It will be appreciated that the hgterm is zero except for subvolumes which carry current, and the hl term is zero except for subvolumes which lie on the fuselink surfaces. When considering the latter, one or more of the thermal conductivity terms is zero.
For a particular current, an iterative process may be used to determine the steady- state temperature distribution, adjustments being made to allow for the resistivity and current distributions. This calculation may be repeated to determine the current level at which the fuse-element restrictions are at their melting-point temperature. The same process may also be used to determine the steady-state running conditions at currents below the minimum fusing level.
2.1.5 Mathematical and experimental studies
Leach et al. [9] used the methods described in the preceding sections to calculate the performances of existing fuselinks so that the accuracy of the results given by them could be assessed. In addition, the ranges over which the various heat-flow assumptions are reasonably valid were studied.
The determination of current-flow patterns within fuse elements forms a fun- damental part of the analytical technique and therefore a study was made of the distributions in a number of elements, each assumed to be operating with a uniform electrical resistivity, the value chosen being that of silver at 20◦C. It is evident that the resistance of a fuse element can be readily calculated once the current-flow pattern is established. This was done for each of the elements considered, and close agreement was obtained, thus verifying the calculations.
In subsequent calculations, the current-flow patterns were determined at various stages during pre-arcing periods and it was found that the patterns do not change greatly as the elements heat up although of course the overall resistance of an ele- ment does rise significantly and the energy inputs rise very greatly in the restricted sections where the resistivities increase to high levels as melting temperatures are approached. Figure 2.7 illustrates the smallness of the change in the current distribu- tion in a typical element between the beginning and the end of the pre-arcing period. The initial distribution (i.e. when cold) is shown in solid lines and the distribution as the restrictions near their melting point temperature is in broken lines.
Pre-arcing times were calculated at extremely high currents for various element forms, in which it is assumed that there is no heat movement, using the method
Figure 2.7 Current-flow lines for uniform and non-uniform temperature distribu- tions
————– for uniform temperature distribution – – – – – – for non-uniform temperature distribution
described in Section 2.1.1. Comparison of the results obtained with those determined experimentally showed that there is close agreement for fuselinks with plain cylin- drical wire elements at currents which give pre-arcing times up to 35 ms. With many modern industrial fuselinks (e.g. those which comply with IEC 60269-2-1), close agreement was only obtained for currents at and above those which give pre-arcing times of about 3 ms, and the situation was even more restricted for fuselinks used to protect semiconductors, satisfactory agreement only being obtained for currents which gave pre-arcing times of 0·3 ms or less. This led to the interesting conclusion that, although in the past the simple treatment, which is possible when heat movement is neglected, was satisfactory for a significant part of the time/current characteristic, this is no longer so, and for many present-day fuselinks it is only applicable for severe short-circuit currents.
Pre-arcing times were calculated allowing for heat movement in the fuselink element but not in the other parts. Close agreement was obtained with experimentally determined values for industrial and semiconductor fuselinks for conditions leading to pre-arcing times up to 5 and 10 ms, respectively.
Pre-arcing times were also determined by calculation assuming heat movement to occur in the fuse elements and the filling material. A calculated characteristic for a semiconductor-protection fuselink, together with the associated experimentally obtained values, is shown in Figure 2.8. It can be seen that the curve begins to diverge for this fuselink at currents which give pre-arcing times of 5–10 s, indicating that the effects of heat losses from the fuselink begin to become significant at times of this order.
