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Author Topic: Ground Breaking Mobile Antenna Articles in QEX  (Read 10366 times)
W8JI
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« Reply #15 on: April 07, 2014, 06:51:51 AM »

1. High Q coils did not outperform a LOW Q coil in field strength tests.(no difference!)


We all know that can't be correct.
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2. Coil positioning or coil distance from the base had more impact on efficiency

Everyone already should know that.

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3. Antenna matching  with a antenna tuner  from  the resonant matched  tuning had a major impact on gain.

That statement is unclear.

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4.  A helical whips  performance on both 80 meter and  20 meters could perform better than  a poorly positioned coil or a based loaded antenna.

So?


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5.  Length of the mast below the coil had a huge impact on performance.

Everyone knows that already.
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6. Size of the  vehicle  has the most impact on ground resistance

Someone would expect otherwise?
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7. Multiple resonators  had no impact on performance.

That isn't true as written.

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8. Magnet mounts  reduced field strength  significantly. If you do use a mag mount use as many magnets as possible.(Mag mount is -2db below direct car connection 14mhz) 80 meters was  4db down.

Who would think otherwise?

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9. Cap hats and coil metal ends had little impact on field strength

That isn't universally true, but it can be true in some cases.

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10. No field strength advantage to using  cap hat loading only over a coil loaded vertical

That's been dis proven hundreds of times by many people.
 
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11. Keeping a cap hat wires horizontal is important

Why? Why can't they slope up?




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12. Alternate coil loading like ferrite shows promising results Ferrite loaded antenna  was 1.3db down from a high q coil antenna!

This argument calls argument 1 a lie.
 
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14. Coil current measurements were performed! "Current tapers  from the bottom to the top of the loading coil"
          Coil Q had little impact on this current taper
          current taper varies depending on what section of the 1/4  wave is being replaced
          There is no link between coil Q and current taper
           The coil current conclusions  are numerous and cant be summarized here that will do the article justice.
I am sure the claims will be very well debated!


The only way current can change is if the coil has displacement currents. It has NOTHING to do with with what amount of a 1/4 wave the coil replaces. That idea is just silly.

73 Tom

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W5DXP
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« Reply #16 on: April 07, 2014, 10:45:52 AM »

The only way current can change is if the coil has displacement currents. It has NOTHING to do with with what amount of a 1/4 wave the coil replaces. That idea is just silly.

Tom, you and I agree on 99.9% of topics and I don't know what the QEX article said about current taper because I am in the act of borrowing those magazines from a fellow ham, have not read them yet, and therefore don't know the details. But it is obvious that if a coil is long enough it can support standing wave current loops and nodes. Otherwise, a helical antenna could not work. The question is: How long must a coil be before the current is not uniform as assumed in a lumped circuit analysis?

It is a well known technical fact that when the electrical length of an inductor results in non-uniform currents through the inductor that the limits of the lumped circuit model have been exceeded and we should switch to the distributed network model just as we do for wires and transmission lines that result in non-uniform currents through them. The distributed network model alleviates the need for the "displacement current" band-aid required by the lumped circuit model. (There are numerous technical papers indicating that displacement current was conjured up because of ignorance of the existence of photons.)

It appears to me that the logical mistake you are making is Petitio Principii, i.e. assuming the proof. You assume that the loading coils must obey the lumped circuit model and therefore they must. However, if the distributed network model is used (or Maxwell's equations) some interesting things appear that do exist in the real world but cannot exist in the lumped circuit model.

Taking the mobile (standing wave) antenna example from The ARRL Antenna Book, 20th edition, page 16-7 where FP is the feedpoint:

FP---15 deg base---+---loading coil---+---15 deg stinger---

We know that the antenna must be 1/4WL (90 deg) long electrically in order to put the reflected wave in phase with the forward wave to achieve resonance and that 90 degrees makes the antenna analysis a distributed network problem. How those phasing components are distributed along the 90 deg antenna is the question. We can do a similar thing with transmission lines to create a shortened 1/4WL (90 deg) stub which also requires a distributed network analysis. For simplicity, let's assume lossless lines with a velocity factor of 1.0.

