Quarter-wave vertical calculator
A quarter-wave vertical is a single λ/4 radiator fed against a ground plane of radials, which supply the missing half of the antenna. The radiator is cut a few percent short to cancel the end effect, so L = k · λ/4 with k ≈ 0.95, while the radials stay near a full quarter wave. This calculator gives both lengths.
The slider covers 1 MHz to 3 GHz. 146 MHz sits in the 2 m amateur band.
0.95 matches the classic cutting rule. Thin wire in the clear runs 0.96 to 0.97, thick tubing lower.
1.00 leaves the radials at a full quarter wave, about 5% longer than a k = 0.95 radiator.
Three or four are enough for an elevated ground plane. A ground-mounted vertical wants 16 or more.
Horizontal radials feed near 36 Ω. Drooping them about 45° brings the feedpoint up to roughly 50 Ω.
A ground-plane vertical seen from slightly above. The bright element is the radiator, the thinner spokes are the radials, and the dashed ellipse traces the circle their tips sweep. Both lengths are drawn at one scale, so on the page too the radials are the few percent longer. Change the droop and they swing down; change the count and the star fills in. Absolute size is not shown: a 1 MHz and a 3 GHz antenna draw the same and only the labels move.
k = 0.95 is a cutting figure, not a measurement. Element diameter, mounting height, the mast under the antenna and nearby metal all shift resonance, usually downward. Cut long, trim to the measured VSWR minimum, and read the impedance figures as indicative rather than predicted.
How it works
- 01
Start from the quarter wave
Find the free-space wavelength, λ = c / f, and divide it by four. At 146 MHz λ is 2.053 m, so a quarter wave is 51.3 cm. That is the electrical target for both the radiator and the radials, before any correction.
- 02
Shorten the radiator for the end effect
The open top end stores charge against its surroundings, and that capacitance makes the element resonate as though it were longer than it measures. Multiplying by k ≈ 0.95 takes it back out, which is where the advice to cut 5% short comes from. Thin wire in the clear runs a shade higher, 0.96 to 0.97. Thick tubing and fat whips hold more end capacitance and want a smaller k.
- 03
Cut the radials a little longer
The radials are the other half of the antenna, not an accessory. Practice leaves them at a full electrical quarter wave, which with k = 0.95 on the radiator makes them about 5% longer than it. Longer radials pull the resonance down and the feed resistance up, so they double as a second tuning control. Some designs run them up to 12% over the radiator.
- 04
Pick the number of radials
Three or four are enough for an elevated ground plane, where each radial is a resonant element sitting in clear air. A ground-mounted vertical is a different problem. There the radials shield the field from lossy soil, and adding wires keeps helping. Sixteen is a sensible amateur floor, and the FCC treats 120 buried radials as an excellent ground system for AM broadcast towers.
- 05
Droop the radials, then trim to resonance
A λ/4 monopole over a perfect ground plane feeds at about 36.5 Ω. Sloping the radials down roughly 45° raises that to near 50 Ω, a direct match for coax, and tips the radiation a little further toward the horizon. Then cut everything 2 to 3% long, watch the VSWR minimum and shorten the radiator. About 1% of length moves resonance by about 1% in frequency.
Formulas
- L — radiator length, m
- k — end-effect factor, ≈0.95 for a wire element
- λ — free-space wavelength, m (λ = c / f)
- c — speed of light, 299 792 458 m/s
- f — frequency, Hz
- Lr — length of one radial, m
- r — radial length factor, 1.00 = a full electrical quarter wave
- at r = 1.00 and k = 0.95 the radials come out 1/k − 1 ≈ 5.3% longer than the radiator
- the long-standing amateur rule for a quarter-wave element
- in metres: L ≈ 71.3 / f (MHz), since 234 · 0.3048 = 71.32
- it implies k ≈ 0.952, a shade longer than k = 0.95 (which gives 71.2 / f)
Worked example
- λ = 299 792 458 / 146×106 = 2.053 m
- λ/4 = 0.5133 m
- radiator: L = 0.95 · 0.5133 = 0.4877 m = 48.77 cm (19.2 in)
- cross-check: 234 / 146 = 1.603 ft = 48.85 cm, within 0.2%
- radials: Lr = 1.00 · 0.5133 = 51.33 cm each, four of them
- → one 49 cm whip and four 51 cm radials, drooped 45° for about 50 Ω
FAQ
- Why does a quarter-wave vertical need radials?
- Because they are the other half of the antenna. A monopole is one leg of a dipole, and the ground plane supplies the mirror image of the second leg together with the return path for the feedline current. Without it that current runs back down the outside of the coax braid, which then radiates, detunes the antenna and puts RF where you do not want it.
- How many radials does a quarter-wave vertical need?
- Three or four for an elevated ground plane, where each radial is a resonant element in clear air. A ground-mounted vertical is different. There the radials shield the near field from lossy soil, and more wires keep helping. Sixteen is a reasonable amateur minimum, and the FCC calls 120 buried radials an excellent ground system for AM broadcast towers.
- Why do drooping radials give 50 ohms?
- A thin λ/4 monopole over a perfect ground plane shows about 36.5 Ω, half the 73 Ω of a dipole, because it radiates into half the space. Sloping the radials downward moves the antenna toward a vertical dipole and raises the feedpoint resistance. Somewhere near 42 to 45° it passes through 50 Ω, which is why the classic ground plane droops its radials.
- Does the diameter of the radiator matter?
- Yes, in two ways. A fatter element carries more end capacitance, so it resonates at a slightly shorter physical length and wants a smaller k. It is also wider in bandwidth, which is why commercial VHF ground planes are built from tubing rather than wire. Diameter has almost no effect on gain or pattern.
- How do I tune a quarter-wave vertical?
- Cut long and trim. Add 2 to 3% to the radiator, mount the antenna where it will actually live, then shorten it a few millimetres at a time while watching the VSWR minimum. As a rule of thumb, taking 1% off the length moves resonance up about 1% in frequency. Trim the radiator first, since the radials mostly set the impedance.
The antenna is cut. What it actually covers depends on where you put it. Waveshed drops your vertical onto real elevation data and draws line-of-sight and coverage, free in your browser.
Map the coverage from your site →Related guides
- Antenna patterns & the sensor panel — a quarter-wave vertical is omnidirectional in azimuth, so the built-in omni pattern fits it
- Heights: AGL vs AMSL — an elevated ground plane goes into the simulator as a height above ground, not above sea level
Sources & further reading
- 47 CFR § 73.189 — Minimum antenna heights or field strength requirements ↗ — the FCC ground-system standard for ground-mounted verticals: buried radials at least a quarter wave long, no fewer than 90, with 120 considered excellent
- The ARRL Antenna Book for Radio Communications, 25th ed. (ARRL) ↗ — the standard practical reference for ground-plane verticals, radial systems and the 234 / f cutting rule, in the chapters on dipoles and monopoles and on the effects of ground
- 1/4 Wave Ground Plane Antenna Calculator — M0UKD ↗ — amateur build practice: velocity factor 0.95, radials cut longer than the radiator, and 45° droop for a 50 Ω feed
- Monopole antenna — Wikipedia ↗ — resonant length 0.95 · λ/4 = 71.25 / f (MHz), 36.5 Ω radiation resistance and 5.19 dBi gain over a perfect ground plane
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