Actually, my first posting was right all along, if Zo is always real.

From Les Besser's Applied RF Techniques:

"For passive circuits, 0=[rho]=1,

And strictly speaking: Reflection Coefficient =
(Zload-Zo*)/(Zload-Zo)

Where * indicates
conjugate.

But most of the literature assumes that Zo is real, therefore
Zo*=Zo."


And then i looked at the trusty ARRL handbook, 1993, page 16-2,
and lo and behold, the reflection coefficient equation doesn't have a
term for line reactance, so both this book and Pozar have indeed
assumed that the Zo will be purely real.

That doesn't mean Zload cannot have reactance (be complex).

Try your calculation again, and you will see that you can never
have a [rho] (magnitude of R.C.)greater than 1 for a passive network.

How could you get more power reflected than what you put in (do
you believe in conservation of energy, or do you think you can make
energy out of nothing)? If you guys can tell us, we could fix our
power problems in CA!

But thanks for checking my work, and this is a subtle detail that
is good to know.


Slick

"Dr. Slick" wrote
Actually, my first posting was right all along, if Zo is always real.

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

What a pity - it never is.

Dr. Slick:

[snip]
What a pity - it never is.

Makes for a good enough approximation for most simulations, though.
:
:
Slick
[snip]

Heh, heh...

No it's not!

Unless perhaps the only use you ever make of transmission line dynamics
Engineering
is for simple narrow band ham radio problems!

Try solving some real [i.e. broadband] problems where Zo is not real, not
even close, and
you'll see how important it is to use the whole danged complex expression.

rho = (Z - R)/(Z + R) is a complex function as are both the driving point
impedances the
load termination Z and the reference impedance of the transmission line R.

rho is complex! Get over it.

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

(Tom Bruhns) wrote in message om...
So, when are you going to start using your own head and not depending
on others? When will you go through the simple set of equations to
find that Vr/Vf = (Zload-Zo)/(Zload+Zo), and there IS NO conjugate on
the Zo in the numerator? I went through all that long ago (several
years) when someone questioned it, and I trust my calcs. It happens
that they agree with those of many, many others.


Who agrees with you? Les Besser is an established authority, as is
the ARRL, and Pozar... i use their examples because I am not yet an
authority myself. Please do some more research before you type.
Hey, we are all standing on the shoulders of giants!

I certainly wouldn't want to stand on YOUR shoulders, though...



Which of the following do you not believe? Because if you believe it
all, then it's a couple VERY SIMPLE algebraic steps to get to the
equation for Vr/Vf.
1. On a TEM line, Vf/If = -Vr/Ir = Zo, which may be reactive.
2. On a TEM line, V=Vf+Vr and I=If+Ir.
3. Where a TEM line connects to a load, the equations in (1) and (2)
hold, and V=Zload*I -- because the net line voltage is the same as the
voltage across the load, and the net line current is the same as the
load current.

Please do us a favor and go do the algebra, and if you get what Besser
teaches, show us the steps, and we'll resolve the difference from what
we get. If you get what we get, then go take it up with Besser.

Cheers,
Tom


I've checked my results with MIMP, and the reflection coefficients
matched the equation:

"For passive circuits, 0=[rho]=1,

And strictly speaking: Reflection Coefficient
=(Zload-Zo*)/(Zload-Zo)

Where * indicates conjugate."


A bit angry aren't we? Typical of one who has absolutely lost an
argument...

When you can show us your free energy device, we would love to see
it!


Slick


"Your Rage Has Imbalanced You!" - Lancelot

It is my opinion that the confusion over whether to use Zo or Zo*
(conjugate of Zo) in the computation for reflection coefficient arises
because there are two different meanings for the reflection
coefficient itself: one applies to voltage or current waves and the
other applies to "power waves." I do not have the Besser text
mentioned above but a 1965 IEEE paper by Kurokawa also uses the Zo*
term to calculate the reflection coefficient. However, Kurokawa makes
it clear that he is referring to "power waves" and not voltage or
current waves.

The Kurokawa paper was given as the justification for what I believe
is an erroneous equation in the 19th edition of the ARRL Antenna Book.
In all previous editions (at least the ones that I have) the formula
for reflection coefficient uses the normal Zo term. In the 19th
edition the formula was changed to use the Zo* (Zo conjugate) term.

