I note that any textbook I pick up shows that VSWR=(1+rho)/(1-rho) where
rho is the magnitude of Gamma (Gamma=(Z-Zo)/(Z+Zo)); rho=abs(Gamma)).

Now, reading TL theory texts can be confusing because of the sometimes
subtle swithes to and from an assumption of lossless line (under which
rho cannot exceed 1).

Since VSWR is the ratio of the magnitude of the voltage at a maximum in
the standing wave pattern to the magnitude of the voltage at a minimum
in the standing wave pattern, if we are to infer SWR at a point on a
line (if that makes sense anyway) from rho (which is a property of a
point on a lossy line), isn't the formula VSWR=abs(1+rho)/abs(1-rho)
correct in the general case (lossy or lossless line)?

Given that rho cannot be negative (since it is the magnitude of a
complex number), the general formula can be simplified to
VSWR=(1+rho)/abs(1-rho).

Seems to me that texts almost universally omit the absolute operation on
the denominator without necessarily qualifying it with the assumption of
lossless line.

If VSWR=(1+rho)/abs(1-rho), then doesn't it follow that rho is not a
function of VSWR (except in the lossless line case where
VSWR=(1+rho)/(1-rho) and therefore rho=(VSWR-1)/(VSWR+1))?

Thoughts?

Owen

Owen wrote:

I note that any textbook I pick up shows that VSWR=(1+rho)/(1-rho) where
rho is the magnitude of Gamma (Gamma=(Z-Zo)/(Z+Zo)); rho=abs(Gamma)).

Now, reading TL theory texts can be confusing because of the sometimes
subtle swithes to and from an assumption of lossless line (under which
rho cannot exceed 1).

To complicate matters, roughly half the textbooks use rho instead of
Gamma for the complex reflection coefficient. You've got to be careful.

Since VSWR is the ratio of the magnitude of the voltage at a maximum in
the standing wave pattern to the magnitude of the voltage at a minimum
in the standing wave pattern, if we are to infer SWR at a point on a
line (if that makes sense anyway) from rho (which is a property of a
point on a lossy line), isn't the formula VSWR=abs(1+rho)/abs(1-rho)
correct in the general case (lossy or lossless line)?

The whole concept of VSWR gets flakey on a lossy line, and really loses
its meaning. It's often analytically convenient to define a quantity at
a point and call it "VSWR", although in the presence of loss it no
longer means the ratio of maximum to minimum voltage on the line. Since
it's lost its original meaning, it comes to mean just about anything
you'd like. And the generally accepted definition then is the equation
you gave in your first paragraph. That is, in the presence of loss, VSWR
is something which is *defined* by that equation, rather than the
equation being a means of calculating some otherwise defined property.

Under the right conditions and if loss is large enough, rho can be
greater than 1, in which case the VSWR as defined by the equation in the
first paragraph becomes negative. Again, this is no longer a ratio of
voltages along a line, but a quantity defined by an equation. If you
alter the equation, you're defining a different quantity. Now, there's
no reason that your "VSWR" definition isn't just as good as the
conventional one (first paragraph equation). But the conventional one is
pretty universally used, and yours is different, so if you're interested
in communicating, it would be wise to give it a different name or at
least carefully show what you mean when you use it.

Given that rho cannot be negative (since it is the magnitude of a
complex number), the general formula can be simplified to
VSWR=(1+rho)/abs(1-rho).

But it can be greater than one. See above.

Seems to me that texts almost universally omit the absolute operation on
the denominator without necessarily qualifying it with the assumption of
lossless line.

If VSWR=(1+rho)/abs(1-rho), then doesn't it follow that rho is not a
function of VSWR (except in the lossless line case where
VSWR=(1+rho)/(1-rho) and therefore rho=(VSWR-1)/(VSWR+1))?

Rho is never a function of VSWR. VSWR is a function of rho. Unlike
actual VSWR (that is, the ratio of maximum to minimum voltage along a
line), the reflection coefficient can be and is rigorously and
meaningfully defined at any point along a line, lossy or not.

Roy Lewallen, W7EL

Roy Lewallen wrote:
Owen wrote:


I note that any textbook I pick up shows that VSWR=(1+rho)/(1-rho)
where rho is the magnitude of Gamma (Gamma=(Z-Zo)/(Z+Zo));
rho=abs(Gamma)).
....
property of a point on a lossy line), isn't the formula
VSWR=abs(1+rho)/abs(1-rho) correct in the general case (lossy or
lossless line)?