FP---15 deg 50 ohm line---X---??? deg 450 ohm line---Y---15 deg 50 ohm line---

When we solve for the number of degrees occupied by the section of 450 ohm line in the center, we come up with a value of 20.8 degrees. So how can 15 degrees plus 20.8 degrees plus 15 degrees add up to 90 degrees? Answer: We have not taken all of the phase shifts into account. There is a negative phase shift at point X where the Z0 changes from 50 ohms to 450 ohms and there is a positive phase shift at point Y where the Z0 changes from 450 ohms to 50 ohms. So there is a total of five phase shifts which when added up equal 90 degrees. The same thing is true for the mobile antenna above.

A wire in space has a characteristic impedance of a few hundred ohms. If the wire is horizontal, we can use the single-wire transmission line equation to calculate that characteristic impedance. For instance, a #14 wire horizontal at a height of 30 ft has a characteristic impedance of 600 ohms. Again for simplicity, let's make the mobile antenna horizontal such that the base and stinger have a characteristic impedance of 500 ohms. We can use the Hamwaves inductance calculator to estimate the characteristic impedance of a 75m loading coil to be in the ballpark of 4500 ohms.

FP---15 deg base---+---??? deg loading-coil---+---15 deg 50 stinger---

One can see the similarity to the shortened stub above. All we have done is multiply the characteristic impedances by 10 so again the loading coil must occupy the same 20.8 degrees that it did in the shortened 1/4WL stub example. And again there are the two phase shifts at the impedance discontinuities that added into the other three phase shifts must add up to 90 degrees.

The Hamwaves inductance calculator is located at:

http://hamwaves.com/antennas/inductance.html

After entering the loading coil parameters, we obtain the characteristic impedance of the coil and beta, the axial propagation factor in radians/meter, for the coil. We take the axial propagation factor in radians/meter and multiply by 1.455 to get degrees/inch of delay in the coil. Multiplying degrees/inch by the length of the coil in inches gives the electrical length of the loading coil in degrees.

Tom, your picture used to be on that web page indicating that you wanted a more accurate inductance calculator but your picture and request seems to have been removed. Could the reason be that the more accurate inductance calculator uses the distributed network model for the coils which is incompatible with a lumped circuit analysis? That inductance calculator agrees in concept with the 75m Texas Bugcatcher loading coil measurements made at Louisiana Technical University.

http://w5dxp.com/coilmeas.htm
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W8JI
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« Reply #17 on: April 07, 2014, 02:20:45 PM »

Cecil,

It is drag racing season, and I am behind on paying work (as usual) so I can't get into long things.

I have the QEX article, but I have not read it in detail. What I have gathered is that Barry seems to have an agenda that ALL loading coils, regardless of dimensions and external termination, behave like the missing section of a 1/4 wave.

I think everyone *sensible* agrees a perfect coil behaves as nothing but a lumped reactance and resistance in series. As the coil is made physically larger, or as shunting capacitance from windings to "ground" is added,  it behaves more and more like a transmission line. As a matter of fact the old TV set delay lines are just like long skinny inductors with a second bifilar winding to make them act like a transmission line. A helical antenna is another example that can be more like a transmission line than a loading coil.

A small dimension toroid with a lot less capacitance from windings to outside world than the whip or hat terminating it acts like a lumped component. I've measured them. They have almost perfectly the same (I can't measure the current change) on each terminal.

Depending on dimensions of what we measure, we can get almost any result we like. Barry obviously measured some stuff where hats did not impact field strength (contrary to hundreds of other measurements) and then summarized that hats NEVER affect FS.



There are probably 50 ways to convey this or express this, but many of the claims Zenki posted are just wrong. Here is some of the text. I can convert the article to plain text with a text scan converter.

Here is part of it:

Plain Language Conclusions:
1. The current tapers from the bottom to the top of loading coils used to resonate shorter than quarter wave length monopoles. The Q of the coil has little to no effect on the drop.

The amount of taper seems related to that portion of the quarter wave that has been replaced by the coil, but that is an over-simplification. The  reason  the  current  tapers,  other  than  a small  amount  of  conductor resistance and radiation, is that  the net current at any point is the “vector” sum of currents at that point. And, at any point along the monopole, or a series inductor,  there is a phase difference  between  the current coming from the source and the current reflected back from the open end or top of the monopole. The resultant net current is less as you move toward the open end of the monopole, where it is virtually zero, because at that end point,  the  forward  and  reflected  currents  are  equal  in  magnitude  and opposite in phase thus superposing to zero.

