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Chapter 6

Design and fabrication of High Q Loading Inductors

6.0 Introduction

From the earliest days of radio inductors for antenna tuning have been recognized as

critical components. Entire books and countless technical papers have been written on

the subject. Earlier chapters have emphasized how critical high QL is for achieving

acceptable efficiency so the focus of this chapter is the design and fabrication of high

Q inductors. While there are many possible constructions, this chapter is concerned

specifically with cylindrical single layer air wound coils using round wire which are the

most common and practical type. There are brief examples of flat spiral or "pancake"

inductors and basket-weave.

This chapter includes both practical advice and a few finer points of inductor design.

However, many users may not want the all the gory details and mathematics. Simple

graphs from which values can be read combined with practical "do this" instructions

can produce a perfectly adequate inductor in many cases but sometimes more

information is needed. For this reason many of the details and justifications for the

advice given have been placed in appendixes A3 and TBD.

Much of the earlier discussion assumed relatively low QL values, 200-400, which

represent small inductors (dimensions of a few inches). However, with careful design,

we can do much better when the dimensions are increased to tens of inches. Such

sizes may not be practical within a transmitter enclosure but usually there will be room

outside at the base of the antenna. Descriptions of large LF antennas often mention

tuning inductor (or "helix") QL of several thousand[1]. Past experience with HF

inductors might make such high numbers seem a fantasy but that's not the case! QL

approaching or even exceeding 1000 is practical for mH inductances at LF or MF

without the use of exotic materials. Most amateur inductors will use materials on hand

or available at a typical hardware store: a plastic bucket or some PVC pipe for a coil

form, salvaged house wiring, etc. The examples keep this in mind.

6.1 How much inductance?

The first order of business is to determine the required inductance (L). Sufficient

inductive reactance (XL) is needed to cancel the input capacitive reactance (Xi) at the

feedpoint, i.e. XL=Xi, where XL is the inductive reactance of the loading coil needed for

resonance:

2

(6.1)

(6.2)

If f is in MHz then L will be in μH.

Figure 6.1 -Tuning inductor inductance for resonance at 475 kHz.

As shown in earlier chapters, Xi can be determined from modeling, calculations or

measurements at the antenna. Figures 3.4 and 3.5 in chapter 3 showed values of Xi

for a range of heights (H) and top-loading wire length (L). To predict the needed

inductance we can convert these values for Xi to inductance in uH as shown in figures

6.1 and 6.2. It should be pointed out that although figures 6.1 and 6.2 assume a "T"

with a single top-wire, the values for the loading inductor would be the same for any

capacitive top-loading structure which provides the same amount of loading (Xt). The

shape of the hat is not what's important, it's the shunt capacitance it adds!

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Figure 6.2 - Tuning inductor inductance for resonance at 137 kHz.

From figures 6.1 and 6.2:

1) At 475 kHz L varies from180μH →2600μH, depending on height and top-loading.

2) At 137 kHz L varies from 2.8mH→30mH.

Potentially, inductances in the range of 0.1 to 30 mH may be needed.

6.2 Symbols and abbreviations

The following is a list of the symbols and abbreviations keyed to figure 6.3:

Aw → cross sectional area of the winding conductor

c → center-to-center spacing between turns, winding pitch

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d → diameter of winding conductor

D → diameter of the winding

f → operating frequency

fr → self resonant frequency

Kp → proximity factor

Ks → skin effect factor

Figure 6.3 - inductor dimensions→

l → coil length

lw→ length of the winding conductor

N → number of turns in the winding

Rdc → DC resistance of the winding conductor

RL → AC winding resistance including Rdc + skin and proximity effects

QL → inductor Q

δ → skin depth

σ → conductivity of winding conductor

Geometric ratios:

(d/c) → conductor diameter/turn spacing ratio

(d/δ) → conductor diameter in skin depths

(l/D) → coil length/diameter ratio

6.3 Inductance calculation

We can calculate the inductance of a coil from its length (l), diameter (D) and number

of turns (N) or, going the other way, given the required inductance, determine the coil

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dimensions, number of turns and wire size. In general we will want to use the simplest

formulas which have adequate accuracy (±5%). One of the most useful equations

comes from Harold Wheeler's 1928 IRE paper[2], equation (6.3). L at low frequencies

of a single layer circular winding is:

[μH] (6.3)

Where l and D are in inches! For dimensions in cm divide eqn (6.3) by 2.54.

Wheeler stated that this equation should be correct within 1% for l/D>0.4 and

experience shows it's still very good for l/D down to 0.2. I have tested this equation on

a wide variety of inductors and found it accurate for frequencies well below the self

resonant frequency fr. This simple equation is the basis for much of the analysis which

follows but we have to understand it's limitations. There is always the problem of

"manufacturing tolerances", i.e. what actually gets built versus the initial design. For

example, we might want to use #12 wire with a diameter of 0.081" and have a center-

to-center spacing ratio (d/c) of 0.5. This means that c=0.162". that's not a very

convenient value when winding many turns. We are more likely to use 0.160", 0.170"

or even 0.200" for c because it's much easier to replicate especially if we're cutting

slots in the coil form.

Another problem relates to the shape of the coil form. Wheeler's equation assumes a

uniform cylindrical coil form. In practice we might use a plastic bucket which has a

tapered diameter. Another option shown in later sections is to use a cage made from

PVC pipe. The cage can be square, hexagonal or octagonal. When we go from

circular to octagonal, to hexagonal, etc, for the same diameter (D) the inductance will

be smaller because the cross sectional area of the coil is less. We have to adjust the

"diameter" to compensate for the change in cross section.

A more serious problem encountered in large inductors is "self-resonance". If you

measure the impedance of an inductor over a wide range of frequencies you will find a

number of series and parallel self-resonances. the inductor behaves very much like a

transmission line. While shape, wire size and spacing have some effect, the primary

determinant of the resonance frequency (fr) is the length of the wire (lw).

