**3.
Current-carrying Capacity of Busbars**

Calculation of Current-carrying Capacity

Approximate dc Current Ratings for Flat and Round bars

The current-carrying capacity of a busbar is usually determined by the maximum temperature at which the bar is permitted to operate, as defined by national and international standards such as British Standard BS 159, American Standard ANSI C37.20, etc. These standards give maximum temperature rises as well as maximum ambient temperatures.

BS 159 stipulates a maximum temperature rise of 50°C above a 24 hour mean ambient temperature of up to 35°C, and a peak ambient temperature of 40°C.

ANSI C37.20 alternatively permits a temperature rise of 65°C above a maximum ambient of 40°C, provided that silver-plated (or acceptable alternative) bolted terminations are used. If not, a temperature rise of 30°C is allowed.

These upper temperature limits have been chosen because at higher maximum operating temperatures the rate of surface oxidation in air of conductor materials increases rapidly and may give rise in the long term to excessive local heating at joints and contacts. This temperature limit is much more important for aluminium than copper because it oxidises very much more readily than copper. In practise these limitations on temperature rise may be relaxed for copper busbars if suitable insulation materials are used. A nominal rise of 60°C or more above an ambient of 40°C is allowed by BS EN 60439-1:1994 provided that suitable precautions are taken. BS EN 60439-1:1994 (equivalent to IEC 439) states that the temperature rise of busbars and conductors is limited by the mechanical strength of the busbar material, the effect on adjacent equipment, the permissible temperature rise of insulating materials in contact with the bars, and the effect on apparatus connected to the busbars.

The rating of a busbar must also take account of the mechanical stresses set up due to expansion, short-circuit currents and associated inter-phase forces. In some busbar systems consideration must also be given to the capitalised cost of the heat generated by the effective ohmic resistance and current (I2R) which leads to an optimised design using Kelvin's Law of Maximum Economy. This law states that 'the cost of lost energy plus that of interest and amortisation on initial cost of the busbars (less allowance for scrap) should not be allowed to exceed a minimum value'. Where the interest, amortisation and scrap values are not known, an alternative method is to minimise the total manufacturing costs plus the cost of lost energy.

**Calculation
of Current-carrying Capacity**

A very approximate method of estimating the current carrying capacity of a copper busbar is to assume a current density of 2 A/mm2 (1250 A/in2) in still air. This method should only be used to estimate a likely size of busbar, the final size being chosen after consideration has been given to the calculation methods and experimental results given in the following sections.

The current that will give rise to a particular equilibrium temperature rise in the conductor depends on the balance between the rate at which heat is produced in the bar, and the rate at which heat is lost from the bar. The heat generated in a busbar can only be dissipated in the following ways:

(a) Convection

(b) Radiation

(c) Conduction

In most cases convection and radiation heat losses determine the current-carrying capacity of a busbar system. Conduction can only be used where a known amount of heat can flow into a heat sink outside the busbar system or where adjacent parts of the system have differing cooling capacities. The proportion of heat loss by convection and radiation is dependent on the conductor size with the portion attributable to convection being increased for a small conductor and decreased for larger conductors.

**Convection**

The heat dissipated per unit area by convection depends on the shape and size of the conductor and its temperature rise. This value is usually calculated for still air conditions but can be increased greatly if forced air cooling is permissible. Where outdoor busbar systems are concerned calculations should always be treated as in still air unless specific information is given to the contrary.

The following formulae can be used to estimate the convection heat loss from a body in W/m2:

where q = temperature rise, °C

L = height or width of bar, mm

d = diameter of tube, mm

The diagrams below indicate which formulae should be used for various conductor geometries:

It can be seen when diagrams (a) and (b) are compared and assuming a similar cross-sectional area the heat loss from arrangement (b) is much larger, provided the gap between the laminations is not less than the thickness of each bar.

**Convection heat loss: forced air cooling**

If the air velocity over the busbar surface is less than 0.5 m/s the above formulae for Wv, Wh and Wc apply. For higher air velocities the following may be used:

where Wa = heat lost per unit length from bar, W/m

v = air velocity, m/s

A = surface** **area per unit length of bar, m2/m

**Radiation**

The rate at which heat is radiated from a body is proportional to the difference between the fourth power of the temperatures of the body and its surroundings, and is proportional to the relative emissivity between the body and its surroundings.

The following table lists typical absolute emissivities e for copper busbars in various conditions. Changes in emissivity give rise to changes in current ratings, as shown in Table 7.

