, and there is something I don't understand. The question is as follows

Find the imput impedance and return loss (in dB) of the loaded line in the last question as seen at the distance l = 5λ/8 from the load

The solution states that "because the input impedance is a periodic function of the line lenth, we need to decuct all multiples of λ/2 and analyse only the remainder (λ/8)..."

Now, I understand that λ/2 is a full rotation of the Smith chart, but why is that deducted? And, if you remove λ\2, how are you left with λ\8 ...from an origianl length of 5λ/8?

The last part may just be some mathematics that is escaping me at the moment, but I just cannot get past this part.

Thanks for any advice.

-S
 
Last edited:
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5/8 - 4/8 = 1/8.

You deduct it so you don't spin around the chart 4 times for nothing.
 
Hello.

Ok, that makes sense ...but, where is the 4 coming from? I mean, why not 6 times?

-S
 
1/2 Lambda = 4/8 Lambda.
 
Ahh, okay ...I am with you now!

That's fantastic. Thanks for your help.

-S
 

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