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Signals & Wireless (RF)

How to calculate antenna length

An antenna works best when its length is matched to the wavelength of the signal. Find the wavelength with λ = c ÷ f, or 300 ÷ f in MHz for metres. A quarter-wave antenna is a quarter of that, trimmed about 5% shorter in practice: about 16.5 cm at 433 MHz, about 3 cm at 2.4 GHz.

Wavelength

A radio wave travels at the speed of light, about 300,000,000 m/s. In one cycle it covers one wavelength:

λ = c over f = 300 over f in MHz metres
FrequencyWavelengthQuarter-wave
27 MHz (CB)11.1 m2.78 m
100 MHz (FM)3 m75 cm
433 MHz69 cm17.3 cm
2.4 GHz (Wi-Fi)12.5 cm3.1 cm

Along a real wire the wave goes a little slower, and the ends add a little, so cut about 5% shorter than the table, then trim for the best match.

Quarter-wave and half-wave

A half-wave dipole is two quarter-wave arms fed in the middle. At resonance it looks like a plain resistance of about 73 Ω. A quarter-wave monopole, a single whip, needs a ground plane beneath it, like a car roof or a board's ground copper, which acts as the missing half. It looks like about 36 Ω.

That resistance, the radiation resistance, is the part of the antenna that sends power into the air. At resonance the antenna's reactance cancels, leaving only it, so the transmitter can match to it.

Live experiment

Tune a short whip

A short whip at 27 MHz looks like 10 pF in series with 10 Ω of radiation resistance. Step the loading coil from nothing to 5 µH: near 3.5 µH it cancels the whip's 590 Ω of reactance and the radiating part gets its share. Then try 30 MHz.

Try it yourself: free lesson

“A short antenna” is lesson 5 of 6 in Carrier Cove, Signals & Wireless (RF) World 1. Free with an account. Lessons unlock in order, so you’ll do the 4 before it first.

Short antennas and loading coils

A quarter-wave at 27 MHz is 2.8 m, far too tall for a car. A 1 m whip is too short to resonate: electrically it's a capacitor, about 10 pF, in series with a small radiation resistance, about 10 Ω. At 27 MHz, that capacitance is about 590 Ω of reactance standing in the way, so almost no power gets through.

A loading coil in series cancels it. It needs +590 Ω of its own:

L = 1 over 4π² f² C ≈ 3.5 µH at 27 MHz with 10 pF

That's the bulge at the base or partway up a CB aerial.

Phones squeeze antennas for several bands into a few centimetres with the same tricks: folding, loading and matching.

Free lesson

Now build it yourself

Each lesson comes in three short parts: see the idea, try it on a live board, then build it yourself, with quick checks as you go.

Try it yourself: free lesson

“A short antenna” is lesson 5 of 6 in Carrier Cove, Signals & Wireless (RF) World 1. Free with an account. Lessons unlock in order, so you’ll do the 4 before it first.

Or start from the very first lesson, no account needed

Work it out

Practice questions

Three quick questions from the lessons. Pick an answer to check it.

  1. Question 1 of 3At 27 MHz a 10 pF whip is about…
    Show the answer

    Answer: 590 Ω of capacitive reactance. XC = 1 ÷ (2π × 27 × 10⁶ × 10 × 10⁻¹²) ≈ 589 Ω. Big, compared with the 10 Ω that does the radiating.

    • Why not “0.59 Ω”? XC = 1 ÷ (2π × f × C) = 1 ÷ (2π × 27 MHz × 10 pF) ≈ 590 Ω, a thousand times more than 0.59 Ω.
    • Why not “59 kΩ”? 59 kΩ is a hundred times too big. At 27 MHz, 10 pF is about 590 Ω.
    • Why not “10 Ω”? 10 Ω is the whip's radiation resistance, not its capacitive reactance.
  2. Question 2 of 3A loading coil makes a short whip work by…
    Show the answer

    Answer: Cancelling its capacitance at one frequency. +jX from the coil against −jX from the whip.

    • Why not “Amplifying the signal”? A coil adds no power. It only cancels the capacitance standing in the way of the radiation resistance.
    • Why not “Blocking DC”? Blocking DC isn't the problem with a short whip. Its capacitance at the radio frequency is.
    • Why not “Making it longer”? A coil doesn't make the whip physically longer. It cancels the whip's reactance so it acts resonant.
  3. Question 3 of 3Move the same loaded whip to 30 MHz. It now…
    Show the answer

    Answer: Is off tune: the coil only cancels the capacitance at 27 MHz. The coil's reactance rises and the whip's falls with frequency, so they only cancel at one frequency.

    • Why not “Works the same”? The coil's and the whip's reactances change with frequency in opposite ways, so they only cancel at one frequency.
    • Why not “Works better”? Moving off the tuned frequency always makes the cancellation worse, not better.
    • Why not “Becomes a long antenna”? The whip's length doesn't change. It's still short, and now it's off tune too.
Keep practising: free lesson

“A short antenna” is lesson 5 of 6 in Carrier Cove, Signals & Wireless (RF) World 1. Free with an account. Lessons unlock in order, so you’ll do the 4 before it first.

Questions people ask

How do you calculate the length of a quarter-wave antenna?

Find the wavelength, λ = 300 ÷ f in MHz, giving metres, and take a quarter. Then trim about 5% because the wave travels a little slower along a real conductor. At 433 MHz, λ is 69 cm, a quarter is 17.3 cm, and a practical whip is about 16.5 cm.

What is the difference between a quarter-wave and a half-wave antenna?

A half-wave dipole is two quarter-wave arms fed in the middle and needs nothing else, with about 73 Ω at its centre. A quarter-wave monopole is one arm over a ground plane, which acts as the missing half; it presents about 36 Ω.

Can an antenna be shorter than a quarter wavelength?

Yes, but a short antenna looks like a capacitor with a small radiation resistance, so on its own it takes very little power. A loading coil in series cancels the capacitance at one frequency, making it resonant. It works, though usually less efficiently than a full-size antenna.

How long is a 2.4 GHz antenna?

The wavelength at 2.4 GHz is 12.5 cm, so a quarter-wave is about 3.1 cm, a little less once trimmed. That's why Wi-Fi and Bluetooth antennas fit inside phones and small boards.

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