Let's run a quick calculation as Tom has suggested several times.

Consider a transmission line with Z0 = 50 - j10 ohms connected to a 50 +
j0 ohm load. As I hope will be evident, it doesn't matter what's
connected to the source end of the line or how long it is. We'll look at
the voltage V and current I at a point within the cable, but very, very
close to the load end. Conditions are steady state. When numerical V or
I is required, assume it's RMS.

I hope we can agree of the following. If not, it's a waste of time to
read the rest of the analysis.

1. Vf / If = Z0, where Vf and If are the forward voltage and current
respectively.

2. Vr / -Ir = Z0, where Vr and Ir are the reverse voltage and current
respectively. The minus sign is due to my using the common definition of
positive Ir being toward the load. (If this is too troublesome to
anyone, let me know, and I'll rewrite the equations with positive Ir
toward the source.)

3. Gv = Vr / Vf, where Gv is the voltage reflection coefficient. This is
the usual definition. Likewise,

4. Gi = Ir / If

5. V = Vf + Vr

6. I = If + Ir

Ok so far?

Combine 1 and 2 to get

7. Vf / If = -(Vr / Ir) = Vr / Vf = -(Ir / If)

From 3, 4, and 7,

8. Gv = -Gi

From 3 and 5,

9. V = Vf(1 + Gv)

and similarly from 4 and 6,

10. I = If(1 + Gi)

From 8 and 10,

11. I = If(1 - Gv)

Dividing 9 by 10,

12. V / I = (Vf / If) * [(1 + Gv) / (1 - Gv)]

Combining 12 with 1,

13. V / I = Z0 * [(1 + Gv) / (1 - Gv)]

And finally we have to observe that, by inspection, the voltage V just
inside the line has to equal the voltage Vl just outside the line and,
by Kirchoff's current law, the current I just inside the line has to
equal the current Il just outside the line:

14. Vl = V and Il = I, where Vl and Il are the voltage and current at
the load.

From 13 and 14, and noting that Zl = Vl / Il, then,

15. Zl = Z0 * [(1 + Gv) / (1 - Gv)]

Now let's test the formulas for Gv that have been presented. We have
three to choose from:

A. The one just posted by Peter, (Zl - Z0conj) / (Zl + Z0conj)

B. Slick's, (Zl - Z0conj) / (Zl + Z0)

C. The one in all my texts and used by practicing engineers, (Zl - Z0) /
(Zl + Z0)

We'll plug the numbers into Peter's (A) first, giving us:

Gv = (50 - 50 - j10) / (50 + 50 + j10) = -j10 / (100 + j10) =
-0.009901 - j0.09901

Plugging this into the right side of 15, we get:

Zl = 46.15 -j19.23

This obviously isn't correct -- Zl is known to be 50 + j0.

Let's try Slick's (B):

Gv = (50 - 50 - j10) / (50 + 50 - j10) = -j10 / (100 - j10) =
0.009901 - j0.09901

from which we calculate, from 15,

Zl = 48.00 - j20.00

Again, obviously not right.

Finally, using the universally accepted formula (C):

Gv = (50 - 50 + j10) / (50 + 50 - j10) = j10 / (100 - j10) =
-0.009901 + j0.09901

from which, from 15, we get

Zl = 50.00 + j0

This is the value we know to be Zl. Getting the correct result for one
numerical example does *not* prove that an equation is correct. However,
getting an incorrect result for even one numerical example *does* prove
that an equation isn't correct. So we can conclude that variations A and
B aren't correct.

------------------

Actually, the accepted formula (C) can be derived directly from equation
15, so if all the steps to that point are valid, so is the accepted formula.

Why aren't Peter's or Slick's formulas correct? The real reason is that
they aren't derived from known principles by an orderly progression of
steps like the ones above. There's simply no way to get from known
voltage and current relationships to the conjugate equations. They're
plucked from thin air. That's simply not adequate or acceptable for
scientific or engineering use. (I challenge anyone convinced that either
of those equations is correct to present a similar development showing why.)

What about the seemingly sound logic that the accepted formula doesn't
work for complex Z0 because it implies that a conjugate match results in
a reflection? The formula certainly does imply that. And it's a fact --
a conjugate match guarantees a maximum transfer of power for a given
source impedance. But it doesn't guarantee that there will be no
reflection. We're used to seeing the two conditions coincide, but that's
just because we're used to dealing with a resistive Z0, or at least one
that's close enough to resistive that it's a good approximation. The
fact that the conditions for zero reflection and for maximum power
transfer are different is well known to people accustomed to dealing
with transmission lines with complex Z0.

But doesn't having a reflection mean that some power is reflected and
doesn't reach the load, reducing the load power from its maximum
possible value? As you might know from my postings, I'm very hesitant to
deal with power "waves". But what's commonly called forward power
doesn't stay constant as the load impedance is changed, nor does the
forward voltage. So it turns out that if you adjust the load for a
conjugate match, there is indeed reflected voltage, and "reflected
power". But the forward voltage and power are greater when the load is
Z0conj than when Zl = Z0 and no reflection takes place -- enough greater
that maximum power transfer occurs for the conjugate match, with a
reflection present.

I'd welcome any corrections to any statements I've made above, any of
the equations, or the calculations. The calculations are particularly
subject to possible error, so should undergo particular scrutiny. I'll
be glad to correct any errors. Anyone who disagrees with the conclusion
is invited and encouraged to present a similar development, showing the
derivation of the alternate formula and giving numerical results from an
example. That's how science, and good engineering, are done. And what it
takes to convince me.

Roy Lewallen, W7EL


Peter O. Brackett wrote:
Slick:

[snip]

[snip]

You are absolutely correct Roy, that formula given by "Slick" is just

plain

WRONG!

rho = (Z - R)/(Z + R)

Always has been, always will be.

[snip]

After consideration, I must agree with Slick.

Slick is RIGHT and I was WRONG!

Slick please accept my apologies!!! I was wrong, and I admit it!

Indeed, the correct formula for the voltage reflection coefficient "rho"
when computed using a "reference impedance" R, which is say the, perhaps
complex, internal impedance R = r + jx of a generator/source which is loaded
by a perhaps complex load impedance Z = ro + j xo must indeed be:

rho = (Z - conj(R))/(Z + conj(R)) = (Z - r + jx)/(Z + r - jx)

For indeed as Slick pointed out elsewhere in this thread, how else will the
reflected voltage equal zero when the load is a conjugate match to the
generator.

Slick thanks for directing the attention of this "subtlety" to the
newsgroup, and again...

Slick, please accept my apologies, I was too quick to criticize!

Good work, and lots of patience... :-)

Regards,

--
Peter K1PO
Indialantic By-the-Sea, FL.