# Voltage along transmission line with scattering parameters and VSWR given

A transmitter is connected to an antenna with a coaxial cable with length $$\l=1.5 \:\text{m}\$$. The characteristic impedance of the cable is $$\Z_0 = 75 \: \Omega\$$ and the scattering parameters are $$\\textbf{S} = \Bigg[ \begin{matrix} 0 & e^{-j27} \\ e^{-j27} & 0 \end{matrix} \Bigg]\$$. The distance from antenna to the first maximum voltage on the transmission line is $$\|z_{max}| = 10 \: \text{cm}\$$.

Plot the magnitude of the voltage along the TM-line normalized to the magnitude of the forward voltage wave along the TM-line when VSWR = 1.5

Comparing the scattering parameters with the general expression of $$\\text{S}\$$ for a transmission line $$\\textbf{S} = \Bigg[ \begin{matrix} 0 & e^{-j\beta l} \\ e^{-j \beta l} & 0 \end{matrix} \Bigg] \$$ we see that $$\\beta = \frac{27}{1.5}=18 \$$.

From the definition of VSWR we can find the reflection coefficient:

$$VSWR=\frac{1+|\Gamma_L|}{1-|\Gamma_L|} \Rightarrow \Gamma_L = \pm 0.2 \: \: \: \: \text{for} \: \: \: \: VSWR=1.5$$

We also know the relation $$|V(z)| = |V_0^+| \cdot |1+\Gamma_Le^{2j\beta z}|$$ which can be manipulated into what we want $$\frac{|V(z)|}{|V_0^+|} = |1+\Gamma_Le^{2j\beta z}|$$

Plotting this function yields: -

I have reason to believe that my method is incorrect. I don't use the information about $$\Z_0\$$ or $$\z_{max}\$$. I also think my determination of $$\\Gamma_L\$$ is incorrect since there is some information lost when we look at $$\|\Gamma_L|\$$ instead of just $$\\Gamma_L\$$.

My question is am I on the right track? Are my determinations valid or am I misunderstanding the transmission line theory?

$$VSWR=\frac{1+|\Gamma_L|}{1-|\Gamma_L|} \Rightarrow \Gamma_L = \pm 0.2 \: \: \: \: \text{for} \: \: \: \: VSWR=1.5$$
This isn't correct. You know $$\|\Gamma_L|=0.2\$$, but $$\\Gamma_L\$$ is not necessarily real. It could be a complex number with any phase. It could be any number with the form $$\0.2e^{j\theta}\$$.
You need to use the information about the distance to the first maximum in the standing wave pattern to work out what $$\\theta\$$ is.