April 26, 2012 (revised May 24, 2022)

Much has been written about the optimal length of a random wire
antenna (sometimes called a long wire antenna), some of which is
contradictory. An antenna that will tune to an acceptable SWR
(without a balun with a large turns ratio as is common with EFHW
antennas) using a common antenna tuner throughout the amateur
radio bands of 1.8-2, 3.5-4, 7-7.3, 10.1-10.15, 14-14.350,
18.068-18.168, 21-21.45, 24.89-24.99, and 28-29.7 MHz would at a
minimum entail using a quarter wave resonance frequency lower than
the lowest operating frequency and avoiding lengths that are
multiples of a half wavelength on any of the higher frequencies
where the impedance at the feed point would be unacceptably high.

I wrote a program in PowerBASIC to calculate
the gaps in the quarter-wavelength frequency spectrum over the
range 0-30 MHz in 10 Hz steps where such resonances would be
absent and the corresponding required electrical lengths of the
antenna. Running the program produced
the following output:

Gap = 4 kHz at 1.07472 MHz (229 feet)

Gap = 24 kHz at 1.15449 MHz (213 feet)

Gap = 7 kHz at 1.241005 MHz (198 feet)

Gap = 13 kHz at 1.256005 MHz (196 feet)

Gap = 4 kHz at 1.27074 MHz (194 feet)

Gap = 33 kHz at 1.366395 MHz (180 feet)

Gap = 12 kHz at 1.39417 MHz (176 feet)

Gap = 15 kHz at 1.492505 MHz (165 feet)

Gap = 23 kHz at 1.543855 MHz (159 feet)

Gap = 33 kHz at 1.666675 MHz (148 feet)

Gap = 58 kHz at 1.720835 MHz (143 feet)

Gap = 114 kHz at 2.20176 MHz (112 feet)

Gap = 62 kHz at 2.30217 MHz (107 feet)

Gap = 14 kHz at 2.482005 MHz (99 feet)

Gap = 26 kHz at 2.512005 MHz (98 feet)

Gap = 88 kHz at 2.58125 MHz (95 feet)

Gap = 119 kHz at 2.74063 MHz (90 feet)

Gap = 41 kHz at 2.990675 MHz (82 feet)

Gap = 83 kHz at 3.069625 MHz (80 feet)

Gap = 376 kHz at 3.311875 MHz (74 feet)

Gap = 436 kHz at 3.930425 MHz (63 feet)

Gap = 352 kHz at 4.341005 MHz (57 feet)

Gap = 125 kHz at 4.604335 MHz (53 feet)

Gap = 100 kHz at 5.00001 MHz (49 feet)

Gap = 175 kHz at 5.1625 MHz (48 feet)

Gap = 860 kHz at 5.792505 MHz (42 feet)

Gap = 752 kHz at 6.62375 MHz (37 feet)

Gap = 1609 kHz at 8.22951 MHz (30 feet)

Gap = 1416 kHz at 9.792 MHz (25 feet)

Gap = 1720 kHz at 11.585 MHz (21 feet)

Gap = 1505 kHz at 13.2475 MHz (19 feet)

In general, wider gaps are preferred. You can think of the gap as the accuracy to which the fundamental resonance frequency of the antenna must be adjusted to avoid high SWR at higher frequencies within the amateur bands.

The shortest antenna that has a chance of tuning all bands has an electrical length of 143 feet and should be trimmed to give a quarter wave resonance at 1.72 MHz. If one were to forgo 160 meter (1.8-2 MHz) operation, an antenna with an electrical length of 74 feet should suffice and should be trimmed to give a quarter wave resonance at 3.31 MHz. These values assume that the antenna tuner is placed at the feed point of the antenna and that a suitable ground or counterpoise is provided.

If there is a feedline between the antenna and the tuner, it is better to design the antenna to be resonant at the lowest desired frequency of operation, and then choose the feedline length between the antenna and the tuner to bring the total electrical length to one of the above values. Suppose, for example that one wants the antenna to operate at the center of the 80 meter band (3.75 MHz). Then the antenna must have an electrical length of 66 feet, and suitable feedlines would have electrical lengths of 8, 14, 16, 24, 25, 41, 46, 77, ... feet. The required physical lengths are shorter as determined by the velocity factor of the feedline.

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