As shown in figure 6.4, if you start at a low frequency and go upward, the measured

inductance initially falls slightly (≈1-2%) but then levels out. However, as you approach

fr the inductance increases rapidly. The design graphs in section 6.4 assume

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operation <0.1fr where the effect is small. fr≈λ/4, at 475 kHz 0.1λ≈200' and at 137 kHz

0.1λ≈700'. The fr limitations due to winding length lw indicated on the graphs are

explained in section 6.14. As the value of L increases fr decreases because there is

more wire in the winding. At some point self resonance begins to seriously reduce QL

(as shown in section 6.5 QL graphs) setting an upper frequency limit at which the

inductor is useful. This effect can be a serious problem because the design becomes

significantly more complicated as explained in section 6.14.

Figure 6.4- Inductance versus frequency.

Besides these problems we need to recognize that we may not even know the exact

value to resonate the antenna! If the antenna has already been built and an accurate

measurement of the input impedance is available, the value for L will known. But the

necessary impedance measuring instrument may not be available or the antenna may

not yet have been built! With careful modeling we can estimate the value for L within

≈5-10% depending on how close the model is to the actual antenna. As was shown in

chapter 3, an approximate value for L can be calculated from antenna dimensions.

Even if we measure the input impedance with a VNA that measurement is only at one

particular time! The short verticals used at LF/MF have high Q's, i.e. very narrow

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bandwidths[3, 4, 5]. They are very sensitive to detuning, particularly as the seasons

change for dry to wet. The shunt capacitance of the antenna will change with soil

conductivity which changes with moisture content. This change in shunt capacitance

will detune the antenna requiring some adjustment of L.

These observations should not be taken as gloom-and-doom! There is a simple way to

deal with these problems: make the inductance larger than the original estimate and

then tap down or prune the inductor to make final adjustments when the antenna has

been erected. Besides tapping there are other options for adjusting inductance as

outlined in section 6.7. As a practical matter any tuning inductor will have to be

adjustable to some extent.

On some occasions a flat or "pancake" winding geometry like that shown in figure 6.5

may be used.

[μH] (6.4)

Where r and t are in inches. For dimensions in cm divide eqn (6.4) by 2.54.

Figure 6.5 - Pancake winding example.

It is also possible to use toroidal windings like that shown in figure 6.6.

[μH] (6.5)

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Where d' is the cross section diameter of the coil and D is the diameter of the axis of

the coil.

Figure 6.6 - Toroidal winding example. From Terman[tbd].

6.4 Inductor design using graphs

Design graphs can be made more general by using geometric ratios as variables:

(d/c) → conductor diameter/turn spacing ratio

(l/D) → coil length/diameter ratio

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Figure 6.7 - Inductance versus diameter.

To illustrate the use of geometric ratios we can assume a given length of wire, lw=100',

of a given size, #12 (d=0.081"). The wire can be wound in many different ways. For

example, we might chose to use a small diameter tube to create a long skinny inductor,

i.e. l/D large, or we could use a much larger diameter form to create a short fat inductor

with small l/D. Besides the l/D ratio we are free to choose the center-to-center turn

spacing (d/c), from a very tight or close winding (d/c=0.9) to a sparse one (d/c=0.1),

i.e. the center-to-center turn spacing varies from a 1.1 to 10 wire diameters. Figure 6.7

illustrates some of the possibilities where the diameter D is the variable and d/c is a

10

parameter (solid contours). The dashed contours correspond to constant values of l/D.

Note that for all values of d/c, maximum inductance occurs at l/D=0.45. In this

example L varies from ≈50 to 460 uH, a range of 9:1 with the same piece of wire!

Figure 6.8 - QL and L versus d/c.

This brings up the question of why we would use values of d/c < 0.90 or even 0.95

since that gives us the maximum inductance for a given length of wire? Besides

inductance we want high QL. We can calculate QL as a function of d/c at a given

frequency (475 kHz) for our 100' of #12 wire with the result shown in figure 6.8. In

figure 6.8 L increases as d/c increases but QL peaks at d/c≈0.35 and then falls rapidly.

Losses due to proximity effect (Kp), which vary with (d/c)2, are the primary culprit. As a

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practical matter, we have to trade inductance for QL and any choice will be a

compromise.

Inductor design begins with the determination of a value for the inductance L. At the

end of the design process to fabricate the inductor we will need to know the coil length

(l) and diameter (D), the number of turns (N), the wire size (d) and length (lw).

The final values for l, D, d, N and lw can be found using the pre-calculated graphs

given in figures 6.9 through 6.24. The graphs are arranged in four groups of four

graphs. Because the required inductance is the starting point, the x-axis on each

graph is inductance in uH. Four graphs in each group have the following y-axis labels:

1. Wire length lw in feet

2. Diameter D in inches

3. Number of turns N

4. QL at 475 kHz

The difference between the graph sets is the value for l/D (0.5, 1 and 2). The value for

d/c in the first three graph sets is 0.5 because this is usually close to optimum. To

design a physically smaller inductor for a given values of L in the last set of graphs d/c

has been changed from 0.5 to 0.8, representing a much closer turn spacing.

Given a value for L, there are many possible starting points using the graphs. For

example you might choose an initial value for QL on the y-axis of graph 4 which in

combination with the value for L on the x=axis will give you the wire size. With the wire

size and the value for L you can find lw, N and D from the other graphs in the set. In

the process you may discover the wire size is impractical or the size of the inductor is

too larger, etc. At that point you can simply repeat the process with a lower value for

QL. If you have a coil form with a given diameter you can select that value for D on

the y-axis of chart 2 as your starting point. Note that you have three choices for the

shape factor (l/D) so you may want to iterate the design with different values for l/D. If

you happen to have a spool of wire with a given size you can use that wire size as your

starting point.

The procedure is straightforward:

1. Draw a vertical line from at desired value for L on each graph, using the graph set

with the desired l/D and d/c ratios.