I did some research on this and exchanged emails with some smart
folks, including Tom Bruhns and Bill Sabin. Then I wrote a note to
Dean Straw, editor of the Antenna Book, explaining why I thought the
new formula in the 19th edition was wrong. Here's a copy of that
note:

===========================================
Email to Dean Straw, 10/5/01

Dear Dean,

A week or so ago I wrote you concerning the formula for rho in the
19th Antenna Book:

rho = (Za-Zo*)/(Za+Zo) [Eq 6, page 24-7]

where Za is the impedance of the load, Zo is the line characteristic
impedance, and Zo* is the complex conjugate of Zo. You replied that
the justification for using Zo* in the numerator is explained in the
1965 IEEE paper by Kurokawa, and that it didn't really make much
difference whether the "classic" formula (Zo in numerator) or the
"conjugate" formula (Zo* in numerator) was used at SWR levels under
100 or so.

I obtained and studied the Kurokawa paper, did some research on the
Internet, exchanged some emails with some folks who know more about
this stuff than I do, and read through all the other technical
literature I have concerning rho. I'm afraid I disagree with both of
your statements (1. Justified by Kurokawa; 2. Doesn't matter for
normal SWR levels). Here's why:

An -infinitely- long line will have zero reflections (|rho|=0). If a
line of -finite- length is terminated with a load ZL which is exactly
equal to the Zo of the line, the situation will not change, there
should still be zero reflections. So if the formula for rho is
rho = (ZL-Zo)/(ZL+Zo)
then |rho| = 0, since the numerator evaluates to 0+j0.
However, if the formula is
rho = (ZL-Zo*)/(ZL+Zo)
then |rho| evaluates to something other than 0, since the
numerator evaluates to 0-j2Xo.

To use an example with real numbers: RG-174, Ro=50, VF=.66, Freq=3.75
MHz, matched line loss = 1.511 dB/100 ft (at 3.75 MHz), which yields a
calculated Zo (at this Freq) of Zo=50-j2.396. If ZL = Zo = 50-j2.396,
then:
|rho(Zo)| = 0 [classic formula, Zo in the numerator]
and
|rho(Zo*)| = 0.0479 ["conjugate" formula, Zo* in the numerator]

Using SWR=(1+|rho|) / (1-|rho|), these two rho magnitude values
evaluate to SWR 1:1 and SWR 1.10:1, respectively. So if a line is
terminated with a load equal to Zo, which is equivalent to an infinite
line, the "conjugate" formula results in a rho magnitude greater than
0 and an SWR greater than 1. This doesn't seem to make intuitive
sense.

This same anomaly may be extended to loads of other than Zo and to
points other than just the load end of the line. Using the Zo for
RG-174 as stated above (Zo=50-j2.396), and an arbitrary but totally
realistic load of ZL=100+j0 ohms (nominal 2:1 SWR), I used the full
hyperbolic transmission line equation to calculate what the Zin would
be at points along the line working back from the load from 0° to 360°
(one complete wavelength) in 15° steps. I then calculated the
magnitude of both rho(Zo) [classic formula] and rho(Zo*) [conjugate
formula] using the Zin values, and plotted the results. Here's the
plot:
http://www.qsl.net/ac6la/adhoc/Rho_C..._Conjugate.gif

(The scale for rho is on the right. The left scale is normally used
for R, X, and |Z|, but those plot lines have been intentionally hidden
in this case just to reduce the chart clutter.) Note that the plot
line for rho(Zo) [classic formula] progresses downward in a smooth
fashion as the line length increases, as expected. The rho(Zo*)
[conjugate formula] swings around, and even goes above the value at
the load point until a line length of about 75° is reached. Again,
this doesn't seem to make intuitive sense, and I can think of no
physical explanation which would result in the voltage reflection
coefficient magnitude "swinging around" as the line length is
increased.

Of course, the same data may be used to calculate and then plot SWR.
Here's the plot:
http://www.qsl.net/ac6la/adhoc/SWR_C..._Conjugate.gif
Again, this doesn't seem intuitive, and this is for a load SWR much
less than 100.