This information may answer the questions we had about the lack of impact of coil Q on field strength and the inability to confirm the published  formulas  to  “optimally”  locate  coils  in  the  mast.  It  may  also explain why capacity only loading is no better than top coil loading, all else remaining the same.


>>>>>>

If what Barry proposed above were true, a toroid would show significant current taper. It does not. Both Roy Lewallen and I have confirmed that.

There have to be displacement currents from the coil to space around the coil for there to be current taper. If there is no place to waves to stand, they will not stand.

73 Tom




 
 

     
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W5DXP
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« Reply #18 on: April 07, 2014, 06:53:50 PM »

What I have gathered is that Barry seems to have an agenda that ALL loading coils, regardless of dimensions and external termination, behave like the missing section of a 1/4 wave.

If you have read my previous postings you will see that you and I agree on a lot of points about what Barry is alleged to have said. I have not read the article so cannot comment yet about what Barry actually said.

Would you agree that in my previous shortened stub example that the 450 ohm line section in between the two 15 degree sections of 50 ohm line behaves like part of the missing section of the 1/4 wavelength stub?

Quote
I think everyone *sensible* agrees a perfect coil behaves as nothing but a lumped reactance and resistance in series.

That is a shortcut method that works for perfect (ideal) coils and near perfect coils. A 75m Texas Bugcatcher is nowhere near a perfect (ideal) coil. The Louisiana Tech grad students measured an electrical length of 41 degrees at 4 MHz. If you wouldn't use the lumped circuit model on a piece of transmission line that is 41 degrees long, why would you use the lumped circuit model on a 75m Texas Bugcatcher loading coil that is 41 degrees long?

The 1/4WL self-resonant frequency of a 75m Texas Bugcatcher loading coil is 8.2 MHz which means it is 90 degrees long at 8.2 MHz. Doesn't it make sense for it to be 41 degrees long at 49% that self-resonant frequency? It certainly doesn't make any sense at all to assert that it is 4.3 degrees long at half of 8.2 MHz where it is a 1/4WL helical monopole.

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A small dimension toroid with a lot less capacitance from windings to outside world than the whip or hat terminating it acts like a lumped component.

I don't doubt it. My argument is that a humongous 75m air-core Texas Bugcatcher loading coil is obviously NOT a small dimension coil and certainly NOT a toroid so that argument is irrelevant.

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If what Barry proposed above were true, a toroid would show significant current taper. It does not. Both Roy Lewallen and I have confirmed that.

Again, my argument is not with Barry because I am suffering from ignorance of what he wrote. Why do you and Roy hide behind small toroidal coils when the subject is large air-core 75m Texas Bugcatcher loading coils? When phase is important, anything longer than about 15 degrees requires a distributed network analysis for valid results. The toroidal coils that you guys tested are probably less than 15 degrees long but a 75m Texas Bugcatcher loading coil is certainly longer than 15 degrees on all HF frequencies.

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There have to be displacement currents from the coil to space around the coil for there to be current taper.

That's only true if one presupposes the assumptions of the lumped circuit model. When the only tool one has is the lumped circuit model, all problems involving inductors look like lumped circuits even when they are nowhere near to being lumped circuits.Smiley Current taper in a standing wave antenna wire is caused primarily by the phasing between the forward current and the reflected current. The same is true for large loading coils. Do you know that the current phase shift over 30 degrees of a small wire dipole is in the ballpark of 1 degree? So why can't the phase shift for a 30 degree long coil mounted in that same dipole also be in the ballpark of 1 degree?
« Last Edit: April 07, 2014, 06:58:10 PM by W5DXP » Logged
W5DXP
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« Reply #19 on: April 08, 2014, 06:38:46 AM »

If there is no place (for) waves to stand, they will not stand.

This is one of those statements akin to, "All an antenna tuner does is make the transmitter happy." It is more of a myth than an accurate technical statement. Any wire that connects a source to a load has a physical length that prohibits RF signals from traveling faster than the speed of light which is a little less than one foot per nanosecond in free space. Even if the transmission line is one foot long, it takes time for the source signal to reach a mismatched load and during that time, there are no reflections. When the signal through that one foot of wire reaches the mismatched load after about one nanosecond, then and only then do reflections occur. During steady-state there is indeed a standing wave on any length of transmission line including short single-wire transmission lines over a ground plane. In the real world, there is no way to avoid a delay in a wire because RF signals cannot travel faster than the speed of light.

