For estimating and education only. Results come from the simplified models described below, are provided as is with no warranty, and are not a substitute for engineering analysis, simulation or measurement. Check them before relying on them in a design, purchase or safety decision. See the terms of use.
Equations
Dipole
L = k\,\frac{\lambda}{2}, \qquad k \approx 0.95≈ 142.6 / fMHz mMonopole
L = k\,\frac{\lambda}{4}Patch
W = \frac{c}{2f}\sqrt{\frac{2}{\varepsilon_r + 1}}\varepsilon_\mathrm{eff} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2}\left(1 + \frac{12h}{W}\right)^{-1/2}\Delta L = 0.412\,h\,\frac{(\varepsilon_\mathrm{eff} + 0.3)(W/h + 0.264)}{(\varepsilon_\mathrm{eff} - 0.258)(W/h + 0.8)}L = \frac{c}{2f\sqrt{\varepsilon_\mathrm{eff}}} - 2\,\Delta LVariables
| Symbol | Meaning | Unit |
|---|---|---|
\lambda | Free-space wavelength | m |
k | End-effect length factor | — |
W,\ L | Patch width, resonant length | m |
h | Substrate thickness | m |
\Delta L | Fringing-field length extension, each end | m |
Worked example
| Input | Value |
|---|---|
| Antenna | Rectangular patch |
| Frequency | 2.45 GHz |
| Substrate | FR-4, εr = 4.4, h = 1.6 mm |
| Result | Value |
|---|---|
| Free-space wavelength | 122.4 mm (4.817 in) |
| Patch length, L (resonant) | 28.81 mm (1.134 in) |
| Patch width, W | 37.23 mm (1.466 in) |
| Effective permittivity | 4.081 |
| Fringing extension, ΔL (each end) | 0.7386 mm (0.02908 in) |
| Minimum ground plane (L + 6h) × (W + 6h) | 38.41 × 46.83 mm |
Treat the result as a starting point. FR-4 permittivity varies enough to move the resonance by tens of MHz; trim the length on a VNA.
Load this example in the calculatorValidity
- The transmission-line patch model works best for thin substrates, h below about 0.05 λ.
- Feed position sets input impedance and is not computed here.
- Dipole and monopole lengths shorten with element thickness; lower k for thicker elements.
Sources
- C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed., Wiley, 2016. Chapter on microstrip antennas.
- E. O. Hammerstad, “Equations for Microstrip Circuit Design,” Proc. 5th European Microwave Conf., 1975, pp. 268–272.
- The ARRL Antenna Book, ARRL (dipole length factor).