Methods for Calculating Loading Coils

Table of Contents

• Method A: K1PLP’s Lookup Chart
• Method B: 66Pacific’s Online Calculator
• Method C: Calculate $X_L$ from $Di{m}_A$ and $Di{m}_B$
  • Example 1: Loads for a full length 17m/20m trapped dipole
    • WARNING
• Comparison

So far, I’ve seen three methods for calculating shortening coils for dipoles. I’ve successfully used the first, K1PLP’s lookup chart, for the design of a homebrew 40/30/20/17m trapped dipole, measuring 50 feet. My results were good, although I intend to build a version 2 of my antenna. The 20m section is quite short, while both the 30m and 40m were longer than I intended by about a meter or so.

I haven’t technically tried Method C, however I get similar reactance values (and subsequently similar inductance and capacitance values) from both Method C and A.

Method A: K1PLP’s Lookup Chart

I simply used the loading coil chart from Chapter 9 of the ARRL Antenna Book, Vol 2 to calculate $X_L$. See figure 9.51. Given $Di{m}_A$ and $Di{m}_B$, the chart will tell you $X_L$, inductive reactance for the traps. This chart is also available in an article from Jerry Hall, K1PLP, in QST magazine, September 1974].

I highly recommend ARRL’s Antenna Book. Chapter 9 in particular has several sections dedicated to dipoles and loading techniques. Here is a link for the full ARRL Antenna Book.

Method B: 66Pacific’s Online Calculator

The math to calculate inductance, given $Di{m}_A$ and $Di{m}_B$ is quite involved (from QST magazine, September 1974):

$$L_{\mu H}=\frac{{10}^6}{68{\pi }^2{f}^2}\{\frac{[ln\frac{24(\frac{234}{f}-B)}{D}-1][(1-\frac{fB}{234}{)}^2-1]}{{\displaystyle \frac{234}{f}}-B}-\frac{[ln\frac{24(\frac{A}{2}-B)}{D}-1][(\frac{\frac{fA}{2}-fB}{234}{)}^2-1]}{\frac{A}{2}-B}\}$$

Where:

• $L_{\mu H}$ = inductance required for resonance
• $f$ = frequency, megahertz
• $A$ = overall antenna length in feet
• $B$ = distance from center to each loading coil
• $D$ = diameter of radiator in inches

Fortunately, 66pacific.com has published a utility for this calculation.

Method C: Calculate $X_L$ from $Di{m}_A$ and $Di{m}_B$

Luiz Duarte Lopes, CT1EOJ , has published an article, Designing a Shortened Antenna in the October 2003 edition of QST.

Example 1: Loads for a full length 17m/20m trapped dipole

For a resonant two-band dipole on 17m and 20m, the inner part of the antenna will be 8.5m (halfwave on 17m). Traps will be placed at ends of the 17m section and since we want at least 1 meter extending past the traps for the 20m section, this means we want very little loading (none if possible) and we would expect a low inductive reactance, $X_L$, and consequently a low inductance for our traps.

This is the basic equation which we will use to calculate the inductive reactance required by the loading coils for our antenna:

$$X=-j{Z}_{0}cot\beta$$

We need to calculate reactance at the point on the antenna where the loading coil begins $X_1$ and ends $X_2$. Coil begin point, $X_1$, is closer to the end and coil end point, $X_2$, is closer to the center/feedpoint.

From $X_1$ and $X_2$, we can calculate $X_L$:

$$X_L=X_2-X_1$$

or

$$X_L=-j{X}_2-(-j{X}_1)$$

Starting with our antenna dimensions for 20m:

• $Di{m}_A$ = 98% (shorten antenna by 2% on 20m)
• $Di{m}_B$ = 85% (because traps will be placed at edge of 17m, and 17m/20m = 0.85)

We must calculate the distance from the end of the antenna to the loading coil point, in degrees.

• ${\beta }_1=90°-90°[Di{m}_B+(1-Di{m}_A)]$
  • ${\beta }_1=90°-90°[0.85+(1-0.98)]$
  • ${\beta }_1=11.7°$
• ${\beta }_2={\beta }_1+90°[1-Di{m}_A]$
  • ${\beta }_2=11.7°+90°[1-0.98]$
  • ${\beta }_2=13.5°$

Next we must calculate the characteristic impedance of a one-wire transmission line, $Z_0$, based on wire diameter and height above ground. Units for $h$ and $d$ can be whatever, as long as they’re the same for both.

• $Z_0=138log\frac{4h}{d}$, where $h$ = electrical height above ground and $d$ = wire diameter
• $Z_0=138log\frac{4\ast 20ft}{d}$
  • $Z_0=138log\frac{4\ast 20ft}{14awg}$
  • $Z_0=138log\frac{4\ast 6096mm}{1.62814mm}$
  • $Z_0=572\Omega$

Now to calculate $X_1$ and $X_2$:

• $X_1=-j{Z}_{0}cot{\beta }_1$
  • $X_1=-j572(cot11.7°)$
  • $X_1=-j2763$
• $X_2=-j{Z}_{0}cot{\beta }_2$
  • $X_2=-j572(cot13.5°)$
  • $X_2=-j2385$

Finally, we can calculate the inductive reactance required by our shortened antenna: ((\X_L\).

• $X_L=-j{X}_2-(-j{X}_1)$
  • $X_L=-j2385-(-j2763)$
  • $X_L=378\Omega$

WARNING

This example gives us the wrong answer for a 17m/20m dipole, b/c it uses the general band lengths as inputs to $X_L$. Instead, we must select resonant frequencies. For example, instead of 20m, we should select 14.055MHz. And instead of 17m, we should select 18.118MHz.

See my calculator for an example. Calculating using resonant frequencies, 18.118MHz and 14.055MHz gives an $X_L$ of 166Ω.

Comparison

It’s helpful to see the inductive reactance, $X_L$, from which you then calculate inductance. $X_L$ is frequency independent, while $L$, inductance, is frequency dependent. Both K1PLP’s chart (method 1) and CT1EOJ’s calculations (Method 2) provider inductive reactance, while 66paficic’s calculator, based on K1PLP’s calculations, does not.

I have only attempted trap dipole design and construction using K1PLP’s lookup chart. I can also verify that Method 3 produces reactance values similar to Method 1. 66pacific’s calculator (Method 2) produces reactance values that are a bit off from my calculations and it’s difficult to evaluate the math, since their calculator only provides inductance and skips over reactance.

Method 3’s main difficulty is with calculating ${\beta }_1$ and ${\beta }_2$ since the geometry can be tricky to get right.

Method 1 is just a lookup chart, which seems accurate enough. But when dealing with extreme inductances/capacitances (at the extremes of low or high inductances), small errors can produce unacceptable deviations in antenna dimensions.