Total Impedance in Cables Calculator – IEC, IEEE

Accurate calculation of total impedance in cables is critical for power system design and analysis. It ensures system reliability, efficiency, and safety.

This article explores total impedance calculation methods based on IEC and IEEE standards. It covers formulas, tables, and practical examples.

Artificial Intelligence (AI) Calculator for “Total Impedance in Cables Calculator – IEC, IEEE”

• Calculate total impedance for a 3-core XLPE cable, 50 meters, 11 kV, 240 mm² copper conductor.
• Determine total impedance of a 1-core paper insulated lead covered (PILC) cable, 100 meters, 33 kV, 300 mm² aluminum conductor.
• Find total impedance for a 3-core armored cable, 200 meters, 0.6/1 kV, 95 mm² copper conductor.
• Compute total impedance of a 1-core XLPE cable, 150 meters, 6.6 kV, 185 mm² copper conductor with concentric neutral.

Comprehensive Tables of Total Impedance Values for Cables According to IEC and IEEE

The following tables provide typical total impedance values for commonly used power cables, based on IEC 60287 and IEEE 835 standards. These values include resistance (R), inductive reactance (X), and total impedance (Z) per unit length, which are essential for system studies.

Fundamental Formulas for Total Impedance Calculation in Cables

Total impedance (Z) in cables is a complex quantity combining resistance (R) and reactance (X). It is frequency-dependent and influenced by cable construction, conductor material, and installation conditions.

1. Total Impedance Formula

Z = R + jX

• Z: Total impedance (Ω)
• R: Resistance of the conductor (Ω)
• X: Reactance of the cable (Ω)
• j: Imaginary unit (√-1)

Resistance (R) accounts for conductor losses, while reactance (X) represents inductive effects due to cable geometry and magnetic fields.

2. Resistance Calculation (R)

R = ρ × (L / A) × (1 + α × (T – T₀))

• ρ: Resistivity of conductor material (Ω·m)
• L: Length of cable (m)
• A: Cross-sectional area of conductor (m²)
• α: Temperature coefficient of resistance (per °C)
• T: Operating temperature (°C)
• T₀: Reference temperature, usually 20°C (°C)

Typical resistivity values at 20°C:

• Copper: 1.724 × 10⁻⁸ Ω·m
• Aluminum: 2.82 × 10⁻⁸ Ω·m

Temperature coefficient α:

• Copper: 0.00393 /°C
• Aluminum: 0.00403 /°C

3. Reactance Calculation (X)

X = 2πfL

• f: Frequency (Hz), typically 50 or 60 Hz
• L: Inductance per unit length (H)

Inductance (L) depends on cable geometry and insulation. For three-core cables, the inductance can be approximated by:

L = (2 × 10⁻⁷) × ln(Dm / r) (H/m)

• Dm: Geometric mean distance between conductors (m)
• r: Radius of conductor (m)

For single-core cables with concentric neutral, the formula adjusts to include neutral conductor effects.

4. Total Impedance Magnitude

|Z| = √(R² + X²)

This magnitude is used in power system calculations such as voltage drop and fault current analysis.

5. Per Unit Length Impedance

For practical calculations, impedance is often expressed per kilometer:

Z (Ω/km) = R (Ω/km) + jX (Ω/km)

Where R and X are derived from manufacturer data or standards like IEC 60287 and IEEE 835.

Detailed Real-World Examples of Total Impedance Calculation

Example 1: Total Impedance of a 3-Core XLPE Copper Cable (11 kV, 240 mm², 50 m)

A 3-core XLPE insulated copper cable with 240 mm² cross-section is installed over 50 meters. Calculate the total impedance at 50 Hz.

• Given:
• Length, L = 50 m
• Conductor area, A = 240 mm² = 240 × 10⁻⁶ m²
• Frequency, f = 50 Hz
• Resistivity of copper, ρ = 1.724 × 10⁻⁸ Ω·m
• Operating temperature, T = 90°C
• Reference temperature, T₀ = 20°C
• Temperature coefficient, α = 0.00393 /°C
• Geometric mean distance, Dm = 0.025 m (typical for 11 kV cables)
• Conductor radius, r = √(A/π) ≈ 8.74 mm = 0.00874 m

Step 1: Calculate Resistance (R)

Resistance at 20°C:

R₀ = ρ × (L / A) = 1.724 × 10⁻⁸ × (50 / 240 × 10⁻⁶) = 0.00359 Ω

Adjust for temperature:

R = R₀ × (1 + α × (T – T₀)) = 0.00359 × (1 + 0.00393 × (90 – 20)) = 0.00359 × 1.275 = 0.00458 Ω

Step 2: Calculate Inductance (L)

