Coil Geometry, Inductance and Wire Length

If you play with HF antennas you are certain to be winding coils, perhaps more than you'd like. They are used to shorten antennas, make traps, as components of impedance matching networks, filters, coax common mode chokes and as a helix can be the antenna itself. Although deceptively simple devices they can be the source of much angst.

• Accuracy: There are a variety of inductance calculators for coils giving different results. I usually ignore these differences since they are typically small and often are minor in comparison to construction variation that in any case requires adjustment in the field. But there are a couple of factors worth considering:
• Coil diameter: What is the true diameter of a coil?
• Insulation: Is there an effect?
• Wire length: How much wire does it take to wind a coil? Of course the trivial answer is that the wire should be long enough to reach from one end to the other! Often you will want a better answer than that when you must cut the wire before winding the coil.
• Efficiency: Diameter, length, gauge, metal, pitch, form and core all have their effects. Lots of good stuff has been written about this so I will direct you there. Excellent resources are W8JI on coil Q, N6LF on effect of wire insulation and K9YC on ferrite cores. 

Rather than deal with the more technical aspects of coils -- which as noted are ably dealt with by others -- in this article I'll address a few geometric parameters that while seemingly trivial are important for building coils. Reviewing elementary knowledge can be helpful.

Coil diameter

What is the diameter of a coil? A common answer is that it is the outer diameter of the coil form. This is only approximately true. As the ratio of wire diameter to the form diameter approaches zero -- thin wire, large form -- it is effectively true. However it is not true when the coil's application is high power or high Q since the wire diameter can be a large fraction of the form diameter. The diameter to use in coil inductance equations is wire centre to wire centre.

The diagram at right illustrates the problem. The form is shown as a solid tube and the wire is shown with insulation, although there may be none. I frequently use insulated wire in my antenna projects for its convenience and to remove the risk of shorted turns in my hand wound coils.

Let's assume the coil form is 2" (5 cm) since that is one I use in some of my projects. AWG 12 wire has a diameter of 0.064" (1.63 mm), which gives a wire to form ratio of 0.032, or 3.2%. This can be quite significant. If the wire is insulated the ratio increases to ~5% (depending on the insulation rating) because the insulation separates the form and wire.

For bare copper the coil diameter is 2.13" and for insulated wire the diameter is ~2.2". If this appears surprising notice that we use twice the wire radius, which is its diameter, and twice the insulation thickness, which is one side of the insulation on each wire. An easy way to do this is to measure the full wire width with calipers and add that value to the form diameter.

Consider a coil built in this fashion that has an inductance of 3 μH for a coil diameter of 2". When you account for the wire gauge and insulation (AWG 12) the true diameter is ~2.1" and the inductance grows to 3.6 μH. That's a big difference! A small difference cause a large inductance change because the inductance of a coil increases with the square of its diameter, for a fixed coil length and turns.

By calculating coil inductance with the correct diameter you won't be for an ugly surprise when you install it in a matching network, and there adjustments will be fewer and finer. For an example look at my article on the 160 meter vertical matching network I built which required no adjustment at all. Accurately measuring coil inductance can be difficult so it's nice to know we can get the inductance right by understanding the impact of coil geometry.

Wire length

Often you'll need to cut the wire before winding a coil. That's when you especially want to accurately calculate the length of wire required. Cut it too short and you run into grief, and waste; cut it too long and some wire is wasted. Copper isn't cheap! I just ordered more radial wire and the price has increased 10% over the past 6 months.

I covered coil diameter first since you need to get that right if you're to correctly calculate the amount of wire in a coil. A naive length calculation of the wire length for a single coil term in the previous example (2" coil form) is 2π = 6.28". However since the true diameter is closer to 2.1" a better estimate of the wire length is 6.59", which is 5% higher. The wire length is proportional to the diameter.

This is significant since for a 10 turn coil on a 2" form you would end up 3.1" short using the naive calculation (65.9" vs. 62.8"). Of course you should not forget to add to the total the length of the coil tails at both ends. I tend to underestimate the tails so you may want to be careful with that.

