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
\delta = \frac{1}{\sqrt{\pi f \mu_0 \mu_r \sigma}}mR_s = \frac{1}{\sigma\delta} = \sqrt{\frac{\pi f \mu_0 \mu_r}{\sigma}}Ω per squareJ(x) = J_s\,e^{-x/\delta}current density at depth xWire, DC
R_\mathrm{dc} = \frac{4}{\sigma\pi d^2}Ω/mWire, RF
R_\mathrm{ac} \approx \frac{1}{\sigma\pi\,(d\delta - \delta^2)}Ω/m, d > 2δVariables
| Symbol | Meaning | Unit |
|---|---|---|
\delta | Skin depth | m |
f | Frequency | Hz |
\sigma | Conductivity | S/m |
\mu_0,\ \mu_r | 4π × 10⁻⁷, relative permeability | H/m, — |
R_s | Surface resistance | Ω |
d | Wire diameter | m |
Conductivity at 20 °C
| Material | σ (S/m) | Note |
|---|---|---|
| Silver | 6.30 × 10⁷ | |
| Copper, annealed | 5.80 × 10⁷ | 100 % IACS |
| Gold | 4.10 × 10⁷ | |
| Aluminum, pure | 3.77 × 10⁷ | |
| Aluminum 6061-T6 | ≈ 2.5 × 10⁷ | about 43 % IACS; varies by temper |
| Brass, C260 | ≈ 1.6 × 10⁷ | about 28 % IACS |
| Tin | 9.17 × 10⁶ | |
| Stainless steel 304 | 1.45 × 10⁶ |
Worked example
| Input | Value |
|---|---|
| Conductor | Annealed copper |
| Frequency | 1000 MHz |
| Wire | 1 mm diameter |
| Result | Value |
|---|---|
| Skin depth, δ | 2.09 µm |
| Surface resistance, Rs | 8.25 mΩ/sq |
| Recommended thickness (5δ) | 10.45 µm |
| Wire resistance, DC | 21.95 mΩ/m |
| Wire resistance at this frequency | 2.632 Ω/m |
| AC / DC ratio | 119.9 |
At 1 GHz the current in copper flows in the outer 2.1 µm. A 1 mm wire has about 120 times its DC resistance.
Load this example in the calculatorValidity
- Good conductors (σ ≫ ωε), at least a few δ thick.
- Smooth surfaces. Roughness comparable to δ adds loss that is not included.
- Non-magnetic metals use μr = 1. Nickel and steel have a μr that depends on frequency and field strength; use measured values.
- The wire formula assumes d ≫ δ and ignores proximity effect from nearby conductors.
- Conductivity varies with alloy, temper, temperature and plating process.
Sources
- D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012. §1.7 (skin depth and surface resistance).
- S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed., Wiley, 1994.
- J. R. Rumble (ed.), CRC Handbook of Chemistry and Physics, CRC Press (electrical resistivity of metals).