Skin Depth Formula
Calculate electromagnetic skin depth using delta = sqrt(2rho/(omega*mu)).
Essential for high-frequency conductor and antenna design.
The Formula
The skin depth formula calculates the depth at which alternating current density drops to 1/e (about 36.8%) of its value at the surface of a conductor. This phenomenon, known as the skin effect, causes high-frequency AC current to concentrate near the outer surface of a conductor rather than flowing uniformly through the entire cross-section.
The skin effect occurs because changing magnetic fields inside the conductor induce eddy currents that oppose the flow of current in the interior. At higher frequencies, these opposing currents become stronger, pushing the current flow closer and closer to the surface. At DC (zero frequency), current flows uniformly through the entire conductor, but at radio frequencies the effective conducting layer may be only micrometers thick.
This has significant practical consequences for electrical and electronic design. At power line frequencies (50-60 Hz), the skin depth in copper is about 8.5 mm, which means standard household wiring is essentially unaffected. However, at 1 MHz the skin depth drops to about 0.066 mm, and at 1 GHz it is only about 2 micrometers. This is why high-frequency conductors are often made as thin tubes or flat strips rather than solid wires, and why the surface finish and plating of RF conductors matters so much.
In the formula, ρ (rho) is the electrical resistivity of the conductor material, ω (omega) is the angular frequency (2πf), and μ (mu) is the magnetic permeability of the material. For non-magnetic conductors like copper or aluminum, μ is essentially equal to μ₀, the permeability of free space (4π × 10⁻⁷ H/m). For magnetic materials like steel, the permeability is much higher, which dramatically reduces the skin depth.
Engineers must account for skin depth when designing transmission lines, antennas, PCB traces, transformers, and any component carrying high-frequency current. Using conductors much thicker than the skin depth wastes material without reducing resistance.
Variables
| Symbol | Meaning |
|---|---|
| δ | Skin depth (meters, m) |
| ρ | Electrical resistivity (Ohm-meters, Ω·m) |
| ω | Angular frequency (radians per second, rad/s) = 2πf |
| μ | Magnetic permeability (Henrys per meter, H/m) |
Skin Depth in Copper at Various Frequencies
| Frequency | Skin Depth |
|---|---|
| 60 Hz | 8.5 mm |
| 1 kHz | 2.1 mm |
| 100 kHz | 0.21 mm |
| 1 MHz | 0.066 mm |
| 1 GHz | 0.0021 mm |
Example 1
Calculate the skin depth in copper at 10 MHz. Copper resistivity = 1.68 × 10⁻⁸ Ω·m, μ = 4π × 10⁻⁷ H/m.
ω = 2π × 10,000,000 = 62,831,853 rad/s
δ = √(2 × 1.68 × 10⁻⁸ / (62,831,853 × 4π × 10⁻⁷))
δ = √(3.36 × 10⁻⁸ / 0.07896)
δ = √(4.255 × 10⁻⁷)
δ ≈ 0.0207 mm (about 20.7 micrometers)
Example 2
Compare the skin depth in copper vs. aluminum at 1 MHz. Aluminum resistivity = 2.65 × 10⁻⁸ Ω·m.
ω = 2π × 1,000,000 = 6,283,185 rad/s
Copper: δ = √(2 × 1.68 × 10⁻⁸ / (6,283,185 × 4π × 10⁻⁷)) = 0.066 mm
Aluminum: δ = √(2 × 2.65 × 10⁻⁸ / (6,283,185 × 4π × 10⁻⁷)) = 0.082 mm
Copper: δ ≈ 0.066 mm, Aluminum: δ ≈ 0.082 mm. Aluminum has deeper skin depth due to higher resistivity.
When to Use It
Use the skin depth formula when you need to:
- Determine minimum conductor thickness for RF and microwave circuits
- Design PCB traces and transmission lines for high-frequency signals
- Choose appropriate wire gauge for power transmission at various frequencies
- Calculate effective AC resistance of conductors (higher than DC resistance)
- Evaluate electromagnetic shielding effectiveness
The effective AC resistance of a round wire can be approximated by considering only the outer shell of thickness δ as the conducting area, which is why AC resistance increases with frequency.