Coil pitch (turns per inch)

A coil turn is a spiral rather than a circle. It should be evident that the length of wire in a spiral is more than in a circle due to the linear displacement of the start and end points. The linear displacement due to the coil's pitch increases the length of wire required for a given coil diameter, pitch and turns count. That is: L > TπD. But how large is the difference?

We can deal with spirals using elementary topology. Notice that a cylinder, like a steel pipe, is a rectangle folded into a circle. Unfolding one turn's worth of the cylinder help to visualize the problem and to accurately determine the length of wire in a one turn spiral.

The spiral turn of wire around the cylinder becomes a diagonal line between opposite corners of a rectangle. The rectangle's height is the the turns pitch; if there are 8 turns per inch the pitch is 0.125". The rectangle's width is the circumference of the coil. Pythagoras comes to the rescue by noting that the diagonal is the hypotenuse of a triangle. The wire length is the square root of the sum of the squares of the pitch and diameter.

Therefore the wire length of for a 10 turn coil is L = T × sqrt(P² + (πD)²). Using a pitch of 0.25" (4 tpi) and a coil diameter of 1.1" (1" form plus wire) the length of wire is 34.7". Even for this rather large pitch the spiral's length is only 0.3% (0.1") more than the 34.6" for a naive calculation assuming the a coil turn is a closed circle.

The naive calculation is perfectly adequate except for coils with an exceptionally large pitch and narrow form. The only time you are likely to encounter this is in a helical antenna. Although not really a coil (inductor) the geometry is the same.

Deformation

When you coil a wire you are often pushing the metal beyond its yield strength to get a non-elastic deformation. That is, the wire doesn't snap back into its pre-coiling shape. Deformation can occur throughout the wire diameter, being stretched into a greater length on the outside and compressed to a shorter length on the inside of the coil. However the former is more likely.

This is not a simple problem to solve, so it is lucky the that the effect is small enough in almost every case that it can be ignored. Since we improved the diameter calculation to use the centre of the conductor and the centre is approximately equidistant from the surfaces where deformation is likely to be greatest we should see little difference in the length requirement. Were the wire centre to stretch we would in fact need a tiny amount less wire to wind the coil.

Insulation adds elasticity to the wire. Bend straight lengths of identical insulated and bare wire and you'll see that the insulated wire rebounds more. The copper is still yielding on the inside but when the force is remove the elasticity of the plastic covering reverses some of that. Since it's the diameter of the conductor that concerns us we can ignore the behaviour of the insulation. It would take an extreme bend radius of a heavy gauge wire for the plastic to ripple on the inside thus increasing its effective thickness.

Toroids

Toroidal and other non-cylindrical forms add another factor for consideration. If the wire is of large enough gauge the coil will tend to follow a circular form rather than seat on the toroid's quasi-square surface. In that case with regard to wire length the diagonal cross-section distance is a good approximation to the coil's diameter, plus of course the wire itself as we saw earlier.

Smaller wire than conforms to the toroid surface can treat the coil form as a square (or rectangle), and therefore those dimension should be used as the basis for determining the length of wire required.

Getting the length right is generally more important for a toroidal coil since in almost every case the wire must be cut beforehand and wound onto a holder that can be threaded through the centre of the toroid. This is where you really don't want to underestimate the length of wire required.

Conclusion

There are several coil and wire geometric parameters than affect inductance and wire length. While some of them are quite interesting there is only one that is significant in nearly every real coil: effective diameter. As we saw, using the coil form diameter for the coil diameter can lead to significant error in the calculated inductance and required wire length.

Well, so much for this little diversion. With less bad luck than I've suffered recently I should soon be back to talking about antennas.

Deadwood

The killing field. The large tree on the
ground is the one that did the deed. The
other leaning deadwood is now cleared.

There's been a lack of articles recently for a variety of reasons. We had a string of springtime snow and ice storms accompanied by two days of high winds (up to 90 kph). Once that was out of the way I had to deal with an impacted wisdom tooth. This brought all antenna work to a halt for nearly two weeks.

