Convert reflection coefficient to Z
Wayne
right references now.
If an antenna reflection coefficient is measured at, for example. 0.333
at -100 degrees, how is Z calculated? I think I can do this with a smith
chart, but the result does not match my attempted calculations.
TIA
Wayne
Walter Maxwell
Hello Wayne,
The first reference I can give you is in "Theory and Problems of Transmission Lines," by Robert Chipman, in
Schaum's Outline Series, Page 128, Eqs. 7.9a and 7.9b.
If you don't have access to Schaum, I'll reconstitute the equations for you in Word, and send them via email.
In addition, I ran across these equations a day or so ago on this NG--I'll try to find them and direct you to
them on this NG.
Walt,W2DU
Owen Duffy
news:0CYQh.11100$P84.5052@trnddc07:
The expression for Gamma that springs to mind is (Zl-Zo)/(Zl+Zo).
Rearranging the terms gives Zl=-Zo(Gamma+1)/(Gamma-1) doesn't it?
Owen
Walter Maxwell
Wayne, I just put the pertinent equations into an email addressed to mygarb...@verizon.net, which I sent.
Is this your correct address? If not, please give me your correct address in an email to me at wa...@w2du.com.
Walt, W2DU
Owen Duffy
61.9.191.5:
>
> Rearranging the terms gives Zl=-Zo(Gamma+1)/(Gamma-1) doesn't it?
That probably looks better written as Zl=Zo(1+Gamma)/(1-Gamma)
Owen
Walter Maxwell
Owen, l just now sent Wayne the equations from Chipman, using the eq editor in Word.
In case others would like the equations I'll try to format them here without the Word editor.
R/Zo = (1 - rho squared)/(1 + rho squared - 2 rho cos phi)
X/Zo = (2 rho sin phi)/(1 + rho squared - 2 rho cos phi)
Walt, W2DU
Owen Duffy
news:ktq813dcbvoq57per...@4ax.com:
Walt, I take that to use rho to mean to magnitude of the reflection
coefficient Gamma.
My formula appears correct for Gamma, and the answer to Wayne's example
is 36-j27.
There is an uglier formula with tanh terms in it that gives the impedance
at a distance along a line with a given propagation constant... but it is
much more complicated to calculate than my expression when the distance
is zero.
Owen
Owen Duffy
...
> R/Zo = (1 - rho squared)/(1 + rho squared - 2 rho cos phi)
>
> X/Zo = (2 rho sin phi)/(1 + rho squared - 2 rho cos phi)
...
These appear to depend on Zo=Ro to be correct, perhaps they would be more
correctly expressed using Ro instead of Zo.
Owen
Wayne
Thanks for the info. Now things are working out right.
--Wayne
"Owen Duffy" <no...@no.where> wrote in message
news:Xns9909846D4C...@61.9.191.5...
Wayne
Thanks very much.
Wayne
"Walter Maxwell" <wa...@w2du.com> wrote in message
news:jmq8139oli0ji3e2f...@4ax.com...
Walter Maxwell
Owen, for historical accuracy, at least in the US, prior to 1950, rho, sigma, and S were used to represent
standing wave ratio. The symbol of choice used to represent reflection coefficient during that early era was
upper case lambda. However, in 1953 the American Standards Association (now NIST) announced in its publication
ASA Y10.9-1953, that rho is to replace upper case lambda as the standard symbol for reflection coefficient,
and SWR to represent standing wave ratio. Most of academia responded to the change, but a few have not. I
don't know about Australia, but in the US lambda is rarely seen as the symbol for reflection coefficient.
WRT Ro vs Zo, I was simply copying directly from Chipman, where Zo is routinely considered the characteristic
impedance of a transmission line, and where it's usually considered sufficiently low loss to the thought of as
Ro.
Walt
Owen Duffy
...
> that rho is to replace upper case lambda as the standard symbol for
> reflection coefficient, and SWR to represent standing wave ratio. Most
Chipman's formulae that you quoted work correctly if rho means the
magnitude of the complex reflection coefficient (rather than the
(complex) reflection coefficient as you say above).
The formulae were probably written when we used slide rules and worked
out the real and imaginary parts separately, whereas today with access to
tools that treat complex numbers as such, we can carry a complex value
through calculations as a single value then separate out the real an
imaginary parts at the end. There is also no real burden in treating Zo
as complex instead of the lossless / distortionless line approximation.
The different notation is painful, isn't it. I write Gamma to mean
uppercase-gamma, and use Gamma for the complex reflection coefficient,
rho for the magnitude of Gamma, lambda for wavelength, don't use Lambda
(I don't think), and gamma for the complex line propagation constant. It
think it is a fairly common convention, but if what you say above is
literally correct, it is not compliant with ASA Y10.9-1953.
