Value of Voltage Reflection Coefficient

[varunag]

hi

i was studying the "wave characteristics of transmission lines". while considering the lines with resistive termination, the term "voltage reflection coefficient(VRC)" came.
With reference to the book, "Field and Wave Electromagnetics", by David K. Cheng,
the following expression was given for the VRC:
$$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} = |\Gamma|e^{j\theta_\Gamma}$$
$$Z_0\text{ is the characteristic impedance, and } Z_L\text{ is the load impedance.}$$

This was when, the distance on the line was measured from the load towards the source.
I wanted to know whether the same formula holds if we measure the distance from source towards the load. I was trying to find the same and would still be trying out, but thought it would be good to know more... (you always get to know more than what you want at PF)

regards,
varunag

 

[maverick280857]

Hi Varun, good to see you on PF. How are you these days? :-)

Well, since you're following Cheng, I will also use the convention he's used:

$$\Gamma(z) = \frac{V^{-}e^{\gamma z}}{V^{+}e^{-\gamma z}}$$

here $z = 0$ corresponds to the load. So at the load,

$$\Gamma_{L} = \frac{V^{-}}{V^{+}} = \frac{Z_L - Z_0}{Z_L + Z_0} = |\Gamma_{L}|e^{j\theta_\Gamma_{L}}$$

Distances towards the source are along negative z, so at a distance z' from the load, the coordinate z = -z' and so

$$\Gamma(z=-z') = \Gamma_{L}e^{-2\gamma z'}$$

which implies that

$$\Gamma(z=-z') = |\Gamma_{L}|e^{j\theta_\Gamma_{L}-2\gamma z'}$$

I guess this is what you were looking for.

Note: We need to be careful when writing $\Gamma(z')$. Depending on the context, it could correspond to a distance z' from the load (which would imply that this is an implicit notation for z = -z' and the minus sign has been suppressed for convenience) or the coordinate z = +z'. Both mean totally different things.

Hope that helps.

 

[varunag]

thanks maverick, for the prompt reply.

a few things i have understood after reading your post and then the book:
1. The VRC I've mentioned, is the "voltage reflection coefficient" of the load impedance $$Z_L$$.
2. The formula does depend on the choice of our z. And I was earlier making some mistake when I was trying to find $$\Gamma$$ using z going from source to load. Although later I was able to find $$\Gamma$$ correctly.