Once I could get outside and do stuff again in the late-to-arrive mild spring weather there were many non-ham jobs that took priority. One of these tasks was to clear away fallen trees and branches due to those storms. Even so I have been able to do some antenna work, in particular making progress on the 80 meter vertical yagi. I'll have more to say on that in about a week.

What I thought would be one of the more mundane tasks was to inspect the northeast Beverage that goes through some heavy bush along its 175 meter length, with the far end at the edge of the swamp (bog). By mid-May this area will be effectively off limits since the growing vegetation will host some nasty wildlife. By this I mean that the bush and hay fields become tick heaven until at least mid-summer.

I had a suspicion this wouldn't be a normal inspection since the Beverage has not been performing well lately. When I approached the feed point my fears were confirmed. The aluminum wire was slack, evidence of a break somewhere. A month earlier all had been well. I waded through the bush along the antenna line looking for the problem.

Wire hanger bent when the tree fell on the Beverage

Approaching the termination the land slopes gently downward into the swamp. With the frost not out of the ground in sheltered areas run off pools on the surface. The ground is very squishy. The problem was discovered just 15 meters shy of the termination resistor.

Presumably during the storms a lot of the dead and dying trees met their fate. Trees in the boggy ground are prey to disease and rot. Many of the softwoods are skinny things with sparse leaves, unable to fare better in the saturated soil. Even so they pack a punch when they come down. It has happened before though without ill effect. Aluminum fence wire is surprisingly strong. However it is not invulnerable.

I counted five of these benighted trees in the space of less than 10 meters that had fallen onto the Beverage wire. One of them was large enough to sever it. Surprisingly a couple of the broken trees were still leaning against the wire which was still under some tension because the big tree was lying on top of it.

Manual splice: ugly but
it works, for now

A severed terminated Beverage becomes an unterminated Beverage. This is still a reasonably good receiving antenna except that it is bidirectional, only rejecting signals off the sides. A bidirectional Beverage has the advantage of covering two directions at once but with the serious disadvantage of poorer performance in the one direction you are most interested in.

The dead trees were pushed aside to rot in peace on the wet ground. After determining that the termination box containing the resistor and the ground rod connection were undamaged I retrieved the broken ends of the wire and manually spliced the break by wrapping 3" of the wires together. It held when I pulled on the wires to test the splice. Although this is not the proper way to splice aluminum wire it is a quick and easy way to temporarily put the antenna back in service.

I was not done since the slack had to be taken out the Beverage wire. All soft drawn wires will stretch a surprisingly large amount when put under high tension. It should be obvious that a wire that has been pushed beyond its breaking strength has also been pushed beyond its yield strength, which is typically ~70% of breaking strength. The 175 meter wire stretched ~60 cm (2').

Since I had only about half that much rope remaining at the termination I returned to the feed point, along the way lifting the wire off the foliage that trapped it while it lay slack. I removed the rest of the slack at the feed point and was pleased to find that my improvised splice held. Again I followed the wire to the termination and then back to the feed point to pull the wire free from twigs it snagged as it was lifted to its original height. Yes, this is a lot of work! Having an antenna farm is not for the lazy.

When night fell I was pleased to discover that the antenna was back to its usual awesomeness. Hopefully it'll survive the summer. In the autumn I may replace this antenna with a bidirectional Beverage although I am still loathe to tamper with an antenna that performs so well to Europe, which is the most productive contesting path.

This tree was too large to remove without assistance. Instead I
defanged the threat by cutting the branches which are long
enough to strike the Beverage wire when the rot progresses
to the point that the tree falls the rest of the way down.

The next day I went back into the bush with a saw and cleared away several dead and dying trees within reach of the Beverage. I want to avoid a repeat. Luckily it is only this area near the swamp that has sick and risky. Elsewhere they're healthy and strong and not too tall or are evergreens that do not have large overhanging branches.

Having lots of trees available as supports is nice provided you account for the risks. Everyone I know who runs Beverages through bush periodically clears deadwood, and yet still suffer breaks from time to time. This is my fate as well. Beverage maintenance will only become more onerous when I put up more of them.