The problem is that the lumped circuit model presumes that electronic/electromagnetic signals travel faster than the speed of light without any real-world delay. If we use the lumped circuit model on a 75m Texas Bugcatcher loading coil, then we are presuming faster-than-light propagation of signals through the loading coil. That presumption is obviously false for all real world inductors. The reason that we get away with that presumption is that the delay through a small real world coil can be negligible and/or irrelevant so the lumped circuit model gives results close enough to reality to be a useful shortcut that avoids some messy mathematics.

It is when speed of light delays become non-negligible and therefore relevant that we need to discard the simple lumped circuit model in favor of the more complex distributed network model which is closer to Maxwell's equations. If both models yield results that are reasonably close to each other, we are free to use the lumped circuit model. If the two results are not reasonably close, the distributed network model results are always the more valid results. It is not simply two ways of looking at the same thing. It is often the choice between a valid model and an invalid model, e.g. the impedance transformation on a transmission line with reflections.

The number of electrical degrees occupied by a 75m mobile air-core loading coil is one of those cases where the results of a lumped circuit analysis differ drastically from the results of a distributed network analysis. Therefore, the results of the distributed network analysis is valid and the results of the lumped circuit analysis are invalid.

A free space distributed network analysis on a 75m Texas Bugcatcher loading coil indicates that it occupies 33.4 degrees at 4 MHz. University lab measurements on my real-world 75m Texas Bugcatcher loading coil indicate that it occupies 41 degrees at 4 MHz. The ~20% difference in those two values is explained by the real-world objects within the field of the coil during the measurements. The propagation delay through the loading coil will always be greater in the real world than it is in free space. Here's an article describing those Louisiana Tech U. measurements.

http://w5dxp.com/coilmeas.htm

Even with band-aid patches on the lumped circuit model, the electrical degrees occupied by the 75m Texas Bugcatcher coil comes out to be in the ballpark of 4 degrees. Since that value differs by a magnitude from the distributed network calculations and lab measurements, it must necessarily be an invalid value.

Depending upon its design, any real world 75m air-core mobile loading coil will occupy ~20-40 degrees of the mobile antenna. The base and stinger occupy some more of the antenna, 30 degrees in the previous example. The phase shifts caused by the impedance discontinuities at the bottom and top of the loading coil make up the remainder of the 90 degree resonant antenna.

Note: this discussion is about large 75m air-core mobile loading coils, NOT about small toroidal inductors. I don't know how many degrees a toroidal inductor occupies. It would be interesting to have a plot of propagation delay vs inductance for toroidal coils.

This is also not a condemnation of the lumped circuit model which is a very useful model. This is simply a request to recognize the system conditions that force us to switch models. One suggested rule of thumb is that when magnitudes are the only important consideration, anything longer than 60 degrees requires a switch to the distributed network model. However, if phase (and phase angles) are an important consideration, anything longer than 15 degrees requires a switch to the distributed network model. With transmission lines, its easy to follow that rule of thumb. With inductors, its not so easy to recognize when to switch.
« Last Edit: April 08, 2014, 06:44:45 AM by W5DXP » Logged
WS3N
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« Reply #20 on: April 08, 2014, 05:14:42 PM »

The distributed network model alleviates the need for the "displacement current" band-aid required by the lumped circuit model. (There are numerous technical papers indicating that displacement current was conjured up because of ignorance of the existence of photons.)

I'm not sure what you're saying here. I doubt that you mean Maxwell was wrong to add the displacement current to Ampere's Law.
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W5DXP
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« Reply #21 on: April 08, 2014, 07:36:19 PM »

I doubt that you mean Maxwell was wrong to add the displacement current to Ampere's Law.

Maxwell was not "wrong" at the time because photons had not yet been discovered and he was understandably ignorant of the fact that electromagnetic fields and waves consist of quantized photon particles. If he had known about photons and their properties, he would never have had to invent the concept of displacement current.
« Last Edit: April 08, 2014, 07:42:57 PM by W5DXP » Logged
WS3N
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« Reply #22 on: April 09, 2014, 04:39:20 AM »

I doubt that you mean Maxwell was wrong to add the displacement current to Ampere's Law.

Maxwell was not "wrong" at the time because photons had not yet been discovered and he was understandably ignorant of the fact that electromagnetic fields and waves consist of quantized photon particles. If he had known about photons and their properties, he would never have had to invent the concept of displacement current.

