i have a pickup that i never really ran much cb stuff in, i always ran my set up in my suburban with the big coax and some pretty big amp's. my mfj meter messed up with me the other day, & i guess what im trying to ask is what is a good length for the jumpers to be i cant get them any closer than 6ft. , the coax is belden rg8x - with a quarter wave ant. i forgot what the small stuffs elec. length was or velocity factor is , & i want to put a set up in my truck. - any help on this will be appreciated thanks c.b.j. :?:
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length of coax
- Thread starter country boy jr
- Start date
the corect length is the amount it takes to go from radio to your antenna. 8)
www.signalengineering.com/ultimate/coax_basics.html
www.signalengineering.com/ultimate/coax_basics.html
you are right i've always cut it what ever it takes to ant. but i was wondering about the jumpers from the boxes to radio etc. , thanks !!!
DXman
Yes, that's 3100 degrees F. Nine yrs of hard work.
country boy jr
You could start with a 3' piece and see how that works, then try a 6' piece.
Everyone has their own opinion as to what length to use and that includes myself.
For my mobile I use a 14' 2" piece from the radio to amp (this is the length needed for a 1/2 wave), radio is only 3ft from amp but I still use the 14'2" length.
My coax from amp to antenna is also 14' 2".
I hear tell, that there is no "hard fast" rules on this subject.
So there ya go, just use what ever length you wish
You could start with a 3' piece and see how that works, then try a 6' piece.
Everyone has their own opinion as to what length to use and that includes myself.
For my mobile I use a 14' 2" piece from the radio to amp (this is the length needed for a 1/2 wave), radio is only 3ft from amp but I still use the 14'2" length.
My coax from amp to antenna is also 14' 2".
I hear tell, that there is no "hard fast" rules on this subject.
So there ya go, just use what ever length you wish
DXman
Yes, that's 3100 degrees F. Nine yrs of hard work.
It working lots better than when I had the 6' length in line.
Two things changed:
While talking with one of our forum members I switch out the 6' to the 14' 2" and the first thing that was noticed on his end was my audio got lots clearer.
Next thing was:
With the 6' my reading were R=62 X=3
With the 14' 2", reading dropped R=53 X=1
So I am happy with the results, even tho not a great difference in the R & X, I am happy that the audio was better.
SWR of course stayed the same 1:1
There is one more thing that changed too.
When I had the 2950 set at 3 DK and 6 footer going the amp, only 3 bars would show when modulating. This is with the amp ON.
But now with the 14' section in and radio still set at 3 DK it will now put the BARS all the way to the top - so radio is also working better now.
Two things changed:
While talking with one of our forum members I switch out the 6' to the 14' 2" and the first thing that was noticed on his end was my audio got lots clearer.
Next thing was:
With the 6' my reading were R=62 X=3
With the 14' 2", reading dropped R=53 X=1
So I am happy with the results, even tho not a great difference in the R & X, I am happy that the audio was better.
SWR of course stayed the same 1:1
There is one more thing that changed too.
When I had the 2950 set at 3 DK and 6 footer going the amp, only 3 bars would show when modulating. This is with the amp ON.
But now with the 14' section in and radio still set at 3 DK it will now put the BARS all the way to the top - so radio is also working better now.
where the feedpoint swr is anything other than 1:1, "........the input impedance of the line depends on the line length and the operating frequency."
The ARRL Antenna Book, 15th Edition, 26-1.
this applies to tx outputs and amplifier inputs as well as tx/amp outputs and antennas. the tuned 1/2 wave line that was used above translated the values of resistance and reactance present at the input of the amplifier to the transmitter input of the feedline. if you want to optimize the connection between the transmitter and amplifier then use a patchcord length of either less than 1/20th of a wavelength or a tuned 1/2 wave line.
the information on Scott's website is in need of some serious correction and updating, especially when you consider the fact that the quoted reference above was published in 1990 and for years previously. the statement that the shortest length that makes it from the radio to the antenna is the correct length is simply wrong whenever the swr at the antenna input is anything other than 1:1, period.
if you want to have some more fun try frequency sweeping the amplifier input with the analyzer and a tuned 1/2 wave line to see what frequency provides the closest reading to R=0 / X=0. random lengths of feedline inserted between the radio and amp at this frequency will make very little difference in the measurements while substantiating the point outlined in the opening of this post. this is the only condition where the statement on Scott's website applies.