Owen
K7ITM
> On Thu, 05 Apr 2007 03:01:11 GMT, Owen Duffy <n...@no.where> wrote:
> >Walter Maxwell <w...@w2du.com> wrote in
About Zo being reasonably approximated by Ro, or not:
I made a note to myself some time ago, and I believe it's reasonably
accurate, that neglecting dielectric loss, for a TEM line, given Zo =
Ro+jXo, then to a good approximation
Xo = -0.180*Ro*A*Vf/f
where A = line attenuation in dB/100ft
Vf = line velocity factor
f = frequency in MHz.
So for small diameter 50 ohm polyethylene dielectric line at 1.8MHz,
the worst case for most ham applications, Xo/Ro is about .18. For
that, I used 2.7dB/100ft for RG174 type line. That's getting to be
pretty significant, a ten degree phase angle away from purely
resistive. As Owen posted, it's so easy these days to deal with
complex numbers that you may as well just carry them all along. Given
the above formula, it's easy to figure the complex Zo for a line where
you know the nominal attenuation, the velocity factor, and the
frequency, and of course the nominal high frequency Zo value.
Cheers,
Tom
Walter Maxwell
Can't argue with your comments above, Tom, but what ham in his right mind would use 100 feet of RG174?
Walt
K7ITM
Stealth, Walt. Stealth. Or--backpacking. Hey, it's less than 3dB
loss. In any event, please note that Zo is independent of length.
It's the same for a 3-foot length of line as for a 10000-foot length,
assuming the line is indeed reasonably uniform. The A100 value is in
the formula only because it's so easy to look up that value directly.
Interestingly, the attenuation of RG174 (because of the thin copper on
the copperweld inner conductor) is relatively constant with
frequency. It doesn't follow the same dB-proportional_to-sqrt(f) law
of line with copper or tinned copper center conductor, at HF
frequencies. So a 30 foot length of it is certainly fine for feedline
to a receiving antenna at HF, and a very acceptable compromise for
many people for foot-transportable or hidden-from-the-neighbors
transmitting work.
Cheers,
Tom
Cecil Moore
>> On 5 Apr 2007 09:49:52 -0700, "K7ITM" <k...@msn.com> wrote:
>>>> So for small diameter 50 ohm polyethylene dielectric line at 1.8MHz,
>>> the worst case for most ham applications, Xo/Ro is about .18. For
>>> that, I used 2.7dB/100ft for RG174 type line.
> Interestingly, the attenuation of RG174 (because of the thin copper on
> the copperweld inner conductor) is relatively constant with
> frequency.
Owen's transmission line calculator gives the Z0 of RG-174
as 50.13-j3.59 at 1.8 MHz. That's an X0/R0 ratio of about
0.07, not 0.18.
--
73, Cecil, w5dxp.com
Roy Lewallen
>
> Can't argue with your comments above, Tom, but what ham in his right mind would use 100 feet of RG174?
I do, for 40 and 80 meter antennas on Field Day since I backpack
everything in. But then I'm often accused of not being in my right mind,
so there still might not be anyone in that category who does.
Roy Lewallen, W7EL
Owen Duffy
news:1175791792.3...@n76g2000hsh.googlegroups.com:
...
> About Zo being reasonably approximated by Ro, or not:
>
> I made a note to myself some time ago, and I believe it's reasonably
> accurate, that neglecting dielectric loss, for a TEM line, given Zo =
> Ro+jXo, then to a good approximation
>
> Xo = -0.180*Ro*A*Vf/f
>
> where A = line attenuation in dB/100ft
> Vf = line velocity factor
> f = frequency in MHz.
>
> So for small diameter 50 ohm polyethylene dielectric line at 1.8MHz,
> the worst case for most ham applications, Xo/Ro is about .18. For
> that, I used 2.7dB/100ft for RG174 type line. That's getting to be
I make the loss/100' of RG174 to be 1.1dB and from that I get Xo=-3.6
ohms. (Did you get loss/100m from somewhere? This is probably the answer
to Cecil's diligent spot of an apparent error.)
> pretty significant, a ten degree phase angle away from purely
> resistive. As Owen posted, it's so easy these days to deal with
> complex numbers that you may as well just carry them all along. Given
> the above formula, it's easy to figure the complex Zo for a line where
> you know the nominal attenuation, the velocity factor, and the
> frequency, and of course the nominal high frequency Zo value.