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2a. If you are starting with a known wire size then the intersection of the vertical line

for the x-axis inductance value and the contour for that wire size gives the values for

lw, N and QL

2b. If you have not preselected a wire size, draw a horizontal line from y-axis value for

the other variable you have chosen, QL, D, etc. the intersection of the vertical and

horizontal lines will give you the wire size from which you can determine the other

variables.

As examples, let's assume L=800 uH and #12 wire (d=0.081") and perform a trial

design using each set of four graphs for comparison. The dashed lines on the graphs

show the solutions for this example and Table 6.1 is a summary of the graphical

results.

Table 6.1 800 uH inductor examples.

l/D= 0.5 1 2 0.5

d/c= 0.5 0.5 0.5 0.8

D= 14.7" 10.7" 8.0" 10.8"

l= 7.35" 10.7" 16.0" 5.4"

lw= 175' 184' 207' 150'

N= 45T 66T 99T 53T

QL= 885 782 730 555

From the table we can see that D, l, lw, N and QL all vary when we chose different

values for l/D and d/c. Of particular interest is how QL varies. Be best value for QL is

865 when l/D=d/c=0.5. A turn spacing ratio (d/c) of 0.5 is very typical and results in

relatively low proximity effect losses. But setting l/D=d/c=0.5 results in a relatively

large inductor, D=14.7" and l=7.4". We could instead wind the wires closer together

with d/c=0.8. This is approximately the value for d/c to expected when we closely wind

insulated #12 THHN wire a coil form. In that case we get a much smaller inductor,

D=10.8" and l=5.4". lw is also reduced from 175' to 150'. However, QL is much lower,

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555 compared to 865! This represents a fundamental trade-off, the larger we make the

inductor the higher QL will be. Note also from the charts, that as we increase the wire

size (d) the inductor gets larger and QL rises significantly.

The graphs for lw, D and N are independent of frequency and winding material,

however the QL graphs are for 475 kHz and copper wire only but we can easily scale

the QL values for other frequencies. As we go lower in frequency both skin and

Proximity effects are reduced. This implies that the winding resistance (RL) is lower,

but XL=2πfL also decreases with frequency. The net effect of decreasing frequency,

with a given inductor, is lower QL. We can express this as a ratio:

(6.6)

(6.7)

For large inductors, with hundreds of feet of wire, the cost of the wire may be a

concern. Up to this point the examples have assumed copper conductors with

σ=5.8x107 [S/m]. Aluminum has somewhat lower conductivity (σ=3.81x107 [S/m]) but

is much less expensive. For example, for large inductors salvaged cable TV coax,

which usually has a solid aluminum jacket, can be used and is frequently free (see

figure 6.38). Skin depth varies as 1/(√ σ), i.e. δ increases as σ decreases. In the same

inductor, replacing copper with aluminum:

(6.8)

QL drops by ≈20% when going from copper to aluminum. The QL graphs scale with

that factor.

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Figure 6.9 - Wire length versus inductance.

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Figure 6.10 - Diameter in inches versus inductance.

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Figure 6.11 - Number of turns versus inductance.

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Figure 6.12 - QL versus wire size.

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Figure 6.13 - Wire length versus inductance.

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Figure 6.14 - Diameter in inches versus inductance.

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Figure 6.15 - Number of turns versus inductance.

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Figure 6.16 - QL versus wire size.

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Figure 6.17 - Wire length versus inductance.

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Figure 6.18 - Diameter in inches versus inductance.

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Figure 6.19 - Number of turns versus inductance.

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Figure 6.20 - QL versus wire size.

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Figure 6.21 - Wire length versus inductance.

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Figure 6.22 - Diameter in inches versus inductance.

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Figure 6.23 - Number of turns versus inductance.

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Figure 6.24 - QL versus wire size.

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6.5 Do it yourself graphs

The graphs in section 6.4 may not be adequate for all designs. You can enter the

equations from which the graphs in section 6.4 were derived into your own

spreadsheet tailored to your requirements. The graphing process proceeds as follows:

[1] Choose values for L, l/D, d/c and d.

[2] Calculate the diameter D from:

[inches] (6.9)

Where D and d are in inches.

[3] Calculate N from:

(6.10)

[4] Calculate lw from:

[inches] (6.11)

[5] Calculate QL from:

(6.12)

Where RL is the AC resistance of the winding at a given frequency.

(6.13)

Where Rdc is the DC resistance of the wire:

(6.14)

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Ks is the skin effect factor which represents the increase in wire resistance with

frequency. Values for Ks for common wire sizes are given in table 6.2.

Table 6.2 - Ks for common wire sizes.

137 kHz 475 kHz

Wire # Ks Ks

8 4.82 8.76

10 3.87 7.00

12 3.10 5.55

14 2.53 4.49

16 2.06 3.61

18 1.15 1.93

Kp is the proximity effect factor which represents an increase in resistance due to

interaction between turns. The current in a given turn produces a magnetic field with

introduces losses in adjacent turns. Table 6.3 gives values for Kp for d/c=0.1→0.9 and

l/D=.3→2.

More detailed explanations for Ks and Kp, along with a number of graphs, can be

found in section 6.11 and appendix A3.

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Table 6.3 - Values for proximity factor Kp.