An interesting effect which was revealed during the calculations was a disconti- nuity in the time/current characteristic for sinusoidally varying currents, that is for currents of the form i= Ipksin ωt. This effect had been noted in the past in experimen- tal results but its cause had not been understood. During periods when the magnitude
10–3 102 103 RMS prospective current A 104 10–2 10–1 pre-arcing time , s 1 10
Figure 2.8 Pre-arcing time/current curve for a 200 A semiconductor-protection fuselink carrying 50 Hz symmetrical sinusoidal current
——— predicted results ◦ ◦ ◦ test results
– – – – curve from adiabatic formula
–.–.– discontinuity discussed in Section 2.1.5
of the current is rising the element heats up rapidly and, if vaporisation does not occur before the current reaches its maximum value, the element temperatures fall during the following quarter cycle as the energy input reduces and heat is conducted away. This behaviour is particularly pronounced with fuselinks which incorporate elements with restricted sections. The maximum temperatures are attained somewhat after the instants of peak current because of the heat conduction. For the fuselinks which were studied in detail, the maximum temperatures were reached between 6 and 7 ms after current zeros (for 50 Hz sinusoidal currents). Fuse operation is only possible in these periods and if it does not occur in them it will not take place until the current is rising again. The effect occurs with over-currents of levels which cause operation in one or two half-cycles, examples being shown in Figure 2.9. Curves of
0 0 200 400 temper ature , ° C 600 800 1000 2 4 6 8 10 time, ms 12 14 16 1550 A RMS 1500 A RMS melting temperature
Figure 2.9 Restriction-temperature/time curves for a semiconductor-protection fuselink carrying two symmetrical sinusoidal currents
the fuse-element-restriction temperature variation with time are shown for two sym- metrical fault currents, one just causing operation within the first half-cycle (6·5 ms) and the other, with only a 3 per cent lower RMS value, causing operation in the second half-cycle (15 ms). Confirmation was obtained from practical tests and, with the fuselink for which the calculations had been made, operation was not obtained between 7 and 14 ms after the application of current. It will be clear that operation in the time range up to about 40 ms is dependent on the current waveform and, for waves which are basically sinusoidal, the behaviour is affected by the instant in the supply-voltage cycle at which a fault occurs and also by the circuit parameters as these affect the transient component of the current wave. At lower currents, for which the operating times are much longer, it is only the RMS value of the current which effectively controls the behaviour and the instantaneous variations are not normally taken into account in theoretical studies.
The temperature distributions in fuselinks can readily be obtained mathematically using the equations given in earlier sections and such information, which cannot be determined by other means, is valuable to designers and application engineers. Calculated distributions over a section of the element of a semiconductor fuselink rated at 200 A when carrying sinusoidal currents of 900 and 1200 A are shown in
916°C 880°C 177°C 205°C 500°C 487°C a b
Figure 2.10 Temperature distribution, just prior to melting over a 200 A fuselink element at two current levels
a 900 A RMS b 1200 A RMS
Figure 2.10 in which temperature is proportional to height. Both the distributions shown apply at times just prior to the melting of the restrictions, the pre-arcing times for the two conditions being 0·7 and 0·11 s. It is interesting to note the generally lower temperatures of the majority of the element at the higher current, this being due to the lower conduction of heat from the restriction because of the shortness of the pre-arcing period.
In addition to the above studies, calculations were done in the manner suggested in Section 2.1.4 to determine the minimum currents at which various fuselinks would
operate, i.e. after an infinite time, and these corresponded well with values obtained experimentally.
Calculations of temperature distributions at rated currents were also done and the effects on them of the cross-sectional and surface areas of the connecting cables were studied and reported on by Leach et al. [9].
Finite-difference methods such as the one described here are often formulated in terms of an equivalent thermal resistance–capacitance network (Beaujean et al. [15]), which allows standard circuit analysis software to be used to obtain the temperature distributions. However, practical fuse designs are truly three-dimensional, with mul- tiple parallel notched elements. A very large and complex mesh of subvolumes is required, with very small subvolumes in the notch zones, and larger subvolumes in the outer regions of the fuse body and terminals. Simplifying assumptions are needed to obtain solutions economically.
Finite-element methods have also been used to model fuse thermal behaviour, but again the models have usually been very much simplified to allow solution with a standard finite-element, FEA, package. However, Kawase et al. [16] wrote a finite-element program to give a true three-dimensional transient model of a semi- conductor fuse with multiple parallel notched elements. Although excellent results were obtained, the computing time was excessive. A significant increase in computing power is required before FEA analysis can be used as a routine design tool.