"Dr. Slick" wrote:

Ohms are still always Ohms, regardless of what you are measuring.
And it's very interesting that the E and H fields have units of
Volts/meter and Ampere(turn)/meter, which when you divide one by the
other, you get basically Volts/ampere, just like you would in a
transmission line.

How do you know when the reduced units of one computation mean the
same thing as another?

An example:
The reduced units of modulus of elasticity (in/in/psi - psi) is
the same as the units for stress (psi) and yet modulus of elasticity
is clearly not stress. And in this case, the unreduced units are
much more descriptive than the reduced units. Reducing discards
information.

On the other hand, Torque (Newton*metres) when multiplied by
Radians (metre/metre) does give Energy (N*m*m/m - N*m), but only
after reduction. And for sure, Torque (N*m) is not the same as
Energy (N*m).

So sometimes it is appropriate to say the reduced results are the
same and some times it is not. Is there a way to know when it is
legal?

What rules have you used to conclude that reducing V/m/A/m to V/A
is appropriate?

....Keith

On Tue, 19 Aug 2003 06:07:19 -0400, wrote:

How do you know when the reduced units of one computation mean the
same thing as another?

They are ALWAYS fungible. You can certainly munge up operations to
prove otherwise, and it is easy to do with some really long chain of
computations.

I would suggest you investigate any of the several really good
Mathematics programs, one being Mathcad which offers a huge repository
of such Units tools that it uses to the enormous and enthusiastic
response by engineers and scientists. There greatest asset is in
allowing, you, the user, to enter your measurement in whatever Units
your profession is comfortable with, and marry them into a novel
situation at the interface to another discipline. How much horse
power generator is needed to supply electrical power that is required
to move a speaker cone how many inches to compress air to what sound
pressure level for it to be just barely discernable to the average
listener? That standard could be described as the force necessary to
move the eardrum the same distance as the diameter of an Hydrogen
Atom.


An example:
The reduced units of modulus of elasticity (in/in/psi - psi) is
the same as the units for stress (psi) and yet modulus of elasticity
is clearly not stress. And in this case, the unreduced units are
much more descriptive than the reduced units. Reducing discards
information.

The Elasticity Modulus is described in kg/mm² which is not quite MKS,
but performing the necessary operation to make it so does not remove
any information whatever. If we stick with electronics and discuss
the stress mechanics of piezos (crystals), then stress can be
described in terms of
(volts/meter) / (newtons/meter²)

On the other hand, Torque (Newton*metres) when multiplied by
Radians (metre/metre) does give Energy (N*m*m/m - N*m), but only
after reduction.

This is negative evidence? It is more a clouded argument.

And for sure, Torque (N*m) is not the same as
Energy (N*m).

You are confusing Work and Rotational Statics as being different.
Can you distinguish between Kinetic Energy and Potential Energy
described in mechanical units? If so, both are used to describe the
complete Work equation:
(KE2 - KE1) + (PE2 - PE1) = 0

So sometimes it is appropriate to say the reduced results are the
same and some times it is not. Is there a way to know when it is
legal?

Strictly speaking from the point of legality, it is demanded of
Professional Engineers by the National Institutes of Science and
Technology (what was called the National Bureau of Standards or NBS
years ago).

This means that ANY P.E. that describes a physical relation that does
not conform to these scientific concepts, and damage results to that
Professional Engineer's customer, then that P.E. is liable in a court
of law. This form of legality is the whole point of being P.E.s and
the government making the demand that P.E.s be part of describing
engineering codes and performing design review.

Anyone here who has put up a tower has had to jump through this hoop.
One of their principle concerns is found in the CM or Center of
Moment. The employment of Units transformation and reduction is part
and parcel to their activity (How tall? Length. How heavy? Mass and
the constant of Gravity).

The ONLY reason the city, or county insists on this report is to have
someone take the insurance hit if there is an error in meeting code.
Was your tower too tall for the weight of that long lever arm (your
antenna boom and elements) that also created a torque? Did it snap
with wind load? Did the guys snap through poor tensioning? Every one
of these uses Units that eventually boil down to one of the
tower/beam/guy specifications expressed in identical Units. There is
no other way for anyone to put their name to a report qualifying your
tower otherwise - why would they want to jack up their malpractice
payments? And it would be their insurance, not the city's if your
tower fell on a citizen and their heirs sued because of the city's
permission to you to erect it. This is called negligence and is why
your homeowner's insurance would walk away from you for a tower
collapse where the tower was not inspected to code.