Using the formula:

L = 2 × 10⁻⁷ × ln(Dm / r) = 2 × 10⁻⁷ × ln(0.025 / 0.00874) = 2 × 10⁻⁷ × ln(2.86) = 2 × 10⁻⁷ × 1.05 = 2.1 × 10⁻⁷ H/m

For 50 m:

L_total = 2.1 × 10⁻⁷ × 50 = 1.05 × 10⁻⁵ H

Step 3: Calculate Reactance (X)

X = 2πfL_total = 2 × 3.1416 × 50 × 1.05 × 10⁻⁵ = 0.0033 Ω

Step 4: Calculate Total Impedance (Z)

Z = R + jX = 0.00458 + j0.0033 Ω

Magnitude:

|Z| = √(0.00458² + 0.0033²) = √(2.10 × 10⁻⁵ + 1.09 × 10⁻⁵) = √(3.19 × 10⁻⁵) = 0.00565 Ω

This low impedance confirms minimal voltage drop and losses over 50 meters.

Example 2: Total Impedance of a 1-Core PILC Aluminum Cable (33 kV, 400 mm², 100 m)

Calculate the total impedance of a 1-core PILC aluminum cable with 400 mm² cross-section over 100 meters at 60 Hz.

• Given:
• Length, L = 100 m
• Conductor area, A = 400 mm² = 400 × 10⁻⁶ m²
• Frequency, f = 60 Hz
• Resistivity of aluminum, ρ = 2.82 × 10⁻⁸ Ω·m
• Operating temperature, T = 75°C
• Reference temperature, T₀ = 20°C
• Temperature coefficient, α = 0.00403 /°C
• Geometric mean distance, Dm = 0.04 m (typical for 33 kV cables)
• Conductor radius, r = √(A/π) ≈ 11.28 mm = 0.01128 m

Step 1: Calculate Resistance (R)

Resistance at 20°C:

R₀ = ρ × (L / A) = 2.82 × 10⁻⁸ × (100 / 400 × 10⁻⁶) = 0.00705 Ω

Adjust for temperature:

R = R₀ × (1 + α × (T – T₀)) = 0.00705 × (1 + 0.00403 × (75 – 20)) = 0.00705 × 1.224 = 0.00863 Ω

Step 2: Calculate Inductance (L)

Using the formula:

L = 2 × 10⁻⁷ × ln(Dm / r) = 2 × 10⁻⁷ × ln(0.04 / 0.01128) = 2 × 10⁻⁷ × ln(3.55) = 2 × 10⁻⁷ × 1.266 = 2.53 × 10⁻⁷ H/m

For 100 m:

L_total = 2.53 × 10⁻⁷ × 100 = 2.53 × 10⁻⁵ H

Step 3: Calculate Reactance (X)

X = 2πfL_total = 2 × 3.1416 × 60 × 2.53 × 10⁻⁵ = 0.00954 Ω

Step 4: Calculate Total Impedance (Z)

Z = R + jX = 0.00863 + j0.00954 Ω

Magnitude:

|Z| = √(0.00863² + 0.00954²) = √(7.45 × 10⁻⁵ + 9.10 × 10⁻⁵) = √(1.655 × 10⁻⁴) = 0.01287 Ω

This impedance value is critical for fault current calculations and voltage regulation in medium voltage networks.

Additional Technical Considerations for Total Impedance Calculations

• Skin Effect: At power frequencies, skin effect causes current to concentrate near the conductor surface, increasing effective resistance. This is accounted for in standards by correction factors.
• Proximity Effect: Magnetic fields from adjacent conductors induce eddy currents, increasing losses and reactance. Cable arrangement influences this effect.
• Frequency Dependence: While 50/60 Hz is standard, harmonic frequencies in power systems require impedance recalculations due to changing reactance.
• Temperature Impact: Cable operating temperature significantly affects resistance; thermal ratings must be considered for accurate impedance.
• Standards Compliance: IEC 60287 and IEEE 835 provide detailed methodologies and correction factors for precise impedance calculations.

References and Authoritative Resources

• IEC 60287 – Electric cables – Calculation of the current rating
• IEEE Std 835-1994 – IEEE Standard Power Cable Ampacity Tables
• National Electrical Manufacturers Association (NEMA)
• CIGRE Technical Brochures on Power Cable Systems

Understanding and accurately calculating total impedance in cables is essential for electrical engineers designing power distribution and transmission systems. Utilizing IEC and IEEE standards ensures consistency, safety, and optimal performance.