Anyone interested in seeing the same derivation in perhaps slightly
different order can review my posting of 8-23 in the thread " A
subtle detail of reflection coefficient. . .". It includes a numerical
comparison of results using the derived formula with results using a
couple of alternative formulas. Of course, you can find a similar
derivation in nearly any electromagnetics or transmission line text. If
you do look it up, please note that I made an error (later corrected) in
stating that conjugately matching the line results in maximum power
transfer to the load. The condition for maximum power transfer for a
given source impedance is of course that the load impedance be the
complex conjugate of the impedance seen looking from the load back
toward the source.

Roy Lewallen, W7EL


David Robbins wrote:
From 'Fields and Waves in Communications Electronics' by Ramo Whinnery and
Van Duzer. section 1.16 and 1.23

start with positive moving wave plus negative moving wave = total to load
for both voltage and current, simple kirchoff's law summations at the
junction of the coax and load.
Vp+Vn=Vload (1)
Ip-In=Iload (2)
note that their convention is that current moving to the 'right' is positive
so the reflected 'negative' current wave is moving left which gives the
negative sign on the second term.

now use ohm's law to rewrite (2)

Vp/Zo - Vn/Zo = Vload/Zload (3)

then solving from (1) and (3) to get Vn/Vp

multiple (3) by Zload on both sides
Vp*Zload/Zo - Vn*Zload/Zo = Vload
substitute this for Vload in (1) to get:
Vp+Vn = Vp*Zload/Zo - Vn*Zload/Zo
group terms:
Vp-Vp*Zload/Zo = -Vn-Vn*Zload/Zo
factor:
Vp(1-Zload/Zo) = Vn(-1-Zload/Zo)
divide out terms
(1-Zload/Zo)/(-1-Zload/Zo) = Vn/Vp
multiply by Zo/Zo
(Zo-Zload)/(-Zo-Zload) = Vn/Vp
mulitply by -1/-1
(Zload-Zo)/(Zload+Zo) = Vn/Vp

therefo
rho = Vn/Vp = Zload-Zo/Zload+Zo

what could be simpler... apply kirchoff's and ohm's laws and a bit of
algebra.