Now if the intent of the "conjugate" formula was to always force rho
to be = 1 and therefore to avoid the "negative SWR" problem, it
appears that this has the effect of "throwing the baby out with the
bathwater." That is, it may make it possible to calculate a rho value
less than 1 and hence a non-negative SWR value in an "extreme load"
situation like ZL=1+j1000 ohms (even though SWR is pretty meaningless
in that case). However, it also changes the rho and SWR values for
completely reasonable loads, such as the example above. At a line
length of 45°, the impedance at the input end of the line is
41.40-j31.29. Using the "classic" rho formula results in calculated
rho and SWR values of
rho=0.3095 SWR=1.90
while the "conjugate" formula gives
rho=0.3569 SWR=2.11
Note that these results are for a perfectly reasonable load on a
perfectly reasonable line at a perfectly reasonable frequency, but the
results differ by an unreasonable amount.

Another point. In the William Sabin article, "Computer Modeling of
Coax Cable Circuits" (QEX, August 1996, pp 3-10), Sabin includes the
Kurokawa paper as a reference. Even with that reference, Sabin gives
the "classic" formula for rho (called gamma in his paper) as Eq 31.
When I asked him recently about this, he stated that the article is
correct and he stands by the given formula for rho.

Given these various intuitive arguments as to why computing rho with
the Zo conjugate formula doesn't make sense, where did it come from?
Well, so far I have two candidates:

1) A QST Technical Correspondence article by Charlie Michaels (Nov
1997, pg 70). Michaels gave a formula for computing the portion of
the loss on a line that is due to standing waves. That loss formula
involves calculating rho by using the "conjugate" formula. The SWR dB
loss result, when added to the normal matched line loss number, gives
exactly the same figure for total power loss as do other formulas that
use completely different techniques (such as in papers by Sabin and
Witt). However, the Michaels QST article never said that the rho
"conjugate" formula should be used to calculate rho in the general
case, only that it should be used as part of an intermediate step to
calculate a dB number.

2) The 1965 IEEE paper by Kurokawa, "Power Waves and the Scattering
Matrix." Kurokawa does indeed show a formula for -a- reflection
coefficient that uses Z conjugate in the numerator. However, in
Section I of his paper he explains that he is talking about "power
waves" and takes pains to explain that these waves are not the same as
the more familiar voltage and current traveling waves. He then goes
on to give a mathematical description of these power waves. In
Section III he defines the power wave reflection coefficient as

s = (ZL - Zi*) / (ZL + Zi)

where ZL is the load impedance and Zi is the internal impedance of the
source. In a footnote he makes it clear that "s" is equal to the
voltage reflection coefficient only when Zi is real (no jX component).
Finally, in Section IX ("Comparison with Traveling Waves") he
explains that when the line Zo is complex the calculations that apply
to voltage and current waves are not the same calculations used to
determine the power delivered. He ends this section with this
statement: "Further, since the traveling wave reflection coefficient
is given by (ZL-Zo)/(ZL+Zo) [note no conjugate] and the maximum power
transfer takes place when ZL=Zo*, where ZL is the load impedance, it
is only when there is a certain reflection in terms of traveling waves
that the maximum power is transferred from the line to the load."

To put some actual numbers with this statement, consider the RG-174
from above (Zo=50-j2.396), with a load of ZL=Zo*=50+j2.396. Then

|(ZL-Zo)/(ZL+Zo)| = 0.0479
= small voltage traveling wave reflection
coefficient

while

|(ZL-Zo*)/(ZL+Zo)| = 0
= zero power wave reflection, meaning maximum
power transfer.

Now it seems to me that this clears up the confusion. It looks like
it is necessary to consider -two different meanings- for rho. One is
for the voltage (or current) traveling wave reflection coefficient.
The classic formula to compute that still holds (as Kurokawa states),
and that is the rho that should be used when talking about voltage (or
current) standing waves on a line. Specifically, that is the rho that
should be used in the formula for SWR,
SWR = 1 + |rho| / 1 - |rho|
possibly with an explanation that this formula is only applicable when
|rho| 1.

The -other- meaning for rho is used when dealing with "power waves" or
with power and loss calculations as in the Michaels QST formula, and
-that- rho (call it rho prime, or maybe some other letter ala
Kurokawa) may be defined as
rho' = (Z-Zo*)/(Z+Zo)
However, rho' does -not- have to do with voltage (or current)
traveling waves, and may -not- be used to compute SWR. If it is
understood that there are two different "reflection coefficients" then
everything starts to fall in place, including the last part of the
Kurokawa quote above saying that there is a situation when the
"voltage" reflection coefficient is slightly greater than 0 while the
"power wave" reflection coefficient is exactly 0. Note that no matter
what combination of values for Zo and ZL are used, |rho'| will never
be greater than 1 (although it can be equal to 1 for purely reactive
loads), thus satisfying the intuitive understanding that there can
never be more power reflected -from- a (passive) load than is
delivered -to- a (passive) load. But at the opposite extreme, as
shown above, "no reflected power" but does -not- necessarily mean "no
standing waves."