(d/c)= 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

l/D Kp Kp Kp Kp Kp Kp Kp Kp Kp

0.3 1.03 1.10 1.22 1.39 1.62 1.92 2.31 2.80 3.42

0.4 1.03 1.12 1.25 1.45 1.72 2.06 2.51 3.06 3.75

0.5 1.04 1.13 1.28 1.49 1.78 2.16 2.63 3.22 3.95

0.6 1.04 1.14 1.30 1.52 1.82 2.21 2.70 3.31 4.06

0.7 1.05 1.15 1.31 1.54 1.85 2.24 2.74 3.35 4.10

0.8 1.05 1.15 1.32 1.55 1.86 2.26 2.75 3.36 4.10

0.9 1.05 1.15 1.32 1.56 1.87 2.26 2.75 3.35 4.07

1 1.05 1.16 1.33 1.56 1.87 2.25 2.73 3.32 4.02

1.1 1.05 1.16 1.33 1.56 1.86 2.24 2.71 3.28 3.96

1.2 1.05 1.16 1.33 1.56 1.85 2.23 2.68 3.23 3.89

1.3 1.05 1.16 1.32 1.55 1.85 2.21 2.65 3.18 3.81

1.4 1.05 1.16 1.32 1.55 1.84 2.19 2.62 3.14 3.74

1.5 1.05 1.16 1.32 1.54 1.83 2.17 2.59 3.09 3.67

1.6 1.06 1.16 1.32 1.54 1.82 2.16 2.56 3.04 3.60

1.7 1.06 1.16 1.32 1.53 1.81 2.14 2.53 3.00 3.54

1.8 1.06 1.16 1.32 1.53 1.80 2.12 2.51 2.96 3.48

1.9 1.06 1.16 1.32 1.53 1.79 2.11 2.48 2.92 3.42

2 1.06 1.16 1.31 1.52 1.78 2.09 2.46 2.88 3.37

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6.6 Inductor fabrication

Figure 6.25 - PVC cage coil form.

How might we fabricate large inductors? PVC pipe and associated fittings provide an

easy, inexpensive and flexible way to construct a coil form of arbitrary size. Figure

6.25 shows a coil form using PVC pipe and standard fittings. The octagonal rings used

1/2" pipe joined with 45° elbows. Because the winding compresses the form it's not

necessary to glue the rings making it much easier to fabricate and adjust them! The

eight vertical supports used 3/4" pipe with slots cut at intervals to hold the wire. Figure

6.26 shows a practical way to cut the slots in the vertical supports. Attaching the

supports to a board makes it much easier to align all the slots and mounting holes.

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Figure 6.26 - Cutting the wire slots.

Figure 6.27 - Winding the inductor.

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Figure 6.27 shows the inductor being wound using a frame made from scrap 2"x4"

lumber. The ends of the coil form were attached to plywood sheets and standard

plumbing hardware used for the axels and crank. While such a frame is not absolutely

required it's assembly is simple and makes the job go much faster with a nicer result!

Figure 6.28 shows the finished inductor. Salvaged wire was used so the wire was a bit

"lumpy". The final inductor required some manual tweaking of the turns and was not

very pretty but perfectly functional.

Figure 6.28 - Finished inductor. PVC cage inductor 2.

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Figure 6.4 is the measured inductance of this coil. At low frequencies L=800 uH. This

drops to 783 uH at 137 kHz and rises to 928 uH at 475 kHz. The measured QL as a

function of frequency is shown in figure 6.29. Note that at 137 kHz QL≈435, the

predicted value was 465. At 475 kHz the predicted value for QL was 865 but the

measured value is ≈680, substantially lower! The values for L and QL, as a function of

frequency demonstrate the effect of self resonance.

Figure 6.29 - Example inductor measured QL.

As an alternative to the PVC cage, a plastic bucket can be used as a coil form as

shown in figure 6.30. The winding has 60 turns of #12 stranded copper wire with THHN

insulation. The winding length is ≈9.75", the average diameter ≈11.3" and l/D≈0.86.

Because the bucket is smooth with some taper, before winding six strips of double

sided mounting tape were attached vertically. The measured inductance was 773 uH.

The measured QL is shown in figure 6.32. At 475 kHz QL≈350. This is a reasonably

good inductor considering the simple construction and materials but the dimensions

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were chosen to accommodate the bucket and the winding spacing is close (d/c≈0.8),

so QL=350 is not very high. The use of stranded wire reduced QL slightly.

Figure 6.30 - Example using a plastic bucket for the coil form.

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Figure 6.31 - Bucket modification for winding.

Figure 6.31 shows 1/2" pipe attached to the top and bottom of the bucket using

standard hardware store stanchions. There are square plywood blocks on the inner

sides of the bucket and the lid. The stanchions are attached with screws through the

bucket into these blocks. The bucket was wound on the fixture shown in figure 6.27.

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Figure 6.32 - QL versus frequency for the bucket inductor.

It is often possible to find large commercial inductors which have been removed from

BC equipment at swap meets and surplus outlets. Figure 6.33 shows a high quality

commercial inductor purchased at an electronics surplus store. The inductor has the

following dimensions:

l=length=20", D=diameter=10" → l/D=2

N=119 turns of #10 wire, d=0.101"

The spacing between the wires is 0.08" → d/c=0.558

Using Wheeler's equation we get L=1.445 mH, the measured value is 1.45mH!

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Figure 6.33 - Large BC inductor.

As shown in figure 6.34, at 137 kHz QL≈450 and at 475 kHz QL≈690. This is quite a

good inductor but it is important to note that it's form (l/D and d/c) is not optimum.

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Using the same length of wire and rewinding it with a more optimum form, a QL

approaching 1000 and higher inductance is possible.

Figure 6.34 - QL versus frequency for the BC inductor.

Instead of a circular solenoid we could use a pancake geometry like that shown in

figure 6.35. Using equation (6.5) with r=5.25", t=4" and N=59 Turns, L=1116 uH. The

measured value was L=1114 uH! Solid #12 THHN was used for the winding. The

center hub is acrylic plastic and the nine radial arms are cut from 3/8" F/G electric

fence wands. The winding is referred to as "basket weave". This type of winding

requires a odd number of support rods. The self resonant frequency was ≈1.3 MHz

and the QL for this inductor is shown in figure 6.36. This is quite low, primarily due to

the significantly higher proximity losses associated with this winding geometry. Inner

turns lie in the fields of outer turns. On the other hand the inductor is very compact for

its inductance (1.1 mH). There may be times when the smaller size makes the lower

QL acceptable.

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Figure 6.35 - Example of a flat or pancake winding.

Figure 6.36 - Measured QL versus frequency for the pancake inductor.

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A clever example of a very light weight inductor (≈18'X18") is shown in figure 6.37.

