As stated earlier it is usually possible to regard a circuit as having a source of EMF and an associated network of series and shunt impedances. In many cases the shunt branches do not have a significant effect, particularly during fault conditions, and the circuit may often be represented simply as a source of EMF es in series with a resistance and an inductance (Rcand Lc, respectively). In these circumstances, the general non-linear equation which would apply during the arcing period would be
cs = i(Ran+ Rc)+d(Lci)
dt + n(Vaf+ Vcf) (3.20)
in which Ra is the resistance of each arc column, Vaf and Vcf are the anode- and cathode-fall voltages, respectively, and n is the number of arcs or restrictions in the fuse element. When step by step computation is to be used to determine the current variation from the end of the pre-arcing period, it is convenient to rewrite eqn. 3.20 in the form
The resistance of the column at the end of any step is given by Ra2= va2
i2 (3.22)
and also by Ra2= l2
σ2Aa2 (3.23)
It must be recognised that eqns. 3.5, 3.7, 3.8, 3.16–3.19 and 3.21–3.23 are independ-ent and include ten unknowns, i2, l2, vol2, Aa2, σ2, ne2, θa2, X2, Ra2and va2. They can therefore be solved if i1, l1, vol1, Aa1, σ1, ne1, θa1, X1, Ra1and va1are known.
For the first time interval, the values of es1and i1must be those which are obtained at the end of the pre-arcing period and the parameters l1, vol1, Aa1, θa1and X1at the beginning of the interval are very small and may therefore be taken to be zero. The only remaining parameter which needs to be known is the voltage along the column.
Oscillograms obtained during many varied tests have shown that the total voltage across each notch of a fuselink element changes abruptly from a small value during the pre-arcing period to about 50 V at the instant of arc initiation. The exact value depends on the dimensions of the restriction, long restrictions having slightly higher initial voltages than short ones. For modelling purposes, the initial value of the voltage across the column was taken by Beaumont and Wright to be 33 V.
Step-by-step calculations of the type described above should enable the charac-teristics during the arcing period to be determined.
Many parameters associated with fuse arcs such as pressure, temperature and length have been calculated using the above model. As an example, the variation of arc temperature with time in a fuselink, for which the current and voltage waveforms were as shown in Figure 3.5, is shown in Figure 3.6. It can be seen that the temperature within the arc varies within the range 10× 103to 15× 103K.
At the time these theoretical studies were undertaken, it was not possible to com-pare the various results with values obtained by experiment because the transducers then available could not be inserted into fuselinks without significantly affecting their performance. Subsequently, however, Barrow and Howe [28,29] conducted laboratory tests on fuselinks into which optical fibres were inserted, the fibres being of similar material to the filling material in the fuselinks, so that they did not appreciably affect the performance.
Initially a row of optical fibres was inserted in each fuselink as described in Reference 28 and shown in Figure 3.7, the inner ends of the fibres being near one face of the element. As a fuselink operated when a high current passed through it, light
b
Figure 3.5 Current and voltage waveforms Where
a waveform of prospective current b waveform of open-circuit voltage c computed waveforms
d waveforms obtained by experiment
20 000
10 000
arc temperature, K
0
1 2 3
arcing time, ms
4 5 6 7
Figure 3.6 Variation of arc temperature with time
five fibres
fuse element
Figure 3.7 Fuselink containing optical fibre
emanating from the arc was transmitted along the fibres and detected and recorded by external equipment. Light was received first from the fibre positioned at the centre of the restriction in the element and then light was received in sequence from the other fibres as the arc extended. It was therefore possible to determine the rate of burnback, i.e. the rate at which the arc extended, and comparisons with values determined experimentally could be made.
In a second series of experiments, described in Reference 29, the light emissions from optical fibres inserted into fuselinks were determined during arcing periods.
A rapid-scanning spectrometer and photomultiplier tube were used to measure the relative intensities of the most prominent wavelengths in the emitted light, and this enabled the arc temperatures to be calculated and compared with theoretically determined values.
Subsequent related experimental and theoretical studies conducted by Cheim and Howe are described in References 30 and 31. It is believed that continuation of such work will enable refined and accurate modelling of fuse behaviour to be achieved and that it will eventually be possible to determine the effects of varying individual parameters, such as the specific heat of the filling material or the amount by which it expands on liquifying. In this way the most suitable materials and optimum dimen-sions of a fuselink for a particular application could be determined. Its performance for a wide range of system conditions including prospective current, instant of fault occurrence in the voltage cycle and fault-circuit power factor could also be computed and then only a small number of experimental checks should be needed to provide confirmation. In this way it should be possible, in the future, to reduce greatly the amount of testing from that needed at present, and significant financial savings may consequently be effected.
Recent work by Rochette et al. [32] introduced a model describing the mechanical interaction and energy transfer mechanisms from the arc column to the porous granu-lated surrounding medium. The heat and mass transfers of the solid, liquid and vapour phases were modelled using Darcy’s and Forchheimer’s laws. The results produced simulated fulgurites with characteristics similar to those observed in practice.
The preceding arcing models are only useful for the interruption of high short-circuit currents, for which multiple arcing occurs in the notch zones, and arc extinction occurs when the limited current reaches zero. In this case the resistive shunting effect of the hot fulgurite minimises the possibility of restriking [33].
For the interruption of low overcurrents the situation is different. Because the heating rate is much slower, the cooling effect of the ends has great effect, and the central region of the fuse element becomes much hotter than its surroundings. A notch in the central part of the fuse melts first, producing a single arc with a relatively low voltage because the current is low. There is little current-limiting effect at first, but the arc voltage increases as the element burns back. When the first current zero is reached, the arc extinguishes, but the gap left may be too short; the arc restrikes, and a second half cycle of arcing begins. This may continue for several half-cycles, until the gap produced by burnback is long enough to withstand the restriking voltage. This process has been studied by Erhard et al. [34], who obtained extensive test data and developed a model which included the effects of sequential burnback and restriking.