Is it worth the trouble? In my experience: yes! Just keep these things in mind:

• It's not a matter of if but when. Have a plan and material on hand to quickly and effectively repair or replace Beverage wires.
• Splicing aluminum wire is difficult and there is likely no mains power nearby. Beverages made from coax that are spliced with connectors cannot take tension so you'll have to use messenger wires or replace the coax.
• Inspect the full length of Beverages twice each year. Once in the spring after winter has done its worst and again in fall before the contest and low band season begins. Remove suspect trees and deadwood that can threaten the Beverage when they come down.
• Be safe! Cutting down trees is dangerous work. It is even worse with deadwood since you will get little warning when a rotten limb or trunk you are sawing snaps. Trees will kick out, twist, break, pivot on obstacles and otherwise behave unpredictably when they are cut, chopped, pulled or pushed. They can be far heavier than you expect since deadwood is often waterlogged. Do not overestimate your ability to outmanoeuver a falling tree!

Get the help of a friend or professional is you are uncertain how to proceed. Don't improvise! As you would for tower work acquire and learn to use the proper tools. Beverage antennas are wonderful things but are not worth the risk of serious injury.

Deadwood cast aside to peacefully decompose on the wet boggy ground

Plaque vs. Mailbox

Antenna and tower articles are in short supply right now because our winter weather is persisting rather than making way for spring. This keeps me more indoors than outside, with projects temporarily on hold. That will soon change. In the interim I thought the following would be of some interest.

I was surprised earlier this week when I went out for my morning run and saw this attached to my mailbox:

Contest plaques are not terribly compatible with mailboxes! Back at my city QTH plaques and parcels were left on the doorstep. In this sparsely populated township the mail carrier never steps outside of her car so you get this sort of thing happening. At least it's better than receiving an invitation to drive 13 km to the local post office to pick up large packages, which happens often.

The surprise mentioned above was not the delivery method but rather that I received a plaque for a contest that I have no recollection of winning. Opening the package I found that it is for 2017 ARRL DX CW in which I won Canada in the low power category. I then learned that my own club, Contest Club Ontario, is the sponsor. Since it is considered polite to thank the plaque sponsor I stood in front of a mirror and thanked my reflection. The plaque is my payback for buying raffle tickets to fund CCO.

Some assembly required

Unlike other plaques I've won that this one comes in kit form. ARRL has one standard plaque to which is affixed a plate bearing the particulars.

The instructions require that I peel the adhesive protective cover on the back of the plate and carefully glue it into the blank rectangle at the lower right.

This is worse than a kit that requires soldering since it is all too easy for the fumble fingered like me to incorrectly centre and align the plate. I doubt the plate could survive removal for a second attempt.

I undertook the task and managed to do a not entirely unsatisfactory job. Although I'd rather ARRL took care of this detail themselves I appreciate the recognition for my contesting effort. Even if I did pay for the plaque by buying all of those raffle tickets over the past several years.

Small Contests and Zombie Contests

I like to participate in several small contests during the year. They are a low pressure way to keep our stations and our skills in good shape, and to try out new equipment and software features. When the major contests come along we'll be better prepared.

As for why they're low pressure, consider the following points:

• They tend to be shorter, and participants are fewer.
• For QSO parties and country-sponsored contests you likely can point all your antennas at the start of the contest and leave them there.
• You can take time off without losing many points since there is a smaller pool of stations to contact and propagation may not be there for part of the contest.

They are also an excellent opportunity to test yourself and your antennas:

• There are many stations from the sponsoring region to test propagation and antennas.
• With all antennas pointing the same direction you can easily compare relative performance at different times (band opening, mid opening, band closing), knowledge you can put to good use at other times.
• Compare fixed yagis with rotatable yagis by seeing how many degrees a yagi can be off the side of the path until it matches a fixed yagi to that region. From here small European contests are most useful since it is common for fixed yagis to be pointed at Europe.
• I like to call the really weak stations to see how often I can complete the QSO. This is good practice for those running low power or QRP, and for QRO operators who get many weak callers during runs. In the small contests there is less rush to move on to the next QSO.