A further example of the importance of making a distinction between
the voltage reflection coefficient and the power reflection
coefficient would be the following: Assume a load of ZL=1+j1000 with
the RG-174 Zo from above, Zo=50-j2.396. Then

|rho| = 1.0047 (voltage wave reflection coefficient)

and

|rho'| = 0.9999 (power wave reflection coefficient)

This shows that the reflected voltage is slightly greater than the
incident voltage, at least at the point of reflection, before the line
loss has caused the calculated (or measured) rho to decay. It further
shows that the reflected power is still less than the incident power,
thus not violating the principal of conservation of energy. Robert
Chipman, "Theory and Problems of Transmission Lines," presents a
mathematical proof of this, including this quote from page 138: "...
a transmission line can be terminated with a [voltage] reflection
coefficient whose magnitude is as great as 2.41 without there being
any implication that the power level of the reflected wave is greater
than that of the incident wave."

In summary, I think a distinction must be made between the "voltage"
reflection coefficient and the "power" reflection coefficient, and
therefore I think the following changes should be made to the 19th
edition of the Antenna Book:

1. Revise the initial equation for rho [Eq 6, page 24-7] back to the
classic "non-Zo*" form, since rho is used in this context as the
voltage reflection coefficient.

2. Equation 11 on page 24-9 is
rho = sqrt(Pr/Pf)
where Pr and Pf are the reflected and forward power levels.
Intuitively this seems to be the "second" definition for rho, namely
the power reflection coefficient, although my math skills are not up
to the task of proving that this formula is the equivalent of the
Kurokawa formula for the power reflection coefficient 's'. Perhaps
this formula should have a footnote indicating that it refers to the
"power" and not "voltage" reflection coefficient, and that the two are
technically equal only when the Xo component of the line Zo is
ignored. (The same point as is made in the Kurokawa footnote referred
to above.) Given the precision to which most amateurs can measure
power, and the fact that under normal circumstances the line loss and
hence the Xo value is much smaller than that of RG-174, of course this
point is moot in a practical sense.

3. If Equation 11 is for the "power" reflection coefficient, then
Equation 12 is a mixing of apples and oranges. Perhaps the second
equal sign could be replaced with an "almost equal" sign.

4. Equation G in Table 2 "Coaxial Cable Equations" on page 24-20
should remain as is, since it obviously is referring to the voltage
reflection coefficient.

Thanks for looking this over, Dean. I would certainly welcome any
comments or feedback you might have.

73,
Dan Maguire AC6LA
===========================================

I don't know if Dean has changed the formula in later printings of the
19th edition or in the upcoming 20th edition. He responded that he
was busy with other matters and would get back to me later. He never
did and I let the matter drop.

I have the Kurokawa paper in pdf format. If anyone would like a copy,
drop me a private email and I'll be glad to send it to you. The two
charts mentioned above were produced with a modified version of the
XLZIZL Excel application. XLZIZL is available free from
www.qsl.net/ac6la.

Dan, AC6LA

(Dr. Slick) wrote in message . com...
....
A bit angry aren't we? Typical of one who has absolutely lost an
argument...

Angry? Perhaps you are (the other part of the "we"), but I'm not.
Disappointed in you that you are unwilling to think for yourself and
go through the simple calculations. Understand what Ian wrote: there
IS no argument!

Cheers,
Tom

(Dan) wrote in message . com...
It is my opinion that the confusion over whether to use Zo or Zo*
(conjugate of Zo) in the computation for reflection coefficient arises
because there are two different meanings for the reflection
coefficient itself: one applies to voltage or current waves and the
other applies to "power waves." I do not have the Besser text
mentioned above but a 1965 IEEE paper by Kurokawa also uses the Zo*
term to calculate the reflection coefficient. However, Kurokawa makes
it clear that he is referring to "power waves" and not voltage or
current waves.


Whether you find the reflection with Vr/Vi, or (Pr/Pi)**0.5, the
impedances should still be the same.