If the P.E. met the standards of transforming between various systems
and observed the Physical Constants defined by NIST, then the P.E. is
NOT liable. If the P.E. is not liable, neither is the city/county.
If the P.E. and government are not liable, you have a problem.


What rules have you used to conclude that reducing V/m/A/m to V/A
is appropriate?

...Keith

Hi Keith,

You probably have no concern for the monolog about P.E.s or you put it
behind you long ago. Or so you and others might presume. That's fine
and this divergence off into mechanics may help some see the relations
but to answer your last question and keep it within the context of the
subject line, we should look at another reference that is less remote
than NIST and closer to antennas:
"Fields and Waves in Communication Electronics," Ramo, et al.

From page 3 (yes, pretty up front):

"Various systems of units have been used, but hat to be used in
this text is the International System (SI for the equivalent in
French) introduced by Giorgi in 1901. This is the
meter-kilogram-second (mks) system, but the great advantage
is that electric quantities are in the units actually measured:
coulombs, volts, amperes, etc."

This reference proceeds to describe those Physical Constants and their
relations that define Permittivity that I have already fully revealed
in a recent posting. If we were to proceed to page 71:

"The quantity known as the magnetic field vector or magnetic
field intensity is denoted H [sound familiar folks?- rwc] and
is related to the vector B define by the force law (2) through
a constant of the medium known as the permeability, µ:
B = µH
...
"In SI units, force is in newtons (N), Current is in amperes (A)
and magnetic flux density B is in tesla (T), which is weber per
meter squared or volt second per meter squared and is 10^4
times the common cgs unit, gauss. Magnetic field H
is in amperes per meter and µ is in henrys per meter. ...
The value for µ for free space is
µ0 = 4 · pi · 10^-7 · H/m"

So, there you have it. Absolutely identical to my other posts. The
RF engineering community's usage of free space Z is in full compliance
with the standards established and maintained by NIST. Both these
sources and standards are employed by commercial engineers and
Professional Engineers alike. It makes no sense to do otherwise
unless you want to start your own system of measurement that allows a
CFA to be 110% efficient. We get many efficiency claims that can ONLY
be judged through these associations described.

The chain of relationships proves that the "ohms" described by the Z
of free space are identical to the "ohms" used for ANY electrical
measurement, among which are the resistance determination of an
antenna (for any feed), or the resistance presented by a carbon
composition resistor.

73's
Richard Clark, KB7QHC

I find this most interesting. As a P.E. licensed by the state of Oregon
(since 1981), I'm aware that I'm subject to state laws governing the
code of conduct of Professional Engineers, and all other applicable
state laws. I didn't realize that I had legal obligations to NIST, or
that any other federal agency has requirements for P.E.s of all states.
Would you please provide some reference where I can further research
this obligation and the rules it has imposed that I'm legally required
to comply with?

Roy Lewallen, W7EL, P.E.

Richard Clark wrote:
. . .
Strictly speaking from the point of legality, it is demanded of
Professional Engineers by the National Institutes of Science and
Technology (what was called the National Bureau of Standards or NBS
years ago).

This means that ANY P.E. that describes a physical relation that does
not conform to these scientific concepts, and damage results to that
Professional Engineer's customer, then that P.E. is liable in a court
of law. This form of legality is the whole point of being P.E.s and
the government making the demand that P.E.s be part of describing
engineering codes and performing design review.

. . .

On Tue, 19 Aug 2003 14:47:58 -0700, Roy Lewallen
wrote:

I find this most interesting. As a P.E. licensed by the state of Oregon
(since 1981), I'm aware that I'm subject to state laws governing the
code of conduct of Professional Engineers, and all other applicable
state laws. I didn't realize that I had legal obligations to NIST, or
that any other federal agency has requirements for P.E.s of all states.
Would you please provide some reference where I can further research
this obligation and the rules it has imposed that I'm legally required
to comply with?

Roy Lewallen, W7EL, P.E.


