Chapter 6 Design and Fabrication of High Q Loading Inductors

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 antenna 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 primarily with cylindrical single layer air wound coils using round wire which are the most common. Examples of flat spiral or "pancake" inductors, toroidal inductors, non-circular coil forms and basket-weave windings are mentioned. This chapter has both practical advice and a few finer points. Even more information can be found in appendices TBD. For those who don't need all the details, several sets of pre-calculated design graphs are provided from which values for coil dimensions, turn number, etc, can be read directly.

Much of the earlier discussion assumed relatively low QL values, 200-400. However, with careful design, we can do much better when the inductor dimensions are increased. Large inductors 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 capacitive reactance (Xi) at the feedpoint, i.e. XL=Xi, where XL is the inductive reactance of the loading coil needed for resonance:

(6.1)

(6.2)

In this equation 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 the resulting Xi!

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.

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 d → diameter of the 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

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

(l/D) → coil length/diameter ratio

(d/ δ)→ wire diameter in skin depths 4

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 dimensions, number of turns and wire size. A very useful equation for a single layer circular winding comes from Harold Wheeler's 1928 IRE paper[2]:

[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 . This 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 taper. 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 in large inductors is "self-resonance". If you measure the impedance of an inductor over a wide range of frequencies there will be a number of self-resonances. As shown in figure 6.4, if you start at a low frequency and go up, the measured inductance initially falls slightly (≈1-2%) but then levels out. However, as you go higher in frequency the inductance increases rapidly. The design graphs in section 6.4 assume operation <0.1fr where the effect is small. The fr limitations due to winding length lw indicated on the graphs are explained in section 6.16. As the value of L increases fr decreases. At some point self resonance begins to seriously reduce QL setting an upper frequency limit at which the inductor is useful. This effect can be a

5 serious problem because the design becomes significantly more complicated as explained in section 6.16.

Figure 6.4- Inductance versus frequency.

Besides these problems we need to recognize that we may not even know the exact value needed 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 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 bandwidths[3, 4, 5] and 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. Besides tapping, there are other options for adjusting inductance as outlined in section 6.9. As a practical matter the 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)

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[6].

6.4 Geometric ratios

Design graphs and equations can be made more general by using geometric ratios as variables. For example: (d/c) → conductor diameter/turn spacing ratio

(l/D) → coil length/diameter ratio d/δ → wire diameter in skin depths

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

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

6.5 Inductor design using graphs

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

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

In each graph the parameter for the different contours is the wire size. The difference between the graph sets is the value for l/D (0.5, 1 and 2). In the first three graph sets d/c=0.5 because this is usually close to optimum. To design a physically smaller inductor for a given inductance, in the last set of graphs d/c= 0.8, representing a much closer turn spacing but with some sacrifice in QL.

For a given value of 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. Alternately, 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 which will give you the wire size. 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. Another approach, if you happen to have a spool of wire of a given size, is to 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.

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 these examples and Table 6.1 is a summary of the 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= 860 782 730 555

From table 6.1 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. The best value for QL is 860 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 in the winding. 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 12 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" and lw is reduced to 150'. However, QL is much lower, 555 compared to 865! This represents a fundamental trade-off, the larger the inductor, the higher QL. Note also, that a larger wire size makes the inductor physically larger. 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 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, in a given inductor, is lower QL. We can express this as a ratio which varies as the square root of the frequency 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 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). In a given inductor, replacing copper with aluminum:

(6.8)

QL drops by ≈20% when going from copper to aluminum. The QL graphs can be scaled by that factor.

Figure 6.9 - Wire length versus inductance.

Figure 6.10 - Diameter in inches versus inductance.

Figure 6.11 - Number of turns versus inductance.

Figure 6.12 - QL versus wire size.

Figure 6.13 - Wire length versus inductance.

Figure 6.14 - Diameter in inches versus inductance.

Figure 6.15 - Number of turns versus inductance.

Figure 6.16 - QL versus wire size.

Figure 6.17 - Wire length versus inductance.

Figure 6.18 - Diameter in inches versus inductance.

Figure 6.19 - Number of turns versus inductance.

Figure 6.20 - QL versus wire size.

Figure 6.21 - Wire length versus inductance.

Figure 6.22 - Diameter in inches versus inductance.

Figure 6.23 - Number of turns versus inductance.

Figure 6.24 - QL versus wire size.

6.6 Do it yourself graphs and calculations

The graphs in section 6.5 may not be adequate for everyone. The equations from which the graphs were derived can be used directly to calculate a particular case or inserted into a spreadsheet. Here are two examples of a direct calculation.

Example 1

[1] Choose values for L, l/D, d/c and d. Let L=800uH, d=0.081" (#12 wire), l/D=d/c=0.5

[2] Calculate the diameter D from:

[inches] (6.9)

→D=14.7", l=7.35"

[3] Calculate the number of turns N from:

(6.10)

→N=45 turns

[4] Calculate wire length lw from:

[inches] (6.11) →lw=175'

[5] Calculate QL from:

(6.12)

Rdc is the DC resistance of the wire:

(6.13)

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

(6.14)

From a wire table Rdc=0.001588 Ω/ft. lw=175'.