The whole concept of VSWR gets flakey on a lossy line, and really loses
its meaning. It's often analytically convenient to define a quantity at
a point and call it "VSWR", although in the presence of loss it no
longer means the ratio of maximum to minimum voltage on the line. Since

Indeed. It occurs to me that if one was to try to measure VSWR (say
using a slotted line with probe) on a lossy line operating at high VSWR,
the best estimate would come from finding a minimum, measuring it and
the adjacent maxima and calculating VSWR=(Vmax1+Vmax2)/Vmin/2 so that
the measurement is biassed towards the notional VSWR at the point of the
minimum (the minimum being much more sensitive to line attenuation than
the maxima).

it's lost its original meaning, it comes to mean just about anything
you'd like. And the generally accepted definition then is the equation
you gave in your first paragraph. That is, in the presence of loss, VSWR
is something which is *defined* by that equation, rather than the
equation being a means of calculating some otherwise defined property.

Noted

Under the right conditions and if loss is large enough, rho can be
greater than 1, in which case the VSWR as defined by the equation in the
first paragraph becomes negative. Again, this is no longer a ratio of
voltages along a line, but a quantity defined by an equation. If you
alter the equation, you're defining a different quantity. Now, there's
no reason that your "VSWR" definition isn't just as good as the
conventional one (first paragraph equation). But the conventional one is
pretty universally used, and yours is different, so if you're interested
in communicating, it would be wise to give it a different name or at
least carefully show what you mean when you use it.

Given that rho cannot be negative (since it is the magnitude of a
complex number), the general formula can be simplified to
VSWR=(1+rho)/abs(1-rho).


But it can be greater than one. See above.

Agreed.


Seems to me that texts almost universally omit the absolute operation
on the denominator without necessarily qualifying it with the
assumption of lossless line.

If VSWR=(1+rho)/abs(1-rho), then doesn't it follow that rho is not a
function of VSWR (except in the lossless line case where
VSWR=(1+rho)/(1-rho) and therefore rho=(VSWR-1)/(VSWR+1))?


Rho is never a function of VSWR. VSWR is a function of rho. Unlike

I mean't function in the sense that there is one and only one value of
f(x) for x, rather than in the causual sense.

actual VSWR (that is, the ratio of maximum to minimum voltage along a
line), the reflection coefficient can be and is rigorously and
meaningfully defined at any point along a line, lossy or not.

Agreed.

Thanks for your exhaustive reply Roy, it is appreciated.

What I glean from this is that although Gamma and therefore rho are well
defined, and both are a function of position on a line in the general
case, the "accepted mathematical definition" of VSWR in terms of rho
does not behave well (eg producing a negative value) in some cases and
is not a good estimator of real VSWR in those cases.

It seems fair to say that the reason that the "accepted mathematical
definition" of VSWR does not behave well is that it depends on an
assumption of a distortionless line (Xo=0). (Lossless lines are
distortionless but the converse is not necessarily true).

I take your point about the need to qualify a different algorithm by a
different name.

Thanks again.

Owen

"Roy Lewallen" wrote in message
...
Owen wrote:

I note that any textbook I pick up shows that VSWR=(1+rho)/(1-rho) where rho
is the magnitude of Gamma (Gamma=(Z-Zo)/(Z+Zo)); rho=abs(Gamma)).

Now, reading TL theory texts can be confusing because of the sometimes subtle
swithes to and from an assumption of lossless line (under which rho cannot
exceed 1).

To complicate matters, roughly half the textbooks use rho instead of Gamma for
the complex reflection coefficient. You've got to be careful.

Since VSWR is the ratio of the magnitude of the voltage at a maximum in the
standing wave pattern to the magnitude of the voltage at a minimum in the
standing wave pattern, if we are to infer SWR at a point on a line (if that
makes sense anyway) from rho (which is a property of a point on a lossy
line), isn't the formula VSWR=abs(1+rho)/abs(1-rho) correct in the general
case (lossy or lossless line)?

The whole concept of VSWR gets flakey on a lossy line, and really loses its
meaning. It's often analytically convenient to define a quantity at a point
and call it "VSWR", although in the presence of loss it no longer means the
ratio of maximum to minimum voltage on the line. Since it's lost its original
meaning, it comes to mean just about anything you'd like. And the generally
accepted definition then is the equation you gave in your first paragraph.
That is, in the presence of loss, VSWR is something which is *defined* by that
equation, rather than the equation being a means of calculating some otherwise
defined property.