The better ones can be fantastic vehicles to encourage non-contesters from the sponsoring country or area to be active, potentially turning some of them into contesters. This weekend I participated in the SP DX contest -- CW, low power category -- and had a great time. They did a fabulous job getting hundreds of stations on the air, many of whom are probably not contesters since they are flagged by the logging software as not present in the master database of contest call signs.

At the other end of the spectrum are what I call zombie contests. These are smaller contests that, although still sponsored, have low activity. There is little or no promotion within the sponsor's country or region, thus only the diehards bother to participate. Like zombies these contests somehow continue to shuffle along year after year despite being as good as dead.

I won't name any of these contests to avoid annoying the guilty parties. Some are so dead that you can work few of the sponsoring region multipliers because no one is active. When the 3830 reports repeatedly note a lack of sponsoring region activity you know that it's a zombie contest. No one enjoys a zombie contest so those stations are unlikely to return next year.

Many zombie contests need to die. However some can be jolted back to life with the backing of an enthusiastic group or club. As for ARRL Field Day and other contests clubs can turn the event into a group activity with a combination of organization, recognizing participation, putting fence sitters in front of radios, and even use gimmicks such as special calls for extra points. It can be done. In the US the California and Florida QSO parties are good examples of small contests with enthusiastic backing and lots of activity.

But are there simply too many small contests on the calendar? Not really since there is no need to participate in all of them. Pick those that appeal to you or take place a week or two before a major contest and go for it. Try one and discover how much fun (and valuable) it can be.

Ice Anchors

Another winter is over and with tower season beginning it is time to prepare for next winter's contests. This is an ideal time to consider an innovation in tower erection: ice anchors. They are fast, low cost and are a fantastic alternative for those that live in cold climates. So what are these ice anchors? I'm glad you asked.

But first some history of the use of ice in major construction projects throughout history. It really is a perfect material for our northern climate.

• Ice roads: Arctic nations such as Canada build ice roads to reach isolated settlements right over top of tundra, bogs, lakes and rivers
• Ice boats: Icebergs naturally float and have inspired the design of ice warships
• Ice buildings: Igloos have been used for millennia and there are even ice hotels
• Ice optics: Ice lenses can start fires and be used in cold weather telescopes

Well, you get the idea. What an exciting opportunity to finally utilize this common material for ham radio applications. Despite being literally in front of most of us all our lives we've overlooked its possibilities beyond hockey and curling. Therefore with no further delay we present the ice anchor!

The ice anchor is simply an ordinary guy anchor for big towers, but one that needs no concrete. It's so simple you'll wonder how it is that no one has thought of it before now. Don't believe me? I'll show you with step by step instructions:

Step 1: Dig a hole. This is exactly the same as the hole you'd need for a concrete anchor.

Step 2: Drop anchor into the hole and aim it at the tower. Use reinforcing since ice can be weak in tension, just like concrete.

Step 3: Fill hole with water. Pump it in or let nature do it for you if the weather cooperates.

Step 4: Sit back and relax! Winter is coming.

Once the anchor is frozen you are ready to attach guys and put up your tower! Soon your tower will be up and your antennas ready for the major contests. You can enjoy months of superior performance. Records will fall and DXCC Honor Roll will soon be yours. Should you ever move the anchors are easily removed for instant reuse at your new QTH.

But wait! There's more!

Of course this is not a year-round system. It is important that you take it all down after the contest season, before spring thaw gets going in earnest. Otherwise you may not be able to reuse the tower and antennas next year.

But surely that's a lot of maintenance to do every spring and fall. Lucky for you there is complementary technology that will convert your seasonal ice anchors permanent fixtures, saving you hours of biannual effort.

Heat pump: Install a closed loop coil system in your anchor holes and you can not only keep your anchors solid longer you can heat your home and your bath water.

XXXL Parasol: Shield your anchors from the hot sun during those warmer months.

Heavy water: Deuterium oxide (D₂O) has a higher melting temperature and more mass to support especially large towers and antennas.

Who knows, someday we may see other application of ice technology in ham radio. Perhaps you'll be the first to design and build effective ice alloys that are suitable for antennas, radial fields or even semiconductors for self-cooled low noise amplifiers (LNA) for EME.

The future is now!

It's also April 1^st.