Here's a numerical example of a transmission line having a complex Z0,
terminated with a load impedance causing the magnitude of the reflection
coefficient to be greater than 1.

For a transmission line, I chose an approximate model of RG-58. I say
approximate, because the conductor loss doesn't include the shield loss,
something I haven't yet accurately included in my calculations. But the
calculated loss and Z0 are at least in the ballpark of what you'd see
with a real transmission line. At a frequency of 10 kHz, my
"pseudo-RG-58" shows a Z0 of 68 - j39 ohms (78.39 at an angle of -29.84
degrees), and velocity factor of 0.492. I chose to analyze a system with
one wavelength of the cable for convenience in doing the calculations.
One wavelength of the cable is 14753 meters, and the matched loss of
that length is 31.60 dB. Other characteristics a

Loss constant alpha * length = 3.637
Propagation constant beta * length = 6.283

For a load, I chose 10 + j50 ohms (50.99 at an angle of 78.69 degrees).
This produces a voltage reflection coefficient of 1.349 at an angle of
115.1 degrees.

Again for calculational convenience, I chose a forward voltage at the
input end of the cable of 1000 + j0 volts. All the results can be scaled
if wished for any other value.

The following uses the notation in _Reference Data for Radio Engineers_:

fE1 = Forward voltage at input end of the line
rE1 = Reverse voltage at input end of the line
E1 = Total voltage at the input end of the line
delta = angle of Z0
psi = half the angle of the reflection coefficient
rho = magnitude of the voltage or current reflection coefficient

And, I'll use Gv for the complex voltage reflection coefficient = -Gi,
where Gi is the current reflection coefficient. Positive reflected
current rI is toward the load. Positive average "reverse power" rP is
toward the source. "" denotes an average value.

ax = alpha * length
bx = beta * length

For current, I is substituted for E, and for the load end, 2 replaces 1.
All voltages, currents, and impedances are complex phasors unless
enclosed in absolute value signs (| |). Values so enclosed are
magnitudes only. All currents and voltages will be RMS. Steady state is
assumed.

Because I've chosen an even wavelength, and calculations are done only
for the ends of the line, the complex propagation constant gamma is
replaced by its real part alpha in all equations below. If other line
lengths are used, or calculations done for intermediate points along the
line, beta will have to be included.

When written in polar notation, A /_ B means "A at an angle of B degrees".

Calculated values a

fE1 = 1000 /_ 0
fE2 = fE1 * exp(-ax) = 26.34 /_ 0
rE2 = fE2 * Gv = 35.53 /_ 115.1
rE1 = rE2 * exp(-ax) = 0.9361 /_ 115.1
fI1 = fE1/Z0 = 12.76 /_29.84
fI2 = fI1 * exp(-ax) = 0.3360 /_ 29.84
rI2 = fI2 * -Gv = 0.4533 /_ -35.06
rI1 = rI2 * exp(-ax) = 0.01194 /_ -35.06

These values allow us to calculate all the voltages, currents,
impedances, and powers at the ends of the line.

E1 = fE1 + rE1 = 999.6 /_ 0.0486
I1 = fI1 + rI1 = 12.77 /_ 29.78
E2 = fE2 + rE2 = 34.11 /_ 70.65
I2 = fI2 + rI2 = 0.6689 /_ -8.033

A quick check shows that the impedance looking into the input end of the
line = E1/I1 = 78.28 /_ -29.73, very nearly the line's characteristic
impedance. This should be expected, considering the line loss. At the
output end, E2/I2 = 50.99 /_ 78.68, which is the load impedance as it
should be.

The average power into the line = E1 * I1 * cos(theta), where theta =
the angle of E1 - the angle of I1 =

P1 = 11080 watts

The average power out of the line at the load end =

P2 = 4.477 watts

So the line loss is 10 * log(11080/4.477) = 33.94 dB. This is a little
greater than the matched loss of 31.60 dB because the line isn't matched.

You must have noticed that the reflected voltage rE2 is greater in
magnitude than the incident voltage fE2 at the load. This doesn't
violate any law of conservation of energy, however -- examples abound of
passive circuits that effect a voltage step-up. But, likewise, the
reflected current exceeds the forward current.

Some posters on this newsgroup are very fond of looking at average
powers calculated from various waves, so let's do those calculations:

fP1 = fE1 * fI1 * cos(delta) = 11070 watts
rP1 = rE1 * rI1 * cos(delta) = 0.009695 watts

Not surprisingly, fP1 ~ P1, so we can't tell much from these. At the
load end,

fP2 = 7.677 watts
rP2 = 13.97 watts

Aha! you say, we've created power!

Well, no we haven't. If you'll recall from the earlier calculation of
P1 and P2, we've lost power, not created it. But the "forward power"
minus the "reverse power" is a negative number! Yes, it is. But if you
bother to go through the math, you'll find that the actual, net power
equals the difference between "forward" and "reverse" power only if Z0
is completely real (or one other special case). The general formula for
total power in terms of "forward" and "reverse" power is:

P = fP - rP + rho * exp(-2ax) * 2 * sin(2bx - 2 * psi) * sin(delta)

delta is the angle of Z0, so the extra term on the right becomes zero
only when Z0 is completely real. Of course, a purely imaginary Z0 (angle
of +/- 180 degrees) would have the same effect, but that can't occur in
a real cable. Interestingly, the right hand term also goes to zero when
2bx - 2 * psi = n * 180, where n is any integer including zero. That
means that, even when Z0 is complex, the average total power will be the
difference between fP and rP at particular points along the line, or at
the input end of particular line lengths.

One of the very important things this example illustrates is the danger
of drawing conclusions from the average powers in the individual forward
and reverse voltage and current wave components. Somewhere, somehow,
you've also got to account for the power in that extra term -- a power
that comes and goes along the cable! I challenge anyone who's fond of
this kind of analysis to explain the component powers on this line.

This analysis has produced a self-consistent set of voltages, currents,
impedances, and (net) power. No physical laws were violated. If anyone
thinks this analysis or its conclusion are in error, I invite you to do
a comparable analysis, starting only with the same assumed transmission
line and load.

I've also run a similar analysis of a hypothetical lossless cable with
the same Z0. Such a cable, as far as I know, can't be constructed. But
if there's enough interest I'll be glad to post that also.

As always, corrections are solicited and welcome.

Roy Lewallen, W7EL