Under the right conditions and if loss is large enough, rho can be greater
than 1, in which case the VSWR as defined by the equation in the first
paragraph becomes negative. Again, this is no longer a ratio of voltages along
a line, but a quantity defined by an equation. If you alter the equation,
you're defining a different quantity. Now, there's no reason that your "VSWR"
definition isn't just as good as the conventional one (first paragraph
equation). But the conventional one is pretty universally used, and yours is
different, so if you're interested in communicating, it would be wise to give
it a different name or at least carefully show what you mean when you use it.

Given that rho cannot be negative (since it is the magnitude of a complex
number), the general formula can be simplified to VSWR=(1+rho)/abs(1-rho).

But it can be greater than one. See above.

Seems to me that texts almost universally omit the absolute operation on the
denominator without necessarily qualifying it with the assumption of lossless
line.

If VSWR=(1+rho)/abs(1-rho), then doesn't it follow that rho is not a function
of VSWR (except in the lossless line case where VSWR=(1+rho)/(1-rho) and
therefore rho=(VSWR-1)/(VSWR+1))?

Rho is never a function of VSWR. VSWR is a function of rho. Unlike actual VSWR
(that is, the ratio of maximum to minimum voltage along a line), the
reflection coefficient can be and is rigorously and meaningfully defined at
any point along a line, lossy or not.

Roy Lewallen, W7EL

Good response, Roy, but concerning rho and gamma to represent reflection
coefficient, I refer you to Reflections, Sec 3.1,

"Prior to the 1950s rho and sigma, and sometimes 'S' were used to represent
standing wave ratio. The symbol of choice to represent reflection coefficient
during that era was upper case gamma. However, in 1953 the American Standards
Association (now the NIST) announced in its publication ASA Y10.9-1953, that rho
is to replace gamma for reflection coefficient, with SWR to represent standing
wave ratio (for either voltage or current), and VSWR specifically for voltage
standing wave ratio. Most of academia responded to the change, but some
individuals did not. Consequently, gamma is occasionally seen representing
reflection coefficent, but only rarely."

Hope this clarifies any misunderstanding concerning the use of gamma for
reflection coefficient.

Incidentally, Roy, I recently mailed you a CD containing Laport's book, "Radio
Antenna Engineering." I'm wondering if you received it, or did it go astray?

Walt, W2DU

Owen wrote:
Thanks for your exhaustive reply Roy, it is appreciated.

It's time to barbecue the virtual reflection coefficient
sacred cow or at least understand that it deviates from
the field of optics and S-parameter analysis.
--
73, Cecil http://www.qsl.net/w5dxp

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Walter Maxwell wrote:
Good response, Roy, but concerning rho and gamma to represent reflection
coefficient, I refer you to Reflections, Sec 3.1,

"Prior to the 1950s rho and sigma, and sometimes 'S' were used to represent
standing wave ratio. The symbol of choice to represent reflection coefficient
during that era was upper case gamma. However, in 1953 the American Standards
Association (now the NIST) announced in its publication ASA Y10.9-1953, that rho
is to replace gamma for reflection coefficient, with SWR to represent standing
wave ratio (for either voltage or current), and VSWR specifically for voltage
standing wave ratio. Most of academia responded to the change, but some
individuals did not. Consequently, gamma is occasionally seen representing
reflection coefficent, but only rarely."

Most interesting. I have on my library shelf 14 texts which deal
primarily or in a major way with electromagnetic waves and/or
transmission lines, and two with microwave circuit design. Of those,

8 (including both microwave design texts) use Gamma
4 use rho
2 use K
2 use k

If NIST's pronouncement had any effect at all, it was the opposite of
what was intended -- the four texts using rho were copyrighted in 1951,
53, 63, and 65; the 8 using Gamma were copyrighted in 1960 - 2000, 6 of
them after 1965. So it appears from my sampling that Gamma is becoming
more, not less, prevalant.

Hope this clarifies any misunderstanding concerning the use of gamma for
reflection coefficient.

I'm afraid it doesn't, unless my collection is very atypical. I don't
think it is, because it includes many of the classics.

Incidentally, Roy, I recently mailed you a CD containing Laport's book, "Radio
Antenna Engineering." I'm wondering if you received it, or did it go astray?

I did indeed, Walt, and please forgive me for not acknowledging your
very kind and thoughtful gift more promptly.

