Busbar Temperature Rise Formulas
Busbar heating formulas: the resistance and heat per metre, the steady temperature rise from the surface cooling and the bar temperature for a current.
The formulas
| To find | Formula |
|---|---|
| Resistance per metre at 20 °C | R20 = ρ / (w × t) |
| Perimeter that cools | P = 2 × (w + t) |
| Steady temperature rise | Δθ = I² × R20 × (1 + α × (θa − 20)) / (h × P − I² × R20 × α) |
| Bar temperature | θ = θa + Δθ |
ρ is the resistivity of copper (1.72e-8 Ω·m), w and t the width and thickness of the bar in metres, α the temperature coefficient of resistance (0.00393 per K for copper), θa the air temperature and h the combined heat transfer coefficient of convection and radiation in W/(m²·K). The formula balances the heat I² × R of the bar, which rises with its temperature, against the heat that the surface sheds. The bar is one bar in free air. Skin and proximity effects, the shading of neighbouring bars, an enclosure, joints and supports are left out, so the real rise is higher in a switchboard. If the denominator is zero or negative the heat grows faster than the cooling, and the bar has no steady state. The value of h = 10 W/(m²·K) and the other inputs are examples. Use the rated currents and the temperature-rise tests of the maker and the standard for a design.
Worked example: a 100 × 10 mm copper bar at 1,600 A, air at 35 °C
- R20 = 1.72e-8 / (0.1 × 0.01) = 17.2 µΩ/m, and I² × R20 = 44.03 W/m
- P = 2 × (0.1 + 0.01) = 0.22 m
- Δθ = 44.03 × (1 + 0.00393 × 15) / (10 × 0.22 − 44.03 × 0.00393) = 23 K
- The bar runs at 35 + 23 = 58 °C, where the heat is 50.61 W/m
Temperature of the 100 × 10 mm bar by current (air at 35 °C)
| Current | Heat at 20 °C | Rise | Bar temperature |
|---|---|---|---|
| 800 A | 11.01 W/m | 5.405 K | 40.4 °C |
| 1200 A | 24.77 W/m | 12.47 K | 47.47 °C |
| 1600 A | 44.03 W/m | 23 K | 58 °C |
| 2000 A | 68.8 W/m | 37.76 K | 72.76 °C |
Frequently asked questions
Why does the formula have a term in the denominator for the resistance?
The resistance of copper rises with its temperature, so the heat grows as the bar warms. The term in the denominator takes that feedback into account in one step.
What limits the temperature of a bar?
The standard of the assembly sets the allowed rise for the bar, the joints and the parts that touch it, and the insulation near it. Use the figure of the standard that you work to.
Why are bars in a board hotter than a bar in free air?
Neighbouring bars and the enclosure cut the cooling and add heat, and the skin and proximity effects raise the resistance. A rated current from the maker covers these.
How is this related to the short-circuit size?
This page is about continuous current. The short-circuit heating and the force follow other formulas, as in the busbar rating and the conductor force articles.
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