Hi Roy,

I am wholly unaware of the full scope of your business and contracts
and I have no interest, nor do I think you would volunteer that
information. I cannot recall a single instance of your relating any
experience of yours that revolved around the matters I have discussed,
nor any matters that were professional beyond your product. I cannot
imagine that your product enters into matters of traceability or
authority when I have seen your disclosures that explicitly remove
yourself from liability:
Legal Disclaimer

The licensee ("Licensee" or "User") acknowledges that the reliability
of any and all results produced by this software are not precise and
are subject to significant levels of variability.
....
LICENSOR HEREBY DISCLAIMS ANY AND ALL WARRANTIES OF
MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.

73's
Richard Clark, KB7QHC

On Tue, 19 Aug 2003 14:47:58 -0700, Roy Lewallen
wrote:

I find this most interesting. As a P.E. licensed by the state of Oregon
(since 1981), I'm aware that I'm subject to state laws governing the
code of conduct of Professional Engineers, and all other applicable
state laws. I didn't realize that I had legal obligations to NIST, or
that any other federal agency has requirements for P.E.s of all states.
Would you please provide some reference where I can further research
this obligation and the rules it has imposed that I'm legally required
to comply with?

Roy Lewallen, W7EL, P.E.

Hi Roy,

"RCW 19.94.150
Standards recognized.
The system of weights and measures in customary use in the United
States and the metric system of weights and measures are jointly
recognized, and either one or both of these systems shall be used
for all commercial purposes in this state. The definitions of
basic units of weight and measure and weights and measures
equivalents, as published by the national institute of standards
and technology or any successor organization, are recognized and
shall govern weighing or measuring instruments or devices used in
commercial activities and other transactions involving weights and
measures within this state."

This is from the state of Washington, I will leave it to you to
research your own particular point of liability in Oregon.

I would add what the IEEE offers into the matter of observing
standards in the development of software and confirming your
disclaimers with:

"The Legal Standard of Professionalism"

"One curious fact from the legal perspective decries a serious
lack: there is no such thing as software malpractice. Why?
A peek into the legal mind provides a disturbing explanation.
There is insufficient evidence to show that programmers
know how to learn from each other, much less from the rest of
the world."

I, for one, could envision you having interest in both, but as I
stated before, I could not see you bothered with the first - seeing
that you have not volunteered any additional details of your trade
aside from software, that stands to good reason.

73's
Richard Clark, KB7QHC

Richard Clark wrote:
On Tue, 19 Aug 2003 14:47:58 -0700, Roy Lewallen
wrote:


I find this most interesting. As a P.E. licensed by the state of Oregon
(since 1981), I'm aware that I'm subject to state laws governing the
code of conduct of Professional Engineers, and all other applicable
state laws. I didn't realize that I had legal obligations to NIST, or
that any other federal agency has requirements for P.E.s of all states.
Would you please provide some reference where I can further research
this obligation and the rules it has imposed that I'm legally required
to comply with?

Roy Lewallen, W7EL, P.E.


Hi Roy,

"RCW 19.94.150
Standards recognized.
The system of weights and measures in customary use in the United
States and the metric system of weights and measures are jointly
recognized, and either one or both of these systems shall be used
for all commercial purposes in this state. The definitions of
basic units of weight and measure and weights and measures
equivalents, as published by the national institute of standards
and technology or any successor organization, are recognized and
shall govern weighing or measuring instruments or devices used in
commercial activities and other transactions involving weights and
measures within this state."

This is from the state of Washington, I will leave it to you to
research your own particular point of liability in Oregon.

Wow, thanks for the heads-up. I'll be more careful to specify circuit
board trace line widths in furlongs, and volumes of radar detection
regions in bushels, those being duly recognized customary units of
measure here in Oregon. I'll no longer use lakj;ofs and mapeurqak!pys,
which I had previously been using.

I would add what the IEEE offers into the matter of observing
standards in the development of software and confirming your
disclaimers with:

"The Legal Standard of Professionalism"

"One curious fact from the legal perspective decries a serious
lack: there is no such thing as software malpractice. Why?
A peek into the legal mind provides a disturbing explanation.
There is insufficient evidence to show that programmers
know how to learn from each other, much less from the rest of
the world."

I, for one, could envision you having interest in both, but as I
stated before, I could not see you bothered with the first - seeing
that you have not volunteered any additional details of your trade
aside from software, that stands to good reason.

