Keith Dysart wrote:
On Feb 20, 12:22 am, Cecil Moore wrote:
The derivation proves the equation to be valid.
The redundancy inside a transmission line of Z0 characteristic
impedance means we don't have to deal with voltages at all

Not quite, since the angle 'theta' is the angle between the voltages,
is it not?

That's part of the redundancy I was talking about. If one
calculates the voltage reflection coefficient at the load
and knows the length of the transmission line, one knows
the angle between those voltages without actually calculating
any voltages. For a Z0-matched system, one only needs to
know the forward and reflected powers in order to do
a complete analysis. No other information is required.

I am pretty sure that it is 100 W. See more below.

Good grief, you are right. I wrote that posting and made
a sophomoric mistake after a night out on the town. I
should have waited until after my first cup of coffee
this morning. Mea culpa.

It does, indeed, converge to 100 W forward but you have to use the
proper equations when summing the powers of superposed voltages.
You can not just add them.

You're right - 20 lashes for me. Some day I will learn not
to post anything while my left brain is in a pickled state.

So forward and reverse powers each converge to 100 W and 200 W is
dissipated in the source resistor, which suggests that your
equation for computing dissipation in the source resistor is
incorrect.

It happened to be correct in the earlier example because
the ratio Rs/Z0 = 1.0, so the equation was correct
*for those conditions*. Rs/Z0 needs to be included when
the source resistor and the Z0 of the feedline are different.
Now that I've had my first cup of coffee this morning, let's
develop the general case equation. Note that we will converge
on the equation that I posted earlier.

50w + 50w + 2*SQRT(50w*50w) = 200 watts

The interference is not between the forward wave and reflected
wave on the transmission line. The interference is actually
between the forward wave and the reflected wave inside the
source resistor where the magnitudes can be different from
the magnitudes on the transmission line.

Note that the forward RMS current is the same magnitude at
every point in the network and the reflected RMS current
is the same magnitude at every point in the network.

Considering the forward wave and reflected wave separately,
where Rs is the source resistance:

If only the forward wave existed, the dissipation in the
source resistor would be Rs/Z0 = 1/2 of the forward power,
i.e. (Ifor^2*Z0)(Rs/Z0) = Ifor^2*Rs = 50 watts.

If only the reflected wave existed, the dissipation in
the source resistor would be 1/2 of the reflected power,
i.e. (Iref^2*Z0)(Rs/Z0) = Iref^2*Rs = 50 watts.

We can ascertain from the length of the feedline and from
the Gamma angle at the load that the cos(A) is 1.0
This is rather obvious since we know the source "sees"
a load of zero ohms.

Now simply superpose the forward wave and reflected wave
and we arrive at the equation I posted last night which I
knew had to be correct (even in my pickled state).

P(Rs) = 50w + 50w + 2*SQRT(50w*50w) = 200 watts

For the general equation: P1=Pfor(Rs/Z0), P2=Pref(Rs/Z0)

P(Rs) = P1 + P2 + 2[SQRT(P1*P2)]cos(A)

And of course, the same thing can be done using voltages
which will yield identical results. What some posters
here don't seem to realize are the following concepts
from my energy analysis article at:

http://www.w5dxp.com/energy.htm

Given any two superposed coherent voltages, V1 and
V2, with a phase angle, A, between them:

If ( 0 = A 90 ) then there exists constructive
interference between V1 and V2, i.e. cos(A) is a
positive value.

If A = 90 deg, then cos(A) = 0, and there is no
destructive/constructive interference between
V1 and V2, i.e. cos(A) = 0

If (90 A = 180) then there exists destructive
interference between V1 and V2, i.e. cos(A) is a
negative value.

Dr. Best's article didn't even mention interference and
indeed, on this newsgroup, he denied interference even
exists as pertained to his QEX article.

The failure to recognize interference between two coherent
voltages is the crux of the problem.
--
73, Cecil http://www.w5dxp.com