The ARRL Antenna Book, 15th Edition, 26-1.
this applies to tx outputs and amplifier inputs as well as tx/amp outputs and antennas. the tuned 1/2 wave line that was used above translated the values of resistance and reactance present at the input of the amplifier to the transmitter input of the feedline. if you want to optimize the connection between the transmitter and amplifier then use a patchcord length of either less than 1/20th of a wavelength or a tuned 1/2 wave line.
the information on Scott's website is in need of some serious correction and updating, especially when you consider the fact that the quoted reference above was published in 1990 and for years previously. the statement that the shortest length that makes it from the radio to the antenna is the correct length is simply wrong whenever the swr at the antenna input is anything other than 1:1, period.
if you want to have some more fun try frequency sweeping the amplifier input with the analyzer and a tuned 1/2 wave line to see what frequency provides the closest reading to R=0 / X=0. random lengths of feedline inserted between the radio and amp at this frequency will make very little difference in the measurements while substantiating the point outlined in the opening of this post. this is the only condition where the statement on Scott's website applies.
DXman
Yes, that's 3100 degrees F. Nine yrs of hard work.
forget all of that..
the correct answer is simply..
use the shortest posible length of coax..
Plain and Simple..
so many people and so many manafactures claim all sorts of nonsense..
use only the shortest you can...
no half waves
no multiples..
Later
the correct answer is simply..
use the shortest posible length of coax..
Plain and Simple..
so many people and so many manafactures claim all sorts of nonsense..
use only the shortest you can...
no half waves
no multiples..
Later
DXMan,
i have taken the figures that you provided in your post and plugged them in to the appropriate formulas along with other pertinent parameters of RG8X feedline to provide you with a synopsis of your transmitter to amplifier input scenario both before (with the 6' line) and afterwards (with the 14'2" line):
6' line:
F. Frequency, megahertz .. 27.185 V. Line Velocity Factor. 0.780
Z. Line Impedance Zo, ohms 50.0 R. Load resistance, ohms 39.8
L. Line length, metres ... 1.83 X. Load reactance, ohms 2.0
D. Inner conductor dia, mm 6.10
(R. Load resistance and X. Load reactance values are present at the input of the amplifier.)
Line series input resistance 62.5 ohms
Line series input reactance 3.5 ohms
(you'll notice that both the line series input resistance and reactance values are to within .5 of your original measurements via the analyzer at the input of the line.)
Equivalent input inductance 0.021 microhenrys
Equivalent input capacitance picofarads
Line length in wavelengths 0.2126 at the line's own velocity
.. .. .. .. 0.1658 at free-space velocity
RF loss resistance 0.175 ohms, for both conductors
Line attenuation 0.0152 decibels when Zo matched
Actual loss in line 0.0157 decibels when SWR is present
Efficiency, Pout/Pin 99.6 percent .. .. .. ..
Reflection Coefficient 0.1159 at load end of line
Angle of Ref.Coeff -12.68 degrees
Standing Wave Ratio 1.26 at load end of line (amplifier input)
14'2" line:
F. Frequency, megahertz .. 27.185 V. Line Velocity Factor. 0.780
Z. Line Impedance Zo, ohms 50.0 R. Load resistance, ohms 53.0
L. Line length, metres ... 4.30 X. Load reactance, ohms 1.0
D. Inner conductor dia, mm 6.10
(R. Load resistance and X. Load reactance values are present at the input of the amplifier.)