Roy Lewallen, W7EL

Walter Maxwell wrote:

Good response, Roy, but concerning rho and gamma to represent reflection
coefficient, I refer you to Reflections, Sec 3.1,

"Prior to the 1950s rho and sigma, and sometimes 'S' were used to represent
standing wave ratio. The symbol of choice to represent reflection coefficient
during that era was upper case gamma. However, in 1953 the American Standards
Association (now the NIST) announced in its publication ASA Y10.9-1953, that rho
is to replace gamma for reflection coefficient, with SWR to represent standing
wave ratio (for either voltage or current), and VSWR specifically for voltage
standing wave ratio. Most of academia responded to the change, but some
individuals did not. Consequently, gamma is occasionally seen representing
reflection coefficent, but only rarely."


Thanks for the information Walter. I must have a few "rare" texts that
use Gamma (Gamma to mean uppercase gamma) for the voltage reflection
coefficient.

I wonder if the recommendation / standard to which you refer is taken up
in any international standard?

I do note that my ARRL Antenna Handbook (18th edition) and ARRL Handbook
(2000) both use rho, however they reckon that rho=(Za-Zo*)/(Za+Zo)
(where Zo* means the conjugate of Zo). They do this without derivation,
and seem to be in conflict with the derivation in most texts. I suppose
the derivation is buried in some article in QST and in the members only
section of the ARRL website.

Back to notation, accepting that the preferred pronumeral for the
voltage reflection coefficient is rho, is there a pronumeral used for
abs(rho)?

Owen

"Owen" wrote in message
...
Walter Maxwell wrote:

Good response, Roy, but concerning rho and gamma to represent reflection
coefficient, I refer you to Reflections, Sec 3.1,

"Prior to the 1950s rho and sigma, and sometimes 'S' were used to represent
standing wave ratio. The symbol of choice to represent reflection coefficient
during that era was upper case gamma. However, in 1953 the American Standards
Association (now the NIST) announced in its publication ASA Y10.9-1953, that
rho is to replace gamma for reflection coefficient, with SWR to represent
standing wave ratio (for either voltage or current), and VSWR specifically
for voltage standing wave ratio. Most of academia responded to the change,
but some individuals did not. Consequently, gamma is occasionally seen
representing reflection coefficent, but only rarely."


Thanks for the information Walter. I must have a few "rare" texts that use
Gamma (Gamma to mean uppercase gamma) for the voltage reflection coefficient.

I wonder if the recommendation / standard to which you refer is taken up in
any international standard?

I do note that my ARRL Antenna Handbook (18th edition) and ARRL Handbook
(2000) both use rho, however they reckon that rho=(Za-Zo*)/(Za+Zo) (where Zo*
means the conjugate of Zo). They do this without derivation, and seem to be in
conflict with the derivation in most texts. I suppose the derivation is buried
in some article in QST and in the members only section of the ARRL website.

Back to notation, accepting that the preferred pronumeral for the voltage
reflection coefficient is rho, is there a pronumeral used for abs(rho)?

Owen

Hi Owen,

From the general use I'm familiar with, rho alone refers to the abs value, while
the two vertical bars on each side of rho indicates the magnitude alone.
However, following Hewlett-Packard's usage in their AP notes, in Reflections I
use a bar over rho for the absolute, and rho alone for the magnitude. However, I
explain the term in the book to avoid confusion.

Walt, W2DU

"Owen" wrote in message
...
.............................
I do note that my ARRL Antenna Handbook (18th edition) and ARRL Handbook
(2000) both use rho, however they reckon that rho=(Za-Zo*)/(Za+Zo) (where
Zo* means the conjugate of Zo). They do this without derivation, and seem
to be in conflict with the derivation in most texts. I suppose the
derivation is buried in some article in QST and in the members only
section of the ARRL website.


Owen,

There was a big discussion about this last year, and somebody posted that
the ARRL was going to eliminate the conjugate reference.

Tam/WB2TT

Back to notation, accepting that the preferred pronumeral for the voltage
reflection coefficient is rho, is there a pronumeral used for abs(rho)?

Owen

Tam/WB2TT wrote:

There was a big discussion about this last year, and somebody posted that
the ARRL was going to eliminate the conjugate reference.

Ok, I take that to mean the ARRL handbooks are in error in stating
rho=(Za-Zo*)/(Za+Zo) (where Zo* means the conjugate of Zo), and that
they will now use rho=(Za-Zo)/(Za+Zo). Kirchoff lives! I guess we wait
and see if it comes to print.

Thanks Tam.

Owen