What do photons have to do with it? Maxwell's equations would not "work" without the displacement current term. There would be no traveling-wave solutions.
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W5DXP
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« Reply #23 on: April 09, 2014, 05:33:20 AM »

What do photons have to do with it?

http://photontheory.com/Kemp/Kemp.html

"Thus the displacement current will be called the photon current, because it too is quantized in nature."

"The displacement current or photon current is also shown to be equivalent to the energy of the photon; thereby making the displacement current the actual photon."

Quote
Maxwell's equations would not "work" without the displacement current term.

That's true, but if Maxwell had known about photons, he would not have named it "displacement" current. The way displacement current was explained to me at Texas A&M in the 50's was that the electrons were displaced around a sneak path while a field was being established between the two plates of a capacitor and I suspect that concept came from Maxwell. Today we know that RF photons flow through the capacitor's dielectric at the speed of light in the medium and electrons cannot move fast enough at RF to traverse that theorized displacement path.
« Last Edit: April 09, 2014, 06:03:19 AM by W5DXP » Logged
WS3N
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« Reply #24 on: April 09, 2014, 06:47:28 AM »

When we solve for the number of degrees occupied by the section of 450 ohm line in the center, we come up with a value of 20.8 degrees. So how can 15 degrees plus 20.8 degrees plus 15 degrees add up to 90 degrees? Answer: We have not taken all of the phase shifts into account. There is a negative phase shift at point X where the Z0 changes from 50 ohms to 450 ohms and there is a positive phase shift at point Y where the Z0 changes from 450 ohms to 50 ohms. So there is a total of five phase shifts which when added up equal 90 degrees. The same thing is true for the mobile antenna above.

This is an interesting way to look at the solution. A more obvious way, at least for me, is to treat it as a simple boundary-value problem. The full antenna has a simple cosine solution between the base at 0 and the tip at 90. We want to remove the center 2/3 of the antenna but we must maintain the continuity of the solution (the current). When the center section is replaced by a coil, we still want the cosine solution in the sections [0, 15] and [75, 90]. At each end of the coil we must enforce the continuity of the current. We also require the voltage to be continuous at these points, but this is equivalent to a discontinuity in the slope of the current. It is these slope discontinuities (equivalent to your phase shifts) that allow the original solutions from the ends to be joined over a much shorter distance.
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W5DXP
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« Reply #25 on: April 09, 2014, 07:28:10 AM »

It is these slope discontinuities (equivalent to your phase shifts) that allow the original solutions from the ends to be joined over a much shorter distance.

As long as we both are using valid models, the results will be the same. I have worked out what I think is an original method of solving these problems using the Smith chart illustrated by the following diagram.



The phase shift at an impedance discontinuity seems to be the difference between the impedance at that point normalized to the two different values of characteristic impedance existing at that point, i.e. the number of degrees between -j2374 ohms normalized to Z01=4747 ohms and Z02=475 ohms is 52.3 degrees. This works for transmission lines when the Z0s are known and it should work just as well for wires and coils if the Z0s are known.
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WS3N
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« Reply #26 on: April 09, 2014, 08:24:41 AM »

What do photons have to do with it?

http://photontheory.com/Kemp/Kemp.html

"Thus the displacement current will be called the photon current, because it too is quantized in nature."

"The displacement current or photon current is also shown to be equivalent to the energy of the photon; thereby making the displacement current the actual photon."

Quote
Maxwell's equations would not "work" without the displacement current term.

That's true, but if Maxwell had known about photons, he would not have named it "displacement" current. The way displacement current was explained to me at Texas A&M in the 50's was that the electrons were displaced around a sneak path while a field was being established between the two plates of a capacitor and I suspect that concept came from Maxwell. Today we know that RF photons flow through the capacitor's dielectric at the speed of light in the medium and electrons cannot move fast enough at RF to traverse that theorized displacement path.

Yes, the name is an artifact of history. I don't remember the details of his thinking, something about vortices (because it's a curl) in the medium. Nevertheless, he wrote his set of equations long before they were shown to be correct for the quantum case, where they describe the photon probability amplitude, similar to the Schroedinger equation for nonrelativistic matter waves.
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W5DXP
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« Reply #27 on: April 09, 2014, 10:08:28 AM »

I received the borrowed copies of QEX and have read the article. The title of this thread is somewhat misleading. Nowhere did Barry, w9ucw, refer to his articles as "ground breaking". IMO, there are two areas that bear some discussion.

