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Following two statements made me think that the transmission line repeates after \$ \frac{ \lambda}{2} \$ distance:

  1. A complete revolution around the Smith chart corresponds to \$ \frac{ \lambda}{2} \$ distance on the line
  2. Distance between two successive voltage Maxima is \$ \frac{ \lambda}{2} \$

In order to proof that the line repeates after \$ \frac{ \lambda}{2} \$ distance, my thoughts are as follows:
Consider a two conductor transmission line as shown below:
enter image description here
The instantaneous voltage expression at a distance \$ z \$ on the line is given as : $$v(z,t)= V_o^+ e^{(- \alpha z)}cos(wt - \beta z) + V_o^- e^{( \alpha z)}cos(wt + \beta z)$$ $$\implies V_s(z) = V_o^+ e^{(- \alpha z)}cos(-\beta z) + V_o^- e^{( \alpha z)}cos(\beta z) \quad \dots (1)$$ Now, \$ cos(\beta z) \$ repeats after a distance of \$ \lambda \$ , then \$ V_s \$ is also supposed to repeat after a distance of \$ \lambda \$;
but according to my conclusion (made in the first line) \$ V_s \$ should repeat after a distance of \$ \frac{\lambda}{2} \$ (as the line repeats after \$ \frac{\lambda}{2} \$ distance)
so, can anyone tell me how \$ V_s \$ repeats after a distance of \$ \frac{\lambda}{2} \$ ?
or any other approach which will prove that the line repeates after \$ \frac{ \lambda}{2} \$ distance

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  • \$\begingroup\$ The zero crossing wavelength.... \$\endgroup\$ Jan 24, 2021 at 15:38
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    \$\begingroup\$ Who says the waveform repeats after a distance of \$\frac{\lambda}{2}\$? \$\endgroup\$ Jan 24, 2021 at 16:15
  • \$\begingroup\$ You need to add where you're getting wrong information from. Perhaps there is some context there, (like |Vs| not Vs per Stefan's answer), or a clarification that you're measuring the sending voltage on a mis-terminated line (hence lambda/2 because you're seeing the out-and-back round trip) that might make this correct information. \$\endgroup\$ Jan 24, 2021 at 17:04
  • \$\begingroup\$ @BrianDrummond A complete revolution around the Smith chart corresponds to \$ \frac{ \lambda}{2} \$ ; also distance between two successive voltage Maxima is \$ \frac{ \lambda}{2} \$ ; so can't we say the line repeats itself after \$ \frac{ \lambda}{2} \$ distance? \$\endgroup\$
    – Suresh
    Jan 25, 2021 at 5:38
  • \$\begingroup\$ It would be kind of the same as saying that a car wheel repeats itself when the car is moving. It is a repeating pattern, but not a repeating wheel. \$\endgroup\$ Jan 25, 2021 at 12:53

1 Answer 1

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The absolute value of the voltage \$V_s(z)\$ repeats after \$\lambda/2\$, not \$V_s(z)\$.

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  • \$\begingroup\$ Could you please append to your answer that is it true to say that "the line repeats after \$ \frac{ \lambda}{2} \$ distance " , if yes , then can you please show its derivation \$\endgroup\$
    – Suresh
    Jan 25, 2021 at 5:56

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