→Rdc=0.278Ω

From table 6.2, at 475 kHz:. →Ks=5.6

From table 6.3: →Kp=1.78

From eqn (6.14): →RL=2.77Ω

From eqn (6.12), at 475 kHz: →QL=860

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 5.6 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. 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 sections 6.12 through 6.14 and appendix A3.

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

Example 2

For this example let's assume we have plastic bucket for the coil form. We know the diameter, D=10.8", the desired inductance, L=800uH and the wire, #12 THHN (d=0.081"). We want to leave the insulation on the wire with the turns close together. This means d/c≈0.8. We need to know the number of turns, N, and the length of the wire, lw.

[1] Calculate N from:

(6.15)

→N=53

[2] Calculate lw:

(6.16) →lw=150'

[3] Calculate l and l/D:

(6.17) →l=5.34" and l/D=0.5

[4] From a wire table Rdc=0.001588 Ω/ft: →Rdc=0.238Ω

From table 6.2, at 475 kHz: →Ks=5.6

From table 6.3:→Kp=3.22

And: →RL=4.29Ω

At 475 kHz: (6.18) →QL=556

6.7 Inductor fabrication

Figure 6.25 - PVC cage coil form. PVC pipe combined with standard fittings provides an easy, inexpensive and flexible way to construct a coil form of arbitrary size as shown in figure 6.25. 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! The eight vertical supports used 3/4" pipe with slots cut at intervals (=c) to hold the wire. Figure 6.26 shows a practical way to cut the slots. Attaching the supports to a board makes it much easier to align all the slots and mounting holes.

Figure 6.26 - Cutting the wire slots.

Figure 6.27 - Winding the inductor. 35

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" requiring some manual tweaking of the turns. Not very pretty but perfectly functional.

Figure 6.28 - Finished inductor. PVC cage inductor 2. 36

Figure 6.4 is the measured inductance of this coil. At low frequencies L=800 uH which drops to 783 uH at 137 kHz and rises to 928 uH at 475 kHz. The measured QL 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 860 but the measured value is ≈680, substantially lower! The changes in L and QL, as a function of frequency, demonstrate the effect of self resonance.

Figure 6.29 - Example inductor measured QL.

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≈0.84. Because the bucket is smooth with some taper six strips of double sided mounting tape were attached vertically before winding to hold the wire in place. 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 were

37 chosen to accommodate the bucket and the winding spacing is close (d/c≈0.84), so QL=350 is not very high. The use of stranded wire also reduced QL slightly.

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

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 bottom and 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.

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 was 1.45mH!

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.

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. 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.

Figure 6.35 - Example of a flat or pancake winding.

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

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.

Figure 6.37 - Pat W5THT, foil wound lightweight inductor.

For 600m and 2200m a foil thickness of 0.005"→0.010" would be close to optimum. One could also purchase thin plastic sheet and roll it to make the coil form.

6.8 Effect of coil form shape

Up to this point a cylindrical coil form has been assumed but other shapes are possible and may be more practical, especially for a 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 to 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':

(6.19)

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

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 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.

6.9 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.

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 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] (6.20)

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.

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.

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 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 a 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 (pitch dark and 48 snowing) many variometers have some form of remote tuning. As figures 6.40-6.51 show there's no shortage of innovation. KB5NJD has the answer shown in figure 6.40: use an inexpensive TV antenna rotor! Some of these examples are almost works of art.

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

Figure 6.40 - Jay, W1VD, WD2XNS variometer.

Figure 6.41 - Jay, W1VD, WD2XNS variometer.

Figure 6.42 - Jay, W1VD, WD2XNS variometer enclosure.

Figure 6.43 - Steve, KK7UV, variometer.

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

Figure 6.45 - Laurence, KL7L, variometer drive.

Figure 6.46 - Laurence, KL7L, variometer enclosure.

Figure 6.47 -Neil, W0YSE, variometer.

Figure 6.48 - W0YSE tuning unit circuit diagram.

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.

As shown in figures 6.50 and 6.51, Ralph, W5JGV, has built lovely variometers using PVC pipe fittings. Inductance of these variometers is not large but they work fine for fine adjustment in combination with a tapped main inductor.

Figure 6.50 -Ralph, W5JGV variometer.

Figure 6.51 - Ralph, W5JGV variometer

6.10 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:

(6.21)

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 (6.21) to create the graphs for Io in figures 6.52 and 6.53. Note that in figures 6.52-6.57, L is the length of the top-wire in feet.

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

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

Vo is the voltage at the feedpoint:

(6.22)

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.

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

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

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 dissipate PL without damage! 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:

(6.23)

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.

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

6.11 Enclosures

Most amateurs will use some form of large plastic box for the tuning inductor enclosure. These are inexpensive, readily available in a very wide range of sizes and have little or no effect on QL. One shortcoming of typical plastic containers is their susceptibility to degradation from the UV in sunlight. 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.

