18. Busbar Calculations

Principle

Busbars are robust conductors—typically made of copper or aluminum—used in switchboards and distribution systems to carry large currents. Designing busbars involves ensuring that the cross-sectional area is sufficient to handle the rated current without excessive temperature rise, keeping voltage drop minimal along the bar, and withstanding mechanical forces during short-circuits.

Key Considerations

  • Current Carrying Capacity (Ampacity): Determined by allowable temperature rise. Busbar ampacity is often expressed as a current density (e.g., A/mm²) or via standard tables. For example, a copper busbar in open air may have a guideline of ~1.6 A/mm² for a ~30°C temperature rise. Thus, a 10×50 mm (500 mm²) busbar might carry approximately 800 A.
  • Voltage Drop: Typically minor in short bus sections but may be significant in long busway systems. The voltage drop can be calculated using the resistivity of the material. For copper, the resistance (R) of a bar of cross-sectional area S (mm²) and length L (m) is approximately:
    R ≈ (0.01724 × L) / S Ω
  • Short-Circuit Stresses: During fault conditions, busbars experience significant magnetic (Lorentz) forces. The force per unit length between two parallel busbars a distance D apart is given by:
    F = (μ₀ / 2π) × (I² / D)
    where I is the peak current. For example, if a fault produces 50 kA rms (with peak currents potentially ~100 kA) and the busbars are 5 cm apart, the force may be on the order of tens of kN per meter—requiring robust support and spacing.
  • Temperature Rise: Busbar temperature rise is calculated by balancing the I²R losses (and effects such as skin and proximity effects) with heat dissipation via convection and radiation. Empirical formulas or manufacturer tables are typically used.

Ampacity Calculation Example

As a rough guide, if a copper busbar in open air is rated at about 1.6 A/mm² and you have a 10×50 mm (500 mm²) bar, its ampacity is approximately:

500 mm² × 1.6 A/mm² = 800 A

Using two bars per phase (total 1000 mm²) would provide a capacity of ~1600 A.

Resistance and Voltage Drop

For a busbar of cross-sectional area S and length L, the resistance is approximately:

R ≈ (0.01724 × L) / S Ω

For example, a 1 m long 500 mm² busbar has R ≈ (0.01724 / 500) ≈ 0.0000345 Ω. For a current of 2000 A, the voltage drop would be minimal:

V_drop = 2000² × 0.0000345 ≈ 0.138 V

Short-Circuit Force Calculation

The force per unit length between two parallel busbars can be approximated by:

F = (μ₀ / 2π) × (I² / D)

For example, with I_peak ~100 kA and a separation D = 0.05 m:

F ≈ (4π×10⁻⁷ / (2π)) × (100,000² / 0.05) ≈ 40 kN/m

This high force emphasizes the need for robust mechanical support and appropriate spacing.

Temperature Rise Calculation

A simplified approach is to calculate the power loss per meter (P_loss = I²R) and equate that to the convective heat loss:

Q_loss = h × A_surf × ΔT

For instance, a 10×50 mm busbar (500 mm²) with an approximate perimeter of 2×(0.01+0.05) = 0.12 m has a surface area (per 1 m length) of roughly 0.12 m². If carrying 800 A with a resistance ~0.000036 Ω/m, the loss is:

P_loss = 800² × 0.000036 ≈ 23 W/m

Assuming a convective heat transfer coefficient h ≈ 5 W/m²K, the temperature rise is:

ΔT ≈ 23 / (5 × 0.12) ≈ 38.3 K

This rough calculation aligns with empirical values, although detailed design would include additional factors such as radiation and mounting conditions.

Niche Considerations

  • Skin Effect: At 50/60 Hz, the skin depth in copper is approximately 8.5 mm. Busbars thicker than ~10 mm may exhibit increased AC resistance. Designers may use several thinner bars in parallel to reduce losses and increase cooling surface area.
  • Proximity Effect: When multiple busbars are installed close together (e.g., in a three-phase arrangement), current distribution may be affected. Detailed analysis may require finite element methods.
  • Dynamic Impedance Under Fault: For very short fault durations, busbars can handle high currents without significant heating. However, the short-circuit mechanical force remains a critical design parameter.

Industry Relevance

Busbars are integral to switchgear and distribution panels. Correctly sizing them ensures that they operate safely under continuous load and survive fault conditions without mechanical failure. In large substations or industrial plants, both thermal and mechanical design of busbars are critical to prevent overheating, insulation breakdown, and structural damage during short-circuits.

Standards

  • IEC 61439-1 & -2: Provide guidelines for low-voltage switchgear assemblies, including temperature rise and short-circuit withstand tests.
  • UL 891: Specifies requirements for dead-front switchboards, including busbar bracing and force calculations.
  • IEC TR 60865: Offers guidance on short-circuit force calculations for HV substations, applicable in simplified form for LV busbars.
  • IEEE Std 605: Provides methods for bus design in air-insulated substations, including thermal and mechanical considerations.

Software Tools

  • Finite Element Analysis (FEA): Tools such as Ansys can model thermal and electromagnetic behavior of busbars in complex enclosures.
  • Ampacity Calculators: Online tools and spreadsheets (often provided by copper associations) allow designers to input busbar dimensions, ambient conditions, and obtain ampacity and temperature rise estimates.
  • Short-Circuit Mechanical Calculators: Some design packages or Excel spreadsheets help estimate mechanical forces and determine support spacing.
  • Busbar Selection Tools: Many switchgear manufacturers provide selection charts and software to determine appropriate busbar size and configuration for a given current and short-circuit rating.

Conclusion

Busbar calculations combine thermal, electrical, and mechanical analyses to ensure that busbars can safely carry the required currents, maintain acceptable voltage drop, and withstand fault-induced forces. By following industry standards and employing appropriate simulation and calculation tools, designers can optimize busbar design for safety, efficiency, and reliability.