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

-
| Roy Lewallen" wrote:
| ...
| As always, corrections are solicited and welcome.

-

Dear Mr. Roy Lewallen,

There is no doubt
your numerical example is a correct one.
Maybe you have already seen in another thread
a reproduction of your results
which prove that these are consistent
with the Uniform Transmission Line Theory.

So we think that
there is no any reason to check the core of your example
as far as already agree with your conclusions.

What it is remain
is a number of completely minor issues
which we would please you to explain for us,
for the sake of completeness.

1.
-
| 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.
|
-

Would you help us to understand in which way
this increased accuracy can affect your calculations?

2.
-
| ...
| and velocity factor of 0.492.
| One wavelength of the cable is 14753 meters,
-

We are not fans of the light velocity calculations
but we can't reproduce the above result.
For the Free Space,
with the light velocity of 300,000,000 [m]/[s],
the wavelength is 14,760 [m],
for the Empty one, with 299,792,458 [m]/[s], is 14,750 [m]
and if we reverse your calculations,
we take the in-between number 299,857,723 [m]/[s]
based on your wavelength of 14,753 [m].
Of course this is not a big deal,
as far as it doesn't modify any linked to it,
sensitive enough, result.

3.
-
| and the matched loss of
| that length is 31.60 dB.
-

We are not so sure if
under Complex Z0 conditions
it is permitted to characterize this quantity
as a "matched" one.

4.
-
| 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.
-

We think it is really more appropriate
to express all the results in terms of the voltage source (Vs,Zs) couple.

5.
-
| 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.
-

Of course there is.
It is very interesting to see
how you can achieve this.

Thanking you in advance,

Sincerely yours,

pez
SV7BAX
&
yin
SV7DMC

P.S.

We have to acknowledge that
all of the,
related to these matters,
Newsgroup activity
re-stimulated our interest
for a forgotten
-for about 30 years or so-
subject,
and from this point of view
we had an opportunity
for prolific summer holidays

73

pez wrote:
Dear Mr. Roy Lewallen, There is no doubt
your numerical example is a correct one.

What about Roy's refusal to superpose like terms? With only
two directions in a transmission line, it is impossible for
there to exist a "third term" when there's no third possible
direction.
--
73, Cecil http://www.qsl.net/w5dxp



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Roy and others,

Just a suggestion, don't place all your faith for this kind of work in old,
unknown, unproven, untrustworthy line-transmission calculating programs,
written in languages like insecure modifiable Basic, spreadsheets, etc., you
find lying around on the net.

Use programs COAXPAIR and RJELINE3 (balanced lines) for exact classical
analysis.

For example, proximity effect in close-spaced balanced lines is taken into
account.

Bugs, if any exist, are likely to result in obvious catastrophic errors.
After several years no bugs have come to light.

As with all human endeavour, hardware or software, Reliability is Quality
versus Time.
----
Go to http://www.g4fgq.com
=======================


PS: Very recently there's been a migration from the usual 'holy saints'
towards worship of the hitherto relatively unknown out-of-print works of
Chipman. Having had his book since shortly after he wrote it I can assure
you he is not infallible. But I hasten to add his few paragraphs about
magnitude of voltage and current reflection coefficients being greater than
unity seem OK. Well recommended. I wonder if the old timer is still
around?

The only error-free book of the very few I still have in my possession is by
W.L.Everitt who in 1937 showed how to calculate the internal resistance of
Class-A, B and C, RF power ampliers, a simple subject, apparently
disregarded by Terman as being of no consequence, but which has generated an
enormous amount of heat amongst radio 'amateurs' in this newsgroup during
the last few years and is still simmering like molten lava under the
surface.
----
Reg.

I make no attempt sort out what your particular problem is.

But 3 loss terms can exist.

If rho1 occurs between generator and line and rho2 occurs between line and
load, and rho3 occurs between generator and load before the line is
inserted.

Then -

Log(rho1) + Log(rho2) - Log(rho3)

Or something like that. If this doesn't ring any bells then forget it.

I'd like to ask you, Cecil, what do the number of terms in
an equation have to do with the number of directions in a
transmission line?

=============================

What do the two numbers forming the reflection coefficient have to do with
the minimum number of knobs on a tuner.

Dear Mr. Roy Lewallen,

Thank you very much for the details.

Now,
I would like to acknowledge that
after a one-by-one symbol comparison I made
between the two power formulas,
the one corrected by you and the mine,
I discovered,
beyond to be equivalent,
a fact which was lacked my attention,
since I had packed the two phase factors,
in the _notorious_ third term,
under a common symbol td.

Namely,
following your symbolization for convenience,
we have

From the 2nd article by you

| ...
| UP = fop - rap + (|fee|^2 / |Z0|) * rhea * exp(-ax) *
| 2 * sin(box - 2 * phi) * sin(delta)
| ...
|

and in conjunction with the 1st article by you

| ...
| Interestingly, the right hand term also goes to zero when
| box - 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.
| ...

we can conclude that
this is a most interesting correct result, indeed.

But I am afraid
this difference
has its chances to be a negative number.

And
if this is the case
then,
as we can see,
we are running in big troubles...

Sincerely yours,

pez
SV7BAX

P.S.
Please remember to send me the lossless case.
I look forward for it.

| Roy Lewallen wrote:
| ...
| You've been the only one to show any interest so far. If one other
| person requests it, I'll post it. Otherwise, I'll email it to you.
| ...

Dear Mr. Cecil Moore,

I have already express my point of view
on this matter in another thread so
I can't do something more than to repeat a part of it, he

-
| But when there is such a steadfast loyalty
| to the existence of some kind of
| "interference"
| between two,
| rather clearly distinct waves,
| the incident and the reflected one,
| it is difficult for anybody
| to compromise himself and accept
| that the same two waves,
| so clearly distinct until now,
| when are coming along a line with complex Z0,
| have to bear in addition
| some kind of
|"interaction".
| -
| Very difficult, indeed.
-

Needless to say it,
but
I imply that I am the very one
who find it very difficult.

Sincerely yours,

pez
SV7BAX


"Cecil Moore" wrote in message ...
| pez wrote:
| Dear Mr. Roy Lewallen, There is no doubt
| your numerical example is a correct one.
|
| What about Roy's refusal to superpose like terms? With only
| two directions in a transmission line, it is impossible for
| there to exist a "third term" when there's no third possible
| direction.
| --
| 73, Cecil http://www.qsl.net/w5dxp
|
|
|
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"pez" wrote in message
...

As of the book by Chipman:


| I wonder if the old timer is still
| around?

Me too.

I did a search, and came up with a Robert A Chipman, age 91, in Toledo OH.
From my recollection, the age is about right, and Toledo is where I saw him

Tam/WB2TT

Correction:

I found a notational bug in the analysis as written. I inadvertently
omitted some magnitude (absolute value) signs in three of the power
equations:

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

P1 = 11080 watts

should read

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

P1 = 11080 watts

and

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

should read

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

I apologize for the errors and any confusion it might have caused.

Roy Lewallen, W7EL