Convert reflection coefficient to Z
 

On Thu, 05 Apr 2007 03:01:11 GMT, Owen Duffy wrote:

Walter Maxwell wrote in
:

...
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

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

Convert reflection coefficient to Z
 

Walter Maxwell wrote in
:

....
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

Convert reflection coefficient to Z
 

On Apr 4, 8:38 pm, Walter Maxwell wrote:
On Thu, 05 Apr 2007 03:01:11 GMT, Owen Duffy wrote:
Walter Maxwell wrote in
:

...
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

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

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

Convert reflection coefficient to Z
 

On 5 Apr 2007 09:49:52 -0700, "K7ITM" wrote:

On Apr 4, 8:38 pm, Walter Maxwell wrote:
On Thu, 05 Apr 2007 03:01:11 GMT, Owen Duffy wrote:
Walter Maxwell wrote in
:

...
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

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

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

Can't argue with your comments above, Tom, but what ham in his right mind would use 100 feet of RG174?

Walt

Convert reflection coefficient to Z
 

On Apr 5, 10:28 am, Walter Maxwell wrote:
On 5 Apr 2007 09:49:52 -0700, "K7ITM" wrote:



On Apr 4, 8:38 pm, Walter Maxwell wrote:
On Thu, 05 Apr 2007 03:01:11 GMT, Owen Duffy wrote:
Walter Maxwell wrote in
:

...
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

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

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

Can't argue with your comments above, Tom, but what ham in his right mind would use 100 feet of RG174?

Walt


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

Convert reflection coefficient to Z
 

K7ITM wrote:
On 5 Apr 2007 09:49:52 -0700, "K7ITM" 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

Convert reflection coefficient to Z
 

Walter Maxwell wrote:

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

Convert reflection coefficient to Z
 

"K7ITM" wrote in
oups.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

Convert reflection coefficient to Z
 

Owen Duffy wrote in
:

....
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

Convert reflection coefficient to Z
 

On Thu, 05 Apr 2007 13:46:14 -0700, Roy Lewallen wrote:

Walter Maxwell wrote:

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

Ok, Ok, I was not in my right mind when I made the comment--apologies to all.

Walt, W2DU