The Kurokawa paper was given as the justification for what I believe
is an erroneous equation in the 19th edition of the ARRL Antenna Book.
In all previous editions (at least the ones that I have) the formula
for reflection coefficient uses the normal Zo term. In the 19th
edition the formula was changed to use the Zo* (Zo conjugate) term.


My 1993 70th ed. of the ARRL handbook assumes the Zo to be always
purely real.

I believe the Zo* version is correct. The purely real Zo version
is correct too, but Zo must be purely real.




An -infinitely- long line will have zero reflections (|rho|=0). If a
line of -finite- length is terminated with a load ZL which is exactly
equal to the Zo of the line, the situation will not change, there
should still be zero reflections. So if the formula for rho is
rho = (ZL-Zo)/(ZL+Zo)
then |rho| = 0, since the numerator evaluates to 0+j0.
However, if the formula is
rho = (ZL-Zo*)/(ZL+Zo)
then |rho| evaluates to something other than 0, since the
numerator evaluates to 0-j2Xo.

To use an example with real numbers: RG-174, Ro=50, VF=.66, Freq=3.75
MHz, matched line loss = 1.511 dB/100 ft (at 3.75 MHz), which yields a
calculated Zo (at this Freq) of Zo=50-j2.396. If ZL = Zo = 50-j2.396,
then:
|rho(Zo)| = 0 [classic formula, Zo in the numerator]
and
|rho(Zo*)| = 0.0479 ["conjugate" formula, Zo* in the numerator]

Using SWR=(1+|rho|) / (1-|rho|), these two rho magnitude values
evaluate to SWR 1:1 and SWR 1.10:1, respectively. So if a line is
terminated with a load equal to Zo, which is equivalent to an infinite
line, the "conjugate" formula results in a rho magnitude greater than
0 and an SWR greater than 1. This doesn't seem to make intuitive
sense.


I think i does make sense in the sense that if the source
(reference) and the load both have reactance, that there WILL be some
reflections.

If ZL=Zo*=50+j2.396, then the capacitive reactance is indeed
canceled, and you are matching a pure 50 ohms to a pure 50 ohms again,
and the numerator will be zero, as it should be.

This is what impedance matching is all about really, not just
getting the real part of the imedance the same, but cancelling any
reactance.

The conjugate formula is correct.



This same anomaly may be extended to loads of other than Zo and to
points other than just the load end of the line. Using the Zo for
RG-174 as stated above (Zo=50-j2.396), and an arbitrary but totally
realistic load of ZL=100+j0 ohms (nominal 2:1 SWR), I used the full
hyperbolic transmission line equation to calculate what the Zin would
be at points along the line working back from the load from 0° to 360°
(one complete wavelength) in 15° steps. I then calculated the
magnitude of both rho(Zo) [classic formula] and rho(Zo*) [conjugate
formula] using the Zin values, and plotted the results. Here's the
plot:
http://www.qsl.net/ac6la/adhoc/Rho_C..._Conjugate.gif

(The scale for rho is on the right. The left scale is normally used
for R, X, and |Z|, but those plot lines have been intentionally hidden
in this case just to reduce the chart clutter.) Note that the plot
line for rho(Zo) [classic formula] progresses downward in a smooth
fashion as the line length increases, as expected.


This would be due to the losses of the line?


The rho(Zo*)
[conjugate formula] swings around, and even goes above the value at
the load point until a line length of about 75° is reached. Again,
this doesn't seem to make intuitive sense, and I can think of no
physical explanation which would result in the voltage reflection
coefficient magnitude "swinging around" as the line length is
increased.


I'm not totally sure if you did this right, but if the
transforming transmission line had reactance in it, and you are
measuring everthing from
Zo=50-j2.396, then i would expect the rho to swing up and down the
same way every 1/2 wavelength, as your data shows.



A further example of the importance of making a distinction between
the voltage reflection coefficient and the power reflection
coefficient would be the following: Assume a load of ZL=1+j1000 with
the RG-174 Zo from above, Zo=50-j2.396. Then

|rho| = 1.0047 (voltage wave reflection coefficient)

and

|rho'| = 0.9999 (power wave reflection coefficient)


If you had used the conjugate Zo* formula, you get

[rho] = 0.99989 which matches the bottom result.


I stand by the Conjugate formula, even more now.


Slick

A big deal is being made of the general assumption that Z0 is real.

