Resonant Lengths
We always have been told that the HF amateur radio bands are harmonics of the 80m-band. But is this really the case?
Let's investigate by determining for every band's center frequency, fc, the harmonic resonant length of a 0.8mm soft-PVC-insulated 4mm² copper wire at 16.75m (55ft) height above a typical city lot (σ = 1mS/m; εr = 5). Bare copper wire at higher heights and above better ground would yield longer lengths, but this does not matter for the validity of the reasoning that will follow.
Even Harmonics
For a reason that will become evident in a brief moment, we will look at the even and odd harmonics in seperate groups, starting with the even harmonics.
What we immediately notice is that the lower band-edges, are exactly harmonical, being integer multiples of 7.000MHz. All even harmonic bands also have similar relative bandwidths; higher harmonic bands become proportionally wider. The geometric center frequencies of the even harmonic bands are therefore also more or less harmonical and their resonant lengths almost equal. (Note: The FM-portion of the 10m-band has been temporarily disregarded.) This is reflected in a low value for the standard deviation (st.dev.) of the resonant lengths. In conclusion, it must be relatively easy to design an off-center-fed dipole for this set of even harmonics, provided we find a feed position with comparable feed impedances for all of these frequencies.
In this example, the geometric mean (as opposed to the arithmetic mean) of the even harmonic resonant lengths is 40.66m, corresponding to a seventh harmonic frequency of 24.912MHz.
band (m) |
fl (MHz) |
fu (MHz) |
fc (MHz) |
harmonic |
length (m) |
Δ (m) |
40 |
7.000 |
7.200 |
7.099 |
2 |
40.52 |
-0.14 |
20 |
14.000 |
14.350 |
14.174 |
4 |
40.75 |
0.09 |
15 |
21.000 |
21.450 |
21.224 |
6 |
40.87 |
0.22 |
10 |
28.000 |
29.200 |
28.594 |
8 |
40.49 |
-0.17 |
|
|
|
|
geometric mean |
40.66 |
|
|
|
|
|
st.dev. |
0.16 |
|
Odd Harmonics
band (m) |
fl (MHz) |
fu (MHz) |
fc (MHz) |
harmonic |
length (m) |
XCL (Ω) |
80 |
3.500 |
3.800 |
3.647 |
1 |
38.68 |
-j91 |
30 |
10.100 |
10.150 |
10.125 |
3 |
42.76 |
+j241 |
17 |
18.068 |
18.168 |
18.118 |
5 |
39.89 |
-j153 |
12 |
24.890 |
24.990 |
24.940 |
7 |
40.61 |
-j12 |
|
|
|
|
geometric mean |
40.46 |
|
|
|
|
|
st.dev. |
1.48 |
|
Let us continue and have a look at the odd harmonics. The whole story is completely different now; Eventhough 3.500MHz is exactly one half of 7.000MHz, the resonant length of the 80m-band is, at its geometric center frequency, much shorter than that of the even harmonic bands. This is due to the fact that 80m is the band with the broadest relative bandwidth (8.2%), easily surpassing the 5.9% of the entire 10m-band (28.000-29.700MHz).
30m is equally problematic, being not even a real harmonic. Its resonant length is much longer than the mean of 40.66m for the even harmonics.
Finally, we see that 17 and 12m that behave very well. The resonant length of the 12m-band actually nearly coincides with the mean resonant length of the even harmonic bands.
Remember the problems that occurred with the OCF antennas discussed in the previous literature survey? This id no longer a suprise as above tables reveal that the non-harmonic frequency bands are the underlying cause for the partial 80m coverage and the complete absence of 30m-band coverage.
Read on to see how both problems can be solved.