Copper busbars are normally part of a larger generation or transmission system. The continuous rating of the main components such as generators, transformers, rectifiers, etc., therefore determine the nominal current carried by the busbars but in most power systems a one to four second short-circuit current has to be accommodated. The value of these currents is calculated from the inductive reactances of the power system components and gives rise to different maximum short-circuit currents in the various system sections.

These currents are very often ten to twenty times the continuous current rating and therefore the transitory heating effect must be taken into account. This effect can, in many cases, lead to dangerous overheating, particularly where small conductors are part of a large heavy current system, and must be considered when determining the conductor size. To calculate the temperature rise of the conductor during a short circuit it is assumed that all the heat generated is absorbed by the conductor with none lost by convection and radiation as for a continuous rated conductor. The temperature rise is dependent therefore only on the specific heat of the copper conductor material and its mass. The specific heat of copper varies with temperature, increasing as the temperature rises. At normal ambient temperatures it is about 385 J/kg K and at 300°C it is about 410 J/kg K.

Short-circuit heating characteristics are not easy to calculate accurately because of complex a.c. and d.c. current effects, but for most purposes the formulae below will normally give sufficiently accurate results:

where t = maximum short-circuit time, s

A = conductor cross-section area, mm2

I = conductor current, kA

q = conductor temperature rise, K

If q = 300°C, then

The value of t obtained from the above equation should always be greater than the required short circuit withstand time which is usually 1 to 4 seconds.

The temperature rise per second due to a current I is given by the following approximate formula:

(I/A) should be less than 0.25 for reasonable accuracy.

The maximum short-circuit temperature is very often chosen to be 300°C for earth bar systems but the upper limit for the phases is normally lower and is dependent on the mechanical properties required and surface finish of the copper material.

**Heating time constant**

The previous section considered very short time effects but in many cases it may be necessary to calculate the temperature rise of a conductor over an extended time, for example the time taken for a conductor to reach normal operating temperature when carrying its rated continuous current. Under these conditions the conductor is absorbing heat as its temperature rises. It is also dissipating heat by convection and radiation, both of which increase with rising temperature difference between the conductor and the surroundings. When maximum operating temperature is reached then the heat loss by convection and radiation is constant and the heat absorbed by the conductor ceases.

The temperature rise after time t from the start of heating is given by the following formula where the change of resistance with temperature can be assumed to be negligible:

where q = temperature rise, °C

qmax = maximum temperature rise, °C

e = exponential constant (=2.718)

t = time, s

t = time constant, s

The time constant can be found using the following formula:

where w = rate of generation of heat at t=0, W

m = mass, kg

c = specific heat, J/kg K

The time constant gives the time taken to reach 0.636 of the maximum temperature rise, qmax.

When a conductor carries a current it creates a magnetic field which interacts with any other magnetic field present to produce a force. When the currents flowing in two adjacent conductors are in the same direction the force is one of attraction, and when the currents are in opposite directions a repulsive force is produced.

In most busbar systems the current-carrying conductors are usually straight and parallel to one another. The force produced by the two conductors is proportional to the products of their currents. Normally in most busbar systems the forces are very small and can be neglected, but under short-circuit conditions, they become large and must be taken into account together with the conductor material fibre stresses when designing the conductor insulator and its associated supports to ensure adequate safety factors.

The factors to be taken into account may be summarised as follows:

a) stresses due to direct lateral attractive and repulsive forces.

b) Vibrational stresses.

c) Longitudinal stresses resulting from lateral deflection.

d) Twisting moments due to lateral deflection.

In most cases the forces due to short-circuits are applied very suddenly. Direct currents give rise to unidirectional forces while alternating currents produce vibrational forces.

**Maximum stresses**

When a busbar system is running normally the interphase forces are normally very small with the static weight of the busbars being the dominant component. Under short-circuit conditions this is very often not the case as the current rises to a peak of some thirty times its normal value, falling after a few cycles to ten times its initial value. These high transitory currents create large mechanical forces not only in the busbars themselves but also in their supporting system. This means that the support insulators and their associated steelwork must be designed to withstand these high loads as well as their normal structural requirements such as wind, ice, seismic and static loads.

The peak or fully asymmetrical short circuit current is dependent on the power factor (cos f) of the busbar system and its associated connected electrical plant. The value is obtained by multiplying the r.m.s. symmetrical current by the appropriate factor given in Balanced three-phase short-circuit stresses.