Line series input resistance 53.0 ohms
Line series input reactance 1.0 ohms
Equivalent input inductance 0.006 microhenrys
Equivalent input capacitance picofarads
Line length in wavelengths 0.4996 at the line's own velocity
.. .. .. .. 0.3897 at free-space velocity
RF loss resistance 0.412 ohms, for both conductors
Line attenuation 0.0358 decibels when Zo matched
Actual loss in line 0.0358 decibels when SWR is present
Efficiency, Pout/Pin 99.2 percent .. .. .. ..
Reflection Coefficient 0.0309 at load end of line
Angle of Ref.Coeff 19.03 degrees
.. .. .. .. 0.3897 at free-space velocity
Standing Wave Ratio 1.06 at load end of line (amplifier input)
0.4996 is as close as the program comes to an electrical 1/2 wavelength and the closest figure on the other side was 0.5007.
notice the reduction in the amplifiers input swr from 1.26:1 to 1.06:1. also notice that the line series input resistance at the transmitter is much closer (instead of R=63/X=3) to 50 ohms with negligible (1 ohm) inductive reactance, facilitating a more efficient transfer of energy not only from the transmitter to the line but also from the line to the input of the amplifier. if the transmitter and amplifier were close enough together to be connected with a jumper approximately < or = 1/20th of a wavelength (about 16") the results would have been pretty much the same, except for Efficiency, Pout/Pin which would increase to 99.9% with the shorter 16" line.
16" line:
F. Frequency, megahertz .. 27.185 V. Line Velocity Factor. 0.780
Z. Line Impedance Zo, ohms 50.0 R. Load resistance, ohms 53.0
L. Line length, metres ... 0.41 X. Load reactance, ohms 1.0
D. Inner conductor dia, mm 6.10
Line series input resistance 53.1 ohms
Line series input reactance -0.9 ohms
Equivalent input inductance microhenrys
Equivalent input capacitance 6274.2 picofarads
Line length in wavelengths 0.0476 at the line's own velocity
.. .. .. .. 0.0372 at free-space velocity
RF loss resistance 0.039 ohms, for both conductors
Line attenuation 0.0034 decibels when Zo matched
Actual loss in line 0.0032 decibels when SWR is present
Efficiency, Pout/Pin 99.9 percent .. .. .. ..
Reflection Coefficient 0.0309 at load end of line
Angle of Ref.Coeff 19.03 degrees
Standing Wave Ratio 1.06 at load end of line (amplifier input)
if you need any further explanation or assistance let me know.
i have taken the figures that you provided in your post and plugged them in to the appropriate formulas along with other pertinent parameters of RG8X feedline to provide you with a synopsis of your transmitter to amplifier input scenario both before (with the 6' line) and afterwards (with the 14'2" line):
6' line:
F. Frequency, megahertz .. 27.185 V. Line Velocity Factor. 0.780
Z. Line Impedance Zo, ohms 50.0 R. Load resistance, ohms 39.8
L. Line length, metres ... 1.83 X. Load reactance, ohms 2.0
D. Inner conductor dia, mm 6.10
(R. Load resistance and X. Load reactance values are present at the input of the amplifier.)
Line series input resistance 62.5 ohms
Line series input reactance 3.5 ohms
(you'll notice that both the line series input resistance and reactance values are to within .5 of your original measurements via the analyzer at the input of the line.)
Equivalent input inductance 0.021 microhenrys
Equivalent input capacitance picofarads
Line length in wavelengths 0.2126 at the line's own velocity
.. .. .. .. 0.1658 at free-space velocity
RF loss resistance 0.175 ohms, for both conductors
Line attenuation 0.0152 decibels when Zo matched
Actual loss in line 0.0157 decibels when SWR is present
Efficiency, Pout/Pin 99.6 percent .. .. .. ..
