Calculate the uniform-current Hertzian dipole approximation from total dipole length, wavelength, and free-space impedance, with results in Ω, kΩ, or MΩ.
Radiation Resistance Formula
This calculator uses the Hertzian dipole approximation: uniform current along a total dipole length much shorter than wavelength, in free space. A practical center-fed short wire dipole has approximately triangular current and a radiation-resistance coefficient one quarter as large for the same total length.
Using the customary approximation Z₀ ≈ 120π Ω (about 377 Ω), the uniform-current formula is approximately:
- Rr = radiation resistance, in ohms
- Z0 = impedance of free space, usually about 377 Ω
- L = antenna length
- lambda = wavelength
- pi = 3.14159…
The calculator converts antenna length and wavelength to meters before applying the formula. It also converts the impedance of free space to ohms if you enter it in kΩ or MΩ. The final radiation resistance is then converted to the output unit you select, such as Ω, kΩ, or MΩ.
Typical Hertzian Dipole Radiation Resistance Values
The values below use Z0 = 377 Ω. They show how quickly radiation resistance changes as the antenna length becomes a larger fraction of the wavelength.
| Antenna length ratio, L/lambda | Approx. radiation resistance | Length ratio context |
|---|---|---|
| 0.001 | 0.0008 Ω | Very small ratio |
| 0.005 | 0.0197 Ω | Very small ratio |
| 0.01 | 0.0790 Ω | Small ratio |
| 0.02 | 0.3158 Ω | Small ratio |
| 0.05 | 1.974 Ω | Larger ratio |
| 0.10 | 7.896 Ω | Formula extrapolation |
Unit Conversions Used
| Input unit | Base conversion |
|---|---|
| 1 ft | 0.3048 m |
| 1 cm | 0.01 m |
| 1 in | 0.0254 m |
| 1 kΩ | 1,000 Ω |
| 1 MΩ | 1,000,000 Ω |
Example Problems
Example 1: Antenna length and wavelength in meters
Find the radiation resistance for an ideal uniform-current Hertzian dipole with a total length of 0.5 m, a wavelength of 10 m, and free-space impedance of 377 Ω.
The radiation resistance is approximately 1.974 Ω.
Example 2: Mixed length units
Find the ideal uniform-current Hertzian dipole radiation resistance for a total length of 12 in, a wavelength of 30 ft, and free-space impedance of 377 Ω.
First convert the length values:
- 12 in = 0.3048 m
- 30 ft = 9.144 m
The radiation resistance is approximately 0.8773 Ω.
FAQ
What is radiation resistance?
Radiation resistance is the part of an antenna’s resistance that represents power being radiated as electromagnetic waves. It is not the same as wire loss or ground loss. A higher radiation resistance usually means the antenna can radiate power more effectively relative to its loss resistance.
Why is the impedance of free space usually 377 Ω?
The impedance of free space is approximately 376.73 Ω, commonly rounded to 377 Ω and approximately 120π Ω. The coefficient becomes approximately 80π² for the uniform-current Hertzian model; rounded 377 Ω gives a slightly different value. A center-fed electrically short wire dipole instead has approximately 20π²(L/lambda)².
When is this radiation resistance formula valid?
This formula assumes a uniform-current electrically short dipole, with total length much smaller than wavelength. Small L/lambda alone does not establish that a real antenna has uniform current. The calculator warns at L/lambda ≥ 0.1; this is a screening cue, not an accuracy guarantee. Use a current-distribution and geometry-specific antenna model for practical center-fed, quarter-wave or half-wave antennas.