Pat, W5THT, fabricated the coil form by wrapping F/G mat around a cardboard tube.

He then impregnated the mat with epoxy and when it had cured soaked the assembly

in water to soften the cardboard for removal, leaving a thin shell on which he wound

1/2" wide copper tape. A protective covering of paint was then applied. The light

weight of the inductor allowed him to hoist it to the top of his vertical where it joined the

capacitive hat. This improved his Rr a bit.

Figure 6.37 - Pat W5THT, foil wound lightweight inductor.

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For 600m and 2200m a foil thickness of 0.005"→0.010" would be close to optimum.

One could also purchase some thin plastic sheet and roll it, joining the ends to make

the coil form.

6.7 Effect of coil form shape

Up to this point a cylindrical coil form, with a circular cross section, has been assumed

but other shapes are possible and may be more practical, especially for large inductor.

It is possible to use almost any shape: triangle, square, hexagon, octagon, etc.

However, with an n-sided polygon the inductance will be lower because of reduced

cross sectional area compared to an inscribed circle. With a polygonal coil form we

need to increase the diameter D' to maintain the same inductance as we would have

had with a circular coil form. Terman[6] suggests the following correction factor for D':

(6.15)

Table 6.4 has values for D'/D for n=3→8.

Table 6.4

n D'/D

3 1.33

4 1.17

5 1.11

6 1.07

7 1.05

8 1.04

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For a triangular coil form the diameter has to be increased by 33% to maintain the

calculated value for L. However, when an octagonal form is used the correction is only

4%. The use of a correction factor assumes you are using relatively small wire which

conforms to the shape of the coil form. If you are using larger wire or tubing that is

more rigid then circular turns may be wound on a triangular or polygonal form as

shown in figure 6.38. This particular example shows wooden strips holding the coil

which resulted in low QL. The wood was replaced with PVC pipe with significant

improvement in QL!

Figure 6.38 - Example of an inductor using Al jacketed coax, triangular form.

6.8 Variable inductors

In most cases the loading inductor will need to be adjustable at least to some degree

and there are a number of possibilities:

1) Make the inductance larger than needed and prune the coil.

2) Make the inductance larger than needed and place taps on the winding.

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3) Use two inductors in series, one with a fixed value for the bulk of the inductance and

the other a smaller variable inductor large enough for the needed tuning range.

Option 1 is very simple but it's a one shot deal. In most cases the antenna will require

some adjustment as the seasons pass making option 2 much more practical. Figure

6.33 is example of extreme tapping. At the bottom of the coil the taps are close

together but for most of the coil the taps a well separated. The idea is to adjust the

wide spaced taps to get into the ballpark and then use the close spaced taps for final

adjustment. Locating taps takes some thought. When l and D are constant L is

proportional to N2. However, when you are adding/removing turns to a winding or

moving between taps the rate of change of L will vary from N2 to N because you're

changing both N and l because l=cN.

[μH] (6.16)

Figure 6.39 illustrates this point. For this example D=3" and the winding uses #18 wire

(d=0.040"). N is varied from 5 to 100 turns for d/c=0.9 (closely wound) and d/c=0.1

(very sparse winding). The dashed lines indicate slopes N and N2.

47

Figure 6.38 - Inductance versus turns number for fixed turn spacing.

With 100 turns, d/c=0.9→l=5" and d/c=0.1→l=40". For small N the rate of change of L

is close to N2 but as more turns are added the rate of change decreases approaching

N for N=100. d/c=0.1 is a very long coil (40") with well separated turns (c=0.4"), in

this inductor L N except at the low end. This behavior is something to keep in mind

when selecting tap locations.

A roller inductor in series with a fixed inductor is a convenient arrangement allowing

fine adjustment. The only practical limitation is that most of the roller inductors found

at ham flea markets tend have <100 uH of inductance which may or may not be

adequate. That problem can be managed by having a fixed inductor with taps for the

bulk of the inductance and a smaller roller inductor in series for fine tuning.

48

Another option is to use a "variometer". This is a form a variable inductor dating to the

earliest years of radio. Current LF-MF operators have shown great creativity in the

design of practical variometers for antenna tuning as the following examples show.

Figure 6.39 - KB5NDJ variometer.

Figure 6.39 is an example fabricated by John, KB5NJD. The base inductor is wound

on the outside of a plastic bucket. Inside the bucket is smaller, rotatable, inductor.

The two inductors are connected in series. By rotating the inner inductor the total

inductance goes from the sum of both to the difference. Usually a variation of 10% is

possible. Given the need for adjustment at inconvenient times many variometers use

some form of remote tuning. As the following examples show (figures 6.40-6.51),

there's no shortage of innovation. While one could be heroic and run out to the

antenna on a cold dark winters night it's much better if the adjustment can be made

from inside a warm shack. KB5NJD has the answer shown in figure 6.40: use an

inexpensive TV antenna rotor! Some of these examples are almost works of art.

49

Figure 6.40 - John, KB5NJD variometer adjustment with a TV rotor.

Figure 6.40 - Jay, W1VD, WD2XNS variometer.

50

Figure 6.41 - Jay, W1VD, WD2XNS variometer.

Figure 6.42 - Jay, W1VD, WD2XNS variometer enclosure.

51

Figure 6.43 - Steve, KK7UV, variometer.

Figure 6.44 - Paul, WA2XRM, variometer enclosure w/remote drive.

52

Figure 6.45 - Laurence, KL7L, variometer drive.

Figure 6.46 - Laurence, KL7L, variometer enclosure.

53

Figure 6.47 -Neil, W0YSE, variometer.

Figure 6.48 - W0YSE tuning unit circuit diagram.

54

Figure 6.49 - W0YSE variometer location.

Neil, W0YSE, has located his variometer just outside a window of the shack.

Adjustment is manual: open window, twist knob, close window.

55

Figure 6.50 -Ralph, W5JGV variometer.

Figure 6.51 - Ralph, W5JGV variometer

56

6.9 Winding voltages, currents and power dissipation

The current at the base of the antenna (Io) is also the current in the inductor. The

voltage across the inductor is the same as the voltage at the base (Vo).