Beyond dealing with transmission lines as Zo=Ro:
A common approximation for Xo is -Ro*alpha/beta and Ro=Ro. This
effectively (approximately) attributes all of the effect of the R element
of and RLGC model to Xo. Tom, your method is equivalent to this when
attenuation is converted to nepers/unit-length and frequency is converted
to radians/unit-length.
My line loss calculator at http://www.vk1od.net/tl/tllc.php takes a
different approach. It computes an estimate of the complex characteristic
impedance implied by the loss=k1*f^0.5+k2*f model, nominal Ro, and
velocity factor using an RLGC model. The model assumes:
* R is proportional to square root of frequency;
* L is constant;
* G is proportional to frequency; and
* C is constant.
These are reasonable assumptions for most practical transmission lines
down to about 100kHz. The assumption to become invalid first is the first
assumption (are you still with me) which depends of fully developed skin
effect, hence the low frequency qualification.
The k1 and k2 values are obtained by regression from published
attenuation figures.
I haven't seen this done in other calculators, so it is one of the
reasons why my calculator will give slightly different results to others
such as TLDETAILS for example.
Owen
Owen Duffy
...
>> So for small diameter 50 ohm polyethylene dielectric line at 1.8MHz,
>> the worst case for most ham applications, Xo/Ro is about .18. For
>> that, I used 2.7dB/100ft for RG174 type line. That's getting to be
>
> I make the loss/100' of RG174 to be 1.1dB and from that I get Xo=-3.6
> ohms. (Did you get loss/100m from somewhere? This is probably the
> answer to Cecil's diligent spot of an apparent error.)
>
Tom, in view of your comment about skin effect not being well developed
in RG174, I went to Belden's data sheet for 8216 (RG174 type) and sure
enough, the loss below 30 MHZ does not track the loss=k1*f^0.5+k2*f
model. Your loss figure of 2.7dB/100' may well be correct, and my line
loss calculator is in error below 30MHz for this particular cable due to
the thin copper plating and steel core of the inner conductor. TLDETAILS
and other calculators based on the same loss model will also be in error.
I note that the ARRL TLW shows 1.8dB/100' for 8216 at 1.8MHz.
This sets me thinking of a way to calculate a lower frequency limit to
the loss model when I generate it, so that I can store that limit in the
database and prevent calculation below that frequency.
Owen
Walter Maxwell
Ok, Ok, I was not in my right mind when I made the comment--apologies to all.
Walt, W2DU
Owen Duffy
> This sets me thinking of a way to calculate a lower frequency limit to
> the loss model when I generate it, so that I can store that limit in
> the database and prevent calculation below that frequency.
I have just analysed the tllc database contents to find cases where the
modelled error is more than 10% different to the data points on which the
regression was based.
There are a few cases, they are all copper clad steel inner conductors
(some of the RG6, RG59, RG174, RG316). I need to implement a lower
frequency limit for model validity for each cable type.
An alternative approach to retain some lower frequency results is to use a
cubic spline interpolation... but it has its own problems.
Owen
K7ITM
> Owen Duffy <n...@no.where> wrote innews:Xns990A4E9D25...@61.9.191.5:
Hi Owen,
Yes, I've played with the same model (k1*sqrt(f)+k2*f) for loss. For
things below 500MHz or so--and generally above--the k2 term seemed to
always be so low as to not be worth including. For one thing, the
high-frequency apparent loss may well NOT be due to increased loss in
the dielectric, but rather to variations in impedance along the line
or some similar phenomenon. Published specs for precision lines seem
to be a lot closer than those for "garden variety coax" to what I'd
expect based on theory and what I believe to be the dielectric's power
factor for polyethylene and PTFE. Assuming that all types of line
from a given reputable manufacturer use the same quality polyethylene
and the same quality PTFE, we should see the same contribution from
dielectric, taking into account solid versus foam, for all types: at
a given frequency, the dielectric loss of the poly or PTFE should be
the same. If that's the case, then for some lines at least, the k2
factor must not be due entirely to dielectric loss.
A note on my simple formula: one of the approximations in it is that
the phase angle of Zo is assumed to be close to zero, so that
sqrt(1+j*x) can be reasonably approximated as 1+j*x/2. For x=+/- 0.2,
the error magnitude is less than 0.005, and the error in the imaginary
part is less than 0.0005. But for line that has a seriously reactive
Zo, the error can be large.
We have Roy to thank for pointing out to me, several years ago, that
characteristic of RG-174. I knew I'd have a chance to thank him in
public for it sometime. ;-)
Cheers,
Tom
Jim Lux
Not the least of which is that you (philosophically) want a model that
is based on the underlying physics (which the sqrt(f), f model is)..
