Frequency-Dependent Current-Carrying Capacity of Cables: Detailed Explanation (Based on IEC 60287)

1. Core Formula for Current-Carrying Capacity (IEC 60287)
The fundamental formula in IEC 60287 is based on thermal limits and electrical losses in the cable:
I = √(Δθ / (R' * (1 + λ1 + λ2)))

Parameters:

• I: Current-carrying capacity (A)
• Δθ: Temperature rise (°C)
• R': Effective AC resistance of the conductor (Ω/m)
• λ1, λ2: Loss factors for dielectric and sheath/armor

2. Calculation of AC Resistance (R')
The AC resistance (R') includes DC resistance (Rdc) and adjustments for skin and proximity effects:
R' = Rdc * (1 + Y)
Y = ks + kp


• Rdc = ρ / A (ρ: resistivity, A: cross-sectional area)
• ks: Skin effect coefficient
• kp: Proximity effect coefficient

3. Skin Effect
Skin effect causes current to concentrate near the surface of the conductor.
Skin Depth (δ):
δ = √(ρ / (π * f * μ))
Skin Effect Coefficient (ks):
ks = 1/2 * (d / δ)^2

4. Proximity Effect
Proximity effect occurs when nearby conductors' magnetic fields distort current flow.
Proximity Effect Coefficient (kp):
kp = μ * f * I / s^2


• I: Current (A)
• s: Distance between conductors (m)

5. Thermal Losses
The total power losses are:
Ptotal = P1 + P2 + P3
P1 = I^2 * R'
P2 = U^2 * ω * C * tan(δ)
P3 = K * I^2

6. Final Formula for Current-Carrying Capacity
Including frequency-dependent effects:
I = √(Δθ / (Rdc * (1 + ks + kp) * (1 + λ1 + λ2)))

Example Calculation

• Assumptions: Copper conductor (ρ = 1.68×10^-8 Ω·m), f = 1 kHz, d = 5 mm
• Skin Depth: δ ≈ 2.1 mm
• ks: ks ≈ 2.83
• kp: Assume kp = 0.5
• Effective Resistance: R' = 0.866 Ω
• Current Capacity: I ≈ 8.14 A

Conclusion
This step-by-step method ensures accurate calculations for frequency-dependent current-carrying capacity. Adjustments like using Litz wires may be required for high-frequency applications.