I messed up who said what. It was not you asking if there was a formula around as I thought. It was theDB that asked that and I replied as if it was you that had asked it. My bad. Too much going on in my personal life right now and no sleep and just started nights. No sleep etc.....yeah excuses but they are honest excuses. LOL
Not to worry CK, I understand you're busy. I figured you just had misread something along the way, just like I thought you failed to read my original post in this thread. I really thank you for talking this out...even though we might disagree on some points. Discussion is the best way to flesh out what is on our minds sometimes...and I consider that good. Most guys seem to just get mad.
Good question. Length to diameter ratio can affect it some. Not only is the resonant length shorter for a fat conductor the impedance will be effected a bit too due to the difference in ohmic losses. I suspect this may be part of the reason why we see a certain value for a 10m antenna and yet a different value for a 2m antenna with the same size material used. There is a difference in terms of wavelength and unless even wire size is scaled the outcome would be different. The material used will affect the outcome as well. Copper, aluminum, and stainless steel all have different electrical properties so that will make a bit of a difference as well. How far the feedpoint is from the groundplane will make a difference too as the ground lead has to be factored in. Something I don't think is being taken into account either is that that figure of 35 ohms, and that is a "nominal" 35 ohms not a hard and fast definite value, (sort of like 50 ohms coax being somewhat near 50 ohms but not exactly) not only assumes a perfect groundplane but if memory serves me correctly it also assumes that perfect groundplane is in free space which eliminates any coupling effects to anything other than the groundplane. Antenna installations are so varied that it is impossible to say something will be X value all the time everywhere in all cases. IMHO worrying about whether it is truly 35 ohms or 27 ohms or 41 ohms is simply a waste of time. The fact remains that it must be matched to 50 ohms for most applications and using the nominal value of 35 ohms is as good a starting point as any.
Just as a real example, back in 1986 I moved an AM transmitter site to a new location. The old site being encroached upon by developments and had the original 1948 tower which was a short top loaded affair that was unstable electrically with changes in the weather. I had a 160 foot steel tower installed with a 24 inch face at the new location on a piece of prime dyke land with very good conductivity. The frequency was 1450 KHz making the 160 foot tower a ful 1/4 wave. The ground system consisted of 120 radials spaced equally around the base and each radial was a 1/4 wavelength of 10 gauge bare solid wire. The radials were flat with the exception that about 1/4 of them sloped downwards slightly for about half their length due to terrain. Directly at the base of the tower was a 20 foot by 20 foot ground mat made from 6 gauge stranded wire with the wires running in both directions and spaced one foot apart and silver soldered with Silfoss wherever they crossed. This was done to maintain a more stable ground in the immediate area of the tower. When all was said and done and the tower self impedance measured we ended up with 38 ohms R and just a few ohms of reactance.This was before the electrical cables and clearance lights were added as well as the Austin toroidal lighting transformer at the base. I merely present this as a real world example of an installation and what the results were. Nothing more.
BTW when I said "And yet there are countless examples all over the internet on countless sites that agree with what I said. Gee ....go figure." it was not meant to be against anything you said but merely an observation of facts. I believe we are on the same wavelength to use a very bad pun.
Below I have tried to duplicate a model at the specifications you noted above. I did not try and duplicate the 20' x 20' screen, but here is what I did to see if my model would prove the results you saw at the time. This is not proof for you, but it might help some take a different track on what I try to do with modeling that nobody else is doing here at the moment. I would like to encourage others to an interest in modeling, but we've heard so many tales that it is hard to sell any idea about modeling. This is not the first time that one of my models came very close to duplicating exactly what another claimed, including LB Cebik who posted in one of his litany of works, as an adviser to Roy Lewallen and his Eznec product, a piece on the 5/8 Wave Mystique.
For those that don't know Eznec, we are limited as to the description we can use. I have tried to describe the models with some understanding, but that can fail for other's that are looking in. This is all to say there is a method to my madness.
Models:
#1. is my attempt to do as you describe in your post. Eznec will not allow us to model on the ground, so this one is at 1' foot. I include for each model the antenna image, the total far field gain and angle report, and the Source Data that shows the feed point impedance values for resistance, reactance, and SWR.
This is just basic stuff and you don't have to know everything there ever was to know about method of moments math. You don't even have to know all the RF theory that you CK, have formerly studied for...as a career choice in real RF work. I think this first model comes pretty close to what you described for the field results you had on that project. There is no mystery here, and this is all the model shows me at least. It probably shows 'Doc a lot more, but that is another story not to be told...for my own self respect.