Reflection Coefficient 0.1159 at load end of line
Angle of Ref.Coeff -12.68 degrees
Standing Wave Ratio 1.26 at load end of line (amplifier input)
14'2" line:
F. Frequency, megahertz .. 27.185 V. Line Velocity Factor. 0.780
Z. Line Impedance Zo, ohms 50.0 R. Load resistance, ohms 53.0
L. Line length, metres ... 4.30 X. Load reactance, ohms 1.0
D. Inner conductor dia, mm 6.10
(R. Load resistance and X. Load reactance values are present at the input of the amplifier.)
Line series input resistance 53.0 ohms
Line series input reactance 1.0 ohms
Equivalent input inductance 0.006 microhenrys
Equivalent input capacitance picofarads
Line length in wavelengths 0.4996 at the line's own velocity
.. .. .. .. 0.3897 at free-space velocity
RF loss resistance 0.412 ohms, for both conductors
Line attenuation 0.0358 decibels when Zo matched
Actual loss in line 0.0358 decibels when SWR is present
Efficiency, Pout/Pin 99.2 percent .. .. .. ..
Reflection Coefficient 0.0309 at load end of line
Angle of Ref.Coeff 19.03 degrees
.. .. .. .. 0.3897 at free-space velocity
Standing Wave Ratio 1.06 at load end of line (amplifier input)
0.4996 is as close as the program comes to an electrical 1/2 wavelength and the closest figure on the other side was 0.5007.
notice the reduction in the amplifiers input swr from 1.26:1 to 1.06:1. also notice that the line series input resistance at the transmitter is much closer (instead of R=63/X=3) to 50 ohms with negligible (1 ohm) inductive reactance, facilitating a more efficient transfer of energy not only from the transmitter to the line but also from the line to the input of the amplifier. if the transmitter and amplifier were close enough together to be connected with a jumper approximately < or = 1/20th of a wavelength (about 16") the results would have been pretty much the same, except for Efficiency, Pout/Pin which would increase to 99.9% with the shorter 16" line.
16" line:
F. Frequency, megahertz .. 27.185 V. Line Velocity Factor. 0.780
Z. Line Impedance Zo, ohms 50.0 R. Load resistance, ohms 53.0
L. Line length, metres ... 0.41 X. Load reactance, ohms 1.0
D. Inner conductor dia, mm 6.10
Line series input resistance 53.1 ohms
Line series input reactance -0.9 ohms
Equivalent input inductance microhenrys
Equivalent input capacitance 6274.2 picofarads
Line length in wavelengths 0.0476 at the line's own velocity
.. .. .. .. 0.0372 at free-space velocity
RF loss resistance 0.039 ohms, for both conductors
Line attenuation 0.0034 decibels when Zo matched
Actual loss in line 0.0032 decibels when SWR is present
Efficiency, Pout/Pin 99.9 percent .. .. .. ..
Reflection Coefficient 0.0309 at load end of line
Angle of Ref.Coeff 19.03 degrees
Standing Wave Ratio 1.06 at load end of line (amplifier input)
if you need any further explanation or assistance let me know.
This all assumes that the velocity factor of the coax is correct, and that the input of the amplifier is close to 50 ohms. If the load is correct (amplifier input), then the length of coax is a non-issue, other than losses due to length.
If you cut two pieces of coax from the SAME reel, and both were exactly the same length, the velocity factor may, and will most likely, be different due to imperfections in the manufacturing of the coax. The only way to know for sure is to use a piece of test equipment. The coax could than be cut for an electrical half wave length, but for only ONE frequency. Once you go above or below that frequency, it is no longer the correct length. Although, this is only a minor issue.
At the end of the day, run the shortest piece of coax you can. 'IF' changing the length of your coax changes your SWR, then you have an issue with the load (antenna or amplifier). Fix that and your 18', 14', 13.5', 7', 5.7689', etc, piece of coax works just fine. Again, its NOT the coax, its the load.....fix that first!