6.12 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! 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.

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

[m] (6.24)

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] (6.25)

At 137 kHz δ=7.03 mils (0.00703") and at 475 kHz δ=3.78 mils (0.00378").

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).

(6.26)

Where:

[Ω] (6.27)

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! You can also find Rdc from a wire table where Rdc is typically listed in Ohms per thousand feet.

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 and δ .

6.13 Skin effect factor Ks

The skin effect factor Ks for an arbitrary conductor diameter can be a bit complicated (see appendix A3) 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:

(6.28) 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.

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

The d/ δ ratio for common wire sizes is summarized in table 6.5.

Table 6.5 -Wire diameters in skin depths (d/δ) frequency= 137 kHz 475 kHz Wire # d [mils] d/δ d/δ 8 128.5 18.28 34.04 10 101.9 14.50 26.99 12 80.08 11.39 21.21 14 64.1 9.12 16.98 16 50.8 7.23 13.46 18 40.3 5.73 10.68

Values of Ks for common wire sizes was summarized in table 6.2.

6.14 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 allows us to separate Ks and Kp in the analysis!

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 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.

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

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

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

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

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

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

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

6.15 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 with single wires this small is high Rdc. 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 6.6.

Table 6.6 - Litz wire examples. frequency equiv. Cir. no. strand nom. Rdc AWG mil strands AWG O.D. Ω/1000' area 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. But there are downsides! The cost is much higher than equivalent solid wire and there is the problem of reliably soldering 1600+ individually 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.

Figure 6.67 - Examples of litz construction.

Litz wire can be useful but we cannot use just any litz. Michael Perry[9] has published a formal analysis of litz wire construction which contains a cautionary tale!

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

Figure 6.68 is taken from Perry. It compares Rac between a solid conductor and a litz conductor which varies with the size of the individual wires (d/δ) or when the frequency is changed. Litz 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 individual strands 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. 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!

6.16 Self resonance

In section 6.3 the variation of inductance with frequency due to self resonance was pointed out. The graphs and equations in sections 6.4 and 6.5 do not take self resonance into account. Up to this point it might appear 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

Figure 6.69-Example of QL variation with frequency.

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. The measured amplitude and phase of the impedance of a 2.2mH inductor is graphed in figures 6.70 and 6.71.

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. The traditional explanation has been that there is stray capacitance across the inductor which resonates with the inductance as shown in figure 6.72B.

Figure 6.70 - Inductor impedance versus frequency .

Figure 6.71 - Inductor phase versus frequency.

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:

(6.29)

It is mathematically convenient to transform the parallel equivalent circuit in 6.72B 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 showing the effect of self-resonance are:

(6.30)

(6.31)

(6.32)

The net effect of adding the shunt capacitance to make the series inductance (Lc), series resistance (RLc) and Q (QLc) vary with frequency as a function of f/fr.

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.

We can illustrate the effect of C by expressing QL in terms of f/fr. QL=QL1 at a given frequency f1 is:

(6.33) equation (6.28) becomes:

(6.34)

Figure 6.74 - F-factor versus f/fr.

Eqn (6.34) is graphed 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 QL,

80 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, equations (6.31-6.33) includes the effect of Ks which changes with frequency.

We can verify that equations (6.25) through (6.32) 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 operating frequency. To determine the effect of self-resonance begin with the values for L and RL calculated as shown earlier. Then recalculate these value using equations (6.30-6.34). fr may be determined from an impedance sweep if 81 the inductor exists but in an initial paper design there's 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. 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.

Even though we've been modeling the self resonance as due to shunt capacitance, a close look at figures 6.70 and 6.71 shows that the impedance behaves more like the input of a shorted transmission line in some combination with shunt capacitance. What seems to matter most is the length of the wire in the winding (lw). The lowest frequency fr occurs when lw approaches λ/4 at the operating frequency, i.e. quarter- wave resonance. The impedance of a shorted λ/4 transmission line behaves like a parallel resonance.

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

(6.35)

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 (6.35) provides a reasonable first guess for fr.

We can use the 800uH example in table 6.1 to demonstrate the effect of self- resonance on L and QL and the use of design iteration to adjust the inductance value.

The goal is 800uh +/- 5% at 475 kHz. From column 1 of table 6.1:

[1] Lw=175' . From eqn (6.35) →fr=1.406 MHz

[2] QL=860. From eqn (6.32) →QLc=762

[3] L=800uH. From eqn (6.30) →Lc=903 uH

At 475 kHz L=903 uH which is about 12% high. At this point we can iterate the design by starting with a lower initial value for L, -10%?

[1] L=725 uH

[2] Using the equations in section 6.5 or the graphs we find that →lw=164'

[3] Recalculating →fr=1.5 MHz, → L=806uH and →QL=757 @ 475 kHz.

The new design meets the inductance requirement. Iteration is easy and usually converges to a useful solution quickly.

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