Owen Duffy wrote:
Roy Lewallen wrote in
:

...
Sorry, I don't think so. The very first equation, Vf=Vs/2, would be true
only if the load = Ro or for the time between system start and when the
first reflection returns. For the specified steady state and arbitrary
load, Vf would be a function of the impedance seen by the source which
is in turn a function of the line length and load impedance.

Vf means the 'forward wave' voltage equivalent voltage at the source
terminals.

I provide the mathematical development of the case that Vf=Vs/2 where
Rs=Ro=Zo at http://vk1od.net/blog/?p=1028 .

Can you fault that development?

Owen

I stand corrected. In the special case of Rs = Z0, the steady state
forward voltage and therefore "forward power" become independent of the
transmission line length and load impedance, just as you said and show
in your analysis.

But this isn't true in the general case where Rs isn't equal to Z0. In
the general case, the forward voltage is

Vf = Vfi * exp(-j*theta) / (1 - Gs*Gl*exp(-j*2*thetal))

where Gs = reflection coefficient at the source looking back from the
line = (Zs - Z0)/(Zs + Z0)
Gl = reflection coefficient at the load = (Zl - Z0)/(Zl + Z0)
theta = distance along the line from the source end
thetal = length of the line in radians
Vfi = initial forward voltage = Vs * Z0 / (Zs + Z0)

At the input end of the line, theta = 0 so

Vf1 = Vfi / (1 - Gs*Gl*exp(-j*2*thetal))

and the reverse voltage at the input is

Vr1 = Gl * Vf1 * exp(-j*2*thetal)

Note that although Vf is a function of Gs, reflected voltage Vr is also
a function of Gs and their ratio is not. So VSWR and Pf/Pr don't depend
on Gs. But the general equation for Vf does include line length and Z0
as well as both source and load Z.

In the example I posted earlier (Transmitter power 100 watts, load Z =
50 + j0, line length 1/2 wavelength), no mention was made of the source
impedance, only the power being delivered by the transmitter, which was
the same for both cases since the impedance seen by the transmitter
didn't change. In the first case, where Z0 = Zl = 50 + j0, the forward
power on the line was 100 watts and reverse power was zero. In the
second case, the line Z0 was changed to 200 ohms. This caused an
increase in forward power to 156.25 watts (calculated from VSWR). From
these we can infer that the forward voltage increased by a factor of 2.5
from sqrt(100 * 50) = 70.71 Vrms to sqrt(156.25 * 200) = 176.8 Vrms.
Let's see if this agrees with the equations.

When the line is a half wavelength long, thetal = pi, so the equation
for Vf1 simplifies to:

Vf1 (half wavelength line) = Vfi / (1 - Gs*Gl)

Expanding Vfi, Gs, and Gl results in

Vf1 (half wavelength line) = Vs * (Z0 + Zl)/(2 * (Zs + Zl))

Various combinations of Vs and Zs can give us the 100 assumed watts, but
we don't need to specify any specific values. Leaving Vs and Zs as
unknown constants and Zl as fixed, we can find the ratio for two
different values of Z0:

Vf1 = Vs * (Z01 + Zl)/(2 * (Zs + Zl))
Vf2 = Vs * (Z02 + Zl)/(2 * (Zs + Zl))

Vf2/Vf1 = (Z02 + Zl)/(Z01 + Zl)

In the example, case 1 had Z0 = Z01 = 50; case 2 had Z0 = Z02 = 200, and
Zl was 50 for both. So Vf2/Vf1 = (200 + 50)/(50 + 50) = 2.5, which
agrees with the calculation based on VSWR.

Once again, the general equations for Vf and Vr and therefore Pf and Pr
include Zs, Zl, line length, and line Z0. While one or more of these
terms might disappear in special cases or ratios, they all have to
appear in any equation of general applicability.

Roy Lewallen, W7EL