This is the first time I've seen anything about 1/20 wave length. Very ineresting. That was also a nice explanation of what is going on every day in a piece of coax, although a little complicated for the hobbiest.
Don't let coax length scare you! Its really not that hard to figure out. Also, don't use coax to fix a problem that resides elsewhere!
Coax length is critical when you want to phase more than one antenna together. In this case, coax is providing a function and not fixing a problem.
If you cut two pieces of coax from the SAME reel, and both were exactly the same length, the velocity factor may, and will most likely, be different due to imperfections in the manufacturing of the coax. The only way to know for sure is to use a piece of test equipment. The coax could than be cut for an electrical half wave length, but for only ONE frequency. Once you go above or below that frequency, it is no longer the correct length. Although, this is only a minor issue.
At the end of the day, run the shortest piece of coax you can. 'IF' changing the length of your coax changes your SWR, then you have an issue with the load (antenna or amplifier). Fix that and your 18', 14', 13.5', 7', 5.7689', etc, piece of coax works just fine. Again, its NOT the coax, its the load.....fix that first!
This is the first time I've seen anything about 1/20 wave length. Very ineresting. That was also a nice explanation of what is going on every day in a piece of coax, although a little complicated for the hobbiest.
Don't let coax length scare you! Its really not that hard to figure out. Also, don't use coax to fix a problem that resides elsewhere!
Coax length is critical when you want to phase more than one antenna together. In this case, coax is providing a function and not fixing a problem.
i disagree with a few points. first of all, the coax did perform a function, that of the transform function. the problem was that without the tuned electrical half wave line we CANNOT KNOW the input resistance and reactance values present at the amplifier.
by inserting a tuned electrical half wave line BETWEEN the transmitter output and the amplifier input we were able to "see" the amplifiers input resistance and reactance at the input end of the line using the analyzer. ideally, sweeping the 1/2 wave line with the analyzer to produce R=50 / X=0 would give us the exact frequency that the tuned 1/2 wave line was cut to and reveal the amplifier input values required to determine whether or not the input to the amplifier was exhibiting the proper values.
the tuned electrical 1/2 wave line also performed a second FUNCTION, that of translating the amplifier input values (since they were close to ideal) directly to the output of the transmitter, facilitating increased power transfer from the transmitter to the line and from the line to the (amp) load, as i mentioned earlier and as was evidenced by DXMan in a previous post.
there was nothing to "fix" because nothing was broken. just like the input of an antenna, the amplifier input values for resistance and reactance will not remain the same as the operating frequency is changed.
simply put, in the absence of any other test equipment we have stated simply that the shortest length that "makes it" from the rig to the amp (aside of the 1/20 wavelength jumper) is NOT ALWAYS the correct length to use, not only from an operating standpoint but also from the perspective that a random length can tell us nothing about the load present further down the line. thus are the properties of the tuned electrical 1/2 wave line.
on the other hand, if the tuned electrical 1/2 wave line was inserted and we saw both resistance and reactance values move even further away from the ideal, (or from what we saw initially with the 6' line) we would then know whether we had a problem in the amplifier input circuit or maybe the frequency in use was not a close enough match for the length of the 1/2 wave line. sweeping the input of the amplifier at this juncture would help us determine whether the line was incorrect in length for the frequency in use or if there was truly a problem with the amplifier input itself.
was the swr anywhere in the line affected? you're damn right it was! if all of this was not so hard to figure out then i wouldn't be posting. the input impedance of the line depends on the line length and the operating frequency" whenever the load (amplifier or antenna input) is anything but 1:1.