Io is determined by the radiated power Pr and the radiation resistance Rr:

(6.17)

At 630m Pr=1.67W and at 2200m Pr=0.33W. Combining the band specific values for

Pr with Rr values we can use equation (6.8) to create graphs for Io as shown in figures

6.52 and 6.53. Note that in 6.52-6.57, L is the length of the top-wire in feet.

Vo is the voltage at the feedpoint:

(6.18)

Figure 6.52 - Io for Pr=1.67W at 475 kHz.

57

Figure 6.53 - Io for Pr=0.33W at 137 kHz.

We can use typical Xi values from chapter 3 to generate values for Vo as shown in

figures 6.54 and 6.55. Despite the low radiated powers (Pr) the voltages at the base

will often be >1kV and can be much higher, particularly when H is small. This must be

kept in mind when selecting a base insulator. A high Vo also means there will be

significant voltage from turn-to-turn in the loading inductor and across matching

network components.

58

Figure 6.54 - Vo for Pr=1.67W at 475 kHz.

Figure 6.55 - Vo for Pr=0.33W at 137 kHz.

59

PL is the power dissipated in the loading inductor, PL=Io2RL. As shown in figures 6.56

and 6.57 this power can easily be >100W (assuming that level of transmitter power is

available). The loading inductor must be designed to dissipate PL without damaging

the coil! In general the larger the physical size of the inductor the better power can be

dissipated. QL=300 is assumed for both 137 kHz and 475 kHz. This is a bit

pessimistic given the earlier discussion since increasing QL reduces PL

proportionately:

(6.19)

In short verticals with limited top-loading Io, Vo and PL can be very high. The key

reducing these values is to use sufficient top-loading.

Figure 6.56 - PL for Pr=1.67W at 475 kHz.

60

Figure 6.57 - PL for Pr=0.33W at 137 kHz.

6.10 Enclosures

Most amateurs will use some form of large plastic box for the tuning inductor enclosure

because these are inexpensive, readily available in a very wide range of sizes and

have little or no effect on inductor QL. One shortcoming of typical plastic containers is

their susceptibility to degradation from UV in sunlight when exposed to the weather. A

simple coat of white house paint is usually enough to allow them to last several years.

Metal enclosures can also be used although large enclosures will usually be custom

fabricated. In general a metal enclosure needs to be substantially larger than the

inductor. In particular the spacing from the ends of the coil to the enclosure wall

should be at least equal to the coil diameter. A conducting enclosure will tend to

reduce both L and QL if it isn't large enough.

61

6.11 Winding resistance

Figure 6.58 - Skin (a) and proximity effects (b).

In an air core inductor most of the loss will be due to winding resistance, RL. As we

transition from DC to higher frequencies, RL increases dramatically due to eddy

currents in the winding conductor, i.e. RL>>Rdc! Two things cause this: skin effect and

proximity effect. The source of these effects is the current flowing in the conductor and

the magnetic fields associated with these currents as shown in figure 6.58. The

dashed lines represent magnetic field lines resulting from the desired current flowing in

the conductor. The solid lines represent currents induced in the conductor. Skin effect

is present in an isolated conductor and in a winding. Proximity effect is present

whenever multiple wires are close together as in a winding.

62

The "penetration" or "skin" depth, δ, in meters is expressed by:

[m] (6.20)

Where:

σ is the conductivity in [Siemens/m].

μ is the permeability of the material which for Cu and Al =4πx10-7 [H/m].

f is the frequency [Hz].

The skin depth in mils for copper is:

[mils] (6.21)

At 137 kHz δ=7.03 mils (0.00703") and at 475 kHz δ=3.78 mils.

The resistance of the winding, RL, can be represented by the product of the DC

resistance (Rdc), a factor attributed to skin effect (Ks) and a second factor associated

with proximity effect (Kp).

(6.22)

Where:

[Ω] (6.23)

Aw is the wire cross sectional area. lw, Aw and σ must be in compatible units which for

σ in S/m would be meters!

Skin and proximity effect terms assume a solid conductor but they can also be used for

tubular conductors. For a tubular conductor use the appropriate conductivity but

assume the conductor is solid, i.e. Aw=πd2/4. This assumption works because the

current in both solid and tubular conductors of practical sizes will be concentrated in a

very thin layer on the outside perimeter. All that matters is the circumference.

63

6.12 Skin effect factor Ks

The skin effect factor Ks for an arbitrary conductor diameter can be a bit complicated

(see appendix TBD) but for the wire sizes typically used in tuning inductors d/δ> 5.

When d/δ> 5, Ks becomes a linear function of d/δ which can be closely approximated

with a simple expression:

(6.24)

which is graphed in figure 6.59. Note that Ks is only a function of the wire diameter d

and the skin depth δ at the frequency of interest (δ 1 √f).

Figure 6.59- Skin effect factor Ks versus wire diameter in skin depths (d/δ).

64

6.13 Proximity factor Kp

Kp is a function of three geometric variables:

l/D - the length/diameter ratio

d/c - the turn-to-turn spacing ratio

N - the number of turns in the winding

Note that Kp is not a function of frequency or skin depth and Ks is not a function of

these three variables. This makes analysis much easier allowing us to separate Ks

and Kp!

As shown in appendix A3, the equations for Kp are complicated so in addition to the

values given in table 6.3 I've elected to show Kp in graphical form from which values of

Kp can be found by inspection. Graphs of Kp versus l/D for different numbers of turns,

N= 10, 20, 30 and 50, are given in figures 6.60-6.66. For N>30, the differences in Kp

are small so the N=50 graphs can be used for N>35. These graphs were created

using Alan Payne's (G3RBJ) analysis[8].

These graphs should be adequate for most design purposes but for those interested in

the math associated with Ks and Kp there is an extensive discussion in appendix A3.

65

Figure 6.60 - Kp versus l/D with N=10.

