How can I calculate the necessary inductance for a half wave helical antenna?

[Timhunt]

I'm extremely interested in the self resonant frequency of an inductor. This is to be used as a half wave helical antenna on our ham bands. The usual helical is operated as a ground plane but I have not seen a complete set of design equations for the former. The initial problem with this is that the self resonant frequency needs to be specified (F mhz) together with the coil's/antenna's diameter (D). The number of turns (N) is required.

However, in order to calculate N, the height of the winding (H) is also required.
Solving for N=29.85(H/D)^(1/5)/[F(D(/5])]*H^(1/5). In order to calculate H, I can assume that the helical is wound with enameled wire of diameter d. Therefore H = Nd (omitting the enamel thickness) is substituted in the right hand side of the above equation giving N = 29.85^(5/4)*(d^(1/4))*(F^5/4)*(D^(3/2) the number of turns to be wound on a piece of conduit of given diameter.

I have tried such a helical, link coupled at the center into 50 0hm coax initially. I was looking for a way to calculate H and it was staring me in the face ! Thank you all for a very interesting introduction to half wave helical antennas. Tim

 

[tech99]

I'm extremely interested in the self resonant frequency of an inductor. This is to be used as a half wave helical antenna on our ham bands. The usual helical is operated as a ground plane but I have not seen a complete set of design equations for the former. The initial problem with this is that the self resonant frequency needs to be specified (F mhz) together with the coil's/antenna's diameter (D). The number of turns (N) is required. However,
in order to calculate N, the height of the winding (H) is also required.
Solving for N=29.85(H/D)^(1/5)/[F(D(/5])]*H^(1/5). In order to calculate H, I can assume that the helical is wound with enameled wire of diameter d. Therefore H = Nd (omitting the enamel thickness) is substituted in the right hand side of the above equation giving N = 29.85^(5/4)*(d^(1/4))*(F^5/4)*(D^(3/2) the number of turns to be wound on a piece of conduit of given diameter. I have tried such a helical, link coupled at the center into 50 0hm coax initially. I was looking for a way to calculate H and it was staring me in the face ! Thank you all for a very interesting introduction to half wave helical antennas. Tim

Not sure if you have made it work or not. I would suggest a greater spacing between turns, so that turn-to-turn capacitance is reduced. It is also possible that the outer ends of the dipole are not working like a helix, but just as capacitive loading, and it is common to replace the outer 25% or so with a metal tube. I would be interested to know the length and frequency and the bandwidth for VSWR=3.

 

[Timhunt]

Many thanks 99, This helically wound quarter wave ground plane has been used by 'hams' for a very long time. The only rule of thumb given is that " A half wavelength of wire, with turns closely wound on a piece of conduit or other material. The ground plane in my case was my station wagon roof. This antenna is not self resonant without the ground plane. I have also made up helically wound dipoles operating as a half wave vertical self resonant dipole which of course do not require a ground plane. The problem there was that no design equations until I found Ed Harris's posting under

"coil self-resonant frequency estimation, theory, and history "

The formula is:
(1/5)
29.85 x (H/D)
F = ------------------- Or
N x D

(1/5)
29.85 x (H/D)
N = ------------------- Where the number of turns N is required. Therefore the frequency F in mhz, the diameter D in meters and the
F x D the height H need to be given/specified. I was looking for a method for calculating H and then it became self
evident. Let H = N*d where d is the center to center turn spacing = the diameter of the wire and if turns are closely
spaced, the thickness of the insulating material can be assumed to be negligible.
This equation then becomes :
(1/5)
29.85 x (N*d/D)
N = ------------------- This needs to be solved for N and the resultant equation is rather messy.
F x D I have not, as yet, used the resultant expression to calculate N and hence design a helically
half wave dipole for any given frequency. When I made up one about 10 years ago for 7 mhz, I did not have
any design equations and used it only for receive. However, using it in this house, signals were very strong.
A link, connected to the coaxial cable,wound on the center of this antenna, was connected to my transceiver.
I will test this design equation for this antenna. This antenna is just a helically wound dipole or a self resonant coil antenna !