As I'm afraid so often happens with your postings, I haven't a clue what
you're trying to say. It sounds vaguely like a complaint, but I can't
for the life of me fathom what about, except that it seems to be some
sort of objection to the legal disclaimers which accompany my software.
Could you please try to rephrase it in a way that can be understood by
an engineer with a sadly deficient liberal arts education?

If you feel that the legal disclaimers which accompany my software are
unduly restrictive or otherwise too onerous for you, or you're not fully
satisfied with EZNEC in any way, all you need do is so state in
peasant-level plain language so I can understand it, and I'll promptly
refund the full purchase price. Just as it says clearly in the EZNEC
manual (Help/Contents/Introduction/Guarantee).

Roy Lewallen, W7EL

On Thu, 21 Aug 2003 20:03:41 -0700, Roy Lewallen
wrote:

Richard Clark wrote:
On Tue, 19 Aug 2003 14:47:58 -0700, Roy Lewallen
wrote:


I find this most interesting. As a P.E. licensed by the state of Oregon
(since 1981), I'm aware that I'm subject to state laws governing the
code of conduct of Professional Engineers, and all other applicable
state laws. I didn't realize that I had legal obligations to NIST, or
that any other federal agency has requirements for P.E.s of all states.
Would you please provide some reference where I can further research
this obligation and the rules it has imposed that I'm legally required
to comply with?

Roy Lewallen, W7EL, P.E.


Hi Roy,

"RCW 19.94.150
Standards recognized.
The system of weights and measures in customary use in the United
States and the metric system of weights and measures are jointly
recognized, and either one or both of these systems shall be used
for all commercial purposes in this state. The definitions of
basic units of weight and measure and weights and measures
equivalents, as published by the national institute of standards
and technology or any successor organization, are recognized and
shall govern weighing or measuring instruments or devices used in
commercial activities and other transactions involving weights and
measures within this state."

This is from the state of Washington, I will leave it to you to
research your own particular point of liability in Oregon.

Wow, thanks for the heads-up. I'll be more careful to specify circuit
board trace line widths in furlongs, and volumes of radar detection
regions in bushels, those being duly recognized customary units of
measure here in Oregon. I'll no longer use lakj;ofs and mapeurqak!pys,
which I had previously been using.

Uh-huh.

....
Could you please try to rephrase it in a way that can be understood by
an engineer with a sadly deficient liberal arts education?

Hi Roy,

Probably not.

73's
Richard Clark, KB7QHC

wrote in message ...
"Dr. Slick" wrote:

Ohms are still always Ohms, regardless of what you are measuring.
And it's very interesting that the E and H fields have units of
Volts/meter and Ampere(turn)/meter, which when you divide one by the
other, you get basically Volts/ampere, just like you would in a
transmission line.

How do you know when the reduced units of one computation mean the
same thing as another?

An example:
The reduced units of modulus of elasticity (in/in/psi - psi) is
the same as the units for stress (psi) and yet modulus of elasticity
is clearly not stress. And in this case, the unreduced units are
much more descriptive than the reduced units. Reducing discards
information.


Not really. Look at this:

http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html

If you notice, the strain is = delta L/ original L, so the strain
is dimensionless. So Young's modulus actually seems to represent the
N/m**2 (PSI) that is required to elongate something to twice it's
original length: delta L = original L, so that the denominator is 1.

interesting that you bring this up.



On the other hand, Torque (Newton*metres) when multiplied by
Radians (metre/metre) does give Energy (N*m*m/m - N*m), but only
after reduction. And for sure, Torque (N*m) is not the same as
Energy (N*m).



Hunh?? how did you get radians = m/m?

Look he

http://www.sinclair.net/~ddavis/170_ps10.html

I admit that this page reminded me that radians are
dimensionless.

So the torque times radians just gives you the work done, which
is in the same units as torque by itself. it's a bit confusing, but
Rotational units are used differently from linear ones (you have the
moment arm), so linear units are force is Newtons or lbs, and work is
in Newton*meters or ft*lbs.

I'm not totally sure, but the reason for this discrepancy seems
to be related to the fact that upon each rotation, you end up at the
same point, so in a certain sense, no work is done. But the crux is
that angles are dimensionless:

http://mathforum.org/library/drmath/view/54181.html

But in either case, rotational or cartesian, the Newton is still
a Newton, and so are the meters.




So sometimes it is appropriate to say the reduced results are the
same and some times it is not. Is there a way to know when it is
legal?

