Quarter-Wave Antenna Length
l = c × VF / (4 × f)
Quick answer
This calculator solves l = c × VF / (4 × f) for Antenna Length. With Frequency 10 MHz, Velocity Factor 50 %, the result is 3.747 m. It solves in 3 directions, so you can enter the values you have and get the one you need.
Calculator
Formula
Description
A quarter-wave antenna (also called a quarter-wave monopole or ground plane antenna) is one of the most common antenna types in amateur and commercial radio. Its electrical length is one quarter of the wavelength at the operating frequency, adjusted by the velocity factor which accounts for the fact that electromagnetic waves travel slower in conductors than in free space. The velocity factor for bare wire is approximately 0.95, while coaxial elements may have VF of 0.66-0.82 depending on the dielectric material.
Variables
- l — Physical antenna length (m)
- f — Operating frequency (Hz)
- VF — Velocity factor (0-1, typically 0.95 for wire)
Practical Notes
A quarter-wave monopole over a ground plane has approximately 36Ω impedance and requires a ground plane or counterpoise for proper operation. For the 2-meter amateur band (146 MHz), the quarter-wave length is about 0.49 m. For HF bands, antennas become much longer: 40 meters band (7 MHz) needs about 10.2 m. Quarter-wave stubs are also used as notch filters and impedance transformers in feed systems.
Symbols and units
| Symbol | Meaning | Default unit | Accepted units |
|---|---|---|---|
| l | Antenna Length | m | mm, cm, m, km |
| f | Frequency | MHz | Hz, kHz, MHz, GHz |
| VF | Velocity Factor | — | —, % |
Step by step
- 1. What you know: Frequency 10 MHz, Velocity Factor 50 %
- 2. The relation: l = c × VF / (4 × f)
- 3. Substitute: f = 10 MHz, VF = 50 %
- 4. Result: Antenna Length = 3.747 m
Worked examples
Minimum range
Given: Frequency 100 kHz, Velocity Factor 2.5 %
Result: Antenna Length = 18.74 m
Low range
Given: Frequency 1 MHz, Velocity Factor 25 %
Result: Antenna Length = 18.74 m
Typical
Given: Frequency 10 MHz, Velocity Factor 50 %
Result: Antenna Length = 3.747 m
High range
Given: Frequency 100 MHz, Velocity Factor 0.8
Result: Antenna Length = 0.5996 m
Maximum range
Given: Frequency 1 GHz, Velocity Factor 8
Result: Antenna Length = 0.5996 m
Solving for each variable
Solve for Antenna Length
Frequency 100 kHz, Velocity Factor 2.5 % → 18.74 m
Solve for Frequency
Antenna Length 1 mm, Velocity Factor 2.5 % → 1.874 GHz
Solve for Velocity Factor
Antenna Length 1 mm, Frequency 100 kHz → 1.33 × 10⁻⁴ %
How Antenna Length changes with Frequency
| Frequency (input) | Antenna Length (result) |
|---|---|
| 1 MHz | 37.47 m |
| 2.5 MHz | 14.99 m |
| 5 MHz | 7.495 m |
| 7.5 MHz | 4.997 m |
| 10 MHz | 3.747 m |
| 15 MHz | 2.498 m |
| 20 MHz | 1.874 m |
| 40 MHz | 0.9369 m |
| 70 MHz | 0.5353 m |
| 100 MHz | 37.47 cm |
The same result in other units
| Unit | Value |
|---|---|
| mm | 3747 |
| cm | 374.7 |
| m | 3.747 |
| km | 0.003747 |
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<p><a href="https://vbatpower.com/formulas/quarter-wave-antenna/">Quarter-Wave Antenna Length</a> Calculator by vbatpower.com</p>Frequently asked questions
How do you calculate Quarter-Wave Antenna Length?
Use l = c × VF / (4 × f). With Frequency 10 MHz, Velocity Factor 50 %, Antenna Length works out to 3.747 m.
What is the formula for Quarter-Wave Antenna Length?
The formula is l = c × VF / (4 × f). A quarter-wave antenna (also called a quarter-wave monopole or ground plane antenna) is one of the most common antenna types in amateur and commercial radio.
How do you find Frequency when you know the other values?
Rearrange for Frequency: with Antenna Length 1 mm, Velocity Factor 2.5 %, Frequency is 1.874 GHz.
How do you find Velocity Factor when you know the other values?
Rearrange for Velocity Factor: with Antenna Length 1 mm, Frequency 100 kHz, Velocity Factor is 1.33 × 10⁻⁴ %.
What units does Quarter-Wave Antenna Length use?
Antenna Length is shown in mm, cm, m, km. Values are held in SI base units internally, so you can mix prefixes without converting by hand.
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