Further comments would be very welcome. With thanks Tim.

 

[tech99]

I think it will require one or two trials to get the number of turns required, because the wire is not exactly half a wavelength long but can vary quite a lot. Some years ago I made a 2 element Yagi antenna for 21MHz which was about half size. To improve efficiency I used three wires in parallel, giving a 3-start helix. The losses in this antenna were about 2 dB and offset much of the the directive gain.

 

[w5dxp]

In 80m mobile antenna shootouts, helical antennas have proven less efficient than center loaded bugcatchers using high-Q loading coils. Here's a useful site:

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

 

[Baluncore]

By helical antenna I assume you expect a vertically polarised antenna that is shortened by winding the wire in a helix around a cylindrical support, rather than a circularly polarised helical antenna.

There are many resonant modes present in such a helically wound antenna. You will normally only excite or use one of the several modes available.

The first model to consider with any coil of wire, such as an inductor, can be seen as a transmission line with the wire working against the local environment, the rest of the wire, and the magnetic core if present. The self resonance of the inductor will be when the wire length is a multiple of half wavelengths. Don't forget to allow for the dielectric constant of the wire insulation when calculating the velocity factor, 1/√Er and so the shorter wire length needed to be resonant.

To be resonant as an antenna above a ground plane, the element would need to be a quarter wavelength whip. But it depends on how you drive it. The roof of a car is not much more than a small plate capacitor at 40 metres so it may be operating to lower the resonant frequency, rather than as a ground plane presenting an inverted image with 3dB gain. For that reason, with a helically wound vertical, you might do better centre feeding a half wave vertical rather than base feeding a quarter wave vertical against the vehicle. You should consider a skeletal capacity-hat if corona becomes a problem at resonance with the vehicle below. You may have entered the mystical territory of the Tesla coil.

I believe that Ed Harris [1996] was considering another model, when a long wound helix becomes a slow-wave transmission line made from many low pass filter elements. The inductance of one turn is the inductance per unit length. Given the dielectric constant of the insulation, the capacitance of one turn to the next is the capacitance per unit length. From that you can compute the impedance and velocity factor of the slow-wave delay line structure, and so the length of wire needed. It is however unlikely that you will use that mode.
"coil self-resonant frequency estimation, theory, and history, Ed Harris, 1996"
https://www.pupman.com/listarchives/1996/june/msg00227.html

"The self-resonance and self-capacitance of solenoid coils", by David W Knight.
http://www.phys.ufl.edu/~majewski/nqr/paper/coil-SRF/self-res.pdf

"H. F. Resistance and Self-Capacitance of Single-Layer Solenoids", R G Medhurst (GEC Research Labs.).
Wireless Engineer, Feb 1947 p35-43, Mar 1947 p80-92.
http://g3ynh.info/zdocs/refs/Medhurst/Med35-43.pdf

 

[tech99]

... The inductance of one turn is the inductance per unit length. Given the dielectric constant of the insulation, the capacitance of one turn to the next is the capacitance per unit length. From that you can compute the impedance and velocity factor of the slow-wave delay line structure, and so the length of wire needed.

I would have thought that the capacitance per unit length would be the capacitance to free space, not the inter-turn capacitance.
I would expect the inter-turn capacitance to shunt the inductance of the turn, and so increase its inductive reactance.
I have found experimentally that many inductors tend to act like transmission lines.

 

[Baluncore]

I would have thought that the capacitance per unit length would be the capacitance to free space, not the inter-turn capacitance.
I would expect the inter-turn capacitance to shunt the inductance of the turn, and so increase its inductive reactance.

There is the difference between simple transmission lines and slow-wave structures. Think of a long coil spring being used as an audio delay line. There are many possible modes.