The ARRL Antenna Book, 15th Edition, 26-1.
and to all of those parroting the plethora of misinformation contained in websites claiming to be "dispelling myths" while repeating the same old tired mantra all over the world wide web, i have this to say to all of you: your ad hominem arguments are an evasion of the law of rationality because they (your arguments) fail to provide relevant evidence against the proposition they seek to disprove. furthermore, if you knew anything about any of this you would spend at least as much time as i do telling us all why it isn't so as i do telling all of you that it is. for as much as some of you know about it, it might as well be garden hose. as i have indicated time and time again, neither of the two propositions are correct 100% of the time. i'll leave it to all of you to figure out which of the two predominates the majority of the time. but that isn't good enough for some of you. lose the "all or nothing" attitude, the one that says it's either one way or the other but not both. both propositions are valid and based on necessary measurements.... those of you who will read this can do one of three things with the information provided here. a few of you will get all butthurt because you think this is "all about you" (and i know some of you do, not my intent) ignore all of it, or you can try any of this out for yourself and correct me here if i'm wrong. i'm done with my rant....
the tuned electrical 1/2 wave line is an invaluable measuring aid and performs a valid function when used with amplifiers and antennas exhibiting values approaching R=50/X=0. the problem is that if you don't know what those values are to begin with then using the 1/2 wave line may or may not be a good idea. (and unnecessary if everything is as it should be for the "frequency in question") as a random length will produce little difference under these conditions. (again i point out that the proposition from the "other camp" is only justified when everything is "perfect".) if however the equipment suffers from no problems, as can be determined by using the 1/2 wave line, then it can be beneficial to allow it to remain as the line in use provided it accomplishes the intended purpose. you may just want to leave it there so you can check the input values of the amplifier input from time to time.
in the case with DXMAn, since R and X were not on the money, the 6' section of feedline served to mask the real values of R and X at the amplifier input. we "unmasked" them using the tuned 1/2 wave line and the analyzer as a troubleshooting tool by taking advantage of the mirroring properties to disclose the real values existing at the amp and delivering the results to the analyzer at the transmitter input end of the line.
i hope i have succeeded in simplifying some of this somewhat for the benefit of the "hobbyist".
by inserting a tuned electrical half wave line BETWEEN the transmitter output and the amplifier input we were able to "see" the amplifiers input resistance and reactance at the input end of the line using the analyzer. ideally, sweeping the 1/2 wave line with the analyzer to produce R=50 / X=0 would give us the exact frequency that the tuned 1/2 wave line was cut to and reveal the amplifier input values required to determine whether or not the input to the amplifier was exhibiting the proper values.
the tuned electrical 1/2 wave line also performed a second FUNCTION, that of translating the amplifier input values (since they were close to ideal) directly to the output of the transmitter, facilitating increased power transfer from the transmitter to the line and from the line to the (amp) load, as i mentioned earlier and as was evidenced by DXMan in a previous post.
there was nothing to "fix" because nothing was broken. just like the input of an antenna, the amplifier input values for resistance and reactance will not remain the same as the operating frequency is changed.
simply put, in the absence of any other test equipment we have stated simply that the shortest length that "makes it" from the rig to the amp (aside of the 1/20 wavelength jumper) is NOT ALWAYS the correct length to use, not only from an operating standpoint but also from the perspective that a random length can tell us nothing about the load present further down the line. thus are the properties of the tuned electrical 1/2 wave line.
on the other hand, if the tuned electrical 1/2 wave line was inserted and we saw both resistance and reactance values move even further away from the ideal, (or from what we saw initially with the 6' line) we would then know whether we had a problem in the amplifier input circuit or maybe the frequency in use was not a close enough match for the length of the 1/2 wave line. sweeping the input of the amplifier at this juncture would help us determine whether the line was incorrect in length for the frequency in use or if there was truly a problem with the amplifier input itself.
was the swr anywhere in the line affected? you're damn right it was! if all of this was not so hard to figure out then i wouldn't be posting. the input impedance of the line depends on the line length and the operating frequency" whenever the load (amplifier or antenna input) is anything but 1:1.