66

Figure 6.61 - Kp versus l/D with N=20.

67

Figure 6.62 - Kp versus l/D with N=30.

68

Figure 6.63 - Kp versus l/D with N=50.

69

Figure 6.64 - Kp versus spacing ratio d/c, l/D 0.3-0.7.

70

Figure 6.65 - Kp versus spacing ratio d/c, l/D 1-5.

71

Figure 6.66 - Kp versus turn number (N) with l/D=0.5

72

6.13 Litz wire windings

For d/δ<1 Ks and Kp are small and it might appear all we have to do to minimize them

is use very small wire. The New England Wire Technologies catalog suggests #40 for

137 kHz and #44 for 475 kHz. The problem of course is in single wires this small Rdc

is large. The solution is to use many small wires in parallel but simply paralleling wires

in a bundle doesn't buy anything because the current still flows on the outside of the

bundle just as it does on a single large wire. In fact ordinary stranded wire has slightly

greater loss than solid wire at RF frequencies. But if we use individually insulated

wires and twist the bundle in such a way that every wire is periodically transposed from

the outside to the inside and then back, the current distribution can be much more

uniform across a cross section of the wire bundle and RL significantly lower. This type

of construction is known as "litz wire". The formal name is "litzendraht", which comes

from German, litzen→strands and draht→wire, "stranded wire". The strands in this

wire are individually insulated and twisted to provide the required transposition. Figure

6.67 (taken from the New England catalog) shows how the strands are assembled.

Initially seven strands are twisted together. To make the wire bundle larger (Rdc

smaller) multiple bundles are twisted together. This process can be extended to have

an Rdc equivalent to a given solid wire as illustrated in table 6.5.

Table 6.5 - Litz wire examples.

frequency equiv. AWG

Cir. mil

area

no. strands

strand AWG

nom. O.D.

Rdc Ω/1000'

137 kHz 12 6,727 700 40 0.118 1.76

475 kHz 12 6,600 1650 44 0.117 1.91

The advantage of litz is that it can substantially increase QL at LF and MF when used

in place of solid wire. It is also very soft and pleasant to work with and wind. But there

are downsides! The cost is much higher than equivalent solid wire and there is the

problem of reliably soldering 1600+ individual insulated wires to make connections at

the wire ends. Soldering can be done but requires a careful choice of wire insulation

and technique. Those interested in using litz should go to the wire manufactures

catalogs and applications notes.

73

Figure 6.67 - Examples of litz construction.

Litz wire can be very helpful but we cannot use just any litz. Michael Perry[9] has

published a formal analysis of litz wire construction which contains a cautionary tale!

74

Figure 6.68 - Rac comparison between solid and litz wire. From Perry[9].

Figure 6.68 is taken from Perry. It shows the how the ratio of Rac between a solid

conductor and a litz conductor can vary as the size of the individual wires (d/δ) is

varied (or the frequency is changed). The litz construction can have a small number of

wires and only a few layers or many wires forming many layers. In general the more

layers and the smaller the wire the greater the improvement. However, there is a trap

here! The reduction in Rac occurs only over a small range of wire sizes at a given

frequency or, for a given wire size, over only a narrow range of frequencies. The key

point shown in this graph is:

If the individual wire size is too large or equivalently if the frequency is

higher than the minimum Rac point, Rac can be much higher using litz than

in an equivalent solid wire!

The following quotation from Perry should be taken to heart if you are considering litz

wire:

"The foregoing analysis indicates some surprising design results which

may directly contradict widely held beliefs regarding ac resistance in wires

and cables. For example, suppose a solid conductor is excited at a certain

frequency which results in a radius which is many times the skin-depth.

75

Then, assume a designer switches to a cable of the same total diameter but

with several layers of stranded wire to reduce losses such that d/δ>2. By

inspection of Fig. 6.68, this process can result in far greater losses than if

the "solid" conductor were employed. Stated another way, an uninformed

design of Litz wire can result in a performance characteristic which is

much worse than if nothing at all were done to reduce losses!

A second and important fact is that the cross-sectional area of a cable

comprising stranded wires is substantially reduced from the conducting

area of a "solid" cable of the same radius. This is due to the fact that each

strand is usually round and insulated with a varnish or other

nonconductor. The round insulated wires in the cable yield a "packing

factor" which reduces the conducting area by a significant fraction, usually

at least 40 percent. The transposition process further reduces the cross-

sectional area available for carrying current. The final result is a

substantial reduction the net savings available in ac resistance by utilizing

Litz wire. Due to these limitations, the Litz wire principle for reducing ac

losses must be thoroughly understood in the context of a specific

application before it should be employed."

Enough said!

6.14 Self resonance

In section 6.3 the variation of inductance with frequency due to self resonance was

pointed out. The design graphs and equations in section 6.4 do not take self

resonance into account. Up to this point it appears that if we make the inductor

physically larger and/or go up in frequency, QL increases . It's tempting to think

this implies we could have really high QL. However, that doesn't happen, in large part

due to self resonance of the inductor. As illustrated in figure 6.69, the measured

values of QL in real inductors initially rises but at some point QL levels out and

then falls as the frequency is increased further. To understand why this happens we

can measure the impedance (Z=R+jωL, ω=2πf) of an inductor over a range of

frequencies. Figure 6.63 is photo of a 2.2 mH inductor. Using a VNA the measured

amplitude and phase of the impedance is graphed in figures 6.70 and 6.71.

76

Figure 6.69-Example of QL variation with frequency.

Up to 200 kHz the impedance behaves like a simple inductor, i.e. Z=2πfL, but at higher

frequencies that is not the case. Multiple resonances are shown, a parallel resonance

at ≈740 kHz, and a more complex resonance at ≈2 MHz which appears to be a series

resonance closely followed by another parallel resonance (i.e. a closely spaced zero-

pole pair). Note that the phase graph is a better indicator for the resonances. The

traditional explanation has been that there is stray capacitance across the inductor

which resonates with the inductance as shown in figure 6.72B.

