What are the essential antenna formulas?
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The core antenna formulas relate wavelength, gain, aperture, impedance match and distance: λ = c/f; G = ecd·D; Ae = λ²G/4π; Pr/Pt = GtGr(λ/4πR)²; VSWR = (1+|Γ|)/(1−|Γ|); and the far-field distance 2D²/λ.[1][2] Each is listed below with its assumptions.
Which formulas describe radiation and gain?
| Quantity | Formula | Notes |
|---|---|---|
| Wavelength | λ = c/f ≈ 300/f(MHz) m | c = 299,792,458 m/s |
| Directivity | D = 4πUmax/Prad | U = radiation intensity (W/sr) |
| Gain | G = ecd·D | excludes mismatch and polarization loss (IEEE 145)[3] |
| Realized gain | Gre = (1−|Γ|²)·G | includes mismatch |
| Effective aperture | Ae = λ²G/(4π) | max effective aperture uses D |
| Directivity from beamwidths | D ≈ 41,253/(θ1θ2) (degrees) | Kraus approximation, single narrow beam[4] |
| Dish gain | G = eap(πD/λ)² | eap typically 0.5–0.7 |
Which formulas describe links and distance?
- Friis: Pr = PtGtGr(λ/4πR)², valid in the far field with matched polarization.[2]
- Free-space path loss: L = 20·log10(4πd/λ) = 32.45 + 20·log10d(km) + 20·log10f(MHz) dB.[5]
- Far-field (Fraunhofer) distance: R ≥ 2D²/λ for apertures large compared with λ.[1]
- EIRP = PtGt; ERP = EIRP/1.64 (−2.15 dB).
- Power density: S = EIRP/(4πR²); field strength E = √(30·EIRP)/R (V/m, W, m).
- First Fresnel-zone radius: r1 = 17.32√(d1d2/(f·d)) m with d in km and f in GHz.
Which formulas describe matching?
- Γ = (ZL−Z0)/(ZL+Z0)
- VSWR = (1+|Γ|)/(1−|Γ|)
- Return loss RL = −20·log10|Γ| dB
- Mismatch loss = −10·log10(1−|Γ|²) dB; VSWR 2:1 gives |Γ| = 1/3, RL 9.54 dB, mismatch loss 0.51 dB
- Quarter-wave transformer: Z1 = √(Z0ZL)
- L-network loaded Q: Q = √(Rhigh/Rlow − 1)[6]
Which formulas describe small antennas and arrays?
- Chu–McLean lower bound on radiation Q: Q ≥ 1/(ka)³ + 1/(ka), with a the radius of the enclosing sphere and k = 2π/λ.[7]
- Bandwidth from Q: FBW ≈ (s−1)/(Q√s) for VSWR ≤ s.[8]
- Short dipole radiation resistance: Rr = 80π²(l/λ)².[1]
- Small loop: Rr = 20π²(C/λ)⁴ per turn, times N² for N turns.[1]
- Uniform linear array factor: |AF| = |sin(Nψ/2)/(N·sin(ψ/2))|, ψ = kd·cosθ + β.[1]
- No grating lobes when d/λ < 1/(1+|sin θ0|).[9]
Frequently asked questions
What is 0 dBd in dBi?
0 dBd equals 2.15 dBi, the directivity of a lossless half-wave dipole.
What is the free-space path loss at 2.45 GHz and 10 m?
About 60.2 dB from 32.45 + 20·log10(0.01) + 20·log10(2450).
What is the effective aperture of an isotropic antenna?
λ²/4π, about 11.9 cm² at 2.45 GHz.
See also
References
Online sources accessed October 21, 2026; printed works cited by edition. See all sources.
- “Antenna Theory: Analysis and Design, 4th ed.”. C. A. Balanis, Wiley. 2016.
- “A Note on a Simple Transmission Formula”. H. T. Friis, Proceedings of the IRE, vol. 34, no. 5, pp. 254-256. May 1946.
- “IEEE Standard for Definitions of Terms for Antennas (IEEE Std 145-2013)”. IEEE Antennas and Propagation Society. 2014.
- “Antennas for All Applications, 3rd ed.”. J. D. Kraus and R. J. Marhefka, McGraw-Hill. 2002.
- “Recommendation ITU-R P.525: Calculation of free-space attenuation”. ITU-R. current edition.
- “Microwave Engineering, 4th ed.”. D. M. Pozar, Wiley. 2012.
- “A Re-examination of the Fundamental Limits on the Radiation Q of Electrically Small Antennas”. J. S. McLean, IEEE Transactions on Antennas and Propagation, vol. 44, no. 5, pp. 672-676. May 1996.
- “Impedance, Bandwidth, and Q of Antennas”. A. D. Yaghjian and S. R. Best, IEEE Transactions on Antennas and Propagation, vol. 53, no. 4, pp. 1298-1324. April 2005.
- “Phased Array Antenna Handbook, 3rd ed.”. R. J. Mailloux, Artech House. 2017.
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