I have found experimentally that many inductors tend to act like transmission lines.

That is correct.
Since terminal capacitance is swamped by the surrounding circuit, the dominant fixed self-resonance is often determined by internal wire length rather than the device inductance with it's own terminal capacitance.

 

[Timhunt]

Many thanks, Extremely interesting. Yes, with very close turn spacing, the relative dielectric constant of the wire covering is going to play a critical role in determining the capacitance per unit length of an inductor. This brings me to the first of my questions.

(1) How might this capacitance be measured ? In the formula quoted neither inductance nor capacitance appear in this formula. In a sense this is not surprising because L and C are circuit theory parameters (for which transmission speed is assumed to be infinite) and we are talking about antenna modes of a coil operating at its major self resonant frequency.

(2) I would very much like to see a derivation.of this formula. Can anyone give me a reference to a derivation of this formula ?

(3) I'm also very interested in finding a transmission line model of an inductor ? I'm very comfortable with the many derivations of the lossles transmission line equations (Two coupled partial DE solution for L and C etc).

I used top loaded vertical antennas in the center of my old Mazda station roof for over 40 years. In particular I operated CW mobile
for 35 years driving from here to Melbourne every working . My top hats were all detachable and ranged in diameter from 3 feet (for 3.5 and 1.8 mhz). All coils were air wound with polystyrene spacers to keep the turns in place). The matching network, containing two three gang capacitors was placed at the base of my antennas. A modified C match was used with the matching inductor being absorbed in the loading coil's inductance. The outside of my coax was soldered to the center of the wagon's roof. I made up a bag of twenty 40 foot radials which I used to try out in the local supermarket's car park after midnight ! On 7 mhz (and higher) the radials made absolutely no difference to my signal (over into Europe and the USA). There was no current in any of the radials. For 3.5 and 1.8 mhz it was a totally different story. I had to drastically change the loading coil's inductance and I got very nasty RF burns when touching the wagons roof. On 7 mhz and higher the wagons roof was totally 'cool'. In spite of these limitations, I used to get quite good reports from the USA on 3.5mhz and also worked Tom W8ji whilst driving through the center of Melbourne to my home in Mt Eliza over a distance of about 40 miles.

I have a photo copy of a journal paper (somewhere). The authors compared two short dipoles both C matched at the center, and measured the radiation efficiencies. Both were short dipoles of length L (I cannot remember the test frequencies) One was a length of tubing with the matching network having to not only resonating the antenna but also having to step up the very low R to 50 ohms.
The second antenna was helically wound for resonance and the C match stepped up R to 50 ohms. The first antenna gave an efficiency of 45 % whereas the second yielded a 95 %. Had the authors let the helical winding absorb the step up role, their radiation efficiency would have substantially exceeded 95 % I have proved this many times by placing my 'loading' coil inside my wagon and it was not much better than a dummy load.

I now plan to make up a rotatable helical dipole for 7 mhz now that I have design equations. All that is required is a variable capacitor
across the center and the two helical arms can be trimmed to give a good match to 300ohm twin line. I have used this approach with a very small 3 foot rotatable (I wish google could spell !)14 mhz dipole. Another network will be required at the TX end. Direct connection of the coax to the center is a poor solution because of its very narrow bandwidth, A second matching network in my house can cope with the 300 + jX to 50 ohms. I have used this approach many times.It allows me to efficiency cover a wide band width.
I have studied David Night's publication, but it does not address the application which interests me.

Many thanks to all Tim (VK3IM)

 

[w5dxp]

(2) I would very much like to see a derivation.of this formula. Can anyone give me a reference to a derivation of this formula ? (3) I'm also very interested in finding a transmission line model of an inductor ?

There are a number of references at the bottom of this page: http://hamwaves.com/antennas/inductance.html

"The calculator returns values for the axial propagation factor β and characteristic impedance Zc of the n=0 (T0) sheath helix waveguide mode for any helix dimensions at any frequency." The velocity factor at any particular frequency can be calculated from the axial propagation factor.