What rules have you used to conclude that reducing V/m/A/m to V/A
is appropriate?

...Keith


Basic algebra and cancellation of units. When have you found it
not to be appropriate? I'll admit that it can be a bit confusing
going from cartesian to rotational, and you have to understand the
context, but the UNITS ARE ALWAYS THE SAME. Isn't this the crux of
science and math? That we have certain standards of measurement, so
when we say it's a meter, it's a meter? God, i hope so.


Slick

"Dr. Slick" wrote:

wrote in message ...
"Dr. Slick" wrote:

Ohms are still always Ohms, regardless of what you are measuring.
And it's very interesting that the E and H fields have units of
Volts/meter and Ampere(turn)/meter, which when you divide one by the
other, you get basically Volts/ampere, just like you would in a
transmission line.

How do you know when the reduced units of one computation mean the
same thing as another?

An example:
The reduced units of modulus of elasticity (in/in/psi - psi) is
the same as the units for stress (psi) and yet modulus of elasticity
is clearly not stress. And in this case, the unreduced units are
much more descriptive than the reduced units. Reducing discards
information.


Not really. Look at this:

http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html

If you notice, the strain is = delta L/ original L, so the strain
is dimensionless.

Yes, and no. It was length per length, not, for example, volt per volt
or
pound per pound or ...

So dimensionless quantities are not all the same, even though they are
all dimensionless.

So Young's modulus actually seems to represent the
N/m**2 (PSI) that is required to elongate something to twice it's
original length: delta L = original L, so that the denominator is 1.

interesting that you bring this up.


On the other hand, Torque (Newton*metres) when multiplied by
Radians (metre/metre) does give Energy (N*m*m/m - N*m), but only
after reduction. And for sure, Torque (N*m) is not the same as
Energy (N*m).


Hunh?? how did you get radians = m/m?

Length of arc divided by radius in MKS units. How quickly we forget when
we get in the habit of leaving out all the units.

After multiplying Torque by Radians, you have computed the length
along the arc through which the force has acted - energy, of course.

Look he

http://www.sinclair.net/~ddavis/170_ps10.html

I admit that this page reminded me that radians are
dimensionless.

So the torque times radians just gives you the work done, which
is in the same units as torque by itself. it's a bit confusing, but
Rotational units are used differently from linear ones (you have the
moment arm), so linear units are force is Newtons or lbs, and work is
in Newton*meters or ft*lbs.

I'm not totally sure, but the reason for this discrepancy seems
to be related to the fact that upon each rotation, you end up at the
same point, so in a certain sense, no work is done.

Actually, you've done 2*pi*radius*force work. Moving one circumference
times the force.

But the crux is
that angles are dimensionless:

http://mathforum.org/library/drmath/view/54181.html

But in either case, rotational or cartesian, the Newton is still
a Newton, and so are the meters.


So sometimes it is appropriate to say the reduced results are the
same and some times it is not. Is there a way to know when it is
legal?

What rules have you used to conclude that reducing V/m/A/m to V/A
is appropriate?

...Keith

Basic algebra and cancellation of units. When have you found it
not to be appropriate?

It is not appropriate to consider Torque and Work to be the same, though
they have the same units.

It is not appropriate to consider modulus of elasticity and pressure
to be the same, though they have the same units after simplification.

But after multiplying Torque times Radians it is necessary to simplify
to discover that Work is the result.

I conclude that simplification is sometimes necessary and appropriate
but other times it is not. I am having difficulty knowing how to know
when it is appropriate.

This brings us back to the Ohms of free space and the Ohms of a
resistor.

While I don't know whether they are the same or not (and opinion seems
divided), it is clear that arguing that they are the same because the
units (after simplification) are the same is quite falacious. On the
other hand if the units were different, it would be clear that they
are not the same.

....Keith

On Tue, 19 Aug 2003 14:35:55 -0400, wrote:

While I don't know whether they are the same or not (and opinion seems
divided), it is clear that arguing that they are the same because the
units (after simplification) are the same is quite falacious. On the
other hand if the units were different, it would be clear that they
are not the same.

...Keith

Hi Keith,

Lets just observe a simple, real situation that any Ham may be faced
with during a power black-out, or during Field Day. Take for instance
a generator. It can give you 1KW of power. You need a gas powered
engine to turn the generator. How much horsepower do you need?