I will say this, so as to speculate without knowing for sure however, That here we see a 1/4 wave at 1.45 mhz showing a natural match that may not be good enough to operate in you line of work without better matching, but this does provide us a distinction, if we care to understand one that I can draw. It suggest why it is not always effective to demonstrate theory for use at very low frequencies...if our interest is higher up in the CB band. To me this is exactly why and maybe how so much CB BS gets started in the CB business.
My reasoning here is, I can model a CB antenna using 5 x 102" elements using similar math and the model will not produce a natural match even close to what we see with the model I made with your limited dimensions. I can give you the math, but then everyone's eyes might go crossed, and they might just give up and leave without a comment or contribution to the discussion.
Instead I will just give you the values that this CB model's source data provides at 1' foot in height, with 1 x 102" radiator, 3 x 102" horizontal radials, and without a mast or a feed line: R=34....X=-30.1....SWR=2.30.
This is more or less the extent of my claim, where I don't see the generally accept idea that a 1/4 radiator will always show us 35 ohms of resistance at resonance, while the idea comes from theory that was likely a source from the area of antenna use...just like your example...broadcast radio.
Beyond this I cannot explain much or why. Maybe your mention of the wavelength to element diameter ratio has something to do with why. I was talking about CBer's and what they have said for years that I consider mostly CB BS concerning there being near 35 ohms showing up on their mobile setup. Maybe when they get their setup close to right...maybe then they will see about 35 ohms...being that close to the ground. That said, I was not referring to mobiles.
#2. is your model #1 that I set over the idea that Eznec uses for a Perfect ground plane, and I removed all losses to make it perfect. We see here the same as the CB model I did at DB's request of a model over an infinite ground plane. There is no change in matching values worthy of note, but we won't ever be seeing this model in our neighborhoods either. But, I would like to know where this idea of testing over Perfect Earth relates to this theory or idea...about the match at resonance for a 1/4 wave radiator being 35 ohms on a consistent bases if not all the time? CK, do you have such a link that is from a reliable source?
#3. is model #1 raised 1/4 wavelength and here we see what I was seeing working with a CB 1/4 wave base antenna to try and understand why guys often fail to get their whips to work right and others say they get their 102"-108"whips to work perfect with a 1.01:1 SWR, with no problem. Here we see the problem I was claiming...a really low resistance, but this model is not resonant, so therefore my idea is not as revealing as I once thought.
I speculate that maybe if we tuned this one somehow that low impedance might shoot right up to 35 ohms, and maybe we could still retain the horizontal radials...that might even work better than slanting the radials down to improve the match. Again IMO, this is where the slanted down radials otherwise radiate some and tend to raise the higher secondary lobe in the pattern. This, IMO, takes away some from the primary lobe no doubt due to the lack of cancellation in the radial field of the slanted down radials...forget the matching benefits...we seldom get something for nothing in antennas.
I posted examples of two mobiles earlier, and it shows why and how this 1/4 wavelength can and will show a near perfect match when mounted correctly on some mobile body designs.
Typically mobiles have a terrible or less than desirable ground plane. Besides this possibility, very few folks can visualize the ground plane effects from a mobile. We most often hear complaints that the GP is not enough. Seldom do we ever hear that the mobile produces way too much of a ground plane relative to what the radiator needs to work effectively and efficiently. IMO, some time the GP can actually produce all the effective radiation, and sometimes that risks being the opposite polarity from the antenna design itself. I can imagine some mobiles potentially doing this.
Those two basic mobile setups I posted earlier are a work of art IMO. How else can a mobile get that match and performance results?
Well, I'll never tell how and what I did to Henry's model. Because when he sent it to me, it was a mess and I almost was never able to get it to work. I think I was just lucky on that model.
The main reason is it is probably not as big of secret as most believe, but more importantly I'm not sure we can duplicate the models results with a real world setup. Some good contesters my agree however that some things work along these lines however.
This is the way I see things, but I'll change my mind in a minute if I ever hear and see something more convincing and if that happened I would feel blessed, because I learned something that I did not know.
I hope I made this clear enough for easy understanding. I choose not wait on DB like I suggested, he has to get up and work for a living, and I have things I just feel like doing as well.
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View attachment Captain's 1.45 mhz project.pdf