If the power factor of the system is not known then a factor of 2.55 will normally be close to the actual system value especially where generation is concerned. Note that the theoretical maximum for this factor is 2Ö2 or 2.828 where cos f = 0. These peak values reduce exponentially and after approximately 10 cycles the factor falls to 1.0, i.e., the symmetrical r.m.s. short circuit current. The peak forces therefore normally occur in the first two cycles (0.04 s) as shown in Figure 13.

In the case of a completely asymmetrical current wave, the forces will be applied with a frequency equal to that of the supply frequency and with a double frequency as the wave becomes symmetrical. Therefore in the case of a 50 Hz supply these forces have frequencies of 50 or 100 Hz.

The maximum stresses to which a bus structure is likely to be subjected would occur during a short-circuit on a single-phase busbar system in which the line short-circuit currents are displaced by 180°.

In a three-phase system a short-circuit between two phases is almost identical to the single-phase case and although the phase currents are normally displaced by 120°, under short-circuit conditions the phase currents of the two phases are almost 180° out of phase. The effect of the third phase can be neglected.

In a balanced three-phase short-circuit, the resultant forces on any one of the three phases is less than in the single-phase case and is dependent on the relative physical positions of the three phases.

In the case of a single-phase short-circuit, the forces produced are unidirectional and are therefore more severe than those due to a three-phase short-circuit, which alternate in direction.

The short-circuit forces have to be absorbed first by the conductor. The conductor therefore must have an adequate proof strength to carry these forces without permanent distortion. Copper satisfies this requirement as it has high strength compared with other conductor materials (Table 2). Because of the high strength of copper, the insulators can be more widely spaced than is possible with lower-strength materials.

**Figure 13 Short-circuit current
waveform**

**Single phase short circuit
stresses**

The electromagnetic force developed between two straight parallel conductors of circular cross-section each carrying the same current is calculated from the following formula:

where Fmax = force on conductor, N/m

I = current in both phases, A

s = phase spacing, mm

The value of I is normally taken in the fully asymmetrical condition as 2.55 times the r.m.s. symmetrical value or 1.8 times the peak r.m.s. value of the short-circuit current as discussed above. It is possible, in certain circumstances, for the forces to be greater than this due to the effect of an impulse in the case of a very rigid conductor, or due to resonance in the case of bars liable to mechanical vibration. It is therefore usual to allow a safety factor of 2.5 in such cases.

**Balanced three-phase
short-circuit stresses**

A three-phase system has its normal currents displaced by 120° and when a balanced three-phase short-circuit occurs the displacement is maintained. As with all balanced three-phase currents, the instantaneous current in one phase is balanced by the currents in the other two phases. The directions of the currents are constantly changing and so therefore are the forces. The maximum forces are dependent on the point in the cycle at which the fault or short-circuit occurs.

The maximum force appearing on any phase resulting from a fully offset asymmetrical peak current is given by

(9

The condition when the maximum force appears on the outside phases (Red or Blue) is given by

(10

The condition when the maximum force is on the centre phase (Yellow) is given by

(11

where Fmax = maximum force on conductor, N/m

I = peak asymmetrical current, A

s = conductor spacing, mm

The peak current I attained during the short-circuit varies with the power factor of the circuit:

Power
factor |
I,
x Irms (symmetrical) |

0 | 2.828 |

0.07 | 2.55 |

0.2 | 2.2 |

0.25 | 2.1 |

0.3 | 2 |

0.5 | 1.7 |

0.7 | 1.5 |

1.0 | 1.414 |

**Correction for end effect**

It has been assumed so far that the conductors are of infinite length. This assumption does not generally lead to great errors in the calculated short-circuit forces. This is not true, however, at the ends of bars where there is a great change in flux compared with the uniform magnetic field over most of the long straight conductor. Where the conductor is relatively short this effect can be considerable, the normal formulae giving overestimates for the forces. To overcome this problem the preceding formulae can be rewritten in the following form:

where Ftot = total force on the conductor, N

L = length of conductor, m

c = constant from relevant previous formula

The following substitution may then be made:

The formula will then be of the form

(12

If

is very large then

is almost equal to

and therefore the modified formula becomes almost identical with the standard formula. In many cases, the following formula is sufficiently accurate:

(13

where Ftot is again the total force along the conductor in Newtons.

Formulae 9 to 11 may be used where

is greater than 20. For values between 20 and 4, is greater than 20. For values between 20 and 4, equation 13 above should be used. For values less than 4, equation 12 should be used.