The ARRL Antenna Book, 15th Edition, 26-1.
and to all of those parroting the plethora of misinformation contained in websites claiming to be "dispelling myths" while repeating the same old tired mantra all over the world wide web, i have this to say to all of you: your ad hominem arguments are an evasion of the law of rationality because they (your arguments) fail to provide relevant evidence against the proposition they seek to disprove. furthermore, if you knew anything about any of this you would spend at least as much time as i do telling us all why it isn't so as i do telling all of you that it is. for as much as some of you know about it, it might as well be garden hose. as i have indicated time and time again, neither of the two propositions are correct 100% of the time. i'll leave it to all of you to figure out which of the two predominates the majority of the time. but that isn't good enough for some of you. lose the "all or nothing" attitude, the one that says it's either one way or the other but not both. both propositions are valid and based on necessary measurements.... those of you who will read this can do one of three things with the information provided here. a few of you will get all butthurt because you think this is "all about you" (and i know some of you do, not my intent) ignore all of it, or you can try any of this out for yourself and correct me here if i'm wrong. i'm done with my rant....
the tuned electrical 1/2 wave line is an invaluable measuring aid and performs a valid function when used with amplifiers and antennas exhibiting values approaching R=50/X=0. the problem is that if you don't know what those values are to begin with then using the 1/2 wave line may or may not be a good idea. (and unnecessary if everything is as it should be for the "frequency in question") as a random length will produce little difference under these conditions. (again i point out that the proposition from the "other camp" is only justified when everything is "perfect".) if however the equipment suffers from no problems, as can be determined by using the 1/2 wave line, then it can be beneficial to allow it to remain as the line in use provided it accomplishes the intended purpose. you may just want to leave it there so you can check the input values of the amplifier input from time to time.
in the case with DXMAn, since R and X were not on the money, the 6' section of feedline served to mask the real values of R and X at the amplifier input. we "unmasked" them using the tuned 1/2 wave line and the analyzer as a troubleshooting tool by taking advantage of the mirroring properties to disclose the real values existing at the amp and delivering the results to the analyzer at the transmitter input end of the line.
i hope i have succeeded in simplifying some of this somewhat for the benefit of the "hobbyist".
My good man, I have sounded the praises of an electrical half wave length a number of times here. I certainly don't disagree with you.freecell said:
It is more symantics than anything else. You said earlier, ".....resistance and reactance values move even further away....." I would argue that nothing actually "moved", but instead we see the actual reading rather than a transformed reading caused by the imperfect load and incorrect length of coax. See, symantics. Just like when I said "function" and you re-interpreted it.
We don't want to transform anything; that equals loss. We want to fix the problem.
In the case of DXman, he didn't have a problem but the load was not perfect. A change in coax length pointed that out. As your well written, long, complicated, "synopsis" pointed out, due to the (slight) imperfections, a different coax length worked better. I'm not disagreeing with that either.
You dropped the name of the ARRL Antenna Book; was there a quote there you were trying to reference? At first I thought it was all that "parroting" stuff, but then realized they would never say that in a publication. I have the 16th and 20th edition of the ARRL's Antenna Book so I couldn't look it up, to be sure.
As to my "hobbiest" comment let me say this: Your posts tend to way heavy on the technical aspect of antennas. You throw complicated formulas and technical terms that most out here don't understand or even have a desire to understand. Not that what you are saying is wrong, but its appears to be more about you showing your intelligence rather than you educating those asking the questions at a level they can understand. If they don't get it, you are wasting everyone's time. I say, make it simple so they can understand. If they want to learn more, point them to a website or suggest a good book ("Reflections" by Walter Maxwell). I see time and again where someone asks a question and then disappears when it has clearly gone into way more detail.
With all that being said, your post to DXman explaining what was happening in his set up with the different coax lengths was educational. He may or may not have understood it, but I enjoyed reading it. I didn't proof your math and am basically taking your word for it, but I see no reason not to believe what you wrote.
I'll finish with this: You said, ".....without the tuned electrical half wave line we CANNOT KNOW the input resistance and reactance values present at the amplifier." What if we used a 1/20th length of coax?