As anyone who has studied transmission lines in any depth knows, Z0 is,
in general, complex. It's given simply as

Z0 = Sqrt((R + jwL)/(G + jwC))

where R, L, G, and C are series resistance, inductance, shunt
conductance, and capacitance per unit length respectively, and w is the
radian frequency, omega = 2*pi*f. This formula can be found in virtually
any text on transmission lines, and a glance at the formula shows that
Z0 is, in general, complex.

It turns out that R is a function of frequency because of changing skin
depth, but it increases only as the square root of frequency. jwL, the
inductive reactance per unit length, however, increases in direct
proportion to frequency. So as frequency gets higher, jwL gets larger
more rapidly. For typical transmission lines at HF and above, jwL R,
so R + jwL ~ jwL. G represents the loss in the dielectric, and again for
typical cables, it's a negligibly small amount up to at least the upper
UHF range. Furthermore, G, initially very small, tends to increase in
direct proportion to frequency for good dielectrics like the ones used
for transmission line insulation. So the ratio of jwC to G stays fairly
constant, is remains very large, at just about all frequencies. The
approximation that jwC G is therefore valid, so G + jwC ~ jwC.

Putting the simplified approximations into the complete formula, we get

Z0 ~ Sqrt(jwL/jwC) = Sqrt(L/C)

This is a familiar formula for transmission line characteristic
impedance, and results in a purely real Z0. But it's very important to
realize and not forget that it's an approximation. For ordinary
applications at HF and above, it's adequately accurate.

Having a purely real Z0 simplifies a lot of the math involving
transmission lines. To give just a couple of examples, you'll find that
the net power flowing in a transmission line is equal to the "forward
power" minus the "reverse power" only if you assume a real Z0.
Otherwise, there are Vf*Ir and Vr*If terms that have to be included in
the equation. Another is that the same load that gives mininum
reflection also absorbs the most power; this is true only if Z0 is
assumed purely real. So it's common for authors to derive this
approximation early in the book or transmission line section of the
book, then use it for further calculations. Many, of course, do not, so
in those texts you can find the full consequences of the complex nature
of Z0. One very ready reference that gives full equations is _Reference
Data for Radio Engineers_, but many good texts do a full analysis.

Quite a number of the things we "know" about transmission lines are
actually true only if the assumption is made that Z0 is purely real;
that is, they're only approximately true, and only at HF and above with
decent cable. Among them are the three I've already mentioned, the
simplified formula for Z0, the relationship between power components,
and the optimum load impedance. Yet another is that the magnitude of the
reflection coefficient is always = 1. As people mainly concerned with
RF issues, we have the luxury of being able to use the simplifying
approximation without usually introducing significant errors. But
whenever we deal with formulas or situations that have to apply outside
this range, we have to remember that it's just an approximation and
apply the full analysis instead.

Tom, Ian, Bill, and most of the others posting on this thread of course
know all this very well. We have to know it in order to do our jobs
effectively, and all of us have studied and understood the derivation
and basis for Z0 calculation. But I hope it'll be of value to some of
the readers who might be misled by statements that "authorities" claim
that Z0 is purely real.

Roy Lewallen, W7EL

One more thing. I've never seen that conjugate formula for voltage
reflection coefficient and can't imagine how it might have been derived.
I've got a pretty good collection of texts, and none of them show such a
thing. If anyone has a reference that shows that formula and its
derivation from fundamental principles, I'd love to see it, and discover
how the author managed to get from the same fundamental principles as
everyone else but ended up with a different formula.

Roy Lewallen, W7EL

Roy Lewallen wrote:
A big deal is being made of the general assumption that Z0 is real.

As anyone who has studied transmission lines in any depth knows, Z0 is,
in general, complex. It's given simply as

Z0 = Sqrt((R + jwL)/(G + jwC))

where R, L, G, and C are series resistance, inductance, shunt
conductance, and capacitance per unit length respectively, and w is the
radian frequency, omega = 2*pi*f. This formula can be found in virtually
any text on transmission lines, and a glance at the formula shows that
Z0 is, in general, complex.

A good approximation to Z0 is:

Z0 = R0 sqrt(1-ja/b)

where Ro = sqrt(L/C)
a is matched loss in nepers per meter.
b is propagation constant in radians per meter.

The complex value of Z0 gives improved accuracy in
calculations of input impedance and losses of
coax lines. With Mathcad the complex value is
easily calculated and applied to the various
complex hyperbolic formulas.

Reference: QEX, August 1996

Bill W0IYH