Bright metal 0.1

Partially oxidised 0.30

Heavily oxidised 0.70

Dull non-metallic paint 0.9

**Table 7 Percentage increase in current rating when e is increased from 0.1 to 0.9 -
three-phase arrangement **

Phase
centres, mm |
|||

No. of bars in parallel |
150 |
200 |
250 |

1 | 23 | 23 | 25 |

2 | 15 | 16 | 18 |

3 | 10 | 11 | 14 |

4 | 9 | 9 | 12 |

5 | 6 | 7 | 9 |

The figures given in Table 7 are approximate values applicable to 80 to 160 mm wide busbars for a 105°C operating temperature and 40°C ambient. The relative emissivity is calculated as follows:

where e = relative emissivity

e1 = absolute emissivity of body 1

e2 = absolute emissivity of body 2

The rate of heat loss by radiation from a body (W/m2) is given by:

where e = relative emissivity

T1 = absolute temperature of body 1, K

T2 = absolute temperature of body 2, K (i.e., ambient temperature of the surroundings)

Radiation is considered to travel in straight lines and leave the body at right angles to its surface. The diagrams above define the effective surface areas for radiation from conductors of common shapes.

The rate at which heat is generated per unit length of a conductor carrying a direct current is the product I2R watts, where I is the current flowing in the conductor and R its resistance per unit length. The value for the resistance can in the case of d.c. busbar systems be calculated directly from the resistivity of the copper or copper alloy. Where an a.c. busbar system is concerned, the resistance is increased due to the tendency of the current to flow in the outer surface of the conductor. The ratio between the a.c. value of resistance and its corresponding d.c. value is called the skin effect ratio (see Section 4). This value is unity for a d.c. system but increases with the frequency and the physical size of the conductor for an a.c. current.

Rate of Heat generated in a Conductor,

W/mm = I2 RoS

where I = current in conductor, A

Ro = d.c. resistance per unit length, W/mm

S = skin effect ratio

also

where Rf = effective a.c. resistance of conductor, W (see above)

**Approximate dc Current Ratings for Flat and Round bars**

The following equations can be used to obtain the approximate d.c. current rating for single flat and round copper busbars carrying a direct current. The equations assume a naturally bright copper finish where emissivity is 0.1 and where ratings can be improved substantially by coating with a matt black or similar surface. The equations are also approximately true for a.c. current provided that the skin effect and proximity ratios stay close to 1.0, as is true for many low current applications. Methods of calculation for other configurations and conditions can be found in subsequent sections.

(a) Flat bars on edge:

(1

where I = current, A

A = cross-sectional area, mm2

p = perimeter of conductor, mm

q = temperature difference between conductor and the ambient air, °C

a = resistance temperature coefficient of copper at the ambient temperature, per °C

r = resistivity of copper at the ambient temperature, mW cm

(b) Hollow round bars:

(2

(c) Solid round bars:

(3

If the temperature rise of the conductor is 50°C above an ambient of 40°C and the resistivity of the copper at 20°C is 1.724 mWcm, then the above formulae become:

(i) Flat bars:

(4

(ii) Hollow round bars:

(5

(iii) Solid round bars:

(6

For high conductivity copper tubes where diameter and mass per unit length (see Table 14) are known,

(7

where m = mass per unit length of tube, kg/m

d = outside diameter of tube, mm

**Re-rating for different current or temperature rise
conditions**

Where a busbar system is to be used under new current or temperature rise conditions, the following formula can be used to find the corresponding new temperature rise or current:

(8

where

I1 = current 1, A

I2 = current 2, A

q1 = temperature rise for current 1, °C

q2 = temperature rise for current 2, °C

T1 = working temperature for current 1, °C

T2 = working temperature for current 2, °C

a20 = temperature coefficient of resistance at 20°C ( = 0.00393)

If the working temperature of the busbar system is the same in each case (i.e., T1 = T2), for example when re-rating for a change in ambient temperature in a hotter climate, this formula becomes

**Laminated bars**

When a number of conductors are used in parallel, the total current capacity is less than the rating for a single bar times the number of bars used. This is due to the obstruction to convection and radiation losses from the inner conductors. To facilitate the making of interleaved joints, the spacing between laminated bars is often made equal to the bar thickness. For 6.3 mm thick bars up to 150 mm wide, mounted on edge with 6.3 mm spacings between laminations, the isolated bar d.c. rating may be multiplied by the following factors to obtain the total rating.

**No. of laminations** **Multiplying factor**

No. of laminations |
Multiplying factor |

2 | 1.8 |

3 | 2.5 |

4 | 3.2 |

5 | 3.9 |

6 | 4.4 |

8 | 5.5 |

10 | 6.5 |