 

[PaulWDent]

I just wound a big helix. 47 turns of 2" wide adhesive aluminum tape on a 4" dia PVC pipe.
Total length of tape 30 meters.
A one turn coupling loop around the middle was coupled to an HP network analyzer which showed the lowest (half wave) resonance to be 17MHz. This is factor of 3.4 times higher than 30 meters stretched straight out.
I have wound smaller helices before and found similarly large factors. I have never figured out how to predict it so I wind a few and measure them.
BTW replacing the ends with pipe is the opposite of what you should do. That would be just tending to a center loaded short dipole which has a triangular current distribution. You should wind the turns tighter at the ends and looser in the middle to make the current distribution more humped in the middle, which gives higher radiation resistance. Or add a capacitive hat on each end in the form of a solid or mesh disc

 

[PaulWDent]

I just wound a big helix. 47 turns of 2" wide adhesive aluminum tape on a 4" dia PVC pipe.
Total length of tape 30 meters.
A one turn coupling loop around the middle was coupled to an HP network analyzer which showed the lowest (half wave) resonance to be 17MHz. This is factor of 3.4 times higher than 30 meters stretched straight out.
I have wound smaller helices before and found similarly large factors. I have never figured out how to predict it so I wind a few and measure them.
BTW replacing the ends with pipe is the opposite of what you should do. That would be just tending to a center loaded short dipole which has a triangular current distribution. You should wind the turns tighter at the ends and looser in the middle to make the current distribution more humped in the middle, which gives higher radiation resistance. Or add a capacitive hat on each end in the form of a solid or mesh disc

I forgot to mention the pipe was 10' ling

 

[PaulWDent]

Many thanks, Extremely interesting. Yes, with very close turn spacing, the relative dielectric constant of the wire covering is going to play a critical role in determining the capacitance per unit length of an inductor. This brings me to the first of my questions.

(1) How might this capacitance be measured ? In the formula quoted neither inductance nor capacitance appear in this formula. In a sense this is not surprising because L and C are circuit theory parameters (for which transmission speed is assumed to be infinite) and we are talking about antenna modes of a coil operating at its major self resonant frequency.

(2) I would very much like to see a derivation.of this formula. Can anyone give me a reference to a derivation of this formula ?

(3) I'm also very interested in finding a transmission line model of an inductor ? I'm very comfortable with the many derivations of the lossles transmission line equations (Two coupled partial DE solution for L and C etc).

I used top loaded vertical antennas in the center of my old Mazda station roof for over 40 years. In particular I operated CW mobile
for 35 years driving from here to Melbourne every working . My top hats were all detachable and ranged in diameter from 3 feet (for 3.5 and 1.8 mhz). All coils were air wound with polystyrene spacers to keep the turns in place). The matching network, containing two three gang capacitors was placed at the base of my antennas. A modified C match was used with the matching inductor being absorbed in the loading coil's inductance. The outside of my coax was soldered to the center of the wagon's roof. I made up a bag of twenty 40 foot radials which I used to try out in the local supermarket's car park after midnight ! On 7 mhz (and higher) the radials made absolutely no difference to my signal (over into Europe and the USA). There was no current in any of the radials. For 3.5 and 1.8 mhz it was a totally different story. I had to drastically change the loading coil's inductance and I got very nasty RF burns when touching the wagons roof. On 7 mhz and higher the wagons roof was totally 'cool'. In spite of these limitations, I used to get quite good reports from the USA on 3.5mhz and also worked Tom W8ji whilst driving through the center of Melbourne to my home in Mt Eliza over a distance of about 40 miles.