**Proximity factor**

**Figure 14 - Proximity factor
for rectangular copper conductor**

The formulae in the previous section used for calculating short-circuit forces do not take into account the effect of conductors which are not round as they strictly only apply to round conductors. To overcome this when considering rectangular conductors, a proximity factor K is introduced into the ordinary force formulae, its value being found using the curves in Figure 14.

Except in cases where the conductors are very small or are spaced a considerable distance apart the corrected general formula for force per unit length becomes:

The value of

is first calculated then K is read from the curve for the appropriate

ratio.

From the curves it can be seen that the effect of conductor shape decreases rapidly with increasing spacing and is a maximum for strip conductors of small thickness. It is almost unity for square conductors and is unity for a circular conductor.

Alternatively, the proximity factor can be calculated using the following formula, from which the curves in Figure 14 were drawn (Dwight 1917). (See Figure 14 for explanation of symbols).

This formula gives the intermediate curves of Figure 14, for s>a, b>0, a>0

**Vibrational stresses**

Stresses will be induced in a conductor by natural or forced vibrations the amplitude of which determines the value of the stress, which can be calculated from the formulae given in Section 8.

The conductor should be designed to have a natural frequency which is not within 30% of the vibrations induced by the magnetic fields resulting from the currents flowing in adjacent conductors. This type of vibration normally occurs during continuous running and does not occur when short-circuit currents are flowing.

The stresses resulting from the short-circuit forces are calculated using the beam theory formulae for simply supported beams for a single cantilever to multispan arrangements, the applied forces being derived from the previous sections. The resulting deflections enable the conductor stress to be calculated and so determine if it is likely to permanently damage the conductor because it has exceeded the proof stress of the conductor material.

**Methods of reducing conductor
stresses**

In cases where there is a likelihood of vibration at normal currents or when subjected to short-circuit forces causing damage to the conductor, the following can he used to reduce or eliminate the effect:

a) Reduce the span between insulator supports.

This method can be used to reduce the effects of both continuous vibration and that due to short-circuit forces.

b) Increase the span between insulator supports.

This method can only be used to reduce the effects of vibration resulting from a continuous current. It will increase the stresses due to a short-circuit current.

c) Increase or decrease the flexibility of the conductor supports.

This method will reduce the effects of vibration due to continuous current but has very little effect on that due to short-circuit forces.

d) Increase the conductor flexibility.

This can only be used to reduce the effects of vibration due to a continuous current. The short-circuit effect is increased.

e) Decrease the conductor flexibility.

This method will reduce the effects of vibration due to either a continuous current or a short-circuit.

It will be noted that in carrying out the various suggestions above, changes can only be made within the overall design requirements of the busbar system.

With very high voltage air-insulated busbars, particularly of the type usually installed out of doors, it is necessary to ensure that with the spacing adopted between conductors of different phases, or between conductors and earth, the electromagnetic stress in the air surrounding the conductors is low enough not to cause a corona discharge. Corona discharge is to be avoided where possible as it creates ionised gas which can lead to a large reduction in the air insulation surrounding the conductor and so can cause flash-over. Should flash-over occur, this will in many cases lead to a short-circuit between either adjacent phases or poles or the nearest earth point or plane. This will cause considerable burning of the conductors and associated equipment together with mechanical damage. Corona discharge can also cause radio interference which may be unacceptable.

To avoid these conditions the busbar system should be free from sharp edges or small radii on the conductor system. If this is not possible then additional equipment will have to be incorporated in the design such as corona rings and stress relieving cones mounted in the areas of high electric stress. The smallest radii required for prevention of corona can be calculated from the formula:

where E = r.m.s. voltage to neutral, kV

r = conductor radius, mm

d = distance between conductor centres, mm

d = air density factor

m = conductor surface condition factor

The values for the factors m and d are as follows:

m = 1 for a polished conductor surface, 0.98 to 0.93 for roughened or weathered surfaces, and 0.87 to 0.80 for stranded conductors.

d = 1 at 1 bar barometric pressure and 25°C. At other pressures and temperatures the value is found as follows:

where b = barometric pressure, bar

T = temperature, °C

At locations above sea level the normal pressure is reduced by approximately 0.12 bar per 1000 m of altitude.

The voltage Ev at which the corona discharge normally becomes visible is somewhat higher than given by the above formula and can be determined as follows:

In bad weather conditions the discharge may appear at a voltage lower than that indicated by the formulae and it is therefore advisable to make an allowance of about 20% as a safety factor.