The problem comes in because sqrt(f) doesn't model skin effect at low
frequencies very well, when the skin depth becomes an appreciable
fraction of the conductor diameter (because the conductor is no longer a
thin wall tube).. the cladding just throws another wrench into the works.
What might work is if you look at the generic curves for Rac/Rdc for
round and tubular conductors. The analytical formulation is quite
complex, but I'm pretty sure there's a simple polynomial approximation.
Jim
Richard Harrison
"The first reference I can give you is in "Theory and Problems of
128 Eqs 7.9a and 7.9b."
Alas, I don`t have Chipman`s book. Another source is Terman`s 1955
"Electronic and Radio Engineering". He solves the differential equations
for a transmission line, starting on page 84 for solution of traveling
wave problems.
Best regards, Richard Harrison, KB5WZI
Owen Duffy
news:ev5q81$rj8$1...@nntp1.jpl.nasa.gov:
Hi Jim,
I did some playing around comparing spline fits with manufacturers data
points. The underlying problem is that the manufacturer might give a data
point at say 50MHz where the skin effect appears well developed (the data
point is a good fit to the simple loss model constructed with that data
point and the ones at higher frequencies), and only one data point much
lower (eg 5MHz) that is not a good fit to the model and suggests that
skin effect is not well developed at that frequency.
The lack of a good number of data points in the region where resistance
is not proportional to f^0.5 prevents accurate modelling. The loss data
doesn't provide enough information to infer the relative diameters of the
high conductivity coating and the low conductivity core.
The approach I have taken with tllc is:
- explain the issue in the usage nots;
- carry into the summarised data, the lowest frequency on which the model
is based so that it can be displayed and users aware when the model
results are an extrapolation;
- the raw data has been analysed to find low frequency data points that
are more than 10% different to forecast by the predicted loss model, and
those points have been excised and the models recreated;
For example, the data for Belden 1189A (a CCS inner conductor) has had a
5MHz data point excised, and the lowest data point used is now 55MHz. The
calculator results shows that frequency, and the user must make his own
mind up about the applicability of an extrapolated result.
I use RG6 coax that has a hard drawn copper centre conductor, and tllc's
results for Belden 1189A (an RG6 type) are probably quite reasonable at
3MHz, but the results would underestimate the loss in real Belden 1189A
because of its use of CCS inner conductor.
An interesting question is ladder lines. Taking Wireman's products, 552
which uses a #16 19 strand copper clad steel conductor or unspecified
coating thickness might well have higher loss than 551 which has a #18
30% single core copper clad steel conductor at sufficiently low
frequency. The question is at what frequency does the effect of the
thinner copper coating of the thicker conductor bundle manifest itself. I
know that Wes measured these lines, and in the article I read he stated
that the measurements were done between 50MHz and 150MHz which would
probably not have shown the effects of the thin coating at low
frequencies.
Owen
Roy Lewallen
the following reasons:
1. Plated or tinned center conductor, where plating or tinning is
thinner than several skin depths at the lowest frequency of analysis.
2. Roughness of a stranded center conductor.
3. Roughness of a braided shield and the necessity for the current to
migrate from wire bundle to wire bundle.
4. Shield thickness which is less than several skin depths at the lowest
frequency of analysis.
5. Tinned shield.
6. Multiple shields.
Numbers 1, 4, and 5 can be calculated, but require modified Bessel
functions and often some mathematical trickery to prevent truncation or
overflow errors even with extended precision calculation. The remainder
are often empirically determined, are complex functions of frequency,
and vary from one cable type or manufacturer to another. All can be
significant with typical cables at HF.
Roy Lewallen, W7EL
Jim Lux
Prevents accurate modeling from measured data.. or, more accurately,
prevents you from validating your model with measured data.
I think you could develop the model from physics (which is how the
sqrt(f),f models were developed), and then determine the coefficients
for a particular type coax by measurement.
I would start by looking at Rac/Rdc for the center conductor for low
frequencies. Some reasonable assumptions are
a) dielectric loss is negligble
b) loss in the shield can be modeled by the infinite plane skin depth
formula.
References such as the ITT Reference Data For Radio Engineers have a
short table from which you can build a piecewise model. The analytic
model is a bit complex (hah.. as I recall, it involves Bessel functions
and elliptic integrals)
Stranded copperclad is a very tricky thing (much like measuring the RF
resistance of braid) because you have both the skin effect in a single
conductor issue and the proximity effect of adjacent conductors, and on
top of that, the current flow among conductors (the inner conductors
carry less current than the outer ones).. Much like trying to analyze
Litz wire.