77

Figure 6.70 - Inductor impedance versus frequency .

Figure 6.71 - Inductor phase versus frequency.

78

Figure 6.72 - Inductor equivalent circuit

Figure 6.72A represents the inductor at lower frequencies where the shunt capacitance

has little effect. Adding shunt capacitance (C) makes the impedance parallel resonant

at fr:

(6.25)

It is mathematically convenient to transform the parallel equivalent circuit in 6.66B back

to an equivalent series impedance as shown in figure 6.72C. In the new equivalent

circuit Lo→Lc, RL→RLc and QL→QLc to reflect the effect of adding C to the circuit.

The transformations are:

(6.26)

79

(6.27)

(6.28)

The net effect of adding the shunt capacitance to make the series inductance

(Lc), series resistance (RLc) and Q (QLc) vary with frequency.

We can demonstrate the reality of equation (6.26) by measuring the inductance over

frequency of some real inductors as shown in figure 6.73. With these particular

inductors Lc is very close to Lo, the low frequency value, at 137 kHz but at 475 kHz Lc

is substantially larger ≈12% to 30%! On 630m we will often have to take fr into

account.

Figure 6.73 - Examples of Lc versus frequency.

80

We can illustrate the effect of C by expressing QL in terms of f/fr. QL=QL1 at a given

frequency f1 is:

(6.29)

equation (6.28) becomes:

(6.30)

Figure 6.74 - F-factor versus f/fr.

We can graph F as shown in figure 6.74 where the dashed line represents the behavior

of QL when resonance is not present. The solid line illustrates the effect of resonance

relative to fr. Note that the maximum point for F, which corresponds to maximum for

81

QL for a given inductor occurs at f/fr≈0.45, roughly half the self-resonant frequency.

The influence of the shunt capacitance becomes significant for f/fr>0.2. Note, equation

(6.30) includes the effect of Ks which changes with frequency.

We can verify that equations (6.25) through (6.30) by comparing calculated values with

measured QL on an actual inductor. Figure 6.75 compares the calculated and

measured QL values. The dots on figure 6.75 are the measured values of QL. If we

insert the measured value for QL=370 @ f1=100 kHz and fr=1.2 MHz into equation

(6.30) we get the dashed line in figure 6.75, i.e. the predicted values.

Figure 6.75 - Measured (dots) versus calculated QL (dashed line).

When designing an inductor or evaluating a existing one, we will want to know L and

QL at the intended frequency. To determine the effect of self resonance inductor we

begin with the low frequency value of L. We then need the value of RL, including the

effects of Ks and Kp, at a given frequency (f1) which is low enough that the self

82

resonance has little effect (i.e. f/fr<0.2). We also need fr which can come from an

impedance sweep like that shown in figures 6.70 and 6.71 if we're working with an

existing inductor but if we are in an early design phase we have nothing to measure.

In fact most amateurs will not have a VNA or other instrument on hand so it may be

necessary to estimate fr from the dimensions of the inductor. There are several ways

to do this, some quite complicated. Appendix A3 has an extensive discussion of the

problems associated with determining fr. It turns out that even when the inductor and a

VNA are available, measuring fr is a tricky business. As a practical matter the best we

usually can do is +/- 10% or so.

Even though we've been modeling the self resonance as due to shunt capacitance, a

close look at figures 6.70 and 6.71 show that the impedance varies more like the input

of a shorted transmission line in some combination with shunt capacitance. In fact

efforts to reduce stray capacitance often do not yield significant increases in fr. What

seems to matter most is the length of the wire in the winding (lw). The lowest self-

resonance (fr) is seen when lw approaches λ/4. A shorted λ/4 transmission line looks

like a parallel resonance.

As a rough estimate of fr we can use the value for a quarter wavelength:

(6.31)

It should be emphasized that this is an approximation, there is significant controversy

as to exactly how to calculate fr. In practice fr is also very sensitive to the details of

inductor lead placement and nearby conductors. Appendix A3 goes into detail on this

subject but for our purposes equation (6.31) provides a reasonable first guess for fr.

We can use the 800uH example from table 6.1 to show the effect of resonance on L

and QL. Lw=175' which, from equation (6.31), says fr=1.406 MHz. For this value of fr,

using equations (6.26) and (6.28), we find that QL=766 at 475 kHz rather than 885. L

at 475 kHz is 903uH, almost 12% higher than the low frequency design value. What

we want is 800uh +/- 5% at 475 kHz. Let's redo the design using a low frequency

value for L about 10% low, i.e. 725uH. Using the equations in section 6.5 or the

graphs we find that lw in now 164' and fr=1.5 MHz. For the new design L=806uH and

QL=757 at 475 kHz. Design iteration is easy and usually converges to a desired value

quickly as it did in this example.

83

References

[1] Watt, Arthur D., VLF Radio Engineering, Pergamon Press, 1967

[2] Wheeler, H.A., Simple Inductance Formulas for Radio Coils, Proceedings of the

IRE, October 1928, Vol. 16, p. 1398

[3] Wheeler, Harold A., Fundamental Limitations of Small Antennas, Proceedings of the I.R.E., vol. 35, December 1947, PP. 1479-1484

[5] Chu, Lan J., Physical Limitations of Omni-Directional Antennas, Journal of Applied Physics, vol. 19, December 1948, pp. 1163-1175

[5] Yaghjian and Best, Impedance, Bandwidth and Q of Antennas, IEEE transactions on Antennas and Propagation , Vol. 53, No. 4, April 2005, pp. 1298-1324

[6] Terman, Frederick E., Radio Engineers Handbook, McGraw-Hill Book Company, 1943. This is a very useful book!

[7] Rudy Severns, N6LF, Conductors for HF Antennas, Nov/Dec 2000 QEX, pp.20-29

[8] Alan Payne, G3RBJ, http://g3rbj.co.uk [9] Michael Perry, Low Frequency Electromagnetic Design, Marcel Dekker, Inc. 1985, pp. 107-116