1. High-Q coils vs low-Q coils - It's obvious that when Barry says "low-Q coil", he is not talking about truly low-Q coils (like a 75m hamstick coil) but is instead talking about relatively high-Q coils that have a lower Q than very high-Q coils. It would have been nice if he had measured and published the Qs of the test coils so we wouldn't be thinking he said that a coil with a Q of 1 is just as good as a coil with a Q of 300.

In one of the 1980s 75m mobile CA shootouts, three mobile antennas tied for top honors. One was a Texas Bugcatcher with a large top hat. One was a screwdriver antenna with a large top hat. The third was my junkbox antenna consisting of a CB whip top-loaded with a long, small diameter loading coil and some short horizontal stingers for a top hat. A picture of that Rube Goldberg junkbox antenna is the second one at:

http://w5dxp.com/shootout.htm

The first picture is a screwdriver antenna with a large top hat similar to the one that tied for top honors in the CA shootout.

The point is that the Texas Bugcatcher coil is a large diameter, large wire very high-Q loading coil. The screwdriver has a lower-Q long, skinny loading coil. As one can see in the third picture, my junkbox loading coil was even longer and skinnier and thus even lower-Q. This evidence from the 1980s CA shootout agrees with Barry's observation about higher-Q and lower-Q coils. But he should not have called them low-Q coils. A low-Q coil is the one in the 75m hamstick used for the CA shootout which was 10dB down from the top antennas. It's too bad that Barry didn't include a 75m hamstick in his mix of mobile antennas.

My guess as to the reason that the lower-Q coils were almost as effective as the high-Q coils is that the low-Q coils, because of their overall longer length compared to the shorter length high-Q coils, is that they have a higher radiation resistance because of their longer length which somewhat compensates for their lower Q. That also would help to explain why screwdriver antennas are as effective as bugcatcher antennas.

2. Current Droop - I agree with Barry that most of the current droop apparent in the measurements are due to phasing between the forward current and reflected current. Anyone who believes that a 75m air-core loading coil is a lumped circuit is confused about the nature of standing wave antennas. A good 75m mobile antenna may have a feedpoint impedance of 15 ohms caused by the in-phase superposition of the forward and reflected waves such that the feedpoint impedance equals (Vfor-Vref)/(Ifor+Iref). For that to be 15 ohms, the magnitude of the reflected wave would have to be about 95% of the forward wave, i.e. the SWR on the standing wave antenna is about 40:1 which makes any phase measurements irrelevant and immaterial because there is virtually no phase shift at all when the SWR is 40:1.

With the exception of Barry not measuring the Qs of his test coils, I think it is a good article.
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« Reply #28 on: April 09, 2014, 03:59:16 PM »

A few things we want to keep in mind about loading coils......

The higher the reactance, the higher the length to diameter ratio. In other words, an 80 meter coil will be much longer than its diameter for optimal Q (≈4:1). On 20 meters, the ratio will be close to 1:1.

Large diameter coils—over about 3.5 inches or so—will have an excessive amount of distributed capacitance. This has two effects. First, the Q is lower, and the self-resonant point is also lower. Equate the latter as excessive loss (lower Q), as the operating frequency nears the self resonant point. If the self resonant point is exceeded, the coil looks more like a lossy capacitor, than an inductor.

Anything, even air, within the coil's field, will effect its Q. This especially includes, large metallic end caps, metallic shorting plungers, and incorrectly mounted cap hats (placed directly atop the coil instead at the top of the antenna).
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WS3N
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« Reply #29 on: April 11, 2014, 08:19:36 AM »

It is these slope discontinuities (equivalent to your phase shifts) that allow the original solutions from the ends to be joined over a much shorter distance.

As long as we both are using valid models, the results will be the same. I have worked out what I think is an original method of solving these problems using the Smith chart illustrated by the following diagram.

The phase shift at an impedance discontinuity seems to be the difference between the impedance at that point normalized to the two different values of characteristic impedance existing at that point, i.e. the number of degrees between -j2374 ohms normalized to Z01=4747 ohms and Z02=475 ohms is 52.3 degrees. This works for transmission lines when the Z0s are known and it should work just as well for wires and coils if the Z0s are known.

I can see how that works for two sections (coil and whip). What do you do for three sections, e.g. your stub made of 50/450/50 sections?
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