Maybe the thing to do is to actually measure some samples and bound the
error.. (probably not worth going farther if the error is 0.1 dB in 1000
ft, for instance).
I'd start by just running the numbers for solid copper and see what the
difference between the "thin walled tube model" (implied in the sqrt(f)
term) and the actual "solid conductor with skin effect"
>
> Owen
Owen Duffy
Jim,
Comments noted.
My objective isn't so much trying to build a better loss model, but
rather to use a simple loss model and build the coefficients from
manufacturer's published data and make that available in the calculator.
Although I calculate the correlation coefficient when do the regressions
on the manufacturers freq/loss tables, and had not used models with poor
r^2, I hadn't explored the lower frequency data points for fit. I did
just that a day or so ago, and interestingly the cases where low
frequency points were not a good fit were all non-homogenous centre
conductors.
I had previously explored thin homogenous centre conductors, and they
should not be an issue for practical cables down to 0.1MHz.
I am also not obsessing about accuracy, because real cables have
tolerances that set a limit to necessary accuracy.
So, to try and persevere with the simple model, I have excluded data
points that appear to be a result of departure from the simple model due
to composite conductor effects, and show the lower frequency that bounds
extrapolated / interpolated results. Extrapolated results are always a
risk, the user must make their own judgements about the applicability.
I noted Roy's comments on braids and composite conductor issues, and I
thank him for his response. I may include some similar explanation in the
calculator's usage notes.
It also occurs to me that some cable construction intended for 500MHz and
above use a plastic film with a very thin metallic coating, and I wonder
about its performance at low frequencies. Perhaps yet another departure
from a simple loss model.
As noted above, the largest departures at low HF frequenciesfrom the
simple loss model were all cables with steel cored inner conductors.
There is a salutory lesson that when buying RG59 or RG6 for HF use, those
cables with 100% copper inner conductors are likely to be better at the
low end of HF. The same may apply for use of such cables for baseband
applications like video.
Owen
Roy Lewallen
> . . .
> My objective isn't so much trying to build a better loss model, but
> rather to use a simple loss model and build the coefficients from
> manufacturer's published data and make that available in the calculator.
Unfortunately, manufacturer's published loss data are often quite
different than actual cable loss. Belden RG cable I measured long ago
was routinely considerably better than the spec -- apparently the spec
was dictated by the MIL SPEC, and the cable was manufactured to never
exceed it. More recently, I've found that in trying to convince rather
naive amateurs to purchase their cable, some manufacturers are claiming
considerably lower loss than the cable actually has. So the bottom line
is that manufacturer's published data are just so many numbers, and
don't necessarily have any direct relationship to any real cable.
Careful scrutiny of real cable will also reveal that the characteristic
impedance varies quite a bit, and the velocity factor of foamed
dielectric cable is even more variable.
Roy Lewallen, W7EL
Owen Duffy
@corp.supernews.com:
> Owen Duffy wrote:
>> . . .
>> My objective isn't so much trying to build a better loss model, but
>> rather to use a simple loss model and build the coefficients from
>> manufacturer's published data and make that available in the
calculator.
> >. . .
>
> Unfortunately, manufacturer's published loss data are often quite
> different than actual cable loss. Belden RG cable I measured long ago
> was routinely considerably better than the spec -- apparently the spec
> was dictated by the MIL SPEC, and the cable was manufactured to never
> exceed it. More recently, I've found that in trying to convince rather
> naive amateurs to purchase their cable, some manufacturers are claiming
> considerably lower loss than the cable actually has. So the bottom line
> is that manufacturer's published data are just so many numbers, and
> don't necessarily have any direct relationship to any real cable.
I understand. One of the cable types that I tried to fit to the loss
model was Davis Bury Flex, and it had the worst regression errors of all
of the 90 line types that I modelled. Of course, some manufacturers data
is an extremely good fit, and I suspect that is a result of fitting their
own measurement data to the same model, then publishing points from the
modelled performance.
>
> Careful scrutiny of real cable will also reveal that the characteristic
> impedance varies quite a bit, and the velocity factor of foamed
> dielectric cable is even more variable.
Agreed... but you have to start somewhere with design, and the
manufacturer's data isn't such a bad place to start. But, I hear your
point that obsessing about model accuracy isn't wise.
Owen
Roy Lewallen
> . . .
> model was Davis Bury Flex, and it had the worst regression errors of all
Ah, yes, that stuff. It's easy to make it fit a model at about 200 MHz
and above -- just flex it a bit until the loss matches your model. At
400 MHz, you'll have even more variation to work with.
Roy Lewallen, W7EL