William E. Sabin wrote:

David Robbins wrote:

"Peter O. Brackett" wrote in message
link.net...



yes, i still hold that they are different. they are from completely
different realms of electromagnetics. in the transmission line
reflection
coefficient you are working with a distributed system that is modeled
with
wave equations. there are delays, waves travel and reflections return
after
a finite delay. in your example you have coerced the voltage and current
waves to be the same in the transmission line, but that is not the
same as
modeling the line itself.


In simulation programs, transmission lines are solved for their two-port
parameters, and are then treated as lumped circuits in the actual
simulation, just like any lumped-element circuit. Which is a good way
to do it.

I notice that in the ARRL Antenna Book, 19th edition , on page 24-7, it
is stated with definite finality that the reflection coefficient formula
uses the complex conjugate of Zo in the numerator.
I also understand that this has been established by a "well-trusted
authority".

I have used Mathcad to calculate rho and VSWR for Reg's example, for
many values of X0 (imaginary part of Z0) from -0 to -250 ohms.

The data follows:

Note: |rho1*| is conjugated rho1, SWR1 is for |rho1*|, |rho2| is not
conjugated and SWR2 applies to |rho2|

X0.......|rho1*|..SWR1.....|rho2|..SWR2
-250..... 0.935...30.0.....1.865...-3.30
-200..... 0.937...30.8.....1.705...-3.80
-150..... 0.942...33.3.....1.517...-4.87
-100..... 0.948...37.5.....1.320...-7.25
-050..... 0.955...43.3.....1.131...-16.3
-020..... 0.959...47.6.....1.030...-76.5
-015..... 0.960...48.4.....1.010...-204
-012..... 0.960...48.9.....0.997....+/- infinity
-010..... 0.960...49.2.....0.990....+305
-004..... 0.961...76.3.....0.974....+76.3
0000..... 0.961...50.9.....0.961....+50.9

The numbers for not-conjugate rho are all over the place and lead to
ridiculous numbers for SWR. It is also obvious that for a low-loss line
it doesn't matter much. But values of rho greater than 1.0, on a Smith
chart correspond to negative values of resistance (see the data).

Something is wrong here that we are overlooking.

The use of conjugate rho is so much better behaved that I have some real
doubts about some of our conclusions on this matter.

What about it folks? How can we get to the bottom of this?

Bill W0IYH


The equation in the ARRL Antenna Book is identical
to the equation for rho that is in the Power Wave
literature (see Gonzalez and also see Kurokawa).
Also, numerous literature sources describe how an
open-circuit generator with internal impedance Z0,
connected directly to load ZL, is actually a power
wave setup that leads to a rho formula that is
identical to the formula in the ARRL Antenna Book.
When calculating rho, it is not necessary to fool
around with the wave equations, because frequency
is constant and everything is steady-state.

Bill W0IYH