The common exchange is 746W per HP for 100% efficient transformation.
Thus you need at least 1.34 HP to obtain that kilowatt. What is a
horsepower (certainly one of the most ancient of units) compared to
these Watts (a relatively modern unit by comparison)? Is there a
direct correlation between the power of a horse, and the power of a
generator? Yes.

First, a word about multiplication by identities. An identity may
also be known in this forum as a conversion factor. One such simple
example is time conversion from seconds to minutes and back through:
(1 · minute) = (60 · second)
the identity is a simple division by one side or the other to leave 1.
A division by minute is a possibility for one identity:
1 = (60 · second) / (1 · minute)
equally valid would be to divide both original sides by (60 · second):
(1 · minute) / (60 · second) = 1
you can confirm there is no hanky-panky by observing the common
expectation that both sides of the equation describe the same thing,
thus the identity of (1) over (1) equals 1 --- both times. In other
words, the identity describes the same thing by different terms, and
those terms are combined to offer a value of 1 (dimensionless).

The process of employing multiplication by 1 (performed below) through
the use of identities with the time example described above (meaning
you have converted to a form of x = 1 or 1 = x) allows for us to
combine and clear terms in shifting from one basis of measurement to
another.

To return to our query about the generator and the engine,
1 Horsepower is 33,000 ft-lb/minute. In the old days, a horse had to
pull against a known load for a know period of time over a known
distance to arrive at this common reference. The popular definition
will allow you to see these units already in place:
33,000 · foot · pound / minute

We begin our trip towards the S of MKS through Units conversions, by
casting out minutes with the time identity multiplying this value:
33,000 · (foot · pound / minute) · (1 · minute) / (60 · s)

Clearing those terms leaves us with:
33,000 · foot · pound / (60 · s)
or
550 · foot · pound / s
when the minute terms are canceled and the equation has been corrected
to using seconds. [I hope many recognize this alternative conversion
factor. It proves that nothing is lost through these conversions.]

Next we move toward the K of MKS by casting out pounds:
550 · (foot · pound / s) · (1 · kg / 2.205 · pound)
This would be tempting to perform, but it would be absolutely wrong!
As far as the expression of power in the original statement goes, the
identity of pounds and kilograms is incorrect. This is because
kilograms express mass and pounds express weight, which is the product
of mass times the acceleration due to gravity. The pounds do cancel
in the equation above, but the statement is incomplete and should be:
550 · (foot · pound / s)
· (1 · kg / 2.205 · pound)
· (9.807 · m / s²)

Combining and casting out terms leaves us with:
2446 · foot · m · kg / s³

Finally, to complete the progress towards MKS, we move toward the M of
MKS by casting out foot using the length identity:
2446 · foot · m · kg / s³ · (0.3048 · m) / (1 · foot)

Combining and clearing terms leaves us with:
745.5 · m² · kg / s³

THIS is the NIST definition for power, but as such it may be
unfamiliar to many (certainly given the angst and denial that attends
this discussion). For the comfort of many, we draw in another
identity that comes closer to expectations.

That is the identity of Power (also in MKS terms) that reveals itself
as joules per second, or newton-meters per second:
(1 · Watt) = (1 · kg · m / s²) · (m) / (s)
or
(1 · Watt) = (1 · kg · m² / s³)
whose identity becomes
(1 · Watt) / (1 · kg · m² / s³) = 1

We apply this to the power equation above:
745.5 · (m² · kg / s³) · (Watt) / (kg · m² / s³)
which (guess what?) reduces to:
745.5 Watts

QED

Rounding introduced 0.5 Watt error (the values provided by NIST to
their complete precision would eliminate that). It also confirms what
we already knew, but few could prove with a linear exercise like this.
That's not uncommon however, because few deal with the Physics of the
terms they are familiar with, this is the provence of the Metrologist
and research scientists, not amateurs.

It is enough to say Watts and Horse Power exhibit a constant of
proportionality, but it is wholly wrong to say that electrical Watts
are somehow different from an animal's work expended over time.

It is equally in error to maintain that the resistance or Z of free
space is somehow remote and different from the resistance of a carbon
composition resistor or Radiation Resistance. ALL terms employed in
the expression of permittivity and permeability conform to these same
linear operations that prove they are congruent.

73's
Richard Clark, KB7QHC