I have a photo copy of a journal paper (somewhere). The authors compared two short dipoles both C matched at the center, and measured the radiation efficiencies. Both were short dipoles of length L (I cannot remember the test frequencies) One was a length of tubing with the matching network having to not only resonating the antenna but also having to step up the very low R to 50 ohms.
The second antenna was helically wound for resonance and the C match stepped up R to 50 ohms. The first antenna gave an efficiency of 45 % whereas the second yielded a 95 %. Had the authors let the helical winding absorb the step up role, their radiation efficiency would have substantially exceeded 95 % I have proved this many times by placing my 'loading' coil inside my wagon and it was not much better than a dummy load.

I now plan to make up a rotatable helical dipole for 7 mhz now that I have design equations. All that is required is a variable capacitor
across the center and the two helical arms can be trimmed to give a good match to 300ohm twin line. I have used this approach with a very small 3 foot rotatable (I wish google could spell !)14 mhz dipole. Another network will be required at the TX end. Direct connection of the coax to the center is a poor solution because of its very narrow bandwidth, A second matching network in my house can cope with the 300 + jX to 50 ohms. I have used this approach many times.It allows me to efficiency cover a wide band width.
I have studied David Night's publication, but it does not address the application which interests me.

Many thanks to all Tim (VK3IM)

Better not to break the helix in the center. Either use a one or two turn coupling loop into 50 ohm coax - no balun needed or put a multi-turn coupling winding around the center, tune that with a capacitor and tap it to get whatever impedance you want 50/75/300

You will find the latter gives a wider, flatter frequency response than the method you were suggesting as a dipole wants to see a shunt tuned circuit at the feedpoint to broaden it while the equivalent circuit of what you were suggesting would be a series tuned circuit.

 

[tech99]

If you use a coupling winding over the centre of a helix you will reduce common mode currents on the feeder. This is highly desirable, because unless the two halves of the helix are identical, a direct feeder connection will have large common mode currents flowing.
Regarding efficiency, remember that it is the overall physical length which will determine radiation resistance, and in a practical case a half size antenna will have a loss of about 3dB. Also bear in mind that a vehicle or ground mounted vertical antenna will tend in many cases to have about 50 Ohms of Earth loss resistance. This makes it pointless to use an exceptionally efficient loading device. For antenna length of, say, 1 or 2% lambda, we might expect losses of 20 to 30dB.

 

[tech99]

I just wound a big helix. 47 turns of 2" wide adhesive aluminum tape on a 4" dia PVC pipe.
Total length of tape 30 meters.
A one turn coupling loop around the middle was coupled to an HP network analyzer which showed the lowest (half wave) resonance to be 17MHz. This is factor of 3.4 times higher than 30 meters stretched straight out.
I have wound smaller helices before and found similarly large factors. I have never figured out how to predict it so I wind a few and measure them.
BTW replacing the ends with pipe is the opposite of what you should do. That would be just tending to a center loaded short dipole which has a triangular current distribution. You should wind the turns tighter at the ends and looser in the middle to make the current distribution more humped in the middle, which gives higher radiation resistance. Or add a capacitive hat on each end in the form of a solid or mesh disc

 

[tech99]

I have found that the end part of the winding acts as a continuous tube, because the inter turn capacitance causes the current to "jump" the turns. This prevents the end part of the winding giving the desired end loading. We want to increase current towards the ends and do not want to increase the hump in the centre.

 

[Baluncore]

When building a compact vertical HF antenna, for example on the roof of a car, there is no real advantage in using a helical vertical. It does not determine the polarisation of the signal.

You will need a capacity hat to react with a loading coil. Once you know the available height, and the area of the capacity hat and car roof you can calculate the capacitance. From that you calculate the inductance of the loading coil for resonance on the band of choice.

One advantage of a separate loading coil is that you can wind it on a suitable ferrite core and so have less copper resistance in what looks at first like series, but is really a parallel LC resonator. You can also couple your feed into the loading coil. It makes no difference where in the vertical you place the loading coil, but the obvious place is at the bottom.